Analysis and Management of Animal Populations
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Analysis and Management of Animal Populations Modeling, Estimation, and Decision Making
Byron K. Williams U.S. Geological Survey Cooperative Research Units Reston, Virginia
James D. Nichols
Michael J. Conroy
U.S. Geological Survey Patuxent Wildlife Research Center Laurel, Maryland
Cooperative Fish and Wildlife Research Unit DB Warnell School of Forest Resources University of Georgia Athens, Georgia
ACADEMIC PRESS An Imprint of Elsevier San Diego San Francisco New York Boston London Sydney Tokyo
Cover images: Top three images, @ 2001 PhotoDisc, Inc. Bottom image, @ 2001, Joe Lange This book is printed on acid-free paper. Copyright 9 2002 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier's Science and Technology Rights Department in Oxford, UK. Phone: (44) 1865 843830, Fax: (44) 1865 853333, e-mail:
[email protected]. You may also complete your request on-line via the Elsevier homepage: http://www.elsevier.com by selecting "Customer Support" and then "Obtaining Permissions". Academic Press An Imprint of Elsevier 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http: / / www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 ,'BY, UK http: / / www.academicpress.com Library of Congress Catalog Card Number: 2001094375 ISBN-13" 978-0-12-754406-9 ISBN-10:0-12-754406-2 PRINTED IN THE UNITED STATES OF AMERICA 06 07 EB 9 8 7 6 5 4
To my parents, Roger S. (deceased) and Mary F. Williams; my wife Genie; and my daughters ]aimin and Shannon. Byron K. Williams
To my parents, James E. and Barbara Irwin Nichols; and to Walt Conley, mentor and friend. James D. Nichols
To the memory of my parents, Edith M. and James R. Conroy. Michael J. Conroy
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Contents
Preface xiii Acknowledgments xvii
2.4.
2.5. 2.6. 2.7. 2.8.
PART
Hypothesis Confirmation 16 Inductive Logic in Scientific Method 17 Statistical Inference 18 Investigating Complementary Hypotheses Discussion 19
18
I FRAMEWORK FOR MODELING, ESTIMATION, A N D MANAGEMENT OF ANIMAL POPULATIONS
CHAPTER
3 Models and the Investigation of Populations
CHAPTER
3.1.
3.2. 3.3. 3.4. 3.5. 3.6.
1 Introduction to Population Ecology Some Definitions 3 1.2. Population Dynamics 4 1.3. Factors Affecting Populations 4 1.4. Management of Animal Populations 1.5. Individuals, Fitness, and Life History Characteristics 7 1.6. Community Dynamics 9 1.7. Discussion 9 1.1.
Types of Biological Models 22 Keys to Successful Model Use 22 Uses of Models in Population Biology 23 Determinants of Model Utility 28 Hypotheses, Models, and Science 30 Discussion 31
CHAPTER
4 Estimation and Hypothesis Testing in Animal Ecology
CHAPTER
2
4.1.
Scientific Process in Animal Ecology 2.1.
2.2. 2.3.
Causation in Animal Ecology 11 Approaches to the Investigation of Causes Scientific Methods 13
4.2. 4.3. 4.4. 4.5. 4.6.
12
vii
Statistical Distributions 34 Parameter Estimation 42 Hypothesis Testing 50 Information-Theoretic Approaches 55 Bayesian Extension of Likelihood Theory Discussion 58
57
viii
Contents CHAPTER
CHAPTER
5
8
Survey Sampling and the Estimation of Population Parameters
Traditional Models of Population Dynamics 8.1.
5.1.
5.2. 5.3. 5.4.
5.5. 5.6.
Sampling Issues 60 Features of a Sampling Design 61 Simple Random and Stratified Random Sampling 62 Other Sampling Approaches 67 Common Problems in Sampling Designs Discussion 76
74
CHAPTER
6
Density-Independent Growth--The Exponential Model 136 8.2. Density-Dependent GrowthmThe Logistic Model 139 8.3. Cohort Models 141 8.4. Models with Age Structure 143 8.5. Models with Size Structure 157 8.6. Models with Geographic Structure 159 8.7. Lotka-Volterra Predator-Prey Models 161 8.8. Models of Competing Populations 164 8.9. A General Model for Interacting Species 170 8.10. Discussion 171
Design of Experiments in Animal Ecology 6.1.
6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9.
Principles of Experimental Design 80 Completely Randomized Designs 83 Randomized Block Designs 89 Covariation and Analysis of Covariance 91 Hierarchical Designs 92 Random Effects and Nested Designs 97 Statistical Power and Experimental Design 100 Constrained Experimental Designs and Quasi-Experiments 102 Discussion 106
CHAPTER
9 Model Identification with Time Series Data 9.1. 9.2.
9.3. 9.4. 9.5. 9.6.
PART
II DYNAMIC MODELING OF ANIMAL POPULATIONS
9.7.
9.8. 9.9.
CHAPTER
7.1.
7.2. 7.3. 7.4. 7.5. 7.6. 7.7.
Model Identification Based on Ordinary Least Squares 174 Other Measures of Model Fit 176 Correlated Estimates of Population Size 178 Optimal Identification 178 Identifying Models with Population Size as a Function of Time 179 Identifying Models Using Lagrangian Multipliers 181 Stability of Parameter Estimates 181 Identifying System Properties in the Absence of a Specified Model 182 Discussion 184
CHAPTER
7
10
Principles of Model Development and Assessment
Stochastic Processes in Population Models
Modeling Goals 113 Attributes of Population Models 114 Describing Population Models 117 Constructing a Population Model 122 Model Assessment 126 A Systematic Approach to the Modeling of Animal Populations 131 Discussion 134
10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9.
Bernoulli Counting Processes 189 Poisson Counting Processes 192 Discrete Markov Processes 197 Continuous Markov Processes 202 Semi-Markov Processes 205 Markov Decision Processes 207 Brownian Motion 210 Other Stochastic Processes 213 Discussion 220
Contents
ix CHAPTER
CHAPTER
11
14
The Use of Models in Conservation and Management
Estimating Abundance for Closed Populations with Mark-Recapture Methods
11.1. 11.2. 11.3.
Dynamics of Harvested Populations Conservation and Extinction of Populations 231 Discussion 237
223
PART
III
Two-Sample Lincoln-Petersen Estimator 290 K-Sample Capture-Recapture Models 296 Density Estimation with Capture-Recapture 314 14.4. Removal Methods 320 14.5. Change-in-Ratio Methods 325 14.6. Discussion 331
14.1. 14.2. 14.3.
ESTIMATION M E T H O D S FOR A N I M A L POPULATIONS CHAPTER
15
CHAPTER
12
Estimation of Demographic Parameters
Estimating Abundance Based on Counts Detectability and Demographic Rate Parameters 334 15.2. Analysis of Age Frequencies 337 15.3. Analysis of Discrete Survival and Nest Success Data 343 15.4. Analysis of Failure Times 351 15.5. Random Effects and Known-Fate Data 361 15.6. Discussion 362 15.1. 12.1. Overview of Abundance Estimation 242 12.2. A Canonical Population Estimator 243 12.3. Population Censuses 245 12.4. Complete Detectability of Individuals on Sample Units of Equal Area 245 12.5. Complete Detectability of Individuals on Sample Units of Unequal Area 247 12.6. Partial Detectability of Individuals on Sample Units 250 12.7. Indices to Population Abundance or Density 257 12.8. Discussion 261
CHAPTER
16 Estimation of Survival Rates with Band Recoveries
CHAPTER
13 Estimating Abundance with Distance-Based Methods 13.1. 13.2. 13.3. 13.4.
Point-to-Object Methods 263 Line Transect Sampling 265 Point Sampling 278 Design of Line Transect and Point Sampling Studies 281 13.5. Other Issues 286 13.6. Discussion 287
Single-Age Models 366 Multiple-Age Models 383 Reward Studies for Estimating Reporting Rates 391 16.4. Analysis of Band Recoveries for Nonharvested Species 398 16.5. Poststratification of Recoveries and Analysis of Movements 402 16.6. Design of Banding Studies 406 16.7. Discussion 414
16.1. 16.2. 16.3.
x
Contents 20.3.
CHAPTER
17
20.4.
Estimating Survival, Movement, and Other State Transitions with Mark-Recapture Methods 17.1. 17.2. 17.3. 17.4. 17.5. 17.6. 17.7.
Single-Age Models 418 Multiple-Age Models 438 Multistate Models 454 Reverse-Time Models 468 Mark-Recapture with Auxiliary Data Study Design 489 Discussion 492
Estimating Parameters of Community Dynamics 561 Discussion 572
PART
IV D E C I S I O N ANALYSIS FOR ANIMAL POPULATIONS 476 CHAPTER
21 Optimal Decision Making in Population Biology
CHAPTER
18 Estimating Abundance and Recruitment with Mark-Recapture Methods 18.1. 18.2. 18.3. 18.4. 18.5. 18.6. 18.7.
Data Structure 496 Jolly-Seber Approach 497 Superpopulation Approach 508 Pradel's Temporal Symmetry Approach Relationships among Approaches 518 Study Design 520 Discussion 522
511
21.1. Optimization and Population Dynamics 578 21.2. Objective Functions 579 21.3. Stationary Optimization under Equilibrium Conditions 579 21.4. Stationary Optimization under Nonequilibrium Conditions 580 21.5. Discussion 581
CHAPTER
22 CHAPTER
Traditional Approaches to Optimal Decision Analysis
19 Combining Closed and Open Mark-Recapture Models: The Robust Design 19.1. 19.2. 19.3. 19.4. 19.5. 19.6.
Data Structure
524 529 Likelihood-Based Approach 535 Special Estimation Problems 538 Study Design 552 Discussion 553
Ad Hoc Approach
22.1. 22.2. 22.3. 22.4. 22.5. 22.6.
The Geometry of Optimization 584 Unconstrained Optimization 585 Classical Programming 593 Nonlinear Programming 597 Linear Programming 601 Discussion 606
CHAPTER
23 CHAPTER
2O Estimation of Community Parameters 20.1. An Analogy between Populations and Communities 556 20.2. Estimation of Species Richness 557
Modem Approaches to Optimal Decision Analysis 23.1. Calculus of Variations 608 23.2. Pontryagin's Maximum Principle 23.3. Dynamic Programming 627 23.4. Heuristic Approaches 638 23.5. Discussion 639
618
Contents
Appendix C Differential Equations 693 C.1. First-Order Linear Homogeneous Equations C.2. Nonlinear Homogeneous Equations m Stability Analysis C.3. Graphical Methods
CHAPTER
24 Uncertainty, Learning, and Decision Analysis 24.1. Decision Analysis in Natural Resource Conservation 644 24.2. General Framework for Decision Analysis 649 24.3. Uncertainty and the Control of Dynamic Resources 650 24.4. Optimal Control with a Single Model 651 24.5. Optimal Control with Multiple Models 652 24.6. Adaptive Optimization and Learning 653 24.7. Expected Value of Perfect Information 654 24.8. Partial Observability 655 24.9. Generalizations of Adaptive Optimization 656 24.10. Accounting for All Sources of Uncertainty 658 24.11. "Passive" Adaptive Optimization 658 24.12. Discussion 660 CHAPTER
25 Case Study: Management of the Sport Harvest of North American Waterfowl 25.1. Background and History 664 25.2. Components of a Regulatory Process 667 25.3. Adaptive Harvest Management 671 25.4. Modeling Population Dynamics 672 25.5. Harvest Objectives 676 25.6. Regulatory Alternatives 677 25.7. Identifying Optimal Regulations 679 25.8. Some Ongoing Issues in Waterfowl Harvest Management 680 25.9. Discussion 684
Appendix A Conditional Probability and Bayes' Theorem
xi
685
Appendix B Matrix Algebra 687 B.1. Definitions B.2. Matrix Addition and Multiplication B.3. Matrix Determinants B.4. Inverse of a Matrix B.5. Orthogonal and Orthonormal Matrices B.6. Trace of a Matrix B.7. Eigenvectors and Eigenvalues B.8. Linear and Quadratic Forms B.9. Positive-Definite and Semidefinite Matrices B.10. Matrix Differentiation
Appendix D Difference Equations 709 D.1. First-Order Linear Homogeneous Equations D.2. Nonlinear Homogeneous EquationswStability Analysis Appendix E Some Probability Distributions and Their Properties 721 E.1. Discrete Distributions E.2. Continuous Distributions Appendix F Methods for Estimating Statistical Variation 733 Distribution-Based Variance Estimation El. Empirical Variance Estimation E2. Estimating Variances and Covariances with the E3. Information Matrix Approximating Variance with the Delta Method E4. Jackknife Estimators of Mean and Variance E5. Bootstrap Estimation E6. Appendix G Computer Software for Population and Community Estimation 739 G.1. Estimation of Abundance and Density for Closed Populations G.2. Estimation of Abundance and Demographic Parameters for Open Populations G.3. Estimation of Community Parameters G.4. Software Availability Appendix H The Mathematics of Optimization 745 Unconstrained Optimization H.1. H.2. Classical Programming H.3. Nonlinear Programming H.4. Linear Programming H.5. Calculus of Variations H.6. Pontryagin's Maximum Principle H.7. Dynamic Programming
References 767 Index 793
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Preface
This book deals with the assessment and management of animal populations. It is an attempt to pull together key elements of what has become a truly overwhelming body of theory and practice in population biology and to add by way of synthesis to our understanding of animal populations and their conservation. Such an effort requires perambulations through the sometimes strange worlds of mathematical modeling, probability theory, statistical estimation, dynamic optimization, and even logical inference. Happily, one need not establish residence in any of these places to absorb what is needed for the journey. On the other hand, one is well served by a visit and by spending at least some time exploring the terrain. The overarching theme of this book is that modeling, attribute estimation, and optimal decision making are linked together in the doing of science-based conservation. Models play key roles in both the science and management of biological systems, as expressions of biological understanding, as engines for deductive inference, and as articulations of biological response to management and environmental change. These roles are supported by the principles of sampling design and statistical inference, which focus on the use of field observations to identify and calibrate models according to their purposes and objectives. Both modeling and statistical assessment are key elements in formal decision analysis, which utilizes model-based predictions along with measures of sampling variation and other stochastic factors to support informed decision making. These thematic elements form the basis for Parts II-IV of the book. We are concerned here with animal populations, recognizing that population biology ultimately must be understood in a broader context of the habitats and communities of which populations are a part. We build on the notion of a population as a partially self-regulating ecological unit composed of potentially inter-
breeding individuals, with characteristics such as birth rate, death rate, age structure, and dispersion pattern through space and time. Our focus is on dynamic populations, in a context of interspecific interactions and environmental influences. It is through a complex network of biotic and abiotic influences that individuals choose and defend territories, select mates and engage in reproduction, compete for food resources, and avoid predators during the course of their life histories. In this book, we emphasize the processes of birth, growth, reproduction, maturation, and death, with the idea that these processes effectively integrate the influences of the biotic and physical environment and thus permit inferences about individual fitness and population status. The combined influences of structure and feedback among species and their habitats can lead to complicated patterns of change, and the attempt to represent these patterns precisely often results in complex models with large numbers of interactions and feedbacks. These in turn can give rise to certain analytic difficulties and an inability to recognize biological features that control population change. In the ensuing chapters, we address the modeling and assessment of animal populations in light of tradeoffs between understanding and complexity, accounting for model realism, precision, and generality. We acknowledge a bias for parsimony in the modeling of biological phenomena, in keeping with the principle that among acceptable alternatives the simpler explanation is preferred over its more complicated alternative. We believe that population modeling is especially useful when models are developed in a context of conservation and management, recognizing that the range of management practices is bewilderingly wide and often difficult to address systematically. Assessment of impacts can be a considerable challenge when the actions involved are as diverse as manipulation of habi-
xiii
xiv
Preface
tats, control of competition and predation, stocking of individuals, and, for those species subject to sport a n d / o r commercial harvest, the regulation of harvests. We provide the reader with examples of population assessment framed in a context of wildlife conservation and management. Animal ecologists often justify their work by claims that more information about animal populations ultimately will lead to better conservation decisions. Although we believe this claim to be true, we also believe that scientists can do much better than simply providing information. Indeed, a key message is that biological information is much more likely to be useful in solving conservation problems if it is collected in the context of a decision-theoretic approach to management. The book is organized into four thematically focused parts supported by a number of technical appendices. Part I sets out a framework for the role of modeling and the treatment of field observations. It begins with an exposition on scientific method in animal ecology, followed by a discussion on modeling in biological investigation and management. The remainder of Part I focuses on statistical estimation, sampling, and experimental design in ecological investigations. This information provides the reader with the conceptual tools needed for the chapters that follow. Part II focuses on modeling approaches for dynamic biological and natural resource systems, using as examples well-known ecological models. We review notation, objectives, and attributes of models of biological populations, discuss the modeling of stochastic influences such as environmental variation and other random factors, and describe some approaches to the identification of model structure based on time-series data. Part II concludes with applications of models in population management, especially harvest management, conservation biology, and experimentation. Part III builds on the statistical framework introduced earlier and treats more formally the problem of estimating population attributes with sampling data. In Chapters 12 through 14 we deal with estimation for "closed" populations, for which individuals neither enter the population through birth and immigration nor leave the population through mortality and emigration over the course of sampling. In Chapters 15 through 19 we discuss parameter estimation for "open" populations, for which population size a n d / o r composition can change during the course of sampling. Finally, in Chapter 20 we address the estimation of community parameters such as species richness, extinction rates, and species turnover rates. Part IV addresses management of biological populations in terms of optimal decision making through time, recognizing that management actions taken at any point in time can influence population dynamics
at subsequent times. We describe and illustrate a number of optimization techniques that originally were designed for nondynamic problems, and then introduce some modern techniques that make explicit the dynamic nature of animal populations. Part IV culminates in a unified framework for optimal management under uncertainty, recognizing multiple sources of uncertainty and accounting for the potential for learning through management. In particular, we describe adaptive optimization as a way to accommodate uncertainties about the structure of biological processes. The book concludes with a case study of modeling, estimation, and management of waterfowl populations. This science-based management system serves as a clear and successful example of how modeling, estimation, and decision analysis can be integrated into a biologically informed, adaptively managed program.
Book Objectives and Intended Audience A rationale for the selection of material covered in this book is given by way of an analogy between books and mathematical models. Many recognize that in a given biological situation, a model that is designed to be general can be less useful in meeting objectives than a model designed specifically for those objectives. The same holds true for a book: it is not possible in a single volume to treat subjects comprehensively and technically, while simultaneously making them accessible to those seeking a less rigorous treatment. Compromise between these competing objectives is always required, and indeed, the level of detail presented in this book is an example of such a compromise. Thus, we have attempted to explain concepts in a relatively straightforward manner, while still providing the background and detail required for more comprehensive understanding. Our primary purpose in doing so is to promote the integration of modeling, estimation, and decision analysis, which we regard as a unique feature of this book. The intended audience for the book consists of graduate students and advanced undergraduate students in animal ecology, biometrics, quantitative ecology, conservation ecology, and fish and wildlife biology; researchers in biology, biostatistics, and natural resource conservation; natural resource conservationists and managers; and libraries and natural resource reference collections. Readers of the book need a working knowledge of probability, statistics, and differential equations, though the subject matter in each chapter is organized so that key messages can be understood without the need for in-depth mathematical study.
Classroom Use The book is designed to be a single reference for modeling, estimation, and decision analysis, with frequent
Preface references in each chapter to supporting materials in other chapters. It would be an appropriate text for a two-semester course for graduate students and advanced undergraduates who have a background in population biology, probability, statistics, and differential calculus. However, the four thematic sections of the book (or combinations thereof) might be useful in a number of different courses. For example, Part I could be used in a course on quantitative methods in population modeling, with a strong emphasis on sampling design and the analysis of biological data. In particular, Chapters 4, 5, and 6 provide a good foundation for the use of statistical methods in the analysis of populations. The material in Part II may be useful in courses on modeling animal populations. For students with limited training in probability, the use of Chapter 10, which provides background material on stochastic processes in population models, can be restricted to the first few sections. We recommend using Chapters 1 and 2 as introductory materials for such a course. Part III could be used in courses on methods of estimating population parameters with count data. The chapters in Part III group naturally into methods for closed populations, which are covered in Chapters 12 through 14, and methods for open populations, which
xv
are covered in Chapters 15 through 20. A course on either topic can be taught with the corresponding chapters in Part III, along with Chapter 4. Of course, a more comprehensive course covering the whole subject matter of Part III could take advantage of materials on both open and closed populations in developing the robust design of Chapter 19. Finally, Part IV may be useful in courses on decision analysis in natural resources. Chapter 3 could help frame the role of models in decision analysis, and the first five sections of Chapter 7 could serve as an aid in describing and formulating dynamic models. The materials in the first section of Chapter 4 on statistical distributions would prove useful as background for decision making under uncertainty. We acknowledge throughout the book that modeling, estimation, and decision making are all very active areas of research in ecology. The necessary framework of theory, methods, and applications already is very broad and in many cases quite elegant, and over the course of writing this book we were both pleased and frustrated that the biological literature was expanding faster than our ability to absorb it. We look forward to continuing developments in these areas and hope that in some small way this book contributes to the effort.
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Acknowledgments
they realized it or not: Carl Bennett, George Burgoyne, Walt Conley, and Glenn Dudderar. Thanks also to an unknown teaching assistant and a pile of gassed Drosophila, in a long ago genetics lab at Michigan State, for convincingly demonstrating the advantages of sampling and estimation over enumeration. Finally, thanks to the many students who have taken my graduate course in population estimation over the years, for their feedback on what "worked" and what didn't, and for innumerable corrections to class notes, which formed the kernel for several sections of the book. I especially thank my current and former graduate students for critical input along the way and for keeping me honest. J.D.N. thanks Christy, Jonathan, and especially Lois for their support and tolerance and coauthors Ken and Mike for their patience, as I was the primary reason for "The Book That Never Ends" almost never ending. I thank my M.S. advisor Bob Chabreck and my Ph.D. advisor Walt Conley for teaching me about wildlife management and science, respectively. My first supervisor, Franklin Percival, provided lots of good advice that has served me well, and he has continued to provide friendship and support. I will follow the lead of Ken and Mike and forego the list of 40+ colleagues and collaborators who have been important influences, but I must acknowledge the special role of Ken Pollock as a friend and collaborator who has shared many ideas with me and patiently listened to mine. Finally, I thank my most frequent collaborator, Jim Hines, the best programmer I know, whose career has been so intertwined with mine that I simply cannot imagine working without him.
We wish to acknowledge the many colleagues, liberally cited in the references, with whom the authors have collaborated over the years on the ideas in the book. Our thanks also for reviews and constructive comments on various portions of the book by Chris Fonnesbeck, Bill Kendall, Clint Moore, Jonathon Nichols, Jim Peterson, Andy Royle, and Nathan Zimpfer. In addition, we thank Jim Hines for computing some of the capture-recapture examples and Shannon Williams for her help with word processing and copy editing. Special thanks from B.K.W. to my coauthors, whom I count myself most fortunate to have as colleagues and friends, and to Fred Johnson, also colleague and friend, who helped to shape many of the ideas expressed in this book about the interface of science and management, especially as concerns the adaptive management of migratory birds. Thanks also to an unnamed faculty member who, during my graduate days many years ago, opined that it was time for me to decide whether I should study math or biology and who thereby started me on a quest to do both, culminating years later in this book. Finally, endless thanks to Genie, who renamed the manuscript "The Book That Never Ends," but who stayed through to the end anyway. With her, the trip is never dull, and she proves daily that what really counts is the going, not the getting there. M.J.C. gratefully acknowledges the love and support of my family, Liz, Mary, and Laura, without whom none of this would have been worth it. Also thanked are key individuals who provided a spark, a kick, or some other form of inspiration at critical moments in the author's early professional development, whether
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PART
I FRAMEWORK FOR MODELING, ESTIMATION, AND M A N A G E M E N T OF ANIMAL POPULATIONS
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C H A P T E R
1 Introduction to Population Ecology
1.1. SOME DEFINITIONS 1.2. POPULATION DYNAMICS 1.3. FACTORS AFFECTING POPULATIONS 1.3.1. Population Regulation 1.3.2. Density Dependence and Density Independence 1.3.3. Population Limitation 1.4. MANAGEMENT OF ANIMAL POPULATIONS 1.5. INDIVIDUALS, FITNESS, AND LIFE HISTORY CHARACTERISTICS 1.6. COMMUNITY DYNAMICS 1.7. DISCUSSION
In this chapter we introduce the concept of a population that changes over time, in response to primary biological processes that influence population dynamics. We discuss the concepts of density dependence and density independence in these processes, and their roles in regulating and limiting population growth. We incorporate these concepts into a biological context of conservation and management of animal populations. The framework of population dynamics as influenced by primary biological processes and their vital rates will be seen to be useful across ecological scales, and in particular will be seen to contribute to a unified frame of reference for investigations at the scale of individuals (evolutionary ecology), populations, and communities.
1.1. SOME D E F I N I T I O N S
A population often is defined as a group of organisms of the same species occupying a particular space at a
particular time (e.g., Krebs, 1972), with the potential to breed with each other. Because they tend to prefer the same habitats and utilize the same resources, individuals in a population may interact with each other directly, for example, via territorial and reproductive behaviors, or indirectly through their use of common resources or occupation of common habitat. Spatial boundaries defining populations sometimes are easily identified (e.g., organisms inhabiting small islands or isolated habitat patches) but more typically are vague and difficult to determine. Spatial and temporal boundaries often are defined by an investigator; however, this arbitrariness does not detract from the utility of the population concept. A key quantity in population biology is population size, which refers to the number of individual organisms in a population at a particular time. In this book, the terms abundance and population size are used synonymously. We reserve the term density for the number of organisms relative to some critical resource. Typically the critical resource is space, so that density represents, e.g., the number of organisms per unit land area for terrestrial species, or the number of organisms per unit water volume for aquatic species. However, the concept of density is sufficiently general that it need not involve space. For example, a meaningful use of the term would be the number of organisms per unit food resource, or in the case of discrete habitat patches, the number of organisms per patch (e.g., the number of ducks per pond on prairie breeding areas). The structure of a population often can be described in terms of the number of individual organisms characterized by specific attributes of interest. For example, the age structure of a population refers to the respective
4
Chapter 1 Introduction to Population Ecology
proportions of individuals in discrete age classes. A population also may be described by its stage structure, with discrete stages defined by variables such as size (the proportions of animals in discrete size classes) (e.g., see Sauer and Slade, 1987a,b), reproductive behavior (e.g., breeders or nonbreeders), or physiological development. In fact, the structure of a population can be described in terms of any attribute thought to be relevant to population dynamics. A common example utilizes the sex ratio of a population, which expresses the proportionate sex composition of a population. 1.2. P O P U L A T I O N D Y N A M I C S Population ecology can be viewed as the study of the distribution of the individuals in a population over time and space. Population ecologists often focus on temporal change in abundance or population dynamics, asking how and why a population changes over time. Temporal population change can be expressed via a simple balance equation that incorporates gains and losses: N(t + 1 ) = N(t) + B(t) + I(t) -
D(t)-
(1.1)
E(t),
where N(t + 1), the population size at time t + 1, is written as a function of population size N(t) at time t, with increases to N(t) during the interval t to t + 1 as a result of reproduction B(t) and immigration I(t), and losses during the interval from mortality D(t) and emigration E(t). The four variables, B(t), I(t), D(t), and E(t), reflect the primary population processes responsible for changes in population size. If an environmental factor or a management action is to influence population size, its influence must be registered through one of these processes. The primary population processes in Eq. (1.1) describe gains and losses in terms of numbers of individual organisms. But births and deaths during the interval (t, t + 1) are likely to depend on the number N(t) of animals in the population at the beginning of the interval. For this reason, it often is useful to rewrite B(t) as B(t) = b(t)N(t), where b(t) is defined as a per capita reproductive rate, or the number of new individuals in the population at time t + 1 resulting from reproduction during (t, t + 1), per individual alive in the population at time t. Similarly, the number of deaths often is rewritten as D(t) = [1 - S(t)]N(t), where S(t) is an interval survival rate, reflecting the proportion of animals alive at time t that are still alive at time t + 1. For populations that are geographically closed (i.e., there are no gains or losses resulting from movement), Eq. (1.1) can be rewritten as N(t + 1) = N(t)[b(t) + S(t)].
(1.2)
For populations that are not geographically closed, it is tempting to write immigration and emigration as functions of N(t). This often is reasonable for emigration, and we can write E(t) as E(t) =e(t)N(t), where e(t) is the proportion of animals in a population at time t that emigrate out of the population by time t + 1. But it is less reasonable for immigration, given that the number of individuals immigrating into the population between t and t + 1 is more likely a function of abundance or density in the source population of immigrants, rather than the size of the recipient population. Immigration thus is treated differently than the other primary population processes, in that it usually is not modeled as a per capita rate based on the recipient population size. Equations (1.1) and (1.2) constitute simple mathematical models of population change, to be discussed in more detail in later chapters. For present purposes, models can be viewed generally as abstractions and simplifications of reality, and in particular, Eqs. (1.1) and (1.2) can be thought of as simple hypotheses about population change. In later chapters we expand and enhance these models, to incorporate a number of biologically relevant factors that influence population change. For example, single-species population models frequently incorporate information about the attributes of individuals in the population, with individuals grouped into classes as defined by variables such as age, size, and sex (e.g., Lefkovitch, 1965; Streifer, 1974; Caswell, 2001). The population then is characterized by a vector specifying the number of individuals in each class or stage. Model enhancements also can include spatial structure, as in Levins' (1970) description of a metapopulation as a "population of populations." Metapopulation models often include different habitat patches that may or may not contain individuals, with reproduction occurring among individuals within a patch and movement of individuals occurring between patches (Levins, 1969, 1970; Hanski and Gilpin, 1997; Hanski, 1999). Metapopulation dynamics are thus a function of both within-patch (reproduction, survival) and between-patch (emigration, immigration) processes. Finally, both single-location and multiplelocation models can be extended to include multiple species and their potential interactions.
1.3. F A C T O R S A F F E C T I N G POPULATIONS Equation (1.1) provides a framework for population change, but carries little information about why populations change. Many questions of ecological and man-
1.3. Factors Affecting Populations agement relevance involve factors that potentially influence the four primary processes driving population change. These can be categorized in many ways, but it often is convenient to think in terms of abiotic and biotic factors. Abiotic factors include physical and chemical characteristics of an organism's environment such as soil type, water availability, temperature, and fire frequency for terrestrial organisms, and water salinity, pH, currents, light penetration, and dissolved oxygen for aquatic organisms. Factors such as these commonly influence population dynamics via multiple rather than single population processes. For example, water and wetland availability on prairie breeding areas in North America can influence duck populations (Johnson et al., 1992) by affecting reproduction (lower probabilities of breeding and increased duckling mortality when conditions are dry), survival of adults (higher mortality of hens associated with predation when nesting during wet years), and movement (increased movement away from relatively dry areas and to relatively wet areas). On the other hand, biotic factors are understood in terms of interactions among members of the same species (intraspecific), or interactions involving species other than that of the population of interest (interspecific). Interspecific factors include vegetative components of the habitat as well as processes such as predation, interspecific competition, parasitism, and disease. Like abiotic influences, they also can affect more than one of the primary population processes. For example, predation clearly influences mortality, but may also influence movement (increased emigration from areas with large numbers of predators) and reproduction (decreased probability of reproducing in response to increased Predation risk). Intraspecific factors involve interactions among the individuals in a population, with potential influences on all of the primary population processes. They often involve direct behavioral interactions, in which some individuals in the population actively exclude other members of the population from habitat patches or deny access to food resources or even to members of the opposite sex. But they also can involve indirect interactions, through the possible depletion of common resources and the occupation of common habitat. Indirect interactions such as these almost always involve other biotic and abiotic factors.
1.3.1. Population Regulation Because population processes are influenced simultaneously by abiotic and biotic factors, there may be only limited value in trying to ascertain which class of factors is most relevant to population change. Never-
5
theless, the history of population ecology has been characterized by repeated arguments about the relative importance of abiotic vs. biotic factors in controlling population dynamics, and the importance of interspecific vs. intraspecific factors (e.g., see Nicholson, 1933; Andrewartha and Birch, 1954; Lack, 1954; Slobodkin, 1961; Reddingius, 1971; Murdoch, 1994). Much of this debate has focused on explanations for the simple observation that populations do not increase indefinitely (Malthus, 1798). The terms population regulation and population limitation refer to concepts that emerge from the impossibility of indefinite population increase. Population regulation refers to the process by which a population returns to an equilibrium size (e.g., Sinclair, 1989). A glance at Eq. (1.1) indicates that in order for a population to grow [i.e., N(t + 1) > N(t)], gains must exceed losses, or B(t) + I(t) > M(t) + E(t). On the other hand, the equilibrium condition N(t + 1) = N(t) is attained when additions to the population equal losses, that is, when B(t) + I(t) = M(t) + E(t). A growing population eventually must reach a state in which the primary population processes change in the direction of equilibrium, that is, births and immigration decrease a n d / o r deaths and emigration increase until gains equal losses. Population ecologists have expended considerable effort in attempting to identify factors that can influence the primary processes of growing populations and thereby produce equilibrium. In reality, such an equilibrium is not likely to be a single fixed population size. Instead, regulation can be viewed as producing a "long-term stationary probability distribution of population densities" (Dennis and Taper, 1994; Turchin, 1995). Murdoch (1994) identified regulation with "boundedness," noting that some cyclic and chaotic populations can also be viewed as regulated.
1.3.2. Density Dependence and Density Independence The debate about population regulation often is framed in terms of density dependence and density independence. Sometimes these concepts are defined in terms of the rate of population change ~'t = N(t + 1) / N(t), although such definitions can become relatively complicated (Royama, 1977, 1981, 1992). Our preference is to define density dependence and density independence in terms of the vital rates associated with the primary population processes. For example, the vital rates associated with a geographically closed population are the survival rate S(t) and reproductive rate b(t) in Eq. (1.2). Though the absolute numbers of births b(t)N(t) and deaths [1 - S(t)]N(t) occurring during the interval (t, t + 1) obviously depend on the population
Chapter 1 Introduction to Population Ecology size at the beginning of the interval [see Eq. (1.2)], density dependence is defined by the functional dependence of a vital rate on abundance or density {i.e., S(t) = fiN(t)] a n d / o r b(t) = g[N(t)]}. Density independence refers to the absence of such a functional dependence. Examples of density dependence might include survival and reproductive rates, which typically decrease as abundance or density increases. The relevance of this concept to population regulation is that regulation requires negative feedback between ~'t (and thus the vital rates that produce kt) and population size at t or some previous period. Finally, we note the possibility of Allee effects, in which survival and reproductive rates may decrease in populations at very low density (e.g., Allee et al., 1949; Courchamp et al., 1999; Stephens and Sutherland, 1999). The concepts of density dependence and density independence provide another means of classifying factors affecting animal populations. Some factors operate as functions of density or abundance (i.e., in a density-dependent manner) and represent dynamic feedbacks. For example, in some rodent populations, intraspecific aggressive behavior among individuals appears to increase as density increases, leading to decreased rates of survival and reproduction (Christian 1950, 1961). Interspecific factors also can act in a density-dependent manner, as when rates of predation or parasitism depend on the abundance of the prey or host population (e.g., Holling, 1959, 1965). On the other hand, some factors act in a densityindependent manner, absent dynamic feedback. When flooding reduces alligator reproductive rates by destroying nests, the magnitude of the reduction in reproductive rate depends on the proportion of nests that are constructed in susceptible locations (e.g., Hines et al., 1968), but not on alligator density. Similarly, severe grassland fires may cause direct mortality of insect and small mammal inhabitants, but the increase in mortality associated with fire events typically is independent of the density of the affected population. In some situations, factors acting in density-dependent and density-independent manners interact, as when density-dependent decreases in reproductive rate occur because of increases in numbers of cavity-nesting birds using a fixed supply of cavities (Haramis and Thompson, 1985).
1.3.3. Population Limitation Every population is restricted in its growth potential, with a range of conditions beyond which the population tends to decrease because of reductions in survival rates, reproduction rates, or both. Consider a population at equilibrium, such that gains equal losses
over time and population size does not deviate greatly from some average or expected value. Limitation refers to "the process which sets the equilibrium point" (Sinclair, 1989) or, more generally, that determines the stationary probability distribution of population densities. Limitation can involve factors that act in a densitydependent manner as well as factors that are density independent. A limiting factor can be defined as one in which changes in the factor result in a new equilibrium level (Fretwell, 1972) or, more generally, a new stationary distribution of population densities. For example, if predation is a limiting factor for a prey population, then a sustained decrease in predation should bring about an increase in equilibrium abundance of the prey. This new equilibrium level would itself be determined by the action of other factors on the primary population processes. Consistent with this definition of a limiting factor is the recognition that populations potentially have multiple equilibria, and a given population may move among equilibria as conditions and limiting factors change (e.g., Hestbeck, 1986).
1.4. M A N A G E M E N T OF ANIMAL POPULATIONS Interest in certain animal populations has led to management efforts to try to achieve population goals. These goals frequently involve a desired abundance and, for harvested species, a desired level of harvest. Some animal species exist at abundances thought to be too great, and management efforts are directed at reducing abundance. These include pest species associated with human health problems [e.g., Norway rats (Rattus norvegicus); see Davis, 1953] and economic problems such as crop depredation [e.g., the use of cereal crops by the red-billed quelea (Quelea quelea) in Africa; see Feare, 1991]. Other species are viewed as desirable, yet are declining in number or persist at low abundance. Relevant management goals for the latter typically involve increases in abundance, in an effort to reduce the probability of extinction in the near future. Such a goal is appropriate for most threatened and endangered species, and methods for its achievement dominate the field of conservation biology (e.g., Caughley, 1994; Caughley and Gunn, 1996). Still other species are judged to be at desirable abundances, and management efforts involve maintenance of population size. Finally, for harvested species, an abundanceoriented goal must be considered in the context of maintaining harvest yield that is consistent with recreational a n d / o r commercial interests (e.g., Hilborn and Walters, 1992; Nichols et al., 1995a).
1.5. Individuals, Fitness, and Life History Characteristics If management is to influence animal abundance, then it must do so by influencing at least one of the four primary population processes in Eq. (1.1). For example, white-tailed deer are judged to be overabundant in portions of eastern North America, and management efforts to reduce abundance have been directed at both increasing mortality (via hunting and culling operations) and decreasing reproduction (via sterilization and chemical contraception) (McShea et al., 1997; Warren, 1997). Management efforts directed at endangered species frequently involve attempts to decrease mortality via predator control, or attempts to influence reproduction, emigration, and mortality by setting aside or maintaining good habitat. For harvested species, the regulation of harvests focuses on both harvest yield (harvest regulations should influence yield directly) and abundance (harvest regulations influence abundance by changing rates of mortality and, sometimes, movement). The concepts of population limitation and regulation underlie population management, especially as they factor into the roles of density dependence and independence. For example, the manager of a threatened or endangered species can utilize an understanding of limiting factors to effect management actions to improve the species status. Many endangered species are habitat specialists that are thought to be limited by the amount of suitable habitat available to them. Thus, the purchase or creation of additional habitat represents an effort to remove a limiting factor and to permit the population to increase to a new equilibrium level commensurate with the expanded habitat. Of course, a population increase occurs because of changes in the primary population processes corresponding to the increase in habitat, and it often is useful to focus on the processes as well as the limiting factors. The concept of density dependence is especially important in management of harvested populations. As a direct mortality source, harvest acts to reduce abundance. However, reduced abundance may lead to increases in reproductive rate or to decreases in nonharvest mortality or emigration, depending on which vital rates behave in a density-dependent manner. For example, much fisheries management is based on stock-recruitment models that incorporate densitydependent reproductive rates (e.g., Beverton and Holt, 1957; Ricker, 1975; Hilborn and Walters, 1992). Management of North American mallard (Anas platyrhynchos) populations is based on competing models that represent different sets of assumptions about the density dependence of survival and reproductive rates (Johnson et al., 1997). Because our definitions of density dependence and independence involve the populationlevel vital rates of survival, reproduction, and move-
7
ment, density dependence again directs the manager's attention to the primary population processes.
1.5. I N D I V I D U A L S , F I T N E S S , A N D LIFE HISTORY CHARACTERISTICS The comments above, and indeed most chapters in this book, focus on the population level of biological organization. However, it is important to remember that the constituents of populations are individual organisms, and the characteristics of these organisms are shaped by natural selection. Characteristics associated with relatively high survival or reproductive rates are favored by natural selection, in that organisms possessing them tend to be represented by more descendants in future generations than do other organisms. Individuals with greater potential for genetic representation in future generations are said to have relatively high fitness. Though they typically are thought to deal with different levels of biological organization, fitness and population growth are closely related. Thus, the growth rate of a geographically closed population is determined by survival rate and reproductive rate, whereas the fitness of an individual organism is determined by its underlying probabilities of surviving from year to year and of producing 0, 1, 2 , . . . offspring each reproductive season. Indeed, fitness associated with a particular genotype can be defined operationally as the growth rate of a population of organisms of that genotype (see Fisher, 1930; Stearns, 1976, 1992; Charlesworth, 1980). An important consequence of the close relationship between population growth and individual fitness is that evolutionary ecologists, population ecologists, and population managers are often interested in the same population processes and their vital rates. Nevertheless, a subtle difference can exist between definitions of survival and reproductive rates at the population and individual levels of organization. We defined the interval survival rate S(t) as the proportion of animals in the population at time t that survives until time t + 1. This quantity is not so useful at the level of the individual organism, because an organism either survives or it does not; however, it can be thought of as having some underlying probability of surviving the interval between times t and t + 1. These two distinct quantities, the probability that an individual survives and the proportion of animals in a population that survive, are closely related. Consider a population of individuals with identical underlying survival probabilities for some interval of interest. The
Chapter I Introduction to Population Ecology proportion of individuals that survives the interval likely is not identical with the underlying individual survival probability. On the other hand, the proportion that survives is expected to deviate little from the individual survival probability. More precisely, multiple realizations of population dynamics over comparable time intervals would produce an average proportion of survivors approaching the individual survival probability. In Chapter 8 we define the terms needed to specify the relationship between population-level survival rate and individual probability of survival. The important point for now is that these quantities are closely related. Throughout most of this book, we will use the terms survival rate and survival probability interchangeably to refer to the underlying individual survival probability. When discussing survival at the population level we will use the term survival rate to denote the surviving proportion of a population or group. Of course, the latter quantity is of interest regardless of whether all individuals in the population have the same survival probability. A similar situation exists for reproductive rate. An individual can produce some integer number of offspring {0, 1, 2 , . . . } during a single reproductive season, but a reproductive rate refers to the number of offspring produced per adult in the population. In essence, this offspring/adult ratio is a population-level attribute. The term reproductive rate could refer in concept to (1) the average number of young produced if we could observe an individual over many replicate time intervals or (2) the average number of young produced per adult in the population if we could observe the population over many replicate time intervals. Our intention here is not to dwell on subtle differences in the terms used for individuals and populations, but instead to emphasize the role of vital rates in determining both fitness and population growth. In the discussion above we suggested that the concepts of population limitation and regulation follow naturally from the simple observation that populations do not increase indefinitely. Similarly, evolutionary ecology is based on the observation that neither species nor populations of genotypes can increase indefinitely, though temporary increases are possible. Species and populations of genotypes must eventually reach a state in which temporary increases and declines in numbers of individuals fluctuate about some equilibrium over time. The necessary balance between average survival and reproductive rates has led to various classification schemes [e.g., r- and K-selected species, "fast" versus "slow" species (Cody, 1966; MacArthur and Wilson, 1967; Boyce, 1984; Stearns, 1992)] for species based on these average values. A basic idea underlying all of
these schemes is that species with high reproductive rates must also be characterized by high mortality rates, whereas species with low reproductive rates must also have low mortality rates. The underlying survival and reproductive rates that apply at each age throughout an organism's lifetime are frequently referred to as life history characteristics (Cole, 1954; Stearns, 1976, 1992). Most discussions of life history characteristics also include features such as age at first reproduction, individual growth rate, body size, and age at which individuals can no longer reproduce (see Chapter 8). However, the relevance of these features to life history evolution involves their relationship to the age-specific schedule of survival and reproductive rates. The magnitudes of survival and reproductive rates throughout the organism's lifetime often are viewed as species-specific characteristics, allowing for variation in survival and reproduction rates among individuals. The expectation is that variation among individuals within a species typically is much smaller than variation among individuals of different species. The suite of life history characteristics is important not only for understanding and predicting population dynamics, but also for managing populations. Consider, for example, the management of two harvested species, one with high mortality and reproductive rates (e.g., several commercially harvested fish species) and one with low reproductive and mortality rates (e.g., harvested whales). Imposition of a fixed harvest rate (proportion of animals in the population harvested) typically has a larger influence on the population dynamics of the species with the otherwise low mortality and the low reproductive rate. In addition to low per capita reproductive rates, such species tend to exhibit delayed sexual maturity, with the consequence that they take longer to recover from decreases in abundance. In summary, there is a close relationship between fitness and population change, despite the fact that these quantities apply to different levels of biological organization. One consequence of this relationship is that even. though population ecologists, population managers, and evolutionary ecologists address different kinds of questions and have different objectives, they are all concerned with population vital rates. Thus, the methods presented in this book for estimating vital rates should be relevant to scientists in these different disciplines. Another consequence is that life history characteristics molded by natural selection are relevant to population dynamics and population management. Knowledge of a species' life history characteristics is of key importance in predicting population
1.7. Discussion responses to management, and thus should play an important role in management decisions.
1.6. C O M M U N I T Y D Y N A M I C S In this book, our focus occasionally shifts to the community level of biological organization, where the term community refers to a group of populations of different species occupying a particular space at a particular time. A community may include all the different plant and animal species represented in the space, or, more commonly, may refer to a subset of species defined by taxonomy (e.g., the bird community of an area), functional relationships (e.g., vegetative or herbivore community), or other criteria that are relevant to a question of interest. One way to model community-level dynamics is to model the population for each species, perhaps linking the models via the sharing of resources to induce interactions. For example, consider a simple model of a single predator species and a single prey species. The survival and reproductive rates of the predator species might be modeled as functions of prey species abundance, such that larger numbers of prey lead to higher survival and reproductive rates of the predator species. In the same model, the survival rate for the prey species could be written as a function of predator abundance, with more predators leading to reduced survival for the prey species. A similar approach frequently is taken for the modeling of interspecific competition. The importance of population-level vital rates is again emphasized in this modeling approach, as the interactions between populations are specified as functional relationships involving the vital rates (or composite quantities that combine vital rates). A less mechanistic and more descriptive approach for community-level modeling does not focus on interspecific interactions. This modeling approach has been used by community ecologists (e.g., MacArthur and Wilson, 1967; Simberloff, 1969,1972) and by paleobiologists (Raup et al., 1973; Raup, 1977) and simply involves models such as those of Eqs. (1.1) and (1.2) shifted to the community level. Thus, instead of projecting changes in numbers of individual organisms within a population, the models specify change in the numbers of different species in the community. The primary population processes and their corresponding vital rates are replaced by analogous processes and vital rates at the community level.
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To see how, let N(t) denote the number of species in the community at time t, with S(t) the species-level survival rate (the complement of local extinction rate) for the interval t to t + 1, and I(t) the number of colonists during the interval (species absent from the community at t, but present at t + 1). Using notation similar to that of Eqs. (1.1) and (1.2), the natural expression for change in the number of species in the community is
N(t + 1 ) = N(t)S(t) + I(t). Consideration of the processes determining S(t) and I(t) again leads back to the primary population processes and associated vital rates. Local extinction rate for a species-population is a function of populationlevel rates of survival, reproduction, immigration, and emigration, and the number of colonizing species is a function of immigration at the population level. The approach of representing a "population" of species via a model for which local extinction plays the role of mortality, and immigration/colonization plays the role of reproduction, is a natural extension of the biological framework portrayed in Eq. (1.1). This analogy has been used in biogeography for many years (MacArthur and Wilson, 1967) and is used frequently in other fields such as conservation biology (e.g., Rosenzweig and Clark, 1994; Russell et al., 1995; Boulinier et al., 1998, 2001; Cam et al., 2000). 1.7. D I S C U S S I O N In this chapter we have introduced the biology of animal populations in terms of the fundamental processes of survival, reproduction, and migration, along with their associated vital rates. These quantities define the balance equation [Eq. (1.1)] by which population dynamics can be investigated, and they also provide a basis for understanding the factors that influence population dynamics. In the chapters to follow we make liberal use of this framework, as we focus on the modeling of populations and the estimation of population attributes. We will see that quantities such as population size, harvest numbers and rates, recruitment levels, and migration patterns are key to an understanding of population dynamics. We focus much of what follows on the use of field data to estimate these and other population parameters. A careful accounting of the statistical properties of these estimates will be seen to be an essential component in the informed conservation of animal populations.
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C H A P T E R
2 Scientific Process in Animal Ecology
2.1. CAUSATION IN ANIMAL ECOLOGY 2.1.1. Necessary Causation 2.1.2. Sufficient Causation 2.2. APPROACHES TO THE INVESTIGATION OF CAUSES 2.3. SCIENTIFIC METHODS 2.3.1. Theory 2.3.2. Hypotheses 2.3.3. Predictions 2.3.4. Observations 2.3.5. Comparison of Predictions against Data 2.4. HYPOTHESIS CONFIRMATION 2.5. INDUCTIVE LOGIC IN SCIENTIFIC METHOD 2.6. STATISTICAL INFERENCE 2.7. INVESTIGATING COMPLEMENTARY HYPOTHESES 2.8. DISCUSSION
ology. Thus, the objective of this chapter is to provide a biological context for scientific methodology, and in so doing to clarify the respective roles of theory development, statistical inference, and the structures of formal logic in animal ecology.
2.1. C A U S A T I O N IN ANIMAL ECOLOGY Science is about the identification and confirmation of causes for observed phenomena, whereby "cause" is meant as a generic explanation of patterns observed for a class of phenomena. The explanatory power of a cause results from the ability to entail many, often apparently disparate, phenomena under its rubric. Causes are recognized as "explanatory" in the context of a scientific theory of which they are components, the theory itself consisting of relatively few causal factors entailing a wide range of phenomena. More formally, causation can be described in terms of antecedent conditions, consequent effects, and a rule of correspondence for their conjoint occurrence. In population biology the "effect" of a cause typically is a biological event (e.g., mortality, growth, population change) that occurs subsequent to the occurrence of some prior condition. Provided the joint occurrence of the prior condition and the subsequent event meet certain theoretical and logical requirements, the prior condition is held to be the cause of the event. The causal linkage between a prior condition and a subsequent effect can be described in terms of the logic of material implication (Copi, 1982). The expression A --~ B describing material implication is taken to mean
However varied the practice of animal ecology, a common feature is the comparison of predictions, deduced from biological hypotheses, with data collected pursuant to the comparison. Much has been written about the testing of biological/ecological hypotheses (Romesburg, 1981; Hurlbert, 1984; Peters, 1991), and specifically about sampling designs and statistical inferences for hypothesis testing (Green, 1979; Hairston, 1989; Skalski and Robson, 1992). However, much of this documentation has focused on the characterizing of biological hypotheses in terms of statistical distributions, and on the investigation of distribution attributes with sample data (Brownie et al., 1985; Burnham et al., 1987; Lebreton et al., 1992). It is useful to consider how these activities fit into a broader context of theory, logic, and data analysis that is definitive of scientific method-
11
12
Chapter 2 Scientific Process in Animal Ecology
that affirmation of the premise A implies affirmation of the conclusion B. However, material implication is silent about tile affirmation of A given that B is affirmed. More formally, material implication establishes the equivalence of A ~ B with the assertion that either A is false or B is true. Thus, one can look to the premise of A ~ B to confirm its conclusion, but one cannot look to the conclusion of A ~ B to confirm its premise. The concept of causation in scientific inquiry is informed by the logic of material implication, by identifying cause (C) and effect (E) as either premise or conclusion. Two distinct definitions of causation can be identified.
2.1.1. Necessary Causation In this case an effect E points to a presumptive cause C, in that the occurrence of the effect guarantees the occurrence of condition C. A logically equivalent argument is ---C --~ ---E, i.e., the nonconcurrence of C guarantees the nonconcurrence of the effect (the symbol ~-in this expression is used to indicate logical negation, so that ---C, which is read "not C," means that the truth of C is negated). Thus, necessary causation asserts that the absence of an effect follows from the absence of the cause. However, it is silent about effect E in the presence of C. Examples of necessary causation might include light as a cause of photosynthesis, Salmonella bacteria as a cause of typhoid fever, and fuel loads as a cause of forest fires. In each example the effect may or may not be present when the presumptive cause occurs; however, the effect is held to be absent when the cause is missing.
2.1.2. Sufficient Causation In this case the presumptive cause C points to the effect E, in that the occurrence of condition C guarantees the occurrence of the effect. Thus, sufficient causation asserts that the occurrence of an effect follows from the presence of condition C. However, it is silent about the effect in the absence of C. Sufficient causation might underlie an argument that heat causes fluid dynamics; that a low level of ambient oxygen during respiration causes the production of lactic acid; that oxygenation of pig iron under high pressure causes the production of steel; that drought causes physiological stress in nonsucculent plants. In these examples the presence of the cause is held to ensure the presence of the effect; however, the effect may or may not be present in the absence of the cause. Sufficient causation is a logically stronger definition than necessary causation, in that necessary causation specifies C as one condition (possibly among many) that must be present to ensure the occurrence of an
effect, whereas sufficient causation specifies that C alone ensures its occurrence. An otherwise necessary cause can be recognized as sufficient by restricting the range of conditions in which it is operative. Thus, a concentrated source of heat (e.g., a lighted match) is a necessary cause of combustion, but a heat source in the presence of combustible material in a cool, dry, oxygenated environment becomes a sufficient cause (under these conditions). The importance of maintaining a clear distinction between necessary and sufficient causation can be illustrated by the controversy about smoking as a potential cause of lung cancer. Advocates for restricting the advertisement and sale of tobacco products base their arguments on the strong statistical association between tobacco use and the occurrence of lung cancer, wherein the great majority of lung cancer victims in the United States also have a history of smoking. On the other hand, opponents of tobacco restrictions have argued repeatedly that the association between smoking and lung cancer is not causal, and cite as evidence the fact that a majority of smokers in the United States do not have lung cancer. Clearly, these conflicting positions (and different assessments of evidence) point to inconsistent uses of the concept of causation. Apparently advocates of tobacco restrictions assume necessary causation, such that a history of tobacco use is inferred from the occurrence of lung cancer. Evidence for smoking as a necessary cause of lung cancer focuses on the fact that lung cancer victims overwhelmingly have a history of smoking, and a key implication is that the avoidance of smoking implies the near absence of lung cancer. On the other hand, opponents of tobacco restrictions appear to use sufficient causation, wherein smoking should lead to the occurrence of cancer. By implication, the absence of cancer therefore should imply the absence of smoking, which is inconsistent with the fact that the overwhelming proportion of smokers have no record of lung cancer. Hence, tobacco is held not to be a cause of lung cancer by opponents of tobacco restrictions. Given the inconsistent uses of causation, it is not surprising that the controversy between advocates and opponents of tobacco restrictions has not been amenable to data-based resolution. Indeed, the evidence likely will continue to indicate that tobacco use is simultaneously a cause of lung cancer (in the necessary sense) and not a cause of lung cancer (in the sufficient sense).
2.2. APPROACHES TO THE O F CAUSES
INVESTIGATION
A definition of cause as necessary often applies to the control of unwanted effects, whereby the elimina-
2.3. Scientific Methods tion of an effect (e.g., typhoid fever) is assured by the elimination of the cause (e.g., destruction of Salmonella bacilli through sterilization). Scientific investigation thus involves a search for conditions that are predictive of the nonconcurrence of an effect of concern. Necessary causation often is implied in population biology when biological effects in the presence of a particular condition are attenuated by the restriction or removal of the condition. A particular example is duck nest predation as a presumptive (necessary) cause of reproductive failure in cultivated prairie lands under nondrought conditions. The implication is that reducing predation will reduce reproductive failure. On the other hand, a definition of cause as sufficient applies to causes (e.g., drought) that guarantee an effect (e.g., physiological stress in plants). Scientific investigation in this case involves the search for conditions that are predictive of the occurrence of an effect. Sufficient causation is implied in population biology when the influence of a prior condition is both direct and adequate to produce an effect of concern. A relevant example is the investigation of sport hunting as a potential cause of declining waterfowl population trends. Thus, heavy hunting pressure is hypothesized to reduce survival and depress population levels, recognizing that population declines can occur even in the absence of hunting. Necessary and sufficient forms of causation share a natural linkage with the experimental elements of treatment and control. A typical experiment investigates the association between some putative causal factor C and an effect E, with the idea that the cooccurrence (along with joint nonoccurrence) of C and E provides evidence for causation. The experiment has treatment C imposed to determine whether effect E occurs in its presence, i.e., to investigate whether C is a sufficient cause of E in the sense of C ~ E. By the rules of logical inference, the occurrence of E is insufficient by itself to support a claim of causation. However, one can infer from an absence of E that the treatment cannot be a (sufficient) cause of E. On the other hand, experimental control allows one to investigate whether the absence of E follows from the absence of C, i.e., whether ---C ~---E. But this is logically equivalent to the assertion of necessary causation, that is, E--~ C. Under experimental control, the absence of effect E is inadequate to support a claim of causation. However, one can infer from the occurrence of E that the treatment cannot be a (necessary) cause of E. It is the coupling of inferences from both treatment and control in an experiment that confers logical rigor to designed experiments. Experimental results in which E occurs in the presence of C but not in its absence provide the evidence for necessary and suffi-
13
cient causation. Under these conditions no other factor than C can cause the effect, for otherwise E presumably would be recorded in the absence of C, in violation of the requirement for co-occurrence. C is therefore recognized as the cause, and the only cause, of E under the experimental conditions. This very high standard illustrates the value (and rigor) in establishing causation through experiment and helps to explain why experimental design is a near-imperative in much of biological science.
2.3. S C I E N T I F I C M E T H O D S A useful context for scientific method involves scientific investigation both before and during a period when it is guided by a recognized theoretical framework. Thus, in its early stages, scientific activity consists of observation guided primarily by intuition, tradition, guesswork, and perceived pattern. Its function initially is to organize observations into coherent categories, to explore these observations for patterns, and to describe the patterns clearly. The process of recognizing the underlying causes of patterns comes as the scientific discipline matures, and a set of relationships, which are accepted as "explanatory," is formulated. These relationships are sometimes called a theoretical paradigm or, more briefly, a theory (Kuhn, 1970). A standard for the operation of science, including biological science, involves a comparison of theoretically based predictions against data, recognizing that a match between data and prediction provides evidence of hypothesis confirmation, and the lack of such a match disconfirms a hypothesis (Hempel, 1965). A somewhat more detailed treatment includes five elements: theory, hypotheses, predictions, observations, and comparisons of prediction against data. 2.3.1.
Theory
First, an explicit statement of a relevant theory is necessary, or at least the reference to it is necessary. The theory is expressed in terms of the axioms, postulates, theoretical constructs, and causal relationships among constructs that constitute the corpus of the theory. This corpus, involving biological elements such as genetics, taxonomy, evolutionary principles, and ecological relationships (Hull, 1974), is operationally accepted as verified and true. A theory is noted in what follows by {T}. Every scientific discipline is founded on an operational theory, which provides a conceptual framework through which the world is observed and facts about the world are discerned. Broadly recognized examples
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might include the theory of relativity, electromagnetic field theory, the theory of plate tectonics in geology, thermodynamic theory, and the theory of evolution by natural selection. An operational theory allows one to recognize patterns among apparently disparate phenomena and to explain relationships among them. It also is the foundation for hypothesis formulation, prediction, and testing. In short, a theory is essential to the conduct of scientific investigation.
amended theory and observed reality. The derivation of predictions is designated by {T} + H --~ P, where P represents a prediction and the arrow indicates logical inference. The notion here is that the addition of H to {T} allows for inferences that otherwise would not follow from {T} in the absence of H. At least some of these inferences are testable, in that they predict observable phenomena that potentially are verifiable with field or experimental data. The key here is that P consists of potentially observable predictions.
2.3.2. Hypotheses Second, a hypothesis that is relevant to the theory is identified, often through field or laboratory observations that appear to be anomalies to the theory, i.e., that appear not to be explained adequately by the theory as it currently is understood. A hypothesis, denoted here by H, asserts a claim about relationships among components of the theory, or about relationships of these components to observed reality, or about relationships among entities in the observed world that are presumed to follow from the hypothesis. An example of the first kind of claim might be the recognition that one component of the theory entails another; an example of the second kind is the predicted existence of heretofore unrecognized sociobiological patterns; and an example of the third kind is the dynamics of dispersal following certain kinds of environmental disruption. We emphasize in what follows the investigation of causal hypotheses, involving antecedent conditions and consequent effects that are identified in a theoretical context. A hypothesis is recognized as potentially true or false. When added to a theory, it renders the theory potentially inconsistent, or potentially false. In what follows, an amended theory is designated by {T} + H, to indicate that H is included as one of the elements defining the amended theory. This notation suggests an attendant increase in theory complexity. Alternatively, H can replace a particular hypothesis H 0 within the body of the theory. This is designated by {T0} + H, where {T0} + H 0 represents the theory before amendment. Scientific investigation then becomes a comparison of the relative explanatory power of the two theoretical constructs {T0} + H 0 and {T0} + H. To simplify notation we use {T} + H to represent both the appending of H to {T} and the replacement of H 0 in {T} b y H .
2.3.3. Predictions Third, potentially observable conclusions are deduced from the amended theory. These follow from logical relationships inherent in the amended theory, or they are derived from relationships between the
2.3.4. Observations Fourth, field or experimental data are collected that are pertinent to the predictions. The investigator's attention is directed to these data by the amended theory, which is used as above to derive predictions for which the data are relevant. Field a n d / o r experimental data, designated by observation O, are essential components by which the amended theory is to be evaluated. Key to successful data collection are statistically sound surveys, experiments, and other data collection instruments.
2.3.5. Comparison of Predictions against Data Fifth, predictions from the amended theory are compared to observations O from the field or laboratory. This comparison is used to determine the acceptability of the amended theory and hence the acceptability of the hypothesis H. If O conforms to P, i.e., if the predicted results of {T} + H are in fact observed, then the investigation provides evidence to confirm H. If O does not conform to P, then the evidence disconfirms H. Statistical testing procedures play a crucial role in the process of hypothesis confirmation. An ideal approach to scientific investigation consists of repeated applications of this sequence across all levels of investigation. Thus, alternative hypotheses often are part of a study design, wherein two or more hypotheses may be considered as alternatives for theory amendment. For a given hypothesis H, numerous predictions may be identified, each worthy of field investigation. For each prediction P, data from several different field and laboratory studies may be appropriate. In addition, studies involving the same hypothesis, the same prediction, and the same kind of data collection often are repeated numerous times, to add to the strength of evidence for confirmation or disconfirmation. We note that in concept, one could identify hypotheses without theoretical justification or guidance, and provided that predictions of the hypotheses are directly
2.3. Scientific Methods measurable, one could collect data that are relevant for testing. However, there are two serious problems with hypothesis testing that is not informed by theory: (1) it is much less likely that one can identify potentially useful and informative hypotheses for investigation, and (2) it is more difficult to determine the appropriate data to collect in support of confirmation or disconfirmation. Theory plays a key role in resolving both these problems, by directing the investigator's attention to theoretically interesting questions, testable predictions, and useful data for comparison against those predictions. Absent a theoretical context for the play of logic in recognizing testable predictions, it becomes much less likely that scientifically meaningful hypotheses can be identified, or that relevant data can be targeted for their testing (e.g., Johnson, 1999).
Example Consider a wildlife species that is exposed annually to sport hunting. A traditional concern in game management is the effect of harvest on future population status, and in particular the effect of harvest on annual survival. Two competing hypotheses have been identified: 1. The hypothesis of additive mortality asserts that harvest is additive to other forms of mortality such as disease and predation. Under this hypothesis the annual mortality rate increases approximately linearly in response to increases in harvest rate. 2. The hypothesis of compensatory mortality asserts that harvest mortality may be compensated by corresponding changes in other sources of mortality. Thus, increases in harvest rate have no effect (up to some critical level c of harvest) on the annual mortality rate. In the standard formulation of the compensatory hypothesis, harvest rates beyond c result in an approximately linear increase in annual mortality. We refer the reader to Anderson and Burnham (1976) and U.S. Department of the Interior (1988) for a more complete development of these relationships. The compensatory and additive hypotheses provide a convenient point of reference for the process of scientific investigation. Research on the effect of hunting is conducted in the context of a theory of population dynamics recognizing structural, functional, and dynamic characteristics of wildlife populations in an ecosystem of interrelated organisms and abiotic processes. Elements of the theory involve reproduction, survival, and migration as influenced by factors such as interspecific interactions, physiological condition, behavioral adaptations, and seasonal habitat conditions. The edifice of concepts, relationships, axioms, and terms relating to populations constitutes the scientific para-
15
digm of population ecology [see Baldasarre and Bolen (1994) for a review of theory and management as concerns waterfowl populations]. It is in the context of this paradigm that the relation between mortality and harvest rate can be investigated. The investigation proceeds with deduction of testable predictions, following from the paradigm along with the compensatory and additive hypotheses. Three general predictions can be recognized for waterfowl populations (Nichols et al., 1984a): 1. The compensatory mortality hypothesis leads to a prediction that there is no relationship between annual survival rate and hunting mortality, so long as harvest rate is less than the critical value defined in the hypothesis. On the other hand, the additive mortality hypothesis suggests that there is negative relationship between annual survival rate and hunting mortality over the whole range of potential harvest rates. 2. Under reasonable conditions, the compensatory mortality hypothesis leads to a prediction that there is a negative relation between hunting mortality rates (during the hunting season) and nonhunting mortality rates (during and after the hunting season). The additive mortality hypothesis leads to a prediction that there is no such relationship. 3. The compensatory mortality hypothesis leads to a prediction that there is a positive relation between nonhunting mortality rate and population size or density at some time in the year. In many circumstances nonhunting mortality rate after the hunting season should be positively related to population size at the end of the hunting season. The additive mortality hypothesis leads to a prediction that there is no relationship between nonhunting mortality rate and population size. These predictions differ considerably in the degree to which they represent explanatory causes of population dynamics, and the difficulty with which data can be collected and used informatively for testing (Conroy and Krementz, 1990). Indeed, it always is an outstanding challenge in scientific investigation to devise ways of collecting data that are pertinent to testable predictions. In this particular case, population surveys (Thompson, 1992), radiotelemetry (White and Garrott, 1990), mark-recapture procedures (Nichols, 1992), banding studies (Brownie et al., 1985), and other field procedures can provide valuable data by which to test the predictions. Such studies can be replicated at different times and different locations, under a variety of different field conditions and different harvest strategies, with a focus on one or any combination of the predictions listed above. Each study adds evidence by which investigators can confirm or disconfirm the
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hypotheses. Replication and redundancy of this kind play an important role in preventing unwarranted generalizations of study results.
In contrast, argument (2) above has a very different logical content. Here the assertion is of the form A--4 B B .'. a
2.4. HYPOTHESIS CONFIRMATION The logic of hypothesis confirmation can be expressed in general terms by means of material implication. The process is denoted by
(1)
{T} + H--~P O---~ --,P {T} O .'. ---H
or
(2)
{T} + H - - 4 P O--4 P {T} O .'. H
In these formulations the first premise asserts that prediction P is a consequent of an amended theory, as described above. The second essentially asserts that P is disconfirmed by observation (argument 1) or that P is confirmed by observation (argument 2). The third premise asserts the truth of theory {T}, and the fourth represents the observed data O. A horizontal line separates the argument's premises and evidence from its conclusion, which is stated on the last line. Again, the symbol --- in argument (1) is used to indicate logical negation, so that the expression O--4---P means "the truth of O implies that P is false" (i.e., the observation indicates that the prediction is incorrect). Though the two arguments above appear to be analogous in their forms, there is a crucial asymmetry in their logical content. Argument (1) is an example of the syllogistic form modus tollens (Copi, 1982), wherein rejection of the conclusion in an argument of material implication implies rejection of the premise: A --4 B ---B .'. ---A
Applying modus tollens to the scientific argument above, the observations O do not correspond to what was predicted; thus, O --4 ---P in the second line of the argument. But ---P implies ---{T} + H from the first line of the argument, which in turn implies either {T} or H (or both) is untrue. Because {T} is assumed in the third line of the argument to be a confirmed and operational theory, this leaves the falsity of H as a conclusion of the argument. Hence the conclusion ---H. Simply put, this argument states that evidence contrary to a hypothesis is logically sufficient to disconfirm the hypothesis.
Thus, the evidence O in argument (2) confirms prediction P, the consequent of {T} + H --4 P. The confirmation of P in turn is held to confirm the amended theory {T} + H. Because {T} + H is held to be true, the component H in particular is presumed to be confirmed. This argument is common in scientific investigation, including research in population biology. Unfortunately, it is logically invalid. Thus, the confirmation of P and the truth of {T} + H--4 P cannot be used to assert the truth of H. Simply put, evidence supporting a hypothesis is logically insufficient to confirm that hypothesis: factors other than H might well lead to confirmation of the prediction P, independent of the truth or falsity of H. The fallacy of affirming the premise of an implication based on its conclusion is an example of the fallacy of false cause, known as affirming the consequent (Copi, 1982). Scientific investigation thus faces an asymmetry in the confirmatory role of experimental or field evidence. On the one hand, a hypothesis can be disconfirmed by evidence contrary to prediction; on the other, a hypothesis cannot be (logically) confirmed by evidence supporting prediction. It is in the context of this asymmetry that scientific hypotheses are held by some to be meaningful only if they are theoretically amenable to disconfirmation (Popper, 1968). The fallacy of false cause can be avoided in argument (2) only if the prediction P can arise in no other way than by the truth of H, i.e., only when P and H have the same truth content (if H is true, P is also; if H is false, P is also). Under this much more restrictive condition the proposition {T} + H ~ P is replaced by {T } + H ~-4 P, whereby the arrow pointing in both directions means that P can serve either as premise or conclusion in material implication. Thus, to avoid the fallacy of false cause all alternative hypotheses must be eliminated through experimental design or otherwise must be identified, investigated, and rejected, so that by process of elimination only the hypothesis H remains as an explanation of a confirmed prediction P. Hypothesis confirmation through the elimination of alternatives was termed "strong inference" in an important paper by Platt (1964). Although relatively simple in concept, such an approach obviously requires thorough field observations as well as careful analysis to identify and properly examine all reasonable alternative hypotheses.
2.5. Inductive Logic in Scientific Method We note that this approach to science includes the essential features of Popper's hypothetico-deductive method of scientific inquiry (Popper, 1963, 1968). However, it differs from Popper's in at least one important feature, namely, the procedure for comparing hypotheses against data. The Popperian model describes a process in which a hypothesis H is tested by experiment to determine its acceptance or rejection, with hypothesis rejection in the event of nonconformance to the evidence, and provisional acceptance otherwise, pending further evidence. The process then is repeated with another hypothesis H', with evidence from another critical experiment leading to acceptance or rejection of H' depending on conformance with experimental data. In this scenario hypotheses are subjected to testing one at a time, with decisions about hypothesis acceptance or rejection made sequentially. Our approach to hypothesis investigation also could be applied one hypothesis a time, as per the Popperian model. However, sequential investigation of hypotheses is only one available option, and not a requirement of the approach. It is possible, and intuitively preferable, for alternative hypotheses to be compared simultaneously against evidence, so as to measure their relative conformance one against the other. Two important benefits accrue from this more comprehensive approach to hypothesis testing. First, it allows every feasible hypothesis to compete in an arena of evidence against all other feasible hypotheses. This is as opposed to the Popperian model, in which previously rejected hypotheses are no longer candidates for comparison against alternatives considered later in the testing process. Second, simultaneous testing allows the process to carry a "memory" of previous test results, via the measures of conformance between individual hypotheses and the evidence. The conformance measures provide a natural mechanism, through the use of updating procedures such as Bayes' Theorem (see Sections 3.3.2 and 4.5, and Appendix A.3), for confirmatory evidence to accumulate as scientific investigation proceeds. The use of weights to express hypothesis likelihoods will be explored in considerable detail in Part IV.
2.5. I N D U C T I V E L O G I C IN SCIENTIFIC METHOD Inductive as well as deductive logic is required for hypothesis confirmation in biological science. That inductive logic is an essential feature of scientific enquiry is seen in the identification of hypothesized biological mechanisms, as well as the testing of these hypotheses with data. Indeed, a key activity in scientific enquiry is to identify, from a limited set of observations,
17
hypotheses that explain more than the particular observations giving rise to them. Thus, a limited body of data generates possible explanations for their occurrence, and these are folded as hypotheses into an extant body of theory for elaboration and testing with additional observations. Because any particular set of data constitutes only a subset of all possible observations that could be used, testing procedures are designed to be robust to inherent variation in the evidence. The formulation and testing of scientific hypotheses, based on only a partial record of potentially relevant observations, render the practice of science inductive. Simply put, causal mechanisms are asserted to hold for a general class of phenomena, based on examination of only limited observations from that class. The inductive nature of the process inevitably gives rise to the possibility of incorrect inference and necessitates the conservative rules of scientific and statistical inference that have been developed to accommodate, and protect against, such a possibility. As described above, the logic of hypothesis confirmation suggests that evidence contrary to a hypothesis is sufficient for its rejection, whereas evidence conforming to the hypothesis is insufficient for its confirmation. In practical applications, however, the situation is less clear-cut. Biological systems are replete with uncertainties, and hypothesized explanations are never wholly sufficient to explain behaviors. Thus, natural variation (and sampling error) can lead to the rejection of a hypothesis that otherwise would be seen as appropriate, just as it can support acceptance of an inappropriate hypothesis. Because biological inferences must be confirmed via inductive logic from particular instances, some of which can be misleading, these inferences lack the logical certainty of deductive arguments such as modus tollens and modus ponens. Biological investigation is by its very nature open to the risk of incorrect inference, which can decline as evidence accumulates but never vanishes. It is the role of probability and statistics in biological science to characterize and account for this risk. Just because the rules of inductive logic are not as prescriptive as in deductive logic, one should not conclude that induction is somehow inferior to deduction. As in all scientific disciplines, both inductive and deductive inference are required in biology. The "truth" of a biological hypothesis can only be confirmed inductively, through an ever-growing body of evidence that lends it credence. But the derivation of observable predictions from hypotheses must be facilitated by deductive argument, building on an extant theory and the evidence supporting it. Indeed, derivation of predictions, rather than the logical confirmation (or disconfirmation) of hypotheses, constitutes the principal role of
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deduction in science. In any observation-based discipline it is the clever interplay of inductive and deductive reasoning that is a hallmark feature of robust scientific investigation.
2.6. STATISTICAL I N F E R E N C E Statistical procedures are involved in hypothesis testing at the point at which data are collected and subsequently used for comparison against predictions. The principles of survey and experimental design serve to improve the efficiency of data collection, and to ensure that the data are relevant and useful in the investigation of predicted responses. Subsequent to data collection, procedures for statistical inference play a key role in determining whether the predicted responses are supported by the data. A correspondence between data and predictions provides evidence for hypothesis confirmation, and the lack of a correspondence leads to hypothesis rejection. Statistical testing procedures often are framed in terms of mutually exclusive and exhaustive "null" and "alternate" hypotheses (Mood et al., 1974). By null hypothesis usually is meant (1) an assertion of extant theory that includes an accepted, sometimes simplified, form of some relevant biological relationship, or (2) a biological relationship per se, to be considered for replacement by an alternate hypothesis. By alternate hypothesis is usually meant a logically distinct, sometimes more complex, and often more appealing biological relationship that potentially can replace a particular null hypothesis. The mechanics of statistical testing involve the matching of observed evidence against predictions based on the null and alternate hypotheses, with the idea that both hypotheses cannot be true, but one must be. Thus, rejection of the alternate hypothesis leads automatically to acceptance of the null hypothesis as its only alternative. Several benefits accrue to the framing of test procedures in this manner. First, one retains the logical consistency afforded by modus tollens, whereby hypothesis rejection is inferred logically from the disconfirmation of a predicted response. For example, a lack of supporting evidence for predictions based on the additive mortality hypothesis leads to its rejection. Second, disconfirmation of predicted responses based on the alternate hypothesis leads automatically to acceptance of the null hypothesis. Thus, the rejection of the additive mortality hypothesis leads to the acceptance of the compensatory mortality hypothesis. Third, confirmation of predictions based on an alternate hypothesis leads automatically to rejection of the null hypothesis, and therefore to acceptance of
the alternate hypothesis. Thus, confirmatory evidence for the additive mortality hypothesis leads to rejection of the compensatory mortality hypothesis and to acceptance of the additive hypothesis. In this case the test discriminates cleanly between hypotheses irrespective of test results, and thereby avoids the fallacy of false cause. Though they appear to be analogous, acceptance/ rejection of the null and alternate hypotheses suffer disproportionate burdens of evidence. Indeed, the use of statistical procedures in hypothesis testing expresses an asymmetry that parallels that of syllogistic logic, based on a requirement that evidence must be quite strong to reject a null hypothesis in favor of the alternate. Thus, testing procedures express a scientific conservatism in which amendment of an extant theory, or acceptance of a favored alternate hypothesis, is to be discouraged without strong evidence that it is warranted. In this sense the asymmetry in statistical testing is analogous to that of logical inference, whereby hypothesis confirmation accrues only through a preponderance of evidence, in striking contrast with the relatively modest evidentiary requirements for hypothesis disconfirmation.
2.7. I N V E S T I G A T I N G COMPLEMENTARY HYPOTHESES Scientific methodology is framed above in terms of hypotheses about alternative mechanisms for an effect of interest, with the idea that only a single hypothesis is operative. Thus, one or more hypothesized mechanisms are considered as potentially explanatory, with repeated use of scientific methodology ultimately identifying the appropriate hypothesis. An underlying assumption is that there is only a single "appropriate" hypothesis, and that other hypothesized mechanisms under consideration will be found to be inadequate through proper use of scientific methodology. Although this scenario no doubt applies to causal mechanisms in many disciplines, it fails to apply to many interesting problems in population biology and ecology. In fact, biological science is replete with examples of complementary factors that interact in complex ways to produce observed effects. For example, it often is less a question of whether interspecific competition, predation, or habitat degradation is the cause of declines in a population, but rather the contribution each factor makes in the declines. In this case all factors may be operating simultaneously, playing important but unequal roles in influencing population dynamics. That issues involving simultaneous complementary
2.8. Discussion factors arise frequently in population biology is indicative of the complexity of the biological systems under investigation. Physical, ecological, and thermodynamic processes simultaneously influence these systems in a complicated network of interactions between populations and the communities and environments of which they are a part. A natural outgrowth of such complexity is the framing of many scientifically interesting issues about cause and effect in terms of the relative contribution of multiple causal factors (Quinn and Dunham, 1984). A useful approach then may involve the estimation of parameters measuring the level of factor influence, based on statistical estimation procedures (see Chapter 4).
2.8. D I S C U S S I O N Some researchers believe too much emphasis is placed on hypothesis testing as a signature feature of scientific methodology (Quinn and Dunham, 1984; Loehle, 1987). This concern is especially prevalent in the use of standard hypothesis testing procedures in statistics (Yoccoz, 1991; Johnson, 1999; Franklin et al., 2001). Quite often much of the information residing in sample data is overlooked in the process of hypothesis testing, because statistical tests address sometimes irrelevant questions about "significant" differences between treatments and controls. The lack of relevance is in large part a consequence of the fact that hypothesis tests often compare hypotheses, one of which (the null hypothesis) is unacceptable by design. Thus, the testing procedure is uninformative, in that it is designed at the outset to confirm what one already knows. The more biologically important information concerning the magnitudes of differences, or the parametric values defining the differences, or the biological structures underlying those differences, remains inadequately treated by statistical testing. The bottom line is that many, arguably most, scientifically interesting questions in biology are addressable by way of the estimation of parameters such as abundance, location, and proportionate influence, or by the selection of alternative models in which these parameters are imbedded. Both parameter estimation and model selection often are handled more effectively outside the context of hypothesis testing. We note, however, that irrespective of statistical method, biological investigation still depends on identification and/or parameterization of theoretically based relationships. It is unclear how such relationships can be recognized, or how assessed, separate from a foundation of theory. It is important to recognize that however hypotheses are investigated, investigation is actually an examina-
19
tion of both the hypothesis and the background theory. This can be seen in the arguments for hypothesis confirmation presented above. Thus, the rejection of predicted response P leads to rejection of the theory {T} as amended by hypothesis H. The argument above concluded that because the theory was assumed to be true, the hypothesis was necessarily false. Of course, it is always possible that the theory itself is false and the hypothesis is true (or both are false). Indeed, the history of science contains many examples of accepted theories that were shown eventually to be false (Kuhn, 1996). This ambiguity likely is an inevitable consequence of scientific methodology, whereby theories are constantly subjected to amendment and revision through the examination of hypotheses. Scientific methodology as described above involves theory amendment either by the addition of hypotheses to a theory, or by the replacement of one hypothesis by another. Standard practices of statistical testing fit well with the latter description, primarily because they are framed in terms of the comparison of null and alternate hypotheses. Two exceptions to this framework should be mentioned. First, it sometimes is the case that of two hypotheses under consideration, neither is easily recognizable as established, and there is a question about which hypothesis is to be identified as the null hypothesis and which as alternate. The decision is obviously of some operational consequence, because of the differential burden of evidence for null and alternate hypotheses. Under such circumstances nonscientific criteria, involving potential costs and benefits of hypothesis acceptance, often influence the decision. When this occurs it is important to recognize, and acknowledge, that the investigation is guided by objectives that go beyond the objective pursuit of understanding. Testing of the compensatory and additive mortality hypotheses provides a good example, with hypothesis acceptance/rejection strongly influenced according to which hypothesis is identified as null, although neither hypothesis is unambiguously recognizable as null. A second exception involves multiple comparisons of more than two hypotheses. Standard statistical procedures such as likelihood ratio testing do not lend themselves to the testing of multiple hypotheses, except with omnibus test procedures such as analysis of variance (Graybill, 1976) or by the comparison of hypotheses taken two at a time (Mood et al., 1974). However, some promising approaches have been identified that allow for the comparison and selection of hypotheses from among multiple candidates. For example, model selection criteria proposed by Akaike (1973,1974) have been used by Burnham and Anderson (1992, 1998) and others in the selection of biological
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Chapter 2 Scientific Process in Animal Ecology
relationships, and adaptive resource management (Waiters, 1986) provides a promising approach to the identification of population models from among multiple alternatives (Williams, 1996a). It is worth emphasizing that scientific methodology as described above is fully complementary to the traditional goals and objectives of population management. Indeed, many of the presumptive causes of biological patterns are recognized from observations made during the course of resource management, and in some instances management has been included in designs for their scientific investigation. The linkage between population management and scientific assessment, in which management both supports and is supported by research, is definitive of an adaptive approach to resource management (Waiters, 1986). Adaptive resource management, in concert with the use of sound scientific methodology, holds great promise for accelerating our understanding of biological processes, while simultaneously improving resource manage-
ment based on that understanding. In the long term, the melding of research and management may offer the only feasible approach to resolving many longstanding problems that confront wildlife and fisheries managers. We deal in considerable detail with adaptive management, and in particular with optimal adaptive decision making, in Chapter 24. We note in closing that population models represent hypotheses to be investigated, with components ranging from those known with great certainty to those derived only from guesses. The challenge is to analyze a model in such a way that the hypotheses strongly influencing model performance can be recognized and scientifically investigated. This task is almost never easy, and becomes increasingly difficult with increasing model size, complexity, scope, and amount of uncertainty as to model components. In Chapter 3 we turn to the relationship between hypotheses and models, and the use of both constructs in the conduct of science.
C H A P T E R
3 Models and the Investigation of Populations
3.1. TYPES OF BIOLOGICAL MODELS 3.2. KEYS TO SUCCESSFUL MODEL USE 3.3. USES OF MODELS IN POPULATION BIOLOGY 3.3.1. Theoretical Uses 3.3.2. Empirical Uses 3.3.3. Decision-Theoretic Uses 3.4. DETERMINANTS OF MODEL UTILITY 3.4.1. Simple versus Complex Models 3.4.2. Mechanistic versus Descriptive/ Phenomenological Models 3.4.3. More Integrated versus Less Integrated Model Parameters 3.5. HYPOTHESES, MODELS, AND SCIENCE 3.6. DISCUSSION
parameterization, and subsequent use of models provide one conceptual thread linking the themes of this book. In this chapter we are concerned with the relationship between theory (and associated hypotheses), as discussed in Chapter 2, and modeling, defined here as the abstraction and simplification of a real-world system (see Chapter 7). Our focus is on scientific models, which are used in the evaluation of hypotheses, and management models, which are used in making management decisions. We limit our discussion to models in population ecology and management, with a focus on model utility and the factors that make some biological models more useful than others. A key point in the chapter is that model utility is strongly influenced by the degree of correspondence between model structure and intended model use. The linkages between structure and function highlight some useful dichotomies in model development, and suggest a classification of models based on their utilization in science and management. The scientific and management literature includes many definitions of theories, hypotheses, and models. Some authors recognize little distinction among these concepts. For example, Neyman (1957) stated that "scientific theories are no more than models of natural phenomena." Hawking (1988) asserted that "a theory is just a model of the universe, or a restricted part of it, and a set of rules that relate quantities in the model to observations that we make." He also wrote that "any physical theory is provisional, in the sense that it is only a hypothesis" (Hawking, 1988). Other authors view the concepts hierarchically. For example, Pease and Bull (1992)stated that "hypotheses
As argued in Chapter 2, models are closely related to hypotheses, and as such are important components of both science and management. Indeed, progress in both science and management depends to a substantial degree on the recognition of a priori hypotheses, along with their articulation and assessment via biological models. The role models play in biological thinking is prominent throughout this book, so much so that the book might well be viewed as an exposition on population models. From this perspective, Part I provides background and a context for models with respect to science and management; Part II concerns the development of population models, with examples of model structures arising in population ecology and management; Part III deals with the estimation of attributes that parameterize population models; and Part IV describes the use of models in making decisions about the management of animal populations. The development,
21
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Chapter 3 Models and the Investigation of Populations
address much narrower dimensions of nature than the models from which they are derived." Hilborn and Mangel (1997) stated that "one can think of hypotheses and models in a hierarchic fashion with models simply being a more specific version of a hypothesis," thereby reversing the hierarchical positions of the terms. We note that the variety of definitions and uses of "theory," "hypothesis," and "model" need not be of great concern, provided the terms are at least operationally defined when they are used. On the other hand, needless confusion and miscommunication can arise in the absence of agreement as to their meanings. In what follows we utilize the conceptual framework developed in Chapter 2, which recognizes hypotheses as identifiable (and testable) elements of a broader scientific or management paradigm.
3.1. TYPES OF BIOLOGICAL MODELS It is a commonplace to identify different kinds of models depending on their uses. For example, a conceptual model refers to a set of ideas about how a particular system works. By translating these ideas into words, we create a verbal model. Similarly, translation of ideas or words into a set of mathematical equations yields a mathematical model. These different model types all correspond to our operational definition of the term model, in that they reflect an abstraction of key features of a system into a simple set of ideas, words, or equations that represents the system. One typically thinks of abstraction in terms of mathematical rather than physical models. However, just as physical phenomena can be modeled by mathematical constructs, so mathematical schemes can be modeled by physical constructs. Skellam (1972) stated that this "reverse modeling" includes "the most powerful instrument known for advancing empirical knowle d g e - t h e designed experiment." Experiments can be viewed as models based on our definition of the term, because they abstract from a real-world situation only a limited number of features to be investigated. In fact, the term empirical model often is used to mean a biological system that is amenable to experimentation. Perhaps the most famous empirical model in animal population ecology is the Tribolium model, a laboratory experimental system developed in the mid-1920s by R. N. Chapman for studying population growth and regulation using flour beetles (Chapman, 1928), and most commonly associated with the later work of T. Park and his students at the University of Chicago (e.g., Park, 1948; Neyman et al., 1956; Mertz, 1972; Wade and Goodnight, 1991). Over the years, work with the
Tribolium model has been characterized by close interaction between empirical and mathematical modeling and has led to strong inferences about many important aspects of population dynamics. A recent example involves prediction of points of transition in parameter values of a nonlinear mathematical model of animal population dynamics (Constantino et al., 1995). The testing of these predictions by altering adult mortality in experimental Tribolium populations produced shifts from point equilibria to stable periodic oscillations to aperiodic oscillations (Constantino et al., 1995). The impressive success of investigations using the Tribolium model highlights the value of investigation that involves the interactive use of mathematical and empirical models. Yet another model type is the physical model (e.g., a scale replica of an individual organism), examples of which have been used to good effect in physiological ecology to estimate heat exchange between organisms and their environments (e.g., Porter et al., 1973; Tracy, 1976). The mechanical model of Pearson (1960) can be viewed as one kind of physical model and is certainly one of the most interesting models ever developed for use in animal population ecology. Pearson's model had the appearance of a large pinball machine, with steel balls (representing individual animals) released onto an incline board. Holes were drilled into the board, and balls falling into these represented deaths. When a ball rolled over pairs of bronze strips, an electric circuit was completed and new balls (reproduction) were released from the top of the board. Pearson (1960) developed an algebraic model to describe the functioning of the mechanical model and utilized both models in producing counterintuitive results that eventually led to an improved understanding not achievable without both approaches.
3.2. KEYS TO S U C C E S S F U L M O D E L USE Just as there are many kinds of models, there also are many ways in which models can be used in the conduct of science and management (see Section 3.3). The large variety of applications precludes specific, detailed instructions about how to build and use a model; however, the following guidelines are offered as keys to successful model use (e.g., see Conley and Nichols, 1978; Nichols, 1987): 1. Clearly define the objectives of the modeling effort; i.e., provide an unambiguous statement of the way the model is to be used in the conduct of science a n d / o r management.
3.3. Uses of Models in Population Biology 2. Include in the model only those system features that are critically relevant to the objectives. Using these guidelines, we discuss below some classes of modeling objectives and provide suggestions for selecting biological features that enhance model utility with respect to objectives. We defer to Chapter 7 a discussion of specifics in developing and assessing a model.
3.3. USES OF MODELS IN POPULATION B I O L O G Y In a restrictive sense, the primary use of mathematical models in population ecology and management is to project the consequences of hypotheses. As noted in Chapter 2 (also see Johnson, 1999), it is possible to distinguish between scientific and statistical hypotheses, and useful to distinguish between their corresponding models. Scientific hypotheses represent stories about how a system works or responds to management actions, and scientific models are used to project the consequences of such hypotheses. For example, we may be able to use a simple conceptual model to anticipate population growth in a stable environment, or track population responses to harvest regulations, or predict species distributions in altered habitats. Because most models are too complicated to project system responses in our heads, scientific models can serve as "calculating engines" (Lewontin, 1963) to project the consequences of scientific hypotheses. Statistical hypotheses are derived from scientific hypotheses and represent stories not just about the system of interest, but also about observable quantities that are relevant to system behavior. A statistical model projects the behavior and values of these observable quantities (data) that are expected if the system is operating in a manner consistent with the statistical, and hence scientific, hypothesis. The construction of a statistical model is based not only on the underlying scientific hypothesis but also on aspects of sampling design and data collection. Thus, scientific models are used to project system dynamics, whereas statistical models are used to project the dynamics of observable quantities under one or more scientific hypotheses. Projections based on statistical models are used to estimate quantities of interest, including parameters of scientific models, and to discriminate among competing hypotheses by addressing the question: "Which statistical, and hence scientific, hypothesis corresponds most closely to the data?" Note that this distinction between scientific and statistical models corresponds closely to the distinction
23
between uses of deductive and inductive logic in the conduct of science (see Section 2.5). Thus, scientific models are used to deduce the consequences of their corresponding scientific hypotheses. On the other hand, statistical models are used to draw inferences about a particular hypothesis and to discriminate among competing hypotheses, through an inductive process based on limited observations. Finally, note that the organization of this book largely follows this distinction between model types, with Part II devoted to scientific models, Part III focusing on statistical models, and Part IV elaborating the interplay between these classes of model for the purpose of managing and understanding system dynamics. In addition to the broad classification of models as scientific and statistical, it is possible to categorize models based on the different classes of problems to which they are applied. In animal population ecology and management it is useful to identify theoretical, empirical, and decision-theoretic uses of models. Empirical and decision-theoretic uses typically require both scientific and statistical models, whereas theoretical uses are largely restricted to scientific models.
3.3.1. Theoretical Uses Here we define "theoretical" model use as the investigation of system responses that are possible under specific hypotheses. Model uses in this context do not entail a comparison of model predictions with data or observations, and indeed, the lack of a confrontation between predictions and data is the distinguishing feature of theoretical model use. The term theoretical as used here is consistent with Lewontin's (1968) view of theoretical population biology as "the science of the possible," and the views expressed in Caswell's (1988) essay on theory and models in ecology. For example, one might investigate with a model whether densitydependent migration can stabilize a particular metapopulation, or whether populations governed by a certain class of nonlinear equations exhibit chaotic behaviors, or whether populations subjected to certain harvest strategies exhibit thresholds in their responses. If such questions are tied to particular a priori hypotheses, then the use of models incorporating these hypotheses constitutes a form of hypothesis assessment and testing. Note that this use of the term theoretical has nothing to do with whether the model is used to address management-oriented questions. In fact, theoretical uses of models can be very important in the management of animal populations. For example, models that exhibit substantial mechanistic differences may lead to very similar management policies. From the perspective of
24
Chapter 3 Models and the Investigation of Populations
the manager, it thus would be unwise to devote resources to learning which model corresponds most closely to reality, because biological distinctions among the models would not be relevant to management decisions (e.g., Johnson et al., 1993; Williams and Johnson, 1995). For this reason one should investigate management implications prior to any effort to distinguish among management-oriented hypotheses and their corresponding models. Even when different models do lead to distinct management actions, it is useful to assess the management value of discriminating among them. It may be that a particular suboptimal policy performs adequately in terms of model objectives (e.g., number of animals harvested), regardless of variation among model-specific optimal policies (Hilborn and Walters, 1992). Modeling can be used to estimate the "expected value of perfect information" (see Section 24.7) as an aid in deciding whether it is worthwhile to expend effort discriminating among competing hypotheses (e.g., Hilborn and Walters, 1992; Johnson et al., 1993). Though modeling exercises of this kind are "theoretical" in the sense that they do not involve a confrontation with data, they nonetheless can be extremely useful from a pragmatic, management perspective.
3.3.2. Empirical Uses By "empirical" uses of models, we refer to predictions of population behaviors for the purpose of comparison with realized population behaviors. The confrontation of model predictions against data in an effort to discriminate among hypotheses (see Section 2.3) is a definitive feature of science-based investigation. Although various authors have identified a number of approaches to science (e.g., Hilborn and Mangel, 1997), here we focus on two generic alternatives and discuss the role of models in each.
Scientific models are used in step 2 to deduce predictions from the scientific hypothesis, whereas statistical models are used in the comparison of test results with these predictions (step 4). Advocates of this approach emphasize that the use of a critical experiment in step 3 is most likely to yield strong inferences (Platt, 1964). However, it is the single a priori hypothesis, rather than the nature of the test, that is the defining feature of this investigative approach. In the situation in which a hypothesis is rejected, there are two options (Fig. 3.1). One is to develop a completely new hypothesis and proceed as above with its investigation. The other is to revise the original hypothesis in a manner that renders it consistent with test results that led to rejection and then proceed with investigation of the revised hypothesis as above. In the event of a failure to reject the tested hypothesis, we again are left with two options (Fig. 3.1). One is to subject the hypothesis to still another test, using either the same or different predictions as those tested initially, recognizing that a hypothesis can be corroborated but can never be "proved" to be true (see Section 2.5) (Popper 1959, 1963). Alternatively, the hypothesis can be extended or otherwise modified, and a test can be formulated that focuses on the extension or modification. Iterative hypothesis testing and refinement as above eventually may identify a hypothesis that survives repeated efforts at falsification and consistently predicts system behaviors. Under these conditions the hypothesis then is accepted as provisionally true, in that we view it as our best approximation of reality (subject, of course, to subsequent investigation and possible refinement).
3.3.2.2. Multiple-Hypothesis Approach This approach usually is traced to a paper by Chainberlin (1897) on multiple working hypotheses (Platt,
3.3.2.1. Single-Hypothesis Approach This approach frequently is associated with the writing of Popper (1959, 1963, 1972) and the influential paper by Platt (1964) on strong inference. The approach is outlined in the following steps, using the elements of the scientific approach identified in Section 2.3: 1. Develop or identify a hypothesis (typically from existing theory). 2. With the help of the associated model, deduce testable predictions. 3. Carry out a suitable test. 4. Compare test results with predictions. 5. Reject or retain the hypothesis.
H
/ H 1
Accept
1
Reject
,/',,, H/1
H//1
H 2
FIGURE 3.1 Schematic representation of the single-hypothesis approach to scientific inquiry. H1 denotes the original hypothesis tested, H{ denotes an extension or elaboration of H1, and H~ denotes a revision of H1 designed to account for those aspects of test results that deviate from predictions of H1. H2 is a new hypothesis.
3.3. Uses of Models in Population Biology 1964). Multiple hypotheses are also an important part of the scientific research programs described by Lakatos (1970). The application described here for biological investigation is adapted primarily from adaptive resource management (Chapter 24; also see Walters, 1986; Johnson et al., 1993; Williams, 1996a), but the joint use of multiple hypotheses is relevant regardless of the motivation for learning. It is outlined in the following steps: 1. Develop a set of competing hypotheses. 2. Derive a set of probabilities associated with these hypotheses. 3. Use associated models to deduce testable predictions. 4. Carry out a suitable test. 5. Compare test results with predictions. 6. Based on this comparison, compute new probabilities for the hypotheses. Mathematical models again are prominent in this approach. Thus, scientific models corresponding to the different hypotheses are used to deduce competing predictions (step 3), and statistical models provide a framework for comparison of test results against these predictions (step 5), leading to new probabilities for the competing hypotheses (step 6). The probabilities in step 2 can be viewed as measures of our relative faith in the different hypotheses. Let P ( H i) denote the probability associated with hypothesis H i, with ~i P(Hi) = 1. Then the comparison of test results (step 5) with predictions of the different hypotheses/models leads to an updating of these probabilities (step 6). We note that this approach is not as widely utilized as the single-hypothesis approach to science, in part because of the need to identify and update hypothesis probabilities or "likelihoods." One approach to probability updating is based on likelihood functions (see Chapter 4) in conjunction with Bayes' Theorem (e.g., see Hilborn and Mangel, 1997). The likelihood function ~s for hypothesis H i describes the "likelihood" (for discrete random variables these are probabilities) of collecting the test data for parameters 0 i of the statistical model corresponding to H i (see Section 4.2.2). The likelihoods corresponding to different hypotheses in the set can be computed directly, using the observations in conjunction with the statistical models associated with H i. Given the set {P[Hi]} of prior probabilities, the test data, and the likelihoods ~(_0i]data) for the different hypotheses, we can compute updated probabilities P' (H i) for the different hypotheses by ~(Oi[data)P(Hi) P' (H i) = ~, ~(Oi]data)P(Hi )
i
(3.1)
25
P(H1)
L1
P(H3) FIGURE 3.2 Schematicrepresentation of the multiple-hypothesis approach to scientific inquiry. H i denotes each of several (e.g., three) alternative hypotheses, with associated probabilities P(Hi). Following an experiment or management intervention, predictions from each hypothesis are compared to observations, to form likelihoods (Li). Bayes' Theorem [Eq. (3.1)] then is used to provide updated values for P(Hi), and the process repeats, now using the updated values P(Hi).
[e.g., see Section 4.5 and Hilborn and Mangel (1997)]. The updated probabilities P'(Hi) then become prior probabilities for subsequent updates with additional data (Fig. 3.2). Likelihood functions, maximum likelihood estimation, and Bayes' Theorem are discussed in detail in Chapter 4, and their application in probability updating is described in Chapter 24. Learning can be thought of as a change over time in the probabilities associated with the different hypotheses (Fig. 3.3). These hypotheses are viewed as competing for our confidence, and each comparison of field data against model-based predictions leads to a change in their probabilities. We expect the probability to increase for the most appropriate hypothesis, and to decrease for the other hypotheses. For example, the accumulation of probability for model 3 in Fig. 3.3 reflects increasing faith in hypothesis 3 as an approximation of reality.
1
-
0.8 A
0.6-
~"
0.4-
--~- M1 1
-__ M 2 ' .
M3
0.2 0
.
0
.
.
.
.
2
.
.
.
4
.
6
8
FIGURE 3.3 Hypothetical changes in probabilities associated with three hypotheses under the multiple-hypothesis approach to scientific inquiry. P denotes probability, and M i denotes the model associated with hypothesis Hi. An investigation (e.g., an experiment) occurs between each pair of steps, and comparison of model-based predictions with test results leads to changes in the probabilities associated with the different models [e.g., using an approach such as Eq. (3.1)].All three hypotheses begin with equal probabilities (e.g., assuming the absence of prior knowledge by which to discriminate among them) at step 1, and investigation leads to high probabilities associated with M 3 and its corresponding hypothesis, H3.
26
Chapter 3 Models and the Investigation of Populations
3.3.2.3. Popper's Natural Selection of Hypotheses In discussing the role of theory in scientific investigation, Popper (1959) wrote that "We choose the theory which best holds its own in competition with other theories; the one which, by natural selection, proves itself the fittest to survive." He later expanded on this analogy (Popper, 1972), noting "the growth of our knowledge is the result of a process closely resembling what Darwin called 'natural selection'; that is, the natural selection of hypotheses: our knowledge consists, at every moment, of those hypotheses which have shown their (comparative) fitness by surviving so far in their struggle for existence; a competitive struggle which eliminates those hypotheses which are unfit." Thus, candidate hypotheses are subjected to falsification tests, and some survive the testing whereas others do not. Popper's analogy between hypothesis testing and natural selection extends easily to the multiple-hypothesis approach to science. Instead of focusing attention on a single hypothesis (analogous to an individual or a genotype) and its survival in the various confrontations with data, our attention is on the hypothesis probabilities P(Hi), which can be viewed as analogous to gene frequencies. Just as selective events bring about adaptive changes in gene frequencies within the population, so do our experiments and tests bring about changes in the probabilities associated with the hypotheses under consideration. Changes in gene frequencies over time reflect the action of natural selection, and changes in hypothesis probabilities reflect the relative predictive abilities of the different hypotheses and their models. The focus is on natural selection and learning, respectively, as the prime determinants of change, recognizing that other sources of variation influence changes in gene frequencies (e.g., environmental variation; "drift" associated with the stochastic nature of fitness components) as well as hypothesis probabilities (e.g., environmental variation; uncertainty about population size).
3.3.2.4. Recommendations Based on the Multiple-Hypothesis Approach The multiple-hypothesis approach to science is not as widely used as the single-hypothesis approach, and as a result, not as much thought has been devoted to it by those interested in scientific methodology. We offer two methodological recommendations. The first is simply to reiterate and reinforce the view long held by scientists, that science is a progressive endeavor. For example, in 1637 Descartes wrote "I hoped that each one would publish whatever he had learned, so that later investigations could begin where the earlier
had left off" [Descartes (translation), 1960]. Modern ecologists often pay only limited attention to the previous work of others, as evidenced by the perfunctory paragraph or so found in introductory sections of most scientific papers (though authors of review papers frequently do attempt to generalize the results of previous work). Our recommendation is to take full advantage of knowledge gained from past work, by accounting when practicable for previous investigation via assignment of probabilities to hypotheses based on past research. A key to this approach is the development of explicit models associated with members of a hypothesis set, which can be used to identify hypothesisspecific predictions from past investigations. Comparison of these predictions with the test results then permits one to update the hypothesis probabilities [e.g., as in Eq. (3.1)]. This approach of course depends on the amount of detail provided in the reporting of past work; but even in cases in which the level of detail is less than optimal, it still may be possible to design and revise hypothesis probabilities, though perhaps less formally. We believe the multiple-hypothesis approach provides a means of better utilizing results from previous investigation, via the updating of prior probabilities. A second recommendation involves study design and statistical methodology, and it emerges from optimal management designs (Part IV) under the rubric of adaptive management (also see Walters, 1986; Johnson et al., 1993; Nichols et al., 1995a; Williams, 1996a). Hilborn and Mangel (1997) note that the historical development of the single-hypothesis approach to science was accompanied by a corresponding development of associated statistical methods. A great deal of thought and effort have been devoted to the design of experiments, with the intent of rejecting or tentatively accepting a priori null hypotheses (Chapters 4 and 6; also see Fisher, 1947, 1958; Cox, 1958). After incorporating the critical design elements (e.g., randomization and replication) for reliable inference, investigators frequently turn their attention to test power, i.e., the probability of rejecting a null hypothesis when it is false (see Sections 4.3 and 6.7). Power frequently is viewed as an optimization criterion in experimental design (e.g., Skalski and Robson, 1992), and efforts are made to maximize power for fixed values for other test characteristics. Under a multiple-hypothesis approach, design criteria based on the rejection of a single hypothesis are no longer relevant. Instead of maximizing test power, the multiple-hypothesis approach seeks to maximize discrimination among models, via sampling and experimental designs for that purpose. Formal, actively
3.3. Uses of Models in Population Biology adaptive management can utilize optimal control methods to identify management policies supporting this objective (Part IV) (see Walters, 1986; Johnson et al., 1993; Nichols et al., 1995a; Williams, 1996a; Conroy and Moore, 2001). Thus, we should be able to use optimization (Chapters 21-23) (see Bellman, 1957; Williams, 1982, 1989, 1996a,b; Lubow, 1995; Conroy and Moore, 2001), in conjunction with objective functions that focus on discrimination among hypotheses, to develop optimal designs. For example, one might use as an objective function a diversity index such as the Shannon-Wiener H' (e.g., Krebs, 1972), computed with the prior probabilities. Diversity indices such as H' are minimized when one of the P(H i) approaches one and the remaining P(H i) approach zero (i.e., when we are confident that one of the hypotheses approximates reality better than all of the others). Regardless of specifics in the investigation, we recommend the use of optimization methods to assist in discriminating among multiple hypotheses.
3.3.3. Decision-Theoretic Uses An important application of models involves projecting the consequences of hypotheses about how a system behaves, for the purpose of identifying appropriate management actions. Just as two approaches to science were discussed under empirical model uses, two approaches to decision-making can be identified here. The following ideas are developed more fully in Part IV.
3.3.3.1. Single "Best Model" Approach This approach to decision-making is common in natural resource management. It relies on a single model that is judged to be the best available for predicting system responses to management actions, and it utilizes (1) an objective function (a formal statement of management objectives), (2) a favorite hypothesis (and corresponding scientific model) for the managed system, (3) a set of available management actions that can be taken to achieve management objectives, and (4) a monitoring program that provides time-specific information about system status and other variables relevant to the objective function. Based on these prerequisites, implementation of the single "best model" approach to management involves the following iterative steps: 1. Observe the current state of the system. 2. Update model parameter estimates, if appropriate, based on current information. 3. Identify an appropriate (or optimal) management action.
27
4. Implement management action and return to step 1. Step 2 usually is based on a statistical model, whereas step 3 typically uses a scientific model of the system. Given the objective function and information on the current state of the system, the scientific model is used in step 3 to identify the management action most likely to meet management objectives. In some cases, the model may be used to project the consequences of a suite of management actions, and the optimal decision is chosen based on the results. Alternatively, optimization algorithms (e.g., Williams, 1982, 1989, 1996a,b; Lubow, 1995) can be used to identify optimal management actions with respect to objectives. In either case, implementation of the management action (step 4) drives the system to a new state, and the process is repeated.
3.3.3.2. Multiple-Model Approach This approach to making management decisions is most commonly associated with adaptive management (Waiters, 1986; Johnson et al., 1993, 1997; Williams, 1996a; Conroy and Moore, 2001). Prerequisites for the approach include the following: (1) an objective function, (2) a model set consisting of the scientific models associated with competing hypotheses about how the managed system responds to management, (3) prior probabilities associated with the different hypotheses (and thus their models) in the model set, (4) a set of available management options, and (5) a monitoring program providing time-specific information about system status and other variables relevant to the objective function. Implementation of the multiple-model approach to management then involves the following iterative steps: 1. Observe the current state of the system. 2. Update model probabilities based on current information. 3. Derive the optimal management action. 4. Implement management action, and return to step 1. The information in step 1 about the current state of the system is provided by the monitoring program, and the estimated state of the system at time t + 1 is compared with predictions made at time t by each of the models as a basis for revising model probabilities (step 2). The updating of model probabilities is accomplished using statistical models with an algorithm [e.g., Eq. (3.1)], whereby probabilities increase for models that effectively track the observations and decrease for models that do not. Derivation of optimal management
28
Chapter 3 Models and the Investigation of Populations
actions is based on the competing scientific models and utilizes optimal control methods (e.g., Bellman, 1957; Anderson, 1975b; Williams, 1982, 1989, 1996a,b; Lubow, 1995; Conroy and Moore, 2001) that account for future effects of present actions. Implementation of the optimal management action (step 4) then drives the system to a new state, and the process is repeated.
benefits to the fisheries of such progress can hardly be exaggerated.
Beverton and Holt clearly recognized fishery biology to be a dual-control problem and recommended an essentially adaptive approach to the management of fishery resources.
3.4. DETERMINANTS OF MODEL UTILITY
3.3.3.3. Learning through Management The growth of knowledge in the field of wildlife management has not been as rapid as many would like (Romesburg, 1981, 1991). One path to faster learning is to make more intelligent use of management for learning (e.g., Holling, 1978; Walters and Hilborn, 1978; MacNab, 1983; Walters, 1986; Murphy and Noon, 1991; Sinclair, 1991; Johnson et al., 1993, 1997; Lancia et al., 1996; Williams, 1997; Conroy and Moore, 2001). Learning through management can occur with either a single-model or a multiple-model approach. Under the single-model approach, predicted system responses to management are compared with the observed (estimated) response. Based on this comparison, the model and its associated hypothesis are either retained for future use, or are rejected and replaced by a new hypothesis and model. Learning thus occurs in the same manner as under the single-hypothesis approach to science, except that here it is an unintended by-product of efforts to meet direct management objectives (see Chapter 24). The multiple-model approach to management also involves a comparison of model predictions with observed system responses, except that the comparison leads to changes in model probabilities (i.e., to learning). For example, "active adaptive management" (Walters and Hilborn, 1978; Walters, 1986; Hilborn and Walters, 1992; Williams, 1996a) uses multiple models to identify optimal management decisions as solutions to the so-called "dual-control problem" (e.g., Walters and Hilborn, 1978) of trying simultaneously to learn (because learning increases our ability to manage in the future) while achieving management objectives. The idea of using management adaptively to discriminate among competing models was articulated in the 1970s by Holling (1978) and Walters and Hilborn (1978). However, their work was presaged as early as 1957 in the pioneering book by Beverton and Holt (1957): It is the changes produced in the fisheries by the regulations themselves...that provide the opportunity of obtaining, by research, just the information that we may have been lacking previously. Thus the approach towards optimum fishing, and the increase in knowledge of where the optimum lies, can be two simultaneous and complementary advances; the
The successful use of models requires clear, unambiguous objectives of the modeling effort, and a focus on biological features of the modeled system that are critically relevant to the objectives. In the previous section we discussed model objectives in population ecology and management in terms of theoretical, empirical, or decision-theoretic uses. Here we focus on the selection of critical system features for a model, with the recognition that this selection ultimately determines model utility. We emphasize three gradients that are especially relevant to model development, which provide a convenient format to illustrate some issues for consideration when one develops a model.
3.4.1. Simple versus Complex Models By definition, the process of modeling involves abstraction and simplification (see Chapter 7), and thus entails a loss of information in the modeling of any real biological system. For that reason every modeler must face a question about model complexity (e.g., Levins, 1966; Walters and Hilborn, 1978). We believe that the modeling process can be usefully viewed as a filter, in which the full complement of information of a real system is passed through the filter and only the system attributes that are essential to the modeling objectives are retained. When the filter is informed by an intended application, the modeling process becomes an effort to match model complexity with model use.
Biologists often overlook the importance of matching complexity to intended use, and indeed, many have a natural tendency to create models that are more complex than necessary. For example, Nichols et al. (1976b) used a detailed simulation model of an alligator population to draw general inferences about the relative effects of size- and age-specific harvest on alligator population dynamics. The model included various components of reproductive and survival rates, but many of the general objectives of the modeling effort could have been met using a much simpler population projection matrix approach (Nichols, 1987). It is not difficult to carry the tendency for complex
3.4. Determinants of Model Utility explanation beyond the point of usefulness. Referring to the science of geographical ecology, MacArthur (1972) wrote that "The best person to do this is the naturalist...But not all naturalists want to do science; many take refuge in nature's complexity as a justification to oppose any search for patterns." Biologists have a natural tendency to focus on complexity. Indeed, the central guiding paradigm of all the biological sciences is Darwinian evolution by natural selection, and the raw material for this process is natural (and heritable) variation. Ecologists thus are taught to focus on differences between individual organisms, between organisms and their behaviors in different habitats, between species, etc., and to build selective stories to explain these perceived differences [see discussion in Gould and Lewontin (1979)]. In an extreme view of variation and complexity, the behavior or fate of an individual organism at a particular point in space and time is a unique event, one which is often of little use in predicting fate or behavior of another (or even the same) individual at a different point in space and time. Under this view, biologists are involved in descriptive work, and perhaps in a posteriori story telling, but not in science. On the other hand, a scientific view searches for generalizations among individual events, in the expectation that at some scale biological phenomena are at least stochastically predictable. This view leads back to modeling and to the recommendation that we incorporate in a model only those aspects of system complexity that are essential for meeting the objectives of the modeling effort.
3.4.2. Mechanistic versus Descriptive/ Phenomenological Models By mechanistic models we mean those that depict causal relationships between variables, in the sense that changes in one variable are directly responsible for changes in another. On the other hand, descriptive/ phenomenological models define statistical relationships between variables, without incorporating underlying mechanisms that are responsible for the relationships. We note that to a certain extent this distinction is in the eye of the beholder, because all models can be viewed as descriptive and phenomenological at some level, and most express at least some degree of biological mechanism. To illustrate the dichotomy, consider the relationship between hunting and population survival rates (Johnson et al., 1993). Different hypotheses about the effects of hunting mortality on annual survival rates of mallard ducks can be incorporated into the equation
29 S i --
0(1 - ~Ki),
(3.2)
where $i is the probability that a bird alive at the beginning of the hunting season in year i is still alive at the beginning of the hunting season the next year (Anderson and Burnham, 1976; Burnham et al., 1984; Nichols et al., 1984d). The parameter 0 usually is viewed as the probability of annual survival in the absence of hunting, and K i is the probability that a bird alive at the beginning of the hunting season in year i dies as a result of hunting during the subsequent season. The parameter ~ denotes the slope of the relationship between annual survival and hunting mortality rate. If = 1, then Eq. (3.2) corresponds to the completely additive mortality hypothesis, under which hunting and nonhunting mortality sources act as independent competing risks. If ~ = 0, at least for some values of K i (e.g., for K i < c, where c reflects a threshold value), then Eq. (3.2) corresponds to the completely compensatory mortality hypothesis under which variation in hunting mortality (below c) brings about no corresponding variation in annual survival (Anderson and Burnham, 1976; Burnham et al., 1984; Nichols et al., 1984d). Chapters 8 and 10 provide more details on the compensatory and additive mortality hypotheses. Recent analyses of band recovery data for North American mallard ducks have produced very different estimates of the slope parameter 13when based on data from different decades. A proposed explanation for this difference identifies density-dependent nonhunting mortality as the most likely mechanism underlying compensatory mortality (e.g., Anderson and Burnham, 1976; Nichols et al., 1984d). Thus, density-dependent responses to changes in hunting mortality would be expected to differ in years of high and low mallard abundance. Johnson et al. (1993) recommended the survival model S i --
with
0i
0i(1 - Ki),
(3.3)
given by ea+bNi(1 -Ki) Oi--
1
+ e a+bNi(1-Ki)"
(3.4)
where N i is the number of mallards alive at the beginning of the hunting season in year i, and a and b are parameters to be estimated. The finding that different estimates of 13 are necessary for different time periods is indicative of the inadequacy of Eq. (3.2) to account for the essential features of the modeled system. Indeed, the density dependence expressed in Eqs. (3.3) and (3.4) guarantees that no single value of ~ in Eq. (3.2) will perform well for populations with widely varying abundances. Given density dependence, Eq. (3.2) might represent survival
30
Chapter 3 Models and the Investigation of Populations
reasonably well over the range of mallard abundance used in estimating its parameters, but it would not be expected to perform well beyond that range (Johnson et al., 1993). Of course, it may be that the true relationship between annual survival and hunting mortality is not well represented by Eqs. (3.3) and (3.4) either, in that they omit some other essential feature of system response to hunting. The main point here is that if density dependence really does underlie the compensatory mortality phenomenon, then Eq. (3.2) inadequately represents the system, and Eqs. (3.3) and (3.4) provide a somewhat more mechanistic, and possibly more useful, model. Our general recommendation regarding this dichotomy is to tend toward more mechanistic models, because they are more likely to provide useful predictions when state a n d / o r environmental variables assume values outside the range used in estimating model parameters. The notion of a mechanistic model is closely related to the idea of extracting essential features of the modeled system, recognizing that mechanism often begets model complexity. Certainly, models that are mechanistic in ways not essential to the purpose of the model (e.g., a model of the physiological death process as steel pellets enter the body cavity of a duck in our example) should be avoided.
rate, K. Some predictions of the hypothesis can be tested using annual survival rate Si, but other predictions can be tested only with estimates of seasonal survival 0 i and hunting mortality K i. Levins (1966, 1968) introduced the term sufficient parameter, as "a many-to-one transformation of lower level phenomena" (Levins, 1966, p. 429), emphasizing its role in integration and aggregation in his discussion of the term. By analogy with "sufficient statistic" from mathematical statistics (e.g., Mood et al., 1974), it seems reasonable to think of a sufficient parameter as one that contains all of the information needed to accomplish the function for which the model is intended. Thus, we return to the second general determinant of model utility and note that the degree to which model parameters are aggregated should reflect the intended model use. For example, if we develop models with the intent of comparing predictions of competing hypotheses under some sort of treatment or manipulation, then the models should include a parameter structure that accommodates the treatment or manipulation, and they must yield predictions that are useful in discriminating among the competing hypotheses.
3.5. HYPOTHESES, MODELS, A N D SCIENCE
3.4.3. More Integrated versus Less Integrated Model Parameters Integration of model parameters reflects the degree to which process components that could be modeled with separate parameters are aggregated into a single parameter. The concept is easily illustrated via an example involving animal population ecology. The finite rate of increase ~ of a population (defined here as the ratio of population sizes in two successive years) is sometimes used to model population change (see Chapters 7-9). We view K as an integrated parameter in the sense that it aggregates effects of survival, reproduction, and movement on the population. In the case of a population closed to movement (e.g., an island population), it sometimes is useful to decompose into two parameters, an annual survival rate and a reproductive rate. For still other modeling purposes, it is better to decompose annual survival rate into component survival probabilities corresponding to different seasons of the year, and to decompose reproductive rate into functional components such as breeding probability, clutch size, hatching success, and brood survival. Assume, for example, that our a priori hypothesis about effects of hunting mortality on mallard survival is given byEqs. (3.3) and (3.4). This hypothesis would be very difficult to test using only population growth
Regardless of the scientific approach, effective learning is conditional on, and accomplished relative to, a priori hypotheses and their associated models. In our view this point is not adequately appreciated by some practitioners of biological science. We emphasize that under a single-hypothesis approach to science (or management), inferences are tied to an a priori hypothesis and its associated model-based predictions, with investigation leading to a decision to reject or provisionally retain the hypothesis. Similarly, a multiple-hypothesis approach to science and management is conditional on a set of a priori hypotheses and their corresponding models. The associated prior probabilities are standardized in the sense that they sum to one over the hypothesis set, and changes in hypothesis probabilities (learning) are entirely conditional on the members of that set. Indeed, the conditional nature of learning holds even if none of the hypotheses under consideration provides a reasonable approximation to reality. These considerations lead to the suggestion that more thought and effort should be devoted to the development of a priori hypotheses and their associated models. It is common for research papers to begin with the statement of a statistical null hypothesis to be tested and thus to give the appearance of scientific rigor. Though the expression of a null hypothesis is not neces-
3.6. Discussion sarily a bad thing (but see Burnham and Anderson, 1998; Johnson, 1999; Anderson et al., 2000), its value to science depends heavily on the nature of the alternative hypothesis. For example, the testing of a null hypothesis of "no difference" or "no variation" against an omnibus alternative of "some difference" or "some variation" is not likely to be useful. At a minimum, testing should be based on numerical, or at least directional, predictions from a priori biological hypotheses and their associated models. As indicated in Chapter 2, competing hypotheses often can be investigated through estimation of parameters reflecting ratios, absolute differences, or other measures of variation in system variables under different treatments. The utility of experimentation is strongly emphasized in ecology, so much so that ecologists tend to view manipulations and perturbations as inherently good and useful to the scientific endeavor. It is true that experimentation can be an extremely powerful means of learning about natural systems (see Chapter 6). However, a priori biological hypotheses are key to informative experimentation, just as they are to more descriptive studies. Manipulations and perturbations conducted simply to "see what happens" are not likely to be nearly as useful to science as those conducted to see what happens relative to model-based predictions. The conditional nature of scientific learning argues that we devote substantial effort in identifying useful
31
hypotheses and developing models corresponding to them. When experience indicates that the hypotheses under consideration are inadequate predictors, it is important to devote additional effort in developing new hypotheses. We believe that the disciplines of animal population ecology and management would be well served by renewed emphasis on the articulation of meaningful hypotheses and their associated models.
3.6. D I S C U S S I O N In this chapter we have focused on the role of models in the conduct of science and management on animal populations. Models are useful, and often essential, in the conduct of science and management, and this theme will be continued in subsequent parts of this book. In terms of model development, we emphasize the importance of specifying objectives of a modeling effort and then tailoring the model to those objectives. This involves an effort to include in the model only those features of the system that are critically relevant to the modeling objectives. In the chapters to follow we describe various model structures for animal populations (Part II), methods for estimating parameters and relationships required for model development (Part III), and applications of models to management decisions (Part IV).
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C H A P T E R
4 Estimation and Hypothesis Testing in Animal Ecology
4.1. STATISTICAL DISTRIBUTIONS 4.1.1. Some Discrete Distributions for Animal Populations 4.1.2. The Normal Distribution for Continuous Attributes 4.1.3. Distribution Parameters 4.1.4. Replication and Statistical Independence 4.1.5. Marginal and Joint Distributions 4.1.6. Conditional Distributions 4.1.7. Covariance and Correlation 4.2. PARAMETER ESTIMATION 4.2.1. Bias, Precision, and Accuracy 4.2.2. Estimation Procedures 4.2.3. Confidence Intervals and Interval Estimation 4.3. HYPOTHESIS TESTING 4.3.1. Type I and Type II Errors 4.3.2. Statistical Power 4.3.3. Goodness-of-Fit Tests 4.3.4. Likelihood Ratio Tests for Model Comparisons 4.4. INFORMATION-THEORETIC APPROACHES 4.5. BAYESIAN EXTENSION OF LIKELIHOOD THEORY 4.6. DISCUSSION
and the scientific process. In both chapters we mentioned uncertainties that result from only partially observing a biological system. However, we are yet to account in a systematic way for this "partial observability," or to describe procedures for using field and experimental observations to parameterize and test models. Ecological systems are variable over space and time, and among individuals, and typically are observed only incompletely, that is, by means of samples. In this chapter we focus on stochastic variation that arises in parameter estimation with sample data, leaving until later a discussion of stochastic effects that arise through random environmental influences. Stochastic influences of the latter kind are distinct from sampling variation, in part because of the manner in which they propagate through time (in that the influence of a stochastic input at time t carries forward to time t + 1 and possibly beyond). The statistical modeling of such phenomena constitutes the discipline of stochastic processes and is discussed in Chapter 10. In this chapter our concern is to investigate the stochastic nature of sample-based parameter estimates, preparatory to an assessment of variation in model predictions. Population parameters typically are estimated by means of a representative sample of field data that are appropriate for the parameter of interest. These data are collected according to some scientifically supportable protocol (see Chapters 5 and 6), by means of which statistical properties of the sample can be ascertained. Mathematical formulas are used to combine the data into parameter estimates, with follow-up analyses of their statistical properties. Because only a sample of potentially available data is used in the process of esti-
In Chapters 2 and 3 we discussed the role of models in evaluating hypotheses about natural processes, and the use of models in making management decisions. Chapter 2 articulated a framework for examining theoretically based hypotheses and highlighted the comparison of predictions from theoretical models against field or experimental data. In Chapter 3 we discussed theoretical, empirical, and decision-theoretic uses of models in natural resources and further elaborated some operational linkages among models, hypotheses,
33
34
Chapter 4 Estimation and Hypothesis Testing
mation, the estimates are subject to sampling variability. The actual or "true" values of the parameters are not known with certainty, and assertions about them must be framed in terms of the statistical likelihood of their values. In this chapter we explore the estimation of population parameters based on maximum likelihood and other estimation procedures. The organizing concepts in the following discussion are (1) a statistical population, from which samples of individuals are to be drawn; (2) a distribution of values associated with individuals in the population; and (3) a formula, or estimator, for combining sample values into a numerical value or estimate of a population parameter. For purposes of this development a statistical population is defined as a collection of individuals that potentially can be sampled in an investigation. A population becomes statistical when (1) the sampling procedure is according to some sampling design whereby probabilities can be ascribed to samples, and (2) specific characteristics of interest (such as sample size, measurements of individuals in a sample, etc.) are recorded for the sample. Statistical populations might include the following examples: 9 White-tailed deer (Odocoileus virginianus) in Georgia. Measurements: length, weight. Sampling design: stratified random sample, with stratification based on sex and age. 9 American black ducks (Anas rupribes) in their breeding grounds. Measurements: age, sex. Sampling design: stratified random sample, with stratification based on geographic location. 9 Wood duck (Aix sponsa) nest boxes in the Missisquoi National Wildlife Refuge, Vermont. Measurement: use/nonuse. Sampling design: simple random sample, with stratification based on geographic location. 9 Striped bass (Morone saxatilis) in Lake Hartwell, Georgia. Measurement: age. Sampling design: stratified random sample. 9 Students at the University of Georgia. Measurement: eye color. Sampling design: simple random sample. 9 Hunting licenses purchased by Vermont residents. Measurement: county of residence. Sampling design: simple random sample. 9 Fish in the Connecticut River. Measurement: species. Sampling design: stratified random sample with stratification based on latitude. 9 Duck nests in North Dakota. Measurement: success/ failure. Sampling design: random sample of nests, followed by repeated visits to individual nests until either fledging or failure of the nest occurs.
9 Harvested waterfowl. Measurement: species, age, and sex. Sampling design: stratified cluster sample, with stratification by state of harvest and clustering by county within state. 9 Trees above 300 m elevation on Camel's Hump Mountain, Vermont. Measurements: tree size, tree height. Sampling design: Cluster sample, with clustering defined by grid points.
4.1. STATISTICAL
DISTRIBUTIONS It is useful to think of natural variation in organisms and their environments in terms of some underlying pattern or distribution of occurrence. To illustrate, consider some observable characteristic of individuals in a population--for example, the size, weight, or age of each organism in a biological population. Ideally the value of the characteristic for any individual can be determined unambiguously, simply by observing (and perhaps measuring) the individual. The relative frequencies with which different values occur in the population constitute a frequency distribution. The frequency distribution reflects a probability distribution for these values when individuals in the population are sampled randomly. In that case, the attribute values for randomly selected individuals occur with probabilities equal to the frequencies of occurrence in the population, so that the frequency and probability distributions are identical. It is useful to distinguish discrete distributions, for which the attribute of interest can assume only a countable number of values, from continuous distributions, for which attribute measures can range over a continuous set of values. Examples of discrete distributions include the following entities: 9 Survival or mortality (two classes). 9 Nest box s t a t u s w u s e d / u n u s e d (two classes). 9 Capture status--captured/not captured (two classes). 9 Taxonomic class of harvested waterfowl (e.g., dabbling ducks, diving ducks, geese). 9 Age/sex class of migrating black ducks (four combinations of age and sex). 9 Atlantic salmon ages (e.g., five age classes). 9 Duck eggs per nest (positive integers). Note that in each of these examples there are only countably many values for the attribute of interest. For continuous distributions the attribute can assume any value over a continuous range. Examples of continuous distributions could include bear weights (positive real numbers), tree heights (positive real numbers), and
35
4.1. Statistical Distributions deviations from average heart rate (positive and negative real numbers).
4.1.1. Some Discrete Distributions for Animal Populations The distribution of a population attribute often can be described with a mathematical function f(x), which allows one to specify with a single formula the frequency of occurrence of any attribute value x. The function f(x) also specifies the probability of occurrence of x for r a n d o m l y sampled individuals, and in that case is called the probability density function. Some important examples follow. 4.1.1.1. Binomial Distribution R a n d o m events in which one of two outcomes can occur are k n o w n as Bernoulli trials. For example, suppose 1 and 0 designate head and tail, respectively, for outcomes of a coin toss. Then the distribution of outcomes can be described by
f(x) = (0.5)x(0.5)1-x. Thus, the frequency of occurrence of heads (x = 1) is f(1) = 1A, and the frequency of occurrence of tails (x = 0) is f(0) = 1A. N o w assume that the head side of the coin is m a d e of lead and the tail side is m a d e of copper. Then the relative frequency of a head is no longer 1/2, but is some general value p. In essence, p is a p a r a m e t e r for the distribution of outcomes (i.e., a constant that provides information about distribution structure). The frequency distribution for this more general case
is
f(xlp) = pX(1
_
p)l-x,
and is k n o w n as a Bernoulli distribution. Note that w h e n p = 1A the general distribution reduces to the distribution for a fair coin. In fact, each value of p defines a different Bernoulli distribution. Instead of considering the outcome of a single coin toss, we can consider the total n u m b e r of heads resulting from, say, two coin tosses. The distribution of outcomes then is given by
f(x)
x!(2
2~ -
X)! (0"5)x(O'5)2-x"
where p = 1A is the probability of getting a head on any single coin toss. Thus the frequency of occurrence of two heads (x = 2) is f(2) = 1/4, the frequency of occurrence of one head (x = 1) is f(1) = 1A, and the frequency of occurrence of no heads (x = 0) is f(0) = 1/~. In contrast to the first example, this distribution has two parameters: the probability of getting a head on
a single toss (p = 1/2), and the total n u m b e r of tosses (n = 2). In general the distribution of the n u m b e r of heads in n tosses, with probability p of a head on any one toss, is described by the probability density function
tnt,x l
x
Each combination of parameters n and p defines a specific binomial distribution, which is designated by B(n, p) to emphasize the role of n and p. The binomial distribution plays a crucial role in the investigation of biological populations. The distribution can be derived as a realization of n i n d e p e n d e n t Bernoulli trials, via the product of separate Bernoulli distributions with c o m m o n parameter p (see A p p e n d i x E). The assumption of independence, a key feature that determines the probability distribution for aggregate data, will be invoked repeatedly in the d e v e l o p m e n t of statistical estimation models. 4.1.1.2. Multinomial Distribution Instead of sampling with dichotomous outcomes (e.g., head or tail), we can consider sampling with several possible outcomes. For instance, a forward pass in football can result in a completion, an incompletion, or an interception. A s s u m e that over the course of last year a quarterback's percentages are Pl = 0.6, P2 = 0.38, and P3 = 0.02, respectively, for these outcomes. N o w consider the distribution of outcomes for any five r a n d o m l y chosen forward passes, based on these frequencies. If x I is the n u m b e r of complete passes, x 2 is the n u m b e r of incomplete passes, and 5 - x I - x 2 is the n u m b e r of interceptions, the distribution of outcomes for the quarterback's passes is given by the probability density function f(x1, X2) =
( 5X2) (0.6)xl(o.38)x2(o.02)
Note that the possible results of five passes include anything from 0 to 5 completed (the latter an unlikely but possible event), from 0 to 5 not completed (extremely unlikely but possible), from 0 to 5 intercepted (just about impossible, but not quite). Note also that two variables (x 1 and x 2) rather than one are required to specify the range of possible outcomes. Finally, note that three parameters are involved in the specification: the n u m b e r of passes (n = 5) and probabilities of two of the possible outcomes (Pl = 0.6, P2 = 0.38). In general the distribution of outcomes for n passes, with probabilities Pl and P2 of a complete and incomplete pass, respectively, is
36
Chapter 4 Estimation and Hypothesis Testing
f(xl, x2ln, pl, P2) =
F/ ~.Xl...x2t 1 ~x1, x 2 / P 1 F 2 ~'
--pl--p2)n-xl-x2.
Each combination of parameters defines a different distribution. Both this distribution and the binomial distribution from the previous example are members of the general class of multinomial distributions, which are appropriate for certain kinds of count data (see Appendix E). Multinomial distributions are especially useful for estimation of biological parameters such as population size, survival rate, and harvest rate. Because these parameters are key to the management of animal populations, multinomial distributions incorporating them are used frequently in the material to follow.
g=-I
0.4
~=0
g=l
0.3
0.2
\
0.1
\ 0.0
0.4
4.1.2. The Normal Distribution for Continuous Attributes 0.3
Many data in biological samples are continuous (e.g., weights, sizes, durations) rather than discrete (e.g., counts, qualities, category memberships). The probability distributions for continuous data are represented by smooth distribution functions over a range of values for the data. By far the most intensively studied and most frequently used continuous distribution is the univariate normal distribution with probability density function
0.2
0.1
~2= 1 0.0 -2
0
2
4
X
f(xllx, r =
1
2x/GGr
exp
-
2\
/
As indicated in this formula, the univariate normal is a two-parameter distribution, parameterized by the distribution mean IXand the variance 0.2 (or the associated standard deviation 0.) (see Section 4.1.3). As shown in Fig. 4.1, it is bell-shaped, symmetric about the mean, and more or less peaked, depending on the variance. The mean IXis a location parameter, in that it specifies the location (but not the shape) of the distribution. Figure 4.1a illustrates the effect of changing the mean, while holding the variance constant. Thus, increasing the mean effectively shifts the distribution to the right along the x axis. On the other hand, the variance is a shape parameter, in that it specifies the shape (but not the location) of the distribution. Figure 4.1b shows the effect of changing the variance, while holding the mean constant. Thus, an increase in the variance leads to a distribution that is less peaked and more evenly spread over the range of x values. Typically the normal distribution is denoted by N(I,, 0.2) or N(xll~, r to emphasize the role of the mean and variance in specifying shape and location of the distribution. As with all continuous probability density
FIGURE 4.1 Normalprobability distribution. (a) Effect of changing the mean (IX)of the distribution. (b) Effect of changing the variance (or2) of the distribution.
functions, the area under the curve is 1 (i.e., f~_oof(x) dx = 1). Furthermore, the area under the curve to the left of any specific ordinate value, say, v, is the probability that a randomly chosen value x from this distribution will be less than or equal to v. The probability is expressed mathematically as
f(u) = P(x<-v) =
f"
N(x]I*,
0 "2)
dx,
--oo
where f indicates the integral of N(x]l,, 0.2) from -oo to the point u (i.e., the area under the curve from -oo to v).
Example We use the distribution of body masses in grams for a hypothetical population of mice (Fig. 4.2) to illustrate several points. First, an individual taken at random from this population will be most likely to have a mass of about 27 g, as represented by the peak (mode) of the probability density function. Second, the probability of a randomly chosen individual having mass between
4.1. Statistical Distributions 25 and 29 g is approximately 0.95. That is, about 95% of the time the mass of a r a n d o m l y selected individual will be between 25 and 29 g. Third, the mean represents the "center of mass" for the distribution, and the variance is a measure of the "spread" or h o w closely distributed around this mean are the masses of r a n d o m l y selected animals. Because of the form of the normal distribution, there is a simple association between the standard deviation r (see Section 4.1.3 for a formal definition) and the area u n d e r the normal curve within I standard deviation of the mean ~. Thus, the area u n d e r the normal curve between the limits tx - r and ~ + r is always 0.68, irrespective of the particular values of the mean and standard deviation (Fig. 4.1). Similarly, the area between the limits ~ - 1.96r and ~ + 1.96r is 0.95, irrespective of the values of tx and r In terms of probabilities, this means that about 95% of all observations from a normal distribution will be within two standard
0.4
0.3
0.2
0.1
37
deviations of the distribution mean and only about 2.5% will be outside this range in either tail of the distribution. Said differently, r a n d o m observations from any normal distribution will be within two standard deviations of the mean with probability 0.95. In fact, there is a specific multiplier of standard deviation associated with every probability. Thus, for any probability P one can determine the multiplier d such that observations from a normal distribution will be within d standard deviations of the mean with probability P (Table 4.1). These multipliers can be found in tables in most introductory statistics textbooks.
Example Consider an estimate of mean bear weight for adult male black bears (Ursus americanus) in Vermont, based on the average weight of five r a n d o m l y chosen bears. An important result of statistics asserts that u n d e r some very general conditions a sample average is effectively distributed normally, with a mean that is the same as the population mean and a variance that is the same as the population variance, divided by the sample size. By this result, which is k n o w n in the statistical world as the Central Limit Theorem (Mood et al., 1974), the expected value of the average of five bear weights is the same as the population mean ~, and the variance of this average is 0.2 times the population variance r Thus, repeated sampling of the population will produce sample means that are within 1.96r of the population mean about 95% of the time. An intuitive extension of the univariate normal distribution involves m e a s u r e m e n t on samples of more than one attribute, i.e., observations are characterized by two or more attributes instead of one. If the corresp o n d i n g r a n d o m variables are normally distributed, then the vector of variables is said to have a multivariate normal distribution. Such distributions are characterized by the population means and variances of the r a n d o m variables, as well as the covariances (see Section 4.1.7) between the r a n d o m variables. A more
b 1.0
0.5
TABLE 4.1
Scaled Distances d for the Normal Distribution and the Corresponding Probabilities a
Distance/probability
d P~
0.0 25
27
Value
1.282 1.645 1.960 2.326 2.576 3.290 0.8 0.9 0.95 0.98 0.99 0.999
29
FIGURE 4.2 Normal distribution for the weights of mice, with = 27 and 0 -2 = 1. ( a ) Probability density function f(x). (b) Cumulative distribution function F(x).
aThe value Pa represents the probability that a randomly chosen observation from a normal distribution is within d standard deviations of the distribution mean.
38
Chapter 4 Estimation and Hypothesis Testing
complete description of the multivariate normal distribution is given in Appendix E. As illustrated in these examples, a complete specification of a frequency distribution requires a mathematical formula for the relative frequencies of an attribute and the parameter values parameterizing that formula. Mathematically, a frequency distribution can be expressed as f(xl0), where _x is the sample value (or values), f indicates the mathematical form of the distribution of values, and 0 indicates the parameter (or parameters) identifying the particular distribution of interest. The variable x often is referred to as the value of a "random variable," because it is determined by the random sampling of a statistical population. The underline of x indicates the "variable" is really a vector of more than one variable (as in the multinomial example above, where x consists of the variables x 1 and x2). The usual convention is to designate a single variable by the letter x without the underline; f(xl0) is often referred to as a joint distribution, to emphasize that more than one random variable is involved. Appendix E describes some distributions that commonly are used in the estimation of parameters for animal populations.
E(x) = (1)(1/6) + (2)(1/6) + ... + (6)(1/6) = 3.5. This result indicates that high and low values from multiple rolls of a die "average out" to a midpoint value of 3.5. By extension, the expected value of any function g(x) of the random variable x is simply the weighted average of the values g(x) can take:
E[g(x)] - ~ x
g(x)f(xlO).
If g(x) = x 2, the same uniform distribution for rolls of a die as in the previous example results in
m
4.1.3. Distribution Parameters Typically populations are described in terms of population means, variances, and other parameters that provide an informative summary of distribution structure. The most commonly used parameters focus on measures of central tendency (e.g., population means, medians, and modes), measures of dispersion (e.g., variances, standard deviations, and maximum and minimum values), coefficients of variation, and other measures of distribution shape such as skewness and kurtosis. Many important population parameters are characterized in terms of a weighted average known as statistical expectation. For example, the expected (or average, or mean) value of a random variable x is just the average of values that x can take, weighted by the frequency of occurrence of those values:
E(x) = ~_, x f(xiO). x
The symbol E(x) is used to denote the expected value, or weighted average, of the variable x. As an example, consider the expected value from a roll of a die (one of a pair of dice). Any integer value between 1 and 6 can be the outcome of a roll, with a uniform frequency distribution for the possible outcomes. Thus the expected value of outcomes is
E(x 2) = (1)(1/6) + (4)(1/6)+--" + (36)(1/6) = 15.17. Many statistical parameters that commonly arise in population biology can be expressed in terms of expected value. Among others, these include distribution properties such as central tendency, dispersion, and population density: 9 The population average or mean value of a characteristic x is given as above, by the expected value E(x) over the population. The expected value typically is represented by the symbol p~. 9 The population variance, designated by 0-2, gives a measure of the dispersion or spread of characteristic values within a population. A small value of 0-2 indicates that values are "clumped" about the distribution mean. A distribution for which 0 -2 is large has values that are widely dispersed. Variance is computed by 0-2 = E[(X~)2]. 9 The population standard deviation is simply the square root of the population variance. As with variance, the standard deviation, denoted by 0-, measures distribution spread. The advantage of standard deviation as a measure of dispersion is that it is measured in the same units as the population mean. 9 The coefficient of variation cv(x) measures dispersion relative to the population mean: cv(x) = 0-/p~. In most situations the parameters of a distribution are not known. Because it usually is impractical or impossible to record attributes for every member of a population, the population is represented by a sample of individuals selected in some representative way from the population. The intent is to characterize the whole population based on attributes of the sample. For example, the sample mean might be used to estimate the population mean; sample variance, to estimate population variance; sample proportions, to estimate population proportions; sample maximum
4.1. Statistical Distributions and minimum values, to estimate population extremal values; etc. The following estimators are frequently used: 9 The population mean ~ can be estimated by the sample mean. If the sample consists of n individuals from the population, the sample mean is given by =
(Xl
+
"'" +
Xn)/n.
9 Population variance 0 .2 c a n be estimated by the sample variance g/
S2-- E ( X i i=1
n-
1
x)2/(n-
1)
x 2 - n~2 "
39
fact that they were generated from the same random process. There is a simple rule for describing the joint distribution of statistically independent random variables. Random variables x I and x 2, each with the distribution fix[0), are statistically independent if and only if they have a joint distribution given by the product
f(xl, x210) = f(xll0)f(x210) of the individual distributions. This rule generalizes in the obvious way to include more than two random variables: n random variables x 1, ..., x n, each with the same distribution, are statistically independent if and only if their joint distribution can be written as the product ///
Note that the divisor in this expression is n - 1 rather than n. Because there are only n - I independent values in the sum (the values in the sum are adjusted by ~), the appropriate divisor is n - 1. 9 The population standard deviation 0. can be estimated by the square root s of the sample variance s 2. 9 The population coefficient of variation 0./1~ can be estimated by the ratio s/Y of the sample mean and sample standard deviation. The classical theory of statistical inference deals with sample-based inferences about the parameters of a statistical population and the degree of confidence with which these inferences can be made. Some procedures for parameter estimation are described in Section 4.3. For reasons to be clarified later, we give special emphasis to the estimation of parameters in the multinomial and normal distributions.
4.1.4. Replication and Statistical Independence In many studies a statistical population is sampled more than once, i.e., the sampling procedure is repeated. These sample "replications" are used not only to better characterize the population, but also to assess the amount of variability in the sampling procedure. Each replication consists of a sample from the same statistical population, and each replication yields a value for the random variable(s) of interest. For example, the experiment described above, involving the number of heads in two coin tosses, could be replicated, say, two times. Thus we have random variables xl and x 2 for the number of heads in each of the two replicates. These random variables are statistically independent, in the sense that the value of one random variable tells us nothing about the value of the other, beyond the
f(xl .....
xnl0) - H f(x~10). i=1
The rule also holds for statistically independent random variables that do not have the same distribution. For example, random variables Xl and x 2, with distributions f1(x1]01) and f2(x2102), respectively, are statistically independent if and only if their joint distribution is given by the product f(Xl, X2]01, 02) = f1(x1101)f2(x2102 ).
This result generalizes to more than two random variables, in the same manner as above.
Example As part of a study of field mice, two researchers independently collect random samples of 10 mice from the same population and measure body length and weight of the individuals in each sample. Let X1 and 91 represent the average length and weight of mice in sample 1, with fl(Xl) and gl (Yl) the respective sampling distributions. Let x2 and 92 represent the average length and weight of mice in sample 2, with distributions f2(x2) and g2(Y2)- Because the samples are collected independently, the joint distribution of average lengths is simply the product fl(Xl) f2(x2) of the two distributions for length. The same product rule applies for the average body weights from the two samples: the joint distribution of average weights is given by gl(Yl) X2(Y2). It also applies for the average length from sample I and average weight from sample 2: the joint distribution is the product fl(Xl) g2(Y2). However, within a given sample the joint distribution of body length and size is not given by a simple product of distributions for length and weight. Because length and weight are measured on the same animals in a sample, these variables are not statistically independent.
40
Chapter 4 Estimation and Hypothesis Testing
4.1.5. Marginal and Joint Distributions In many studies more than one attribute is recorded for each sampling unit. As an illustration, the age x 1 and sex x 2 for each individual in a sample of small mammals might be recorded, so that every individual in the sample is characterized by a pair (x 1, x 2) of values. Random sampling of the population induces a probability distribution f(x 1, x2), such that the probability of choosing an individual at random from the population with particular age and sex values x I and x 2 is f(x 1, x2). For example, the probability distribution for a randomly sampled population with unit per capita production (equal numbers of adults and juveniles) and equal sex ratio (equal numbers of males and females) in each age class is f(x 1, x 2) = 0.25 for each age-sex class, i.e., for each combination of x 1 and x 2. The notion of a probability distribution for two attributes extends easily to multiple attributes. Thus, if x 1, ..., x k represent k attributes for individuals in a population, random sampling of the population induces a probability distribution f(xl, ..., Xk), such that the probability of choosing an individual at random from the population with attribute values x I through x k is f(x 1, ..., Xk). Such a distribution is called a joint probability distribution, to emphasize the multivariate aspect of the probabilities. Often it is useful to determine the statistical distribution of a single random variable from the joint distribution of two or more random variables. Assume, for example, that the joint distribution of x I and x 2 is specified by f(xl, x2), where f(x 1, x 2) is the probability mass for the pair (x 1, x 2) of values for the random variables. To determine the probability mass for Xl, without reference to x 2, it is necessary to aggregate the values f(xl, x 2) across all the values that x 2 can assume. For random variables corresponding to discrete distributions like the binomial, this consists of a simple summation: fl(Xl) = ~ f(xl, x2), X2 where the subscript in fl(x1) indicates a marginal distribution of x 1, derived from the joint distribution f(xl, x2). For random variables corresponding to continuous distributions like the normal, the marginal distribution is produced by integration over x2:
can be obtained by aggregation of probability mass over all values of Xl" f2(x2) = E f(xl, x2) Xl for discrete random variables, and f2(x2) = f
x1
f(xl, x2) dXl
for continuous random variables. The specification of a marginal distribution can be generalized to multivariate distributions of dimension greater than 2. Thus, if x' = (Xl, ..., x k) is a vector of random variables with joint probability density function f(x 1, ..., Xk), the marginal distribution of xi is given by aggregating probability mass over all values of the other variables: fi(xi) -- ~ ... ~ ~-, ... ~, f ( x 1, ..., X k) Xl xi-1 Xi+l Xk
for discrete distributions, and
fi(xi)=f Xl "'f Xi-1 f Xi+l ""f Xk f(Xl,...,X k) d x 1 ... d x i _ 1 dXi+l ... d x k
for continuous distributions.
Example Consider the above example with the first sample of length and weight from 10 mice, with fl(Xl) and gl(Yl) the respective sampling distributions for length and weight. If we wish to obtain the marginal sampling distribution of lengths, for each value of length we integrate the joint distribution over all weights, that is fl(Xl) ----- f
x2
f ( X l ' X2) dx2.
Likewise, to obtain the marginal distribution of weight, for each value of weight we integrate the joint distribution over all lengths: f2(x2) = f
Xl
f ( X l ' X 2) d x 1.
4.1.6. Conditional Distributions fl(Xl) "- f
x2
f ( X l ' X2) dx2
Thus, the distribution of x I for both discrete and continuous distributions is obtained from the joint distribution of x I and x 2 by aggregating, for each value of Xl, the probability mass f(xl, x 2) corresponding to all values of x2. In analogous fashion, a marginal distribution of x 2
Irrespective of their statistical independence, the joint distribution of two random variables can be expressed in terms of the conditional distribution of one of the variables, given a value for the other. The joint distribution of x I and x 2 can be expressed as f ( X l , X2) = fl(Xl) f2(X2[Xl)
4.1. Statistical Distributions where f2(x2]x1) is read "the probability distribution of x2 given Xl." The latter distribution is said to be "conditional" in that knowledge of Xl allows us to update the distribution of x 2. If x I and x2 are statistically independent, of course, knowledge of x I provides no information about the distribution of x 2. In that case the conditional distribution of x2 given x I is simply f2(x2), and the joint distribution is given by the product
f(Xl,
41
age at death). Random sampling of the population defines a joint distribution for these variables; however, the event of surviving to age Xl is subsumed in the event of surviving to age Xl and then dying later at age x 2. This means that the joint distribution x I and x 2 can be expressed in terms of the distribution of x 2 alone:
f(xl, X2) -- f2(x2)" X2) = fl(Xl) f2(x2)
for independent random variables. As indicated in the preceding example, measurements made jointly on the same sample organisms typically are not independent. The conditional distribution of one variable will depend on the value of the other variable, and their joint distribution must be expressed in conditional terms, as above.
Example Consider a situation in which two measuresmfor example, height and w e i g h t - - a r e identified for each individual in a population. Random sampling of the population defines random variables x I and x 2 for height and weight, with a joint distribution describing the frequency of occurrence of their values. The conditional distribution of x 2 given x 1, f(x21271), incorporates the tendency of weight to be associated with height, i.e., tall individuals tend in general to weigh more than short individuals. This association is expressed by a shift in the conditional probability distribution f(x21x 1) toward larger values of x 2 when x I is large and a shift to smaller values of x 2 when x I is small.
Example Let x I be an indicator of sex in humans, with 271 0 if an individual is male and x I = 1 if the individual is female. Let x 2 be an indicator of the sex chromosome, with x 2 = 0 for the X chromosome and x 2 = 1 for the Y chromosome. Random sampling of individuals in a population, followed by random selection of one of the pair of sex chromosomes for each individual, defines a bivariate distribution of the random variables x I and x2. The frequency of occurrence of values for x 2 clearly is influenced by the value of x 1" if x I = 0 (the individual is male) then x 2 is equally likely to be 0 or 1 (because males possess both chromosomes), but if Xl = 1 (the individual is female) then x 2 = 1 with probability 1 (because females possess only the X chromosome).
Example Consider the monitoring of a sample of individuals in a population from a particular point in time until the time of death. Let Xl represent current age at the start of the monitoring and x2 represent life-span (i.e.,
Therefore the conditional distribution of age at death, x 2, given survival to age x 1, is found from f(x 1, x 2) = f2(2721271)f1(271) to be
f(x21Xl) = f2(x2)/f1(271). In words, the probability of an individual dying at age x 2, given that it has survived Xl years already, is the probability of having a life-span of x 2 years divided by the probability of survival to age x 1. This particular example is important in the analysis of age-structured population models (see Section 8.4). In that application the probability of surviving to age x is denoted by lx, and the probability of a life-span of x years is denoted by Px. It follows that individuals of age a have a lifespan of x years with probability Px/la. The definition of conditional distribution leads to a very useful relationship known as Bayes' Theorem. Because the joint probability density function f(x 1, 272) can be expressed either by
f(xl, X2) = f1(271) f2(X2]Xl ) or by
f(Xl,
X2) -- f2(x2)
fl(XlIX2),
equating these two expressions and solving for (say)
f2(X21X 1) yields f2(X21X1) =
f2 (X2) fl(XllX2) fl(Xl) '
providing a standard form for Bayes' Theorem. This expression is closely related to the event-based form
P(E21E1) =
P(E2)P(E1]E2) P(E1 )
(4.1)
of Bayes' Theorem, where P denotes probability and E 1 and E2 represent two chance outcomes (see Appendix A). As seen in Section 4.5, Bayes' Theorem can provide a powerful method for updating models with data from experiments or monitoring programs, an important feature of adaptive resource management.
42
Chapter 4 Estimation and Hypothesis Testing
4.1.7. Covariance and Correlation Though the random variables corresponding to replicated samples are statistically independent by design, many random variables are not. For example, the two variables characterizing the passing success of an allpro quarterback are not statistically independent: knowledge of his completions (x 1) in 10 passes enables us to better predict his incomplete passes (x2). A standard measure of statistical dependence is covariance, defined by
COV(X1, X2) =
E[(Xl --~L1)(X2
--~2)]-
In essence, covariance tells us about the tendency of two random variables to "covary" from sample to sample, i.e., to take values that are associated.
Example Consider the number of heads in 10 coin tosses (X1) and twice the number of heads in 10 tosses (x2 = 2Xl). Replicated samples of this experiment will yield perfect covariance between these two variables, in that knowledge of the value of one completely specifies the value of the other. For this example it follows that if x I is large for a given sample, x 2 is sure to be large as well; if x I is small for a sample, x 2 also will be small. An equally trivial example involves the number of heads in 10 tosses (x 1) and the number of tails (x2 = 10 - x 1). These random variables also covary perfectly, but in opposite directions: if x I is large, x 2 is sure to be small; if Xl is small, x2 is sure to be large. For both examples it should be clear that information about the value of one random variable is informative (in these cases completely informative) about the value of the other. This is of course the defining characteristic of statistical dependence.
Example A less trivial example of covariance concerns the statistical dependence between estimates of harvest and survival rates, based on analysis of banding data (see Chapter 16). In the past the techniques used to analyze banding data produced mean survival and harvest rate estimates with high levels of negative sampiing covariance. Thus, low estimates of mean survival were associated with high estimates of mean harvest rate, and high mean survival estimates were associated with low mean harvest rates, irrespective of the actual biological situation. Because of the negative sampling covariance between mean survival and harvest rate estimators, areas with high survival rate estimates were likely to have low harvest rate estimates, and vice versa, even if there was no biological relationship whatever between the actual rates. Such misinterpretations
of a biological situation can have [and in fact did have; see Anderson and Burnham (1976)] major consequences for the management of biological populations. In large measure the modern methods of analysis of banding data, many of which are covered in the chapters of Part III, are designed to redress the problem of sampling covariance among estimators of biological parameters. The covariance between two random variables often is more informative if it is referenced to the underlying variation of the variables. A closely associated parameter is correlation, which essentially uses standard deviation to scale the covariance to a value between - 1 and +1. To calculate the correlation of two random variables, one simply divides their covariance by the standard deviations of each: c o r r ( x 1, x 2) =
COV(X1, X2) SD(xl)SD(x2)"
Values of correlation close to 1 indicate that x I and x 2 are strongly associated in a positive direction, values close to - 1 indicate a strong negative association, and values close to 0 indicate little or no association between Xl and x 2. For example, the correlation between the number of heads (x 1) in 10 coin tosses and twice the number of heads (x2) is 1, indicating perfect association in a positive direction. The correlation between the number of heads and the number of tails in 10 tosses is - 1 , indicating perfect negative association. In biologically meaningful applications, of course, the associations among variables are less than perfect. A correlation of 0.8, for instance, often is indicative of strong association between biological variables.
4.2. PARAMETER ESTIMATION The subject matter of statistics includes both probability theory and statistical estimation. These subjects both address random phenomena, and each relies on an operational framework that includes (1) a definable population of interest, (2) random sampling of individuals from the population, (3) measurement of attributes for the sample, and (4) a probability distribution for the attribute measures that is based on the sampling distribution. Despite these similarities, however, probability theory and statistical estimation focus on somewhat different questions:
1. The probability question. The study of probability is primarily an inquiry into patterns of distribution: for a given probability distribution fix]0) with known parameter 0, what are the probabilities of occurrence of
4.2. Parameter Estimation
43
the values of x? Here the mathematical form of the distribution is assumed to be known, and the value of parameter 0 is specified. The focus is therefore on patterns of the distribution frequencies and how they change with changing parameter values. The probability question presupposes that 0 is fixed and x is variable, and asks "for a specified value of 0 what values of x can I anticipate when I randomly sample individuals from the population? That is, how frequently will I observe different values for a given form and parameterization of the distribution function?" The study of distributions as they are influenced by their parameters is the subject matter of probability theory. The function f(xl0) is called a probability density function, describing the probability distribution of a random variable that assumes different values x. 2. The estimation question. Rather than focusing on the frequencies with which data values arise from known distributions, statistical estimation focuses on databased inferences about structural features of the distribution. The question here is "given an observed value for x, what is the distribution from which this value arose?" In practical applications the question usually takes a narrower form: given an observed value of x, what is the corresponding value of 0 parameterizing the distribution giving rise to x? Here the roles of parameter and variable have been reversed: now x is assumed known (having been obtained by sampling the population) but 0 is u n k n o w n and thus considered variable. The challenge is to use information about the known value of x, along with other information about the form of the distribution of x, to estimate the value 0. That is, we wish to use the value x obtained by sampling the population to make inferences about population parameters. An estimator of 0 will be designated by 0 to indicate that it is sample-based and therefore different from the true population parameter. Because it is based on the random variable x, the estimator 0 is subject to random variation and-thus is a random variable, with its own distribution inherited from f(xl0). The derivation of estimators and the study of their statistical properties is the subject matter of statistical estimation theory.
were to be obtained by repeated sampling of the population, and each sample were used to estimate the parameter of interest, we would find variation in these estimates. Because any given estimate of the parameter is the result of a random process, the estimate will differ from other randomly generated estimates and from the "true" value of the parameter of interest. This variation is characterized by a probability distribution for the estimator. The question then arises: how good is an estimator, in the sense of being "close" to the parameter of interest? An approach to this question is to identify measures of statistical behavior to use in comparing estimators. Three such measures are estimator bias, precision, and accuracy, each of which is based on the fact that random sampling imparts random variation to a sample-based estimator.
We will be especially concerned with estimation procedures for multinomial probability distributions, since they arise frequently in the modeling of wildlife and fish populations.
Example
m
m
D
4.2.1.1. E s t i m a t o r Bias
Let g(010) represent the distribution of an estimator 0, where g(010) is derived from the sample distribution f(x[0). For discrete sampling distributions such as the multinomial, the expected value of 0 is given by E(0) = ~
0g(010),
m
where the summation is over all possible values of 0. If E(0) ~= 0 the estimator 0 is said to be a biased estimator of 0, with the difference E(0) - 0 measuring the estimator bias. In words, bias expresses the tendency of replicated parameter estimates to differ systematically from the parameter value. If E(0) - 0 = 0 the estimator is said to be unbiased.
Example Consider the distribution of the number x of heads in n tosses of an unfair coin, for which the probability is p of getting a head on any random toss. Because the expected value of x is E(x) = np, x/n is an unbiased estimator of p (i.e., E(x/n) = p). On the other hand, (n + 1) / (x + 1) is a biased estimator of l / p , because E[(n + 1)/(x + 1)] ~ 1/p.
One could estimate variance with a sample of size n by n
~y2 __ 1~.~
4.2.1. Bias, Precision, and Accuracy Because an estimator of a population parameter is based on random sampling, the estimator is a random variable, with its own distribution. If replicate samples
?/.
(X i __ ~)2.
l = 1
It can be shown that this estimator is biased, in that E(6-2) ~ o-2. However, the bias is a function of sample size and decreases asymptotically to 0 with increasing
44
Chapter 4 Estimation and Hypothesis Testing
sample size. In this particular instance the bias can be eliminated simply by replacing n in the denominator b y n - 1"
of these counts is used to estimate N, the result will be biased: E(~I) = E
H
S2 -- n -
1 ~.
Ci
(xi -- 2)2.
l=1
Y/
= ~
Then the expected value of S 2 c a n be shown to coincide with the value it is intended to represent, namely cr2. The replacement of n in if2 by n - 1 is a relatively minor adjustment, and its importance diminishes as sample size increases and ~.2 and s 2 both converge to cr2.
E(Ci)
i=1 H =
i=1 = ~N,
Example
so that E(/~/) ~ N.
Systematic bias that is independent of sample size can occur when one assumes a probability distribution that is inappropriate for the random variables under investigation. Suppose there is a consistent bias in the readings of a mass scale used to weigh organisms. Each observation then underestimates the weight x i of an organism by some amount d (i.e., the scale registers low by d grams), and the sample mean ~ underestimates of the average organism weight ~ for the population by the amount d: E(~) = E ~ ( x i / n ) i=1 /I/
= ~,~ E ( x i ) / n i--1 =
lln
~_j(~i - d )
]
-i=1
= ~-d. Note that the bias is unaffected by increases in sample size. In the absence of information about the magnitude of d, it is not possible to eliminate the bias in this estimate.
Bias is introduced in both the above examples because an invalid assumption is made about the relationship of the sample to the underlying probability model and its parameters. Thus, the scale readings are assumed incorrectly to represent actual organism weights, and plot counts are assumed incorrectly to measure actual abundances. Note that these biases persist irrespective of sample size, and adjustments to eliminate them are not possible unless the bias factors d and ~ are known or can be estimated. A key source of model bias in animal population estimation is the inadequate treatment of organism detectability in the sampling units. The problem of detectability will be addressed in more detail in Part III. 4.2.1.2. Precision
An estimator 0 is subject to random variation, in that repeated sampling yields estimates that vary from sample to sample. The tendency of replicated estimates to be dispersed is an expression of estimator precision, which is measured by the variance of the estimator. Estimator variance is defined as var(0) = E{[0 - E(0)] 2} = E(0 2) -
Example
Suppose we wish to estimate the abundance N of a population in an area consisting of n study plots. Because of our sampling procedure, on average we overlook 10% of the animals present on each plot. The count Ci on plot i therefore relates to the actual abundance N i by E(Ci) = ~Ni,
where ~ = 0.9 is the expected proportion of animals detected on any given plot. If the summation N = ~ i C i
(4.2)
[E(0)] 2,
where E(0 2) -- ~
6 2 g(610),
with the estimator distribution g(010) inherited from the sampling variation. When estimate values exhibit a large amount of dispersion (i.e., when the estimator variance is high), the estimator is said to have low precision. Conversely, when dispersion is low (i.e., the estimator variance is low), the estimator precision is said to be high. Another measure of precision is the
4.2. Parameter Estimation square root of var(0), referred to as the standard error of the estimate. Derivation of var(0) with Eq. (4.2) requires knowledge of g(010), which in turn requires knowledge of the parameter 0 being estimated. As an alternative to derivation based on Eq. (4.2), one can estimate var(0) with replicated samples. Let 0i be an estimate of 0 based on sample i, i = 1.... , n. With these replicates var(0) can be estimated by
v~(O) =
n-
1
n 1 E(6i i=1
_0)2,
where 0 = (61 if- ".. -ff O,)/n is the mean of the n estimates. The value of v~(0) is low whenever the replicated e s t i m a t e s 0i are similar and it increases with increasing variability among them. Of course, a perfectly repeatable estimation procedure would have identical values among samples, yielding v~(0) = 0. In practice, repeated samples on which to base multiple estimates are rarely available, and var(0) must be obtained in other ways, usually with data in a single sample. We deal at length with techniques for estimating variance in Part III and Appendix E
45
circumstances var(/~) ~ 0 but E(/~) = f~N with f~ < 1, irrespective of sample size n.
Example A sample of n observations of a continuous measure x provides an unbiased estimate ~ = ~i xi/n of the population mean ~ of attribute x. The variance of this estimate is var(~) = 0-2/n, where 0 -2 is the population variance. Thus the variance of a sample mean is inversely proportional to the sample size and declines asymptotically to 0 as sample size increases. As above, the variance of the estimate ordinarily would be estimated from sample data, for example by sa/n. The square root of this quantity is the standard deviation of the mean, also known as the standard error.
4.2.1.3. Estimator Accuracy Accuracy combines both bias and precision in an assessment of estimator performance. An accurate estimator is one that is both unbiased and precise, whereas an inaccurate estimator is either imprecise or biased (or both). One measure of estimator accuracy is mean squared error (MSE), defined as MSE(0) = E(0 - 0) 2
Example
= E{0 - E(0) + E(0) - 0] 2
Let x be the number of heads in n tosses of an unfair coin. The variance of the estimator/~ = x/n of the proportion of heads is given by the expression var(j~) = p(1 - p)/n (see Appendix E). Because the variance of j6 is inversely related to sample size n, estimator precision can be doubled (that is, the variance can be reduced by one-half) by doubling the sample size. Indeed, the variance of ]~converges asymptotically to 0 as the sample size increases. An intuitive estimator of var (/~) is given by substituting the estimate ]~ for the population parameter p in the variance formula: v~(]~) =/~(1 - ~)/n.
= vat(0) + bias(0) 2.
(4.3)
The concepts of bias, precision, and accuracy are displayed graphically in Fig. 4.3. Note that an accurate estimator is both precise and unbiased, but an inaccurate estimator can be the result of either large bias or large dispersion (or both). Under certain conditions (e.g., a specified parametric sampling distribution) it is possible to determine MSE exactly, using Eq. (4.3). Otherwise, it must be estimated. If one knows 0 and also has access to n repeated samples from the population, MSE can be estimated as n
Example It is not uncommon for precision and bias to be unrelated, in that a precise estimate can be biased, and an unbiased estimate can be imprecise. An example involves counts of animals that are highly repeatable (i.e., low intercount variability) but still underrepresent the true number of animals present. Let Ci be the count of organisms on n plots of unit size, with 1~ = ~i Ci/n an estimator of the average density N of organisms in the area. With systematically dispersed organisms the variability among counts can be vanishingly small, even though each count underrepresents the number of organisms on the corresponding plot. Under these
MS'E(0) =
1
n-1
E(~i
_ 0)2"
/=1
Absent a knowledge of 0 a n d / o r the availability of replicate samples, computer simulation of the sampling process offers an approach (among others) to the estimation of MSE. We will discuss the tradeoff between variance and bias further when we consider model selection in Section 4.4.
4.2.2. Estimation Procedures To answer the probability question, one must identify the population of interest, as well as the mathemati-
46
Chapter 4 Estimation and Hypothesis Testing a
b
with the subscript indicating that multiple attributes are recorded for a sample. The value of 0 for which one or more of these equalities hold is the method-ofmoments estimate of 0. Thus the functions 1.1,i(0), ~2i(0), a n d / o r ~ij(O)essentially determine the estimate 0. The "knowns" in this approach are sample values Xil, ..., xin, and the unknown is 0. A value for 0 is to be chosen so that the population moments are equal to the sample moments.
Example e
XX
ct
0
F I G U R E 4.3 Estimator bias, precision, and accuracy. Sample estimates 0 (denoted by x) compared to parameter value 0. (a) Precise and biased. (b) Imprecise and unbiased. (c) Imprecise and biased. (d) Precise and unbiased.
You wish to estimate the annual survival rate S and recovery rate f for mallards banded on their breeding grounds in the preseason and subjected to harvest during the ensuing two hunting seasons. By recovery rate is meant the probability that a banded bird is harvested and the band is reported to a central repository. A proportion Pl = f of bands is expected to be recovered in the year of banding. Because birds must survive until year 2 in order to be harvested that year, the proportion of bands expected to be recovered in the second year is P2 - Sf (assuming the same recovery rate in both years). If 1000 birds are banded in the preseason of year I and 90 and 60 bands are recovered in years 1 and 2, respectively, an estimate of S and f can be obtained with the method of moments by pI(S, f) = = 90/1000
cal form of the underlying probability distribution f(xl0) and the value of the distribution parameter 0. On the other hand, statistical estimation may or may not presume knowledge of the mathematical structure of the distribution function, and in any case the parameter value 0 is unknown. The investigator therefore must estimate 0 by sampling the population and using the sample data along with other available information about the distribution. Two procedures for estimation are the "method of moments" and maximum likelihood estimation. 4.2.2.1. Method of Moments If the parameter 0 can be expressed in terms of population means, variances, and covariances, and no other information is available beyond the sample values, a common procedure (though certainly not the only one) is "method-of-moments" estimation. In this case the population moments are estimated with the sample moments, e.g., ~bi(0) = Xi ~'i(0) 2 = S2
G j(0) = sij,
= 0.09 and A
p2(S, f ) =
Sf
= 60 / 1000 = 0.06, from which we get f = 0.09 and S = 0.67. Note that the estimates of f and S are derived without any information about the underlying distribution of the recovery data, beyond the mathematical assumptions about the return proportions. Thus, the information necessary to assess the reliability of the estimates and to perform tests of statistical significance is missing. 4.2.2.2. Maximum Likelihood Estimation A procedure used throughout this book to estimate parameters is maximum likelihood estimation. The procedure requires knowledge of the underlying distribution of a random sample, as well as the actual sample values. Thus, one presumes to know the mathematical form of the distribution function f(x]0), but not to know the value 0 for the distribution. Under these conditions
4.2. Parameter Estimation one estimates 0 by sampling the population and using the distribution function as a likelihood function. The difference between maximum likelihood estimation and other estimation techniques concerns the use of the likelihood function. Instead of choosing 0 to equate sample and population moments, for example, here we choose 0 to maximize f(xl0). Note, however, that the roles of parameter and variable have been switched: we wish to choose a value for the "variable" 0 to maximize f(xl0), which is "parameterized" by the sample value x. It is in this sense that the distribution function becomes a likelihood function. The usual convention is to express the likelihood function as L(01x) to emphasize the changed roles of 0 and _x. The usual method for determining maximum likelihood estimates is based on differentiation of the likelihood function. Assume for now that the 0 is the maximum likelihood estimator of a single parameter 0. Under certain mild conditions it can be shown that the value 0 maximizing L(01x) can be obtained by differentiating log(L) with respect to 0 and setting the derivative to 0:
1, 0, 0, 0, 1, 1}, or six successful and four failed nests. On assumption that the nest fates are independent and identically distributed, this is an outcome of a random sample of size 10 from a Bernoulli distribution, with unknown probability p of nest success. We can model the probability associated with y successes out of 10 nesting attempts with the binomial distribution:
f(ylp) = (lO) pY(l - p) l~ Y The likelihood function corresponding to six successes is therefore
L(ply=6)=
:0.
(10)p6(1 _p)4.
The maximum of this likelihood can be obtained by taking the first derivative of the logarithm of the likelihood with respect to p, setting the result equal to 0, and solving for p. After some algebra this results in the equation 6 p
d[log L(OIx)] dO
47
4 1-p
m
0I
or
The value of 0 that satisfies this equation (which is known as the likelihood equation) also maximizes L and therefore is the maximum likelihood estimator (conditional on x). In the case in which there are, say, k parameters in O, then k likelihood equations are defined by partial differentiation of the log likelihood with respect to each parameter:
]5 -- 0.6. To show that this estimate maximizes the likelihood, we can construct the likelihood L(ply = 6) for the sample data and then substitute different values for p (Fig. 4.4). For any other value of p than 0.6 the likelihood will be seen to have a lower value than L(0.61y = 6).
8[log L(__elx)] :
30;
0i 0.25
i = 1, ..., k. Simultaneous solution of the likelihood equations yields the vector 0 of maximum likelihood estimates for B__.Appendix H provides background and techniques for optimization of expressions such as the likelihood function. E
Example We can illustrate maximum likelihood estimation by means of a Bernoulli distribution. Recall that the Bernoulli distribution specifies the probabilities of binary attributes, e.g., coin tosses (heads or tails), sex character (male or female), physiological condition (e.g., alive or dead), or capture status (captured or not captured). Suppose that we have a random sample of n bird nests, and x -- 1 represents nest success (at least one bird fledges) and x = 0 represents nest failure (all eggs or nestlings are destroyed). Assume that 10 nests are observed, with the sample outcome x = {1, 0, 1, 1,
0.20
0.15
0.10
0.05
i
0
0.6
1
F I G U R E 4.4 Example of m a x i m u m likelihood estimation with Bernoulli distribution, p = 0.6. Plots of the likelihood function L(p x) = (~0)p6(1 _ p)4 computed for sample data x' -- (4, 6) in the text example, over a range of values for the parameter p. The m a x i m u m value of the likelihood function is attained by/~ = 0.6, the m a x i m u m likelihood estimate for the data x. m
48
Chapter 4 Estimation and Hypothesis Testing
lO/p - 20/(1 - p) = 0,
Example The results from the example above can be generalized by allowing for an arbitrary number n of independent Bernoulli trials and any number y of successes. The appropriate statistical distribution for the situation is the binomial, Y pY(1 - p)n-y, and the corresponding likelihood function for, say, y = Y0 successes is
which has the solution/~ = 0.33. Based on data from the study and the binomial structure of the probability density function, the most likely value for mortality rate under the study conditions is 0.33. Note again that the estimate ~3depends on the particular values of Xl, x2, and x3. If the investigation were run again, different values for these random variables likely would be obtained, resulting in a different value for the estimate. It is in this sense that ~3can be thought of as a random variable, with its own distribution of values that is inherited from the distribution f(xl, x2, x3).
Example
Maximizing a likelihood function involves the choosing of an estimate for each parameter in the probability distribution. For example, there are two parameters for the distribution of our quarterback's passing success for five randomly chosen passes. Maximizing the corresponding likelihood function involves the estimation of both parameters, based on the same data for passing success. Because the same data are used to estimate both parameters, the estimators typically are not statistically independent. As a general rule, the estimation of multiple parameters with the same data results in covariation among estimators. Maximum likelihood estimation has some very strong advantages over other estimation techniques:
A laboratory study involves testing the impacts on wildlife of the agrochemical diazinon. The objective of the study is to determine the mortality of passerine birds exposed to a particular concentration of diazinon. The investigation involves the use of American robins (Turdus migratorius) as a representative species, with an aim of estimating the probability p that a randomly chosen robin, when thus exposed, will die. The study design involves the exposure of 10 robins to diazinon, with the number of deaths recorded. The study is replicated three times, resulting in 3, 4, and 6 deaths. From probability theory we recognize that the appropriate distribution function for each replication in the experiment is the binomial B(10, p), where p is the probability of death. The replications are independent of each other by design, so the joint distribution for Xl, x2, and x3 is the product
9 The maximum likelihood estimator 0 has an approximately normal distribution for large sample sizes. Furthermore, its distribution converges asymptotically to a normal distribution as sample sizes increase (Appendix E). 9 Though the estimator 0 may be biased, it is asymptotically unbiased in the sense that the expected value of 0 converges to the parameter 0 as sample sizes increase. 9 The variance of the estimator 0 is asymptotically minimum, in that 0 has the least variance of all unbiased estimators of 0 when sample size is large. 9 With some mathematical manipulation the variances and covariances of maximum likelihood estimators can be approximated directly from the likelihood function, using the "information matrix" (Appendix F).
L(p]y = Yo) = \yo/PyO(1 (n t _p)n-yo As before, the maximum of this likelihood can be obtained by taking the first derivative of the logarithm of the likelihood with respect to p, setting the result equal to 0, and solving for p. This leads to the equation
Yo/P - (n - y0)/(1 -p) = 0, from which we get the intuitive maximum likelihood estimator
= yo/n.
f ( X l , X2, XB]p ) = ( 1 0 ) ( 1 0 ) ( 1 0 ) p~iXi(1 _ p)30-:~ixi, X1 X2 X3
with a likelihood function of L(p[2, 3, 5 ) = 110) (10 t tlO)p1~
- p) 2~
for this problem. Differentiation of the log likelihood yields the likelihood equation
It is because of these and other useful properties that we focus on maximum likelihood estimation in this book.
4.2.3. Confidence Intervals and Interval Estimation Because an estimator of a population parameter is based on a random sample, the estimator is a random
4.2. Parameter Estimation variable, with its own distribution. A key question is how good the estimator is, in the sense of being "close" to the parameter of interest. At issue is the relative confidence one has that an estimate obtained from random sampling accurately represents the parameter. Approaches to this question constitute the subject matter of confidence intervals and interval estimation. Interval estimation is based on knowledge about the estimator's distribution, which can be obtained from the underlying sampling distribution. We focus here on the normal distribution, because maximum likelihood estimators (MLEs) are approximately normally distributed, asymptotically unbiased, with variances and covariances that can be derived directly from the likelihood function. Thus if 0 is an MLE with approximate variance v~(0), its distribution is approximately normal with distribution N(0, v'~ (0)). Furthermore, the frequency of occurrence of values for 0 in an interval (a, b) is simply the area under the corresponding normal curve between a and b. A "confidence interval" for a parameter 0 utilizes the distribution of 0. The idea is to identify an interval for values of 0 that is, say, 95% certain to include 0. Thus, to specify properly a confidence interval, both the interval length and the probability level associated with the interval are required. The procedure is first to identify the desired probability (the "confidence level") of including 0 in the interval. Then the corresponding range of values for 0 is expressed in terms of 0. Finally, the expression is mathematically manipulated to identify confidence interval bounds on 0. For example, one might seek a 95% confidence interval for 0 based on the MLE 0. As indicated above, MLEs are approximately normally distributed estimators that are asymptotically unbiased. Therefore the MLE 0 is within 1.96 standard deviations of 0 with probability 0.95, i.e., 0 - 1.96X/v~(6) < 6 < 0 + 1.96V'v"dr(0), (4.4) with probability 0.95, where both 0 and v~(0) are determined from the likelihood function. After some algebraic manipulation these inequalities can be rewritten as
{} - 1.96X/v"d'r(0) < 0 < 0 + 1.96V'v~(0), (4.5) which exhibits a 95% confidence interval
49
expressions correspond to the same probability, so the interval in expression (4.5) is 95% certain to include 0. Said differently, replicated confidence intervals constructed as in expression (4.5) will include the parameter 0 with 95% frequency. If one wishes to be even more certain that the interval includes 0, one can increase the interval length; i.e., the value 1.96 can be replaced by a larger number. The appropriate value for a given confidence level can be found in standard lookup tables in most statistics textbooks. In the above example, confidence intervals are obtained by invoking the asymptotic properties of the MLEs, in particular the property that the estimates are asymptotically normally distributed. Small sample sizes and other factors can result in estimates that are not well represented by normal theory. Then confidence intervals based on a normal approximation may overrepresent the frequency with which the parameter is included in them. For some statistical models (e.g., the binomial) exact methods are available to compute confidence intervals in lieu of a reliance on normal theory. A more general methodology, applicable to all MLEs, is the method of profile likelihood. In profile likelihood, the likelihood function is used to calculate a confidence interval on a parameter 00 based on the function ~p(00) = 2 In LL(--~0,O) '
L(00,__6)
where is the likelihood function evaluated at the MLEs for all parameters, and L(00, _0) is the likelihood function evaluated at the MLEs of the other model parameters _0 and with 00 varying over its admissible range. It can be shown that the random variable ~p(00) is asymptotically distributed as chi-square with one degree of freedom (see Appendix E for a discussion of the chi-square distribution). A (1 - c~) confidence interval on 00 may be obtained by solving for 00 in q~(00) = X2(o0,
(4.6)
where X12(o~)is the (1 - oL) percentile of the chi-square distribution with one degree of freedom (Buckland et al., 1993). There typically are two solutions to Eq. (4.6), and the confidence interval consists of all values 00 between them.
Example 6 - 1.96X/v~(0), 0 + 1.96V'v~(0)) for the parameter 0. Note the distinction between expression (4.4), in which a fixed interval contains the random variable 0, and expression (4.5), in which a random interval contains the fixed parameter 0. Both
Mule deer (n = 100) are outfitted with radio transmitters to estimate survival over winter (a 90-day study period). Of 100 deer, 10 animals die during the study period, no radio transmitters fail, and all the animals remain in the study area. The likelihood for the survival parameter p is therefore
50
Chapter 4 Estimation and Hypothesis Testing
with a m a x i m u m likelihood estimate for p of ]~ = 0.9. The profile likelihood interval is obtained by
true value of 0 is given by H 0. A straightforward method for testing H 0 comes directly from the procedure for confidence intervals: if 00 lies within the 95% confidence interval for 0, then H 0 is accepted (at the 5% significance level).
2 In [L(/~, L(p, ~)] ~) = X12(0.05)
4.3.1. Type I and Type II Errors
L(p]10) =
(100)p9~176
or
2[-14.118 - In
L(p, ~)]
= 3.841,
which has solutions for p at 0.788 and 0.966. Therefore the profile confidence interval (0.788, 0.966) is 95% certain to contain the parameter p. In contrast, an asymptotically normal confidence interval for p is formed by
f~ -- z(0"05)~/~(1n- j~)' where z(0.05) = 1.96 is the 0.05 ordinate for a standard normal distribution [i.e., P(z > 1.96) = 0.025 if z --- N(0, 1); see Table 4.1]. This in turn yields the 95% confidence interval (0.9 - (1.96)(0.03), 0.9 + (1.96)(0.03)) = (0.84, 0.96). Thus the profile likelihood interval is a more conservative (i.e., wider) confidence interval than that produced by the normal approximation. As expected, the two intervals converge as sample size increases.
4.3. H Y P O T H E S I S
TESTING
Closely associated with confidence interval estimation is the statistical testing of hypotheses, with an objective of determining whether parameters differ from hypothesized values. A testing procedure can be framed in terms of the comparison of null and alternative hypotheses. The null hypothesis H0:
REJECT
DO NOT REJECT
0 = 00
specifies some parameter value that is assumed prior to the test to be operative, but that is to be considered for possible rejection depending on the test results (see Chapter 2). The alternative hypothesis Ha:
The usual procedure for hypothesis testing requires specification of both the null and alternate hypotheses, as well as the specification of the significance level of the test. The test is essentially a binary decision process, in that the result is either to accept H 0 (over H a) or reject H 0 (in favor of Ha). Thus there are two ways in which a testing procedure can reach a correct decision and two ways that it can make an error (Fig. 4.5). The correct decisions, of course, are to accept H 0 w h e n it is true and to reject it w h e n it is false. There are two types of incorrect decisions: rejecting H 0 w h e n it is true (type I error) and accepting H 0 w h e n it is false (type II error). The probabilities of making these two types of error are conventionally denoted as o~and ~, respectively. One can guard against type I errors by increasing the significance level of the test. For instance, increasing the size of the confidence interval (i.e., by increasing the significance level of the test) makes it more likely that 00 will be included in the interval w h e n H 0 is true. In general, decreasing the probability significance level of the test will decrease the probability of a type I error. However, larger confidence intervals also are more likely to include 00 even if H 0 is false. Thus, a decrease in the probability significance level increases the chances for a type II error at the same time that it
0
=
TYPE I ERROR
CORRECT DECISION
CORRECT DECISION
TYPE II ERROR
0a
specifies a second parameter value 0a, to be considered as an alternative to 00 in the event the latter is rejected. As written, these hypotheses propose one of two values for the distribution parameter 0, with the true value of 0 assumed to be either 00 or 0 a. The objective of testing is to determine whether, based on the data from a r a n d o m sample, it is reasonable to conclude that the
FIGURE 4.5 Possibleoutcomes for a simple hypothesis test. Offdiagonal entries correspond to correct inferences; diagonal entries correspond to incorrect inferences.
4.3. Hypothesis Testing decreases the chances for a type I error. Clearly, there is a tradeoff between the two types of error that must be considered when establishing the significance level of a hypothesis test. The testing of a simple null hypothesis (i.e., H 0 specifies a single value for 00) against a simple alternative hypothesis (i.e., H a specifies a single value for 0a) can be generalized to allow for composite null and alternative hypotheses, either or both of which can include a range of parameter values. For example, one might specify a null hypothesis that includes any value for 0 that is less than some specified quantity, say 0", with the alternative hypothesis consisting of any parameter value larger than 0". In particular, it is useful to consider simple null hypotheses against one-sided alternatives. As an example, one might investigate whether the body masses of males and females differ, on assumption that if there is a difference, it favors larger males. The null hypothesis for this situation specifies that the mean body mass of females is equal to that of males, with an alternative that the body mass of males is greater: H0:
51
0.4
0.3
0.2
1.645
0.4
0.3
0.2
~males-- ~females, .
Ha:
~males > ~females.
Thus the test includes a one-sided alternative, with H 0 rejected only if the test results indicate that ~males is larger than ~females by some minimal amount. In general the testing procedure for simple versus one-sided hypotheses involves rejection of the null hypothesis if the test statistic exceeds some threshold value. The rejection region is associated with only one tail of the probability distribution of the test statistic. In contrast, one could consider a simple null hypothesis against a two-sided alternative. Using the example involving body mass of males and females, the alternative model could allow the average body size of females to exceed that of males, and vice-versa: H0: Ha"
~males -- ~females, ~males 5h ~females.
Rejection occurs if the mass for either sex exceeds that of the other by some critical amount, so that the rejection region is associated with both tails of the probability distribution of the test statistic. The critical values of a test statistic signifying rejection will differ under one-sided versus two-sided alternatives, because the probability of type I error in the former case is allocated to one tail, whereas in the second case it is allocated to both tails (Fig. 4.6). This allows for a lower rejection threshold for the test statistic under a one-tailed test. It therefore is easier to recog-
----
.
0.1
-1.96
0
X
1.96
FIGURE 4.6 Hypothesis test with probability oL= 0.05 of a type I error in detecting a specified difference in population means. (a) One-sided alternative. (b) Two-sided alternative. Test statistic is distributed as N(0,1).
nize statistically significant differences with one-sided alternatives. This conclusion supports the intuitively appealing idea that biological structure is easier to recognize when its investigation is limited to a few, substantially different, alternatives.
4.3.2. Statistical Power The ability of a test to reject false null hypotheses refers to the power of the test. Formally, test power is defined as 1 - P (type II error), so that powerful tests are unlikely to result in type II errors. The power of a test typically is high when (1) hypothesized parameter values in H 0 and H a are quite different from each other, (2) the underlying sample distribution has low variance, a n d / o r (3) the testing procedure is based on a sample of large size. Thus a strategy to protect against both types of errors is to set the probability significance level high (protecting against type I error) and to sample intensively enough to control against type II
52
Chapter 4 Estimation and Hypothesis Testing
error. We will focus on sample sizes that maintain statistical power when we consider experimental design (Chapter 6).
4.3.3. Goodness-of-Fit Tests Often it is important to determine the adequacy of a statistical model in characterizing field data. The idea is that if field data reflect an assumed statistical distribution it should be possible, by proper choice of distribution parameters, to demonstrate that the data "fit" the distribution. That is, it should be possible to choose parameter values such that the sample data conform to the statistical distribution underlying them. If the data reflect a distribution different from the one assumed, then there should be a "lack of fit," irrespective of parameter choice. Goodness-of-fit procedures can be placed in the context of statistical testing, whereby the null hypothesis is that a particular model fits a set of field data and the alternative hypothesis is that the model does not fit the field data. However, there is an important difference between goodness-of-fit testing and the standard parametric testing procedures discussed earlier. Goodness-of-fit procedures use sample data to investigate the mathematical structure of a distribution, rather than specific values for its parameters. This is in contrast to the usual statistical procedures for parameter estimation and testing, which assume a known form of the distribution under investigation and utilize sample data to investigate specific values of the distribution parameters. Goodness-of-fit testing is especially appropriate for multinomial distributions, for which there is a welldeveloped theory involving the use of maximum likelihood estimation. For problems for which the multinomial distribution is an appropriate statistical model the procedure is as follows: 1. The cell probabilities Pi, i = 1, ..., k + 1, of a multinomial distribution with k + 1 cells are modeled in terms of a vector _0' = ({)1, ..., Or) of parameters. The vector 0_ typically contains fewer parameters than the number of cell probabilities. Cell probabilities for the model are designated as Pi = pi(O) 9 2. Maximum likelihood procedures are used to estimate the parameter 0_. Let 0_ represent the MLE for _0. 3. The expected cell frequency E(x i) = npi for each cell of the multinomial distribution is approximated with npi = rlpi(~), and the statistic k+l
(yl~ i __ Xi)2
i=1
npi
X2--
is calculated, where x i is the observed cell frequency.
This statistic is asymptotically distributed as a chi -~ square random variable with k - r degrees of freedom, under the assumption that the mathematical form of the cell probabilities is correct. 4. The value X2 is compared to a tabulated chisquare value for the specified significance level of the test. A significance level of 0.95 often is chosen, so that a computed value of X2 exceeding the 0.95 chi-square quantile for k - r degrees of freedom results in a rejection of the model. Model rejection essentially means that the multinomial model with cell probabilities parameterized by pi(O) is inadequate to characterize the data, i.e., the model is held not to "fit" the data. Several alternatives to the above approach for assessing model fit exist and may have advantages in particular situations. The deviance, defined as
- 2 ln[L@x) /L(~saturatedlX)], describes the fit of a candidate model compared to a model containing as many parameters as independent observations, and is distributed approximately as chisquare with degrees of freedom equal to the difference between the candidate and saturated models (Agresti, 1990). The deviance is used in programs such as MARK (White and Burnham, 1999) to provide some indication of model fit. However, the chi-square approximation for either of these statistics is frequently poor in practice, especially when sample sizes are small, suggesting the need for approaches such as parametric bootstrapping (Appendix F) to more accurately assess model fit.
Example The goodness-of-fit procedure can be illustrated with an example involving the banding of mallards in the preseason of year 1 and the subsequent recovery of bands during the next three hunting seasons. Assuming that survival and recovery rates are constant over time, the appropriate model for this situation is
f(xlO) = ( _
n
) p~lp~2p~ 3
X1 , X2, X 3
_
3< (1 =
(
-
Pl
-
P2 -
P3 ) n - x l - x 2 - x 3
t/ tfXl(fS)X2(fS2)X 3 x 1, x2, x3j
• (1 - f -
fS - fS2) n - x l - x 2 - x 3 ,
where n is the number of birds banded and f and S are the recovery and survival rates, respectively. Four multinomial cells are defined for this model, with recoveries (x 1, x2, x3, n - Xl - x2 - x3) and expected recoveries [npl, rip2, np3, n(1 - P l - P 2 - P 3 ) ] - N o w
53
4.3. Hypothesis Testing assume that a total of 1000 mallards are banded in the preseason of year 1, and recoveries over the next three hunting seasons are x I = 95, x 2 = 62, and x 3 = 39. The corresponding likelihood function is 1000
~
L(f, Six) = 95, 62,39J
95 62 39fl P l P2 P3 '~ -- Pl -- P 2 -
804 P3)
1000 ~f95(fS)62 (fS 2) 39 \95,62,39 / x (1 - f - fS - fS 2) 804, from which the m a x i m u m likelihood estimates f = 0.095 and S = 0.64 are obtained. Thus the actual cell counts (95, 62, 39, 804) correspond to expected cell counts of (95, 61, 39, 805). Using these values in the goodness-of-fit statistic gives a value of • = 0.013, with k - r = 1. When compared against the 0.05 significance value of 3.84 for a chi-square distribution with one degree of freedom, the model is seen to fit the data exceptionally well. Band recovery models of the kind used in this artificial example are developed in greater detail in Chapter 16.
Example In an effort to determine the size of a population of fish, electroshocking sometimes is used in a removal experiment. A proposed model for this situation incorporates the assumptions that all fish have the same probability of removal and removal probability is constant over time. To test these assumptions for a particular species, 100 fish are subjected to an electroshocking experiment over four periods. The appropriate statistical model is
____
(l~176p Xl[(1 X [(1 -
-
p)p]X2
p)ap]x3
• [(1 - p)3plX4
4
x [(1 - P) - ,4,100~ x J j=l I, where p is the probability of removal of any randomly selected fish. The expected cell frequencies for this model are E(xj) = 100p (1 - p~-l. Based on a sample with (x 1, x 2, x3, x4) = (42, 23, 13, 10), the likelihood function is f(xl0)
=
(10xO)p42[(1-p)p]23[(1-p)2p]13 m
X [(1 -- p)gp]10[(1 -- p)4112,
with MLE/~ = 0.41. Using E(xj) = 100(1 - p)j-1 the expected cell counts for the model are approximately
(41, 24, 14, 9, 12). That the expected cell counts closely resemble the actual counts is confirmed by the goodness-of-fit statistic, which gives a value of • = 0.49 with k - r = 3 degrees of freedom. When compared against a 0.95 significance level of 7.81 for a chi-square distribution with three degrees of freedom, the model is seen to fit the data. Removal studies of the kind used in this example are described in more detail in Chapter 14. It is important to recognize the relationships between goodness of model fit and the probability significance of the goodness-of-fit test statistic. From the computing formula for step 3 above, it is clear that the goodness-of-fit statistic varies in magnitude to the extent that "expected" cell frequencies npi deviate from "observed" cell frequencies x i. On assumption that the model is appropriate for the data, large deviations occur only infrequently, according to a chi-square distribution with k - r degrees of freedom. Thus, the larger the computed value of X2, the smaller the probability that randomly collected data will generate a value that large or larger. The mathematical relationship between the magnitude of X2 and the probability significance level is parameterized by the chi-square "degrees of freedom" (see Appendix E). Most of the statistical models we will use for estimation can be formulated in terms of cell counts for a multinomial distribution, and most have associated goodness-of-fit testing procedures. In particular, Part III presents goodness-of-fit tests for a class of statistical models that are useful for estimating population size and other population parameters.
4.3.4. L i k e l i h o o d Ratio Tests for Model Comparisons In the development of statistical models we are concerned not only with the adequacy of a model in characterizing data, but also with a comparison of the model with other models that differ in their parametric structures. Like goodness-of-fit testing, model comparison procedures can be seen as an example of a hypothesis test. The difference is that the general alternative of a goodness-of-fit test is replaced with an alternative of a specific model. Thus the test compares the fit of a hypothesized model versus the fit of an alternative model. Specifically, the null hypothesis is that the hypothesized model fits the data as well as the alternative model; the alternate hypothesis is that the alternative model fits the data better. Typically the alternative model is more general in its parametric structure than the model of the null hypothesis, so that the null hypothesis can be couched
54
Chapter 4 Estimation and Hypothesis Testing
in terms of restrictions on the parameter structure of the alternative model. The objective of a model comparison is to determine whether the fit of a hypothesized model can be improved by relaxation of its parameter restrictions. For example, the two-parameter model above for band recovery data can be generalized to include time-specific survival rates, and a comparison of the restricted and generalized models would give an indication of the importance of the additional parameters in characterizing the data. As with goodness-of-fit testing, maximum likelihood estimation theory^provides a mechanism for model comparisons. Let 0 be the MLE corresponding to a model with parameterization 0_ [e.g., _0' = ($1, ..., Sk, f)], and _00be the MLE for a model with 0 restricted [e.ig., 0' = (S, f) with S 1 . . . . . Sk = S]. Le(-L(fl]x) and L(0_0lx~ represent the likelihood function evaluated at 0 and 00, respectively. Then the likelihood ratio statistic
X2
=
-2 ln[L(~olx)/L@x)]
(4.7)
is asymptotically distributed as a chi-square random variable, on condition that the restricted model is correct. The degrees of freedom h for this statistic are given by the difference in the number of independent parameters in _0 and _00 (in our example h = k - 1 degrees of freedom, because _0 and _00 contain k + 1 and two parameters, respectively). A procedure for comparing models is as follows: 1. Goodness-of-fit procedures are used to identify a general statistical model "fitting" the data. Model generality is defined here in terms of the number of independent parameters, with the most general model defined by a lack of restrictions on the model parameters. Denote the corresponding parameter vector by 0_ and its MLE by _0. Let kg denote the dimension of 0 and L(~lx) represent the likelihood function evaluated at ~0. 2. Restrictions are imposed on the parameters in O, to produce a restricted parameterization _00 for the model. The corresponding^likelihood function L(0_0lx) is evaluated at the MLE _00 of 00. Denote by kr the dimension of the reduced parameter 00. 3. The likelihood ratio statistic X2 is calculated as above. This statistic is asymptotically distributed as a chi-square random variable with kg - kr degrees of freedom, under the assumption that O0 is the appropriate parameterization. 4. The value X2 is compared to a tabulated chisquare value with kg - kr degrees of freedom. If • 2 exceeds the tabulated value for a specified significance level, the model with the more general parameterization is held to improve the fit of the model over that of the more restricted parameterization.
Example In a study of harvest rates for mallards, an investigator releases 100 each of banded male and female birds prior to the hunting season, with the intent of examining harvest recoveries from each release sample. The objective is to determine if there are differences in harvest rates between the sexes. Let Pm and pf represent the harvest probabilities for male and female birds respectively. Assuming that the harvest of males is independent of that of females, an appropriate statistical model is f(Xm, x f ) =
/100\
Xm
~xm)Pm
(100~ xf (1 - pm)100--Xm_ Xf JPf
X (1 - pf)100-x,. If (Xm, Xf) = (30, 17) the likelihood function is
L(pm, pf[30, 1 7 ) =
(lOg)p3m~
pm)70(lO0) pJ717
x (1 - pf)83, and the maximum likelihood estimate for p is ~ = {0.30, 0.17}. Substituting these values into the likelihood function yields m
L(/~m, ~f130, 1 7 ) = (1Oo)(0.30)3~176
x
(,00
17 / (0.17)17(0.83)83.
Maximum likelihood estimates can be obtained in a similar way under the hypothesis of equal harvest rates, i.e., Pm = Pf = P" Under these conditions the likelihood function is L(pl30, 17) =
(13o)
(100,171 ]P
p30(1 _ p)70 17
t - p)83,
with maximum likelihood estimate ~ = 0.235. Substituting this value into the likelihood function yields L(~I30, 17) = (1OO)(0.235)3~176
x (1OO)(0.235)17(0.765)83, and Eq. (4.7) gives a likelihood ratio statistic of X2 = - 2 ln[L(_Oolx)/L(~alX)] = 4.749. Comparison of this value with a tabulated chi-square value of 3.84 for one degree of freedom and 5% signifi-
4.4. Information-Theoretic Approaches
55
cance indicates that the less restrictive parameterization significantly improves the fit of the model (at the 5% significance level). Thus, the study results provide evidence that there are differences in parameters for the two models, and in particular, that the harvest rates differ between sex categories.
and
The above sequence of steps began with an initial goodness-of-fit test for the more general model. In fact, the theory underlying likelihood ratio testing is based on the assumption that the more general model (corresponding to the alternate hypothesis) provides an adequate fit to the data. If the goodness-of-fit test for the most general model under consideration provides evidence of lack of fit, then the procedure for testing between models must be modified. Lack of fit often is a result of overdispersion of the data, in which case the goodness-of-fit statistic can be used to compute a "variance inflation factor," which in turn can be used to translate the likelihood ratio test statistic into a new test statistic distributed as F (see Section 17.1.8) (Lebreton et al., 1992).
Though the parameterizations _01 and 0 2 a r e subsets of _0 and can be formed from _0 by imposing constraints on 0, neither 01 n o r 02 can be formed by constraining the parameter space for the competing model. Attempts to apply Eq. (4.7) would result in a chi-square statistic with zero degrees of freedom, because each model has the same number of parameters. An alternate approach based on information theory addresses the tradeoff between model fit (which favors more parameters) and estimator variance (which favors fewer parameters) in an optimization rather than hypothesis-testing framework. The approach is based on a statistic known as Akaike's information criterion (AIC) (Akaike, 1973), which utilizes the likelihood for each model via the term - 2 ln(L) and a penalty term for the number of parameters in the model:
4.4. I N F O R M A T I O N - T H E O R E T I C APPROACHES Though appropriate for many situations, the likelihood ratio testing procedure is not always satisfactory for model selection. First, there are philosophical problems with treating model selection as a hypothesis testing problem versus an estimation problem, particularly when the data have not been collected under an experimental design (e.g., Burnham and Anderson, 1992). Second, model comparisons based on likelihood ratio tests can only be used when the parameter space under one likelihood is a nested subset of that of a more general alternative. We will encounter instances later in which the parameter spaces of two competing models, fit to a common data set, will not be nested. To illustrate, suppose that data in the previous example are collected by both age and sex, so that a general parameterization would allow for both age and sex variation in harvest rates: O' = (Pam, Pym, Paf, Pyf), where the subscripts a and y now denote age-specific (adults and young) harvest rates within each sex. Two restrictions on this parameterization are 0_~ = (Pam, Pym, Pf), where Paf = Pyf = Pf,
O~ -- (Pm, Paf, Pyf) with Pam = Pym = Pm"
AIC = - 2 In(L) + 2q,
(4.8)
where L is the likelihood for a model under consideration and q is the number of parameters in the model. The idea is to select the model for which AIC is minimum. Although Eq. (4.8), especially the "penalty," seems somewhat arbitrary, AIC has a strong theoretical basis in information theory (Burnham and Anderson, 1998). In addition to expression (4.8), other forms for AIC also can be used; these incorporate a correction for small sample size (AIC C) and a "quasilikelihood" adjustment (QAIC) for extra binomial variation (Section 17.1.8) (see Burnham and Anderson, 1998). Model selection based on minimization of an appropriate one of these information measures will account for the bias-variance tradeoff in model parameterizations and if properly applied should result in the selection of a "best approximating model," i.e., the best data-based approximation to "full reality" (Burnham and Anderson, 1998).
Example Consider the previous waterfowl harvest example, in which data are collected on both age and sex of the harvested animals, and 100 individuals of each age-sex stratum are banded and released prior to harvest. Let {Paf, Pam, Pyf, Pym} represent the probabilities of harvest for adults and young (subscripts a and y) of both sexes (subscripts m and f), with {Xaf, Xam, Xyf, Xym} the numbers harvested in each category. Then an appropriate statistical model is
56
Chapter 4 Estimation and Hypothesis Testing
[100~ Xa f f(Xaf , Xam, Xyf. Xym) -- ~ Xaf/paf (1 - paf) 100-xaf • (100' nXam(1 __ Pam)100_Xa m \Xam ' ram
• (100~.Xy f Xyf/ry f (1 - pyf)100-Xyf
and log likelihood of In L1 = -12.695265. The corresponding value of AIC for this parameterization is AIC = -2(-12.695265) + 6 = 31.3905. By comparison, the parameterization
O~ = (Pal. Pam. Py)" • ( 1 0 0 / p y y m ( I - Pym)100--Xym. \Xym./ If (Xaf, Xam, Xyf, Xym) -- (35, 20, 47, 44) the likelihood function is
yields the likelihood function
C2(Paf. Pam, py.35, 20, 47, 4 4 ) = (1OO)p3~(1- paf) 65 (1OO)p2O ( 1 - Pam)80
L(paf. Pam.Pyf. Pym135.ao. 47. 44) = ( lOO)p35(1- paf)65 /100'~ 20 X ~ 20/Pam (1 --Pam )80
X
(lOO) pyo (1
fl00~ 47 X ~ 47 Jpyf(1 - pyf)53 X (1OO)Pym(1 -- Pym)56.
(100~ 47 47/PY (1 - py)53
56
-- py) ,
with maximum likelihood estimate ~ = (0.35, 0.2, 0.455)
with a maximum likelihood estimate 6 = {0.35, 0.20, 0.47, 0.44}. These estimates result in a value for the log likelihood function of In L = -9.8464 and AIC value of B
AIC = - 2 1 n L + 2 q = -2(-9.8464) + 8 = 27.6928. On the other hand, the parameterization
O~ = (Pa. Pym. Pyf) results in the likelihood function
g1(Pa, Pyf, Pymlg5, ao, 47, 44) = (100)p35 (1- pa)65 • (1OO)p2~ (1 - pa)80
~100~ 47 • ~ 47 J p yf (1 -- pyf)53 X (1OO)py44m(1 -- Pym)56, with the maximum likelihood estimates 6~ = (0.275, 0.47, 0.44)
and log likelihood of In L2 - -9.93716. The value for AIC in this case is AIC = -2(-9.93716) + 6 = 25.874. Based on AIC values of 27.6928, 31.3905, and 25.874 for the parameterizations _0,_01,and 0_2,respectively, we conclude that parameterization 0 2 provides the best variance-bias tradeoff among the three alternatives. Some important points about AIC should be noted. First, AIC is appropriate only for comparison among models that all have been fit to a common set of sample data; comparison among AIC values from models fit to different data sets is meaningless. Second, a hypothesis testing framework for model selection may be preferable in those situations in which an experimental or quasiexperimental design provides a context for testing predictions based on theory, models, or both (see Chapters 2 and 3). Third, whereas AIC can be used to rank a number of competing, nonnested models, it does not always result in clear selection of a single model. Small differences in AIC can be expected to occur by chance and thus are indicative of virtually identical information content in the competing models. Burnham and Anderson (1998) advocate the computation of "model weights" based on the difference between each model's AIC value and that of the lowest
4.5. Bayesian Extension of Likelihood Theory ranked model. These weights, which are normalized to sum to 1 over all models considered, are roughly interpretable as the probability a given model is the best approximation to truth among the models considered. Closely ranked models with high weights (e.g., >0.25) should be retained for further consideration with other model selection criteria. Alternatively, the AIC weights may be used to compute weighted averages of parameter estimates across all the models considered. Buckland et al. (1997) and Burnham and Anderson (1998) recommend taking into account the (weighted) deviation of model-specific estimates from such a weighted average and inflating variance estimates accordingly. This approach accounts for the uncertainty in estimation induced by the process of model selection and seems preferable to the usual approach of reporting only sampling variances, conditional on an assumed true model. In Section 17.1.8 we discuss in greater detail the computation of model weights and corresponding variance components for capture-recapture models.
4.5. B A Y E S I A N E X T E N S I O N OF LIKELIHOOD THEORY
P(Hi)P(x]Hi) P(x) "
(4.9)
The value P(H i) in this expression is called the prior probability for hypothesis Hi, because it precedes the collection of the sample data x, and P(Hi]x) is called the posterior probability for H i by reason of its being computed posterior to data collection (see Appendix A). Assuming that H' = {H1, ..., Hn}
represents the set of all possible alternative hypotheses under investigation, by Bayes' Theorem we can rewrite P(x) as n
P(x) = ~, P(x]Hm)P(H m) m=l
and therefore Eq. (4.9) can be expressed as n
P(Hi]x ) = P(Hi)P(xIHi)/ s
P(Hm)P(x]Hm).
(4.10)
m=l
Equation (4.10) now gives us a procedure for evaluating the probability of any hypothesis in the hypothesis set based on the prior probabilities P(H i) and the likelihoods of each model given the sample data.
Example Suppose there are three competing biological models (i.e., three hypotheses H 1, H 2, and H 3) for a system of interest, to which are assigned the prior probabilities P(H1), P(H2), and P(H3), respectively, based on previous information. Additional sample data x are collected in a field study, and these are used to obtain maximum likelihood estimates under each model. When evaluated at their respective maximum likelihood estimates, the three likelihood functions take values of P(x]H1) = 0.10, P(x]H2) = 0.20, P(xlH 3) = 0.15. These values then can be used to compute the posterior probability of each hypothesis using Eq. (4.10). For example, equal prior probabilities result in m
3
There is yet another important application of likelihood theory in statistical estimation and hypothesis testing, which is especially useful for updating one's understanding of biological processes via predictive models (Section 3.3.2; see also Chapter 24). This application utilizes Eq. (4.1), along with a Bayesian definition of probability as "a measure of the degree of belief in an outcome" (Lee, 1992). On substitution of hypothesis H i for event E2, and x, a sample outcome, for event E1 in Eq. (4.1), one has
P(Hi]x) =
57
P(H1]x) = P(H1)P(x]H1) / ~, P(Hm)P(x_.IHm) m=l 1/3(0.10) 1/3(0.10) + 1/3(0.20) + (1/3)0.15 = 0.22. This same approach can be applied to determine P(H2]x) = 0.44 and P(H31x) = 0.33. The new triple {0.22, 0.44, 0.33} of hypothesis probabilities reflects the fact that hypothesis H 2 fits the data _xbetter than the other two hypotheses, and hypothesis H 1 fits the data more poorly than the others. These posterior probabilities now can serve as prior probabilities, to be updated with additional data in subsequent investigations. In many situations the set H' of hypotheses consists of specific values 0 i that a parameter 0 may assume. Then Eq. (4.10) can be written as P(0
= 0iIx)--
P(Oi)P(xlOi)/ ~ P(Om)P(xlOm). m--1
If the prior probability distribution for 0 is characterized by a continuous probability density function f(0), the updating process is governed by f(Olx) = f(O)f(xlo)/f o f(v)f(xlv) dr. In either case, once we have obtained updated (posterior) probabilities for the parameter values under consideration, these in turn can be used as new prior
58
Chapter 4 Estimation and Hypothesis Testing
probabilities, in anticipation of another round of data collection and posterior updating (see Appendix A). This provides a powerful procedure for sequential updating, as we will see in Chapter 24 (also see Section 3.3.2).
4.6. D I S C U S S I O N In this chapter we have provided basic principles for estimation and hypothesis testing of parameters for animal populations. In Chapters 5 and 6 we explore how surveys and controlled experiments can be designed to assure that data are suitable for estimating
model parameters and making comparisons of parameters over space or time, or with respect to other attributes. This background will be utilized in Part III to focus on statistical modeling techniques that are appropriate for animal populations, taking into account the conditions under which data from populations and communities are collected. These conditions affect the nature of inferences that can be made and make it necessary to develop specialized statistical models. In Part IV we explore the use of both deterministic and stochastic population models for optimal decision making in an adaptive framework, wherein monitoring and estimation contribute to the simultaneous pursuit of understanding and management of animal populations.
C H A P T E R
5 Survey Sampling and the Estimation of Population Parameters
5.1. SAMPLING ISSUES 5.1.1. Temporal Heterogeneity 5.1.2. Spatial Heterogeneity 5.1.3. Sampling Variability 5.1.4. Detectability 5.2. FEATURES OF A SAMPLING DESIGN 5.2.1. Replication 5.2.2. Randomization 5.2.3. Control of Variation 5.3. SIMPLE RANDOM AND STRATIFIED RANDOM SAMPLING 5.3.1. Simple Random Sampling 5.3.2. Stratification and Stratified Random Sampling 5.4. OTHER SAMPLING APPROACHES 5.4.1. Cluster Sampling 5.4.2. Systematic Sampling 5.4.3. Double Sampling 5.4.4. Adaptive Sampling 5.5. COMMON PROBLEMS IN SAMPLING DESIGNS 5.5.1. Failure to Define Target and Sampled Populations 5.5.2. Lack of Replication and "Pseudoreplication" 5.5.3. Misinterpretation of Pattern as Cause and Effect 5.6. DISCUSSION
the necessary data are available from random sampling. We highlighted bias, precision, and other measures of confidence for parameter estimators, recognizing that estimator accuracy is a function of temporal, spatial, and sampling heterogeneity. Of course, the relative importance of these features depends to a large degree on study objectives and operative constraints on the study design. Inferences about population parameters depend on samples that represent the population at large and thus on the random sampling of observations from the field. At several points in Chapter 4 we stressed the fact that data-based estimators are random variables that inherit their probability distributions from the data used to construct them. Of course, these data have inheritable probability distributions because of random sampling. In this chapter we focus on the process of obtaining samples from a population for use in estimating its parameters. We emphasize the importance of randomization and replication as part of sampling designs for biological investigation, leaving to Chapter 6 the issue of randomized assignment of treatments in an experimental context. Our emphasis on randomization and replication is motivated by the fact that in their absence, there is no statistically reliable way in which inferences can be made to the population under investigation. In what follows we examine a number of schemes for the random sampling of a population. We pay particular attention to sampling designs that partition a population into groups, thereby taking advantage of the population's structure a n d / o r physical distribution. The logic for sampling from groups within a population is that it often is more efficient to collect samples within groups, compute group-specific values, and
Estimating population parameters requires representative sample data of sufficient quantity to produce credible and useful parameter estimates. In Chapters 1 and 3 we noted that forecasting with biological models depends on the accuracy of parameter estimates in the models, and in Chapter 4 we focused on parameter estimation and testing procedures, on assumption that
59
60
Chapter 5 Survey Sampling and Population Parameters
then combine these values into an estimate for the whole population, than it is to collect samples randomly from the whole population in the absence of grouping. Stratified random sampling, cluster sampling, and multistage sampling all are examples of this approach.
however, that the resulting statistical inferences apply only to the cohort under investigation and not to the population at large. Questions regarding the population at large can be addressed only if the investigation addresses variability among spatial cohorts. This in turn requires the selection of a sample of cohorts, with follow-up assessment to determine the characteristics of each.
5.1. S A M P L I N G ISSUES In what follows we address some sampling issues that are especially relevant in field biology, preparatory to the development of estimation models and their data structures for animal populations (these will be addressed in some detail in Part III). We focus here on populations that are distributed over heterogeneous habitats and are subject to temporal, spatial, and sampling variability.
5.1.1. Temporal Heterogeneity Populations change through time in response to changes in habitats, community interactions, management, and other factors. Thus, data-based inferences about population dynamics are tied to the time periods during which data on the population are collected. It is important (but often quite difficult) to account for temporal variation in one's field sampling. This is true even if one wishes only to determine population impacts of a particular management action, because its impacts are almost always manifested over some period following the action. In the absence of some means to address temporal variability, management effects are confounded with environmental conditions and other factors that are specific to the time period. A difficult but important problem is to distinguish systematic population changes from those occurring as a result of random environmental variation. An obvious approach is to replicate a population study at randomly selected times over the time frame of interest. Of course, such an approach increases the amount of field effort by the number of replicates and therefore is often unrealistic for more than a few replication times.
5.1.2. Spatial Heterogeneity Many, perhaps most, populations are found in areas of considerable heterogeneity and thus exhibit variability in their spatial distributions. In some instances this "patchiness" induces isolation among spatial cohorts of a population, with the result that an investigation may focus exclusively on a single cohort. In this case the cohort should be treated as a separate subpopulation and analyzed as such. It is important to realize,
5.1.3. Sampling Variability Given a sampling design that addresses both spatial and temporal factors, there still remains a need to consider the amount and pattern of variability arising from randomly selecting individuals. For example, the change in size of a population might be investigated with four spatially distributed cohorts of the population that are selected at random, with each cohort followed over a period of 5 years. For each cohort, variation in the estimate of cohort size depends on the number of individuals selected from the cohort. Unless the design allows for sufficient numbers of individuals to be sampled, the resulting estimates may be too imprecise to be useful, even if spatial and temporal heterogeneity are accounted for in the design.
Example A land management practice is to be implemented in a region that overlaps the range of a local wildlife population with numerous geographically defined subpopulations. Management impacts on the population are to be investigated through a comparison of population densities before and after implementation of the management practice. A design for this situation might include the following features: 9 Random selection of population cohorts over the range of the population, in areas not subjected to the management practice and in areas that are. 9 Selection of periods prior to and subsequent to implementation of the management practice. Because the most important aspect of the investigation is a contrast of population status prior and subsequent to initiation of the practice, measurements are to be made on the cohorts immediately before and after its implementation. Because there is no randomization in the selection of sampling times, results of the investigation apply only to this limited time frame. 9 The use of mark-recapture techniques (see Part III) to estimate the density of each cohort at each of the times prior to and after implementation of the management practice. This involves the design of a trapping program over the area of each cohort.
5.2. Features of a Sampling Design The sampling of cohorts prior to and after initiation of the management practice allows one to assess management impacts by comparing earlier against later density estimates. The sampling of cohorts not subjected to the practice allows one to adjust the difference for temporal effects that are unrelated to the practice. Of course, the design requires a considerable trapping effort in multiple areas, in order to get sufficiently precise estimates of density to be informative about management impacts.
5.1.4. Detectability A distinctive feature of sampling animal populations is that random sampling often is insufficient by itself to guarantee reliable sample-based estimators. One reason is the tendency for animal counts from sampled units to underrepresent the true number of animals present on these units, resulting in systematic errors in sample-based estimators. To illustrate the problem, let detectability ~i represent the proportion of N i individuals on sampling unit i that one can expect to include in a count Ci for the unit: E(Ci) = ~iNi .
Any one of three cases may occur: 1. Complete detectability: ~i = ~ -- 1. In this situation individuals are completely detectable over time, space, or other dimensions, so that the sample count is actually identical to N i. Sample counts Ci then produce error-free comparisons of size in space and time (i.e., across samples), and they can be used for estimation of parameters requiring unit-specific numbers. 2. Less than complete but constant detectability: ~ i - 1. The count Ci is a biased estimate of N i by the factor ~i, but the bias is uniform over time, space, or other dimensions. If ~ can be estimated, the counts can be adjusted to provide unbiased estimates of abundance. In any case, the counts may be used as an index for comparisons of abundances over space and time. 3. Variable detectability: ~i < 1 and ~i ~ ~j" Counts represent biased estimates of N i, and the bias is nonuniform over time, space, or other dimensions, adding variability to the estimates in addition to bias. In particularly difficult cases, ~i may be influenced by experimental treatments or management interventions, and the failure to account for this influence can result in the masking of true experimental effects or can induce the illusion of real effects when there are none (or both).
In our experience, by far the most common situation in sampling animal populations is for detectability to vary over time and space, in response to environmental and ecological factors. Unfortunately, the issue of vari-
61
able detection rates has received insufficient emphasis in both observational and experimental studies in ecology. We focus on detectability, and on approaches for dealing with it, in Part III (see also Thompson, 1992; Lancia et al., 1994; Wilson et al., 1996, Thompson et al., 1998).
5.2. FEATURES OF A SAMPLING DESIGN The framework for an investigation involving estimation a n d / o r hypothesis testing builds on a probability model linking samples to the population from which they are drawn. Key elements of the framework are a clear idea of the population to be investigated and a sampling plan that utilizes replication, randomization, and control of variation. Here we take the population of interest to be composed, at least conceptually, of an identifiable list of sample elements (e.g., license holders, vegetation quadrats), recognizing that the members of biological populations rarely are recorded on actual lists. Often it is important to distinguish between the target population, the object of an investigation, and the sampled population from which samples are actually taken (Fig. 5.1). The target and sampled populations are ideally, but not necessarily, identical. If the sampled population differs from the target population, for example, because of a restriction on sampling effort to only a subset of a target population (Fig. 5.1b), then certain elements of the target population have zero probability of appearing in the sample, and samplebased inferences are not applicable to the target population. An example might involve the investigation of hunting that targets hunters in a state, though sampling is restricted to only those hunters who purchase licenses and respond to a voluntary survey. Another example might involve the study of vegetation, with restrictions on the placement of quadrats so that they are within a specified distance from a forest road. In such cases, inferences from the sample will apply only to the sampled population, but not to the entire target population. When there is a disparity between sampled and target populations, inferences to the target population depend on auxiliary information (e.g., a followup nonresponse survey, an off-road vegetation survey) that links the sampling frame and the target population. In the absence of such information, inferences to the target population are statistically unreliable.
5.2.1. Replication By replication is meant the selection of multiple samples from a sampled population, pursuant to the
62
Chapter 5 Survey Sampling and Population Parameters element in confidence interval estimation and statistical testing.
5.2.3. Control of Variation
\
/i
T
/
""
FIGURE 5.1 Conceptual relationship between target population Y (boundary denoted by dashed lines), sampled population S (boundary denoted by solid line), and sample s. (a) Effective coincidence between target and sampled populations. (b) Disparity between target and sampled populations. Sample-based inferences do not extend to the area of the target population ouside the area of the sampled population.
estimation and testing of population parameters. Along with randomization, replication is necessary to assess the variability of sample estimates. It is only with a measure of variation, of course, that one can assign a level of confidence to estimates of parameters. As noted in Section 5.5.2, care must be taken in the definition and selection of sampling units, to avoid problems of pseudoreplication that result in biased inferences.
There are essentially two reasons to control variation in a sampling design. First, reducing variation increases the precision of parameter estimates. Second, reduced variation will result in an increase in the power of hypothesis tests. The idea is to eliminate as many sources of variation as possible by sampling design; i.e., by accounting for each source of variation so as to reduce its influence on parameter estimates. There are several ways to control variation, including (1) the use of stratification or blocking to eliminate systematic variation, (2) the use of ancillary covariates to eliminate nuisance sample-to-sample variation, and (3) increasing sample size so as to increase estimator precision. Of course, the control of variation should be considered against a background of replication and randomization, because in combination these design elements largely determine how informative a study will be. In what follows we consider a finite population of N sampling units (e.g., individual organisms, kinship units, plots of land), each of which is characterized by some measurable attribute Yi. Thus, the population is represented by the set {Yl, ..., YN}. Sampling produces a sample set of these values, which then can be combined into estimates of population parameters. We focus below on the population mean N
W ~- ~_j Y i / N , i=1
the population total Y = NY,
the population variance N
0"2 = ~ (Yi -- Y ) 2 / N , i=1
5.2.2. Randomization Random selection of sampling units from a population protects against the systematic influence of unrecognized sources of variation. The term randomization refers to both the random selection of samples from a population and the random assignment of treatments (including controls) to samples. Randomization is an essential component of any sampling scheme that involves estimation of parameters and testing of hypotheses about them. It is required for inferences about the population from which samples are taken, and it allows for the estimation of sample-to-sample variance, a key
and population attributes incorporating these parameters.
5.3. S I M P L E R A N D O M AND STRATIFIED RANDOM SAMPLING Although much of the theory of population estimation is based on the notion of sampling from a population of unknown size, there are many instances in which samples are drawn from a list of the entire population. Examples might include surveys of sportsmen
5.3. Simple Random and Stratified Random Sampling from a complete list of hunting or fishing license holders and surveys of animal or plant abundances from areas that are divided into contiguous quadrats. In what follows we describe some sampling protocols of primary importance in sampling animal populations of finite size, recognizing that the protocols apply equally well to arbitrarily large populations. Interested readers are referred to Cochran (1977) and Thompson (1992) for more detailed treatments.
5.3.1. Simple Random Sampling In sampling from a population of finite size, n sampling units are selected from a list of N total units in the sampled population. This list, referred to as the sampling frame, includes the whole population under investigation, and there are several ways to select sample units from it. A straightforward and frequently used sampling protocol is simple random sampling, in which the units are drawn so that each unit has the same probability n/N of being selected. We assume here that the population consists of discrete sampling units that do not overlap, i.e., the population is partitioned by the sampling units. Simple random sampling can be with or without replacement. In sampling without replacement, a unit, if selected once, cannot be selected again and thus can occur only once in the sample. Under sampling with replacement, a unit, having previously been selected, may appear again in the sample. Simple random sampling with replacement has a simpler expression for variance than does simple random sampling without replacement. On the other hand, sampling with replacement typically yields less precise estimates (see below).
5.3.1.1. Estimation under Simple Random Sampling An unbiased estimator of the population mean Y = (Yl + "'" + yN)/N is the ordinary sample mean
~ = ~ yi/n i=1
with the usual estimate tl
$2=
s
Y ) 2 / ( n --
var(f) = or2/n,
(5.1)
an unbiased estimator for which is va"~(f)
=
S2/n
(5.2)
1).
i=1
for the population variance or2. Under sampling without replacement, y is an unbiased estimate of Y, but the variance of y is somewhat more complicated than Eq. (5.1), essentially because of restrictions on the number of ways a sample can be drawn without replacement from a finite population. It can be shown that the variance of y in this situation is
2(N)
var(y) = m
nN-1
(1 - n/N).
(5.3)
To simplify notation, we define the term $2 =
N
0 -2 '
N-1 so that Eq. (5.3) can be expressed as S2 var (y) = m (1 - n/N). tl
(5.4)
The term 1 - n/N accounts for the finite size of the population, reducing the variance of y as the sampling proportion n/N increases. The term is appropriately called a finite population correction (Cochran, 1977; Thompson, 1992). An unbiased estimate of the sampling variance in Eq. (5.4) is S2
v"~r(~) = -- (1 - n/N). Y/
(5.5)
The sample mean can be used to estimate the population total Y by = ~
(5.6)
var(Y) = N 2 var(y),
(5.7)
with variance
where var(y) is computed as in Eq. (5.1) or (5.3) depending on whether sampling is with or without replacement. An unbiased estimator of var(~') is va"~(9)
from a simple random sample of size n. On assumption that sampling is with replacement, the variance of ~ is simply
63
= N 2 v~(y),
(5.8)
with v ~ ( y ) given by Eq. (5.2) or (5.5) for sampling with or without replacement, respectively. A comparison of Eqs. (5.1) and (5.3) shows that y is a more precise estimator when based on simple random sampling without replacement. The gain in precision is effectively (1 - n/N), and it is more pronounced as a sample includes more units of the population (i.e., the proportion n/N increases). On reflection this pattern
Chapter 5 Survey Sampling and Population Parameters
64
makes sense. As the sample size increases, sampling without replacement more closely approximates a census of the whole population, for which sampling variation necessarily vanishes. On the other hand, sampling variation is always present when sampling is with replacement, irrespective of the sample size and even if the sample size exceeds the population size. It thus is reasonable to expect more precise estimates for sampling without replacement and to expect increases in the relative precision as sampling intensity increases.
2. An expression of sample size n as a function of estimator variance, coefficient of variation, confidence interval length, or some other measure of reliability. This in turn requires an analytic expression for the variance of the estimate, or at least a probability model for the estimator. 3. Specification of one or more parameter values such as a population mean a n d / o r variance, based on the results of a pilot study, on literature values, or sometimes on best guesses.
Example
A typical application involves the determination of the required sample size to ensure that a sample mean is within a specified distance of the true mean, oLproportion of the time. This can be expressed by
Counts on sample quadrats are used to estimate abundance for cottontail rabbits (Sylvilagus floridanus) on a 1000-ha study area. The area is divided into 1-ha plots, and 100 plots are selected at random. Each plot is surrounded by a wire barrier, and field workers drive the rabbits into an enclosure on the plot, where a complete count for the plot is made. The results from the sample plots provide a sample mean y = 16 and variance s 2 = 40. Application of Eq. (5.6) produces an estimate for total abundance of
9= = 1000(16)
P ( l Y - YI
= 16,000,
P
(
~ / V ~ < cr/X/-ff < ~
rL)
= 1 - oL
with variance given by Eq. (5.7)" va'-?(~') = (1000 - 100)(1000)(40) 100 = 360,000. Because of the large sample size, an approximate 95% confidence interval for Y can be based on a normal distribution (see Section 4.2.3): [ ~ " - Z0.osV'v~ (~'), ~" + Zo.osV'v~ (~') ] = (14824, 17176).
5.3.1.2. Sample Size Determination under Simple
Random Sampling An important consideration in sampling design is the selection of the appropriate sample size n. Very small samples result in estimates with poor precision, but very large samples are wasteful of effort that could be expended more productively elsewhere. An analysis of sample size depends on three factors. 1. A goal for estimator precision, often stated in terms of deviations from the estimator mean. The goal also can be expressed in terms of minimizing estimator variance subject to cost constraints or minimizing costs subject to precision constraints.
with
rY
/ x/-n
z /2
for the standardized normal. It follows that n =
(Z~2)2 CV
,
(5.9)
where CV = ~r/Y is the population coefficient of variation and z~/2 is the upper oL/2 point of the standard normal distribution. If sampling is without replacement, the appropriate sample size is given by adjusting n by
n' = n/(1 + n/N).
(5.10)
The effect of the adjustment is to reduce the required sample size, in response to the finite population correction (1 - n/N). Of course, the adjustment becomes negligible if N is very large compared to n.
Example An investigator wishes to estimate the mean number of plants per 0.1-ha plot on a 91-ha study area. There are N = 910 plots from which to take a simple random sample of size n. The goal is to collect enough samples
5.3. Simple Random and Stratified Random Sampling
65
m
to estimate Y within 10% of the true value (c~ = 0.05). Estimates based on previous studies in similar habitats indicate that the population coefficient of variation is CV ~ 0.486, so Eq. (5.9) gives a first approximation to sample size as n = (1.96/0.10) 2 (0.486) 2
tum i. The population variance is a weighted sum of stratum variances and deviations among stratum means: 0-2= ~ ( y i j - y)2 N i,j m
Ni
~91, i
with an adjustment for the influence of a finite population according to Eq. (5.10)"
]
(5.11)
-- ~-J. WiI0-2 4- (Wi-~')21,
n' = n / ( 1 + n / N )
1
= 91/(1 + 91/910) 83. Therefore, 83 of 910 plots, or approximately 9% of the population, must be sampled to achieve the desired goal of precision.
5.3.2. Stratification and Stratified Random Sampling Often a heterogeneous population can be divided into more or less homogeneous subpopulations, which then can be used to advantage in estimating population parameters. The idea is to partition the population into groups or "strata" according to some grouping principle (e.g., age, sex, geographic location, morphology) so that variation within groups is relatively low, and population variability primarily reflects group-togroup differences. Assume that a population is divided into I strata of size N1, ..., NI, such that N 1 4- ... 4- N! -- N. Then a stratified sample for the population involves the random selection of a sample of size n i from stratum i, i = 1, ..., N. It is the independent sampling within each stratum that clearly distinguishes this approach from simple random sampling, where a sample of size n = n 1 + --- + n I is selected randomly from the population as a whole. The population mean for a stratified population can be expressed as
with 0-2 the variance of sampling units in stratum i. If the sampling units within strata are similar but sampling units across strata are not, the stratum variances 0-/2 will be small relative to the differences Y i - Y among stratum means. In that case the overall population variance 0 -2 primarily reflects across-stratum differences, rather than within-stratum variances.
5.3.2.1. Estimation under Stratified Random Sampling An estimate of the population mean from a stratified random sample is the weighted average I Yst = ~ i=1
(5.12)
WiYi
of the sample m e a n s Yi for each stratum, where the weights are the proportionate sizes W i = N i / N of each stratum. The variance of the sample mean is given in terms of stratum variances, by var(ys t) = ~
W2 var(~i)
i
(5.13)
1
s/2
-- N2 ~ Ni(Ni - ni) i Yli (assuming sampling without replacement), with an estimate of variance given by A 1 var(Yst) = ~-E ~ N i ( N i i
,s/2 ni)--, Y/i
(5.14)
where
ni S2-- ~ ( Y i j -
Yq
i j Ni
m
Ni ( Y i - y)2 4- ~. -N ~ Ni l /
(yij- Yi) 2
y i ) 2 / ( n i - 1).
j=l
yij
=E-dE i j Ni -- ~_j WiWi, i
where Yi = Y~jYij/Ni is the mean for stratum i and W i = N i / N is the proportion of the population in stra-
Equation (5.14) simplifies t o ~i W 2i (s 2i/ni) w h e n t h e values N i are large relative to n i or when sampling is with replacement. Example
Consider the previous problem in which plant density is estimated on a 91-ha area. Assume that the area
Chapter 5 Survey Sampling and Population Parameters
66
is stratified into four habitat types containing 90, 100, 400, and 320 0.1-ha plots, respectively, with n = 83 samples allocated approximately equally to each stratum (Table 5.1). The stratified sampling estimate of the mean plant density is
n'), then Eq. (5.12) for the sample mean reduces to the estimate from simple random sampling: I Yst = 2 W i y i i=1
Y s t - - 0.10(20.5) + 0.11(15) + 0.44(30) + 0.35(21)
I l ~ yF/Pi j ,
= 24.25
= 9
with estimated variance
_
j = l r/
1
- n'I ~.,Yij. v~(Yst)
= (1/910)
z,]
2 =y.
(4731.4 + 3385.7 + 64971.4 + 19136.0) = 0.11. The high precision of this estimate is a consequence of both stratification, which results in small stratum variances, and large sample sizes relative to the withinstratum variation. Under mild conditions on the similarity of units within strata, the estimate of a population mean or total from stratified random sampling can be shown to be at least as precise as that from simple random sampling (Cochran, 1977). The potential advantage of stratification is seen in Eq. (5.13), which shows the variance of a sample mean to be based solely on withinstratum variances_ absent any reference to the stratum differences Yi - Y. An effective stratification loads the population variance into stratum differences as per Eq. (5.11), leaving within-stratum variances, and thus var (y), small. Absent the stratification (i.e., with simple random sampling) the variance of a sample mean is based on the larger population variance O"2 in Eq. (5.3), which includes stratum differences as well as withinstratum variation. Note that if the stratum sizes are all equal (Wi = W = 1//) and stratum sample sizes are all equal (n i --
TABLE 5.1
Stratum (i)
1. Pine forest
Example of Estimation U s i n g Stratified Random Sampling a N i (ha)
Ni/N
ni
--
2
Yi
si
90
0.10
21
20.50
16.0
2. Oak savannah
100
0.11
21
15.00
9.0
3. Bottomland
400
0.44
21
30.00
9.0
4. Agricultural
320
0.35
20
21.00
4.0
Total
910
1.00
83
a Stratum sizes N i are n u m b e r of 0.1-ha plots available in each of four vegetative cover types. In each stratum, n i sample plots are selected at r a n d o m and the density of plants Yij is measured on each plot (j = 1..... ni).
Though the two estimators are identical in their computation formulas, they nevertheless can differ greatly in their variances because of the differences in their underlying sampling designs. The relative efficiency of Yst depends of course on the effectiveness of stratification in reducing within-stratum variance. 5.3.2.2. S a m p l e S i z e D e t e r m i n a t i o n u n d e r S t r a t i f i e d
Random Sampling Whereas a decision about sampling effort in simple random sampling involves little more than the application of Eq. (5.9) or (5.10) to determine sample size, in stratified random sampling one must determine not only the overall sample size but also the allocation of samples among strata. Proportional allocation and optimal allocation are two commonly used allocation protocols. Under proportional allocation n sample units are allocated according to the relative sizes of each stratum, that is n i --
n(Ni/N),
(5.15)
for i = 1, ..., I. Here we assume that the sample units are each the same size or that variation in size among units is unimportant. An alternate form for proportionate allocation is based on the area A i of each stratum relative to the total area A: ni = n(Ai/A),
i = 1, ..., I. Area-based proportionate sampling is especially useful when the proportions N i / N are u n k n o w n or difficult to determine. Though simple to apply, proportional allocation often fails to produce estimates with m i n i m u m variance or m i n i m u m total cost. In particular, suboptimal allocation occurs if variances or per-unit sampling costs vary among strata. Optimal allocation, the second general protocol mentioned above, takes stratum variances, sampling costs, and stratum size into account. The idea here is to allocate samples to strata in a man-
5.4. Other Sampling Approaches ner that minimizes var(y), given an overall cost constraint C = C1Y/1 q- . . . q- C~n~ for n = n I + ... + n~ available samples. This is a constrained optimization problem (see Chapter 22), with an optimal solution of
i~=1 X i ~
where Ci is the cost of sampling per unit in stratum I. If unit costs are constant across strata (Ci = C), Eq. (5.16) simplifies to gli--
n~ gi~ ~ k~. Nicri]"
It is clear from Eq. (5.16) that the relative sample size ni/n for stratum i increases as the stratum size N i and stratum variance cr2i increase. It also is clear that ni/n decreases as the relative cost of sampling within the stratum increases.
Example Consider the previous 91-ha area, composed now of four strata containing 90, 100, 400, and 320 0.1-ha plots, respectively. We can use Eq. (5.15) to perform a proportional allocation of the previously determined sample size of n = 83 (Table 5.2). Thus, the protocol allocates 8, 9, 37, and 29 samples to the strata of size 90, 100, 400, and 320 sampling units, respectively. If estimates of stratum variances are available from, e.g., a previous study (Table 5.3), and costs of sampling per unit are the same for each stratum, an optimal allocation of 12, 10, 40, and 21 samples to these strata is obtained by applying Eq. (5.17). The shift in allocation reflects the influence of stratum variances. Thus, a relatively large variance in stratum 1 leads to an increase in the stratum sample size, whereas a relatively small variance for stratum 4 leads to a proportionate decrease in its sample size.
TABLE 5.2 Stratified Random Sampling with Proportional Allocation for Plant Density Problem in Table 5.1 a
Ni
Ni/N
n i = n(Ni/N)
1
90
0.10
8
2
100
0.11
9
3
400
0.44
37
4
320
0.35
29
Total
910
1.00
83
aAllocation based on stratum size
N i.
Ni
Si
SiN i
90
4.00
360.00
2
100
3.00
300.00
10
3
400
3.00
1200.00
40
4
320
2.00
640.00
21
Total
910
2500.00
83
Stratum
aAllocation based on stratum size N i and standard deviation per-unit sampling costs assumed constant among strata.
ni
12
Si;
(5.17)
l
Stratum
TABLE 5.3 Stratified Random Sampling with Optimal Allocation for Plant Density Problem in Table 5.1 a
1
/
ni nI
67
5.4. O T H E R S A M P L I N G APPROACHES 5.4.1. C l u s t e r S a m p l i n g
A second approach to partitioning a population involves the grouping of sampling units into "clusters" of dissimilar individuals, with the idea that variation within the clusters is to be high relative to variation across clusters, so that the overall population variance primarily reflects within-cluster variation. A key difference between this approach and stratified random sampling is in the nature of the partitioning. Recall that the partitioning in stratified random sampling is designed to group similar sampling units into strata, so as to reduce the within-stratum variation among sampiing units and thereby represent the population variance primarily in terms of differences among strata. The effect of clustering can be seen in the decomposition of population variance [Eq. (5.11)]. Efficient clustering yields relatively large values of within-cluster variance ~r2i and only a relatively small co mpon__ent of variance associated with the differences Y i - Y. This reverses the pattern of variation in stratified random sampling, wherein efficient stratification leads to relatively large differences across strata and relatively small variances within strata. An important benefit of clustering is that one need only investigate a sample of the clusters in order to estimate population parameters. That is, the clusters become (primary) sampling units. This feature distinguishes cluster sampling from stratified random sampling, in which all strata must be sampled. In what follows we denote by M the number of clusters in a population, to distinguish that number from the population size N. We also denote by m the size of a sample of clusters. In general, a cluster (or "primary sampling unit") is composed of some number N~ of secondary units within cluster i, i = 1, ..., M. A key feature of cluster
68
Chapter 5 Survey Sampling and Population Parameters
sampling that distinguishes it from multistage sampling (Thompson, 1992) is that once a sample of primary units is selected, all the secondary units from each primary unit are included in the sample. Even though it is the secondary units from which measurements are taken, random selection occurs only at the level of the primary units. Thus, the clusters essentially become the sampling units, with cluster attributes that are aggregations of attributes for the sampling units within the clusters. There are several reasons w h y cluster sampling may be appropriate in ecological sampling. First, there may be no effective way to obtain a list of the secondary units of interest, or, even if theoretically obtainable, the list may be extremely large. For example, it may be possible to obtain a list or map of ponds on which waterfowl nest, but impossible to obtain a list of the nests. If we wish to estimate the number of duck nests for a species known to nest on the margins of ponds, we could first obtain a list of M suitable ponds and select a sample of size m from that list. We then could locate all nests for each pond. If the primary units are selected by simple random sampling, a simple estimator of the population total Y = Yl 4- "'" 4- YM is obtained by applying Eq. (5.6) to the cluster totals: m
(5.18)
9 -- M ~_j Yi
m i=1
Myyct
where yc is the mean number of nests per pond and
sampling probabilities that are proportional to the size of the clusters, require application of Hansen-Hurwitz, Horvitz-Thompson, or similar estimators (Thompson, 1992). Example
Suppose that we are interested in estimating the total number of eggs produced in duck nests in an area containing M = 50 ponds of a certain characteristic. We select a sample of m = 5 ponds and count the eggs in every nest around each pond (Table 5.4). Application of Eq. (5.18) yields 5 i=1
= (50/5)(334) = 3340. From Eq. (5.19) an estimated variance of this estimate is va'r(~') = M ( M -
m)(S2c/n)
= 50(50 - 5)(243.2/5) = 109,440 with an estimated standard error of V'109,440 = 330.81. Note that every nest on each of the ponds is assumed to be observed, i.e., we have included all sampling units in each of the m clusters. Because this assumption is almost certainly invalid, we need to account for nest detectability in our estimation process (see Section 15.1.4, Part III).
NI
5.4.2. Systematic Sampling
Yi = ~_~ Yij j=l
is the total for the N I secondary units measured in cluster i, i = 1, ..., m. Assuming samplin~g is without replacement, the estimated variance for Y is given by Eq. (5.8): va'-~(9) = M ( M - m)(s2/m), C
In some circumstances in which random sampling may be difficult or impossible, systematic sampling may TABLE 5.4
Example of Cluster Sampling" Number of eggs
(5.19) Pond (i)
N;
Number of nests (N~)
(Yi = ~'j =1 Yij) 60
where m
$2c = m 1- 1 /~1 ( Y i "~_
Yc)2"
An unbiased estimator of the mean per primary unit is Yc = "~/M, with variance var(~) = (1/M 2 var(~'). On the other hand, an unbiased estimator of the mean per secondary unit is ~ = Y/N, with var (y) = (1/N) 2 var (9), where N = N 1 + .-. 4- N M is the total number of secondary units in the population (Thompson, 1992). More complicated sampling schemes, for instance with
a
1
10
2
12
72
3
15
90
4
8
48
5
14
64
Total
59
334
Five ponds are randomly selected, and at each pond i, all N~
N;
nests are examined and the total number of eggs (Yi = ~'j =1 Yij) are counted.
5.4. Other Sampling Approaches
69
be a practical alternative. In systematic sampling, sample units are placed in an ordered list, e.g., alphabetically or by some identifying number. To obtain a sample of size n from a population of size N, an initial unit is selected, typically at random, from the first k = N/n units. Thereafter, every kth subsequent unit is selected from the list, with a total of n units. Systematic sampling is equivalent to random sampling if the ordering of the individuals is independent of the attribute being measured and the ordering does not result in the selection of a nonrepresentative sample. For example, if one is conducting a survey of sportsmen and has available a complete alphabetized list of 3000 license holders, then selecting every tenth name from an alphabetized list results in a random sample of 300 sportsmen, provided there is no correlation between the attribute being measured (e.g., hunter satisfaction) and the alphabetical order of the names. Assume, however, that names are systematically interleaved in such a way that the 300 selected names are only from A-K and the remaining L-Z names are associated with a different segment of the population (e.g., nationality). Then the sample might no longer be representative of the target population. Systematic sampling must be used with caution in ecological populations, because it is often impossible to rule out nonrandom ordering of the sample units. For example, measures for samples taken along linear transects can easily be correlated with environmental gradients such as elevation, solar insolation, moisture, and salinity, resulting in correlated measures among individuals from adjacent segments of the transect. In that situation, sample variances will tend to underestimate the population variance, compared to estimates from a sample drawn completely at random. In the extreme case, a systematic sample is equivalent to a cluster sample with a single primary unit, effectively without any replication for estimating a variance. Under these circumstances it is possible to estimate variance only by making unjustified assumptions, e.g., by treating the systematic sample as if it is a simple random sample. Because of the importance of representation, randomization, and variance estimation in sampling biological populations, we strongly recommend sampling designs in which the sample units are selected at random and suggest that systematic sampling be used, if at all, with caution.
easier to measure. In this situation, double sampling may be an efficient approach. Double sampling takes its name from the fact that two samples are collected, one a subsample of the other. Thus, we measure an auxiliary variable (x i) on a sample of n' units and measure the primary variable (Yi) on a subsample of size n of these units, where n typically is much smaller than n'. Ratio or regression methods then can be used to predict values of Yi for the larger sample, and if x i and Yi are highly correlated, the precision of population estimators based on the predicted values can be improved substantially. There are a number of important applications of double sampling to natural resource problems. For example, it is expensive and time consuming to estimate timber volume of forest stands on the ground. Estimation of stand volume with aerial photos constitutes an inexpensive alternative, but double sampling is needed to calibrate the volumes measured from photographs (X i) to those measured from ground surveys (Yi). Once the relationship between the photographic measures and volume on the ground is established, the calibration can be used over a much larger area at much lower cost than could be accomplished by ground surveys alone. Another example involves breeding ducks, which can be counted rapidly and efficiently over a large part of their breeding range using low-flying aircraft. However, aerial counts (xi) are less accurate than ground surveys, and the detectability of ducks varies by species, so the counts must be calibrated by ground surveys (Yi) on a subsample of the aerial sample (Smith, 1995). The relationship between xi and Yi then can be used to adjust aerial counts over the much larger geographic coverage of the aerial survey.
5.4.3. Double Sampling
b = i=~l(Yi - Y)(Xi - x)/i=l~ (Xi - ~)2
Often the variables of interest in a sampling design are difficult or costly to measure, but correlated auxiliary variables can be identified that are cheaper or
is the estimated regression slope between x and y in the subsample (Cochran, 1977). An estimated variance for Yreg is
5.4.3.1. Regression E s t i m a t o r Under an assumed linear relationship between the auxiliary variable (x) and the variable of interest (y), linear regression can be used with data from a double sample to obtain an estimate Yreg = Y q- b(x' - X)
(5.20)
of the population mean Y, where ~' is the mean of the auxiliary variable in the larger sample, ~ and y are the sample means for the subsample, and F/
F/
(5.21)
Chapter 5 Survey Sampling and Population Parameters
70
!X' -- 2) 2
va'r (Yreg)= Sy,x[ l +
]
n
= 19.1 + 1.902(9.3- 10.2)
(5.22)
Z (Xi -- 2) 2
= 17.39.
i=1
2_ Sy,x2
Sy2 N'
Sy
n'
The estimated variance of this mean is provided by Eq. (5.22) as
where
(X' -- 2) 2 ]
va"~r(Yreg)-S2y, x [1
~ ( X i - ~f2J
1l
Sy,x 2 =
l
Z ( Y i - Y)(Xi- 2) i=1
2 q- Sy2 _ Sy,x
n-1
n'
(Cochran, 1977). Regression estimates of the population total Y follow from Eqs. (5.6) and (5.8), substituting Yreg and v~(Yreg) for ~ and v ~ ( ~ ) , respectively. The regression estimator has the m i n i m u m variance of all linear unbiased estimators, on condition that (1) the relationship between x and y is linear and (2) the variance of Yi about the regression line is proportional to x i.
Example A population of mallards (Anas platyrhynchos) is sampled via a r a n d o m sample of 20 5-ha plots from a total study area composed of 100 such plots. Aerial counts via helicopter are m a d e from all 20 sample plots. A subsample of 10 plots is r a n d o m l y selected, and immediately following the aerial counts, each of these is intensively searched by ground observers to obtain complete counts, resulting in the s u m m a r y data in Table 5.5. The subsample provides an estimate of the linear relationship between aerial and ground counts from Eq. (5.21) as
b = ~, ( Y i i=1
Y)(Xi-
2)/~
Sy2 N
i=1
(x i -
2) 2
147511+9 102,217714 (2824.9/9) 100
(2824.9 - 1467.5)/9 20 = 20.9.
Estimates for the total and its variance are provided by Eqs. (5.6) and (5.8) as Wreg = Nyreg = 100(17.39) = 1739 and var (Wreg) = X 2 v ~ (Yreg) = 1002(20.9) = 209,000. By comparison, an estimate for the total population based on the aerial counts alone, unadjusted for detectability, is
1467.5 771.4
~" = M2' = 100(9.3)
= 1.902,
= 930,
which is used in conjunction with the aerial survey data from the overall sample (n' = 20) to estimate m e a n per-plot duck numbers, adjusted for detectability:
Yreg = Y q- b(2' - 2)
TABLE 5.5 Sample
Aerial counts only (n' = 20) Aerial plus ground counts (n = 10)
and an estimate based solely on the ground counts is )" = 1,910 with var(~') = 282,500. Thus, the regression estimator is superior to one based on aerial counts alone, which is biased, and also is superior to one
S a m p l e Statistics for a D o u b l e S a m p l e of Mallard D u c k s a ~
~-~i(Yi -- ~)2
~,i(Xi __ ~,)2
~'i(Yi -- Y)(Xi -- "X)
2824.9
771.4
1467.5
9.3
10.2
19.1
a Ducks were counted on 20 5-ha plots involving aerial counts (n = 20) and ground counts (n' = 10).
5.4. Other Sampling Approaches based on ground counts alone, which is unbiased but imprecise (see Section 4.2.1). 5.4.3.2. Ratio Estimator
Under the further assumption that the functional relationship defining the regression estimator passes through the origin, we obtain a simple ratio estimator of the mean as Y r a t i o - - R~' =
(y/~)~,
(5.23)
(Cochran, 1977). An approximate variance for this estimator is given by va"r (Yratio) ~ (N - n') S 2
N
n'
,In n]n
(5.24)
n'n-(n -- 1) ~_j(Yi -- axi )2, i=1
where sy2 is the sample variance of Yi in the subsample. On condition that the relationship between primary and auxiliary variables passes through the origin, both the regression estimator in Eq. (5.20) and the ratio estimator in Eq. (5.23) can be used to produce effectively unbiased estimates. However, the ratio estimator is somewhat more efficient, in that the variance in Eq. (5.24) is smaller than that in Eq. (5.22). This essentially follows from the fact that a ratio estimator relies on a parametrically simpler model, for which the model parameters can be estimated more efficiently.
5.4.4. Adaptive Sampling Because animals frequently are distributed in an uneven (e.g., patchy) manner, animal population surveys that utilize conventional sampling schemes such as simple random and stratified random sampling often result in an inefficient allocation of sampling effort and highly variable estimates. For instance, in aerial counts of waterfowl during the nonbreeding season, many sampling units (e.g., square or rectangular plots) contain few or no animals, and a few units might contain thousands. Obviously, estimates based on such data will have very low precision. For example, a sequence X = {1, 0, 0, 1100, 20, 0, 0} of aerial counts might involve the expenditure of considerable effort in searching for animals in each of the sampling units, but only the count in the fourth unit contributes substantial numbers of birds to an estimate of the population total. We may well have missed other
71
sampling units, perhaps in the vicinity of the fourth unit, with high counts that would have contributed better information to the estimate of the population total. A natural inclination of biologists, having finally found a plot with large numbers of animals, is to use this information to reallocate sampling effort. Classical survey sampling lacks a theoretical framework to support such an adjustment, except to suggest that it can lead to counts that are ad hoc, nonrandom, and unuseful for estimating population parameters. Fortunately, developments under the rubric of adaptive sampling (Thompson, 1992; also see Thompson and Seber, 1994, 1996) provide a statistical framework to adjust the sampling procedure based on interim survey results, i.e., to adjust the sampling design adaptively. The basic idea is to alter sampling probabilities during the course of a survey, as a function of previously collected sample values. Adaptive sampling allows for unbiased estimates of population means, totals, and other parameters, based on auxiliary information with which the selection probabilities for the sampling units can be updated. A variety of adaptive sampling designs have been developed (Thompson, 1992), but the simplest, which we outline here, is adaptive cluster sampling. As with other sampling procedures, adaptive cluster sampling involves the selection of sampling units from a population consisting of N units with associated values {Yl, ..-, YN}. The obiective is to estimate a population mean or total based on a sample of these values. As before, an adaptive sampling design assigns a probability to every possible sample. However, the new feature with adaptive sampling is that the selection probabilities at each point depend on the values for previously selected sampling units. The adjustment of selection probabilities is based on the concept of a neighborhood for each sampling unit. Typically the neighborhood for a unit represents geographic proximity to the unit [e.g., the neighborhood A i of quadrat i might consist of all neighboring quadrats sharing any perimeter with it (Fig. 5.2)], though noncontiguous neighborhoods also can be defined. The neighborhood concept can be used to alter one's sampling plan adaptively, i.e., as the results of sampling accumulate. Thus, one includes all sample units in a neighborhood of unit i into a sample according to whether sample i satisfies a condition C based on the value Yi. Typically, the condition is that the observed value of Yi exceeds some threshold value: y -> C. Some of the neighborhood units may satisfy the condition, in which case their neighborhood units also are included. To apply adaptive sampling, an initial sample of size n is drawn at random from the population, and
72
Chapter 5 Survey Sampling and Population Parameters
for each unit i the selection criterion C is applied. The collection of units that are included in the sample as a result of the initial selection of unit i is effectively a cluster. Clusters typically include some elements that meet the condition C and some elements that do not. The group of sampling units within a cluster that meet the condition is called a network. Units in the cluster that do not satisfy C are called edge units, and it is convenient to think of them as representing networks of size one. The grouping of sample units into networks in this manner constitutes a partitioning of the population, based on the initial sample of size n. Because the inclusion of any unit in the network means that all units in the network also are included, the extended sample now effectively consists of a collection of networks. Estimation requires incorporation of the inclusion probabilities of the sample units within each network.
A modified Horvitz-Thompson estimator (Thompson, 1992) of the population total is K
9 =
~.
ykz___ k, k=l
(5.25)
Pk
where y~ is the total of the y values observed in the kth network, K is the number of networks, and z k is an indicator variable, i.e., z k = 1 if any unit of the kth network is in the initial sample, and 0 otherwise. The parameter Pk represents the inclusion probability for the kth network and is given by
pk=l - (NnNk)/(N) under sampling without replacement and Pk = 1 -- (1 -- N k / N ) n
F I G U R E 5.2 Adaptive sampling of a population with a total Y = 487 individuals distributed over N = 100 sampling units. (a) Initial random sample of n = 10 units. (b) Adaptive sample with networks defined by initial sample and criterion C = {y: y - l } . Shaded areas indicate adaptive sampling clusters, dark shading indicates networks, and bordered areas without numbers indicate cluster and network size of 1 (no individuals detected in initial sample).
5.4. Other Sampling Approaches for sampling with replacement, where N k is the number of units in network k. The corresponding estimator of the per-unit population mean is simply Ya -- ~ ' a / N
= 1
N
y zk k=l
Pk
An unbiased variance for this estimator is v~(ya)
1[1;;2
=
y2zk
KK( +~~ 1 k=l h*k PkPh
(5.26) YkYhZkZh
,
where Pkh is the probability that networks k and h are included in the initial sample:
73
TABLE 5.6 Example of Estimation Using Adaptive Cluster Sampling and the Modified Horvitz-ThGmpson Estimator for Data in Fig. 5.2 Unit (k)
Network size (N k)
Count (Y~)
Inclusion probability (Pk)
Y~z~
Pk
1
1
0
0.100
0.000
2
1
0
0.100
0.000
3
1
0
0.100
0.000
4
4
34
0.348
97.598
5
1
0
0.100
0.000
6
2
6
0.191
31.430
7
1
0
0.100
0.000
8
4
135
0.348
387.520
9
1
0
0.100
0.000
10
2
24
0.191
125.720
Ya
6.423
9a
642.268
Pkh = a In this example, z k
1-
(5.27)
-
-
1, k = 1.... ,10.
K
Wa -- ~.~ YkZk
k--1 Pk when initial sampling is without replacement and
= 97.60 + 31.43 + 387.52 + 125.72
Pkh = 1 --([1 -- Nk/N] n + [1 - Nh/N] n -[1
- ( N k + Xh)/X] n)
= 642.3 (see Table 5.6), and the estimated per-plot count is
when sampling is with replacement (Thompson, 1992).
Y a = Y a/N
Example
= 642.3 / 100
A population of mule deer (Odocoileus hemionus) of size 756, distributed in a very heterogeneous manner (Fig. 5.2), is counted by helicopters on a 100-km 2 study area that is divided into 1-km 2 plots. An initial random sample of 10 plots is selected, and aerial counts are obtained via helicopter. Clusters are defined by the condition C = {y:y -> 1}, where y is the sample count on a selected plot. There are thus four clusters of more than a single sampling unit (Fig. 5.2a), and these include networks of m i = 4, 2, 4, and 2 units for which cluster totals are Yi = 34, 6, 135, and 24 individuals each. The remaining six "clusters" consist of m i -- 1 unit each, with cluster totals Yi = 0. The inclusion probabilities reported in Table 5.6 are computed by
= 6.423. Variances are estimated by application of Eq. (5.26), where Pkh --
(10010 Nk) + (100--Nh)_10 ( lO0-Nk-Nh)lO 1--
t11000t
from Eq. (5.27). Substitution of the sample counts and network sizes (Table 5.6) provides an estimated variance for the estimated per-plot m e a n Ya of va'r (Ya) = 14.88,
10 From Eq. (5.25) we obtain an estimate of the population total as
with corresponding variance for Ya that is estimated by va"~(~a) = 14.88(100) 2
74
Chapter 5 Survey Sampling and Population Parameters = 148,814.
A 95% confidence interval for the total based on the Student's t distribution (see Appendix E) is
~'a -+- t0.05,9~//V~(~'a ) = 642.3 + 385.76(2.262)
and sometimes badly misleading inferences. Unfortunately, sampling designs, and the estimators based on them, all too often are inappropriately used in the ecological disciplines. Here we address some common pitfalls in applying the sampling schemes described above.
= (-230.4, 1514.9). By comparison, an estimate based only on the initial sample (n = 10) would have provided an estimated total population of Y = Ny = 100(1.2) = 120 with variance v~(~') = 1002(0.3511) = 3511, providing a 95% confidence interval on the total of 120 + 59.25(2.262) = (-14.0, 254.0). The adaptive sampling estimate does a better job of representing the true population size (Y = 756) than does the estimate based on simple random sampling, which severely underestimates the mean and total. Of course, both are unbiased estimates of the respective parameters, but the adaptive sampling estimator provides better interval coverage when the population is patchily distributed, as in the example. The efficiency of adaptive sampling relative to simple random sampling depends on the spatial pattern of animal distribution, the thresholds used for determining neighborhoods, the initial sample size, and other factors. In general, adaptive cluster sampling is more efficient than simple random sampling if the within-network variance of the population is sufficiently large. This is consistent with similar patterns that were noted in Section 5.5.1 for cluster sampling. The comparative influence of design factors on the performance of adaptive sampling is a topic of active investigation (e.g., Smith et al., 1996).
5.5. C O M M O N PROBLEMS IN SAMPLING DESIGNS Care must be taken to adhere to the principles of sampling design, if sampling data are to be useful for statistical inferences about animal populations. Inattention to these principles can result in unreliable parameter estimates and thus to uninformative models
5.5.1. Failure to D e f i n e Target and Sampled Populations As noted in Section 5.2, a sampled population (i.e., the portion of the population from which samples are taken) ideally should be identical to the target population of interest. However, a number of practical (e.g., logistic) factors often lead to sampling restrictions, with the result that sampled and target populations differ. Moderate differences usually are not of great concern, but extreme differences can render sample-based inferences inapplicable to the target population. This situation can easily occur when investigators select a sample in an ad hoc fashion, e.g., all sample units are taken near roadsides. Given a mismatch between sampled and target populations, there are three possible courses of action. One approach is to redesign the survey so that the sampled and target populations correspond. In the case of roadside surveys, this means including areas distant from roads in the sampled population, possibly after stratification. Another approach is to use auxiliary information to establish a predictive relationship between the sample elements in the target population and the sampled subset. This is really just a special case of double sampling (Section 5.4.3), in which all portions of the target population are ultimately involved in the sample. Yet another approach is simply to redefine the target population. For instance, the roadside survey sample can be thought of as representing a target population of habitats near roadsides. Obvious limitations are (1) roadsides by themselves are unlikely to be of primary interest and (2) the temptation to "extrapolate" results to nonroadside habitats may prove irresistible. What must be avoided in any approach is ad hoc selection of the areas to be sampled and ad hoc selection of the sampling units within areas. Otherwise, the resulting estimates are unlikely to bear any meaningful relationship to the target population and its parameters, and even if they do, it will not be possible to confirm that relationship through statistical inference. It is important to recall that in most cases, reliable parameter estimates and associated hypothesis tests depend on the assumption of a valid sampling design. Absent a valid design, the mechanics of summarizing data and computing estimates may superficially re-
5.5. Common Problems in Sampling Designs semble those based on legitimate sampling schemes, but the underlying estimation theory that relies on valid sampling no longer applies, and the resulting estimates are of unknown reliability.
5.5.2. Lack of Replication and "Pseudoreplication" An important issue in sampling design and estimation is to determine an appropriate amount of replication, so as to represent the variability of field data and to control that variability in parameter estimation and hypothesis testing. Sampling designs lacking proper replication will not accomplish these tasks and may result in biased estimates, inaccurate measures of precision, and the commission of statistical hypothesis testing errors (especially type I errors). No reputable ecologist would take a single sample (e.g., a count of animals from a single plot) and use the resulting statistic as a basis for statistical inference about a widely distributed population. Yet, there are several ways in which ecologists can effectively do the same thing. First, there is the common practice of measuring a single point in space over time, computing a mean and variance, and then claiming that variability in the attribute has been fairly measured. For instance, a single 1-m 2 plot might be measured for biomass at randomly selected days during the growing season. Leaving aside the obvious fact that growth is likely to occur over the season, a sample statistic thus computed contains no information about spatial variation that is relevant to a spatially distributed target population. One could select a second 1-m 2 plot, measure its mean and variance over time, and compute a t test of the "hypothesis" of no difference between the two plots, though the result would be meaningless. Another type of such "pseudoreplication" (Hurlbert, 1984) occurs when a single sample is divided into multiple subsamples. For example, a single 100m transect along which animal counts are made could be divided into 10-m intervals, with a "variance" computed based on this "replication." Again, the exercise would be meaningless in a broader geographic context: there is but one sample unit, the transect. Here it is useful to remember how we make use of the properties of statistical samples for estimation. Typically, we assume that (1) a sample is composed of separable, individual sampling units, (2) the units share an identical underlying distribution and parameters (or could be stratified so as to be identical), and (3) they are statistically independent. These three features of sampling--identity, individuality, and independence (i,i,i)--are ideals. Though not always
75
perfectly met, they nevertheless should be goals of both sampling and experimental designs (Chapter 6).
5.5.3. Misinterpretation of Pattern as Cause and Effect In this situation, two or more variables are measured, either jointly as part of a common sampling design (e.g., abundance of animals and estimates of survival) or separately as elements of independent sampling schemes (e.g., survival rates from a sampling design, along with weather characteristics from public records). Subsequent to the collection of these data, analyses are performed to examine covariation in the variables, with the objective of forming explanatory a n d / o r predictive models describing the pattern of association. Typically the analyses include linear or nonlinear regression for univariate "responses," or multivariate analysis such as principal component analysis (PCA) or canonical correlation analysis (CCA) when "responses" are multivariate (Krzanowski and Marriott, 1994). There are at least two potential problems with this approach. First and most obvious, sampling designs ordinarily are not set up as experiments, in which causative factors (e.g., environmental conditions) are under the investigator's control and subjects are assigned to different combinations of the predictive factors. The essential difference between an experiment and a sample survey is that with experimentation, causal factors usually are under investigator control and experimental design can be used to assess biological responses to them. With proper randomization, replication, and control, these responses can be interpreted in terms of causal inference (see Chapters 2 and 6). On the other hand, in a sample survey both the "response" and the putative causal factors are random variables. Covariation between the two may, but may not, imply a cause-and-effect relationship. Cause and effect are especially difficult to infer when the sampling design has compromised the random selection of sampling units from a target population. For instance, suppose we systematically establish transects along an elevational gradient and record the abundance of two bird species. Suppose further that species A utilizes low-elevation shrubs and species B prefers higher elevation conifers. As the elevation increases, shrubs will become scarcer and conifers more abundant, with a resulting increase in species B and decrease in species A. A correlation analysis between the counts of species A and B will reveal a strong negative correlation, but this correlation cannot be taken as evidence of a negative interaction between the species (e.g., competitive exclusion): a negative cor-
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Chapter 5 Survey Sampling and Population Parameters
relation would be expected simply because of variation in habitat, irresvective of species interactions. Neither the collection of habitat data nor the use of multiple a n d / o r partial regression analysis is likely to solve this problem. Without some kind of control, we cannot know whether species A would have occurred in the higher elevations had species B been excluded or if habitat manipulations (e.g., planting shrubs) would have rendered the higher elevations more suitable to species A. Though it contains some subtleties, this example nevertheless incorporates the basic fallacy of false causation (Chapter 2). The take-home message is that correlation does not imply causation and correlated patterns of change cannot by themselves confirm a causal relation among ecological factors. Another misinterpretation occurs when two or more sample estimates are computed with the same sample data and correlation between resulting estimates is interpreted as implying a correlative (often causal) relationship between the parameters being estimated. For example, suppose that the abundance of a population is estimated from a series of successive counts/~/t just before the breeding season, for t = 1, ..., k years. Suppose also that an investigator is interested in examining the relationship between abundance and population growth rates, to discover whether the population appears to be growing according to a density-dependent model (Section 1.3.2). A natural estimator of the finite rate of increase Kt is based on the successive population estimates:
~t = l~i+l/l~t" The investigator might examine, say, 10 years of (/~t, Kt) pairs and, observing a negative correlation coefficient, conclude that there is evidence of density dependence. The difficulty is that the estimators of the quantities of interest ( N t and ~t) have a built-in statistical association with one another, because they share data (the counts used to estimate Nt). There may (or may not) be a biological relationship between N t and Kt, but even if there is not, there will be a statistical relationship between/~t and )~t.This generally is the case whenever the same data are used to estimate two or more parameters: the resulting parameter estimates will be statistically dependent, irrespective of any biological association. Statistical independence of the estimates will be assured only if they are based on independent samples (e.g., density estimated with counts, survival with mark-recapture). Of course, even if one guards against misleading statistical covariation (e.g., by using statistically independent estimates), the ability to make valid causal inferences cannot be assured unless the principles of experimental design are followed. Rather than simply
recording population growth rates at uncontrolled abundance levels, it becomes necessary to control abundance and then observe (and compare) growth rates. Naive interpretation of correlations between sample estimates is so common that an admonition to avoid interpreting correlations as causal bears frequent repetition. This point will be given special emphasis in Part III when we consider sampling designs and statistical models for estimating abundance and demographic parameters.
5.6. D I S C U S S I O N The proper design and execution of sample surveys is critical in parameter estimation and hypothesis testing. Classical sampling theory provides a framework for the design of sampling efforts and mechanisms for accounting for spatial, temporal, and other sources of variability. We have given special emphasis in this chapter to the need for random selection of samples and for sufficient replication to ensure credible and useful estimates of population parameters. We have focused on the estimation of population means, totals, and variances, in addition to other parameters based on these. The principles and procedures of sampling were introduced in terms of simple random sampling, and extended to a number of other designs that utilize population structure (stratified sampling, cluster sampling, adaptive sampling) and auxiliary information (double sampling). Decisions about which design to use, and the appropriate intensity and allocation of sampling effort, depend on the parameter(s) to be estimated, the cohort structure of the population, the amount of inherent variation among sample units, detectability of organisms, cost constraints, precision requirements, availability and cost of ancillary information, and a number of other factors. Obviously, no single approach to sampling design can provide efficient and effective sampling plans for all situations. It is important to recognize that the subjects covered in this chapter represent only a very small fraction of the issues that could be covered. The field of survey sampling is quite mature, and there is a huge body of theoretically deep, often highly specialized, literature on the subject. Our intent in this chapter is not to recapitulate this literature, nor even to review all aspects of survey sampling that are relevant to population biology. Our goal is rather to highlight some sampling issues and concerns, building on the statistical principles elaborated in Chapter 4 and incorporating them into some familiar sampling protocols in population biology. We think it important for the reader to understand the underlying principles as well
5.6. Discussion as the potential pitfalls in field sampling, preparatory to more in-depth discussion of specialized sampling and estimation in Part Ill. The focus now shifts from field sampling (the emphasis of this chapter) to experimental analysis and testing of patterns of variation, including assessment of causation. Randomization, replication, environmental heterogeneity, and other issues will be extended in Chapter 6 to the situation in which control occurs in
77
an experimental or quasiexperimental context. We consider there the investigation of treatment effects via data that are obtained under an experimental design, where sample units in the population are subjected to some manipulation (treatment) and their responses are compared to those of other sample units that did not receive the treatment (i.e., controls). We will see that with a proper experimental design, strong inferences about causal relationships are possible.
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C H A P T E R
6 Design of Experiments in Animal Ecology
6.8. CONSTRAINED EXPERIMENTAL DESIGNS AND QUASI-EXPERIMENTS 6.8.1. Impact Studies 6.8.2. Observational Studies 6.9. DISCUSSION
6.1. PRINCIPLES OF EXPERIMENTAL DESIGN 6.1.1. Control 6.1.2. Replication 6.1.3. Randomization 6.1.4. Experimental Error 6.2. COMPLETELY RANDOMIZED DESIGNS 6.2.1. Single-Factor Randomized Designs 6.2.2. Multifactor Randomized Designs 6.2.3. Statistical Models for Multifactor Randomized Designs 6.2.4. Associations among Models 6.2.5. Testable Hypotheses in Randomized Experiments 6.2.6. Hypothesis Testing 6.3. RANDOMIZED BLOCK DESIGNS 6.3.1. Restricted Randomization of Treatments 6.3.2. Statistical Models for Randomized Block Designs 6.3.3. Estimation and Testing 6.4. COVARIATION AND ANALYSIS OF COVARIANCE 6.4.1. Statistical Models for the Analysis of Covariance 6.4.2. Parameter Estimation and Testing 6.5. HIERARCHICAL DESIGNS 6.5.1. Split-Plot Designs 6.5.2. Crossover Designs 6.5.3. Repeated-Measures Designs 6.6. RANDOM EFFECTS AND NESTED DESIGNS 6.6.1. Statistical Models for Nested Designs 6.6.2. Estimation and TestingmFixed Effects 6.6.3. Estimation and TestingmRandom Effects 6.6.4. Associations with Hierarchical Designs 6.7. STATISTICAL POWER AND EXPERIMENTAL DESIGN 6.7.1. Determining Sample Size Based on Power
We turn our attention in this chapter to experimentation as a mechanism to explore biological relationships. By an experiment is meant an investigation under controlled and repeatable conditions that focuses on a prospective association between treatment factors and population responses. An experimental design is a protocol for allocating treatments to a collection of experimental units representing a population of interest. What makes the design "experimental" is that the allocation of treatments is under investigator control, and indeed, it is the assignment of treatments to experimental units that most clearly distinguishes experimental design from the r a n d o m sampling designs in Chapter 5. Ideally, two randomizations are involved in experiments: (1) the r a n d o m selection of experimental units from a population of interest, as per the sampling designs of Chapter 5, and (2) the r a n d o m assignment of treatments to experimental units. Methods for the r a n d o m assignment of treatments are discussed further below. In some experimental designs, treatments may be characterized in terms of a single factor of interest, as with different levels of an environmental indicator (e.g., ambient temperature). In others, treatments are described in terms of multiple treatment factors, as in a factorial design involving two different environmental
79
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Chapter 6 Design of Experiments
indicators. Treatments may consist of discrete levels of essentially continuous factors such as temperature, soil moisture, or population density, or they may consist of fundamentally categorical factors such as dietary preference or species identity. For either scenario, in what follows we use alphabetic subscripts to designate factor levels, as with the letter i for levels of design factor A, and (i,j) for the combination of levels i and j for factors A and B, respectively. This notation generalizes naturally to multiple design factors. In experiments certain attributes of the experimental units are identified as response variables, with the idea that treatment responses are to be measured by attribute differences across treatments. Typically, though not necessarily, responses can vary over a continuous range of values. We designate experimental response with the variable y, with appropriate subscripts to denote treatments and replications. Thus, Yi(k) represents the response of experimental unit k to level i of treatment A, and Yij(k) represents the response of unit k to the combination of levels i and j of factors A and B, respectively. Parentheses are used in these expressions to indicate the nesting of replicates within treatments. Again, this notation extends naturally to more than two design factors. Like sampling designs, experimental designs can be described in a context of modeling, estimation, and hypothesis testing. The issue is to estimate the mean of a conceptual population consisting of the population of interest, but with every member thought to have been subjected to a particular treatment. It is assumed that treatment effects are registered through their influence on the means of the corresponding statistical populations and that population variances are unaffected by treatments (or if the variances are affected, the effect can be recognized and accommodated via statistical adjustment; see Section 6.4.2). Estimates of the population means and experimental variance can be used to compare and contrast population parameters and to test hypotheses about them. In this manner one can investigate patterns of association between treatments and population responses, with the possible imputation of causal relationships. It is the ability to assign treatments to experimental units that allows for causal inference. Experiments offer the potential to determine not only whether a response occurs in the presence of a given factor (sufficiency), but also whether the response only occurs given the presence of the treatment (necessity). Causation that entails both conditions (see Section 2.2) requires the assignment of treatments to some experimental units and the assignment of controls to other units. Of course, the statistical testing of a contrast between treated and untreated populations utilizes an estimate
of population variance (often referred to as experimental error), which in turn requires replication in treated and control groups. The random assignment of treatments to groups of experimental units avoids systematic and often unrecognized biases in the estimation of treatment effects and population variances.
6.1. PRINCIPLES OF EXPERIMENTAL D E S I G N The features of control, replication, and randomization from Chapter 5 also apply to experimental designs, regardless of the treatment structure and response measures. In some experiments, it may be appropriate to use simple counts, indices, or other measures taken on the experimental units, as the response variable. In others, more complicated measures are appropriate-for example, when there is variation in detection rates and particularly when unadjusted measures result in the confounding of experimental responses and rates of detection. Nevertheless, the key features of experimental control, replication, and randomization of treatments apply irrespective of the experimental response. We focus below on these features, deferring until later a discussion about detection rates, treatment responses based on subsampling, and other issues of importance in experimentation.
6.1.1. Control The term control in experimentation is used in a context of treatments that are applied to experimental units, with the idea that treatment effects are registered through the comparison of treated and untreated samples. The inclusion of a control establishes a baseline for the population of interest, against which these comparisons can be made. However, the control population may involve a baseline level of some manipulation, with the experimental "treatments" representing an alteration of the level of manipulation. The point of experimental control is not so much to measure attributes under pristine, unperturbed conditions, as it is to establish this baseline for comparison. For example, an experiment might involve exposing animals to herbicides, to determine the effect on animal weights. The experimental control might consist of a low level of herbicide that currently is applied to a forest in which the animals are trapped. In essence, the control mimics a maintenance herbicide regime to serve as a basis for comparison against other herbicide treatments in the experimental design. The point here is that an appropriate control, like the treatment structure, is defined by the goals of the experiment and must be
6.1. Principles of Experimental Design chosen by the investigator along with the treatment structure as a part of the experimental design. The notion of experimental control plays prominently in the assessment of causal associations. Recall from Chapter 2 that causation can be described in terms of necessity and sufficiency, with different kinds of evidence supporting necessary and sufficient causes. Thus, a factor a can be a sufficient cause of the response b, provided b occurs whenever a does, recognizing that other factors also could lead to b. Other factors can be eliminated as potential causes of b by establishing that a is necessary for b, i.e., the absence of a leads to the absence of b. Under conditions that are both necessary and sufficient, the response b follows from a, and only from a. Necessary and sufficient causation can be usefully investigated with experimentation by means of the treatment of subjects with factor a and by the use of controls that are absent factor a. Through the treatment one can evaluate whether a putative causal factor is sufficient as an explanation for a given experimental result, following a sufficiency argument of the form "if the treatment is present, then the predicted response will occur." A consistently observed response provides evidence (though not proof; see Section 2.4) for the factor as a sufficient cause, and a lack of response leads to a conclusion that the treatment is not a sufficient cause. However, even if there is a consistent response the possibility always exists that it is a reaction to some other, perhaps unrecognized, causative agent. The possibility of alternative explanations is particularly acute in so-called mensurative experiments in which no actual manipulation occurs, but subjects are simply observed at various levels of the treatment. Controls allow one to investigate whether the response would have occurred in the absence of the treatment factor and thus give additional strength to a causal inference above and beyond what is possible with nonexperimental investigation. Two additional points should be emphasized about experimental controls. In ecological investigation, "control" sometimes is used to signify "pristine conditions" or "the absence of human intervention (management)." Some practitioners therefore claim that controls are either rare or impractical, or would provide results that are of little real-world significance. We reiterate here that controls are a matter of definition to be tailored to the study goals, and as such they are common, practical, and important elements in the investigation of management interventions. The second point is that the term control often is used to connote a reduction of experimental error, through the removal or stabilization of "nuisance variables" that are not of primary interest or that may
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confound interpretation of experimental results. It is in this sense that ambient conditions are said to be "controlled" (i.e., standardized) in laboratory studies. To avoid confusion we use the word control in what follows to mean a component of the treatment structure of an experimental design, and we apply the terms standardization and stabilization to the maintenance of uniform conditions across the experiment.
6.1.2. Replication Replication in an experimental context is the assignment of the same treatment to more than one experimental unit. Together with randomization, replication allows one to estimate experimental error, which then can be used to separate natural or background variation (i.e., not related to the treatments) from "treatment effect" that is the object of experimentation. As a general rule, increasing the number of replicates per treatment increases the precision of the estimates of treatment effect and increases the power of statistical tests (see Sections 4.3.2 and 6.7). As in sampling, there often is confusion about what constitutes an "experimental unit" or "replicate," and thus there is a potential for "pseudoreplication" (Hurlbert, 1984). In general, true replication involves experimental units that are physically separable, which allows treatments to be assigned independently. The lack of separability leads to pseudoreplication, which occurs frequently in ecological studies. For example, assume that two animals are selected from a population to which inferences are to be made. One animal receives a treatment and another remains untreated, with responses from the two individuals recorded through time. The monitoring of these animals over time may be important in elucidating temporal patterns of response to the treatment, but a temporal sequence of data on them gives no information about variability among similar animals not included in the experiment and no information about variability in treatment responses. The data thus are incapable of providing estimates of experimental error. A second example involves a study area that is divided into two parts, with each part randomly assigned either a treatment or a control. Several samples are taken from each part, and a test statistic is computed based on the sample means and variances. Again, there are but two experimental units. The samples from each unit simply provide more precise estimates of a "response" (treatment or control) for these two units, but no information can be forthcoming about whether this response would have occurred on similar experimental units. Although a statistical test could be computed, the "hypothesis" for such a test would be restricted to
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Chapter 6 Design of Experiments
the potential difference between the two experimental units. It is not possible to investigate with these data whether the populations of treated and untreated study areas differ. A third example involves treatments and controls that are assigned at random to each of 10 randomly selected animals, with each animal receiving both the treatment and the control at different times. At one level there is true replication in this design, because the subjects represent a random sample and each is independent of other subjects in the sample. However, the design involves an application of both treatment and control to every experimental subject. Nonindependence in the assignment of treatments means that sample-based estimates for treatment and control are statistically correlated, and this correlation must be accounted for in the design and analysis (see Section 6.5.2).
6.1.3. Randomization Randomization is a protocol for assigning experimental subjects to treatments so as to avoid bias. Randomization ensures that experimental results are free of an investigator's preconceptions (conscious or otherwise) in the assignment of treatments and avoids the confounding influence of uncontrolled variables on the experimental results. It also promotes the proper dispersion of treatments among subjects and the independence of treatment effects (i.e., treatments applied to units in one treatment group do not affect responses in a different group). For example, an investigation of the effect of timber harvest on small mammal populations might involve 10 forest stands, five of which are to be harvested and five of which are to serve as controls. Clearly, it would be a poor design to select for the treatment group five stands with highly suitable small-mammal habitat and as controls, five with poor or unsuitable habitat, because the results of such an "experiment" might have more to do with pretreatment habitat and population conditions than with effect of harvest. A less obvious influence might arise from arranging the stands systematically along a spatial (e.g., elevational) gradient that strongly influences habitat characteristics, with the first five stands along the gradient assigned to the treatment and the second five to the control. In this case, variation within the two groups may well be reduced from that expected under random assignment, and thus experimental error may be underestimated. More importantly, an extraneous factor has now been introduced (the elevation gradient) with the potential either to mask true treatment effects or to be falsely interpreted as an effect where none exists. Although
this type of influence can be controlled to some extent by blocking (see Section 6.3) or the use of covariates (see Section 6.4), it often is impossible to anticipate all the factors that can intervene to confound the results of an experiment. Randomization is thus not simply an exercise to satisfy the statistician; it is a bulwark against misinterpretation of experimental results and misguided decision-making based on them.
6.1.4. Experimental Error A key assumption in the analysis of experiments is that observation variances are all identical. There are at least three circumstances in which this assumption can be violated. First, there may be a mathematical relationship f(p3 = 0"2 between population means and their corresponding variances. It is not uncommon, for example, for variation in organism sizes to be proportional to mean organism size. If such a relationship between the mean and variance can be ascertained, a variance-stabilizing transformation of the data can produce unbiased estimation and testing procedures. Second, heterogeneous variances can be introduced by way of subsampling. If subsamples Xijk of a sampling unit are averaged to produce a response value Yiq) ~k Xijk/nij for that unit, the corresponding variance is inversely proportional to subsampling intensity: var(yi(j))
= (y2/nij ,
where nijk is the number of samples used to calculate Yi(j). A simple corrective for nonconstant subsampling is to weight each sampling unit by the number of subsamples included in it and then proceed with the standard estimation and testing procedures. Third, variance heterogeneity can be introduced in the choice of the sampling frame. This occurs, for example, in comparisons of taxa with greatly varying taxonomic diversity, in contrasts across areas of greatly varying geographic extent, in studies involving widely varying magnitides of environmental fluctuations, and so on. A general expression for such heterogeneity is (y 2 _
W i(y2r
where w i expresses the relative variance for population i. On assumption that this variance heterogeneity can be quantified, an appropriate procedure is to weight the experimental means with the terms wi. There are in fact many sources of variability in biological studies that can result in nonconstant variances among the sampling units or populations. It usually
6.2. Completely Randomized Designs is wise to test for their occurrence at the outset of an analysis of variance [see, e.g., Brown and Forsythe (1974) and Milliken and Johnson (1984) for testing procedures].
Example Consider a forest that is about to be treated with herbicide to thin its understory. Two different herbicides are being considered, with treatments that may consist of a combination of both. There is concern about possible impacts of the herbicides on the physiology of a species of animals inhabiting the forest, and an experiment has been designed to investigate the issue. The experiment involves trapping and relocation of animals into individual cages, where ambient conditions can be standardized, herbicide treatments can be imposed selectively, and animal weights can be monitored. Herbicide A is to be applied at a single dosage level in the experiment, with a corresponding control (the absence of the herbicide). Herbicide B is less expensive and also more effective, so treatment levels are to include low and high dosage levels, as well as a control. The treatment structure for the experiment is thus a 2 • 3 crossed design, in which each of the two levels of herbicide A is to be applied with each of the three levels of herbicide B. A total of six statistical populations is defined, each characterized by specific combinations of herbicide A and herbicide B. The idea is to measure the response, in this case the change in weight after some predetermined period, of individuals subjected to the six treatment combinations, and then to use these measurements to examine differential impacts of the herbicides. Of interest is the effect on animal weight when herbicide application consists of herbicide B alone, herbicide A alone, or a combination of herbicides A and B. To avoid the possibility of systematic but unrecognized bias in applying the herbicides to individual animals, the treatment combinations are to be assigned to animals randomly. Because the testing of treatment comparisons requires an estimate of experimental error, more than one animal must be subjected to the same treatment for some (hopefully all) of the treatments. Cell means for the 2 • 3 layout, with corresponding sample numbers, are shown in Table 6.1. From Chapter 5 an estimate of the population mean ~q for population (i, j) is Yij, i = 1, 2 and j = 1, 2, 3. On assumption that the treatments affect population means but not population variances (i.e., ~ = or2), an estimate of experiment-wide variance is given in terms of the sample variances:
~. (nij - 1)s~. ~ 2 = l,j n-6
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with ~i,j nij = n. A judicious use of the estimates yq and @2 allows one to examine the differential effects of various herbicide combinations. In what follows we discuss in more detail the identification and testing of biologically informative hypotheses in multifactor experiments.
We describe below the elements of randomized designs, including treatment structures, statistical models, and hypothesis tests. For the remainder of the chapter we include controls as a part of the treatment structure and use the term "treatment" to refer to both experimental manipulations and baseline controls.
6.2. C O M P L E T E L Y RANDOMIZED DESIGNS Completely randomized designs frequently are used when the population being investigated is absent the kind of systematic structure normally accounted for in stratified random sampling. In a completely randomized design, experimental units are assigned to treatments and controls completely at random, according to design specifications for the number of replications for treatments and controls. The unrestricted allocation of treatments is definitive of a completely randomized design, just as the unrestricted selection of samples defines simple random sampling. If factors inducing systematic variation in the experimental units can be identified and measured, a restricted randomization of treatments through blocking often is effective in reducing experimental variation and improving the power of tests (see Section 6.3).
6.2.1. Single-Factor R a n d o m i z e d D e s i g n s The simplest completely randomized design involves a single experimental factor with two experimental conditions: presence of a treatment at some level of interest and absence of the treatment (control). In this scenario a set of some n experimental units is divided randomly into groups of, say, n I and n 2 = n - n I units, with units in the first group assigned the treatment and units in the second group assigned the control. If the two groups are of equal size, i.e., n i -n / 2 , the design is said to be balanced. One method for allocating treatments to experimental units is to assign a number to each experimental unit, reorder the list of numbers randomly, and assign the first n I units on the reordered list to the treatment and the remaining n 2 units to the control. Equivalently, n random numbers from {1, 2,..., n} can be drawn in sequence (without replacement) from a uniform proba-
84
Chapter 6 Design of Experiments TABLE 6.1
Cell M e a n s and S a m p l e Sizes for a 2 x 3 Crossed D e s i g n a
Population means
Sample sizes
1
2
3
[-I,11
b1,12
b1,13
)/'21
~22
~.1
~.2
1
2
3
P-1.
//11
//12
//13
//1.
b1'23
I-L2.
//21
//22
/123
//2.
~.3
~..
n.1
n.2
n.3
// ..
a R o w a n d c o l u m n totals of cell m e a n s are g i v e n b y ~i = Ej ~ij a n d [.l,.j = ~,i bl,ij, respectively. R o w a n d c o l u m n totals of s a m p l e sizes are g i v e n b y hi. = ~,j//ij a n d / / . j -- ~i//ij, respectively.
bility distribution, and the first n I numbers (and thus their corresponding units) that are drawn are assigned to the treatment, with the remaining n 2 assigned to the control. Both procedures ensure that every possible grouping of experimental units into nl and n 2 individuals is equally likely. Randomization and allocation of treatments generalize naturally to designs with multiple treatments. Thus, n experimental units can be divided into groups of n 1, ..., n k units at random, with units in the first group assigned treatment 1, units in the second group assigned treatment 2, and so on. A balanced design has equal numbers of experimental units in each group, i.e., n i = n/k. The process of assigning individuals randomly to groups proceeds as in the two-group case.
until fledging or failure, and an estimate of nesting success is computed for each stand based on the Mayfield estimator (Chapter 15). One-way analysis of variance on the logit-transformed daily survival estimates can be used to test the null hypothesis of no treatment effect.
6.2.1.1. S t a t i s t i c a l M o d e l for the Single-Factor Randomized Design A model for the single-factor randomized design includes a population mean for each of the treatment levels and a random term representing the randomization process:
Y~q) - ],l,i q- ~,iq),
Example Interest centers on the effect of ingestion of lead shot pellets on survival of mallard ducks. One hundred female mallards are captured and outfitted with radio transmitters. On capture each duck is assigned a radio frequency at random (numbered 164.010, 164.020, ..., 165.000). All ducks receiving frequencies numbered 164.010-164.500 are administered number 2 lead shot; those receiving numbers 165.510-165.000 are given a placebo but otherwise are treated identically. All birds are released and monitored for 90 days or until mortality or disappearance of the radio signal, and the hypothesis of no treatment effect is tested using a logrank statistic (Chapter 15).
where again the subscript parentheses connote the nesting of replicate j within treatment i. Here it is assumed that the random variables 8,i(j) a r e independent for all i and j, and that each has a normal distribution with mean 0 and u n k n o w n variance or2. Then the expected value of Yi(j) is
E(yir
[,1, i
and an unbiased estimator of [,1,i is [~i -- Yi.
= E Yi(j)/ni 9 J
Example Researchers are interested in testing the effects of three rates of herbicide on nesting success of songbirds. Thirty forest stands, each with similar composition and management histories, are randomly ordered, and the first 10 are assigned as controls (no application of herbicide, but all other management identical), the second 10 are assigned to moderate levels of herbicide, and the third 10 are assigned to high levels of herbicide. All stands are searched for nests, nests are monitored
=
The difference Yi -- Yi: between treatment means has a normal distribution with mean E(yi.-
with variance
var i
Yi:)--
[,l,i- [,l,i,
6.2. Completely Randomized Designs An unbiased estimator of the experimental variance 0-2 in this model is given by ~2=
~, (n i -- 1)s 2i i
!
n --a
where s 2i is the sample variance for replicates within treatment i, a is the number of treatments, n i is the number of experimental units assigned to treatment i, and ~ i Y/i -- t/. The difference between treatments i and i'. is estimated by the difference Yi. - Yi:. in treatment means, and an appropriate test of no treatment effect is based on the within-treatment e r r o r ~ 2 (Graybill, 1976).
6.2.2. Multifactor Randomized Designs It is straightforward to extend the one-factor design to situations with two or more treatment factors operating at different levels. In particular, cross-classified or factorial designs involve two or more factors of interest in which replicates are assigned to each combination of the treatment factors. If there are m factors under investigation with k i levels for each factor i, then there are K l-Iik i combinations of treatment factors. In a balanced design, n = rK experimental subjects are required for r replicates per treatment combination. Clearly, even moderately sized factorial experiments can demand large numbers of subjects. For instance, a three-factor experiment with only three levels for each factor would require 33 = 27 subjects with no replication (r = 1) and 4 • 27 = 108 subjects with four replicates for each treatment combination. For this reason it is common in large factorial experiments to use multiway interaction terms in lieu of experimental error based on replication. However, this approach obviates the possibility of testing for higher order interactions and potentially inflates the estimate of experimental error, thereby reducing the power of hypothesis tests. As an alternative to large factorial experiments, investigators can pursue a series of smaller studies that include only some of the experimental factors of interest. The latter approach sacrifices one's ability to investigate certain factor interactions, in order to achieve sufficient replication per treatment combination so as to allow estimation of a "true" experimental error. =
85
tions: three levels of harvest (no harvest, moderate harvest rates, and high harvest rates) and two levels of feeding (no feeding and ad libitum feeding via automatic feeders). On each management area, 50 deer are captured, radio collared, and monitored to obtain an estimate of survival. Factorial analysis of variance can be used to test hypotheses about harvest effects, feeding effects, and interactions between harvest and feeding.
6.2.3. Statistical Models for Multifactor Randomized Designs In what follows we use a two-factor crossed design to illustrate some options for modeling responses in a multifactor completely randomized design. Notation for the two-factor design includes the letter i as an index for design factor A, the letter j as an index for design factor B, (i,j) to indicate the combination of levels i and j for factors A and B, respectively, and Yij~k~ to represent the response of replicate k to the combination of levels i and j of factors A and B. Table 6.1 displays population means and sample sizes for a twofactor experiment involving three levels of one factor and two levels of the other. 6.2.3.1 Cell Means Model
Perhaps the simplest expression for the two-factor design is the cell means model (CMM) Yij(k) = P~ij + ~i/(k),
where Yij(k) is the kth observation for cell (i,j) in the experimental design, ~ij is the population mean for cell (i,j), and ~ij(k) is a random error term associated with the random assignment of treatments to experimental units. The model is so named because it describes the cell means in terms of individual parameters, one for each cell. On assumption that there are a and b possible levels of the design factors and all combinations are represented in the design, a total of ab parameters is required to specify the population means. Errors typically are assumed to be independent and normally distributed with a mean of 0 and unknown variance 0 "2. Estimation of the cell means and experimental variance is straightforward and utilizes ~ij -- Yij.
Example
Interest focuses on the compensatory relationship between harvest and other mortality sources for whitetailed deer under varying levels of supplemental feeding during winter. Eighteen wildlife management areas are randomly assigned each of six treatment combina-
= ~, Yijk/nij k
for cell-specific sample means and ~,(nij - 1)s 2ij .. 4 2--
(6.1)
"1
n - ab
Chapter 6 Design of Experiments
86
namely, those that can be expressed as linear combinations of the cell means.
for the experimental variance.
Example The artificial data in Table 6.2 offer a specific illustration of the estimation of cell means and experimental variance for the CMM. Thus, a 2 • 3 crossed design involving the application of two herbicides has sample sizes varying from two observations when herbicides A and B both are applied at level 1, to six observations when herbicides A and B are applied at control levels 2 and 3, respectively. A straightforward computation of sample means produces the parameter estimates (~11, ~12, ~13, ~21, ~22, ~23) = (20, 60, 80, 80, 60, 20), and Eq. (6.1) for the experimental variance yields ~ 2 = 9.6. With these values it is possible to test whether, e.g., the application of herbicide A has an effect at the control level of herbicide Bmthat is, whether [,1,13 - -
6.2.3.3. Restricted Parameterization Model
['1'23 = 0.
and
A third commonly used parameterization is specified by the restricted parameterization model (RPM) Yij(k) = ~* if- 0~ if- ~
where the a, [3, and F parameters are constrained by the "E-restrictions"
i
E J
6.2.3.2. Fully Parameterized Model
E i
An unconstrained parameterization (6.2)
Yij(k) = tx + ~ -[- ~j nu Fij + 8ij(k)
is perhaps the most familiar form for the two-factor model. In this form (xi expresses an "overall effect" for level i of factor A, [3j expresses an analogous effect for level j of factor B, and Fij expresses an interaction between the two factors. The meanings of these parameters can be defined in terms of the means txij of the cell means model, as discussed below. Though at most, ab parameters are necessary to specify cell means for a two-factor crossed design, the fully parameterized model (FPM) contains (a + 1)(b + 1) parameters. Thus, the model is said to be "overparameterized" or "less than full rank." An important consequence is that unique estimators of the model parameters cannot be obtained. However, it is possible to estimate certain linear combinations of parameters,
TABLE 6.2
= o,
= E j
= 0
The practical effect of the E-restrictions is to eliminate redundancies in the parameter structure. For example, the E-restrictions for a 2 x 3 crossed design yield ~ - -~, ~
=
-N-
~,
1-'~2 ----- - F ~ 2 ,
F~I = -F~I , 1-'~3 = --1-'~3 = F~I + F~2. From these equations it can be seen that there is only one independent oL*,only two independent f~*,and only two independent F*. Thus the E-restrictions reduce to six the number of parameters needed to specify the model: once Ix*, ~ , f~, f~, F~I, and F~2 have been estimated, the estimates for the remaining parameters are obtained directly from them.
Artificial Data Set for a Herbicide Study" 6.2.4. A s s o c i a t i o n s a m o n g M o d e l s
Herbicide B
1 Herbicide A
q- r~]. q- 8ij(k),
2
3
18, 22
57, 63
76, 78, 82, 84
78, 80, 82
58, 60, 62
16, 17, 19, 21, 23, 24
a The study involves three levels of one herbicide and two levels of another in a simple crossed design. Six populations are defined, with varying numbers of experimental units from each population.
Though the three models described above have different mathematical expressions, in fact they are merely alternative representations of the same set of statistical populations. Because the same cell mean can be expressed in three different ways, certain equivalences exist among the parameterizations. For example, the RPM and CMM are related by ~ij
~l,* q- (X i
~j + Fij.
6.2. Completely Randomized Designs For a 2 x 3 design these equations can be expressed in matrix form by ~11
1
1
1
0
1
0-
~b12
1
1
0
1
0
1
1
~13
1
-1
-1
-1
1
-1
1
0
~22
1
-1
0
1
_~23.
1
-1
~21
=
-1
-1
r,j =
~/j = ~1, -}- O~i -}- ~j q- F/j.
0 -1
1
1
_F~2.
~..,
(6.3)
-~.j if- -~...
(6.4)
=
and r~]. = ~l,ij -
-~i. -
These relationships provide reasonable interpretations for the parameters in the RPM. Thus, the "main effect" parameters a *i and ~j* correspond to the row and colu m n averages ~i. and ~.j, whereas the interaction ~q -- J'l'ij' -- ~i'j q- ~'I'i'j' is equivalent to F ;'1. - F*,j - F i~" -}- F*,j,. For example, in the two-factor design involving two herbicidal factors, the main effect oL*for treatment level i of herbicide A corresponds to the average (~il -}- ~i2 if- ~i3)//3 across all three levels of herbicide B. The main effect ~ for treatment level j of herbicide B corresponds to the average (~lj + ~2j)/2 across both levels of herbicide A, and contrasts among the F i~"are equivalent to contrasts among cell means for the corresponding levels of the two herbicides. In a similar fashion, associations between parameters in the RPM and FPM are given by ~ * q- OL~ nu ~
q- F~. = ~1, q- oLi q- ~j -Jr- F ij ,
which, if there are no missing treatment combinations, reduces to the simple expressions
~*=~+~
+ ~. + F.., - F..),
(6.5)
F~. = F q - F i . - F.j + F...
(6.6)
O~t = O~i "-t'- F i . =
(~
r , , , j - r i*j + r,,y.
Finally, the association between parameters in the CMM and FPM is given by
which, assuming all factor combinations are included in the design, reduces to
OLt ~- ~ i . -
parameters in the FPM. They also indicate a one-toone relationship between interaction parameters F~ and Fq in the two models, so that F/j - Fry - Fij, +
-1
-1 0
87
+
and
These relationships make explicit the fact that row and column averages, as represented by O~*i and ~ , correspond to combinations of main effect and interaction
Because the FPM is overparameterized, there is no unique representation of its parameters in terms of the cell means. Indeed, infinitely many combinations of FPM parameters can satisfy this equation. This lack of uniqueness is a direct consequence of overparameterization. Notwithstanding the equivalences among model forms, the CMM has certain practical advantages over the FPM and RPM for crossed designs, especially in the absence of constraints on the cell means. It has the simplest mathematical form, with single parameters for each of the cell means. Any linear combination of these parameters is estimable, including individual cell means. If there are no model restrictions, the estimates themselves have an intuitive form and are easily computed. Finally, the CMM is the simplest model with which to express biologically meaningful hypotheses.
6.2.5. Testable Hypotheses in Randomized Experiments It is not uncommon for biologists to test hypotheses that are either irrelevant or uninterpretable. This is so primarily for two reasons. First, biologists often fail to specify hypotheses in a way that corresponds to the sampling design of the study. For example, simple expressions such as "factor A has no effect" and "there is no main effect for A" are insufficiently specified. Unambiguous hypotheses are best expressed in terms of the design parameters. Second, computing procedures that often are used, though convenient, are inappropriate for the hypothesis of interest. Such errors result from the application of computing procedures without adequate attention to their corresponding hypotheses. General computing algorithms are available for testing hypotheses about parameters in any linear model. It is necessary only that a hypothesis be expressible as a linear constraint on model parameters and that it satisfy a general "testability criterion" [see, e.g., Graybill (1976) and Searle (1971)]. Here we describe three hypotheses of general interest and identify equivalent expressions for them in terms of parameters in each of the models mentioned above.
88
Chapter 6 Design of Experiments TABLE 6.3 Equivalent Hypotheses for Average Main Effects and Interactions for a Two-Factor Crossed Design a
6.2.5.1. Average Main Effects It often is of interest to know whether one factor has an effect when averaged over all levels of other factors. To illustrate, consider the effect of factor A when averaged over all levels of B, recognizing that this situation is entirely analogous to the effect of B when averaged over the levels of A. For factor A the hypothesis of interest is
Ho"
Fi. = Fi'.,
where ~i. is the average of cell means for level i of factor A. Using the crossed herbicide experiment as an illustration, we may wish to examine differential weight changes for treatments with herbicide A when averaged across all levels of herbicide B. The relevant hypothesis is No"
(I,1,11 q- I-1,12 if- I-1,13)/3 = (],1,21 nt- I-1,22 if- 1,1,23)/3.
Differential effects for levels of A across all levels of B address what typically (but not exclusively) is identified as the "main effect of A." It is termed here an average main effect to emphasize that it is indeed an average effect. Note that the hypothesis does not "average out" the effects of B, because the possible effect of interactions between A and B is still present. Nor does the hypothesis specify that factor A has no influence on the pattern of cell means, again because of the possible influence of interactions. In fact, the hypothesis specifies nothing more than the equivalence of row averages. Stronger assertions concerning the complete lack of influence of factor A require different, more complex hypotheses. These points often are overlooked by the users of analysis of variance. The hypothesis for average main effects can be expressed in terms of the RPM by means of Eq. (6.3) as No:
=
and from Eq. (6.5) it is characterized in terms of the FPM by m
H 0.
oLi + Fi. = oti,
+
Fi, ..
The latter form expresses directly the influence of interactions, thus alerting the investigator not to overinterpret H 0. Again it is emphasized that all three expressions for H 0 are equivalent, though their mathematical forms differ (Table 6.3). 6.2.5.2. Interactions
Another hypothesis of general interest concerns the relative effect of factor B as influenced by factor A (or vice versa). The hypothesis of interest is
Ho" P~ij-
P'i'j = [l'ij' -- P~
CMM b
RPM a
FPM c m
A effect
~i.--
B effect
P-.j = P,.j'
Interaction
~l, ij -
[/,i'.
~
+ Fi.
! =
~
q-
m
~Lij' = ~ i ' j -
~.l,i'j'
Fi'. --
. Ot i - -
0
:6
~i + F.j = [3j, + F.j,
~j = 0
Fij
F~
= 0
= 0
a Expressed in terms of parameters in the cell m e a n s model, the fully parameterized model, and the restricted parameterization model. It is a s s u m e d that there are no e m p t y cells and that i 4: i'
and j 4: j'. bCMM, Cell means model. cFPM, Fully parameterized model. e RPM, Restricted parameterization model.
specifying that there is no interaction. The issue here is whether differences between means for two levels of factor A are specific to the level of factor B. If so, the pattern of cell means is said to display interaction. The interaction hypothesis specifies that there is no interaction. From Eq. (6.4) an equivalent form for the RPM is H0:
F~. = 0,
and Eq. (6.6) yields the analogous form for the FPM: H 0" F/j = 0 (Table 6.3). Hypotheses for average main effects and interactions are of course only three of many hypotheses that could be addressed with a crossed design. However, they do address structural features of a completely randomized experiment that usually are of interest to biologists, and they typically are the hypotheses intended when "main effect" and "interaction" tests are conducted. They also are the hypotheses tested by many standard computing packages (Milliken and Johnson, 1984; Williams, 1987).
6.2.6. H y p o t h e s i s T e s t i n g
Testing procedures for the multifactor randomized design are based on sample contrasts of treatment means. For example, a test of the average main effect of A in a two-factor crossed design is given in terms of the contrasts
J
J
J
J
6.3. Randomized Block Designs of average main effects. The mean of this contrast is
E ( ~ , Yij. - E J
Yi'j.) = E J
~l'ij -- E J
["Li'j J
= ~i.- ~i'., and its variance is given by
var(yi.--yi,.) = E (n!.j+ 1to2. j
Hi'j ,/
The experimental error 0"2 is estimated by the withintreatment variances according to ~2=
~_, (nq - 1)s 2q i,j n - ab '
where s i2:.is the sample variance for replicates within treatment combination (i, j), nq is the number of experimental units assigned to (i,j), and ~,i,j nij = n. An appropriate test of no treatment effect is based on the withinteatment error 42 (Graybill, 1976). Similarly, a test of interactions is based on the interaction contrasts ~ij -
~i'j -
~ij' -]- ~i'j' = Yij - Yi'j - Yij' -]- Yi'j'
with mean E(Yij.
-
Yi'j. - Yij'. q- Yi,j,.) -- ~ij-
Poi'j-
~ij' q- ],l,i'j'
and variance
var (Yij.
-
Yi'j. - Yij'. -}- Yi,j'.)
are found in randomized block designs. Blocking, like stratified random sampling, involves the stratification of similar subjects into groups (either by choice or necessity), potentially reducing experimental error and increasing the power of statistical tests. As with stratification, the blocking factor is assumed to be known, recognizable, and applicable for aggregation of experimental units. Just as stratification in sampling aims at reducing variation within strata, blocking in experimental design aims at reducing experimental error, leading to more precise treatment means and hypothesis tests. Finally, randomized block designs and stratified sampling both can be based on a simple stratification, as in geographic location, age, or species identity, or more complicated stratifications, as in the combination of age and species identity. In either case, treatments are randomly assigned to experimental units taken from each of the strata. Randomized block designs essentially describe a replicated experiment, in which the assignment of treatments within each block constitutes an experimental trial. Experimental errors for any trial can arise only from sources of variation affecting units within that trial. Therefore the error associated with treatment comparisons over a number of trials must arise from within the individual replicates. Systematic differences across trials do not contribute to these errors; thus, a proper analysis of variance removes block-to-block differences, retaining only the within-block variation as a basis for estimating experimental error for hypothesis testing (Cochran and Cox, 1957).
(ni,1 + ni'J1+ nij1+' ni'j1 ' t 2 6.3. R A N D O M I Z E D BLOCK D E S I G N S Randomized block designs are appropriate when a population of experimental units has recognizable structure and one can (or must) utilize that structure in assigning treatments. Properly used, randomized block designs can be effective in reducing experimental error, in the manner of a reduction of variance in stratified random sampling. Recall from Chapter 5 that stratification applies when the individuals of a heterogeneous population can be aggregated into more or less homogeneous groups, so that variation across the population is expressed primarily through group-togroup differences. In this way estimates based on stratum subsamples can produce more precise estimates of population parameters (see Section 5.4.2). Many features of stratified random sampling also
89
6.3.1. Restricted R a n d o m i z a t i o n of Treatments Randomized block designs are examples of restricted randomization, in that the allocation of treatments to experimental units is not completely random. In the case of randomized complete block designs, treatments are allocated within blocks so as to ensure that each block receives all treatment combinations in the experimental design. In this way it is possible to utilize treatment contrasts within the blocks to estimate experimental error and thereby to test hypotheses about treatment effects.
Example Considering the earlier herbicide example, suppose that the 30 stands are composed of 10 different combinations of species groups (pine vs. hardwood) and age (five age classes), with three stands in each combination (block). A total of 10 blocks is thus defined, with three replicate stands in each block. The three treatments (control, medium, and high levels of herbicide)
90
Chapter 6 Design of Experiments
are then assigned randomly to stands within the blocks, with the randomization occurring independently for each block. The experiment then proceeds as before, with nesting success estimated for each stand. Because there is a single stand per treatment in each block, tests of the average main effect of the herbicide treatment depend on the assumption of no interaction between blocks and the herbicide treatments (see below). Randomization within blocks can of course occur even if there are insufficient replicates within blocks to accommodate all treatment combinations. Randomized incomplete block designs describe the allocation of treatments under these conditions. The protocols for incomplete block designs (Cochran and Cox, 1957) still allow for inferences about treatment effects based on treatment contrasts, but the absence of all treatments in all blocks results in the loss of precision in parameter estimates, and the loss of power in hypothesis tests. An important consequence of restricting the random assignment of treatments is that they often can be applied more efficiently, especially if the blocking is spatially determined. This gain often, though not always, is coupled with a gain in precision attendant to reduced experimental error. However, ineffective blocking can actually lead to the loss of statistical power relative to that of a completely randomized design with the same sample size.
6.3.2. Statistical Models for Randomized Block Designs The CMM, FPM, and RPM all can be extended to allow for blocking. As with completely randomized designs, the mathematical expressions of these models differ substantially, even though they all represent the same statistical populations. Chief among the differences is the manner in which restricted randomization is handled via constraints on the model parameter structures. A consequence of restricted randomization is that it induces constraints on the model structure for block designs, with implications as to model representation and estimation. To illustrate, consider again the 2 • 3 layout, except design factor A now is a blocking factor representing three forest types, and the treatment consists of a herbicide to be applied at some appropriate level. Randomization of treatment and control is to occur within each forest type, so there are three independent randomizations, one for each of the forest types. Paired replicates are available in each forest type; thus, the treatment and control are both to be applied in all three forest types. Every cell in the 2 • 3 layout therefore is repre-
sented, and the layout is superficially similar to the 2 • 3 completely randomized design. However, an important difference between the two designs is that the randomization here only occurs across the treatment factor, but not the blocking factor. For the example, we let Yij represent the treatment response for block i, i = 1, 2, 3, and treatment j, j = 1, 2. The restricted randomization and lack of replication require us to use the contrasts Y i l - - Yi2 as replicates for the difference between treatment and control, based on the assumption that there is no treatment by block interaction. For the 2 • 3 crossed design, a lack of interaction is expressed by --
~12
--- ~b21 - -
~22
(6.7)
~/'21 - -
~22
--
~32"
(6.8)
~11
and ~1'31 - -
But these restrictions in turn imply that the cell means of the CMM are not independent, i.e., bt'21 - -
bit'11 - -
~12
q- ~ 2 2
~31
nt- ~ 3 2 "
and ~/'22 -~ ~ 2 1
--
Thus, knowledge of {[/'11, ~12, ~31, ~ 3 2 } is tantamount to knowledge of the full set of six cell means. From Eqs. (6.7) and (6.8) we have treatment contrasts that are constant across blocks, i.e., ~/,il - - ~/'i2 = ~ for i = 1, 2, 3. It is easy to show that this condition is satisfied if and only if the cell means can be represented as an additive combination of row and column factors: ~/.q--
~1, q
O~i -}- ~ j .
An additive structure for the cell means is confirmed through the effect of the interaction constraints on the FPM and RPM. In particular, the interaction constraints can be shown to correspond to the constraints Fq = 0 and F i~" - - 0 in the FPM and RPM, respectively. Thus, the appropriate model is
Yij
= ~ + ~
if- ~j if- ~.ij,
(6.9)
where o~i now expresses the effect of a blocking factor A, ~j expresses the level of treatment factor B, and ~q is the error associated with block i and treatment j. This model is similar to the two-factor model in Eq. (6.2), except for the lack of replication, the absence of any interaction terms, and the identification of A as a blocking factor. Because of its simpler representation when there are constraints, we utilize the FPM to represent randomized block designs. Model (6.9) generalizes naturally to situations with crossed blocking structures and crossed treatment
6.4. Covariation and Analysis of Covariance structures. For example, the factor o~i could easily be replaced by OLi q- "Yk q- (O~'~)ik representing the ith and kth levels of crossed blocking factors, and ~j could be replaced by ~j + ~1 + (~8)jl, representing a treatment structure with two treatment factors. However, the key feature in any application is the lack of an interaction between treatments and blocks.
6.3.3. Estimation and Testing The assumed lack of interaction between blocks and treatments is necessary to identify an experimental error for testing treatment effects. In the example above, the contrasts Yil - Yi2 for the three blocks serve as replicates for the difference between treatment and control. Assuming the absence of a block by treatment interaction, the contrast replicates can be used to test a hypothesized difference. This follows directly from model (6.9), because Zi--
Yil -- Yi2
-- ( ~ if- OLi q- ~1 if- 8il) -- ( ~ if- OLi if- ~2 q- 8i2) = (~1 -- ~2) if- (8il -- 8i2),
and therefore
E(zi)
= ~1 -- ~2"
Then the average ~_j Z i / 3 = ~_, (Yil -- Y i 2 ) / 3 i i
of block-specific contrasts can be used to estimate the treatment contrast ~1 -- ~2, and the sample variance O.z'2 = ~ ( Z i i
_ 2)2/2
can be used to test the hypothesis of no treatment effect. In the event that the "no-interaction" assumption is incorrect, systematic block-to-block differences are included in the error term, which then is inflated to the detriment of statistical power in the hypothesis test. If one determines that there is a nonzero interaction [see, e.g., Graybill (1976) for the appropriate statistical test], then it becomes necessary to replicate treatments within blocks in order to obtain an unbiased estimate of error. Assume, for example, that paired treatment replicates are obtained within each block in the randomized block design above, with the replicates represented by Zi(k) -~ Y i l ( k ) - - Y i 2 ( k ) . Here the subscript k represents the kth replicate of the contrast within
91
block i. The replicate contrasts Zi(k) n o w to estimate experimental variance by ~.2=
i
can be used
~(Zi(k ) _ ~i)2 k
~,ni_ i
3
"
where Zi is the average of the treatment contrasts within block i and n i is the number of replications in block i. Because it is based solely on contrast variation within blocks, this estimate is absent any assumption about treatment by block interaction. Of course, a substantially larger experiment is necessary to obtain it.
6.4. C O V A R I A T I O N A N D A N A L Y S I S OF C O V A R I A N C E Along with blocking, the use of covariates is an effective approach to reduce experimental error. A covariate x can be useful for reducing error when there is a mathematical relationship between the response y and x such that the conditional distribution of y given x has smaller variance than the distribution of y alone. To illustrate the idea, consider a single-factor completely randomized design in which three herbicide treatments are to be applied in an experiment to examine herbicidal effects on forest-dwelling organisms, with weight change as the response. The analysis of data from the experiment might proceed as before, based on the design model for a single treatment factor (see Section 6.2.1). But assume that the investigator records several measures of organism size at the beginning of the study, including organism length. On assumption that organisms of the same initial weight but different lengths respond differently to the herbicide treatments, the investigator can include organism length as a covariate, anticipating that the overall effect of the herbicide can be measured more effectively by adjusting for organism size. Basically, the size covariate is used to eliminate extraneous variation in weight changes that otherwise tends to mask the effect of the treatment. The covariate essentially "adjusts" the response to account for organism length, after which the test for an effect of herbicide treatment can proceed by way of a standard analysis of variance.
6.4.1. Statistical Models for the Analysis of Covariance A statistical model for covariates analysis involves conditional probability distributions, described here in terms of the bivariate distribution of a response variable y and a covariate x. Random sampling yields a
92
Chapter 6 Design of Experiments
random bivariate variable [x y]' with a joint probability distribution characterized by mean [~x ~y]' and dispersion = I ~2 r
=
2_
0- y
2/r
(6.11)
0- xy
(see Section 4.1.6 for a discussion of conditional distributions and their parameters). The presence of nonzero covariance 0-xybetween x and y ensures that the conditional variance 0-~jxis less than the unconditional variance 0-~. Under these conditions the use of a covariate can improve the precision of estimates and the power of hypothesis tests for the experiment. Allowing ~x(i), ~y(i), and E(ylxir to represent the means of x, y, and y lxir respectively, for the ith treatment, the appropriate model for the analysis of covariance is
Yir
E(ylxir
+ Gi(j)
--- {~y(i) +
(0-xy/0-
2 x)[Xi(j)
-
~x(i)]} if-
~'i(j)
or
Yir
(0-xy/0-2x)[Xi(j)-
~x(i)] = ~y(i) + ~ir
Thus, the covariance model reduces to a simple randomized design model, but with a new response variable
Yi(j)
--
(0-xy /
0- 2x)EXi(j)
_
(@xy/~2)[xi(j)
~(i)l
-
Yi(j)
--
~..(Xi(j ) _ l,j
-- Yi.)
~i.)2
(6.12) m (Xi(j) -- Xi.)"
(see Section 9.1 for a discussion of model fitting with least squares). The design model then becomes
zi(j) = ~y(i) + ei(j), (6.10)
~x~
and variance 2=
= Yir -
z,I
~y + (0-xy / ~ x)~Xi(j)
0- y lx
zir
~..(xi(j) - -Xi.)(Yir
r ~"
It can be shown that the conditional distribution of y Ix (i.e., the distribution of y when x is restricted to a specific value xy) has mean
E(ylxy) =
the method of least squares, to produce the new response variable
and the analysis of variance proceeds in the standard way for a one-factor design model. The procedure for analysis of covariance thus can be decomposed into two steps. The first step uses leastsquares estimation to estimate the linear relationship shown in Eq. (6.10) between the response and covariate and then fashions the adjusted response shown in Eq. (6.12). Step 2 uses the standard methods for analysis of variance to complete the parameter estimation and hypothesis testing for the design model, based on the adjusted response. Computer programs for analysis of covariance typically combine these two steps into a single analysis. There are additional costs associated with the monitoring of covariates as well as responses on experimental units, and the means, variances, and covariances in the relationships above must be estimated with sampling data so as to enable the adjustment of responses. The benefits of including a covariate in the analysis must be balanced against these additional burdens. Often, however, the marginal cost of recording a covariate in an experiment is minimal when compared to the cost of collecting the experimental units. As a general rule, it is appropriate to include covariates when the reduction in variance compensates for the loss of precision in estimating additional parameters in the conditioning functions. Scheff6 (1959) discusses further the relative efficiency of analysis of covariance.
~x(i)].
From Eq. (6.11), the experimental variance for this new 2 2 2 model is 0-y - 0-xy/0-x.
6.4.2. P a r a m e t e r E s t i m a t i o n a n d T e s t i n g
Parameter estimation and testing with covariates involves the measurement of both a response variable and a covariate on each experimental unit in an experiment and the use of these values to estimate the means and variances of the covariance model. The linear relationship shown in Eq. (6.10) is estimated by means of
6.5. H I E R A R C H I C A L D E S I G N S
The designs discussed thus far have involved the assignment of treatments to a sample of experimental units by means of a single randomization, either experiment-wide or within blocks. Though we considered designs with multifactor treatments and incorporated restrictions on the assignment of treatments in randomized block designs, in each case we assumed that an experimental unit is assigned a treatment, or a combination of treatments, through the process of a single randomization.
6.5. Hierarchical Designs We generalize the standard application of treatments in this section, by allowing a hierarchical assignment of treatments, as when different treatment factors are applied at different spatial or temporal scales. The application of treatments is thus separated into two distinct components, each with its own randomization, so that additional random factors must be included in the corresponding statistical models. As a result, experimental error must be handled in a somewhat different manner than with completely randomized and randomized block designs.
6.5.1. Split-Plot Designs A frequently used design in life sciences involves the division of an experimental unit into subunits, as in a plot of land being divided into subplots, with certain treatment components applied to the entire unit and other components applied to the subunits. Designs of this sort often are referred to as split-plot designs, because they are frequently applied in agricultural and land management experiments involving units of land. The experimental units in such an experiment are called whole plots to signify the larger entity to which some treatment is to be applied, with the subunits called split-plots to signify the fact that each of them is obtained by "splitting" a whole plot into parts. Every whole plot is assigned a whole-plot treatment and also is assigned the split-plot treatments over some part of it. Every split plot within a whole plot is assigned the whole-plot treatment and also is assigned one (and only one) split-plot treatment. Implicit in the split-plot design is the idea that different treatments are appropriate at different scales, but the investigation of treatments across scales can be combined into a single experiment. The design is most applicable across geographic scales, wherein larger units of land can be subdivided into smaller units.
Example A simple example of a split-plot design is provided by the application of herbicide to a study area. Suppose that herbicide A and a control are randomly assigned to 10 stands. Each stand is divided into three subunits, which then are assigned at random one of three experimental treatments (control, medium, or high herbicide levels) for herbicide B. Thus, the stands are experimental units for a whole-plot experiment that focuses on herbicide A, but they also are part of an experiment involving the application of herbicide B to subplots. The subplots are experimental units for a split-plot experiment that focuses on herbicide B, but they also are part of an experiment involving the application of herbicide A.
93
Like the randomized block design, a split-plot design involves a restriction on the random assignment of treatments to experimental units. Though a completely randomized design is used in this example to assign whole-plot treatments to the plots, and a randomized design also is used within each plot to assign splitplot treatments to the subplots, the overall assignment of the combination of whole-plot and split-plot treatment factors clearly is not completely random. Because the design calls for all split-plot treatments to be assigned to every whole plot, at the scale of the subplot the design bears a resemblance to a randomized block design, with whole plots playing an analogous role to blocks. The difference, of course, is in the random assignment of whole-plot treatments, which adds another scale of experimentation in split-plot designs that is not shared with randomized block designs. 6.5.1.1. Statistical Models
Because there are two levels of randomization in the assignment of treatments, two random factors are required in the statistical model for split-plot designs. To simplify notation, we assume that a completely randomized design is used for whole plots, with a single treatment factor A that is represented by o~i with a treatment levels. We also assume a single treatment factor B for the subplots, represented by ~k with b levels. Then the appropriate model is
Yqk = I~ + oLi q- Tli(j) if- ~k q- Fik + 8ijk,
(6.13)
where Fik represents the interaction between wholeplot and subplot treatments. The distinctive feature of this model is the presence of an additional random factor T]i(j ) corresponding to the random assignment of whole-plot treatment i to plot j. Note that the subscript for the error term Tliq) includes parentheses, whereas the subscript for 8ijk does not. Parentheses are used here to connote the nesting of one factor within another, as with the nesting of replicate j within the whole-plot treatment i. Because the same split-plot treatments are applied in all whole-plot replicates, they are not nested within them. Hence the use of ~ijk for the subplot error. 6.5.1.2. Estimation and Testing
An intuitive estimator for the contrast of wholeplot treatment levels utilizes the average of subplot treatment responses
~tij. = ~k Yijk /b = ~ + c~i + "qi(j) + ~. + Fi. + -~ij..
94
Chapter 6 Design of Experiments
When averaged across all replicates, the contrast n
Yi..-
Yi'.. = (IX if- Ri q- ~i. q- ~. if-- (t x + ~ = [(OLi q-
Fi. + ~i..)
+ ~i'. + ~. + Fi'. + ~i'..)
That this is an unbiased estimator of the interaction can be seen from E(yi.k-
Yi'.k- ~[i.k' q- Yi'.k') -- (~ + ~k q- Fik) -- (O~i'-}- ~k -}-
Fi.) - (o~i, -}- Fi,.)] -}- [(~i. q- ~i..)
-- (Oti-Jr- ~k' q-- (~i'.
q- ~i'..)]
-}- (OLi' q- ~k' q- ri'k')
of these responses provides an unbiased estimate of the whole-plot effect: Yi'..) = ((xi + Fi.) - (o~i, + Fi,.).
E(yi..-
Note that the whole-plot treatment difference expresses the average main effect of A, averaged over all levels of B. The appropriate test for the absence of a whole-plot treatment effect is based on the whole-plot error, expressed in terms of the average of subplot treatment responses Yij.:
Fi'k) Fik')
= F i k - Fi, k - Fik, + Fi,k,. The appropriate test for the absence of an interaction is based on the subplot error
,, 2
E E ( Y i j k - Yi.k )2 i,k j n (nij-
ab
1)s 2ik
i,k E
E i
( Y i j . - Yi..)2
j n --a
E =
(hii H --a
1)$2i
n -
,
(6.14)
where n i is the number of replications of treatment i and n = ~i ni is the total number of whole plots. Note that this testing procedure is nothing more than the procedure for a completely randomized design, based on the whole-plot response variable yq.. On the other hand, a test of the split-plot treatment effects relies on a different error term. An intuitive estimator for the difference among subplot treatment levels is based on the average of the differences Y i j k - Yijk ' = [(~k + Fik) -- (~k' + F/k')] + [~'ijk- gijk'] for replicate j of treatment i. Then the contrast ( Y i j k - Yijk') -- (Yij'k- Yij'k') = (~k -- ~k') q_ [(E,ijk n E,ijk,) __ (~'ij'k- E'ij'k')], across replicates j and j' within treatment i has an expected value of 6k -- 6k'. The average of these differences across all replicates for all treatments i provides an estimate of the split-plot treatment effect. The appropriate test for the absence of a subplot treatment effect is based on variation of these replicate contrasts within a treatment. The testing procedure mimics the procedure for a randomized block design, where the whole plots within a treatment serve as blocks in the estimation and testing of subplot treatment effects. Finally, a test of the interaction is based on yet another error term. An intuitive estimator for the contrast Fik - Fi, k - Fik, + Fi, k, averages the differences Yijk -Yi'jk -- Yij'k + Yi'j'k across all replicates, to produce the estimator Y i . k - Yi'.k -- Yi.k' nu Yi'.k'"
ab
"
properly adjusted to account for sample sizes in the contrasts.
6.5.2. Crossover
Designs
Another scale-related assignment of treatments to experimental units involves a temporal rather than spatial scale. Here an experimental subject receives multiple treatments in a predetermined sequence, with the idea that the sequencing of treatments can induce systematic variation in treatment responses. In some instances systematic variation is considered a nuisance; in others it is an important element of the treatment response. Designs that account for systematic response to the sequencing of treatments are called c r o s s o v e r d e s i g n s , so named because the sequencing of treatments is crossed in the design (i.e., some subjects get treatment A followed by B; some get B followed by A). The key element of crossover designs is the random sequencing of different treatments (e.g., a drug and placebo control) to the same subject. Crossover designs eliminate effects of variation between experimental units, by assigning all treatments to each of the units. However, other problems arise in the form of carry-over effects, which occur when the effect of a previously applied treatment has not worn off by the time a later treatment is applied. If the lingering effect of an earlier treatment interferes with the response of the unit to a later treatment, then there is a residual or carry-over effect on the later treatment, which should be accounted for via experimental design. Thus, the crossover design includes a treatment effect, time effect, carry-over effect, and two random
6.5. Hierarchical Designs terms, one for replication and one that accounts for treatment sequencing. Example
An investigator wishes to determine the effect of diet on the behavior patterns of small mammals. Two different diets are to be investigated, representing different ecological settings. From a pilot study it is known that the behaviors of individual organisms vary widely, and this variation potentially could mask any effect of diet. To control for the organism-to-organism variation, a crossover design is to be used for the investigation, recognizing the potential for carry-over effects from the first diet to the second. A group of 20 organisms is to be randomly divided into two groups of 10 subjects per group, with group 1 receiving diet 1 followed by diet 2 and group 2 receiving diet 2 followed by diet 1. The behaviors of individuals are to be recorded for a fixed period of time while each is on the first diet, and immediately thereafter the subjects are to be shifted onto the second diet and their behaviors observed over another period of equal length. The data for the study consist of paired behavioral responses for each animal, along with an indicator specifying which diet was applied in period 1 and which was applied in period 2. Analysis of these data includes the estimation and testing of treatment effects as well as carry-over effects.
95
for n 2 subjects receiving treatment 2 prior to treatment 1. Here o~i represents the effect of treatment level i, Tt represents the effect of time t, h i represents the carryover effect as a result of applying level i initially, T]ik represents the kth replicate subject receiving treatment level i initially, and g,itk represents random error for subject k receiving treatment level i at time t. Note that the model includes a carry-over effect only for the second time period, as there is no possibility of carryover in time 1 because no treatment has yet been applied to the subject at that time. Note also the presence of two error terms, one (Tlik) that establishes a statistical association between the two measurements on a given subject and o n e (E,itk) that is associated with the response to individual treatments and is independent of the other error term. Several contrasts are potentially of interest, based on the differences (Y22k -- Y11k)/n1 = Y 2 2 . - Yll.
s k
= (~
- ~
q- ( ~ 2 2 . -
The simplest crossover design involves the application of a treatment and control that are to be applied to each subject in the experiment. The pair of responses for the treatment and control are correlated, because both are measured on the same subject. The level of response is assumed to depend on whether the treatment is applied prior to the control or subsequent to it. For this experiment the subjects essentially play the role of blocks, with treatment and temporal factors jointly assigned to the subjects. We use Yitk to represent the response of subject k to treatment i applied at time t. Assuming two treatments and two times, the corresponding model is given by
Y11k = ~ q- OL1 -+- T1
-+- Tllk q- 811k,
Y22k -- ~ q- O~2 q- 3"2 nt- ~'1 nt- T]lk q- 822k,
(6.15)
for n 1 subjects receiving treatment I prior to treatment 2, and
Y21k -- ~ q- OL2 q- T1
( Y 1 2 k - Y21k)/n2 = Y 1 2 . - Y21.
s k
-" (OL1 -- 52) if- ('I"2 -- "I"1) q- ~'2 ~21.)"
For example, the sum of these contrasts can be used to investigate temporal effects: ( Y 2 2 . - Y11.) if- ( Y 1 2 . - Y21.) -- 2('r2 - "I"1) q- (K1 q- ~'2) q- ( ~ 2 2 . - 811.) q- ( ~ 1 2 . - 821.),
with E ( Y 2 2 . - Yll. q- Y 1 2 . - Y21.) = 2(T2 -- "1"1) q- (K1 -ff ~.2)
and variance
var(Y22.- Y11. if- Y 1 2 . -
Y21.) "-
(6.17) 2[ 1 q _ ~ ]
2 0"~~
/'/1
The difference between the contrasts can be used to estimate treatment effects, by ( Y 2 2 . - Yll.) -- ( Y 1 2 . - Y21.) -- 2(~ q- ( ~ 2 2 . -
-- ~
~11.) -- ( ~ 1 2 . -
if- (~'1 -- ~-2)
~21.),
with expected value
q- T]2k q- G21k,
Y12k = ~ q- 0~1 q- T2 q- )k2 q- "q2k q- G12k,
~11.)
and
-+- ( 8 1 2 . -
6.5.2.1. Statistical Models and Parameter Estimation and Testing
q- ('I"2 - "rl) q- K1
(6.16)
E(Y22.- Y 1 1 . - Y12. q- Y21.) = 2(0t2 -- Ogl) q- (~1 -- ~-2)
96
Chapter 6 Design of Experiments
and the same variance shown in Eq. (6.17). Tests for treatment and time effects thus require an estimate of 2 which can be expressed in terms of the differences Zlk -- Y22k -- Y11k for subjects initially receiving treatment 1 and the differences Z2k = Y12k -- Y21k for subjects initially receiving treatment 2. Because the withinsequence sum of squares for Zlk and Z2k has expected value
[
E E(Zik-
zi.)2/(Yll q-
n 2 -- 2)
i,k
]
= 2or2,
the mean square on the right-hand side of this expression can serve as an error term for tests of treatment and temporal effects. On the other hand, carry-over effects can be investigated via the aggregate responses for each subject, because E(Yllk k
+
Y22k)/n1
--
E(Y21 k + k
Y12k)/n2
= (Y11. q- Y22.) - (Y12. -}- Y21.) = (hl -- h2) + 2 ( ~ 1 . - ~2.)
Example
if- (811. if- 822.) -- (821. if- 8"12.) with expected value E(Y11. + Y 2 2 . - Y 2 1 . - Y12.) = )kl - )k2 and variance
var(Y11. + Y22.- Y21.- Y12.)
= 2[n~+ 1--](2cr2 n2 + if2)" A test for the carry-over effect thus requires an estimate of 2~ 2 + (r~, 2 which can be expressed in terms of the aggregates Wlk = Y22k + Y11k and W2k = Y21k + Y12k for subjects in the experiment. It can be shown that
E[~,
(Wik--
Wi.)2/(//1-}-//2--2)]
i,k
subsequent to treatment. For repeated-measures designs, time is a design factor, albeit not one that is applied randomly. The design can be thought of as a standard experiment with multiple (temporal) responses for each unit, the goal of which is to investigate temporal pattern in the response through the estimation of pattern parameters and the testing of hypotheses about them. Response trends for individual experimental units can be used to estimate treatmentspecific trends and experimental error, and thus can be used to test hypotheses about treatment effects. The distinction between repeated-measures designs and crossover designs is that with repeated measures, an experimental unit is subjected to a single treatment rather than multiple treatments, so that the temporally indexed responses for a particular experimental unit all correspond to the same treatment. A typical experiment might involve the random assignment of various levels of some treatment to replicate experimental units and the tracking of unit responses over time. Thus, each experimental unit receiving treatment level i has corresponding to it a vector y'q = [Yijr..YijT] of temporally ordered responses.
(6.18)
= 2(2o.12 + (~2), so that the mean square in this expression can serve as an error term for the carry-over test.
6.5.3. Repeated-Measures Designs Repeated-measures designs focus on temporal responses to treatments. In this case treatments are randomly assigned to experimental units, and unit responses to the treatment are recorded at several times
An investigator is interested in the capacity for learning about food availability as a function of the pattern of stimulus signals, recognizing that the clearer and more consistent the linkage between an auditory or visual stimulus and the availability of food, the more rapid will be the rate of learning about that linkage. The study is to focus on the pattern of association between stimulus and response, with the idea that variation in responses over time under different stimulus regimes is indicative of differential learning capacity. The experiment involves the capture of organisms from a target population and their random assignment to individual pens. Treatments consisting of different patterns of stimulus then are assigned randomly to the organisms, and individual responses to the stimulus are recorded each day over the 5-day duration of the investigation. The treatment structure consists of an auditory stimulus only, a visual stimulus only, a combined auditory and visual stimulus, and a control. In each case the signal is to be followed by immediate availability of food, and the response consists of the length of time required for an individual to begin feeding. It is anticipated that the response time will decrease over the study irrespective of the stimulus, but that the rate of decrease will vary depending on the pattern of the stimulus. The data for the study consist of a vector of response times for each animal, one response time for each day of the study, along with an index for the stimulus regime for that animal.
6.6. Random Effects and Nested Designs
6.5.3.1. Statistical Models Given that there is no randomization associated with the temporal factor in a repeated-measures design, a straightforward way to model the design is in terms of the multivariate model Yi(j) = P~ if- ~i( j),
with E[Yijt] = ~ + OLi -}- Tt q- (O~T)it. Here the term oLi represents the average effect of treatment level i, Ct represents the average effect of time t, and (O~T)it represents the interaction of treatment and time. The dispersion ~ for Yi(j) consists of the covariances 0-t't = COV(Yijt, Yijt'), which express the fact that multiple observations are recorded on the same individual through time and thus are correlated. A simple and frequently assumed covariance structure for repeated measures is compound symmetry, expressed as m
m
1
p
p
...
p
p
1
p
p
1
... ...
p
p
p
9
o
0 -2
97
In the absence of compound symmetry, several approaches to the analysis of repeated measures have been developed. An approach described by Box (1954) essentially reduces the degrees of freedom for the associated test statistics, based on a measure of how far the covariance matrix deviates from compound symmetry. A more formalized approach is to treat the time vector of observations for each subject as a multivariate response and use multivariate analysis of variance to examine the data (Morrison, 1976). In the specific case of autocorrelated observations, it can be shown that an exact test of treatment effects (but not temporal effects) is obtained by the same testing procedure as with compound symmetry (Milliken and Johnson, 1984). For investigation of temporal effects, an approach described by Albohali (1983) utilizes a maximum likelihood estimate of p and least-squares estimates of the errors to "filter" the data, i.e., remove the autocorrelation between time intervals before carrying out a within-subjects analysis. Then the standard repeated-measures analysis for temporal effects can be carried out on the filtered data.
(6.19)
!
m
p
p
p
...
6.6. R A N D O M NESTED
1
where 0-t't is given by 0-t't = p0-2. A univariate statistical model that corresponds to this structure is Yijt = po + OLi if- TIi(j ) if- T t nt- (OLT)i t if- 8ijt,
with ~qi(j) a n error term for replicate subjects, and 8ij t a n error term that corresponds to the response of subject j to treatment level i at time t. With this model it is easy to show that the observations {Yijl, Yij2, ..., YijT} are all correlated, with a correlation structure as in Eq. (6.19). A comparison of this model and the model (6.13) for split-plot designs reveals that the two designs are essentially identical, so that the estimation and testing procedures for split-plot designs are applicable to the repeated-measures design without alteration. More complicated analyses are required when the covariance structure is more complicated. For example, autocorrelation among the repeated measures (see Section 10.8.4) leads to the covariance matrix 1
p
[32
...
pT-1
p
1
p
...
p2
p
1
...
pT-2 pT-3 0-
1
w
P
T-1
p T-2
~) T - 3
..i
1
2
_p2"
EFFECTS AND DESIGNS
In the designs discussed thus far, we have focused on treatment factors that are fixed and constant across the experiment. For example, a two-factor crossed design involves factors A and B, with the same levels of A applied for every level of B, and the same levels of B applied for every level of A. The constancy across treatment combinations is definitive of a crossed design. In this section we relax the requirement that design factors are crossed in an experiment and consider designs in which one factor is "nested" within another. A nested design has at least one design factor for which the factor levels are not replicated across the experiment. Thus, the nonreplicated treatment factor is essentially nested within other treatment factors.
Example To test the capability of three analytic facilities to meet established standards in their assessment of environmental contaminants, the facilities are sent samples of known concentrations of a contaminant. Each facility is to conduct five analyses on each of three analyzers at the facility. If factor A represents facility and factor B represents analyzer, a standard cross-tabulation of the data for this experiment shows five replications for each combination of A and B. The layout of the data might suggest a two-factor crossed analysis of variance; however, such an analysis would be inappropri-
98
Chapter 6 Design of Experiments
ate, because the same analyzers were not used at each facility. In fact, factor B is nested within factor A, in that the levels of B (the particular analyzers used) are specific to the level of A (the facility where the analyses are conducted). In many applications of nested designs, the levels of a nested factor may represent specific instances of a design factor, as with the particular analyzers in this example, and inferences from the experiment are made only to those particular instances. In such a case the nested factor is appropriately considered to be fixed for the factor levels, and the experiment is an example of a fixed-effects nested design. In other applications, the factor levels are thought to represent a population of potential levels, and the levels in the experiment are essentially replicate samples from a population of factor levels. In the latter case, the nested factor is appropriately modeled as a random design factor, and the experiment becomes an example of a random-effects nested design. In what follows, we consider nested designs with both random and fixed effects.
6.6.1. Statistical Models for Nested Designs To simplify notation we restrict our attention here to a two-factor design, in which factor B is nested within factor A. The statistical model for this situation is
which can be estimated by Yij. -" Ix + OLi + ~i(j) + -~ij."
The common variance ~2 for the model can be estimated by
E E(Yij(k)- Yij.)2 ~2=
k n - ab
It is easy to see that the differences between means within a treatment are given by Ixi(j)- Ixi(j') = ~ i ( j ) -
~i(j'),
which can be estimated by Yij. - ~]ij: -- ~i(j) -- ~i(j') + (-~ij. - -~ij:),
with variance ~]ij'.)(tl-~.]. +
[email protected],) 2
Far (~q.
A test of no difference among levels of factor B is based on the estimated variance =
On the other hand, contrasts between levels of treatment A are based on the averages
Yijk = Ix + ~ + ~i(j) + ~'ij(k)"
Here we use a fully parameterized model to represent the design because it allows for a simple display of the nesting of factors. Thus, o~i represents the ith level of design factor A, ~i(j) represents the jth level of factor B nested within the ith level of factor A, and 8ij(k ) represents the random error for replicate k of the combination (i,j) of factors A and B. On assumption that factor B is fixed, the levels ~i(j) a r e nonrandom and there is only the single random variable eij(k) in the model. If, however, ~i(j) is considered random, then there are two random variables in the model, with implications for analysis that are highlighted below. In either case the notation makes clear the nesting of factor level j within i and the nesting of replicate k within the combination (i,j).
i,j
Yi.. = Ix + ffi + ~i. + ~i.., with E(~i..) = tx + oq + ~i.
and v a r i a n c e o2/ni... Then the contrast between level i and i' of factor A is estimated by m
D
Yi..- Yi:. = (0~i + ~i.) -- (0Li' + ~i:) + ( 8 i . . - ~i:.), with variance var(~i..- ~i:.) = (n@.. + 1 ) 0 . 2 . A test of no difference among levels of factor A is based on the estimated variance va"? (Yij.
6.6.2. Estimation and Testing---Fixed Effects The nested model for fixed effects has expected value Ixi(j) = Ix + OLi + ~i(j),
Estimation and testing with the fixed-effects nested model thus proceeds in a straightforward way that is analogous to the treatment of the two-factor crossed design.
6.6. Random Effects and Nested Designs
6.6.3. Estimation and T e s t i n g ~ Random Effects
has expected value
To express the fact that levels of factor B are random, we alter the notation of the model slightly, replacing the fixed level f3ir with the random level ~ir Then the nested model Yij(k) -= P" 4- ~ 4- 'rli(j) 4- ~'ij(k)
(6.20)
for random effects has expected value ~i = ~ 4- OLi,
which can be estimated by Yi.. = ~ 4- O~i 4- ~i. 4- ~i..
with variance 2 var(yi..) = 0-02 / b 4- 0-~/ni..
To simplify notation, we assume here that the design is balanced in its fixed treatment factor, with b replicates for each treatment level. Contrasts between levels of the fixed treatment factor thus can be estimated by Yi..-
Yi:. = (OLi- O~i') 4- ( ~ i . -
99
~i:) 4- ( ~ i . . -
~i:.),
with
I~_, ~ nij(Yij. i
j
yi..)21 --
(6.24)
0-2[~/( o hi. _ ~. Ha ij/ni. )]+ 0-2[a(b]
1)],
a rather complicated linear combination of the two components. Indeed, every sum of squares involving the fixed treatment yields a combination of the components of variance, but only in special circumstances do they provide a form that is appropriate for a test of treatment effects. Milliken and Hartley (1984) discuss in some detail the estimation of components of variance in random-effects models and mixed models with both random and fixed effects and also describe procedures for constructing tests in multifactor experiments with unbalanced designs. We note that with balanced designs, for which nij = k for all treatment combinations, Eq. (6.21) becomes v a r (Yi.. - yi:.) =
20-o2 / b
4- 2 0-~2 / b k
= (2/bk)(k0- 2 + o.2) and Eq. (6.24) simplifies to
[ ~ ~, k(~tij _ E i Ja(b- 1)
E(yi.. - yi:.) -- o L i - OLi'
and
var,i
1l+ni
0-2 8o
(6.21)
A test of the hypothesis of no effect for the nonrandom treatment factor thus requires individual estimates of 2 and 0-~, 2 so that they can the variance components 0-~ be combined into the appropriate linear combination 2 can be for an unbiased test. As before, the variance 0-~ estimated by the within-treatment sum of squares --
-
i,j
2
Yij.),
(6.22)
k
because
E(62) = E[~i,j,
Z(Yij(k)k
Yij.)2 / (n -- ab) ]
2 = k0- 2O 4- 0-~.
(6.25)
Thus, for balanced designs, the across-treatments mean square can be used as an error term to test for differences among treatments. Unfortunately, no such convenient computing form is available for unbalanced designs (Milliken and Hartley, 1984). In addition to testing for treatment effects for the nonrandom treatment, it also is possible to test for effects in the random factor. A test of differences in the levels of the random treatment Tli(j ) in a mixed 2 = 0. T h e model essentially tests the hypothesis H0: 0-~ within-treatment and across-treatment mean squares shown in Eqs. (6.22) and (6.23) can be used for an unbiased test of this hypothesis, as a function of the ratio of the mean squares in these equations.
6.6.4. Associations with Hierarchical D e s i g n s
_ 0-2
However, the usual sums of squares for treatment ef2 and fects all correspond to linear combinations of 0-~ 2 2 that depend on the sample sizes, rather than on 0-0 0-n alone. For example, it can be shown that the sum of squares ~.~ Z i j
~i..)2]
Ylij(Yij. - ~i..)2
(6.23)
A comparison of models (6.13) and (6.20) reveals that the split-plot design is an example of a mixed model, with the added feature of a treatment structure imposed on the subplot replications. The whole-plot replications correspond to the random treatment in a mixed model, and the subplot treatments correspond to the mixed-model replications. Because the subplot
100
Chapter 6 Design of Experiments
treatments are imposed essentially as a r a n d o m i z e d block design, the subplot design is balanced, with the same n u m b e r of subplots for each whole plot. Under these circumstances the appropriate error term for whole-plot treatment contrasts is given by Eq. (6.25), which is identical (up to a constant) to Eq. (6.14). Likewise, crossover designs can be viewed as examples of mixed models, as is evident w h e n one compares models shown in Eqs. (6.15) and (6.16) with model (6.20). The sequencing of treatments can be viewed as a fixed effect, with the replicate subjects representing a r a n d o m effect. Measurements of the response to treatment levels on individual subjects correspond to the mixed-model replications. Because the subjects all have an identical n u m b e r of responses (in this case k = 2), the design is balanced within subjects and the appropriate error term for the contrast of carryover effects is given by Eq. (6.25), which is identical (up to a constant) to Eq. (6.18). Finally, on assumption that the covariance structure of observations on individual subjects follows comp o u n d symmetry, the models for repeated-measures designs are essentially the same as those for split-plot designs and thus can be thought of as examples of mixed models. As before, the treatments are assumed to be fixed design factors and imposed on the experimental subjects, which can be thought of as r a n d o m design factors. The appropriate error for testing treatment effects thus is given by Eq. (6.14) or (6.25), because these are identical (up to a constant).
6.7. S T A T I S T I C A L P O W E R A N D
EXPERIMENTAL D E S I G N Though varied in their model features and operative constraints, all the experimental designs above are formulated to describe the structural features of biological populations and to compare population models so as to determine which model is most appropriate. The ability to recognize structure with statistical design is directly associated with the notion of statistical power, described here in terms of potential errors in the contrasting of null and alternate hypotheses. Recall from Section 4.3 that a comparison of null and alternate hypotheses presents two w a y s in which a testing procedure can reach a correct decision and two ways that it can make an error (Fig. 4.5). The correct decisions are to accept H 0 w h e n it is true and to reject it w h e n it is false. Incorrect decisions are to reject H 0 w h e n it is true (type I error) and to accept H 0 w h e n it is false (type II error). Statistical power is defined in terms of type II error, in that it focuses on the ability of a test to reject false null hypotheses (or equivalently, to accept alternative hypotheses w h e n true).
Test power can be defined formally, if s o m e w h a t obscurely, as 1 - P(type II error). An equivalent and s o m e w h a t more accessible definition of power is simply the probability of recognizing Ha as true, w h e n it is. That these definitions are equivalent follows from the fact that conditional acceptance of a hypothesis is the complement of conditional rejection, i.e., Power =
is acceptedlH a is true) = 1 - P ( H a is rejectedlH a is true) = 1 - P(type II error). P(H a
(6.26)
To frame the issue of p o w e r operationally, consider an experiment involving a single treatment with m e a n i.gl and a control with m e a n i~2. Assuming a c o m m o n variance 0-2 for treatment and control populations, we use the sample m e a n s x 1 - ~-~j X l j / n l and 2 2 -~,.! x2./n2, along with the sample variance s2 = J
~,i,j(xi j _
~i)2 / (n I + n2 _ 2), a s e s t i m a t e s
of the relevant
population parameters. To simplify notation we assume below that sample sizes for treatment and control are equal, with n 1 = n 2 = n. Then a test of the null hypothesis H0:~1 - - ~ 2 z 0 against the alternative hypothesis Ha: laq - Ix,2= /~ is based on the test statistic y = n(~ 1 - - 2 2 ) 2 / ( 2 S 2 ) , which is distributed as central F with 1 and 2(n - 1) degrees of freedom under the null hypothesis (see Appendix E). An s-level test is to accept H 0 if y -< F1_~(1, 2n - 2) and to reject H 0 (and thus accept H a) if y > F1_~(1, 2n - 2), where FI_~ (1, 2n - 2) is the 1 - oL quantile of the F distribution with 1 and 2n - 2 degrees of freedom. This test has a type I error rate of e~u n d e r the null hypothesis, because the test statistic y exceeds the critical value Fl_~ (1, 2n - 2) cx proportion of the time by chance alone, even though the treatment and control populations do not differ in their means. One w a y to lower the Type I error rate is simply to choose a smaller value of oL, resulting in a larger value Fl_~(2n - 2) and thus in a reduced probability that y will exceed that value by chance alone. On the other hand, a consideration of the power of the test requires that we account for the difference/~ = I~ - 1~2, in addition to sample size n and test size o~. On assumption that H a is true, the test statistic y is distributed as a noncentral F, with noncentrality parameter 2
X=2(~- )
(6.27)
(see Appendix E). Thus, the m a g n i t u d e of X is increased w h e n e v e r the variance 0 -2 is reduced, the sample size n is increased and the difference A between the means of treated and control populations is increased. In Appendix E we discuss the effect of X on a noncentral F distribution, pointing out that the distribution is in-
6.7. Statistical Power and Experimental Design creasingly skew to the right for larger X. Thus, greater probability mass is found in the region beyond the quantile F1_~(1, 2n - 2), thereby increasing the probability of rejection of H 0. It follows that the power of the test is enhanced when (1) hypothesized parameter values in H 0 and H a are quite different from each other (i.e., & is large); (2) the underlying sample distribution for the test statistic has low variance (i.e., 0 - 2 is small); a n d / o r (3) the testing procedure is based on large samples (n is large). These factors combine to make more likely the rejection of H 0 (and thus the acceptance of Ha). In particular, large differences in &, as specified under hypothesis H a, lead to a greater tendency to accept H a, i.e., to greater test power. In general, an attempt to guard against type I errors by increasing the significance level of the test increases the exposure to type II errors. For instance, increasing the size of the confidence interval (i.e., by decreasing the interval significance level) makes it more likely that a test parameter will be included in the interval when H 0 is true. However, larger confidence intervals also are more likely to include the parameter even if H 0 is false. Thus, a decrease in the probability significance level increases the chances for a type II error at the same time that it decreases the chances for a type I error (Table 6.4). Thus, one must account for tradeoffs between the two types of error when establishing the significance level of a hypothesis test. From the discussion above it is clear that, with a given rejection criterion for the test, both types of error can be reduced by
T A B L E 6.4
T y p e II Error and Power for Various Effect Sizes (Mot) for a S t a n d a r d Normal Test a
Ot
Z~
0.025
1.96
0.05
0.10
1.64
1.28
~k/or
~b
1-
~c
0.5
0.93
0.07 0.17
1
0.83
2
0.48
0.52
3
0.15
0.85
0.5
0.87
0.13
1
0.74
0.26
2
0.36
0.64
3
0.09
0.91
0.5
0.78
0.22
1
0.61
0.39
2
0.24
0.76
3
0.04
0.96
a Based on the normal distribution N(0, 1). b Probability of accepting null hypothesis when the alternate is true. c Power of the test.
101
increasing experimental sample sizes, so that a strategy to protect against both types of errors is to set the probability significance level low (protecting against type I error), but to sample intensively enough to control against type II error.
6.7.1. Determining Sample Size Based on Power The power of a statistical test sometimes can be calculated using formula (6.26). Consider, for example, the test described above for the difference of treatment and controls means, with n I = r/2 = n. The critical value for an c~-level test is F1_~(1, 2n - 2), which is available from standard tabulations of the central F distribution. Then the power of the test is given by Power = 1 - P(accepting H0[Ha) (6.28) =
f ( x " 1, 2n - 2, X) dx, FI-~(1, 2n-2)
where f ( x : 2 n - 2, X) is the probability density function for a noncentral F distribution with 2n - 2 degrees of freedom and noncentrality parameter X. Because X = (n/2)[~/0-] 2, the specification of n, A, and 0- yields the value X and thus the particular density function for the study. Then the integral in Eq. (6.28) can be obtained from standard tables of the F distribution. Rather than determining power based on a known sample size n, one often wishes to determine n for a desired level of power. Assuming values for oL and 0-, one can determine the sample size necessary to achieve a specific level of power to detect the difference A between means. From Eq. (6.27), one can calculate a value for X for a given value of n and then use the corresponding density function in Eq. (6.28) to compute the power of the test. Each sample size n generates a distinct value of power, allowing one to choose the sample size that is appropriate for the level of power that is desired (see, e.g., Cohen, 1977a). Example
An experiment is planned in which captive-raised mallards are to be randomly assigned, half to receive a sublethal dosage of an organochlorine pesticide, the other half to receive a placebo. The response to be measured is grams of loss in body mass following treatment. The null hypothesis of no effect is H0:P,1 - ~2, where ~1, ~2 are the mean mass loss for the treatment and control, respectively. A 10-g mean weight loss due to the treatment is considered biologically important; thus Ha: ~b1 - - ~1, 2 = / k = 10. A pilot study indicates an experimental error of 0- = 10 can be antici-
102
Chapter 6 Design of Experiments
pated for the study. For n = 10 the noncentrality parameter Eq. (6.27) provides n ( &)2
10(10~2
=5~
which is used in Eq. (6.28) to calculate power as
~oo
f(x "1, 18, 5.0)dx = 0.562.
d E 0 .95(1,18)
Similarly, power for n = 20, 25, and 30 can be computed as 0.869, 0.933, and 0.967; thus n ~ 25 is needed to provide power of 0.90 or greater. Distribution-based methods such as this provide exact values for power, under the assumption that the test statistic follows a known statistical distribution. However, in many instances a proposed test statistic has no known distribution (e.g., because distributional assumptions are not known or are suspected to be violated). Monte Carlo methods utilize the proportion of experiments that, if repeated under identical conditions, would result in rejection of a false null hypothesis. A general procedure for Monte Carlo estimation of power is as follows: 1. Specify a test statistic y = g(x), a critical value Yc for rejection, a distribution fa(X) for x under the alternative hypothesis and a sample size n. 2. Generate n pseudorandom values of x from fa(X) and compute y from these values. 3. Reject H 0 in favor of H a if y > Yc. 4. Repeat steps 2-3 for m trials. 5. Count the number of rejections r in m trials and estimate power as r/m.
Example A Monte Carlo experiment was performed to evaluate the power of the mallard dosing experiment, when the observations were generated from a mixture of normal and exponential distributions. This situation might arise when unaccounted heterogeneity occurs in the observed treatment response, for instance, because subjects are differentially detectable. In this example, application of the F-test in 10,000 Monte Carlo trials resulted in 5864 rejections for n = 25, indicating a power substantially lower than that obtained under assumptions of normality.
6.8. CONSTRAINED EXPERIMENTAL DESIGNS A N D QUASI-EXPERIMENTS The strongest inference to be obtained from experimental designs occurs when (1) there is a manipulative
treatment with experimental controls, (2) experimental units are assigned at random to the treatments and controls, and (3) the experiment is replicated, i.e., there are multiple experimental units in each treatment. Ecological experiments often require that compromises be made in one or more of these features, with the result that the ensuing experiment may be less than optimally effective in exhibiting biological patterns. For example, it may be impossible (or unethical) to assign experimental units at random to treatments, or there may be an insufficient number of units available to replicate the experiment. In some cases, the "experiment" is an event beyond the control of the investigator, and it may not be possible to anticipate the event and establish experimental units as controls. Nonetheless, in many situations, inferences still can be made about the effect of putative causal factors, even if the experiment is severely compromised in one or more of its design features. In these cases, however, investigators must recognize the inferential limitations that the design restrictions entail. The most frequent restriction is a lack of randomization, which occurs when treatment conditions are replicated in some manner, but the assignment of the experimental subjects to treatment groups (and controls) is beyond the control of the investigator. Under these conditions, inferences about the causal effects of the treatments are still possible, if responses corresponding to a control are available for comparison to the treatment units, both before and after the treatment occurs. However, a lack of randomization has the potential of confounding design factors with other unrecognized causal influences, leading to improper causal inference. Typically one thinks of an "experiment" as involving the deliberate manipulation of experimental units by an investigator. A more flexible definition of experiment (perhaps better termed a quasi-experiment) would allow for manipulations to be outside the control of the investigator, so long as temporal and spatial controls are available. Either spatial or temporal controls (ideally both) are key elements of an experiment (quasior otherwise), and the absence of control or baseline units essentially turns an "experiment" into what R. A. Fisher (1947) termed an "experience." Although it may be possible to use data collected in the absence of controls to make valid inferences, we believe that such "experiments" (including most so-called natural experiments) might be better classed as "surveys" or "monitoring studies" (Green, 1979). There is no hard and fast line between constrained or quasi-experiments and monitoring or sample surveys. In general, the former include some elements of classical experimental design such as control and
6.8. Constrained Experimental Designs and Quasi-Experiments replication, but may rely heavily on modeling or novel analyses for inference. Typically the strength of causal inferences is intermediate between purely descriptive (e.g., from monitoring) and causative (from classical experimentation). The goal of the investigator should be to incorporate in a study as many elements from classical design as feasible, recognizing that analysis and interpretation of results must account for the limitations of the study design. In many cases neither rigid application nor cavalier disregard of the "rules" for experimental design is appropriate or of practical value. A substantial literature is available on these issues as they pertain to ecological experimentation, and readers wishing a more thorough discussion of them are referred to texts such as Green (1979), Skalski and Robson (1992), Scheiner and Gurevitch (1993), and papers by Hurlbert (1984), Carpenter et al. (1989), and Eberhardt and Thomas (1991). We distinguish in what follows between two kinds of quasi-experiments, namely impact and monitoring (observational) studies. Impact studies are distinguished from observational or monitoring studies by the presence in the study design of a treatment structure. Thus, an impact study (Green, 1979), which seeks to determine the causal linkage between a particular factor (an "impact") and changes in a population or community, could otherwise involve classical experimentation and the design elements of control, randomization, and replication. In an impact study, the nature of the impact and the fact that it has occurred, or will occur, are both known. But it may or may not be possible to measure the system both before and after the impact occurs (temporal control), and an adequate number of subjects may or may not be available to act as controls. In contrast, the goal of a monitoring study is simply to detect change from some present (possibly baseline) state, and there is no a priori notion of a causative factor or impact under investigation. Impact studies can be particularly useful as a gauge against which to compare the results of monitoring and to provide a basis for monitoring so as to detect future changes.
6.8.1. Impact Studies An optimal impact study design (Green, 1979) has four prerequisites. First, the impact to be evaluated must not have already occurred, so that baseline data can be collected (temporal control). Second, the type, time, and place of the impact must be known so that sample data can be collected to test appropriate hypotheses. Third, relevant measures of biological and environmental variables must be obtainable from all the experimental units. Fourth, spatial controls must be
103
available from, e.g., areas, individuals, or other subjects not receiving the impact. The first element (temporal controls) and fourth element (spatial controls) merit further discussion. Randomization ordinarily will not be possible in an impact study, because the investigator does not control which experimental units receive the impact, even if (ideally) that information is available prior to the study. Thus, spatial controls are needed to avoid misinterpreting chance environmental influences as related to the impact: if units lacking the impact and units subjected to the impact both change in the same manner over time, some other factor than the impact is a likely cause of the change. Temporal controls are needed precisely because the experimental units usually differ from each other prior to the impact occurring. If this difference is measured only following the impact, then persistence of preimpact differences could be misinterpreted as signifying an impact. The natural analysis of an impact design is a treatment • time factorial design, with the test impact obtained by testing the null hypothesis of treatment • time interaction. This analysis can be investigated formally with statistical models and analysis of variance, but results often can be interpreted readily in graphical form, even where a statistical analysis is infeasible. For example, Fig. 6.1 illustrates possible impact study outcomes where there are two subjects (a treatment and a control) measured at two times (before and after impact). The parallel responses in Fig. 6.1 are clearly interpretable as "no impact" responses, whereas nonparallel responses are indicative of either a positive or a negative treatment effect. A major difficulty with this approach is that an optimal impact study design often will contain no true replication, so that an assessment of interaction effects by analysis of variance is not possible (Eberhardt and Thomas, 1991). We have already seen instances of experiments that lack replication--for example, a twofactor factorial experiment in which there is but one subject per treatment combination. Because hypotheses about interactions require an estimate of experimental error that is based on treatment replicates, the absence of replication renders the testing of the interaction terms impossible. One solution for a lack of replication involves randomized intervention analysis (Carpenter et al., 1989), in which randomization tests are used to derive the distribution of the test statistic under the null hypothesis. In randomized intervention analysis, a test statistic (e.g., based on pre- and posttreatment differences between experimental and control subjects) is computed. Subjects then are reassigned at random to pre- or posttreatment, and the statistic is recomputed. The random-
104
Chapter 6 Design of Experiments a
at some future time. In this way inferences are possible that, although weaker than under a classical design, are stronger than possible under a purely observational, retrospective analysis.
b
x(t)
Example
x(t)
x(t)
x(t)
tI
t2
t~
t2
F I G U R E 6.1 Graphical representation of possible results from an impact study with a single experimental unit that is treated and a single unit that serves as an experimental control, with measurement of a response x ( t ) on each unit prior to (t 1) and after (t2) the impact. (a) Preimpact measure on the control unit is greater than for the treatment unit. (b) Preimpact measure on the control unit is less than for the treatment unit. Rows 1 and 2 exhibit parallel time response between the treatment and control, indicating no impact. Rows 3 and 4 exhibit nonparallel time response between the treatment and control, indicating an impact.
ization process is repeated to calculate either an exact probability distribution (if the number of treatment permutations is small) or an approximation based on Monte Carlo simulation. Finally, the sample test statistic is compared to the resulting distribution to evaluate the probability that the result could have occurred by chance (i.e., type I error). Opportunities for applying variants of optimal impact study design are common in natural resource management. In most situations, managers have a knowledge of the timing and location of planned treatments (for instance, proposed forest cutting operations). The design of an optimal impact study requires application of this knowledge so as to select appropriate spatial controls and a scheme for monitoring both before and after the anticipated management intervention. Controls may in fact be experimental units that will be subject to the treatment under investigation
Investigators were interested in the impact of forestry practices on the vital rates of forest-dwelling birds and in particular on the impacts of thinning and prescribed burning on the survival and nesting success of wood thrushes (Hylocichla mustelina) (Powell et al., 2000b). It was known in advance that two forest compartments would undergo thinning and burning during the winter of 1994-1995, and that three other compartments would not be treated during the course of the study. All five compartments were included in the study, and measurements on habitat characteristics, bird survival, and nest success were taken during pretreatment (1993 and 1994) and posttreatment breeding seasons. This design provided the essential elements of an optimal impact study. Because measurements were taken on > 1 compartment in each treatment category, over a period of 2 years of pre- and postimpact, both spatial (among compartments) replication and repeated measurement were included in the design, allowing for greater flexibility in modeling and assessment. Powell et al. (2000b) performed randomized intervention analysis, in addition to more conventional analyses of time • treatment interactions with analysis of variance (Green, 1979) and tests of odds ratios for survival rates (Skalski and Robson, 1992). Results from all analyses supported the conclusion of no treatment impact, i.e., no shift in the relative response of treatment and control study plots after the intervention. In many situations, compromises often are imposed on what otherwise would be an optimal impact study design. For instance, spatial replication may be lacking in a situation in which the treatment is either unique or is impractical or unethical to replicate (e.g., the installation of a power plant). In the case in which spatial replication is lacking, one often can utilize multiple control areas, allowing one to distinguish temporal variation at the treatment site from variation among the control sites due both to temporal and spatial variation (Green, 1979). When temporal "replication" (i.e., before/after comparisons) is not available (for instance, in the event of an accidental oil spill), it may be possible to establish spatially replicated "control-treatment" pairings (for instance, sites at varying distances from the accident), in order to assess the probable impact of the event. Such heavily constrained designs typically require prior knowledge about the nature of the potential impact. In turn, the analyses and resulting infer-
6.8. Constrained Experimental Designs and Quasi-Experiments ences are more dependent on the impact assumptions than with more conventional analyses based on completely randomized designs and analysis of variance (Skalski and Robson, 1992).
Example We use an example of a hypothetical accident assessment presented by Skalski and Robson (1992). In this example, managers are presented with an accident (e.g., an oil spill) that has just occurred within a specific, geographically defined area (Fig. 6.2). "Treatment" (i.e., within the impact area) and "control" (outside the area) plots are selected at random, and one or more population responses (e.g., abundance, survival, natality) are measured through time. Skalski and Robson (1992) suggest a multivariate, repeated-measures analysis to test for the hypothesis of parallelism (no treatment impact) between treatment and control plots over time. Alternatively, randomized intervention analysis could be used for the same purpose. The most parsimonious interpretation of a nonparallel response between treatment and control areas would be that a treatment impact occurred, although the lack of pretreatment monitoring prevents one from ruling out preexisting differences as having induced such a response. An unavoidable consequence of this "design" is that immediate positive or negative impacts with no residual effect will be indistinguishable from preexisting differences between the areas not associated with the accident impact. Nevertheless, postaccident monitoring on both impact and nonimpact areas could enable inferences not possible by simply measuring system response on only the accident site.
l cl C
FIGURE 6.2 Accidentassessment scenario. Following the occurrence of an accident and the delineation of an impact area (shaded), treatment (T) and control (C) plots are randomly selected from within and outside the impact area. A time series of observations on the T and C plots are compared over time for evidence of a nonparallel response (after Skalski and Robson, 1992).
105
6.8.2. Observational Studies None of the above approaches, including optimal impact study design, is ideal with respect to drawing inferences about causal relationships, but each offers some of the elements of experimental design, mainly the presence of some type of experimental control against which to compare a treatment or impact. When experimental controls (either spatial or temporal) are completely absent, the term experiment is no longer applicable. Nevertheless, we suggest that such observational studies can still be considered in a hypothetico-deductive framework, and they can prove useful in both the advancement of scientific knowledge and management. In principle, the steps are the same as under any scientific investigation: (1) a body of knowledge is used to develop a research hypothesis; (2) deductive logic is used to obtain predictions (test consequences) that must be true for the hypothesis to hold; and (3) data are collected and the results examined with respect to their agreement or disagreement with the predictions of the theory. The asymmetry between confirmation and disconfirmation noted in Chapter 2 exists with respect to the interpretation of possible outcomes, with disagreement between predictions and evidence generally providing stronger inference via the application of the logical form modus tollens. As in experimentation, the need exists to use statistical methods to distinguish "disconfirmation" from random variation among the subjects observed. As with sampling for descriptive purposes, one still needs to be able to use sample data to make inferences about a "target population" of interest. This in turn requires the definition of a target population and sampling frame, the random selection of units from the population, and methods for unbiased estimation of the quantities of interest (abundance, survival, etc.). We distinguish here between two situations in which the investigator has different vantage points with respect to the timing of events. In the first, which we term a retrospective study, the events already have occurred, and the investigator assesses them in the light of predictions made as described above. For instance, a reasonable prediction is that for a population not density regulated, neither survival rates nor birth rates should vary in a density-dependent fashion (see Section 2.4). Having made this prediction, the researcher then examines 20 years of survival, reproduction, and density data, and formulates tests for correlations among estimates of these quantities. The motivation for this particular study, and for the examination of these specific statistical hypotheses, is with respect to theory disconfirmation:
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Chapter 6 Design of Experiments
negative correlations between either survival rates or birth rates and density may be taken as evidence that the research hypothesis (no density regulation) is false. We offer two caveats to such an interpretation. First, inferences from the study are necessarily weak, primarily because density and the vital rates might have been influenced similarly by an unrecognized covariate, thereby producing a pattern of covariation that is noncausal. Second, hypotheses in a retrospective study often are identified as a result of "data mining," in which patterns in the data themselves are used to formulate a hypothesis that then is tested with the same data. At a minimum one should avoid this logical circularity, and the invalidity of assertions arising from it, when analyzing data from a retrospective study. In contrast, a prospective study occurs when the investigator, based on prior theory, analyses, or both, makes a prediction about events that have not yet occurred and then proceeds to collect data to test these predictions as the events occur. In the density regulation example, the prediction of no correlation between density and survival or reproduction rates would be formulated before any observations are made about the future state of the system, and the data, once collected, then would be used to confirm or disconfirm these predictions, with the same logical and statistical asymmetries as under a retrospective study. Clearly neither prospective nor retrospective studies contain the power of a controlled experiment, and both are subject to abuse and misinterpretation, but prospective studies can be free from the potential of "data dredging" when subsequent "hypothesis testing" is based on having seen the results--or worse, on selectively retaining those hypotheses that support a favored direction. More interesting than distinctions about investigators' viewpoints, however, is the connection of a prospective study and the notions of forecasting, updating, and adaptation. Here the idea is that our provisional understanding (which may be viewed as a research hypothesis) can be used to make a forecast (i.e., a prediction about the future state of the system, possibly under a specific management action), which then is compared to data as they become available. Deviations of observations from predictions can be a basis for updating the hypothesis (typically encoded as a model) and, to the extent that management actions depend on future model predictions, adapting the decision-making process to new information. We discuss procedures for incorporating information adaptively when we consider adaptive resource management in Chapter 24.
6.9. D I S C U S S I O N The models of experimental design can be viewed as special cases of the general linear model y=XB+~, where each row of the matrix equation represents an observation from the study, the vector ~ includes the design parameters, X is a matrix of zeros and ones, and the random vector 8 has mean 0 and dispersion E__(Graybill, 1976). For example, the components of for the cell means model of a two-factor crossed design include the cell means ~/j, and the dispersion matrix = 10-2. Ordinary least-squares (OLS) procedures (see S-ection 9.1) can be used to estimate the model parameters, and assuming the random component 8 is normally distributed, the resulting estimates ~ are normally distributed and maximize the corresponding likelihood function (see Section 4.2.2 for a discussion of maximum likelihood estimation). For example, OLS estimation for the cell means model of a two-factor crossed design leads to maximum likelihood estimates ~ij = Yij., and substitution of ~ij into Eqs. (6.3) and (6.4) produces the maximum likelihood estimates for the corresponding restricted parameterization model. The linkage between OLS and maximum likelihood estimation provides a straightforward procedure for testing hypotheses about model parameters, based on the theory of likelihood ratio testing described in Section 4.3.4. In this procedure the hypothesis of interest is used as a constraint on model parameters, effectively reducing their number, and the test consists of a comparison of the OLS estimates of 0 -2 for the constrained and unconstrained models (Graybill, 1976; Rao, 1965; Searle, 1971). Indeed, the mean square statistics of analysis of variance are variance estimates under different assumptions about the parameters to be included in the design model. The testing procedures described in the preceding sections of this chapter all can be derived in this manner. Because experimental design can be described in terms of the general linear model, there is a very deep theory for the subject, which we have mostly avoided in this chapter. Nor have we attempted to describe all the designs that are relevant to ecological investigation or to deal comprehensively with the very broad suite of special design issues that arise in ecology. As indicated in Section 6.8, the complexity that is a hallmark of ecological systems often requires special adaptations of experimental designs, which complicate the analysis and limit the strength of inference. For example, the higher order interactions among system components
6.9. Discussion often are key to system behaviors, and these somehow must be targeted in the study design while avoiding an unfeasibly large and expensive investigation. Because organisms die, migrate, or otherwise are unavailable, there may well be missing cells in what otherwise is intended to be a crossed experimental design, which require adjustments in the testing procedures (Williams, 1987). Often there are numerous responses of interest in an experiment, and it is important to account for their biological associations as well as their statistical relationships through multivariate analysis of variance. A proper accounting of design factors in an ecological investigation can involve rather complicated mixed models of random and fixed factors, often with multiple covariates and multiple responses, occasion-
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ally with both categorical and continuous response variables. A theoretical framework and practical guidelines are available to handle these and many other issues in experimental design, though obviously they are beyond the scope of this book. Our purpose here has been to provide a basic framework for experimental design in ecological investigations as it relates to modeling, estimation, and sampling, preparatory to the estimation of population parameters in Part III. Readers wishing a more thorough treatment should consult textbooks on experimental design such as those by Cochran and Cox (1957), Cox (1958), Federer (1955), Fisher (1947), Gill (1987), and Kendall and Stuart (1966).
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PART
II D Y N A M I C M O D E L I N G OF ANIMAL POPULATIONS
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C H A P T E R
7 Principles of Model Development and Assessment
7.1. MODELING GOALS 7.2. ATTRIBUTES OF POPULATION MODELS 7.2.1. Quantitative Approaches 7.2.2. Discrete and Continuous Models 7.2.3. Deterministic and Stochastic Models 7.3. DESCRIBING POPULATION MODELS 7.3.1. Mathematical Formulation and Notation 7.3.2. Model Components 7.3.3. Canonical Processes 7.4. CONSTRUCTING A POPULATION MODEL 7.4.1. Hierarchical Modeling Strategies 7.4.2. Encoding Relationships among Model Variables 7.4.3. Incorporating Stochastic Factors 7.4.4. Translating Differential Equations into Dynamic Models 7.5. MODEL ASSESSMENT 7.5.1. Verification and Validation 7.5.2. Sensitivity Analysis 7.5.3. Identifying Model Equilibria 7.5.4. Stability in Model Behaviors 7.5.5. The Influence of Initial Conditions B Model Ergodicity 7.6. A SYSTEMATIC APPROACH TO THE MODELING OF ANIMAL POPULATIONS 7.7. DISCUSSION
There are no definable limits on the kinds of objects that can be modeled and no limits on the kinds of models that can be employed. Thus, a physical object can be represented by means of another physical object, in the way a model airplane represents an actual aircraft. A dynamic natural process can be represented by means of controlled laboratory apparatus, in the way that airflow in a wind tunnel "models" air movements around a physical object in nature. An intellectual activity can be represented by way of a general conceptual framework, in the way that a description of scientific method "models" the doing of science. Of special relevance in this book, the dynamics of biological entities such as animal populations can be represented by means of mathematical models. The development of any model involves the twin activities of abstraction and symbolic representation. By abstraction is meant the highlighting of system features considered important to the modeler, features chosen from the unlimited variety that potentially could be highlighted. For example, a model of a biological population might incorporate age composition along with the processes of mortality and birth, while ignoring genetic composition, social interactions among individuals, size and stage structure in the population, fluctuating environmental influences, communitylevel interactions, and a host of other features. Indeed, the system components expressed in a model are always fewer by far than the number of identifiable components that are omitted. In large measure the "art of modeling" consists of choosing which system features to highlight and which to ignore. This goes to the scientific goals of parsimony and elegance of explanation, as expressed by Occam's Razor: that representation is
Whatever its focus, a model is by definition an abstract representation, an expression that "stands for" or symbolizes something else. The expression is symbolic in that it characterizes its object in a way that highlights but does not attempt to recreate the object. It is simplified, in that the model captures key features of what it represents, while ignoring other features.
111
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preferred which requires the fewest assumptions and the least complexity (Palmer, 1988). The process of abstraction advances these goals, by limiting one's focus to only those system features thought necessary to capture the "essence" of a system. The second feature common to all models is the use of symbolic representation, by which is meant the representation of an object (or idea) by signs, concepts, or other objects that are distinct from what is symbolized. In our case the objects of interest are animal populations and (possibly) their environments, which are represented with equations, stochastic processes, and statistical distributions. These elements, incorporating the appropriate biological interactions, management controls, and other factors, define the population model. A useful classification of population models is based on the biological understanding incorporated in them and the amount of data on which they are based. As shown in Fig. 7.1, a cross-classification with these two factors leads to four model categories: (1) models incorporating good biological understanding and supported by a strong data base, (2) models based on substantial biological understanding, but unsupported by a strong data base, (3) models incorporating little biological understanding but supported by a strong data base, and (4) models based primarily on speculation, incorporating little biological understanding and supported with few data. Though models in the first category are ideal, they are the exception rather than the rule in population biology. The second category is exemplified by many theoretical population models and by process-oriented mechanistic models for which specific model parameters are not known. The third
III
I
IV
II
r./3 o
. v.-,i
f
Biological Understanding F I G U R E 7.1 Cross-classification of population models based on biological understanding invested in a model and empirical evidence for the model. Quadrant I represents an ideal situation in which the model is based on a firm biological understanding of the population and is strongly supported by data.
category includes regression and other statistically based models that are "fitted" to data, without accounting of biological mechanisms. Finally, and unfortunately, many models of biological populations fall into that least desirable fourth category defined by insufficient data and insufficient biological understanding. In this book we use mathematical models as devices to improve our knowledge of the biology of animal populations. Populations are characterized in terms of the population state, the biological processes influencing the population, the environmental milieu of the population, and the management regime to which it is subjected. Because the population processes of interest are dynamic, we focus here on dynamic models, i.e., models that allow for change through time. In the sections below, the modeling process is described in terms of the mathematical specification and refinement of a model, as guided by model goals and objectives. At several points in this process a model is subject to revision, in which previously overlooked factors can be incorporated and inadequately treated processes can be refined. One potential result is an accretion of complexity, as more realism and biological detail are included in the model. The behaviors of only the simplest biological models can be investigated solely by means of mathematical analysis. It is not difficult to show that even very simple models can exhibit quite complicated behaviors, behaviors that are influenced by zones of stability in the state variable space, critical points in state variable values corresponding to trajectory bifurcations, sensitivities to system initial conditions, and other factors that generate highly complex behaviors (e.g., May, 1974). The addition of stochastic effects, cohort structures, size dependencies in survival and recruitment, and nonlinearities in model structures adds to this complexity, further limiting the value of analytic procedures. With only a few of these complicating factors, even apparently simple models can become effectively unanalyzable without the aid of a computer. A source of complexity in model behavior lies in the nature of "feedback loops" that arise in the accretion of structural complexity. A feedback loop is simply the linkage of model components whereby one component influences other components, which in turn influence the first. A simple example is the logistic growth model, wherein population size influences the per capita rate of growth, which in turn influences population size, which again influences per capita rate of growth, and so on. In this case it is straightforward to recognize the feedback loop and to predict its consequences for model behavior. However, for models with more complicated structures, it can be difficult even to recognize all the feedback loops and quite impossible to assess
7.1. Modeling Goals their impacts without the aid of a computer. For example, there may be interlocking feedback loops with variables participating in several loops simultaneously, or feedback loops may be nested within other feedback loops, or feedbacks may occur in daisy chains and other configurations. Of course, combinations of these configurations may be operating simultaneously, so that the linkages may be obscure, especially if model components are many and/or highly interconnected. Clearly, it is important to limit the complexity of a model by including only those features that are necessary to characterize system behaviors of interest. Key to this effort is recognizing the purpose of the modeling effort.
7.1. M O D E L I N G GOALS It is a commonplace that the structure of a model is (or should be) tailored to its objectives (Chapter 3). Even a cursory review of the literature indicates considerable variation in model structures and functions, with quite different population models designed for broadly different purposes. However, some general patterns can be seen.
1. Model generality. Many population models are designed for generic applicability, with model performance measured by the ability to highlight general patterns of population dynamics for a broad range of species and environments. Such models are characterized by model simplicity, a lack of biological detail, and low precision in representing particular biological systems. Examples include the logistic and LotkaVolterra models, which are discussed in some detail in Sections 8.2 and 8.7-8.8. 2. Model realism. Many population models focus on biological mechanisms and thus incorporate highly detailed descriptions of biological processes. These models are characterized by precise mathematical description of biological mechanisms; however, the high degree of resolution limits their generality (see Levins, 1966) and often induces imprecision in the estimation of model parameters. An example might be the modeling of the reproductive process for waterfowl, which can be disaggregated into, e.g., mate selection, nest site selection, nesting and egg laying, brood rearing, and fledging, with the component processes influenced by competition, predation, habitat conditions, and other factors. It is easy to see how the data requirements for highly detailed modeling of these processes could reach beyond the limits of available data. 3. Model accuracy. Often a population model is developed for predictive purposes, i.e., to predict accu-
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rately population changes under a specific range of conditions. Model predictability usually is obtained by limiting its operating range, at a cost of both realism and generality. Predictive models typically include regression equations, time series models, and other statistical forms. Examples include certain bioenergetic and biological production models. 4. Identification of information needs. The focus of some modeling efforts is to explore the adequacy of biological data bases and identify biological information needs. Models developed for this purpose often are broadly conceptual and sometimes consist of diagrammatic and/or logical representations of biological interactions. Examples include large-scale ecosystem models, which often are most useful in identifying what is not known about key ecological interactions. 5. Management. Management-oriented models attempt to forecast the biological impacts of management decisions, accounting for both population effects and management costs/benefits. Distinctive model characteristics include the incorporation of decision variables that influence population dynamics. Good examples include harvest and stocking models for fish and wildlife populations (e.g., see Beverton and Holt, 1957; Getz and Haight, 1989; Williams and Nichols, 1990; Hilborn and Walters, 1992). In developing a population model, one faces tradeoffs among the possible objectives, which prevents one from meeting all objectives simultaneously (Levins, 1966). For example, a goal of developing a highly general model, one that applies to numerous species under a variety of environmental conditions, is incompatible with the goal of developing a model that is rich in specific detail and highly accurate in predicting population behaviors. It thus is important to decide at the outset what one wants a model to do, and based on that decision, to identify the appropriate model resolution, time frame, system boundaries, and data requirements for construction and evaluation of the model. The purposes listed above contribute to the key tasks of systems analysis, which can be defined in terms of system inputs, system outputs, and system design and configuration. In a general systems context, models can be represented in terms of linkages among system inputs (environmental influences, management controls, and other exogenous factors), system structures and functions (e.g., populations and population cohorts, interspecific interactions, biological processes such as reproduction, mortality, and migration), and system outputs (e.g., biological productivity, harvest yields, and long-term sustainability). These components define the fundamental tasks of systems analysis (Bossel, 1994), one of which is system design. The design
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task builds on specified input scenarios and output prescriptions for a system, and focuses on design (or characterization) of the system in a way that accommodates the inputs and tracks the output prescriptions. Thus, the intent is to identify the system structure for a given input and desired output. For ecological systems this may translate into an identification of structural features for a model, given that the histories of both environmental factors and the behaviors of key system indicators are known. Important steps in such an effort include specifying the range of model applicability and degree of mechanism in model design, as well as determining the criteria by which to assess conformance of model behavior to the output prescriptions. We address design issues in chapters on model identification and assessment in Part II, and the whole of Part III focuses on data-based parameterizations of biological populations. The second fundamental task is path analysis, whereby the behavior of a system is to be determined on assumption that a system model is available and (stationary or time-dependent) inputs to the system are given. Basically, the task here is to determine the output for a given system in response to a given input scenario. In the case of population ecology this might involve simulation gaming with a population model, as part of an effort to analyze model behaviors, assess and evaluate variability in systems output, and highlight information needs for further model refinement. We deal in considerable detail with issues related to path analysis in the chapters in Part II. The third task may be termed input or policy analysis, so named because it focuses on role of inputs (especially management inputs) in influencing a system pursuant to performance goals. Here the idea is to determine appropriate inputs for a given system so as to produce a desired output. It is within this task that management purposes are expressed, via techniques such as risk analysis, decision theory, and optimization. We address conservation and management issues in some of the chapters in Part II and explore the application of optimal decision-making in considerable detail in Part IV.
7.2. A T T R I B U T E S OF POPULATION MODELS
7.2.1. Quantitative Approaches Several quantitative approaches to population modeling can be taken, depending on model objectives and data availability. These include the following approaches:
1. Analytic models. Mathematical analysis of certain population models can be useful for assessment of population dynamics. However, mathematical analysis requires that a model be fairly simple in its mathematical structure, with limited feedbacks among components and simple (or no) stochastic variation. In Chapter 8 we use conventional analytic procedures to analyze some traditional ecological models, preparatory to more complicated population modeling and estimation. 2. Computer simulation. Many population models are so complex that their investigation requires computerbased numerical techniques to simulate population dynamics. Because of the power of computers to handle enormous amounts of data, simulation models can be structurally complex, including multiple population cohorts, stochastic effects, and virtually unlimited biological and mathematical detail. Computer simulations often are conducted with computer programs known as simulation languages. A simulation language automatically handles much of the "overhead" and "bookkeeping" responsibilities associated with computation, thereby enabling the modeler to concentrate on model structures and functions rather than computer coding. Thus, a simulator might generate mathematical formulas for changes in state variables based on user-defined relationships among model components and apply the formulas to compute rates of change at each point in simulated time. Parameters, system variables, driving variable records, and controls then can be updated, relevant summary statistics saved, and the process repeated for the next time period. The management of computer resources [e.g., memory and central processing unit (CPU) management], logical sequencing of computing operations, and organizing of data in the simulation can be handled automatically by the computer. The usual tradeoffs between flexibility and ease of use are found in simulation languages. For example, general-purpose languages such as SIMPAS for TurboPascal provide great flexibility in formulating models, though at considerable cost in the amount and complexity of programming that is required. Specialized simulation languages such as CSMP and DYNAMO offer the advantage of relatively simple computer programming, in a language designed specifically for simulation models; however, a substantial investment in time still is required to learn the language, and applications are restricted to the programming structures and functions imbedded in the language. Object-based simulation languages such as STELLA have been used for ecological modeling (e.g., Bossel, 1994; Grant et al., 1997) and show great promise in bridging the gap between flexibility and ease of use.
7.2. Attributes of Population Models 3. Statistical models. Population models typically
include parameters that are not known with certainty m for example, population size, survival rate, reproduction rate, and migration patterns. Depending on the sampling procedures used in data collection, models incorporating statistical assumptions about parameters can be useful in estimation of these parameters. In Part III we describe procedures for developing and analyzing statistical models for use in identifying biological structures and process parameters. 4. Dynamic optimization. In many cases management-oriented population models are used to assess the consequences of management activities, with the aim of identifying optimal long-term strategies for managing a population. Factors such as harvest yields, conservation costs, and long-term biological consequences can be incorporated in the evaluation of different management strategies. Dynamic optimization procedures, which are based on feedbacks between population models and evaluation criteria, can be useful in determining the most appropriate management strategy. Each of these modeling approaches is explored in some detail in this book. The remainder of this chapter focuses on development and assessment of population models, with emphasis on analytic and simulation models. In Chapter 8 a number of traditional population models are treated mathematically. Chapter 9 focuses on the use of population data in model identification, and stochastic processes are discussed in Chapter 10. Chapter 11 is devoted to the application of biological models, namely, the management of biological harvest and the conservation of populations and their habitats. Part III describes sampling and estimation procedures for particular population parameters, and Part IV investigates dynamic optimization in the context of population management.
7.2.2. Discrete and Continuous M o d e l s In some cases the distinction between discrete and continuous forms of a population model can be important. The mathematical expressions of discrete and analytic models can differ substantially, and the mathematical tools for analyzing them are quite different. Occasionally, analogous models in discrete and continuous time exhibit unanticipated differences in behaviors. The mathematical relationship between continuous and discrete model formulations can be viewed from either of two points of view. A continuous model can be thought of as the limiting case of a discrete formulation, in which the period between times in the time
115
frame becomes vanishingly small. This can be illustrated with the exponential model N(t + 1) = N(t) + rlN(t),
where the parameter r I indicates a rate of growth per unit of time. The solution of this difference equation is N(t) = N(t0)(1
+
rl) t
(see Section 8.1 and Appendix D), with N(t) exhibiting explosive, unregulated growth through time if r I ~ 0, and asymptotic declines to 0 if r I < 0. Note that the model expresses a change in population status relative to a change of one unit of time: N(t + 1) - N(t)
(t+l)-t
= rlN(t ).
This formulation can be modified to allow for increments of time that differ from 1" N(t + A ) - N(t) = raN(t) (t + A ) - t or
[N(t + a) - N ( t ) ] / & = raN(t),
where & is the interval between times in the discrete time frame and ra is a constant rate of growth over &. The smaller time step allows for "compounding" of the process within each unit of time, much as interest on a multiyear investment security can be compounded at intervals smaller than 1 year. As with compound interest, the effect of a smaller time step is to increase the population growth over a unit of time. If we now let A become vanishingly small, we have the differential equation dN N(t + A ) = lim dt a-,0 & = rN(t),
N(t)
the continuous form for the population model with instantaneous growth rate r. The solution of this differential equation is N(t) = N(to)e rt,
which exhibits the same general form of explosive growth (for r > 0) or asymptotic declines (for r < 0) as does the discrete model (see Appendix C). Another way to view the relationship of discrete and continuous models is to recognize a discrete model as an approximation, in which the discrete model is used to approximate the values of the continuous model at certain points of time over the time frame.
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To see this, consider the solution of the differential equation dN/dt = rN
for the continuous exponential model over the increment of time [t, t + k]N(t + A)=
N(t)e ra
= N ( t ) + [e ra - 1]N(t).
(7.1)
By the mean value theorem of calculus, there is a value t* in the interval [t, t + A] such that N ( t + A) = N ( t ) + r N ( t * ) k ,
with the difference t* - t dependent on the interval size A. For r > 0, the approximation Na(t + A) = N ( t ) + r N ( t ) A
(7.2)
of N ( t + A ) i s biased, in that Na(t + A ) < - - N ( t + A); however, the approximation error is of order (rA) 2, which is negligible for small A. A comparison of Eqs. (7.1) and (7.2) shows that a small increase in the intrinsic rate of growth r can compensate for this bias. Thus, if ra = e ra - 1
is used in place of r in Eq. (7.2), the discrete and continuous models produce identical values over the discrete time frame: Na(t + A ) = N ( t + A). This example illustrates several noteworthy points. First, the model forms for discrete and continuous models are analogous, often with easily identifiable relationships among model components. Second, the parameters in discrete and continuous models are mathematically related, with relationships that often are uncomplicated. Thus, a model with a 1-year time step and intrinsic growth rate r 1 has the same general pattern of growth as a continuous model with intrinsic growth rate r, on condition that the parameters r and r I satisfy e r = (1 + r 1)
or r = ln(1 + rl). Third, the two model formulations generally require the same amount of information, in terms of model structure, parameterization, and initial conditions. Fourth, the behaviors of analogous discrete and continuous models often are similar, at least in their general forms [though not always: discrete models sometimes exhibit unstable a n d / o r chaotic behaviors not seen with their continuous analogues--see May (1974)]. Notwithstanding these similarities, there are note-
worthy differences between continuous and discrete model formulations. Continuous models often are used for discerning general patterns of behavior, which sometimes are easier to recognize via analysis of differential equations. For example, one sometimes can solve the transition equation for the population trajectory of a model with unspecified population parameters and then investigate the impact of parameter changes on the trajectory. The example above suffices to illustrate: a solution N ( t ) = No ert to the continuous time transition equation gives population size as a function of time, parameterized by the initial population size and intrinsic growth rate. Using this function, one can examine the impact of different growth rates a n d / o r initial conditions on the population trajectory, determine population doubling times under different parameter assumptions, etc. In fact, it may not be necessary actually to solve the transition equations in order to extract information from them. For example, the "sensitivity" 3 N / 3 0 i of N ( t ) to variation in a model parameter 0 i c a n be obtained without actually obtaining a mathematical solution of the transition equation (see Section 7.5.2). On the other hand, it often is more natural to model wildlife and fish populations in discrete time. The life cycles of many natural populations are organized according to the seasons of the year, with reproduction and or mortality occurring at particular times (e.g., a time-limited "birth pulse"--see Section 8.4), so that an annual increment of time is appropriate for their transition equations. Thus, reproduction often occurs in the spring, with brood rearing through the summer and early fall. North American migratory species usually migrate south in the fall, to return north in the late winter/early spring to initiate nesting. These and other biological events are accommodated naturally in a discrete time model with annual time increments. There are other advantages for models with discrete time frames. For example, the analytic requirements for assessment of discrete time models often are less than for continuous models. In essence, the difference equations are sometimes less difficult to analyze than differential equations. Indeed, numerical approaches to the solution of differential equations utilize discrete time approximations in their solution algorithms. In addition, the data on which biological models are based almost always are collected at discrete times through the biological life cycle, making it appropriate to model these data with models having discrete time frames.
7.2.3. Deterministic and Stochastic Models An important distinction is between deterministic models and those containing stochastic elements. A
7.3. Describing Population Models
deterministic model contains no random variation in its mathematical structures; once the model form is specified, its parameters are identified, and its driving variable and control trajectories are incorporated, the behavior of a deterministic model is completely determined (that is, completely predictable). In essence, certain knowledge of model components leads to the certain predictability of population size at any point of time. Stochastic models contain structural uncertainties, i.e., the value a n d / o r behavior of at least one model component is not known with certainty. Thus the trajectory of population size is not completely predictable. In such a situation one can only make probabilistic statements about population size at any point in time. The distinction between deterministic and stochastic models can be illustrated with the simple exponential model N(t + 1) = N(t) + rN(t) with constant per capita population rate of change. With constant, nonrandom parameters r and N(t0), the future history of population size for this model is completely determined. For example, if one assumes that r = 0.04 and N(t o) = 100, then the population size after, say, 4 years is 117 individuals. This population size is predicted with certainty because the model structure is assumed to be identified correctly, the parameter values are assumed constant and known with certainty, and the model contains no random environmental influences. Now consider a model for which the parameter r is subject to random variation, i.e., r cannot be specified with certainty. For simplicity, assume that r is either 0.04 or 0.1 and that either value is equally likely. The randomness in r essentially defines two models:
N(t + 1) = N(t) + (0.04)N(t) and
N(t + 1) = N(t) + (0.1)N(t), defining equally likely population transitions. If we assume that N(t o) = 100 and r remains fixed, then the population size after 4 years is either 117 or 146, with either outcome equally likely. On the other hand, if r can switch randomly between 0.04 and 0.1 from year to year, then the population size after 4 years is given by N(4) = 100(1.04)t(1.1) 4-t, with t the number of years of growth at a rate of 0.04. This defines five different outcomes, with probabilities depending on the value of t. Thus, the certain outcome of the deterministic model has been replaced by a probab~listic outcome for the stochastic model. This multi-
117
plicity of possible behaviors, with corresponding probabilities of occurrence, expresses the fundamental difference between deterministic and stochastic models. Stochastic effects can enter into a population model in any number of w a y s - - f o r example, (1) imperfect knowledge about biological mechanisms, (2) lack of certainty about effects of management actions, (3) unpredictability of a stochastic environment, (4) imprecise identification of key population parameters, a n d / or (5) limitations on population and environmental monitoring capabilities (see Chapter 24). The added complexity in analyzing stochastic models (see Chapter 10) militates against their use unless important patterns in population dynamics are lost by the failure to account for stochasticities.
7.3. DESCRIBING POPULATION MODELS We consider here the description of both continuous-time and discrete-time population models. As indicated in the previous section, continuous-time models are based on a time frame for the model that is continuous, with system dynamics described in terms of differential rates of population change (see Section 7.2.2). On the other hand, discrete-time models utilize a discrete time frame, and system dynamics are modeled via instantaneous changes in system state at discrete points in the time frame. This approach is known as discrete-event simulation, and the changes in system state are called events; hence the discrete-event descriptor. There are basically two ways to handle time in discrete-event simulation. The first way is known as next-event advance (Law and Kelton, 2000), wherein the simulation clock at any point in time is advanced to the time of occurrence of the most imminent future event, followed by an updating of the system state to account for the fact that an event has occurred. The process of temporal advance and state updating is repeated until the end of the time frame is reached. With state changes occurring only at event times, periods of inactivity are essentially skipped as the simulation clock advances from one event to the next. An alternative to next-event advance is known as fixed-increment advance (Law and Kelton, 2000). In this case, the simulation clock is advanced in increments of exactly &t time units, after which a check is made to determine if any events occurred during the interval immediately past. If so, all events in that interval are treated as if they occurred at the end of the interval, and the system state is updated accordingly. The process of time advance, followed by interrogation of the most
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Chapter 7 Principles of Model Development and Assessment
recent interval and state updating based on events included in the interval, is repeated until the end of the time frame is reached. We note that operationally, fixed-increment time advance and next-event time advance are identical for discrete-event models with events scheduled to occur only at the endpoints of regularly spaced intervals. These conditions apply for most of the discrete-time models in this book.
7.3.1. Mathematical Formulation and Notation Our concern here is with dynamic population models, i.e., models that exhibit change in population structure a n d / o r size over some (continuous or discrete) time frame T. The time frame is delimited by an initial time t 0, which may or may not be specified, and a terminal time tf, which may be infinite. As above, it is convenient to think of discrete time frames as consisting of a sequence of equally spaced points in time between to and tf. Continuous time frames consist of all points along the time continuum between to and tf. We refer to the variables characterizing population state at each point in time as state variables. For example, the size of a population with only one age class can be described in time by a single state variable. The value of the state variable gives a "snapshot" of the population at each point in the time frame, so that population changes can be tracked through time. We use N(t) to denote population size at time t, thereby emphasizing its time-specific nature. For populations with age cohorts or other structural features we use Ni(t) to denote the size of population component i. The sequence of time-specific state variable values defines a state variable trajectory {N(t) 9t e T} of the population over the time frame. Of course, cohort models have a collection of such trajectories, one for each cohort. We use the notation {x(t) 9t ~ T} for the trajectory of a general vector x(t) of state variables, possibly including multiple populations, multiple cohorts, habitat elements, a n d / o r other time-varying system features. A second group of exogenous or driving variables represents factors that influence, but are not influenced by, population dynamics. Typical examples of driving variables are environmental influences such as daily weather patterns, annual rainfall and temperature regimes, and other climatic patterns influencing population dynamics. These variables are "exogenous" in that their values at each point in time are extrinsic to the model, rather than being generated from within the model structure (like the state variables). Thus, driving variables characterize the "system environment," within which population dynamics are expressed. We use Z(t) to denote the value of a driving variable at
time t. If more than one driving variable is represented in a model, Zj(t) denotes the value of driving variable j at time t. As before, the dynamic nature of these variables defines a driving variable trajectory {Z(t) 9t T}, with separate trajectories for each driving variable in the model. Yet a third class of variables includes management or control variables, representing mechanisms by which managers can influence population dynamics. For example, management control often is modeled in terms of reduction (harvest) or enrichment (stocking) of a population at specific points in the time frame. Other control variables might consist of actions to improve habitats or actions to control specific processes such as mortality or migration. We use U(t) to denote the value of a control variable at time t, emphasizing the temporal nature of these values. If more than one control variable is necessary for the model, Uk(t) denotes the value of control variable k at time t. A control variable trajectory {U(t)" t ~ T} is defined as above. State, control, and driving variable trajectories interact through a set of transition equations that express population dynamics over the time frame. In discrete time, the general form of the transition equation for a population with one age class is
N(t + 1) = N(t) + f(N, Z, U, t).
(7.3)
Thus, population size N(t) at time t is updated by means of the transition equation to produce the population size N(t + 1) at time t + 1. The term f(N, Z, U, t) represents the change N(t + 1) - N(t) in population size between t and t + 1. It incorporates biological processes of reproduction, migration, and mortality, which can be modeled separately or aggregated into a single expression. In continuous time the general transition equation has the analogous form
dN/dt = f(N, Z, U, t),
(7.4)
where f(N, Z, U, t) now expresses continuous rather than discrete changes in population size. The transition equations can be used to determine population dynamics over the time frame, subject to an initial value N(t o) for the population size. Assuming a discrete-time formulation of the model, the transition equation, along with population size N(t o) at the start of the time frame, produces the value N(t o + 1) for population size after the first time interval. With the updated value of population size for t o + 1, the transition equation again is used to produce the population size N(t o + 2) after the second time interval. The process is repeated for each time up to tf, producing timespecific values of the population size. The sequence of these values constitutes the population trajectory {N(t): t = t 0, ..., tf}.
7.3. Describing Population Models
Example Let N(t) be the number of deer in the state of Vermont in the preseason of each year, with a time frame T = {0, 1, 2, ...} starting in a particular year in which population size is approximately known. A crude model with which to project population size would allow per capita population growth rate to be constant through time:
119
fected by population size. This is of course a defining property of a driving variable. Finally, management impacts can be added to the model by incorporating a control variable U(t) for the annual harvest: N(t + 1) = N(t)
[ N(t) l_U(t)" + rN(t) 1 - K [ 1 - I 1 Z(t)/Zmaxl ] -
[N(t + 1) - N(t)]/N(t) = r o
r
N(t + 1) = N(t) + rN(t) = (1 + r)N(t). Assuming r > 0, this model exhibits unregulated population growth, in which additions to the population each year are fixed proportions of an ever-increasing population:
N(t) = N(t0)(1 + r) t. Because the number of deer cannot increase indefinitely, a somewhat more realistic model might incorporate an upper bound on the population level through, for example, density dependence in the per capita rate of growth:
With the addition of U(t), there now are three factors influencing population dynamics: (1) factors intrinsic to biological populations such as reproduction, mortality, and migration, which are incorporated in the model through mathematical expressions involving the parameters r and K and the state variable N(t); (2) an extrinsic factor, in this case the pattern of annual precipitation, which is incorporated in the model by means of the driving variable Z(t); and (3) a factor under direct management control, in this case the annual harvest, which is expressed in terms of the control variable U(t). For any choice of the growth parameter r, carrying capacity K, and initial condition N(t0), and for each driving variable and control trajectory, a trajectory for population size is obtained. A wide array of population trajectories can result from various combinations of these factors.
N(t + 1) -- N(t) + rN(t)[1 - N(t)/K]. In this formulation the per capita rate of growth is approximately r for small populations, because is 1 N(t)/K is approximately 1. As the population increases in size, the factor I - N(t)/K decreases linearly to 0, so that the population rate of growth decreases to 0 as the population gets close to its maximum value of K. Additional realism can be incorporated by allowing the population bound K to be influenced by environmental conditions. For instance, we might think of K as the carrying capacity of the habitat, which fluctuates with annual precipitation. This situation might be modeled with a driving variable Z(t) representing the spring-summer rainfall each year. Thus K would be replaced by, say, K[1 - ]1 Z(t)/Zmax]], where ~ is the long-term average rainfall. Under these conditions, optimal habitat conditions prevail when the amount of rainfall is the long-term average, and less than optimal conditions prevail when rainfall is other than the longterm average. The model now has the form -
[ N(t) ] N(t + 1) = N(t) + rN(t) 1 - K[1 - I 1 Z(t)/Zmaxl] ' -
for which the pattern of population change depends on the trajectory of rainfall amounts. Note that annual rainfall, which influences population size, is not af-
7.3.2. Model Components Modeling of dynamic animal populations, and in particular simulation modeling, is based on concepts and terminology inherited from the field of systems analysis. To represent populations and (possibly) their environments, the following components are needed: 9 Accumulators that act as reservoirs or accumulation points, much as a stock inventory is an accumulation of items of stock. These earlier were called state variables (in our case, representing population levels and other habitat indicators). 9 Flows between accumulators whereby accumulator levels are updated, much as inflows increase reservoir levels and outflows lead to depletion. Flows represent directional movement of material between accumulators, in which one accumulator is depleted and one is increased. In the case of population models, flows represent the operation of processes such as birth, death, migration, and the transfer of individuals among cohorts. 9 Sources for movement of material from outside the system boundaries into an accumulator within the system, much as precipitation represents inputs into a reservoir from outside a system of reservoirs. A source variable is essentially an undepletable stock that is
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outside the system boundaries, which serves as the origin of a flow across system boundaries. In the case of biological populations, a source variable provides inputs of individuals into the system via immigration or birth. 9 Sinks for movement of materials from within system boundaries to accumulators outside the system, much as evaporation represents outflow from a reservoir to a point beyond the system boundaries. In the case of biological populations, a sink variable provides for population losses from death and emigration. 9 Flow regulators that regulate the magnitudes of flows between accumulators, much as a valve controls the flow of water in a water pipe. System dynamics result from the flow of material (e.g., individuals in a population) between system accumulators (e.g., size cohorts), whereby the rate of each flow is essentially controlled by a flow regulator. The term converter also is used, to emphasize the "conversion" of inputs (from an accumulator or a source variable) into outputs (to another accumulator or a sink variable). In general, a converter is a model component other than a stock variable that influences a flow, either directly or indirectly (through another converter). A converter that varies with the system state sometimes is called an intermediate variable (Bossel, 1994). 9 Exogenous factors that influence the movement of material across system boundaries and among state variables within the system. Exogenous variables are by definition outside the system boundaries and therefore influence, but are not influenced by, system dynamics. Exogenous variables may represent sources or sinks for material flows across system boundaries (e.g., precipitation), or nonmaterial inputs of information into certain converters (e.g., ambient temperature as it affects reproduction). 9 Controls by which system dynamics can be artificially influenced. Control variables can represent flows across system boundaries, whereby stocks are either enriched or depleted, or are nonmaterial impacts on flow rates, whereby flow rates can be altered at certain times in the time frame. In the case of biological populations, controls often are expressed in terms of harvest and stocking (e.g., depletion of stocks through removal of individuals by sport hunting), or in terms of habitat management (e.g., alteration of flow rates for reproduction). 9 Connectors that indicate the influence of one accumulator, converter, or parameter on another. Connectors essentially represent directed information (as opposed to material) flows among system components, and they are used to indicate that the value of one component influences the value of another. An example is the feedback of a state variable into the flow
regulator controlling flows into that variable. In this case, information rather than material is "connected" from the state variable to a flow, for the purpose of regulating the flow of material into the state variable. These components are the primary building blocks of population models. They are illustrated in Fig. 7.2, representing a model of a population subject to birth and death. In this case the model includes one accumulator for population size, two flows for birth and death, three converters for carrying capacity and per capita birth and death rates, five connectors for controls on birth and death flows, one source for reproduction, and one sink for mortality. The actual process of modeling is focused (and constrained) by the specification of model objectives and the identification of the system to be modeled. For example, the system might include populations of species in a predator-prey system, but not include the geochemical cycles supporting the species. In this case, the populations would be included within system boundaries, and geochemical impacts would be incorporated in the model (if at all) as exogenous variables.
7.3.3. Canonical Processes Many of the processes in population models have canonical forms, i.e., forms that are structurally invariant and occur repeatedly in the model formulation. The following processes are of special interest in population modeling:
1. Compounding process. In this case, source and accumulator variables are linked via a flow, with the flow regulator controlled through connectors from the accumulator and a rate parameter (Fig. 7.3). The mathematical form of the flow regulator is a simple product of the rate parameter and a (possibly scaled) accumulator variable. The behavior of the process is one of accelerating growth of the accumulator, with the rate of accumulation at any given time dependent on the size of the accumulator. The simple exponential model is an example of this formulation. Note that the form of
source
stock
sink
growth
per capita birth rate
carrying capacity
per capita death rate
F I G U R E 7.2 C o m p o n e n t s of a model of a harvested population with d e n s i t y - d e p e n d e n t recruitment net of population losses.
7.3. Describing Population Models
121
stock
Stock flow
0
stock
1-
per capita flow rate F I G U R E 7.3 Compounding process characterizing net increases in population size over time.
the process is quite general, and covers simple birth processes and other phenomena involving proportionate increases of stocks through time. 2. Draining process. In this case, sink and accumulator variables are linked via a flow, with the flow regulator again controlled by way of connectors from the accumulator and a rate parameter (Fig. 7.4). The mathematical form of the regulator is the same as that for the compounding process, except that movement of material is out of, rather than into, the stock. The behavior of this process is one of steady depletion from an initial level, with the rate of depletion dependent on the size of the accumulator at any given time. The formulation is appropriate for simple death processes and other phenomena involving proportionate drawdown of stocks. 3. Resource-based production process. In this process, the flow between a source and an accumulator is regulated by means of connectors from a rate parameter, another resource stock, and (possibly) the accumulator itself (Fig. 7.5). A standard mathematical form for the flow is the product of the rate parameter and (possibly scaled) values for the accumulator and the resource stock. Typical behaviors show an increase in accumulation, the rate of which is moderated by the availability of the resource stock. An example is a plant-herbivore system, in which herbivore growth is dependent on a dynamic stock of forage.
0 F I G U R E 7.5 Resource-based production process characterizing the flow for a population as influenced by a dynamic resource, rl and r2 parameterize the stock on resource flows.
4. Stock-adjustment process. In this process, the flow between a source and accumulator variable is regulated by means of connectors from a rate parameter, the accumulator, and a target parameter for the accumulator (Fig. 7.6). A standard mathematical form defines per capita rate of change as a product of the rate parameter and the difference between the accumulator and its target. Typical behaviors for this model show a reduction in accumulator size if it is larger than the target level and growth in the accumulator if it is smaller. Thus, differences between the accumulator and its target are eliminated by appropriate changes in the flow rate. Of course, the process allows for tracking of a time-varying target as well as convergence toward a stationary target. 5. Implicit stock-adjustment process. Once again, a source and accumulator variable are linked with a flow, which is regulated via connectors from a rate parameter and the accumulator. However, in this case, the flow
stock flow -1-
stock
49 target
per capita flow rate F I G U R E 7.4 Draining process characterizing net losses in population size over time.
F I G U R E 7.6 Stock-adjustment process characterizing the flow for a population as influenced by the disparity between population size and a target size. r parameterizes the stock flow.
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is controlled via a connector from a stock-adjustment factor expressing the difference between a second stock variable and its target (Fig. 7.7). When the discrepancy between the second stock and its target is zero, there is no flow to the first accumulator; when the discrepancy is large, the flow is large. Feedback between the accumulators is established via a connector from the first accumulator to the flow regulator of the second, so that increasing the level of the first accumulator increases the flow to the second, thereby reducing the discrepancy between the level of second accumulator and its target. In essence, differences between the second accumulator and its target lead to changes in flow rates for the first accumulator, influencing its size and thereby influencing the flow rates for the second accumulator. In this way the second accumulator tracks its target indirectly, by way of flow adjustments through the first accumulator, hence the term implicit. 6. Co-flow process. This process again involves two stocks, but in this case, the flow of one stock is connected directly to the flow of the other (Fig. 7.8), hence the name co-flow (for coincident flow). Such a process is appropriate for systems in which the accumulation of two different materials are tracked as they move together within a system. An example might be the tracking of both numbers and biomass of individuals accumulating in a population stock. Note that the key difference between resource production and co-flow processes is that the former connects one stock to a flow associated with another stock, whereas the latter connects a flow of one stock to a flow of another.
stock 1
flow to stock 1
_f'-~
stock 1
(?
oow t stock,
,
I
stock 2
0
0 r2
F I G U R E 7.8 Co-flow process characterizing the flow for one population as influenced by the flow for another population.
Though obviously not inclusive of all possible modeling forms, the processes listed above provide a useful and easily recognizable format for many of the mathematical structures found in population models. They serve to highlight the ways in which the model building blocks identified earlier can be used to construct models. For example, a simple birth and death model includes compounding and draining processes, and a logistic model with carrying capacity includes both compounding and resource-based processes. Many traditional predator-prey models include combinations of draining, compounding, and resource-based processes, and models of grazing herbivores often include co-flows along with other structures.
7.4. C O N S T R U C T I N G A POPULATION MODEL target parameter
(2
7.4.1. Hierarchical Modeling Strategies
stock adjustment factor
stock 2 flow to stock 2
F I G U R E 7.7 Implicit stock-adjustment process characterizing the flow for a population as influenced by the disparity between the population size and a target size for a second population.
Along with the identification of goals and objectives, initial steps in model development involve identification of the system of interest and its boundaries. The "system" consists of the complex of components and flows among components that, taken as a whole, exhibits behaviors of importance to the investigator. For example, the dynamics of a population of herbivores may be of interest to a modeler, but not the dynamics of their parasites. A predator-prey system may be of interest, but not the forage base of the prey, nor the biology of secondary predators. The physiological ecol-
7.4. Constructing a Population Model ogy of a reptile species may be of interest, but not the mechanics of solar radiation illuminating the reptile's environment. In each case the system is defined by an investigative focus, which identifies some things as "within the system" and other things as "outside the system." Simply put, those components requiring state variables, flows, and mechanisms for regulation of flows define the system. Everything else is outside the system, and the distinction defines the system boundaries. Of course, system boundaries can be revised during the course of model development, as the need to incorporate additional system features (or the opportunity to exclude unnecessary features) is recognized. Two approaches to model identification can be recognized, based on a hierarchical view of biological systems. Thus, system components at each level in the hierarchy are aggregated into modules that interact at the next higher level. On the other hand, the components at each level represent aggregations of interacting components at the next lower level. From such a hierarchical perspective, a "top-down" strategy for modeling biological populations starts by establishing the biological boundaries imposed by environmental and management conditions and by describing the constraints on biological processes that these boundaries represent. The biological model is organized into modules (e.g., primary producers, secondary producers, decomposers) that are influenced by the milieu. Within each module, interactions among components at the next lower level (e.g., herbivores, carnivores, secondary carnivores) are tailored to produce the patterns exhibited at the next higher level. This hierarchical scheme continues downward through the system to the level of the individual state variable (e.g., population cohorts of particular species). The actual process of modeling moves downward through the hierarchy, with model components and information/material flows identified at the highest level of organization first. "Bottom-up" modeling moves in the opposite direction, from the lowest level of the hierarchy upward. In this case, individual components and flows between components are modeled in detail and then are coupled together with other components and other flows to effectively build the model "from the ground up." Bottom-up modeling has the advantage of incorporating considerable detail and realism into the model structure as it evolves. Its problem is that the component parts of the model are not designed to fit together and hence the model is likely to have problems at the interface between modules. Both top-down and bottom-up approaches ultimately require the identification of state or accumulator variables and the flows among them. Population models include accumulators for population cohorts
123
and possibly for other variables representing habitats and other environmental factors. Accumulators for populations often track population numbers, whereas accumulators for habitats and other factors often track biomass a n d / o r other biotic and abiotic materials. For example, the amount of surface water available to waterfowl on the spring breeding grounds might be modeled as a state variable that impacts habitat carrying capacity. It is at the point of identifying accumulators and flows that the processes of abstraction ("one from many" feature selection) and aggregation ("many into one" combining of features) come into play. Having identified the system boundaries, the relevant state variables, and the flows among state variables, it is necessary to model the flow rates. This is essentially a two-step process, the first of which is to identify the system components influencing each flow. Here one uses connectors to represent a "connection" between a given flow and another system component (either an accumulator or a converter). The second step is to describe mathematically the influence of these components on the flow. This part of the modeling process results in mathematical formulas (or tabulations) describing material increases in a receiving accumulator and decreases in a donating accumulator. These formulas are based on the magnitudes of certain state variables, on exogenous inputs and system control variables, and on other flows, converters, and parameters in the system. Finally, having identified the flows of material and information among system components, one must incorporate system initial conditions into the model formulation, as well as any terminal conditions and other constraints. Once this step is complete, all the basic components of the model are in place, and processes such as verification, validation, and model refinement can be initiated. Because of the large amount of information that must be tracked, computer programming is key to many modeling efforts. A computer program sequentially simulates population dynamics, using values of the state variables at a particular time in combination with data for the exogenous and control variables, to update the system state at the next time. The sequence of operations involves (1) the updating of driving variable control records, followed by (2) the updating of time-dependent parameters, followed by (3) the calculation of flow rates, and finally (4) the updating of state variable values in accordance with the calculated flow rates. The simulation begins with initial conditions for the system state and proceeds step by step through the time frame until the terminal time is reached. Clearly, one's ability to perform these operations without the aid of a computer, or to actually solve the system of
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transition equations in closed form, is severely limited for all but the most elementary models.
7.4.2. Encoding Relationships among Model Variables As mentioned earlier, the "art" of population modeling largely consists of recognizing those features of a population that are necessary to capture its dynamics, while ignoring others that are not (Section 3.4.1). This involves the representation of directed information linkages among system components. The influence of one component on another can be expressed in several ways: (1) analytically, by way of a mathematical formula expressing one component as a function of other components that influence it, (2) graphically, whereby a graphical representation is used to portray the relationship of influencing and response components, or (3) by means of tabular input, wherein the relationship between influencing and response components is established through the entries in a look-up table. These three methods are illustrated for the model displayed in Fig. 7.9, which portrays a logistic population model with a single age class. The per capita rate of growth in the model is influenced by the population size N(t), the intrinsic growth parameter r, and the carrying capacity K.
1. Analytic approach. In many cases the relationships among model variables can be expressed analytically, by means of a mathematical expression. For example, a converter that controls the flow of material into or out of a state variable is described as a function of other converters a n d / o r state variables in the model. The per capita growth rate for the model in Fig. 7.9 might be expressed as Growth = r[1 - N(t)/K], the logistic form for growth of a population. In this case, per capita growth is expressed analytically as a function of population size and per capita growth rate, which can be computed from the size of the population.
flow
stock
0 |
/K
'
F I G U R E 7.9 Representation of a model with logistic growth. The flow rate for the model is r[1 - N(t)/K].
The relationships among variables in many population models can be expressed in "closed form" with a mathematical formula. These formulas are derived from theory, from field sampling, from experimentation, or from intuition. 2. Graphical approach. Sometimes one can establish a relationship among model variables through the use of graphical representation. Here the influence of one variable on another is expressed by a graph displaying their association in a Cartesian coordinate system. The first value of a coordinate pair (the "x coordinate") corresponds to the model component from which the connector originates, and the second value (the "y coordinate") corresponds to the model component to which the connector points. The locus of points described by these coordinate pairs describes the relationship between the model components. In the case of the model in Fig. 7.9, we might describe the relationship between per capita growth rate and population size by means of a straight line between the points (0, R) and (K, 0), where the x coordinate represents population size and the y coordinate represents per capita growth rate. In this simple example, the linear graphical representation is equivalent to the formula for logistic growth exhibited above. There are many situations, however, where the geometric pattern of a relationship between model components is approximately known, even though a closed mathematical form for the relationship is not. Under these circumstances it is useful for a computer simulator to have the capability of incorporating graphical relationships. 3. Tabular approach. The relationship among model components also can be expressed through tabular arrays. In this case, separate columns of numbers can represent the controlling and recipient variables in the relationship, with, for example, each row of the table representing an (x, y) coordinate pair for the relationship. The table corresponds to an array of such points, and an interpolation procedure is required to generate values for y when the corresponding value of x is not specified in the table. For Fig. 7.9, we could describe the relationship between population size and per capita growth with a table, the entries of which are (0, R), (0.5K, 0.5R), and (K, 0). Because the three points are collinear and coincident to the line described above, a linear interpolation procedure simply reproduces the mathematical formula for logistic growth. However, when tabulation and linear interpolation are used with a nonlinear relationship, the interpolated values will not be coincident with the values produced by the mathematical formula. The degree of variation will depend on the distance between tabular points relative to the degree of nonlinearity in the mathematical relationship.
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7.4. Constructing a Population Model
7.4.3. Incorporating Stochastic Factors Uncertainties about system structure, function, and representation often necessitate the incorporation of stochastic effects in a population model. We emphasize randomness resulting from imperfect knowledge of biological processes, from limitations in the monitoring of population status, or from lack of predictability in a stochastic environment. Such variation is characterized by means of random variables, values for which can be generated by computer with a random number generator. For illustration we characterize stochastic influences with a generic random variable ~ with an unspecified distribution. Example distributions that often arise in population modeling are the symmetric bell-shaped normal distribution, the continuous uniform distribution, and the multinomial distribution (see Appendix E). As above, the inclusion of random variables in the model structure renders model behavior only stochastically predictable, that is, predictable only within limits determined by the influence of the random components. Typically random variables are incorporated in the model structure as multiplicative or additive factors in the flow equations, system initial conditions, a n d / o r system parameterizations.
Example Consider a simple birth-death model of a singleage population, in which reproduction is influenced by habitat conditions in the breeding season, with habitat conditions in turn influenced by precipitation in the late winter and spring. Because precipitation is effectively stochastic, habitat conditions and thus reproduction also are stochastic, with reproduction reduced under adverse habitat conditions by as much as 50% below what it would be under normal conditions. On the other hand, advantageous habitat conditions can lead to a 50% increase in reproduction. One way to model this situation is to modify reproduction by scaling the flow with a trinomial random variable 8 that takes a value of I (representing normal habitat conditions) with probability p, a value of 0.5 (representing poor habitat conditions) with probability q, and a value of 1.5 (representing good habitat conditions) with probability I - p - q. The challenge, of course, is to choose reasonable values for the probabilities p and q.
Example Again consider a simple birth-death model of a single-age population, which now is subject to predation. Assume that predation has a (potentially) major impact of survivorship, but because the predator population is not monitored, the extent of the impact at any
point in time cannot be determined. Assume also that predator populations fluctuate through time, though without any recognizable pattern in the population size. As a consequence there are substantial (but unpredictable) fluctuations in prey survival. One way to model this situation is to scale the maximum annual survival rate for the prey with a random variable that is uniformly distributed over the range (0, 1), so that annual survival fluctuates stochastically between 0 and the maximum survival rate. Because the behavior of such birth-death models is strongly influenced by the death parameter, it clearly is important to estimate accurately the maximum prey survival rate.
Example Consider a species that is subjected annually to regulated sport hunting. The regulation of hunting often has a goal of some annual harvest yield, say, H. If, however, a regulatory strategy is used that limits the take per hunter but not the number of hunters, then total harvest will fluctuate, depending on level of hunter participation in a given year. A way of modeling this situation would be to model the annual harvest flow from the population as a targeted harvest level H, to which is added a random number 8 with a (truncated) normal distribution with mean 0 and some appropriate variance. The effect of such a modification would be to allow for total harvest both above and below the target level, with the dispersion of harvest amounts depending on the variance of 8. In order to ensure that harvest is nonnegative, it would of course be necessary to truncate the distribution of below - H .
7.4.4. Translating Differential Equations into Dynamic Models Many population models are expressed in the form of differential or difference equations. By following a few simple rules these models can be put in the form of a systems model, with accumulators, flows, converters, connectors, and source/sink variables. Model translation involves the following actions, taken more less in sequence: 9 Depict state variables in the differential or difference equations as accumulator variables. 9 Express higher order differentials or differences in terms of first-order differential or difference equations. This requires additional accumulators for second-order and higher order differences and differentials. For example, the second-order differential equation d2x/dt
2 = ax 2
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Chapter 7 Principles of Model Development and Assessment
is resolved into two first-order equations, dx/dt
= x1
and dxl / d t = ax 2,
with connectors linking the two accumulators Xl and x. 9 Depict first-order differentials or differences as flows, with the flow arising at a source variable and terminating at the state variable accumulator represented in the differential or difference term. 9 Describe the flow regulators in terms of the state variable stocks and other exogenous and control variables arising in the differential or difference equations. 9 Set the initial values of the stocks in terms of initial conditions of the system of equations. 9 Incorporate any other system constraints into the mathematical specifications of the system converters. 9 Choose the time step for the simulation, and the method of computing flows over each time interval. 9 Simulate system dynamics over the time frame specified for the differential or difference equations. Example
To illustrate the translation of a differential equation into a simulation model, consider the logistic equation d N / d t = rN(1 - N / K ) , N(O) = N o,
with population growth governed by an intrinsic growth rate r and carrying capacity K. Because the model is expressed as a first-order differential equation, it is not necessary to define additional variables for higher order differentials. The state variable N is identified as an accumulator, essentially tracking the "stock" of individuals in the population. A single flow from a source variable to this stock is identified, with the flow regulator expressed in terms of a product of an intrinsic growth factor r and a factor involving carrying capacity (1 - N / K ) . These factors are expressed in terms of converters, the latter incorporating population size via a connector from the population stock. Both converters in turn provide information to the flow regulator to define the flow rate. After initial conditions N(0) = N O are specified and a time step chosen, the simulation model can be run over the time frame of interest. The diagram of this model is shown in Fig. 7.9.
7.5. MODEL A S S E S S M E N T The initial construction of a model is only the first of many steps that must be taken in the modeling
process. Stages in the process subsequent to model construction involve the analysis of model performance, as well as the biological interpretation of output. Even within the construction stage there are several steps subsequent to the initial model development.
7.5.1. Verification and Validation Subsequent to initial model development, it is important to ensure that the model structure is as intended and that it conforms to the data on which it is based. Procedures to "check out" the model, to verify that the model's mathematical structure is what was intended and its performance reflects the data used in its development, are k n o w n collectively as model verification (Law and Kelton, 2000). Verification is to be distinguished from validation, which occurs at a subsequent stage and involves comparison with data other than those used in building the model. An important component of verification involves the documentation of computer code and the cross-checking of mathematical formulas and parameter values for accuracy and appropriateness. This review, which all too often is overlooked or downplayed, can make a crucial difference in both the efficiency of all subsequent work on the model and the validity of all applications of it. The other major component of verification, namely, the evaluation of model performance against data used to construct the model, ensures that the model "fits" the data. If so, the investigator has reason to anticipate that the biological information represented in those data is implicit in the structure of a model that tracks them. The verified model represents a "testable hypothesis," in the sense that its structure conforms to the investigator's intentions and its behavior accords with the data on which it is based. Note that verification, as described here, does not refer to a correspondence between the model and reality, but to a correspondence between the model and investigator intention. Having verified that the model possesses the intended structure and that it faithfully reproduces the dynamics of data on which it was built, the researcher now must "validate" the model. Models typically are validated by exercising the model under a broad range of initial conditions and driving variable and control trajectories. Thus, patterns of model behavior under certain control and environmental conditions are matched against data collected independent of model development and verification. In a sense, the validation step can be seen as a check against "prediction bias" that arises in statistical regression when models are overspecified or "overfitted" to data and therefore pro-
7.5. Model Assessment duce misleading results when used with independent data. Here we restrict the use of the term "validation" to apply specifically to a comparison of model outputs to output data from the actual system under investigation, with the idea that a close resemblance of the former to the latter supports a conclusion that the model is valid for its intended purposes. We emphasize again that no model is "valid" in the sense of its comprehensively representing a system; indeed, the point of modeling in the first place is to configure those system structures and functions that are relevant to system behaviors, pursuant to the goals of the model. As described here, model validation can be recognized as an application of scientific hypothesis testing, in that the model is essentially used as a testable hypothesis wherein model behavior is compared with predictions other than the reference behaviors it was designed to reproduce. However, this essentially describes a single-hypothesis approach to investigation (see Chapter 3), and as such, is inadequate to capture the full extent of evaluation that should occur in the process of model development and assessment. Indeed, Anderson and Burnham (2001) have opined that the "concept of validation is of relatively little worth in the empirical sciences." On the other hand, the notion that models should be subjected to additional assessment above and beyond their development and verification should be strongly emphasized, precisely because in practice it often is not. The prevalence of the term "validation" in the modeling literature provides a useful venue for placing that emphasis. It should be noted that the processes of verification and validation (follow-up investigation with additional data) essentially compose an iterative sequence of model testing and refinement. For example, a verified model may be found to be inadequate in representing data not included in the development phase. Refinements therefore are incorporated in the model to address these inadequacies, and the refined model is verified as now fitting the extended data base. Then more data, yet unincorporated in the model, are used to evaluate the refined model. If the model continues to be inadequate in representing the extended data base, further model refinements are implemented, leading to further verification and the collection of additional data for further evaluation. Iterations of this process continue until the investigator is satisfied that the model is properly validated over the pertinent range of operating conditions.
7.5.2. Sensitivity Analysis A useful examination involves assessment of the sensitivity of model behavior to changes in system
127
initial conditions, parameter values, and structural features of the transition equations. A conventional formulation of sensitivity analysis is to assess variation in population size in response to parametric variation. A mathematical approach for continuous systems can be expressed in terms of the model dN/dt = f(N, Z, U, tl0),
shown here as parameterized by the parameter 0. Then the sensitivity of population size N(t) to the parameter 0 is obtained by differentiation of N(t) with respect to 0, or ON/O0. From the chain rule of calculus it can be shown that
•
aN
-d-F
oN\a0/'
which, on interchanging the order of differentiation, becomes d ON ~
Of + . O0 0N\ O0 ]
(7.5)
Thus the sensitivity is obtained as a solution of the ordinary differential equation, Eq. (7.5), in 3 N / 3 0 (see Appendix C). Note that this approach extends readily to systems with multiple components a n d / o r multiple parameters. In this case a system of sensitivity equations in the sensitivity coefficients 3Ni/30j is produced. Of course, there are attendant mathematical difficulties in solving this more complicated system of equations. An analogous formulation of sensitivity analysis for discrete systems is based on the discrete model N(t + 1) = N(t) + f(N, Z, U, t[0).
As in the continuous case, the sensitivity of N(t) with respect to 0 is obtained by differentiation of the transition equation: ON
= t+l
ON
"~
Of +
+
t
O0
.
ONKO0]
t
This describes the sensitivity ON/O0 at each point in time as a solution to the difference equation xt +l =
+ x t 1 + ON t
t
(see Appendix D), with {xt: t 9 T} defining a trajectory of sensitivity values ON/O0 over the time frame. Again, the approach extends readily to more complex systems with multiple components and multiple parameters. An alternative approach to sensitivity analysis involves computer simulation and typically consists of simulating system dynamics over a range of values for a parameter or group of parameters thought to be
128
Chapter 7 Principles of Model Development and Assessment
important in affecting population dynamics. Factorial designs (see Section 6.2) often are used in identifying parameter combinations, with high and low parameter values bracketing a median value. If the system contains many parameters, fractional factorial designs (Cochran and Cox, 1957) can be used to specify important parameter combinations. Again, the result of the sensitivity analysis is to identify parametric and structural features of the system to which system behavior is most sensitive. The resulting information can be useful in ensuring that a model is adequately specified and its output appropriately interpreted. The definition of sensitivity can be generalized to include variation in any attribute of system dynamics. For example, it might be informative to investigate the sensitivity of a population's growth rate to variation in one or more of its vital rates, or the sensitivity of terminal population size to variation in population initial conditions, or, more generally, the sensitivity of the maximum of a state variable over some time frame to a system parameter of interest. One particularly useful variant of sensitivity is the concept of elasticity or proportional sensitivity, defined as the proportionate change in system response with respect to a proportionate change in a system parameter. For example, the elasticity of population size Nf with respect to parameter 0 is oN,/00
Nf/O
time frame T. Thus, the value x* is said to be an equilibrium if, having attained the value x(t*) = x* at some time t* in T, the system is maintained at _x* over the remainder of the time frame if unperturbed. For discrete systems an equivalent definition is expressed by the stationarity requirement x(t + 1) = x(t). From Eq. (7.3) this means that x(t + 1) = x(t) + [(x, Z, U, t) = mx(t), or
[(x, Z, U, t) = 0.
Similarly, an equilibrium condition is defined for continuous systems [e.g., Eq. (7.4)] by the absence of differential change: dx/dt = f(x, Z, U, t) =
Or
resulting in the same stationarity condition fix, Z, U, t) = 0
as with discrete systems. Assuming no variation in exogenous and control variables, this condition can be simplified to f(x,t) = 0_,
oXf/Xf 00/0 "
(7.6)
with further simplification to
Elasticity can be defined similarly for discrete systems,
[(x) = 0
e.g.j,
ANf/A0 XflO
ANf/Nf
A0/0 "
Because it focuses on proportionate change, the concept of elasticity can be especially useful in comparing sensitivities for parameters that differ widely in scale (e.g., survival rates vs. reproduction rates). In Chapter 8 we discuss elasticity as it applies to the dynamics of age-structured populations. In Chapters 22 and 23 we extend the concept of sensitivity to optimal decisionmaking for dynamic resource systems.
7.5.3. Identifying Model Equilibria The investigation of system equilibria is a key first step in model assessment. Here we define system equilibrium (or system steady state) at x* if a system ceases to change once x* is attained. Simply put, an equilibrium x* is defined for an unforced system if x(t) = x* for all t -> t*. A formal definition is given in terms of the change in a system trajectory {x(t): t e T} over the
for autonomous systems (see Appendix C). The identification of equilibria is thus an algebraic problem of identifying zeros for the function y = [(x, Z, U, t). Example
Consider the simple logistic model, dN/dt = rN(1 - N/K), N(O) = N o,
of population size over the continuous time frame T = {t -> 0}. Equilibria for this population are given by the solutions of rN(1 - N/K) = 0,
which is satisfied by N* - 0 and N* = K. Once the population attains either of these values, it ceases to change from that time on. The equilibrium value N* = K specifies that the population is at its "carrying capacity" (see Section 8.2) and no further growth is
7.5. Model Assessment possible. On the other hand, the equilibrium condition N* = 0 is tantamount to population extinction. From Eq. (7.6) it is clear that stationarity can be sustained by the application of an appropriately chosen control trajectory. To illustrate, consider a population with growth potential in the absence of harvest, where the harvest level U(t) at time t is an additive factor in population change:
129
a Taylor series expansion. To illustrate, consider a system of m populations (or population cohorts) N(t) that experience change through time according to the equation d N / d t = F(N),
(7.7)
where F(N) expresses differential change in population status at any particular point in time. A Taylor expansion about the equilibrium value N* is
x(t + 1) = x(t) + F(x, t) - U(t). Fk(N * + n ) = Fk(N *)
If the amount of harvest at each point in time is chosen according to
m
U(t) = F(x, t),
then reduction of the population size by harvest is compensated by intrinsic population growth F(x, t), and the population is maintained in equilibrium. For more complicated multivariate systems the concept of equilibrium can be extended to include other attributes besides population size. Populations with age-specific reproduction and mortality rates offer a case in point. Consider a population consisting of, say, k age classes, with each age class a characterized by its own annual survival probability Sa and per capita rate of reproduction ba. These parameters define the wellknown Leslie matrix model (see Section 8.4.2), which can be used to project changes in a population's size and age composition through time. A population described by a Leslie matrix model exhibits a transition period during which there is temporal variation in population attributes, followed by a period of stability in which the population maintains a constant rate of growth and a stable age distribution (Section 8.4.2). For such a model it is reasonable to extend the concept of equilibrium, so as to encompass stationarity in other attributes (such as age composition) besides population size. In the case of a Leslie matrix model we thus can have a population that attains equilibrium in its age distribution, while exhibiting explosive population growth once that stable age distribution is reached.
7.5.4. Stability in Model Behaviors The concept of stability is closely related to equilibrium, in that system perturbations around a stable equilibrium are eliminated over time as the system returns to equilibrium. More formally, a dynamic system x(t) is held to be stable about an equilibrium x* if a "small" perturbation from _x* induces a trajectory {x(t)} that converges to x*. Issues of interest include the maximum size of allowable perturbations and the path of convergence back to x*. A general approach to stability involves the use of
OF k
+ Z
9
ni-~ii(N)
i=1
+
(7.8)
m n 2 32Fk,N,)
t_
i=1
32Fk ninj c~Xic~lkl(X*)-}-"l
+ ~
i~j
"'"
for k = 1 .... , m, with n = N - N* representing deviations in population sizes from the equilibrium values in _N*. For small deviations the higher degree terms in Eq. (7.8) are of negligible importance, and the equation reduces to
m Fk(N * + n ) ~
~
OFk
9
n i - ~ i ( N ).
i=1
We can write d(N)/dt as dN/dt = d(N* + n ) / d t = dn/dt, m
so that Eq. (7.7) for population dynamics can be expressed in terms of the deviations n = N - N*: m OFk dnk/dt = Z n i - ~ i ( N
9 )
i=1
for k = 1, ..., m. This provides a linear approximation of the transition equations [Eq. (7.7)] in a neighborhood of N*. A matrix formulation is d n / d t = J(N*)n,
(7.9)
where n' = (n 1, ..., n m) and n
[• J(N*) =
LaN,(N*) ]
(Appendix C). Equation (7.9) can be used to describe stability about N* in terms of the trajectory {n(t)} of deviations. Thus, a trajectory that converges to 0 defines a stable equilib-
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Chapter 7 Principles of Model Development and Assessment
rium (i.e., convergence of _n(t) = _N(t) - _N* to 0 implies convergence of N(t) to N*). On the other hand, divergence of {n(t)} from _0 defines an unstable equilibrium [because divergence of n(t) = ~N(t) - m N* from 0 implies divergence of N(t) from N*]. Finally, a stable pattern of oscillations of {n(t)} defines neutral stability [i.e., the oscillation of n(t) about 0 corresponds to the oscillation of N(t) about N*]. It is shown in Appendix C that deviation trajectories for dn/dt = J(N*)n in a neighborhood of N* are controlled by the eigenvalues of J(N*). If all eigenvalues of J(N*) are negative, the trajectory {n(t)} converges to O; if at least one eigenvalue is positive, the trajectory diverges from 0; and if there is at least one pair of complex conjugate eigenvalues, the trajectory exhibits oscillatory behavior. Example
Consider again the logistic model dN/dt = rN(1 - N/K)
from the previous example, with equilibria N* = 0 and N* = K. Here we assume a positive intrinsic rate of growth r for the population. Because the model is univariate, the matrix J(N*) consists of the single element
ON
For N* = 0 it is easy to see that this expression is positive, and therefore N* = 0 is an unstable equilibrium: small deviations n = N - N* result in divergence of the deviation trajectory from 0, and in consequence, N(t) diverges from N* = 0. On the other hand,
ON =-r, which establishes N* = K as a stable equilibrium: deviations n = N - N* result in convergence of the deviation trajectory to 0, so that N(t) converges to K. Stability can be similarly defined for discrete systems (see Appendix D). For example, consider a single population with transitions given by N ( t + 1) = F(N),
where F(N) is a differentiable nonlinear growth function. As with continuous systems, population dynamics can be expressed in terms of a Taylor series expansion of F about an equilibrium N*: N ( t + 1 ) = F(N* + n t)
n2t d2F dF = N* + n t --d~(N*) + 2 d N 2(N*) + ...
with n t = N ( t ) - N*. The higher degree terms are of negligible importance and may be discarded for "small" deviations about N*, leading to the linear expression N ( t + 1 ) = F(N* + n t) dF = N* + n t -d~(N*).
Expressing N ( t + 1) = N* + nt+l, we then have dF N* + nt+ 1 = N* + n t-d-~ (N*),
so that the equation for population dynamics can be written in terms of deviations: nt+l
=
dF nt - ~ (N*).
In this way a nonlinear transition equation can be approximated by a linear difference equation in a neighborhood of N*. It follows that N* is a stable equilibrium if IF'(N*) I < 1 (because the trajectory of deviations exhibits exponential decay a n d / o r d a m p e d oscillations) and N* is an unstable equilibrium if IF'(N*) I > 1 (because the trajectory of deviations exhibits exponential growth a n d / o r increasing oscillations). Appendix D provides a more comprehensive treatment of stability for discrete systems.
7.5.5. The Influence of Initial C o n d i t i o n s ~ M o d e l Ergodicity From Eqs. (7.3) and (7.4) it should be clear that the behavior of a model is influenced, at least in part, by its initial conditions. Two interesting possibilities can be identified as to long-term system dynamics: (1) the asymptotic behavior of x(t) is influenced by x(t0), or (2) the asymptotic behavior of x(t) is independent of x(t0). The latter case describes model ergodicity (a literal meaning is "recurring states," in this case recurring with respect to different initial conditions; see Section 10.3.2). System ergodicity anticipates biological processes that are asymptotically insensitive to accidents of environment and other factors affecting the starting point for system dynamics. Time-varying attributes of such systems typically converge to a single stationary value, irrespective of where the system trajectory originates. The Leslie matrix model, referred to in Section 7.5.3 and described in more detail in Section 8.4, is a case in point, in that the trajectory of a Leslie matrix
7.6. A Systematic Approach to the Modeling of Animal Populations model converges asymptotically to a stable age distribution irrespective of the initial age distribution. For ergodic systems, any variation among the long-term behaviors necessarily represents differences in system processes, rather than differences in system initial conditions. The other possibility for system behavior is that the system starting value x(t0) does influence long-term system dynamics. This influence defines nonergodic systems, for which variation in asymptotic behaviors can be induced by simply changing the system initial conditions. A nonergodic system is intrinsically nonlinear, typically possesses multiple equilibria, and often exhibits patterns of local stability whereby perturbations in local zones are followed by a return to local equilibria. The assessment of nonergodic systems is complicated by the fact that observed variation in longterm system behaviors may be induced by differences among systems in their initial conditions or by differences in their system processes. A trivial example is a logistic model of population dynamics for each of two geographic areas in the absence of migration between them. In this case the asymptotic behavior of the aggregate system varies depending on whether initial population size on each of the areas is nonzero. Computer simulation offers one approach to the investigation of initial conditions. The following steps are a possible sequence: 9 Identify the system attribute(s) to be investigated (e.g., population rate of growth, sex ratio, age distribution). 9 Select an appropriate range of initial conditions {x(t0)} to be investigated. 9 For each particular set of initial conditions x(t0), simulate system dynamics over some extended time frame and record the resulting values of the attributes of interest. 9 Assess the variation among recorded values of the attributes. An absence of variation in attribute values is indicative of ergodicity. On the other hand, a clustering of attribute values into discrete groupings may suggest further inquiry into the structure of the system processes inducing nonergodicity.
7.6. A S Y S T E M A T I C A P P R O A C H T O THE M O D E L I N G OF ANIMAL POPULATIONS Having introduced the major components in modeling and some tools for model assessment, we now can describe systematically the process of development
131
and application of models of animal populations. As mentioned earlier, the process begins with identification of goals and objectives for the effort and leads systematically to model identification, computerization, and testing. The modeling process includes the following key steps:
1". Establishment of model goals and objectives. Because the purposes for which a model is to be developed determine in large measure its structure (and thus its dynamic behaviors), this crucial step should begin the process. As discussed earlier, goals and objectives often are associated with properties such as model generality, model realism, model accuracy, model identifiability, and potential uses of the model for management or other purposes. Depending on its goals and objectives, a population model can have very different structural features and can exhibit substantially different behaviors. Given the importance of establishing model goals and objectives, it is surprising how frequently this step is overlooked (or inadequately reported). 2. Identification of system features and system boundaries. Early on it is necessary to determine what is to be included in a model and what is not. This decision establishes which population features to characterize, which environmental and management variables to include, and what biological mechanisms to model. Identification of model components includes (1) state, control, and exogenous variables, (2) flows among state variables, (3) intermediate variables and parameters, (4) information connectors among state variables, intermediate variables, flows, and model parameters, and (5) mechanisms by which flows are regulated and intermediate variables are updated. These features are captured in a diagram of the system that characterizes system components by means of, e.g., stocks, flows, and information connectors, and recognizes system boundaries by means of sources and sinks that are associated with elements outside (but connected with) the system. An intuitive order in identification of model features starts with the state variables, then includes the flows among state variables, followed by intermediate variables, parameters, and connectors among the information components. Exogenous influences are added as flows across system boundaries (e.g., precipitation events) or as converter components (e.g., temperature regimes). Control variables also are included as flows (e.g., stocking or harvest) or as converter components (e.g., alteration of mortality rates). Those model parameters that are likely to be altered as the model is investigated should be represented with converters. Through the process of identifying system features and boundaries, model goals and objectives often are
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Chapter 7 Principles of Model Development and Assessment
clarified. System identification forces the modeler to consider the feasibility of developing a population model in terms of, e.g., spatial and temporal comprehensiveness, level of biological aggregation, and the degree of biological mechanism to be included. These considerations often lead in turn to revision and refinement of the model goals and objectives. Indeed, model identification and the establishment of goals and objectives can be seen as an iterative process, with goals and objectives shaping the model structure, with model identification in turn helping to refine model goals and objectives, with the refined goals and objectives further influencing the model structure, and so on. In practice, iterations of this process can continue for the duration of the modeling exercise.
3. Development of the mathematical~simulation model. It is useful to think of the mathematical formulation and coding of a model as occurring subsequent to identification of its features. However, the process of mathematical formulation actually begins as its features are being identified. Indeed, feasibility issues in model identification often relate to mathematical feasibility, in that mathematical relationships among system components must be known (or at least be estimable), to be included in the model. On assumption that the identified model features can be characterized mathematically, a natural order for model development is (1) mathematical description of functional forms for the flows among state variables, (2) mathematical description of functional forms for the intermediate variables, (3) incorporation of values for the parameters identified as converters, and (4) identification of initial values for the stock variables. If the model requires simulation, a computer program corresponding to the mathematical model also must be developed. The simulation model includes the following important features: (1) Time specifications for the simulation. These include the length of the time frame, the time step, and the output interval. (2) Specification of the format of the desired output (e.g., data protocols, graphical a n d / or tabular formats). This includes identification of system variables to be displayed, scaling of graphical displays, and the layout and labeling of output. (3) Documentation of the computer code. The latter step, which often is overlooked, helps to ensure that the model can be understood by others not involved in program development. Documentation of computer code involves both a review and description of the computing logic and thus is an integral part of the verification process. It is important to recognize that mathematical and computer formulations of a biological model can be mutually informative. On the one hand, simulations
with a computer model can be used to good mathematical effect, e.g., to highlight inconsistencies in the mathematical formulation of the model, to focus on important structural features that control model behaviors, or to suggest interesting mathematical features worthy of further exploration. On the other hand, the mathematical formulation of a model can focus one's attention on model parameters or other features that can be explored usefully by means of computer simulation. Indeed, the interaction between these two approaches (mathematical analysis and computer simulation) really constitutes an iterative process, in which analysis is used to direct one's efforts in simulation, the follow-up simulations suggesting relationships among system features to be explored analytically, the follow-up analyses in turn suggesting further simulations, and so on. The interplay of simulation and analysis thus provides insights about system structure and function that extend beyond either approach considered alone. 4. Model sensitivity analysis. As described above, sensitivity analysis involves an assessment of variation in model behavior, with the idea of varying some component of the model and evaluating the impact on model performance. For all but the simplest models it is not possible to conduct a comprehensive sensitivity analysis. For example, if we consider only two levels for each model parameter in a deterministic model, the number of simulations required to comprehensively examine a model with k parameters is 2k. Thus, a simple Lotka-Volterra model for three competing species (see Section 8.8) involves 2 1 2 - - 4096 simulations. Clearly, it is necessary to devise strategies for sensitivity analysis that avoid most parameter combinations and yet focus on combinations of importance. Often one can adapt certain experimental designs from statistics, e.g., stratified or fractional factorial designs (see Section 6.2) to aid in this effort. Nevertheless, the choice of parameters remains largely a matter of "probing" over the set of potential parameters, aided by intuition, previous investigation, and luck. We note that sensitivity analysis, like verification and validation, is neither a one-time exercise nor an ending point in model assessment. Indeed, sensitivity analysis is perhaps most useful in highlighting model features that should be identified with a high degree of accuracy. Thus, it serves as a guide for the allocation of effort in model development, as well as model revision and refinement. 5. Model verification. As mentioned earlier, verification consists of a review of the model structure and computer code, as well as an evaluation of model performance with data used in model development. The purpose of the review is to ensure the model "looks"
7.6. A Systematic Approach to the Modeling of Animal Populations the way it is intended to look, in the sense that the mathematical forms of the relationships are as intended, the parameter values are correctly specified, the sequence of logic in the computer code is as intended, and so on. The evaluation of model performance ensures that the model adequately represents important patterns in the data used to create it. 6. Model validation. Validation extends the assessment of model behavior to include evaluation of model performance based on data not used in its development. The idea is to test whether the model remains "valid" for representing independent, representative data. In this sense the model acts as a complex hypothesis, to be evaluated by comparison with observations of the system. A correspondence of model predictions and independent data over the intended range of operation of the model supports the validity of the model for its intended purposes. If the model fails to correspond with independent data, further model refinement is necessary. Typically this involves retention of the independent data in the data base, refinement of the model based on this extended data base, verification of the refined model, and comparison of predictions from the refined model against additional independent data. Thus, validation and verification are not one-time activities, but instead are part of an iterative process by which a model evolves in its structure and function. The process is driven by the comparison of field data against model predictions, and it is a part of model assessment and evaluation. 7. Stability analysis. In addition to sensitivity analysis and verification/validation, it is useful to determine the equilibria of a system and to explore equilibrium stability. Stability properties for both discrete and continuous systems can be investigated by means of a first-order Taylor series expansion of the transition equations, with stability determined by the eigenvalues of a matrix of transfer function derivatives. Similarly, an investigation of initial conditions can prove helpful in anticipating their influence on both transient and asymptotic patterns of population change. Computer simulation is one way to explore the effect of initial conditions on population dynamics.
8. Application to management of animal populations. Population models developed as management tools ultimately are used to provide information to managers about the population consequences of management. Pursuant to this goal, models are used in essentially two ways: (1) to play "what-if" games, wherein potential management policies are imposed on the model (through identification of a control variable trajectory) and the model is used to simulate population dynamics under the policy, and (2) to
133
identify optimal management policies, based on some well-defined measure of model performance. For example, models of a harvested population sometimes can be used to identify optimal harvests through time, based on an objective of maximizing total harvest over an extended time frame. The use of models in dynamic optimization is discussed in some detail in Part IV. A point worthy of strong emphasis is that the modeling process does not end with validation and application to population management/assessment. Models represent biological systems that change through time in response to ecological, environmental, and management factors. As these systems evolve, the models representing them should incorporate new information about the structure and function of the system. The information on which a model is based is sometimes in the form of data and sometimes in the form of intuition, theory, or anecdotal evidence. In all cases, the information base grows as system changes are observed, and the system model can be updated as additional information becomes available. The need for adaptive updating is especially important for management-oriented models. Even if they are carefully constructed and properly verified and validated, such models nevertheless are useful only over a limited range of values for the biological system under investigation. This range often is defined by "normal system conditions" or by behaviors in an unperturbed state. On the other hand, the management of biological systems almost always involves considerable perturbation, which often tends to move the system outside of its normal operating range. For this reason, models of managed biological systems, to retain their usefulness, must be updated as new information becomes available. An ongoing cycle of management, monitoring, and model revision therefore is prescribed, including the following activities: 9 As the modeling process progresses, the model is verified, validated, and analyzed based on available population data. 9 The model is used to assess the consequences of management decision-making (e.g., population harvest or stocking). 9 Management decisions lead to population changes and updated information about population status (and the impact of management decisions on population status). 9 The updated data base is used to revise and refine the model, through the processes of model identification, verification, validation, etc.
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Chapter 7 Principles of Model Development and Assessment
The revised model is again used to inform decisionmaking, leading to further changes in population status and further refinements in the model. A key point in this sequence is that modeling is (or should be) an evolutionary process that is ongoing throughout the useful life of the model. At no point can one stop the iterative refinement with an assurance that, because the model now represents the past adequately, it is certain to represent the future adequately. The embrace of uncertainty, along with the iterative refinement of management-oriented models with new information as it becomes available, defines an adaptive approach to management. The term adaptive is used to characterize management approaches that account for learning, i.e., that focus on the biological information obtained through management and use this information in future decision-making. On condition that information for model refinement and improvement is simply an unintended by-product of management decision-making, the approach is described as passive-adaptive management (Waiters, 1986). If, on the other hand, management actively seeks decisions that are informative of system structure and function, the approach is described as activeadaptive management. We discuss adaptive management in the context of dynamic optimization in Chapter 24.
7.7. D I S C U S S I O N
In this chapter we have presented a systemsanalytic view of the modeling of animal populations. The framework for much of the discussion is borrowed from systems engineering, which has a long and distinguished history in the modeling and analysis of dynamic systems. Animal ecology has benefitted substantially in recent years from the approaches and techniques of systems modeling and analysis, and increasingly the ecological literature documents this work. Nevertheless, the notational density and mathematical rigor exhibited in the systems literature continues to be a barrier to understanding for many ecologists. The field of dynamic modeling is truly huge and includes quite sophisticated treatments of subjects such as systems identification, systems analysis, and optimal control (Luenberger, 1979; Palm, 1983; Flood and Carson, 1988; Dorf, 1989; Bertsekas, 1995). We have touched only briefly here on these concepts, especially in such complex areas as stochastic differential modeling and assessment and the mathematical treatment of systems control. We discuss these and other concepts in some detail in the following chapters and in the appendices, recognizing that a comprehensive treatment is well beyond the scope of this book. We leave it to the interested reader to explore further this very rich body of knowledge.
C H A P T E R
8 Traditional Models of Population Dynamics
8.1. DENSITY-INDEPENDENT GROWTHwTHE EXPONENTIAL MODEL 8.1.1. Habitat Effects 8.1.2. Harvest Effects 8.1.3. Perturbations 8.2. DENSITY-DEPENDENT GROWTHmTHE LOGISTIC MODEL 8.2.1. Incorporating Harvest 8.2.2. Incorporating Time Lags 8.3. COHORT MODELS 8.3.1. Populations with Independent Cohorts 8.3.2. Transitions among Cohorts 8.4. MODELS WITH AGE STRUCTURE 8.4.1. Life Tables 8.4.2. Discrete-Time Models with Age Cohorts 8.4.3. Continuous-Time Models with Age Cohorts 8.4.4. Characterizing Populations by Age 8.5. MODELS WITH SIZE STRUCTURE 8.5.1. Discrete-Time Models with Size Cohorts 8.5.2. Continuous-Time Models with Size Cohorts 8.6. MODELS WITH GEOGRAPHIC STRUCTURE 8.7. LOTKA-VOLTERRA PREDATORPREY MODELS 8.7.1. Continuous-Time Predator-Prey Models 8.7.2. Discrete-Time Predator-Prey Models 8.8. MODELS OF COMPETING POPULATIONS 8.8.1. Lotka-Volterra Equations for Two Competing Species 8.8.2. Lotka-Volterra Equations for Three or More Competing Species 8.8.3. Resource Competition Models 8.9. A GENERAL MODEL FOR INTERACTING SPECIES 8.10. DISCUSSION
Some familiar population models have been analyzed in the literature in considerable detail, especially as concerns the influence of model parameters, the interactions among model components, and in some cases, the projected impacts of management actions on model behaviors. In particular, there is a large literature on the logistic, Leslie matrix and other single-species models, and on certain multispecies models that account for competition and predation. In this chapter we describe some of these models, beginning with examples that are biologically and mathematically simple. Additional complexity is incorporated gradually, with the addition of environmental factors, management effects, cohort structures, and other components that enhance biological realism (and also complicate model analysis!). We restrict attention here to deterministic models and defer to later chapters the treatment of statistical uncertainties and stochastic model behaviors. To help frame the discussion, it is useful to consider the nature of density dependence in population growth. Density is described here as population size divided by the area occupied by a population (see Section 1.1 for a generic definition). If the area under consideration is constant over time, population density is proportional to population size, and the influence of density on biological processes can be expressed in terms of population size. Because our focus here is primarily on population dynamics rather than fluctuations in available area, we make the convenient assumption that the area associated with a population is constant over time. Density dependence can be introduced via the bal-
135
136
Chapter 8 Traditional Models of Population Dynamics
ance equation (1.1) from Chapter 1, which expresses population change over a unit time step in terms of birth, death, and migration. The balance equation also can be written in terms of per capita rates, by N ( t + 1) = N(t) + B(t) + I(t) - D(t) - E(t) =
[1
+
bt +
it -
d t -
et]N(t)
= )ttN(t),
which provides the simplified form N ( t + 1 ) = )ttN(t)
to describe population dynamics. The parameter )~t, called the finite rate of population growth, expresses the per capita change in a population over a unit of time by N ( t + 1) - N(t) N(t)
=
)~t - -
1.
It is clear from this equation that a population increases, decreases, or remains constant over [t, t + 1] as )kt ~ 1, ~kt ~ 1, o r )k t = 1 . Density dependence is established by the influence of population size on the population rate of growth, that is, by ~kt - - M N ( t ) ) at each point in time. On the other hand, density independence obtains when the population rate of growth is independent of N(t). A familiar form of density-dependent growth has M N ( t ) ) decreasing monotonically in N(t), so that a larger population has a lower population growth rate, and population losses--for example, through removal of individuals through harvest--lead to an increase in the population growth rate. In the latter scenario, increases in growth rate are said to "compensate" for population losses. By inducing compensatory changes in growth rate, monotonic density dependence attenuates the effects of environmental variation, harvest management, and other influences, and thereby promotes stability in population dynamics. Of course, other forms of density dependence also are possible. For instance, h(N(t)) could be monotonically decreasing in N(t) for large values of population size, but monotonically increasing for small values (the Allee effect of Chapter 1) (Allee et al., 1949; Courchamp et al., 1999; Stephens and Sutherland, 1999). In this situation an increase in the size of a depauperate population (for example, through stocking of individuals) would actually lead to an increase in the population rate of growth. Such an effect might arise through an increased frequency of mating, as a result of additional mating opportunities. On the other hand, the removal of individuals from a depauperate population would
lead to declining growth rates, ultimately resulting in population extinction. In the following sections we discuss the forms and effects of density-dependent growth, preparatory to the consideration of other factors such as population structure and interspecific interactions. We begin with the exponential model and variations of it that do not include density dependence, and compare behaviors against models such as the logistic equation that do include density-dependent factors.
8.1. D E N S I T Y - I N D E P E N D E N T GROWTH~THE EXPONENTIAL MODEL The exponential model is perhaps the simplest of all models for population dynamics. It is used to describe population growth that is directly proportional to population size at each point in time, in the absence of mechanisms for regulating growth. Typical derivations of the model incorporate a number of restrictive biological assumptions: 9 Generations are either nonoverlapping (as in annual plants), or else surviving offspring reach sexual maturity within one time step. 9 All individuals in the population have the same reproductive potential and the same survival probability. 9 The per capita rate of growth for the population is not subject to temporal variability. 9 There are no density-dependent mechanisms that alter the population rate of growth in response to changing population densities. These assumptions result in an exponential model with constant per capita growth rate over the time frame of the model (see Figs. 7.3 and 7.4). The discrete-time form of its transition equation is N ( t + 1) = N(t) + rN(t),
(8.1)
which yields a population trajectory with the elements N(t) = N(t0)(1 + r) t.
The continuous-time analog is d N / d t = rN(t),
(8.2)
corresponding to a population trajectory given by N(t) = N(to)e rt
(see Chapter 7 for mathematical associations between discrete and continuous forms of the exponential model). If the rate parameter r is positive, the trajec-
8.1. Density-Independent Growth--The Exponential Model tories of both continuous and discrete models exhibit uncontrolled, explosive population increases known as exponential population growth. For negative values of r, the population trajectory exhibits exponential declines, leading asymptotically to population extinction. For the continuous model, population change is smooth over the course of the time frame, with the rate of change determined by the initial population size N ( t o) and the maximum growth rate r. Population change for the discrete model occurs in steps as time increases in discrete increments. Again, the rate of change is controlled by N ( t o) and r. These patterns of unregulated change exemplify density-independent population dynamics (Fig. 8.1). Often the growth rate r in the exponential model is disaggregated into birth and death components" N ( t + 1) = N(t) + rN(t) = N(t) + ( F -
d)N(t)
= FN(t) + (1 - d)N(t) = FN(t) + SN(t),
where S is the single-period survival rate and F is the net reproduction rate (also referred to as fecundity). Because net reproduction includes both birth and survival, the parameter F can be further disaggregated into a product of survival and birth rate. Assuming reproduction occurs at the end of [t, t + 1], only S x N(t) individuals are available to produce b[S x N(t)] offspring, which then are recruited into the population: N ( t + 1 ) = (bS)N(t) + SN(t)
= S(1 + b)N(t),
(8.3)
137
Steady-state conditions dN/dt = 0 and N(t + 1) = N(t) for the exponential model can be attained only on condition that r = 0 or N(t) = 0. The former condition eliminates all population dynamics, reducing the population size to a time-invariant constant across the time frame of the model. The latter condition is tantamount to there being no population at all. In either case, population dynamics are trivial, with N(t) = N o. For r ~ 0 and N O ~= 0, from Eqs. (8.1) and (8.2) it is easy to see that the population grows or declines depending on the sign of the rate parameter r. Obviously, no population can continue to increase exponentially over an indefinite period; there are limits to the growth of all populations (see Chapter 1) (Malthus, 1798; Lotka, 1956). Thus, an exponentially increasing population eventually must approach the limits of the resource base needed to support it, and the tendency toward ever-increasing growth leads to resource depletion and population collapse. Thus, population trajectories for exponential populations might be expected to follow a cyclic pattern of explosive growth and population collapse (Fig. 8.2). It is necessary to modify the model accordingly, to allow not only for growth but also for periodic population crashes. Suitably modified, the exponential model sometimes is used to describe the growth of insect populations and other opportunistic species with high reproductive potential. The key biological features in most applications are short generation times, high numbers of offspring, and the lack of any mechanism to regulate population size. In the case of insect infestations, artificial means such as application of insecticides sometimes are used to limit population size or to hurry along the periodic population crashes.
with b the per capita reproduction of survivors.
1800 1600
1400
1400
1200
1200 1000 1000 800
800
600
600 400
400
200 200 10
2'0
30
40
50 20
FIGURE 8.1 Exponentialpopulation growth. At each point in time, the rate of population growth is proportional to population size.
2'5
30
t
FIGURE 8.2 Exponential population growth for a population with periodic population crashes.
138
Chapter 8 Traditional Models of Population Dynamics
8.1.1. Habitat Effects Habitat conditions can be included in the exponential model by means of a variable E(t) that is (possibly) subject to management control through time. Habitat and environmental effects can be registered either through the reproduction process or through survivorship (or both). For example, the influence of E(t) on reproduction might be expressed as a linear function b = f(E(t)) = E o + cE(t),
so that a change in the amount or quality of available habitat leads to a proportionate change in per capita birth rate. Such a formulation might be used to record the deterioration of a resource base through time, with concomitant declines in population birth rate and eventual population extinction. The model also could express the potential for population growth as a result of management efforts to increase the amount and quality of available habitat. Alternatively, habitat and environmental effects might be registered through the influence of habitat on survivorship. For example, the relationship of habitat and survival rate might be modeled by the function St =
E(t) + K
where K is the amount of habitat at which survivorship is half the maximum survivorship S0. In this case the value So is approached asymptotically as the habitat measure E(t) is increased. Again, a declining resource base through time would lead to lower survivorship and thus to declines in the population growth rate. On the other hand, efforts to improve the habitat base would result in improved survivorship and increased potential for population growth.
8.1.2. Harvest Effects
decline over time. Note that the sustainable harvest rate is independent of population size, in that the same rate applies to the population irrespective of its size. An alternative approach to the modeling of harvest is to express harvest impacts through a relationship between harvest and survival rate. The compensatory mortality and additive mortality hypotheses described earlier offer two expressions for such a relationship. If harvest mortality simply adds to other sources of mortality such as disease and predation, the relationship between survival and harvest takes an approximately linear form. Strict additivity has an increase in harvest rate leading to a corresponding decrease in survival rate: St
=
S0[1
-
h(t)],
where So denotes the probability of survival that would exist in the absence of hunting mortality. This relationship assumes that harvest and nonharvest mortality act in the manner of independent competing risks (e.g., Berkson and Elveback, 1960; Chiang, 1968) and is known as the additive mortality hypothesis (Anderson and Burnham, 1976). The effect of the relationship is essentially to add a harvest component to nominal mortality, thereby decreasing the survival rate St to S011 - h(t)]: N ( t + 1) = N(t) + bN(t)
- {1 - S011 - h(t)]}N(t). On the other hand, changes in harvest mortality may be compensated by corresponding changes in other sources of mortality (e.g., increases in harvest may bring about decreases in risks associated with nonhunting mortality). A simplified expression for compensatory harvest mortality has survival rate remaining unchanged over a range of values for harvest rate up to some compensation limit and declining thereafter as harvest rate increases:
Harvest can be included in the exponential model by means of a control variable H(t) specifying the level of harvest at time t. In discrete time a harvest model might be described by N ( t + 1) = N(t) + rN(t) - H(t),
where H(t) represents the postreproduction harvest of individuals in the population at time t. The population remains unchanged through time if H(t) = rN(t), i.e., if the harvest rate h(t) = H ( t ) / N ( t ) is identical to the population rate of increase. This is the maximum harvest rate that allows for a sustainable population; any harvest rate in excess of r causes the population to
h(t) > C h(t) <- C.
Thus, compensatory harvest has no effect on population dynamics if the harvest rate is sufficiently small, but is additive if the harvest rate is in excess of C (see Section 11.1.3 for a more detailed treatment of the additive and compensatory mortality hypotheses). With additive harvest, the rate h=l
1-b So
8.2. Density-Dependent Growth--The Logistic Model produces an equilibrium population, whereas with compensation the harvest rate h--1
(1 -
b)(1 -
C)
So
yields an equilibrium. Note that the equilibrium harvest rate does not include population size for either additive or compensatory mortality. However, the biological mechanisms giving rise to compensatory mortality are density dependent. A model that incorporates this density dependence explicitly was presented in Section 3.4.2, and the importance of density dependence to the concept of compensatory mortality is discussed at several points in later chapters.
139
it often is appropriate to model population growth with a stochastic model in which the growth rate r(t) is assumed to be a random variable. For example, an application might involve choosing a value for r(t) at each time t from a specified probability distribution and then updating the population size for time t + 1 based on N ( t + 1) = [1 + r(t)lN(t). Repetition of these steps over the model time frame produces random population trajectories, which inherit their stochasticity from the probability distribution of r(t). We discuss stochastic processes in more detail in Chapter 10.
8.2. D E N S I T Y - D E P E N D E N T GROWTH~THE
LOGISTIC MODEL 8.1.3.
Perturbations
Though structurally simple, the exponential model can exhibit rather complicated dynamics when pulse events are used to alter the model parameters. Assume, for example, that periodic and precipitous environmental declines result in immediate reductions in population size. Then the population exhibits "sawtooth" dynamics, with periods of exponential increase punctuated by periodic population collapse (Fig. 8.2). If adverse conditions result in a change from a positive rate of growth to one that is negative, the population exhibits a pattern of exponential growth followed by exponential decay (Fig. 8.3). Of course, both kinds of perturbation can be incorporated into the same model, whereby perturbation events result in both an immediate reduction in population size and a change in the rate of growth. Because birth and death rates can be sensitive to environmental factors that vary randomly over time,
140 120 100 80 60
The logistic model of population growth was developed by P. F. Verhulst in a series of three papers published from 1838 to 1847 [see historical notes of Hutchinson (1978)]. The logistic model incorporates in a simple manner an intrinsic regulatory response to population size, through the depression of growth rates as populations increase (see Fig. 7.9). Thus, the per capita growth rate of the logistic model declines from r for small population sizes, to zero when the population size is K. In discrete time the model is N ( t + 1) = N(t) + rN(t)E1 - N ( t ) / K ] ,
(8.4)
with an analogous continuous form of dN/dt = rN(t)[1 - N ( t ) / K ] .
(8.5)
The parameter K, known as the carrying capacity, specifies the maximum sustainable size to which a population can grow (i.e., the "carrying capacity" of the resource base supporting the population). The model allows for the initial population to exceed K; however, the growth rate then is negative, and population size is decremented as the population asymptotically declines to K. Starting with a small population, growth rates for the logistic model increase with population size up to some maximum rate and then decrease to zero as the population approaches its carrying capacity (Fig. 8.4). Assuming t >- to and N ( t o) = N 0, the trajectory
40
N(t) =
K 1 + Ce -r(t-t~
20
10
2o
3o
4o
t
FIGURE 8.3 Exponential population dynamics, with periodic changes between positive and negative rates of change.
with C = K/No - 1 solves the continuous logistic equation (8.5) (see Appendix C). If N O< K / 2 , this solution has the familiar $ logistic shape, with monotonic increases in population size over t >- t 0, an inflection point for N ( t ) = K / 2 , and asymptotic convergence of
140
Chapter 8 Traditional Models of Population Dynamics Steady-state conditions obtain when harvest exactly balances population growth, that is, when
100
H(t) = rN(t)[1 - N ( t ) / K ]
80 or 60
h(t) = H(t) / N ( t )
= r[1 - N ( t ) / K ] .
40
J 2o
40
60
80
F I G U R E 8.4 Population dynamics for a logistic population. Population rates of growth are low when population size is near zero or K. The m a x i m u m population rate of growth occurs when population size is half the carrying capacity.
N ( t ) to K as t --~ oo. This pattern of change is an example
of density-dependent population growth, with populations that are attuned to their resources and population dynamics that tend toward resource-based equilibria. A key to the pattern is density dependence in reproduction a n d / o r survivorship. Indeed, the logistic model can be seen as a simple modification of the exponential model, to include linear density dependence in the rate of growth r -- b - d. Equilibrium states for the logistic model can be found by setting d N / d t = 0 or N ( t + 1) = N ( t ) . In either case the resulting equation is
This is the maximum harvest rate allowing for a sustainable population; any larger harvest rate causes the population to decline. Note, however, that the sustainable harvest rate is dependent on population size. One implication of this dependence is that, within certain limits, each level H(t) of annual harvest corresponds to an equilibrium population size that can sustain it. A question of traditional interest to managers concerns the "maximum sustainable harvest," i.e., the maximum level of annual harvest that can be maintained over time. This harvest level corresponds to the equilibrium population size that is given by dH/dN
= r-
2r(N/K)
=0 or
N* = K / 2 .
The maximum sustainable harvest then is H* = rN*(1 - N* / K)
= rN*(1 - 0.5K/K) = (r/2)N* = rK/4,
rN(t)[1 - N ( t ) / K ] = 0,
from which it follows that N* = 0 and N* = K are population steady states. The equilibrium condition N* = 0 is shown in Appendices C and D to be unstable, in that small population sizes lead to population increases. On the other hand, N* = K is a stable equilibrium, in that small deviations of population size from K* are eliminated through time. The population level corresponding to maximum growth is found by simple differentiation of the growth rate rN(t)[1 - N(t)/K]. After some arithmetic it can be shown that the population grows most rapidly at half the maximum population size, or K/2.
8.2.1. Incorporating Harvest As with the exponential model, harvest can be incorporated in the logistic model by means of a variable H ( t ) specifying the postreproduction harvest of individuals at time t. A discrete-time model that includes harvest is N ( t + 1) = N ( t ) + rN(t)[1 - N ( t ) / K ] - H ( t ) .
and the optimal per capita harvest rate is given by h*= H*/N* = r/2.
Note the following conditions: 9 The optimal per capita harvest rate h* is simply one-half the maximum rate of growth and is not influenced by the carrying capacity K. To determine the optimal per capita harvest rate it is necessary only to know the rate r. 9 The optimal sustained population size N* is onehalf the carrying capacity K and is not influenced by the rate of growth r. To determine the optimal sustainable population size it is necessary only to know the carrying capacity for the population. At a population size of one-half the carrying capacity the population grows as rapidly as possible, and the harvest of this growth maintains the population in optimal equilibrium. 9 The maximum sustainable harvest H* is the product h'N*, or H* = r K / 4 . To determine the maximum sustainable harvest it is necessary to know both the rate of growth r and the carrying capacity.
8.3. Cohort Models
8.2.2. Incorporating Time Lags Lags can be incorporated in the logistic model by including a lag parameter -r in the density-dependent term (e.g., see Hutchinson, 1948; Wangersky and Cunningham, 1957; Caswell, 1972). Thus N ( t + 1) = N ( t ) + rN(t)[1 - N ( t - "r ) / K ]
for the discrete model, with an analogous form for the continuous model: dN/dt
= rN(t)[1 - N(t - "r)/K].
The effect of a lag is to accelerate the growth of the population to the carrying capacity. To see why, assume that a population is below its carrying capacity at times t and t - r with the population size at time t - r less than the population size at time t. Then the damping effect of the carrying capacity is not as great if N ( t - T) is used in place of N ( t ) , and consequently, population growth is more robust. One result is that the population eventually grows beyond the carrying capacity, resulting in a population size in excess of K. Population growth beyond the carrying capacity continues until N ( t - ~) = K, at which time, population growth ceases and then becomes negative. This leads to a downward trajectory of the population, which reduces the population to a level below the carrying capacity. Population reductions continue until the lagged population size reaches carrying capacity, at which time the population begins to increase again. These oscillations, which are a direct result of a lag in adjustment for the carrying capacity, are larger in amplitude as both the lag time and the maximum rate of growth increase. Within certain parameter limits, they eventually damp out, approaching the population carrying capacity asymptotically over time. Example
The effects of the parameters r, K, and T can be illustrated with a sensitivity analysis of the logistic model that is parameterized by r = 0.3, K = 3000, r = 4, and N O = 10. Figure 8.5a displays trajectories for this model for a 50% proportionate change in r, from 0.15 to 0.45. Note that larger values of r lead to more rapid growth toward the carrying capacity, more extreme amplitudes in the oscillations, and a longer transition period before the oscillations damp out. Figure 8.5b displays trajectories for the model based on a 50% increase in K to 4500 and a 50% decrease to 1500. Larger values of carrying capacity again lead to larger amplitudes in the oscillations and an extended transition period until stabilization. However, the effect of a proportionate change in carrying capacity is not as severe as the same proportionate change in the intrinsic rate r. Figure 8.5c displays model trajectories for a change
141
in the lag r of one time step. Note that larger lags have a strong effect on the oscillation amplitudes and lead to substantially greater transition times until stabilization. As the lag becomes larger the oscillations essentially become nondamped and the population exhibits stable oscillatory behavior throughout the time frame of the model. The logistic equations described above represent only a few of the expressions that have been used to model density dependence in single-species population dynamics (May, 1972, 1974a,b, 1975, 1976; May et al., 1974; May and Oster, 1976). In particular, difference equation models can exhibit bifurcations, whereby the dynamical behavior moves from a stable point to stable cycles of differing period and finally to a regime of apparent chaos (e.g., May, 1976; May and Oster, 1976). Quite complicated dynamics of this sort can be produced by simple model structures, with very different behaviors resulting from different parameter values. 8.3. C O H O R T M O D E L S If there is substantial variation in reproduction or survivorship among individuals in a population, it often is useful to aggregate individuals into population segments or cohorts, with cohort-specific parameters controlling reproduction and mortality. The idea is to capture heterogeneity among individuals in a population by stratifying the population into groups of individuals that are homogeneous in reproduction and survivorship. Depending on the nature of the cohort structure, it often is necessary to account for transitions among cohorts.
8.3.1. Populations with Independent Cohorts In some cases, a population can be represented with cohorts that are independent, in that there are no transitions among them. Genotypic variation in a population of asexually reproducing individuals provides an example. Consider a population that is partitioned genetically into, say, k cohorts, each characterized by its own genetically based maximum growth rate r i and its own initial size Ni(O). A continuous-time exponential model for such a population is d N / d t = r l N 1 4- raN 2 4- ... 4- rkN k
= [~iPi(t)ri]N(t) = ~(t)N(t),
142
Chapter 8 Traditional Models of Population Dynamics
a
FIGURE 8.5 Population dynamics for a logistic population with per capita growth that includes a time lag in population size. The time lag induces oscillations that damp out over time. (a) Effect of a 50% increase and 50% decrease in r. (b) Effect of a 50% increase and 50% decrease in K. (c) Effect of an increase and decrease on one time step in the lag. The first column shows standard parameterization, the second column displays effects of decreasing parameter values, and the third column displays effects of increasing parameter values.
N(t)
b N(t)
/
/
c
N(t)
where N(t) = ~ i N i is the aggregate population size and ~ (t) = ~ i pi(t)ri is an average of cohort growth rates, w e i g h t e d by the cohort proportions Pi = Ni/N. Change in the average population rate of growth through time reflects the change in these proportions:
d~ -
piri]
of change of fitness (measured here by the rate of growth ~) in a population at any instant in time is equal to the variance of fitness a m o n g genotypes in the population at that time (e.g., see C r o w and Kimura, 1970). The cohort with the largest rate of g r o w t h eventually dominates such a population, with convergence of the population growth rate to that of the d o m i n a n t cohort.
Example
= [ N ~, rdNi i---~
--
(~i riNi,]-~j-~ ~dN11
The change in cohort proportions can be illustrated with a model consisting of only two cohorts, with initial cohort sizes NI(0) and N2(0). Let
p(t) = Nl(t)/N(t),
l
1 = N ~ r2Ni- y2N2]N2
where N(t) = Nl(t) + N2(t). The change in proportions is given by
i
= ~ pir2 _
~2.
i
This is a special case of Fisher's (1930) f u n d a m e n t a l theorem of natural selection, which states that the rate
dt
=
~L-N/ dN1
=
_ dN1 1
N--d- f- - N,-~JlCr
8.4. Models with Age Structure
= [N(rlN1) - N I ( r l N 1 + r2N2)]/N 2 = (r I - r 2 ) N I N 2 / N 2 = (r 1 - r2)p(1 - p). For r I > r 2 the proportion p(t) exhibits logistic growth, increasing asymptotically in time to 1. This confirms the tendency of the cohort with largest growth potential to dominate other cohorts. For the two-cohort model, the pattern of change is logistic, with asymptotic convergence of p(t) to unity. In the behaviors of these simple cohort models one can observe certain properties that otherwise might not be anticipated. For example, though all the cohorts exhibit exponential growth, the growth of the population as a whole is not exponential (because the population intrinsic rate of growth is not constant). This illustrates the concept of an emergent property, whereby patterns of change are manifested at one level of ecological organization but not at others.
143
what follows, we use the index a to denote age, as in the reproduction rate ba for animals of age a. When convenient we also use the index i to characterize age for discrete age classes, as in the survival rate S i for animals in age class i.
8.4.1. Life Tables A traditional approach to age structure organizes age-specific model parameters into a life table. There are two key parameters involved in life table analysis. The first is a survivorship function la, defined as the probability of survival from birth to age a. To illustrate, assume that individuals mature continuously over time, so that a continuous survivorship function can be expressed as la = e-f~ "(~)dv where l0 = 1 and bL(v) is the instantaneous risk of mortality to an animal of age v, i.e.,
ix(v)dv = Pr[death in (v, v + dv)lsurvival to age v]
8.3.2. Transitions among Cohorts Many populations have cohorts that are not independent. Perhaps the most familiar example involves models incorporating age structure in a population, wherein one age class matures into the next older class. Models that include size structure generalize this situation by allowing individuals in a size class to remain in the class or to transfer into a larger (or smaller) class over time. Models that incorporate geographic structure generalize the situation yet further, by allowing individuals to transfer among classes that lack the natural ordering of age and size. For models with interacting cohorts such as these, it becomes necessary to account for the transfer of individuals by means of cohort-specific transition equations. In the sections to follow, we investigate the behavior of models with age, size, and geographic structure. We will see that the trajectories of these more complicated models generalize the simple growth patterns discussed thus far, but at some considerable cost in mathematical complexity.
8.4. MODELS WITH AGE STRUCTURE Demographic parameters for many populations vary with the age of individuals in the population. It is useful under these conditions to model the population as an aggregation of age classes, with distinct survival a n d / o r reproduction rates for each class. In
(Caswell, 2001). Here we assume that instantaneous risk can be aggregated from birth to any particular age a of an individual. The survivorship function can be used to describe specific survival probabilities for populations having two different temporal patterns of reproduction. The first is known as birth flow (Caughley, 1977; Caswell, 2001), in which reproduction occurs continuously over the interval [t, t + 1]. Then the probability S a that an individual of age a at time t survives the interval is approximated by
fi+llvdv Sa ~
a
f
lv dv a-1
la+ 1 + la la + la-l" where the numerator and denominator approximate the average l v for animals in age classes a and a - 1, respectively (Caswell, 2001). The second pattern is known as birth pulse, in which reproduction is concentrated in a relatively short breeding season. Age-specific survival in discrete time is obtained in a more straightforward manner for birth pulse populations. Thus, age-specific survival is defined by
Si-- li+l/li. Population growth over each interval may be considered from times either immediately prior to (prebreed-
144
Chapter 8 Traditional Models of Population Dynamics
ing census) or following (postbreeding census) reproduction. We note that for postbreeding censuses, survival over the first age interval is given by So = l ( i ) / l ( O ) and covers the period from birth until age 1 year. On the other hand, this mortality component for prebreeding censuses is included in the reproductive parameters (see below). In either case, age-specific survival determines the number of animals in an age class that survive to the subsequent age class over [t, t + 1]. For example, Ni+l(t + 1) = N i ( t ) S i, where N i ( t ) is the number of animals of age class i alive at time t. As with survival, reproduction also may be considered a continuous function of age, according to a maternity function b a. Continuous forms for this function are considered in Section 8.4.2; here we consider reproduction assuming discrete time. Computation for birth flow populations is complicated by the fact that the average number of births occurring over an interval must be approximated, and several ways of doing so are described by Caswell (2001). For a birth pulse population, an age-specific reproductive or fecundity rate can be calculated as F i = S ibi+l
for a postbreeding census, where b i is the per capita number of age 0 animals born to individuals of age class i. Here F i represents the per capita number of offspring the following year, from individuals of age i in the current year. This definition reflects the fact that an animal of age i following breeding in year t must survive the year until the next breeding period in order to reproduce. In contrast, individuals of age class i in a prebreeding census reproduce and a portion of their offspring survives to the subsequent year, to become members of
TABLE 8.1 i
0 1 2 3 4 5 6 6+ a
that year's prebreeding population. Recruitment under this scenario is defined as F i = Sobi,
where the only survival rate at issue is that for the newly born animals in their first year of life. Thus F i reflects the number of young in the prebreeding period of year t + 1 per animal of age i in the prebreeding period of year t. Unless otherwise noted, in what follows we describe population dynamics in terms of postbreeding census times. Table 8.1 illustrates reproduction and cohort aging for a hypothetical cohort of 1000 newborn animals that is followed until all are dead following 6 years of age. We note that for sexually reproducing organisms, definitions of parameters such as b i and F i lead to a tendency to use age-specific population models that follow the female component of the population rather than both males and females. In most of the following discussion, we simply refer to individuals, but the reader should note that the ideas apply most naturally to females. It is possible to develop two-sex models that incorporate different vital rates for the sexes. These models will not be described here, but the interested reader is referred to Caswell (2001; also see Keyfitz, 1968, 1972; Pollard, 1973; Yellin and Samuelson, 1977; Schoen, 1988).
8.4.2. D i s c r e t e - T i m e M o d e l s with Age Cohorts
For discrete-time models of populations with age structure, one must include transition equations as above for each age cohort in the population. Thus, surviving individuals in any age class except the last automatically transfer into the next age class. The last age class can be modeled in either of two ways. In some formulations (e.g., Leslie, 1945), individuals in
Life Table Data for Hypothetical Cohort of 1000 Animals a
li
Si
N i
bi
Yi = Sibi+ 1
Bi = NiF i
1.000 0.250 0.162 0.114 0.080 0.040 0.024 m
0.250 0.650 0.700 0.700 0.500 0.600 0.000 m
1000 250 163 114 80 40 24 0
0.50 1.00 2.00 2.00 2.00 2.00
0.125 0.650 1.400 1.400 1.000 1.200 u
125 163 228 160 80 48 0
Followed from birth (age class i = 0) until all have died (age class i = 6).
8.4. Models with Age Structure the final age class k at time t are all assumed to die before reaching time t + 1. In the other formulation, the oldest age class represents all individuals in the population of age k or older, and surviving members of the cohort remain there. Recruitment for such agestructured models are given as an aggregate of agespecific reproductive efforts, based on cohort sizes at the time of reproduction (Fig. 8.6). A conventional model for this situation includes age-specific survival and reproduction rates, which are assumed for now to be constant over time. The transition equation for each age cohort except the first and last is
Ni+ 1 (t + 1) = SiNi(t) , with S i representing the probability of survival from t to t + I of individuals in age cohort i. Because surviving individuals from cohort i at time t are recruited into cohort i + 1 in time t + 1, both the subscript and time index in this equation are incremented. Updating the oldest cohort involves the addition of surviving individuals from the oldest and next oldest cohorts:
Nk(t + 1 ) = SkNk(t)
+ Sk_lNk_l(t
) .
Reproduction in each time period can be handled in one of two ways, depending on the census time. For populations censused just after breeding occurs, reproduction is based on the reproduction rates of surviving individuals from the previous time period:
145
time period as the number of surviving newborns from the previous time period:
[k
]
Nl(t + 1 ) = S O ~_, biNi(t) . i=1
Example A simple extension of the discrete model [Eq.(8.3)] allows for two age classes: a "birth-year" or juvenile class that survives at rate Sj over [t, t + 1], but does not reproduce during that time, and an "adult" class that survives at rate Sa and produces b young per adult. At the beginning of each year (in this development we assume a postbreeding census) the population is of size N(t) = Na(t) + Nj(t), where Nj(t) and Na(t) are the number of birth-year and adult (i.e., breeding age) animals, respectively. The transition equations for adults and juveniles are
Na(t + 1 ) = SaNa(t) + SjNj(t) and
Nj(t + 1) = [SaNa(t) 4- SjNj(t)]b, respectively. The finite rate of increase for each age class is )~a(t)
=
Na(t + 1)/Na(t)
= S a 4- S j [ N j ( t ) / N a ( t ) ]
and )~j(t) = Nj(t + 1)/Nj(t)
k
No(t + 1 ) = ~ biNi(t + 1)
= +(SaNa(t) \ N---~ Sj)b,
i=1 k-1
= ~ bi+lSiNi(t) i=0
4-
bkSkNk(t).
On the other hand, reproduction for populations censused just before breeding is carried forward at each
and the population rate of increase is given in terms of these cohort rates:
)t(t) = N(t + 1)/N(t) = [Na(t + 1) + Nj(t + 1)]/N(t)
= [)ta(t)Na(t) + )tj(t)Nj(t)]/N(t).
F I G U R E 8.6 A g e - s t r u c t u r e d m o d e l w i t h t w o age classes. The c o n v e r t e r s B 1 a n d B2 are g i v e n b y B i = biSiNi, i = 1, 2, w i t h S i = 1 - d i. The i n p u t f l o w for N 1 is the s u m of B 1 a n d B2, a n d the i n p u t f l o w for N 2 consists of the n u m b e r of s u r v i v o r s f r o m N 1.
Because the factor N j / N a can vary over time, the rates )k a and )tj can as well, and thus the population rate )~ changes as the population grows. A constant rate of growth for the population requires a stable age distribution, that is, a constant proportion of animals in each age class. If the population is not at stable age distribution, growth rates will change every year until a stable age distribution is achieved, even with constant survival and reproduction rates. Once a stable age distribution is attained, the growth rates of the two age classes become equal: )k a = )kj = )k. Of course, age stability is reached quickly for a simple two-cohort population (Fig. 8.7).
146
Chapter 8 Traditional Models of Population Dynamics ing at time t to postbreeding at time t + 1. Combining Eqs. (8.6) and (8.7) in sequence results in
No(t + 1) Nl(t + 1) N2(t + 1)
/
3o
(8.8)
jf J J 1
1
/
J
\
~
.Nk(t + 1).
Nl(t) N2(t)
'\\//
]
-No(t) N~(t)
= FSoblS;b 25 1. ""SkolbkS!.
,
t F I G U R E 8.7 Dynamics of a prebreeding population model with two age cohorts and constant per capita birth and survival rates. 0
For a multicohort model, both survivorship and reproduction can be expressed in terms of matrix multiplication. Assuming a postbreeding census, the product
m
0
0
0
0
0
$1 0 ...
..
No(t)
No(t) S0 0 0 . . .
Nk-l(t) | Nk(t) J
J
Sk
which tracks k + 1 age classes in N(t) = [N0(t), Nl(t), ..., Nk(t)]' through time. One also can track the transitions for a prebreeding census, simply by switching the order of the matrix multiplications shown above. Thus,
B
Nl(t + 1) N2(t + 1)
Sk-1
Nl(t) N3(t)
Nl(t) N2(t)
m
bl 1 0
b2 ... bk 0 ... 0 1
...
0
0
0
...
1
,,.
m
Nl(t) N2(t) (8.9)
(8.6) 0
Nk(t + 1)
0
0 ... Sk_ 1
Sk
Nk(t)
Nk_l(t) Nk(t)
represents survival and aging from the period immediately after breeding in year t to immediately before breeding in year t + 1. In turn,
characterizes the transition from prebreeding to postbreeding at time t, whereby No(t) newborns are added to the population. Then m
[Nl(t + 1) N2(t + 1
-No(t + 1)" Nl(t + 1) N3(t + 1)
Nk(t)
0 0...
0
fl
S1 0
0
o
9
9
. . . . . . . (8.7)
0 0
lNk(t + 1
i
."
Sk-1 k
m
bl
b2
bk- -Nl(t + 1)-
1
0
0
1
0 0
9
o
0
0
1
N2(t + 1)
accounts for reproduction of surviving individuals at time t + I and completes the transition from postbreed-
Nk(t)
m
represents survival and aging of the population cohorts until just prior to breeding at time t + 1. The application of Eqs. (8.9) and (8.10) in sequence produces
Nl(t + 1)
I
rSobl Sob2 ... Sobk Sobk
N2(t. + 1)
Nk(t + 1)
(8.10)
Nk_l(t ) n
_Nk(t + 1).
m
N0(t) Nl(t) N2(t)
LNk(t + 1)
$11 $20 =
.
O0
O0
.
.
Nl(t)
. 9 (8.11)
|Nk-l(t) | 0
sk_~ sk
LG(t)
J
There are some noteworthy differences between the postbreeding transitions of Eq. (8.8) and the prebreed-
8.4. Models with Age Structure ing transitions of Eq. (8.11). First, the biological time reference differs for the two models, with Eq. (8.8) tracking population status just after breeding each year and Eq. (8.11) tracking population status just prior to breeding. Second, the survival parameters used to compute reproductive input differ between the two models, with age-specific parameters used in Eq. (8.8) and a single survival rate So used in Eq. (8.11). Note that when survival rate Sk for the oldest age class is zero, the reproductive contribution Skbk in Eq. (8.8) vanishes and the final column of the projection matrix consists entirely of zeros. This is not the case with model (8.11). Third, the vector of age cohorts in Eq. (8.8) includes the cohort No(t) of newborns, whereas the vector in Eq. (8.11) does not. Similarly, the (k + 1)-dimension projection matrix in Eq. (8.8) accounts explicitly for newborns, whereas the k-dimension matrix in Eq. (8.11) does not. In essence, the number of young in the postbreeding model [Eq. (8.8)] is treated as a state variable, along with the other cohort counts. In the prebreeding model [Eq. (8.11)], it is treated as an intermediate variable. The matrix A of age-specific constants for survival and reproduction in Eqs. (8.8) and (8.11) is known as a population projection matrix. A standard form for the postbreeding model, Eq. (8.8), is ..
No(t + 1) Nl(t + 1) N2(t + 1)
,,.
Nk(t + 1)
F~_I ik 51... 0 0
Sk-1
9
Nk-l(t)
Nk(t)
..
LNk(t + 1)
FIs1 F20 "'" FO-1 i k Nl(t)
0
0 5k-1
skJ
= ~ hNi(t) i = )~ ~ Ni(t) i
Example (8.12)
m
-- 0052
i
= KN(t).
where the parameter Fi (for fecundity) represents the number of young produced by survivors who were in cohort i at time t. For a prebreeding census, a standard expression for model (8.11) is
FNI(tN2(t++ 11))
N(t + 1)= ~Ni(t + 1)
m
Nl(t)
SkJ
.,
animal population ecology and in applied areas dealing with management and conservation. In what follows we refer to age-specific projection matrices of the general form of Eq. (8.12), with Sk = 0, as Leslie matrices. The above matrix projection model can be applied iteratively to determine cohort trajectories. Starting with an initial vector N(0) of cohort sizes, application of Eq. (8.12) or (8.13) yields the vector N(1) at time 1. Application of Eq. (8.12) or (8.13) a second time, using N(1) for input, yields N(2) at time 2. This process can be repeated indefinitely, with cohort sizes used as input to produce new cohort sizes the next time. It can be shown that repeated application of the model in this manner eventually leads to a stable age distribution for the population, i.e., an age distribution for which Ni(t + 1 ) = )~Ni(t). The parameter )~ = 1 + r specifies the population growth rate r for the population once it has achieved a stable age distribution:
No(t)
F~ F10 "'"
=
147
IN/l(t)|
(8.13)
LNk(t) J
where F i now represents the number at time t + 1 of surviving young that were produced at time t by individuals in cohort i. Population projection matrices with this general form (but with Sk = 0, indicating a final age after which all individuals die) were developed independently by Bernardelli (1941), Lewis (1942), and Leslie (1945, 1948). These models saw little use in animal population ecology until the 1970s (Caswell, 2001), but now are widely used in studies of
Consider a population with four age cohorts and age-specific survival rates of S' = (0.5, 0.65, 0.85, 0.4). Assume that the age-zero cohort consists of (nonbreeding) immature organisms and that reproduction rates for the other three cohorts are age specific: b' = (0, 1.0, 2.0, 3.0). The corresponding projection model is
No(t+ Nl(t + N2(t + N3(t +
I
1) 1) 1) 1)
=
0 0.5 0 0
0.65 0 0.65 0
1.7 0 0 0.85
1.2 0 0 0.4
~No(t)- ] |Nl(t) | /N2(t) | " LN3(t)_]
Figure 8.8 shows the trajectories of each cohort in the population starting with initial age distribution _N(0)' = (10, 100, 200, 500). Note that the cohorts exhibit variation early on in their trajectories, but gradually a stable age distribution is attained and all cohorts expand exponentially at the same per capita rate of growth. This behavior is indicative of Leslie matrix models. 8.4.2.1. Stable Age Distribution and Rate of Growth
Convergence of projection matrix models to a stable age distribution follows from the mathematical structure of the matrix A. In particular, the lead right eigenvector of the matrix specifies the stable age distribution
148
Chapter 8 Traditional Models of Population Dynamics
Nl(t)
2000
for every age cohort but the first. Assuming a stable age distribution, dynamics for the zero-age cohort are given by
1500
No(t) - ~kaNo(t - a)
/ / / ~/"
~- 1000
500
/\/k/~ / ~N , / V \ / ~ . , / ./
"
/
___/~/ /'----- /. / ~
'X,>~'/-
/
N2(t)
~.N3(t) N,(t)
or
No(t - a) = ~k-aNo(t).
Substituting these expressions into the transition equation for the zero-age cohort leads to k
No(t ) = ~ , baNa(t) 2
4
6
8
a=l
10
t
k
F I G U R E 8.8 Dynamics of a prebreeding population model consisting of four age cohorts, with constant per capita birth and survival rates.
= ~ , l a b a N o ( t - a) a=l
or k
of a population, and the lead eigenvalue is the population rate of growth X = 1 + r, assuming stable age distribution (see Appendix B for a discussion of eigenvectors and eigenvalues). Both the lead eigenvalue and elements of the lead right eigenvector are positive (Gantmacher, 1959). The lead eigenvalue and right eigenvector of A can be determined by solving the characteristic equation A P = XP
(8.14)
m
for X and P. Starting with any nonzero vector, iterative application of Eq. (8.8) or (8.11) eventually produces numerical values corresponding to both P and X. Alternatively, X (and therefore P) can be obtained as a solution of the well-known Euler-Lotka equation [Euler, 1970 (1760); Lotka, 1907, 1956] k
1 = ~
~k-abala,
a=l
where
No(t) = ~ , X-al~b~No(t), a=l
and division of both sides of this equation by No(t) produces the Euler-Lotka equation. The Euler-Lotka equation makes explicit the influence of survivorship and reproduction on the population growth rate. For example, the same value of L can be produced by a population with high cumulative survivorship and low reproduction, or a population with low survivorship and high reproduction. Clearly, if X > 1 the corresponding population trajectory will exhibit a pattern of exponential increase as it attains a stable age distribution, whereas the trajectory will show an exponential decrease if X < 1. Of course, if = 1, the population remains unchanged after the stable age distribution is attained. The Euler-Lotka equation can be expressed in terms of the parameters of a Leslie matrix, as in Eqs. (8.12) and (8.13) with Sk = 0. In terms of the postbreeding parameters in Eq. (8.12), the Euler-Lotka equation is k
a-1 la= H Si
1 = ~ , ~k-ala_l(Sa_lba ) a=l
i=0 k
and l0 = 1. The Euler-Lotka equation is really just a combined form of the transition equations, assuming a stable age distribution. A derivation (e.g., see Mertz, 1970) is based on
= K-1 ~
K-(a-1)la_lFa_l,
a=l
from which we get k-1
Na(t)-- S a _ l N a _ l ( t - 1) -- Sa_lSa_2Na_2(t-
K = ~ , X-alaFa . 2)
(8.15)
a=0
With Eq. (8.15) we can show that a vector P with components = laNo(t - a)
Pi = )t-(i-1)li-1,
8.4. Models with Age Structure i = 1, ..., k + 1, is the lead right eigenvector for a postbreeding projection matrix with Sk = 0. Thus,
149
with a-1
l* = II Sa*
-F o So 0
0
F1 0 S1
0
... ... ...
...
Fk_ 1 0 0
Sk- 1
O0 0
i=0
"k-1 " i~,=oh-ilif i
1 U-1ll
a-1
= ca[i.o
ll h-ll2
and h* the rate of growth for a population with agespecific birth rates b a and survival rates cS~. But
u-(k--'l)lk_l u-klk
0
= Cala
m
h-(k'-l)lk ,n
k
1 = ~ , (h*)-abal'~
with the lead term in the resultant vector equal to h from Eq. (8.15). Factoring h out of each of the terms in the resultant vector produces UP and demonstrates that P is an eigenvector of A. In terms of prebreeding parameters in Eq. (8.13), an expression of the Euler-Lotka equation is obtained by multiplying both sides by So:
a=l k
-- ~
(U*) -a ba(c a la )
a=l k
-- ~_~ (U*/c)-abala, a=l
which is satisfied by the unique rate of growth h corresponding to birth rates ba and survival rates Sa. Thus,
k
S O = ~_j h-ala(Soba) a=l
(8.16)
h*/c =
k or
= ~_~ h-alaFa 9 a=l
U* = c h .
Equation (8.16) can then be used to show that a vector with components Pi = u-(i-1)li,
i = 1,..., k, is the lead right eigenvector for a prebreeding projection matrix with S k = 0: .. k ~_, h-(i-1)liFi S1
0
...
0
]
12
OoS2"" 0i il |X-~k-2'/k-1 " _ 9 iii 0
...
Sk_ 1
L ~k-(k-1)lk
Example
i=1
h - 12
.
h-(k--2)lk
This result indicates that the scaling of survival rates induces an equivalent scaling of the population rate of growth, provided birth rates remain unchanged. For example, a 50% reduction of all the cohort survival rates results in a 50% reduction in the population rate of growth.
.
Factoring h out of the lead term of the resultant vector produces SO = lI from Eq. (8.16), so that the resultant vector can be expressed as UP and thus recognized as an eigenvector of A.
Consider two populations with age-specific survival rates that are related by S* = cS a, as in the previous example. Assume that the population with reduced survivorship also has geometrically larger birth rates, according to b* = b,,/c a. The Euler-Lotka equation for the latter population is k
1 = ~ , (h**)-a(ba / Ca) (Ca Ia) a=l k
Example
Tradeoffs between survivorship and reproduction can be illustrated by the scaling of age-specific survival rates. Suppose that each of the parameters S~ in a Leslie matrix is reduced by a positive constant c < 1, i.e., Sa is replaced by Sa* -- CSa. From the Euler-Lotka equation we have
-- ~_j (U**)-abala, a=l
which again is satisfied by the unique rate of growth U corresponding to birth rates b a and survival rates Sa. We therefore have U ~ "~b ~
U I
k
1 = ~,~ (h*)-abal*a, a=l
demonstrating that the two populations have identical rates of growth and confirming the fact that a geometric
150
Chapter 8 Traditional Models of Population Dynamics
scaling of birth rates "compensates" for the constant scaling of survival rates. Note that the scalings of survival and birth rates are reciprocal, in that a decrease in survivorship requires an increase in birth rates and vice versa. For example, a 50% reduction in survival rates requires a geometric doubling of birth rates in order to maintain the population growth rate. On assumption that a stable age distribution has been attained, it is straightforward to show that the pattern of relative cohort sizes at any point in time is determined by survivorship. Thus, a stable age distribution requires that Ni+l(t 4- 1 ) = KNi+l(t) ,
which, when combined with the cohort transitions Ni+l(t 4- 1 ) = SiNi(t) ,
produces Ni+l(t) Ni(t)
_
Si
-
--.
)~
(8.17)
It follows that the relative sizes of adjacent cohorts in stable age distribution vary with survival rates but not with birth rates. As argued below, this property can be used to advantage in determining recruitment to the population based on cohort-specific harvests.
parameters in a projection matrix model. Recall from Section 8.1 that N ( t + 1) = N(t) + rN(t) - (F + S)N(t) = KN(t)
for the exponential model, with F and S the per capita net reproduction rate and survival rate for an exponential model. Thus the factor )~, which scales N(t) to produce N ( t + 1), is simply the sum of the net reproduction and survival rates. In words, an exponential population at time t + 1 consists of those organisms alive at t that survive to t + 1, along with the offspring produced by surviving organisms. A population for which the sum F + S exceeds unity expands exponentially; a population with F + S less than unity declines exponentially. Now consider an age-structured population that has achieved its stable age distribution, with Pi = N i ( t ) / N(t) the proportionate representation of cohort i in the population. From the cohort transition equations in the projection matrix model, Eq. (8.8), we have k
N(t + 1)= ~ N i ( t
+ 1)
i=0 k
= N o ( t + 1) + ~ N i ( t
Example
+ 1)
i=1
Consider two populations with the same birth rates and with age-specific survival rates that are related by S* = cSi as in the previous example. Because the scaling of survival rates by a constant induces the same scaling of the population rate of growth, we have N*+l(t) _ S* N*(t)
)~* _ cSi cK _ Ni+l(t) Ni(t ) '
so that the relative sizes of adjacent cohorts are unaffected by constant scaling of survivorship, and both populations have the same stable age distribution. Thus, the scaling of survival rates affects population growth rate but not stable age distribution. An implication is that the pattern of age distribution in a population is not diagnostic of the potential for population growth. Indeed, the same stable age distribution can apply to populations that are increasing, decreasing, or stable. It is instructive to consider the relationship between the growth parameter )~ and the birth and survival
k-1
k
= ~
bi+lSiNi(t) + ~
i=0
=
SiNi(t) + bkSkNk(t)
i=0
Pifi + ~ -
PiSi N(t)
i=O
= (F + S)N(t) = KN(t).
As with single-age exponential populations [Eq. (8.3)], an age-structured population with stable age distribution exhibits exponential growth at a rate that depends on net reproduction and mortality. However, the reproduction and mortality parameters of the single-age model are replaced here with average reproduction and mortality rates, in which age-specific values are weighted by cohort proportions in the stable age distribution. Long-term population increases occur if the average reproduction and survival rates sum to a number in excess of 1, and long-term population decreases occur if the sum is less than 1. Thus, the same patterns are found for models with and_ without age structure, and the weighted averages b and S for the projection matrix model reduce to the population reproduction and survival rates for the single-age
8.4. Models with Age Structure model. Indeed, the Leslie matrix model can be described as a multivariate analog to the univariate exponential model.
8.4.2.2. Sensitivity Analysis A matter of some interest is the sensitivity of the population growth rate k to variation in survival and reproduction rates. At issue is the change to be expected in the asymptotic population growth rate k in response to a corresponding change in one of the vital rates. This issue has been addressed numerically (e.g., Cole, 1954; Lewontin, 1965; Mertz, 1971b; Nichols et al., 1980) and, for specific characteristic equations, by implicit differentiation (Hamilton, 1966; Demetrius, 1969; Goodman, 1971; Mertz, 1971a). For purposes of illustration, define 0 to be some component of a population projection matrix, i.e., a matrix element aq, a component of a matrix element, or a parameter that appears in multiple elements. The sensitivity of k with respect to 0 is the change in k that accompanies a small change in 0, or
151
models. Note also that they are equivalent (up to a constant) to reproductive values for individuals in age class/[see Eq. (8.20)]. A formula for the sensitivity of growth rate to changes in survival and fecundity can be expressed in terms of the components Pi and qj of the right and left eigenvectors of A (Caswell, 2001). Taking the differential of both sides of Eq. (8.14) produces (dA)P + A(dP) = (dk)P + k(dP),
and multiplication by the left eigenvector Q yields Q(dA)P + Q A(dP) = (dk)Q P + kQ(dP).
Substituting Q A = kQ into this expression and simplifying, we have Q(dA)P = (dk)Q P or
dk = Q ( d A ) P / Q P.
s = 0k/00. In particular, the sensitivity of k to changes in survival and fecundity rates can be expressed in terms of the eigenvectors of A. Recall that the rate of growth k is given by the characteristic equation A P = kP,
where k and P are the dominant eigenvalue and associated right eigenvector for the projection matrix A. From above, the lead eigenvector for a prebreeding projection matrix A with Sk = 0 has components
In the case of differential change in a single element aij of A, one therefore obtains the useful formula 3k / Oaij = q iPj/ Q P,
which asserts that the sensitivity of growth rate to aij is proportional to the product of the reproductive value of the ith cohort and the relative size of the jth cohort in stable age distribution. For example, the sensitivity of k to F i is 3k
Pi = K-(i-1)li,
3Fi
i = 1..... k. It can be shown (see Appendix B) that the lead left eigenvector Q of matrix A corresponds to the same eigenvalue as does the lead right eigenvector: Q A = kQ.
(8.18)
qlPi QP K-(i-1)li Qp "
so that
Direct substitution of the components qi of Q into Eq. (8.18) produces qi = K - l ( q l F i + Siqi+l),
3Fi+1
qlPi+l = )k/S i.
and choosing ql = 1 results in qi = K-1Fi + )k-lSiqi+l
qlPi
n
3Fi
Thus, the sensitivity of k to fecundity is monotone decreasing in age (assuming k>0). On the other hand,
)ki_ 1 k
= l,
E. .
]=l
,-Jljfj.
Note that these eigenvector components apply equally for both prebreeding and postbreeding Leslie matrix
c~)k/cgSi = qi+lPi/Q P
~ E k S i j=i+l
-Jlf j/
QP,
152
Chapter 8 Traditional Models of Population Dynamics MATLAB code to compute the matrix of sensitivities for any projection matrix.
so that 0k /
0k
qi+lPi
c~Si
cgSi+1
qi+2Pi+l
J
Jlj j
Sij=i+l
j=i+2
Si+ 1//S i.
Thus, the sensitivity of k to survival is monotone decreasing if survival rates increase with age. A measure of sensitivity that is useful for some comparative purposes is proportional sensitivity or elasticity, defined in Section 7.5.2 by e =
a~/ao x/o ax/x aOlO
8.4.2.3. D e m o g r a p h i c R e l a t i o n s h i p s f o r A g e - S t r u c t u r e d M a t r i x Models
Age-structured matrix models can be used to draw a variety of inferences about the populations that they characterize. Many of these inferences can be viewed as "asymptotic" in the sense that they apply to a population exposed to the same survival and reproductive rates (i.e., the same projection matrix) every time step. Such a population can be described in terms of its stable age distribution and asymptotic growth rate k. An inference that requires survival rates that do not vary with time, but does not depend on a stable age distribution, involves the expected life-span remaining to individuals in cohort (age) a. This quantity is given by
_ 0 log k,
Ea -- ~ a l x / l a , x~a
a log 0 where 0 is either survivorship Si or a reproductive parameter (F i or b i) for a projection matrix model. Elasticity is found by dividing this expression by k/0. For example, the elasticity of population rate of growth with respect to the survivorship and fecundity parameters is given by Ok / Oa q _ q iPja q k/aij
k Q P"
with aij = S i or F i. If aij = F i this expression becomes 3k/OFi k/F i
qlPiFi k Q P = k - l l i F i / Q P,
and if aij
=
S i
the expression is cOk/3Si m qi+lPiSi k/Si k Q P
with l x the probability of surviving to age x. To see why, note that the probability of surviving to age x is also the probability of dying at age greater than x. Thus, if Px is the probability that a newborn individual survives to age x and then dies, lx can be expressed as lx = P x + 1 4- P x + 2 4- "'" 9Furthermore, the probability is Px/la that an individual of age a survives to age x (with x > a ) and then dies (see Section 4.1.6). It follows that the average number of years remaining to individuals of age a is E(x - a]a) = ~ (x - a ) P x / l a x~a
= [Pa+l 4- 2Pa+2 4- 3Pa+3 4- "'" ]/la --[Pa+l 4- Pa+2 4- Pa+3 4- "'" 4-
Pa+2 4-
Pa+3 4- "'"
4-
Pa+3 4- "'" o
j=i+l
Although the above formulas are relevant to the Leslie matrix model, general expressions for stagebased projection matrix models (see Section 8.5; also see Lefkovitch, 1965; Caswell, 2001) have been derived for sensitivity (Caswell, 1978, 2001) and elasticity (Caswell et al., 1984; de Kroon et al., 1986; van Groenendael et al., 1988; Caswell, 2001). These expressions permit simultaneous changes in several life history parameters, but they simplify considerably for the case of changes in a single parameter. Caswell (2001) provides
.]/l a .
Thus, the average life-span remaining to individuals in cohort a can be expressed rather simply, in terms of the survival factors l x 9 E a = E(x - a]a)
- (/a+l + la+2 4- la+ 3 4- " " ) / l a -- ~ a lx/la" x~a
8.4. Models with Age Structure
153
so that
Example Assume that the survival rate for a cohort transition between successive ages is S, irrespective of cohort age. The probability of an individual surviving to age a under this assumption is la = ~
E(a) = ~
aP a
a~O ~laba~ : a>O ~ a \ x ~ o lxbx '1
Px
x~a
= ~
sx-l(1-
S),
x~a
which, after some algebra, simplifies to la = S a. The expected life-span remaining to individuals in cohort a is therefore Ea--
~lx/l
a
x~a
-- ~_j S x / S a x>a =
s/(1
-
s).
Thus, the average life-span remaining to an individual is S/(1 - S), no matter what its age. On reflection, this apparently counterintuitive result makes sense. Whether young or adult, an individual is assumed to survive from one year to the next with probability S. Under these circumstances all individuals alive at a given time are equivalent in their survival probabilities and therefore have the same expected life-span from that time on. A related demographic measure is the average age of reproduction, denoted here by E(a). E(a) characterizes the mean age at which individuals reproduce, based on age-specific birth rates and assuming stable age distribution. Leslie (1966) noted that E(a) provides a measure of generation time (cohort generation time) of a population characterized by a specific projection matrix. Let Bo(t ) = ~
Na(t + a)b a
a>O -- ~ No(t)laba a>O
represent the total number of offspring produced by individuals born at time t. Then the proportion of total offspring that are produced at age a is Pa =
Na(t + a)b(a) Bo(t) No(t)laba
~, No(t)lxb x
x>O
laba
~, lxb x"
x>O
is the average age of reproduction based on these proportions. Reproductive rate, another important measure for age-structured matrix models, focuses on the amount of age-specific reproduction rather than age. Consider the future production of offspring for Na(t) individuals in cohort a at time t, Ba(t ) = ~, Na+x(t + x)ba+ x x>>_0 = ~, No(t - a)la+xba+ x x>>_o
= No(t-
(8.19)
a) ~, lxbx. x>_a
Then the per capita future production of offspring or reproductive rate for individuals in cohort a is R(a) = Ba(t)/Na(t) = N o ( t - a) ~ lxbx/Na(t) x:> a
= No(t - a) ~, lxbx/No(t - a)l a x>__a
= ~, lxbx/la. x>~a
In particular, the net reproductive rate is the average per capita production over the lifetime of offspring: R o = R(O)
= ~ lxbx. x-O A comparison of R0 and ~,x K-Xlxbx shows that the net reproductive rate must exceed unity for the population to grow (i.e., for )~> 1). Conversely, the population declines ()~<1) when the net reproductive rate is less than unity. Finally, Fisher's reproductive value of individuals accounts not only for the number of offspring, but also for population growth over time (Fisher, 1930). We note that the accumulator Ba(t) in Eq. (8.19) assumes the equivalence of all future offspring, irrespective of when they are born. Alternatively, future offspring can be discounted by the population growth rate, to account for variation in the reproductive value of offspring born at different times in the future. Discounting by the population growth rate yields the present value of future offspring for cohort a, Na(t)(ba + K-1Saba+l if- K-2SaSa+l ba+ 2 q- ...)
154
Chapter 8 Traditional Models of Population Dynamics
and thus a per capita reproductive value
v(a) = b a 4- K-1Saba+l 4- K-2SaSa+ 1 ba+ 2 4- "'"
(8.20)
Ka
= la E, a Xtxbx ,
x>~a
for individuals in cohort a, with reproductive value for individuals of age 0 set equal to 1, v(0) = 1. These expressions capture the intuitive notion that future reproduction is less important than present reproduction in a growing population, whereas future reproduction is more important in a declining population (also see Mertz, 1971a). If the population is in equilibrium, then k = 1 and the reproductive value v(a) for individuals in cohort a is simply their reproductive rate R(a). For any age- or stage-based projection matrix, the vector of age-specific reproductive values is given by the left eigenvector associated with the dominant eigenvalue of the matrix (e.g., Leslie, 1948; Caswell, 2001). Reproductive value can be interpreted in various ways (see Caughley, 1970; Stearns, 1976) and can be usefully viewed as the number of animals alive at some future time that descended from an animal currently of age a, expressed relative to the number of animals at the future time that descended from an animal currently aged 0 (newborn), where v(0) = 1. Thus reproductive value for animals of any age a, v(a), is the value of an animal of age a to population growth, expressed relative to the value v(0) of an animal of age 0. This quantification of the relative worth of individuals of different age to population growth is relevant to questions in fields as diverse as evolutionary ecology and harvest management. For example, an important theoretical result in the field of life history evolution is that the maximization of fitness can be recognized as equivalent to the maximization of reproductive value at every age (Schaffer, 1974; Taylor et al. 1974; Caswell, 1980, 2001; Yodzis, 1981). Reproductive value is an important quantity in several derivations of optimal age- and stage-specific harvesting strategies (e.g., MacArthur, 1960; Cooch et al., 2002) and has been used in preliminary investigations of restocking quotas for young animals as a function of the harvest of older animals (Nichols et al., 1976b). Reproductive value is also relevant to arguments about colonization rates and age classes that are most likely to be successful colonists (e.g., MacArthur and Wilson, 1967).
1967; Beddington and Taylor, 1973; Beddington, 1974; Doubleday, 1975; Rorres and Fair, 1975; Reed 1980, 1983; Getz and Haight, 1989). Unlike the case with the single-age model, inclusion of harvest in a projection matrix model requires a harvest variable Hi(t) for each age cohort. Assuming that harvest is proportional to cohort size [i.e., Hi(t) = h(t)Ni(t)], the harvest model is expressed in vector notation as
N(t + 1) = A [1 - h(t)] N(t).
For a harvested population with stable age distribution, transition Eq. (8.21) becomes
N(t + 1) = k (1 - h)N(t) = [(1 + r)N(t)] (1 - h). It follows that h = r / k is the maximum sustainable harvest rate for a population with stable age distribution; harvest rates in excess of r will result in population declines. Note once again that this harvest rate is independent of the actual population size: a population of any size can sustain a maximum harvest rate of r without declining, as long as its age distribution is stable. It is not difficult to show that the growth parameter k* for a population subject to age-independent harvest is related to the growth parameter k for the unharvested population by k* = k(1 - h). Furthermore, the same stable age distribution applies to both the harvested and unharvested populations. The lack of a harvest effect on the stable age distribution confirms that age distribution alone is inadequate as an index of overharvest. As indicated earlier, a given age distribution can be associated with increasing (k > 1), decreasing ()~ < 1), or stable (~ = 1) populations. In fact, uniform harvest pressure across cohorts can be used to control population growth, while maintaining a preferred age structure for the population. A generalization of the harvest model allows for age-specific harvest rates hi, with model equations Ni_~l(t 4- 1 ) =
Si(1 - h i ) N i ( t )
and k-1
No(t + 1 ) = ~
bi+lSi(1 - h i ) N i ( t ) .
i=0
Under conditions of stable age distribution we have Ni + l(t)
Si(1 - hi)
Ni(t)
k
8.4.2.4. Harvest
Harvest can be incorporated into projection matrix population models and used to study effectiveness of various harvest strategies and even to derive optimal strategies (e.g., Darwin and Williams, 1964; Lefkovitch,
(8.21)
from Eq. (8.17), or (1 - hi)Ni(t) = hg-Ni+1 (t),
(8.22)
8.4. Models with Age Structure with )~ corresponding to a Leslie matrix with survival parameters Si(1 - hi). Rewriting Eq. (8.22), we can describe the pattern of cohort sizes in stable age distribution as )t Ni(t) = -~-Xi+ 1 (t) + hiNi(t) '-'i
(8.23)
= ~iNi+l (t) + Hi(t).
A useful result of this relationship is that population recruitment can be identified without accounting for age-specific reproduction rates. Assume that the oldest cohort is harvested at each time period: h k = 1 and therefore Hk(t) = Nk(t). From Eq. (8.23) we then have Hk(t) + Hk-l(t)
Nk-l(t)
-
Nk-2(t)
= Sk_lSk_2 ~Hk(t)
Sk-1
+
-1 (t)
155
8.4.3. Continuous-Time Models with Age Cohorts There is a rather complicated analog of the Leslie matrix model for populations in continuous time. Development of continuous-time models preceded that of discrete-time models, with the latter frequently viewed as approximations to the former. Goodman (1967) and Keyfitz (1968) reconciled the two approaches, and Caswell (2001) presents a readable discussion of this reconciliation. Here we use N(a, t) to denote cohort size for a continuous-time model, where a is the (continuous) cohort age, and age-specific instantaneous reproduction and mortality rates are denoted by b(a) and d(a), respectively. As with the Leslie matrix model, two transition equations are required for the population, one for reproduction and one for survival/maturation. Transitions among age cohorts are given by dN(a, t)/dt = -d(a)N(a, t)
q- Hk_ 2 (t) or
(8.24) ON(a, t)/Ot + ON(a, t)/Oa = -d(a)N(a, t),
k
N~
)k i
/~0=--iTiHi(t)"
Equation (8.24) demonstrates that if (1) the population is known to be in stable age distribution and (2) the population growth rate )~ is known, then reproduction can be determined on the basis of age-specific harvests. If survivorship is constant across cohorts, i.e., if Si = S, Eq. (8.24) reduces to
(8.25)
where ON(a, t)/Oa and ON(a, t)/Ot denote the partial derivatives of N with respect to a and t, respectively. Equation (8.25) is known as the von Foerster equation, and it essentially expresses a continuous decline in cohort size as cohorts age through time. This is analogous to the aging process in the discrete-time Leslie matrix population model, with mortality losses at each stage of the process. It is relatively straightforward to show that the von Foerster equation has the solution N(a, t) = N(a - t, O) exp
[ft -
d(a - t + x) dx
]
(8.26)
o
No(t) = ~
Hi(t),
if t
i=0
and assuming the population is in equilibrium, the latter formula reduces further to k Hi
No = i ,o In theory, Eq. (8.24) can prove to be useful, because it sometimes is more difficult to measure reproduction than to determine harvest and survivorship for a population. Given estimates of cohort-specific harvest and survivorship along with some indication of population rate of growth, Eq. (8.24) allows one to determine reproduction without having to measure it directly. Of course, the data requirements for this approach can be quite difficult, and often impossible, to meet.
Ira
N(a, t) = N(0, t - a) exp -
d(x) dx
o
]
(8.27)
if t>a. The solution for tKa reflects mortality losses for individuals that were of age a - t in the population at time 0. The solution for t>a represents the die-off over (t - a, t) of individuals born into the population at time t - a. To determine a solution in the latter case, the number of births N(0, t - a) at time t - a is required. Reproduction is given by the renewal equation N(0, t) =
f
oo
b(a)N(a, t) da.
(8.28)
o
This equation is analogous to age-specific reproduction in the Leslie matrix, except that summation over a
156
Chapter 8 Traditional Models of Population Dynamics
discrete number of age cohorts is replaced by integration over a continuous age distribution. The limits of integration encompass all possible ages, shown here as ranging from 0 to oo. At any particular point in time the actual range of ages for the population is limited by the initial age distribution and the value t. As with the Leslie matrix model for discrete age cohorts, the age distribution for continuous time stabilizes to a pattern of exponential growth for all age cohorts, with the same growth rate across cohorts. For population size
and foo
= bN(t), where birth rates b(a) = b and death rates d(a) = d are independent of age. From Eqs. (8.26)-(8.28) a solution is given by
N(a, t) = N ( a - t , O)e -dt if t
N(t) = f~ N(a, t)da and age distribution
ft(a) = N(a, t ) / N ( t )
N(a, t) = N(0, t - a)e -da if t>a. Under the condition of stable age distribution, survivorship has the form l(a) = exp(-da), so that the Euler-Lotka equation is
at time t, asymptotic stability is characterized by
1 = b f ~ e -a(r+d) da
lim ft(a) = f(a).
= b/(r + d)
t - + oo
Under conditions of stable age distribution, the intrinsic rate of growth, the age-specific life-spans, and the reproductive value all can be expressed in forms that are similar to those in the Leslie matrix population model. For example, the continuous-time analog of the Euler-Lotka equation is 1 = f ~ e-rab(a)l(a) da, where
and the population intrinsic growth rate is r = b - d. Thus, the population increases if birth rate is greater than death rate, declines if death rate is greater than birth rate, and remains constant if birth and death rates are equal. These patterns are equivalent to patterns described earlier for the simple exponential model and demonstrate the rather obvious fact that if mortality and birth rates are age independent, accounting for age in a population model is unnecessary.
Example l(a) = exp [ - f oad(x) dx ]
is the probability of a newborn individual surviving to age a, and r is the instantaneous rate of growth for the population in stable age distribution. For any agespecific birth and mortality functions b(a) and d(a), it can be shown that only one value r satisfies the Euler-Lotka equation. As with the Leslie matrix model, this parameter governs the exponential growth of each cohort in the population, once the stable age distribution is attained.
Example Consider a continuous-time population with constant birth and mortality rates for all ages at all times in the time frame. The transition equations in this situation are
ON(a, t)/Ot + ON(a, t)/Oa = - d N ( a , t)
Let the birth function in the previous example be given by b(a) = be ka, with b the rate of birth for 0-aged individuals and k a constant that takes positive or negative values. As above, trajectories N(a, t) for the age-specific cohorts are given by Eqs. (8.26)-(8.28). However, the Euler-Lotka equation now takes the form 1= b f ~ e -a(r+d-k) da
= b/(r + d -
k),
so that r = b + k - d. A comparison with the previous example shows that age-specific increases in birth rate result in a larger value for r than is the case with constant birth rate. Thus, if birth rates increase with age (k>0), the constants b and k jointly compensate for the effect of mortality, leading to an increased value of r. However, age-specific decreases in birth rate result in a smaller value for r, because the difference (rather than the sum) of the constants b and k is used to compensate for the effect of mortality.
8.5. Models with Size Structure As with the Leslie matrix model, age-specific harvest can be added to the continuous-time model. To illustrate, consider a model with instantaneous harvest rates h(a) that (possibly) vary among age cohorts. Then the model can be expressed as aN(a, t)/at + ON(a, t)/aa = -[d(a) + h(a)]N(a, t) and N(0, t) --
f
oo
b(a)N(a, t) da. o
In this formulation the age-specific harvest rate h(a) adds to other sources of mortality, resulting in the survivorship curve
Lf a[d(x) + h(x)]dx ] .
l(a) = exp -
0
This in turn leads to a stable age distribution and intrinsic rate of population growth that reflect the impact of harvest, through its depressing effect on the survivorship curve l(a). For example, assuming a birth function of the form b(a) = beka and constant harvest rate leads to 1=
b f ~ r -a(r+d+h-k) da
= b/(r + d + h -
k),
so that r = b + k - d - h. This makes explicit the depressing influence of harvest on the population rate of growth. It follows that a population of size N can be maintained in stable-age distribution by choosing the harvest rate h = b + k - d. Harvest rates of smaller magnitude allow for population growth, and rates of larger magnitude lead to population declines.
8.4.4. Characterizing Populations by Age The examples above illustrate that the dynamics of age-structured populations are determined by agespecific reproduction and survivorship parameters, with each combination corresponding to a different stable age distribution for the population. For both discrete and continuous models, population dynamics typically can be divided into two phases. Thus, a transition phase is characterized by transient dynamics in which the distribution of age cohorts differs from, but converges to, the stable age distribution. Convergence to the stable age distribution is irrespective of initial distribution; however, the cohort rates of growth in the transition phase, and the rate of convergence to stable age distribution, all depend on initial population structure. After stable age distribution is attained, the population grows at a constant rate that is given by
157
the solution of the Euler-Lotka equation. The age distribution is maintained throughout this phase, with each age cohort growing at the same rate. Thus, the population as a whole, and every cohort in it, exhibits unregulated exponential growth. Even with the added flexibility and realism that age structure can bring to a population model, for a number of reasons it may not be advantageous to characterize population structure in terms of age. In many instances, size is a more important factor than age in expressing demographic variability within a population. This is the case whenever survival and reproduction are closely associated with size, but not necessarily with age, a situation that appears to exist in a number of animal populations (e.g., Hughes, 1984; Crouse et al., 1987; Hughes and Connell, 1987; Sauer and Slade, 1987a,b; Nichols, 1987). In other cases, both size and age are relevant to variation in vital rates (Slobodkin, 1953; Law, 1983; Law and Edley, 1990). Of course, factors other than size can introduce heterogeneity in the structure of a population--for example, the simultaneous occurrence of multiple reproductive a n d / o r survival strategies in the population (e.g., see Schaffer and Rosenzweig, 1977; Pugesek and Wood, 1992; McNamara and Houston, 1996). It is common for some animals in a population to forego breeding in a particular year, such that some individuals of a particular age breed and others do not (e.g., Nichols et al., 1976b, Newton 1989; Pugesek and Wood, 1992; Cam et al., 1998). Age alone is not sufficient to characterize variation in such situations. The point here is that in many cases it is sensible to characterize animals by state variables other than age (Houston and McNamara, 1992; McNamara and Houston, 1996), and more general models may be needed to deal with such state specificity of vital rates. Stage-based projection matrix models were introduced by Lefkovitch (1965; also see Goodman, 1969; Houllier and Lebreton, 1986) and are now widely used in animal population ecology (Caswell, 2001).
8.5. M O D E L S W I T H
SIZE STRUCTURE As noted above, in many cases it is the size of an individual rather than its age that determines reproductive success and survivorship. If the size of individuals is determined exclusively by age, whereby size y is given in terms of age x through a monotonic growth function y = f(x), then a population model based on size cohorts has essentially the same structure as one based on age (up to the labeling of the cohort index). Typically, however, size is not uniquely determined by
158
Chapter 8 Traditional Models of Population Dynamics
age. Thus, a given age cohort may contain individuals of more than one size, and, similarly, a particular size cohort may contain individuals of more than one age. If demographic factors are associated more directly with size than with age, then size structure and the parameters necessary to account for transitions among size classes provide a more appropriate structure for the population transition matrix.
8.5.1. D i s c r e t e - T i m e M o d e l s w i t h Size Cohorts Consider a population in which the sizes of individuals can be divided into discrete categories, ranging from 0 (the smallest cohort) to k (the largest cohort). Assume also that individuals can either remain in their size class or grow into the next larger class over the course of one unit of time. If Pi(t) is the proportion of individuals in cohort i at time t that grow into cohort i + 1 at time t + 1, then the cohort transition equation is
where bi(t) is the per capita reproduction rate for cohort i at time t. With constant parameters, the model is written in matrix form as No(t + 1) Nl(t + 1) N2(t + 1) N3(t + 1)
.Nk(t + 1).
(1 - Po)So + PoSobl
$162
S2b3
...
Sk_l"bk
PoSo 0 0
(1 - P1)S1
0
...
0
0
Nk(t + 1 ) = Sk(t)Nk(t) + Pk_l(t)Sk_l(t)Nk_l(t). On condition that the transition and survival probabilities are constant through time, the transition equations for populations with size structure reduce to
Ni+l(t 4- 1) = [1 - Pi+l]Si+lNi+l(t) 4- PiSiNi(t) and
Nk(t + 1) = SkNk(t) 4- Pk_lSk_lNk_l(t). As with populations with age structure, reproduction is modeled simply by aggregating the reproductive contribution from each cohort: k
No(t + 1) = ~ biNi(t 4- 1) + (1 - Po)SoNo(t) i=1
k-1
= ~ bi+l{Pi(t)Si(t)Ni(t) i=0
+ [1 - Pi+l(t)]Si+l(t)Ni+l(t)}4- bkSk(t)Nk(t) + (1 - Po)SoNo(t),
(1
0
-- P 2 ) 5 2
...
0
P252
... ...
0
... ...
(1 - P k _ l ) S k _ l Pk_lSk_l
0
I
i! I
Sk
where bi represents the average reproduction at time t + 1 for surviving individuals from size class i - 1 at time t:
Ni+l(t 4- 1) = [1 - Pi+l(t)]Si+l(t)Ni+l(t) 4- Pi(t)Si(t)Ni(t) for all but the smallest and largest cohorts. Note that if Pi(t) = 1 for all cohorts, this transition equation has the same form as the age-structured model. As with one of the age-structured models, updating the largest cohort involves the addition of surviving individuals from the largest and next largest cohorts:
PIS1 0
Skb k" r N n ( t ) l i Nl(t) I 0 I N-,(t) i 0 I N~(t) I 0
m
bi =
(1
-
Pi_l)bi_l
4- P i _ l b i .
Note that the principal differences between this model and the model for age-structured populations [Eq. (8.8)] are the averaging of reproduction rates, along with the occurrence of transition elements on both the diagonal and lower off-diagonal of the matrix. If the transition probabilities Pi in the size model are mall unity, then the average reproduction rates reduce to b i = b i + l , all diagonal elements except the first and last vanish, and the mathematical form of the size model is identical to that of model (8.8) with age structure.
Example Consider a population with four size classes, for which the reproductive rates are (1.0, 2.0, 3.0), survival rates are (0.5, 0.65, 0.85, 0.4), and transition probabilities are (0.75, 0.55, 0.35), respectively. The transition equations for this population are
No(t+ 1) Nl(t 4- 1) N2(t + 1) = N3(t + 1)
I
0.5 0.375 0 0
1.0 0.2925 0.3575 0
2.0 0 0.5525 0.2975
1.2 0 0 0.4
FNo(t)-] /Nl(t)/ /N2(t)/" LN3(t)j
These reproduction and survival parameters were used in a previous example of an age-structured population. Here we simply redefine the cohort index to represent size rather than age and incorporate the nonzero probability of individuals remaining in a cohort longer than one time period. A comparison of the behavior of this
159
8.6. Models with Geographic Structure model (Fig. 8.9) and that of the corresponding agestructured model (Fig. 8.8) reveals that a principal effect is to reduce the transition phase of the model with size structure, and retard growth of the largest size class below that of the oldest age class. However, the model with size structure shows the same general pattern of convergence to a stable distribution among cohorts, followed by exponential growth for each cohort in the population.
As with age-structured models, there is a continuous-time analog for population models in discrete time (see Sinko and Streifer, 1967; Streifer, 1974). We may think of N(s, t) as characterizing the number of individuals of size s at time t, with a population size of
~
oo
N(s, t) ds
o
and a distribution ft(s)
N(s, t) / N ( t )
=
of sizes in the population at time t. As above, two transition equations are required, one for reproduction and one for physiological development and survivorship. Transitions among size classes are given by dN(s, t)/dt = -d(s)N(s, t) or
ON(s, t)/Ot + g(s, t) 0IN(s, t)]/Os = -d(s)N(s, t),
Nl(t)
2000
/
/
1500
Ndt) ~- looo ~/ 500 ,, "k~
Z\ 0
/"- ~
~~" ~
.
~
/
N3(t)
~
. ~ "
/
_. N,(t)
i
i
I
2
4
6
8
N(O, t) =
b(s)N(s, t) ds. o
The function g(s, t) in the first equation is the growth rate for individuals at time t, i.e., ds/dt = g(s, t). Thus the term g (s, t) 0IN(s, t)]/Os represents the growth into and out of cohort s. Size-specific harvest can be added to the model by including an instantaneous harvest rate h(s), so that ON(s, t)/Ot + g(s, t) 0IN(s, t)]/Os = -[d(s) + h(s)]N(s, t) .
8.5.2. Continuous-Time M o d e l s with Size Cohorts
N(t) =
and reproduction is given by
1'0
t
F I G U R E 8.9 Dynamics of a prebreeding population model consisting of four size classes, with constant per capita birth and survival rates and constant transition probabilities among size classes.
8.6. M O D E L S W I T H GEOGRAPHIC STRUCTURE In both the age-structured and size-structured models discussed above, there is a natural order in the cohort indices. Thus, cohort 0 is the youngest (or smallest) cohort, and cohorts increase in age (or size) with increasing indices. However, the cohort structure of a population need not follow such an ordered pattern. In some cases a natural progression is indeed carried in the indices for stage structure, as in larval and instar stages of development in the life cycles of certain insect species. In others the cohort index may not denote a sequential process of physiological or morphological development, so that a natural progression in indices is absent. For example, there is no natural ordering for populations consisting of geographically identified cohorts. A well-known application is in the field of island biogeography, in which migration rates among geographic cohorts are modeled as functions of island size, distance to a mainland, and sizes of mainland and island population units (MacArthur and Wilson, 1967). Migration rates and sources of variation in these rates are relevant to modeling in population genetics [e.g., island versus stepping stone versus more general isolation-by-distance models; see Crow and Kimura (1970)]. Current interest in metapopulation dynamics (Hanski and Gilpin, 1997; Hanski, 1999), source-sink models (Pulliam, 1988), and the general topic of dispersal (Clobert et al., 2001) have also sparked interest in estimating and modeling migration rates. As is the case with many aspects of populationdynamic modeling, human demographers were the first to incorporate multiple locations into projection matrix models (Rogers, 1966, 1968, 1975, 1985, 1995; Le Bras, 1971; Schoen, 1988). These so-called multiregional matrix models now are being applied in animal ecology (e.g., Fahrig and Merriam, 1985; Lebreton and Gonzalez-Davilla, 1993; Lebreton, 1996; Lebreton et al., 2000). To illustrate, consider a population consisting of three
160
Chapter 8 Traditional Models of Population Dynamics
age cohorts in each of two regions, with N 1, N 2, and N 3 representing juveniles, subadults, and adults in region 1, N 4, N 5, and N 6 representing juveniles, subadults, and adults in region 2. Here the transition among cohorts consists of the processes of aging and movement of subadults between regions. For individuals remaining in region 1, the transition matrix is
Nl(t) ~ooo a
/
/
800 ~
600
/ N3(t)
/
//
/
I
Fs0~ F2
0
N2(t)
F3 ]
400
$3
200
/
0 ,
52
and the corresponding matrix for individuals remaining in region 2 is
[i:
~ooo b
0 $5
N4(t)
S6J
Combining both subpopulations in the absence of migration produces a transition matrix of the form
800
600
200 7 /
hL
N3(t + 1) = (1 - P2)S2N2(t) -F PsS5Ns(t) + S3N3(t) and
N6(t + 1) = (1 - Ps)S5Ns(t) + P2S2N2(t) -I- S6N6(t), and the transition matrix now has the form F3 0
0
(1 - P2)$2
0
0
0 _0
0 0
0 0
00
53
0
P5S5
0
0
F4
F5
F6
0
0
P2S2
0
$4 0
0 0 (1 - P5)$5 56.
Note that the cohort indices for this model carry a natural sequence only within each region, but not across regions. The behavior of this model is displayed in Fig. 8.10 for F' = (0, 0.8, 1.6, 0, 0.8, 1.6), S' = (0.5, 0.65, 0.85, 0.4, 0.55, 0.75) and (P2, P5) - (0.4, 0.15). Note that both B
/./"
400
Migration of subadults between regions is incorporated by, e.g., incorporating a parameter P2, for the proportion of subadults migrating from region 1 to region 2, and a parameter P5, for the proportion of subadults migrating in the opposite direction. The corresponding transition equations are
F2 0
/
/
-F 1 F 2 F 3 0 0 0 S1 0 0 0 0 0 0 S2 53 0 0 0 0 0 0 F4 F5 F 6 0 0 0 54 0 0 _ 0 0 0 0 S5 $6_
"F 1 51
/ Ns(t)
/
~,,/"
,.,......_.-__ ~
...-- /
/
/
N6(t)
.v
,
2
4
'6
'8
t
FIGURE 8.10 Dynamics of a prebreeding population model consisting of two subpopulations with three age classes each. Each subpopulation has constant per capita birth and survival rates, and rates of migration between subpopulations are constant. (a) Cohort dynamics for subpopulation 1. (b) Cohort dynamics for subpopulation 2.
subpopulations exhibit "Leslie matrix" behaviors, in that both show a nonequilibrium transition phase with rates of change that are specific to cohort age, followed by an equilibrium phase with constant rates of change within the subpopulations. Note also that the two subpopulations have the same asymptotic rates of growth, even though subpopulation I has higher survival rates. This is a direct result of migration rates linking the two subpopulations. The asymptotic rate of growth for the whole population is in some sense an average of the rates of growth for the subpopulations considered separately, weighted by the migration rates. Thus, migration from subpopulation I to subpopulation 2 compensates for lower survival rates in subpopulation 2 and results in a higher asymptotic growth rate for subpopulation 2 than would be the case in the absence of migration. On the other hand, the loss of animals from subpopulation 1 via differential migration retards its
8.7. Lotka-Volterra Predator-Prey Models growth to a rate below what would be the case in the absence of migration. Eventually the growth rates for the two populations become identical, as additions and losses between the subpopulations from reproduction, mortality, and migration come into balance. This pattern occurs irrespective of the migration rate from subpopulation 1 to subpopulation 2, as long as there is some movement from one area to the other: large numbers of migrating animals, produced by a large subpopulation 1 with its higher growth potential, contribute to the growth of subpopulation 2, elevating the growth of the latter to that sustained by the former. We note in closing that the addition of cohort structure adds substantially to the burden of identifying the parameters controlling cohort transitions. This burden increases as one includes size structure with the attendant cohort growth functions, and geographic structure, which requires migration rates among geographically distinct cohorts. In Part III we describe statistical models, field protocols, and data requirements for these situations. In particular, we highlight some of the advances in areas such as sample survey methodology, tag-resighting approaches, and other estimation techniques, which provide enhanced capabilities for model development and analysis. Nevertheless, it will be clear in later chapters that data requirements and mathematical complexities can quickly overwhelm an investigation of these parameter-rich models.
8.7. L O T K A - V O L T E R R A PREDATOR-PREY MODELS Lotka-Volterra models (Volterra, 1926, 1931, 1937; Lotka, 1932) explicitly incorporate predation via state variables for both predators and prey, and two transition equations are necessary to track changes in both predator and prey population levels. The models assume that predators influence prey populations through prey mortality, whereas the prey influence predator populations through predator reproduction. Predator-prey interactions can be addressed in both discrete and continuous time, though there are important differences in mathematical behaviors between the two models.
8.7.1. Continuous-Time Predator-Prey Models If Nl(t) and N2(t) are prey and predator population levels, respectively, the continuous form of the Lotka-Volterra transition equations is d N 1 / d t = [r I - dlN2(t)~Nl(t )
161
and d N 2 / d t = [b2Nl(t) - d2~N2(t ).
Here r I represents the (constant) per capita rate of growth for prey in the absence of predation, whereas the mortality rate d i N 2 is a linear function of the number of predators. On the other hand, the per capita mortality rate d 2 for predators is constant, and the birth rate b2N 1 is linear in the number of prey. Thus the coefficient d I expresses the (negative) impact of predators on prey, and b2 expresses the (positive) impact of prey on the predators (Fig. 8.11). Here the predation rate is assumed to be proportional to the rate of encounter of predators and prey, with predation modeled as a simple product of population sizes, scaled by a speciesspecific parameter d 1. The coefficient d I in the transition equation for prey represents the proportion of prey taken by each predator, whereas the coefficient b2 in the transition equation for predators represents the "efficiency of conversion" of prey into predators. In the absence of predators [N2(t) = 0], the prey population grows exponentially according to the equation d N 1/dt = rlNl(t),
and in the absence of prey [Nl(t) = 0], the predator population declines exponentially according to d N 2 / d t = -d2N2(t).
Equilibrium conditions for the Lotka-Volterra predation model are given by setting the transition equations to 0 and solving the resulting equations. A quick inspection indicates that there are two equilibrium conditions: (N~, N~) = (0,0) and (N~, N~) = (d2/b 2, rl/dl). Thus, the model has the rather odd property that the nonzero equilibrium size for the predator population is independent of the birth and death rates for predators, and the equilibrium population size for prey is independent of prey growth and death rates. As argued in Appendix C, _N* = _0 is an unstable equilibrium, in that deviations in a neighborhood of 0 exhibit growth away from 0. On the other hand, population dynamics near (N~, N~) = (d2/b 2, r I / d 1) exhibit stable oscillations m
k2,,,'
FIGURE 8.11 Lotka-Volterrapredator-prey model. Output flow for prey N 1 is influenced by predator population size N2. Input flow for predators N2 is influenced by prey population size N 1.
162
Chapter 8 Traditional Models of Population Dynamics
about the equilibrium. This allows one to partition the "phase plane" of points (N1, N 2) into four quadrants defined by the perpendicular lines N1 = N~ and N 2 = N~_,with different population behaviors in each quadrant (Fig. 8.12). The patterns of population change for predators and prey are distinct in each quadrant:
populations, and the oscillations are stable, i.e., there is no tendency for the populations to converge to equilibrium (Appendix C). The recurring pattern of oscillation in population dynamics shown in Fig. 8.12 is known as neutral or cyclic stability (Edelstein-Keshet, 1988) (see also Section 7.5.4 and Appendix C.2.2).
Quadrant h This region of the phase plane is defined by Nl(t) > N~ and N2(t) > N~. Under these conditions the per capita growth rate b2N 1 - d 2 of predators is positive [because N 1 > N~, b 2 N 1 - d 2 > b 2 N ~ - d 2 (-- 0)], and the per capita growth rate of prey is negative [because N2(t) > N'~, r 1 - d i N 2 < r I - d i N ~ (= 0)]. Therefore the predator population increases in quadrant I, whereas the prey population decreases. Quadrant Ih This region is defined by predator and prey values such that Nl(t) < N~ and N2(t) > N~. Under these conditions, the per capita growth rates of predators and prey are both negative, and therefore both populations decline in quadrant II. Quadrant III: This region is defined by predator and prey values such that Nl(t) < N~ and N2(t) < N~_.Here the per capita growth rate of predators is negative, and the per capita growth rate of prey is positive. Therefore the predator population decreases in quadrant III, whereas the prey population increases. Quadrant IV: This region is defined by predator and prey values such that Nl(t) > N~ and N2(t) < N~. Here the per capita growth rates of predators and prey are both positive, and therefore both populations increase in quadrant IV.
Example
Consider the dynamics for populations governed by the Lotka-Volterra predator-prey equations, with an initial predator population of 210 individuals and initial prey population of 900 individuals. Per capita reproduction rates are (0.0001)N 1 for the predator population and 0.25 for the prey population, and per capita mortality rates are 0.1 and (0.001)N2, respectively. Figure 8.13a displays the population dynamics for this system. Both populations exhibit stable oscillations, with the same oscillation period but different amplitudes and phase shifts. Figure 8.13b exhibits a phase
a
1000
800
600
prey predators
400
These behaviors induce continuous oscillations about the nontrivial equilibrium point. The magnitude of the oscillations depends on initial conditions for the
200
400lb 300
350
I I I
II
i
I
280
300 or) 0
r r L
260
~
240
250
t_ 200
150 220
IV
III
100 400
600
800
10;0
1200
prey
200
800
900
1000
11;0
12;0
prey F I G U R E 8.12 Phase diagram of a Lotka-Volterra predation system. Predator and prey populations oscillate in a stable pattern,
without any trend toward equilibrium.
F I G U R E 8.13 Dynamics of a Lotka-Volterra system with one predator and one prey species. Population dynamics for both predator and prey are characterized by stable oscillations. (a) Time series trajectories for predator and prey populations. (b) Phase diagram of predator-prey dynamics.
8.7. Lotka-Volterra Predator-Prey Models diagram of these same population dynamics, with quadrant-specific increases and decreases in population sizes. The oscillation amplitudes for both predators and prey are determined by the degree of displacement of initial population sizes from equilibrium.
Both the size and shape of the oscillations in a predator-prey system are dependent on the location of the equilibrium point (N~, N~_) in the phase plane and the initial population sizes NI(0) and N2(0) relative to (N~, N~). The influence of the location of N O relative to N* is shown in Fig. 8.14, where initial population sizes close to equilibrium result in oscillations of small amplitude, and initial sizes that are distant from the equilibrium result in large fluctuations about N*. The "shape" of the oscillations also is influenced by the relative positions of N O and N*, with a more nearly elliptical phase diagram for N O close to N* (Fig. 8.14). The influence of the absolute position of N* on oscillation size and shape is shown in Fig. 8.15, which displays the phase diagrams for two predator-prey systems with different equilibria. The two systems differ only in that growth and death parameters r I and d 2 for one system in Fig. 8.15 are twice those for the other, so that steady-state population sizes are twice as large. Initial population sizes were chosen so that the distance between (NI(0), N2(0)) and (N~, N~_)is the same for both systems. Note that the phase diagram corresponding to the larger equilibrium is more nearly
bL )
200
150 t~ 0 L Q.
8.7.1.1. Oscillation Size and Shape
163
100
~oo
.oo
,'oo
~oo
~ooo
prey
FIGURE 8.15 Phase diagrams for two Lotka-Volterra predator-prey systems corresponding to different equilibria. (a) Equilibrium condition N* = (N~I,N~2)is D units from the origin. The distance between equilibrium N* and initial condition NOdetermines the amplitudes and period of oscillations. (b) Equilibrium state is 2D units from the origin; initial condition NOchosen to maintain the same distance to the equilibrium state N* as in (a). _
m
elliptical, as indicative of symmetric oscillations about (N~, N~). On the other hand, oscillations for the system nearer the origin are less symmetric, with larger amplitudes. The oscillation period is determined by the factor (rid2) 1/2, so that high prey growth rates and predator death rates accelerate the cyclic changes in population status. Thus, the oscillation period of the system with larger equilibria is 50% that of the system with smaller equilibria. 8.7.1.2. L o g i s t i c E f f e c t s
Density dependence can be incorporated into the Lotka-Volterra equations by modifying the prey a n d / o r predator reproduction functions. For example, logistic effects in prey reproduction lead to the system of equations
100
dN1/dt
= r1N1(1 - N 1 / K ) - d l N I N 2
and dN2/dt
0
200
400
600
800
prey
FIGURE 8.14 Phase diagram of predator-prey dynamics for a Lotka-Volterra predator-prey system, starting at different levels of initial population size. Trajectories correspond to an initial predator population size of 30 and initial prey population sizes of 100, 150, and 200. Trajectories exhibit stable oscillations with differing periods and amplitudes.
= b 2 N I N 2 - d2N2,
with the per capita rate of prey reproduction decreasing logistically with prey population size. As before, equilibria for this system are defined by d N / d t = 0, or r1N1(1 - N 1 / K ) - d l N I N 2 = 0 and b 2 N I N 2 - d 2 N 2 -- O.
164
Chapter 8 Traditional Models of Population Dynamics
These equations are satisfied for N* = 0 and 1200
:
LNt/
bI
d2/b2 1 b,a2 l,
1000
bdlKl
with d2 < b2Ka necessary condition for N~ to be positive. It is shown in Appendix C.2.2 that N* = 0 is an
unstable equilibrium, near which the prey population grows and the predator population declines. However, the system no longer exhibits neutral stability about the nontrivial equilibrium N* = (d2/b2, b l / d I - bld2/ b2dlK)'. As a result of the logistic modification, this equilibrium is now stable, with small deviations resulting in d a m p e d oscillations as population sizes return to N*. The stability of the nontrivial equilibrium obtains no matter how minor is the logistic adjustment. However, the time required to approach equilibrium very much depends on the size of the carrying capacity K.
8.7.2. Discrete-Time Predator-Prey Models The transition equations for the Lotka-Volterra predator-prey system also can be expressed in discrete time: Nl(t + 1) = Nl(t) + [r I - dlN2(t)]Nl(t)
and N2(t + 1) = N2(t) + [b2Nl(t) - d2]N2(t).
Equilibrium conditions are found by equating population sizes in successive time periods, which leads again to the equilibrium conditions N~ = d2/b 2 and N~ = r 1/d 1. As with the continuous-time model, this results in a partition of the plane of predators and prey into four quadrants, defined by the lines N 1 - N~ and N 2 = N~ that intersect at the point (N~, N~). The patterns of population change for predators and prey are specific in each quadrant, leading to oscillatory behavior. However, oscillations for the discrete-time system are unstable, with steadily increasing population sizes and more dramatic population reductions through time (Fig. 8.16). The cause of this instability is tied directly to the discrete nature of the time step, which effectively induces a time lag into the transition equations. Thus, the population at time t + 1 is determined by growth rates that are based on population sizes at time t. As with lag effects in the logistic model, lags in the Lotka-Volterra system cause predator and prey populations to "overshoot" what would otherwise be their maximum and minimum population sizes. Assume, for example, that the prey population is large and increasing slowly and the predator population is small and increasing rapidly. The depressing effect of preda-
800
~.._
600 400 200
20
40
6'0
80
100
120
140
FIGURE 8.16 Trajectoriesfor a Lotka-Volterra predator (---) and prey (m) system in discrete time. Oscillations increase in magnitude over time, in contrast to the continuous model.
tors on the growth of prey over [t, t + 1] is based on predator population size at time t, which is substantially lower than the predator population size at t + 1. This allows the prey population to continue to increase above what would otherwise be the case with a smaller time step, in turn inducing more rapid growth in predators as more prey are available. On the other hand, if the prey population is small and decreasing slowly whereas the predator population is large and decreasing rapidly, then prey population reductions over It, t + 1] are driven by larger predator population sizes at time t than would be the case with a smaller time step. The result is more dramatic reductions in both predators and prey than would be exhibited with a smaller time step. These effects are manifested at each cycle of oscillation, leading to ever-increasing population sizes at their peaks and ever-decreasing population sizes at their nadirs (Fig. 8.14).
8.8. M O D E L S OF COMPETING POPULATIONS In this section we introduce two models for competition among populations, one that is appropriate for interference competition and one that applies to exploitation competition. In the former, the competitive impact of one species on another is registered directly, through the use of "competition coefficients" that essentially depress the population rate of growth in the manner of a carrying capacity. In the latter, competitive impacts are registered through the exploitation of a shared resource, whereby resource consumption by one species leaves a reduced resource base for the other. The distinguishing feature for these models is whether
8.8. Models of Competing Populations there is mediation of species interactions through a shared resource. 8.8.1. Lotka-Volterra Equations for Two Competing Species
We first consider a system of two competing species with density-dependent population growth rates in the absence of competition. Competition between the species influences growth rate by adding to the effect of density dependence. The continuous-time model is
165
of each is retarded by the presence of a competitor (Fig. 8.18). The logistic form of growth and the damping effect of the interaction for both species distinguish the Lotka-Volterra competition model from the LotkaVolterra predator-prey model. Equilibrium conditions for this system are given by setting both of d N i / d t to 0, which results in N'; = K 1 - a12N~2, N ~ = K 2 - a21N ~.
A rearrangement of terms leads to the matrix equation dN1/dt
= rlNI[K 1 - N 1 - a12N2]/K 1
dN2/dt
= r2N2[K 2 - N 2 - a21N1]/K2,
where r i and K i are the growth rate and environmental carrying capacity for population i in the absence of competition. The coefficients a12 and a21 represent competitive interactions between species, whereby the growth rate of one species is depressed because of the presence of the other. The coefficient a12 is a nonnegative competition coefficient specifying the per capita impact of species 2 on species 1, whereby the carrying capacity of species 1 is effectively reduced to K1 a 1 2 N 2 in the presence of N 2 individuals of species 2. Similarly, the coefficient a21 specifies the per capita impact of species 1 on species 2, so that the carrying capacity of species 2 is effectively reduced to K2 a21N1 in the presence of N1 individuals of species 1 (Fig. 8.17). This model is known as the Lotka-Volterra competition model, in reference to the fact that it characterizes direct competition between two competitors through linear terms in the transition equations. Here the competition is couched in terms of interference competition, wherein two species, through direct contact, negatively interact or "interfere" with each other. Under the model, each population is assumed to grow in a logistic fashion in the absence of the other, and the effect of competition is essentially to lower the carrying capacities of both species. Thus, the growth
[K1] = I 1 a121 [ g ~ l K2 a21 1 LN~__I with solution
[1 a12]IK1] --1
LN~_J
a21
1
K2
1000
800
fl
7
600
II I / / ' / 400
N~(t) N2(t) 200
1000
b
/
800
f
600
400
Nl(t) N2(t)
200
0
FIGURE 8.17 Lotka-Volterramodel for two competing species. Input flow for each population is influenced by the size of the other population.
,0
do
~0 t
do
.0
6o
FIGURE 8.18 Population dynamics for two populations described by the Lotka-Volterra competition equations. (a) Dynamics of population one with the competition coefficient a21 = 0 and a12 = 0.25. (b) Dynamics of population two with a12 0 and a21 0.25. =
=
166
Chapter 8 Traditional Models of Population Dynamics
or
N~
l
= 1/D
E--a21 1 lI1 ll a12
K2 '
where D = 1 - a12a21 (see Appendix B.4 for a discussion of matrix inverses). Thus the equilibrium population sizes are K1 N~ = 1 -
-
a12K2 a12a21
and K2 -
N~ = 1
-
8.8.1.2. Competitive Exclusion
a21K 1 a12a21
For analysis of population dynamics it also is useful to describe population sizes for which dNi/dt = 0 for one but not both of the populations. Setting dN 1/dt = 0 yields N 1 = K 1 - a12N2,
which describes combinations (N 1, N 2) of population sizes with 0 growth rate for population 1. The corresponding line describes a null cline (see Appendix C.3), so called because of the absence of growth in population I along it. Similarly, setting dN2/dt = 0 yields the null cline N 2 = K2 -
a21N 1
for population 2. The point of intersection of the null clines defines equilibrium population sizes, because the growth rates for both populations are 0 there. The numerators of the equilibrium formulas above define conditions for population coexistence, and graphs of the null clines can be used to highlight directions of population change toward equilibrium. Three possibilities arise: stable coexistence, competitive exclusion, and unstable population equilibrium. 8.8.1.1. Stable Coexistence
If the numerators of the equilibrium formulas are both positive, i.e., if K 1 > a12K 2
rying capacities K1 and K2, reduced by amounts a12K2 and a21K1, respectively, and scaled by 1 - a12a21. Both populations converge to the equilibrium population levels irrespective of the initial population sizes. Initial population sizes larger than N~ and N~ lead to population declines toward the steady-state values, and initial population sizes lower than N~ and N~ lead to increases toward the steady-state values. These tendencies are shown in Fig. 8.19a, which displays the population equilibria and null clines and indicates with arrows the direction of population change at any point in the phase plane.
(8.29)
and
If only one of the two conditions in expressions (8.29) and (8.30) is met, the corresponding population eventually approaches its carrying capacity and the other population is driven to extinction. Thus, if K1 > a12K2 and K2 ~ a21K1, then species 2 is excluded and species 1 converges to K1 (Fig. 8.19c). Convergence to the carrying capacity is independent of population initial conditions. If K 2 ~ a21K 1 but K 1 ~ a12K2, species 1 is excluded and species 2 converges to K2 (Fig. 8.19b). Again, convergence of population 2 to its carrying capacity is irrespective of population initial size.
8.8.1.3. Unstable Population Equilibrium If both conditions in expressions (8.29) and (8.30) are met, the equilibrium population sizes describe an unstable equilibrium. It is easy to show that K 1 ~ a12K2
and K 2 ~ a21K 1
are equivalent to a12a21 ~ 1, or D = 1 - a12a21 ~ 0. In this case, populations with initial population sizes of N~ and N~_ will be maintained at equilibrium levels indefinitely, but initial population sizes other than N~ and N~ result in the extinction of one of the populations. Thus, K2 - a21K1 N2(0) > Kll K 2a12 ~NI(O) leads to the extinction of population 1, and K2 a21K 1 N 2 ( 0 ) < Kll - a12K---~2NI(O) -
K 2 > a21K 1,
(8.30)
then the populations can coexist in equilibrium. In this case, steady-state population levels are simply the car-
leads to the extinction of population 2. Null clines and direction arrows indicating population changes for this situation are shown in Fig. 8.19d.
8.8. Models of Competing Populations
167 F I G U R E 8.19 Phase plane diagram for the Lotka-Volterra competition equations for two species, exhibiting isoclines and zones of coexistence, competitive exclusion, and unstable equilibrium. (a) Population coexistence. (b) Extinction of species 1. (c) Extinction of species 2. (d) Species extinction depends on population initial conditions.
K~
K1/a12~
Kl/a12
K2 N2*
~.
N1*
K1
K1 K21a21
K2/a21
K1/a12 c K~
K11a12 N2*
K2/a21
K1
Example The Lotka-Volterra competition model can be illustrated with a discrete-time model of two competing species, with transition equations
Nl(t + 1) - Nl(t) + 0.2 Nl(t)[700 - Nl(t) - 0.25 N2(t)]/500 and
N2(t + 1) = N2(t) + 0.3 N2(t)[1000 - N2(t) - 0.5
Nl(t)]/lO00.
Species 2 in this model has a higher rate of growth than species 1 (r 2 > r 1) and higher carrying capacity (K2 > K1). On the other hand, species 1 has a stronger competitive effect on species 2 (a21 > a12). Figure 8.20a displays trajectories for each species under these parametric conditions, starting with initial population sizes of NI(0) = 25 and N2(0) = 25. For comparative purposes the population trajectories in the absence of competition are shown in Fig. 8.20b. Note that competition lowers the effective carrying capacity of each population. However, the impact of competition is disproportionate between the populations, as a result of differences in the population parameters. The effect of competition on species 1 is to lower the effective carrying capacity by about 20% and to slow the rate of
NI*
K2/a21
convergence to this size limit. Similarly, the effect of competition on population 2 is to reduce its carrying capacity by about 20%, even though there are large differences between the two populations in the sizes of their competition coefficients and carrying capacities.
Example If the competition coefficients a12 and a21 in the previous example are increased to a12 -- 0.75 and a21 = 1.5, the system becomes unstable, with equilibrium conditions depending on initial population sizes. Instability results from the strong competitive interactions, expressed by a12a21 -- (0.75)(1.5) > 1.0. Under these conditions, one or both of the populations are driven to extinction, depending on the initial sizes of the populations. For NI(0) - 700 and N2(0) = 800, population declines result in the extinction of population 1, after which population 2 converges to its carrying capacity. For N 1(0) = 900 and N2(0) = 800, population 2 is eliminated and population I converges to its carrying capacity. For N 1(0) = 800 and N2(0) = 800, both populations are driven to extinction. 8.8.2. Lotka-Volterra Equations for Three or More C o m p e t i n g S p e c i e s The Lotka-Volterra competition equations for two species can be extended to include three or more spe-
168
Chapter 8 Traditional Models of Population Dynamics
1000 /
..,.,.._ ~
.
.
.
As before, rearrangement of these equations leads to the matrix equation
.
/
Nl(t) N2(t)
800
I
i
..
B
600
/
/
/
/
K1 K2
/
1
a12
a13
...
alto
a21
1
a23
...
a2m
B
J
I]
m
N~ N~d .
I
|
I
400
//
am1
1. LN*m_
am2
200
with solution .,.,
~o
do
30
40
m
I> NI~ N2(O
6 0 0
/
/
/
t
Nm*
.=
..
1
a12
a13
...
a21
1
a23
...
am1
am2
alm] -1 -K 1a2m I K2.
This matrix equation defines m equilibrium conditions, one for each species. If m = 3, for example, steadystate population sizes are
........--t
1
m
N~
1000
800
..,
N~
50
I
N~ = [(1
-
a23aB2)K1 -
a12(K 2 -
aagK3 )
(8.32)
a 1 3 K 3)
(8.33)
a12K2 )
(8.34)
400
-- a 1 3 ( K 3 - a B 2 K 2 ) ] / D ,
/
//
N~ = [(1
200
.j/
-
F I G U R E 8.20 Trajectories for two species with d y n a m i c s given by the Lotka-Volterra competition equations. (a) Population trajectories for a12 = a21 = 0. (b) Population trajectories for a12 = 0.25 a n d a21 = 0.5.
cies. The competition equations for, say, m competing species are
dt
= riNi
Ki -
Ni -
~
j=l
j,i
a Nj q
/Ki,
(8.31)
with species index i = 1.... , m. Equilibrium population sizes N~, ..., N* are given by the solution of
N~ =
K1 -
a12N'~ -
a13N ~ .....
almNr~ ,
N~ =
K2 -
a21N ~ -
a23N ~ .....
a2mNr~ ,
9
N *m = K m -
~
amlN'~ -
o
am2N ~ .....
am,m_lNr~_ 1 .
-
a13a31)K2 -
a23(K 3 -
a21(K 1 -
a31K1)]/D ,
and N~ = [(1 -
-
a12a21)K3 -
a32(K 2 -
a31(K 1 -
a21K1)]/D,
where D is the determinant of the competition matrix. As with a two-competitor system, a stable equilibrium is assured if D > 0. In addition, species coexistence is assured by the three equilibrium conditions N* > 0, i = 1, 2, 3. Thus, the populations either coexist or are driven to extinction, depending on the sizes of the competition coefficients and population carrying capacities. The following patterns are noted: 9 The equilibrium population sizes are given by the corresponding carrying capacities, reduced by amounts that account for competitive interactions among species. For example, the equilibrium size N~ for population 1 is the carrying capacity K 1, scaled by 1 -a23a32 and reduced by a sum of terms for species 2 and 3. These terms are products of the appropriate competition coefficients and respective carrying capacities, with the latter adjusted to account for competitive interactions between species 2 and 3. 9 Under certain conditions, the steady-state equilib-
8.8. Models of Competing Populations rium for a population is expressed in an additive form that is analogous to the two-species case. If, for example, a23 = a 3 2 - - 0 , the steady-state size for population 1 is N~
=
K1
-
a12K2
1 - a12a21
-
a13K3
-- a13a31
9 If m = 2 the steady-state population sizes, and the conditions for positive equilibria, reduce to the equilibrium conditions described above for two competing species. Example
Consider three competing populations with population dynamics specified as in Eq. (8.31). Assume that the populations each have the same carrying capacity (K i = K, i = 1, 2, 3) and all competition coefficients are identical (aij = a, i = 1, 2, 3 and j = 1, 2, 3). The determinant D = (1 - a) 2 (1 4- 2a) of the competition matrix is positive for all a ~ 0 except a = 1, so the system possesses stable equilibria for all levels of competition except a = 1. The coexistence conditions (8.32)-(8.34) reduce to 0 < [(1 - a 2) - a(1 - a) - a(1 - a)] = (1 - a) 2, which again is satisfied for values a ~ 1. Thus, nonunity competition coefficients lead to coexistence of all three species. For example, a = 0.2 and K = 1000 yields an equilibrium size for all three populations of 714, whereas a = 0.8 yields equilibrium population sizes of 385.
8.8.3. Resource Competition Models A second class of competition models expresses competition through the sharing of a resource by two or more competitors. To illustrate, consider a community of herbivores that utilize the plant biomass in an area for food (Tilman, 1980, 1982). Let R represent the available biomass of the food resource, subject to herbivory and regeneration over time. Here it is assumed that herbivore reproduction rates are influenced by availability of the food resource, but mortality rates are not: d N i / d t = b i N i [ R / ( R + Hi)] - mini,
with H i the amount of food resource necessary to sustain a reproduction rate for species i that is one-half the maximum reproduction rate b i. Thus, the reproduction rate for each herbivore population increases asymptotically from 0 to b i as R increases. In contrast, the popula-
169
tion mortality rate m i is constant irrespective of the availability of food. The dynamics of the food resource reflect the fact that each herbivore population depletes the resource according to the food requirements of individuals in that population. Regeneration of food is modeled in terms of growth to a maximum supply of food, with the rate of regeneration dependent on the difference between actual and potential supply: dR dt = a(S - R) - ~ ,
i
Ci f(Ni),
where S is the maximum amount of food that is potentially available and the parameter a represents the rate at which food supply is replenished. Depletion of the resource is a function of consumer populations through a consumption function f ( N i) that is scaled by speciesspecific terms c i. Exploitation competition is effected through the second term of this equation, wherein interspecific exploitation of food resources results in fewer resources and thus in lower growth rate than would be the case in the absence of competing species. Equilibrium conditions for this system are obtained by setting the transition equations to 0 and solving for the equilibrium levels of R and Ni:
biN~[ R* ] N*= mi R* + Si and
R*- S-
~ ciX~/a i
for f ( N i) = N i. From these equations an equilibrium value of R can be defined for each population considered in isolation. The solution for the complete system of equations leads to an equilibrium value R* such that biR*/(R* + H i) - m i vanishes for one competitive population and is negative for all others. This in turn leads to the eventual elimination of all other consumer species except that corresponding to the 0 growth rate. Example
Exploitation competition is illustrated in Fig. 8.21 with a discrete-time model involving two species that are competing for a common forage resource. Mortality losses for both competitors are described by simple death processes, with per capita death rate of 0.5 for each species. Reproductive success is modeled as a simple birth process (intrinsic birth rate is 1 in both cases) that is scaled by the factor R / ( R + H i) expressing the availability of food resources (H 1 = 400 and H2 = 500 for species I and 2, respectively). Thus, the amount
170
Chapter 8 Traditional Models of Population Dynamics
16
N~(t)
14
the population sizes. By extension, we can express a general model for m interacting species as -dN1/dt"
12
aN2/at
10 8 6 ,~ ~.. /
dNm/dt
\
4
"~
2
- all(N)
a12(X)
a13(X)..,
a21(X)
a22(X)
a23(X)..,
alm(X)- -Nl(t)N2(t) a2m(~
-.._.
40
9
F I G U R E 8.21 Population dynamics of two species that are competing for a common resource. Depending on the relative consumption efficiencies Hi, one species eventually is driven to extinction.
of food resource at which species 1 grows at one-half its maximum rate is less than the amount required for species 2. The stock of food resources is depleted in proportion to the sizes of the competitor populations, with species-specific scaling factors of cI = 10 and c2 = 10. The level of food resources is constrained by the maximum resource level of S = 1000, with replenishment occurring at a rate that is one-half the unmet potential. Note that species 2, with a lower efficiency in the transformation of resources into reproduction, declines asymptotically to extinction. On the other hand, species 1 and the resource asymptotically approach nonzero equilibrium states.
8.9. A GENERAL M O D E L FOR I N T E R A C T I N G SPECIES As indicated above, both predation and competition can be modeled with linear combinations of terms that express the species interactions. Thus, the Lotka-Volterra competition equations express competition by scaling the per capita population rates of growth with linear functions
E
Ki
-
-
N i - "= aq
K---~
that incorporate population sizes of the competitors. Similarly, the Lotka-Volterra predator-prey model expresses predation by scaling per capita rates of birth and death with scaling factors that also are linear in
~
_aml(~ arn2(~ am3(~ ... a m m ( ~ .
Nm(t)
where aij(N) is a (usually differentiable) function expressing the impact of population j on population i. A useful special case of the general model is defined by the linear forms (8.35)
a i j ( ~ = bij + c ijNi
for the model coefficients. By restricting the choice of the coefficients cij and bij, the model can be used to characterize interspecific interactions as described earlier. For example, if
i=j ir
and cij =
-ri/K i riaq/K i
i = j i ~ j
then Eq. (8.35) is identical to Eq. (8.31) describing multispecies Lotka-Volterra competition. Though some models discussed in this chapter satisfy a linearity requirement as in Eq. (8.35), in general, population models do not. Obvious extensions include quadratic, cubic, and higher degree equations, sinusoidal and other periodic relationships, multispecies functions aij(~_ that include nonlinear terms in the population sizes, and other mathematical characterizations. Population dynamics for these more complicated systems can be described by a linear model in a neighborhood of system equilibrium, via a linear approximation of the transition equations. Under fairly mild differentiability conditions, the function F i (-~ in the model d N / d t = F(N) can be expressed as Fi( ~
-- Fi(N* ) -t-- ~ L a N j ( N * ) 1
] (Nj-
N 7)
8.10. Discussion in a neighborhood of an equilibrium _N* (see Appendix C). Thus, the system dynamics of d N / d t = F(N) in a neighborhood of N* can be approximated by the linear system dt = F(N*) +
(N*)
( N - N*),
(8.36)
where the matrix d F ( N * ) / d N is defined as in Appendix B.10. On condition that species interactions are symmetric [i.e., 3Fi/ON j = 3 F j / 3 N i for every species pair (i, j)l, d F ( N * ) / d N can be expressed in terms of the singular value decomposition dF
aN(N_*) = e
e',
with _ha diagonal matrix of eigenvalues for C, and P an orthonormal matrix (i.e., P P' = P ' P = / ) with columns consisting of the corresponding eigenvectors (see Appendix B.7 for a description of singular value decomposition and other matrix procedures). Recognizing that d N / d t = d ( N - N * ) / d t and that F(N*) = 0 for the equilibrium point N*, we therefore can express Eq. (8.36) as dn/dt
= [P )~ P']n,
where n(t) = N(t) - N*. Multiplication of both sides of this system of equations by P' yields d m
d~ [P'n(t)] = k P ' n ( t ) or
dZ/dt
= k Z(t),
with Z(t) = P ' n ( t ) and n(0) = N(0) - N*. This reduces to m independent equations d z i ( t ) / d t = hiz(t)
in the synthetic variables zi(t) , with solutions zi(t) =
Zi(O) exp()~it). Back-transformation of Z ( t ) by P Z ( t )
= P P ' n ( t ) = n(t) then produces the population dynamics
for each population. A general solution is given by the solution based on zi(t) - Zi(O) exp ()~it), combined with the particular solution Ni(t) = [Fi(N*)]t , i = 1, ..., m (Rainville et al., 1996).
8.10. D I S C U S S I O N In this chapter we have described some models that traditionally have been used in ecological and biologi-
171
cal sciences. We began with simple expressions for the exponential and logistic models and added structure throughout the chapter to account for various biological features. In the appropriate context, each of these models can provide useful insights about population dynamics. As discussed in Chapter 7, there is an inevitable tradeoff between the generality provided by relatively simple models lacking specific, detailed mathematical structures, and the realism and precision that can be attained by more complicated and biologically rich models. We have seen that the incorporation of additional biological structure and function into a population model quickly leads to difficulties in interpreting model behaviors. For example, a complete sensitivity analysis of a Leslie matrix model with four age cohorts would involve the assessment of eight reproduction and survivorship parameters (four per capita birth rates and four survival rates), requiring a sensitivity analysis for each parameter at a minimum of 27 different combinations of values of the other parameters (assuming only two values for each parameter). Thus, even without accounting for the influence of system initial conditions, this relatively simple model requires a total of 256 different sensitivity assessments. Clearly, there is strong incentive to include only the features in a model that are essential in meeting its objectives. As indicated in earlier chapters, the biological justification for a model and the interpretation of evidence for it are key to its usefulness. Consider the observation of a sequence of population abundances, to be fitted with a model of population growth (Fig. 8.22). These data appear to support the assumptions of the logistic model, which imply that per capita birth rates, death rates, or both decrease with increasing density. Assume for now that abundance at each time step is estimated perfectly, so that statistical sampling error is not at issue. As shown in Fig. 8.22a, a discrete logistic model with parameters r = 0.2, K = 500 seems to fit the data, so we might conclude that the logistic model "explains" temporal variation in abundance. But other biologically reasonable models fit the data as well. For example, an alternative model assumes the population is growing in a density-independent fashion up to an absolute limit K (e.g., as determined by available space), and then all excess individuals above K either die or migrate. The predicted trajectory for this model (Fig. 8.22b) is similar to that for the logistic, yet the model invokes no assumptions about density dependence. A second alternative might involve growth that is density independent up to K, but that K, rather than being fixed, varies randomly (e.g., because of annual precipitation factors). Again, the predicted trajectory
172
Chapter 8 Traditional Models of Population Dynamics FIGURE 8.22 Comparison of three models againstfield data. (a) Discretelogisticmodelwith r = 0.2 and K = 500. (b) Exponential model, truncated at K = 500.(c) Exponential model, truncated at randomly varying K.
N(t)
-%-.
a
500
._
oooeeee
c
00000
o 9
450
400
350
3OO
0
;
10
is
20
fs
0
5
10
-- 15
(Fig. 8.22c) is similar to that of the logistic model and resembles the observed abundances. In fact, the available field data are unuseful in discriminating among these three candidate models for population growth. Any of the three models could have generated the observations, and thus it is not possible to validate one particular model based on the data. In Chapter 9 we consider the identification of models with time-series data, and Part III deals in considerable depth with the use of sampling data to estimate specific model parameters. Here we simply emphasize that the use of biologically based models, combined with good experimental designs and careful inference, can help to avoid an unjustified connection between observations and underlying premises. In this example, improper inferences about density dependence could lead to erroneous predictions about the impacts of harvest on population growth and thus to faulty management recommendations. Well-designed monitoring efforts and careful assessments help to avoid such errors. We also emphasize the need to account for sampling variablility when population inferences rely on vital rates that are sample-based estimates rather than exact parameter values. Suppose we use demographic data from two populations to obtain estimates of survival and reproductive rates, which in turn are used in Leslie matrix models to determine the population growth rates of 1.05 and 0.99. Because these growth rates rely on estimates of survival and reproduction rates that are based on sampling data, the growth rates inherit randomness from this sampling variability. Replicated sampling of the populations would produce different values for the growth rates, according to a probability distribution that depends on the sampling scheme. A
0
5
10
1'5
20
2'5
number of questions therefore arise about the magnitudes and differences between the population growth rates, assuming one of the underlying models is appropriate. Inferences about population vital rates, and thus the growth rates, must account for sampling variation in estimates that are based on sample data. Methods for estimating sampling variances of asymptotic growth rates include both bootstrap approaches and delta method approximations (e.g., see Lenski and Service, 1982; Lande, 1988; Alvarez-Buylla and Slatkin, 1991, 1993, 1994; Brault and Caswell, 1993; Franklin et al., 1996; Caswell, 2001). This variation must be considered when one asks questions about the magnitude and direction of population changes and about differences in growth rates between the two populations. The usefulness of modeling procedures, especially in decisionmaking, is likely to be improved by smartly designed sampling efforts supporting models that avoid unnecessary complexity and by continual comparison of model predictions to observations, where possible in an experimental framework. Finally, it should be emphasized again that, though none of the models discussed in this chapter is "correct" in the sense of capturing all the features of a population, no model is necessarily "incorrect." In fact, no model is capable of a comprehensive characterization of a real biological population. Put differently, all models are "wrong," in that all models leave out far more about population structures and functions than they incorporate. It is the role of the biologist, modeler, and analyst to determine what level of biological detail is necessary (and feasible!) for a model to meet its objectives and, having made that decision, to find informative ways to investigate model behaviors pursuant to those objectives.
C H A P T E R
9 Model Identification with Time Series Data
9.1. MODEL IDENTIFICATION BASED ON ORDINARY LEAST SQUARES 9.2. OTHER MEASURES OF MODEL FIT 9.3. CORRELATED ESTIMATES OF POPULATION SIZE 9.4. OPTIMAL IDENTIFICATION 9.5. IDENTIFYING MODELS WITH POPULATION SIZE AS A FUNCTION OF TIME 9.5.1. Model Identification in One Dimension 9.5.2. Model Identification in Two Dimensions 9.5.3. Model Identification in Three or More Dimensions 9.6. IDENTIFYING MODELS USING LAGRANGIAN MULTIPLIERS 9.7. STABILITY OF PARAMETER ESTIMATES 9.8. IDENTIFYING SYSTEM PROPERTIES IN THE ABSENCE OF A SPECIFIED MODEL 9.9. DISCUSSION
ulation model, based on available data and other relevant information. Typically the information used in model identification comes from a wide range of sources, including laboratory experiments, field studies, anecdotal information, historical information that is documented in the published literature and in field notes, and other sources. The process of identification involves the use of this information in recognizing model features and estimating model parameters. Three elements in the process can be recognized (Ljung, 1999): 1. An available set of information, including extant data bases and data collected in the field pursuant to the identification of model structures and parameters. 2. A set of candidate models, from which to identify that model which is most appropriate for its intended uses. 3. A rule for comparing and contrasting models, to serve as an aid in selecting a "best" model as guided by data and other information.
In Chapter 8 we discussed a number of population models with parameters such as initial population size, survivorship, and reproduction rates, having outlined in Chapter 7 a process of model development that incorporates these parameters and assesses their importance in influencing model behavior. However, the actual procedures by which model features can be identified a n d / o r estimated with data are yet to be developed. In this chapter we describe some techniques to "identify" a model of a particular mathematical form, utilizing data to guide the identification process. This is preparatory to a comprehensive treatment in Section III of statistically based approaches to parameter estimation. The overall objective of model identification is to specify the structural features and parameters of a pop-
Model identification can be recognized in the approach described in Chapter 7 for model development, especially in the processes of verification, validation, and identification of system features and boundaries. Several kinds of activities are involved, including the following approaches: 9 Initial specification of model equations, parameters, operating constraints, the model time frame, and so on. Initial model development is based on biological theory, mathematical analysis, intuition, expectations about model performance, and other sources of infor-
173
174
Chapter 9 Models and Time Series Data
mation. The process of model development was described in Chapter 7. 9 The "fitting" of models to time series data, in the sense of identifying structural features a n d / o r parameters of a model through a comparison of model trajectories against sequences of field observations. In this case, model identification occurs indirectly, through, e.g., the choice of parameter values that provide a good "match" between the model trajectory and field observations. 9 The estimation of parameters such as population size, density, and survivorship with data collected for the purpose of estimating particular population parameters. Estimation in this context typically involves data collection according to a sampling design that targets the parameters of interest and data analysis based on probability models that include these parameters. In this chapter we focus on the "fitting" of models to time series data, through a comparison of model trajectories against a sequence of field observations. This activity is closely associated with model verification, which also involves a comparison of model performance with observed patterns in population dynamics. Model identification also can be seen as part of an adaptive process of model development, application, and refinement (see Chapter 24). A conceptual framework for model identification includes a population model
N(t + 1) = N(t) + f(N, Z, U), N(O) = N O and data that are collected at various discrete times (typically at each time) over the course of the time frame. The model describes population dynamics in terms of population size N(t), environmental influences Z(t), and (possibly) management actions U(t). Its mathematical form is assumed to be well defined, but some of the parameters in the model are not known and therefore must be identified. Here we use N(t) to indicate a model-based prediction of population size, with the actual population size to be estimated with data. The data collection focuses on population size and possibly other population attributes at various times in the time frame. Let S = {tI .... , tk} represent times at which an estimate IQ(t i) of population size is available. The notation N(t i) is used for estimated population size to emphasize that these values are based on data that are subject to sampling~ variability. The amount of variation in N(t i) depends on the sampling design and sampling effort, as discussed in Chapter 5. Assuming the samples are representative of the population and the estimation procedure is unbi^
ased, the accuracy of 1Q(ti) increases asymptotically with sampling effort.
9.1. M O D E L I D E N T I F I C A T I O N BASED ON ORDINARY LEAST SQUARES For illustrative purposes, we begin with a simple exponential model over a discrete time frame T, along with a set {N(ti): t i ~ S} of estimates of population size. The estimates in this "observation set" correspond to a set S = {t1, ..., t k} of times that are distributed over the time frame. In the absence of additional information about birth/death rates or population growth rates, model identification consists of choosing model parameters based on the set of population estimates. From Section 8.1 we know that the behavior of an exponential model with net per capita growth r is given by
N(t + 1) = N(t) + rN(t), N(O) = N 0, with a trajectory determined by the two parameters N Oand r. To identify the model, it therefore is sufficient to estimate these parameters. A method for doing so consists of comparing predicted and estimated (or observed) population sizes at each point in the time frame for which data are available and of choosing values for N Oand r to ensure the best possible match of model output and data. A standard index by which to measure such a match is "mean squared error:" F(N 0, r) = ~
[N(t i) - ]Q(ti)]2/k,
ti~S
where N(t i) is the predicted population size from the model and N(t i) is the estimated (or observed) population size based on field data (Rawlings, 1988). Because the parameter values N Oand r influence the predicted values, they also influence the mean squared error function. We describe 1Q(ti) in what follows as an observed population value at time t i, recognizing that the "observations" are based on data with which population size is estimated. Note that mean squared error is small to the extent that the model represents the observed population values, with a limiting value of zero in case the model fits the observations exactly. On the other hand, mean squared error is large to the extent that the model fails to represent the observations. The effect of squaring the deviations between observed and predicted values is to give very large weight to large deviations. Thus the mean squared error, which is greatly inflated by
9.1. Model Identification Based on Ordinary Least Squares large deviations, can be reduced dramatically by reduction (or elimination) of these deviations. Of course, this reduction is obtained through the choice of values for the parameters No and r. Because the mean square error depends on the pair (N 0, r) of parameter values, the fitting of the model to data can be seen as an optimization problem, wherein the pair (N 0, r) is to be chosen to minimize F(N0, r): minimize F(N o, r) = ~ [ N ( t i) - l~](ti)]2/k NO, r
175
in Fig. 9.1. Mean squared error, represented by the vertical axis, is shown as a function of the parameters N O and r, represented by the two horizontal axes. At some point in the parameter plane, the mean squared error assumes a minimum value. Provided the error function is minimum for positive values of N o and r, its partial derivatives both are zero at the minimum point. Geometrically, this means that the tangent plane for the error function is horizontal at (N~, r*) (see Appendix H).
tieS
Example
subject to N(t + 1) = N(t) + rN(t), N(O) = No.
Because N(t) = N0(1 + r) t for the exponential model, the transition equations can be incorporated directly into the objective function: F(N o, r) = ~ [ N o ( 1
+ t") ti -- l ~ ( t i ) ] 2 / k .
ties
Necessary conditions for a nonzero solution to this optimization problem are
aF/aNo
Consider a population of rodents that were introduced into a previously uninhabited habitat. Resource managers are concerned about the rapid growth of this population and need to predict population size as they consider potential control programs. Population growth has been tracked each year since the time of introduction with population surveys, producing the estimates [/~/(1),/~/(2), N(3), N(4)] = (20, 35, 68, 121) of population size. Because introduction of the species occurred only recently and as yet there are no indications of declining population rate of growth, an exponential model is used to describe the population. Identification of the model involves estimation of per capita growth rate r and initial population size N 0, based on the available survey data. Optimal estimates of these parameters can be obtained by minimizing the
where the partial derivatives are given by OF~ONo = 2~[No(1
+
F) ti --
/Q(ti)](1 + r)ti/k
ties
and OF~Or = 2 N 0 ~ ti[No(1
+ r) ti -
/~/(ti)](1
+ r)ti-1/k
ties
(see Appendix H). Thus the optimality conditions are equivalent to gl(N0, r ) = ~[(N0(1 + r) ti - N ( t i ) ] ( 1
+ t') ti = 0
ties
and g2(No, r ) = ~
ti[No(1 + y ) t i _ ]Qti](1 + r ) t i - l =
O,
tieS
and fitting the exponential model to the data set {/~(ti)" t i ~_ T} reduces to a problem of finding zeros for the two functions gl(No, r) and g2 (No, r) that are defined by partial derivatives of the mean squared error function. Model identification through the minimizing of an error function is illustrated for the exponential model
F I G U R E 9.1 Geometry of mean square error for the exponential model N(t) = N0(1 + r) t. The error function F(N 0, r) = ~i[N(ti) /~(ti)] 2 is minimized for values (N~, r*) of the model parameters
(No, r).
176
Chapter 9 Models and Time Series Data
mean squared error, subject to the model transition equations. As above, this is equivalent to finding the zeros of gl(N0, r ) =
~[N0(1
+
r) ti -/Q(ti)](1 + r) ti
ti*S
= [N0(1 + r ) - 2 0 ] ( 1 + r) + IN0(1 + r) 2
_
35](1
+ r) 2
4- [N0(1 4- r) 3 - 68](1 + r) 3 + [No(1 + r) 4 -
121](1 + r) 4
and g2(X0,
r) = X tiN0(1 + r)ti
__
Xtil(1 + r)ti-1
ti~S
= EN0(1 + r) - 20] + 2IN0(1 + r) 2
_
35](1 + r)
+ 3[N0(1 + r) 3 -
68](1 + r) 2
+ 4[N0(1 + r) 4 -
121](1 + r) 3.
Application of a gradient search procedure (see Appendix H) yields the values N~ = 11 and r* = 0.83 that minimize the mean squared error for this population. Thus, the population model is
N(t) = N~(1 + r*) t = 11(1.83) t, and the predicted population size for year 5 is 226. Though uncharacteristically simple, this example of model fitting is nevertheless informative of a general approach to model identification. Key components of the approach are as follows: 9 Description of the problem in terms of constrained optimization, with mean squared error as the objective function to be optimized and the system transition equations representing constraints on the choice of parameter values (see Chapter 21). A general statement of the problem is minimize F(a) = ~ [ N ( t i) - N(ti)]2/k a
tieS
subject to
N(t + 1) = N(t) + f(N: a), N(0) = No, where _a is a vector of model parameters (perhaps including N 0) that are to be identified, and f(N:a) specifies the predicted change in population size through time. As a matter of notational convenience, environmental
and control variables Z(t) and U(t) are suppressed in this formulation. 9 Incorporation of the transition equations into the mean squared error objective function. For simple systems, this sometimes can be accomplished by actually solving the transition equations, so that N(t) can be expressed as a function of the model parameters. In the example above, N(t) is given in terms of the parameters N O and r, by N(t) = N0(1 + r) t. In most cases an analytic expression for N(t) cannot be obtained, and the transition equations must be incorporated by means of "Lagrangian multipliers." The use of Lagrangian multipliers is described in some detail in Section 22.1 and Appendix H. 9 Differentiation of the objective function (as modified by incorporation of the transition equations) with respect to the model parameters. This defines a system of functions in the model parameters. 9 Determination of the zeros for these functions (i.e., the parameter values for which the functions have a value of 0). The zeros can be determined by numerical methods or, in a few instances, by mathematical analysis.
9.2. O T H E R M E A S U R E S OF M O D E L FIT Though mean squared error is the most common measure by which to judge the fit between data and a model, it is by no means the only measure. Another that sometimes is used is mean absolute error:
F(a) = ~,lN(ti) - /~(ti)l, tieS
where IN(t) - ~l(t) I is the absolute value of the difference N(t) - ~l(t). This measure is less sensitive than mean squared error to large deviations between predicted and estimated population sizes. Nevertheless, its value is large when deviations are large and small when deviations are small, with a lower limit of zero as deviations approach zero. Other mathematical forms can be used to measure the importance of deviations, and other factors can be included in the objective function to account for, e.g., patterns of variation in the estimates N(t). For example, both the mean squared error and mean absolute error functions can be modified so that the deviations are scaled with weights that decrease with increasing variation in the estimates /x/(t). The logic for such a weighting scheme is that population estimates with large variance are not as informative of the true population size as estimates with small variance. Under these circumstances, N(t) - ~l(t) only imprecisely represents
9.2. Other Measures of Model Fit the deviation between actual and predicted population sizes, making it more difficult to ascertain the "best" parameter values with which to represent the population. It therefore is reasonable to weight deviations with small variance more heavily, because they better represent differences between actual and predicted population sizes, and to weight deviations with large variance less heavily (Rawlings, 1988). A generalized expression for the identification problem is minimize F(a) = ~ g[N(t i) - ~l(ti)]/k a
ti*S
subject to
N(t + 1) = N(t) + f(N: a), N(0) = N 0, where a is the vector of model parameters to be identified anti giN(t) - / Q ( t ) l is a monotone increasing function of the deviations N(t) - [q(t). In the case of mean squared error,
g[N(t) -/~/(t)] = wt[N(t) - /~/(t)]2, where w t is the weight assigned to deviation N(t) /Q(t) and Xt~s wt = 1. In the case of mean absolute error,
ity). An experiment to investigate this hypothesis involves several populations of fruit flies that are subjected to different temperature regimes under controlled experimental conditions. A small (but unknown) number of fruit flies is released at the beginning of the experiment into each of several growth chambers that are regulated for temperature, and daily estimates of population size are recorded for 5 consecutive days thereafter. The data subsequently are used to fit a series of logistic models of continuous population growth under the different temperature regimes. Model parameters for each of the populations are identified by means of an error function with components wt[N(t) - 1Q(t)]2, with the weights based on (1) an indication from the data that variation in the population estimator increases with population size, and (2) improvement of the investigators' counting skills through the course of the experiment, so that later counts are less subject to counting error than are earlier counts. Identification of model parameters for each experimental population is obtained through an optimization process that accounts for these features: 5
minimize F(N 0, r, K) = ( 1 / 5 ) ~ wt[N(t) -/Q(t)] 2 No,r,K
g[N(t) - / ~ ( t ) ] = [N(t)
-/Q(t)]
3/2,
which is influenced by large deviations to a lesser degree than mean squared error, but to a greater degree than mean absolute error. It also would allow for weighting schemes that include other factors besides variation in the population estimates. For example, it often is reasonable to emphasize the fit of the model to data of more recent vintage. A weighting scheme that emphasizes the value of more recent data over [0, 1.... , T] is w(t i) = ti/~, j tj, for which weights decline linearly with the age of the data.
Example Temperature is hypothesized to influence the growth of fruit fly (Drosophila spp.) populations through its effect on both the rate of population growth and the population potential (i.e., the carrying capac-
t= 1
subject to
g[N(t) -/Q(t)] = wt]N(t) - ~l(t) I. This formulation of the identification problem is quite general, in that it can accommodate any deviation function, so long as it is monotone increasing in N(t) - /Q(t), and any weighting scheme, so long as the weights are nonnegative and their sum is 1. For example, the formulation would allow for the deviation function
177
dN/dt = rN(1 - N/K), N(0) = N 0, where {var[/Q(t)]} -1 W t
s
1{var [/Q(t)l }-1
with var[1Q(t)] the sampling variance of the estimator N(t). In this particular case, var[l~(t)] can be approximated by N(t)/t, so that the weights are
t/1Q(t)
wt
-
~t=15 t/1Q(t)"
The continuous logistic model has solution
N(t) = 1 +
K Ce -rt'
with C = K/N o - 1 (see Section 8.2), which can be incorporated directly into the objective function:
5 [ F(N 0, r, K) = ( 1 / 5 ) ~
Wt
N(t) = 1 +
Ce -rt
t
9
t=l
As before, necessary conditions for a nonzero solution to this problem are given by partial differentiation of
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Chapter 9 Models and Time Series Data
this function, so that the problem reduces to finding the zeros of a system of equations in N 0, r, and K.
9.3. CORRELATED ESTIMATES OF POPULATION SIZE Depending on the sampling and estimation procedures, the estimates of population size used in model identification can be correlated, in that the estimates in successive (and possibly other) time periods have a nonzero sampling covariance. Both the sampling variances and covariances can be accounted for in the weighting scheme of the objective function (Seber and Wild, 1989). Assume, for example, that estimates of population size are obtained in successive years of a multiyear study, and the estimates are subject to both sampling variability and covariation. Assume also that the estimator variances and covariances are known (or can be estimated). Let 0-i2 represent the variance of the ith estimate and 0-/j represent the covariance between estimates for periods i and j. An appropriate form for the error function that "adjusts" for the correlation structure is k
F(a)
=
-
,Y__, 0-q[N(i) -
l~( i) ][N( j)
-/~(j)],
i,j=l
where/q' = [/~(1),/~(2) .... ,/~/(k)] is a vector of population estimates in successive years, and 0-zjis the element in the ith row and jth column of the inverse of the dispersion matrix of variances and covariances. . .
Example
Assume that the estimates of population size in k successive time periods all have the same variance and that the correlation of the estimates decreases exponentially with the time between estimates" 0-i2 -- 0 -2 and 0-q = O"213 Ii-jl. Under these circumstances, the inverse of the dispersion matrix is composed of the elements ~1/(1 - [32)0- 2 .. J(1 + 132)/(1 -o"] = ~-p/(1 - 132)0.2
Lo
132)0"2
i = j = 1 or k j = 2 .... , k - 1 [i- jl = 1
i =
otherwise
(Graybill, 1969) and a quadratic error function that adjusts for the correlations is given by F(a) = [N(1) -/~/(1)] 2 + [N(k) -/~/(k)] 2 k-1
+ ~ (1 + p2)[N(t) - /~/(t)]2 t=2 k-1
- 2p ~, [N(t + t=l
1) -/~/(t + 1)][N(t) -/~(t)].
On examination, similarities can be seen between this somewhat complicated function and the simpler error function F(a) = ~ t [ N ( t ) - N ( t ) l 2 for uncorrected data. For example, both retain a sum of squared error terms. However, the more complicated quadratic error function also includes cross-product terms that are associated with the nonzero correlation p.
9.4. OPTIMAL IDENTIFICATION There is a large, mathematically sophisticated literature on optimization of multidimensional functions. Here we discuss three general approaches to the problem of finding points at which a smooth (differentiable) function assumes a minimum value. To restrict attention to optimization methods that are appropriate for model identification, we assume the following: 9 The only feasible parameter values are positive. Because population initial conditions, rates of growth, carrying capacities, competition coefficients, predation coefficients, and the like are positive (or can be reparameterized to be positive), this assumption is not biologically limiting. 9 The error function is everywhere differentiable over the set of feasible parameter values. Again, this assumption is unlikely to be limiting for the usual measures of identification error. 9 The error function has a minimum value for some unique point in the set. Geometrically, this means that the error function is "downward sloping" toward a single minimum value over the range of parameter values (e.g., Fig. 9.1). 9 The population transition equations are incorporated into the objective function either directly, by solving for the population size as a function of time, or indirectly, by adding the transition equations to the objective function by means of Lagrangian multipliers (see Appendix H). In either case the problem is one of minimizing a function (either the objective function or the Lagrangian function) of the parameters of interest. We illustrate three data-based approaches to the identification of model parameters, each of which is distinguished by its computational and analytic requirements. We first discuss their application when population changes through time are incorporated directly into the objective function and then consider applications when population change is accommodated by means of Lagrangian multipliers. The approaches first are described in terms of models containing a single unknown parameter and then are generalized to account for two or more parameters.
9.5. Identifying Models with Population Size as a Function of Time 9.5. I D E N T I F Y I N G M O D E L S W I T H P O P U L A T I O N SIZE AS A FUNCTION OF TIME We begin with a description of methods for which parameterized forms of population size can be incorporated directly into an objective function, utilizing a closed form for population size as a function of time. For example, population dynamics for the exponential model can be described by N(t) = N0(1 + r) t, and this function can be substituted directly into the error function prior to its being minimized. The result is a minimization problem involving the two parameters r and N 0. Similarly, population dynamics for the continuous logistic model can be described as above by
N(t) =
1 +
Cr - r t
which can be substituted directly into the objective function to be minimized. The result is a minimization problem in the three parameters N 0, r, and K.
9.5.1. M o d e l Identification in One D i m e n s i o n A biological example of model identification with a single parameter might involve the fitting of an exponential model for which initial population size is known with certainty. The problem then reduces to finding a value for intrinsic rate of growth so that the model optimally fits a set {l~(ti): t i ~_ S} of data. In general terms, the optimization problem is
179
9 In the event that the derivative of F can be derived but zeros of the resulting equation cannot be obtained analytically, numeric procedures can be used. A standard approach is Newton's method, in which the derivative of the error function is used in an iterative search procedure (see Appendix H). Newton's method utilizes the derivative dF/da at some starting value a 0 to determine the tangent line of the objective function at a 0. The zero of this line is used as an updated value a 1 for a, and the derivative of F at a I is used to determine a new tangent line with a zero that defines yet another value for a (Fig. 9.2). The updating process continues iteratively until no further change is found in the value of a. Note that the derivative of the error function must be evaluated at each iteration. 9 In case the error function a n d / o r transition equations are so mathematically intractable that derivatives cannot be obtained or Newton's method is computationally burdensome, one can use directed search procedures, in which some initial value a is updated through evaluation of the error function at points in either direction from a (Appendix H). The initial value is replaced with a new value that gives the largest reduction in the value of the objective function. This process continues iteratively until reductions in the error function cease.
9.5.2. M o d e l Identification in Two D i m e n s i o n s An example of model identification in two dimensions is the fitting of an exponential model in both
minimize F(a) = ~, g[N(t i) - l~l(ti)]/k a tieS subject to
N(t + 1) = N(t) + f(N: a), N(0) = N 0, where a is a single model parameter to be identified. Approaches to this problem include the following considerations: 9 Solving the equation obtained by equating the derivative of the objective function to zero. Because an optimal value a* must satisfy dF/da = 0 for a > 0, solving the equation identifies candidates for minim u m error identification. A sufficient condition for minimization is d2F/da 2 > 0 (see Appendix H). Note that this approach requires F to be differentiable, and the equation dF/da = 0 must have a solution. Either or both these requirements may fail to be met for a particular problem.
(.9
6
/
a2
a1
80
F I G U R E 9.2 N e w t o n ' s m e t h o d for finding the m i n i m u m of a differentiable function F(a), given that dF/da = G(a) v a n i s h e s at a m i n i m u m . Starting at an initial v a l u e a 0, the zero of the t a n g e n t line to G(a) at a 0 is u s e d as an u p d a t e d value a 1. This v a l u e is u s e d in t u r n to d e t e r m i n e a n e w t a n g e n t line at a], w i t h a zero that defines yet a n o t h e r value a 2. The u p d a t i n g process continues iteratively until the values of a cease to change.
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its parameters, as discussed above. The corresponding optimization problem is expressed by
function. This process continues iteratively until reductions in the objective function cease.
minimize F(a) = ~, g[N(t i) - l~(ti)]/k a
--
ti*S
9.5.3. M o d e l Identification in Three or More D i m e n s i o n s
subject to
N(t + 1) = N(t) + f(N: a), N(O) = N 0, where the vector _a consists of the two parameters r and N O. Approaches to this problem include analogs to those for the one-dimensional problem: 9 Solving the system of equations obtained by equating the partial derivatives of the objective function to zero. A necessary condition for a positive value of a* is that a* must satisfy OF/Oa = 0. Thus, solving the system of equations identifies candidates for m i n i m u m error identification. Sufficient conditions for minimization are
32F/3a 2 < 0 for i = 1, 2, and
32F/ Oa2
c32F/ Oa13a2
32F/Oa10a2
c92F/Oa2
<0
(Appendix H). Note that this approach requires the differentiation of F with respect to both parameters, as well as the solution of a system of equations in the derivatives. Either or both of these requirements may be difficult for a particular problem. 9 In case the derivatives of F can be obtained but the resulting equations cannot be solved analytically, one can use iterative gradient search procedures to identify the m i n i m u m point of the error function. Thus, an initial value a 0 is updated with a new value in the direction of the gradient (3F/Oa1, OF/Oa2) of the objective function at a. The new value of a is chosen to minimize the error function in the direction of the gradient (see Appendix H.1). The partial derivatives of F are evaluated at the new value, and a new search is initiated. This process continues iteratively until no further change is found in the value of a. 9 To avoid differentiability problems, derivativefree search procedures also can be used. Thus, an initial value a 0 is updated through evaluation of the objective function at other points near a 0. Because of the addition of a second dimension in the parameter space, at each iteration it is necessary to evaluate the objective function at a number of different points in the two-dimensional space defined by a. The value a 0 is replaced with a new parameter value that most reduces the objective m
An example of model identification in three dimensions is the fitting of a logistic model in all three of its parameters, as discussed above. The corresponding optimization problem is expressed as in the twodimensional problem, except that the vector _acontains k parameters, with k -> 3. Approaches to the multidimensional problem include analogs to those for the two-dimensional problem: 9 Solving of the system of equations obtained by equating the partial derivatives of the objective function to zero. A necessary condition for a positive value of a* is that a* satisfy OF/a_ = 0. Solving the system of equations identifies candidates for m i n i m u m error identification. Sufficient conditions for minimization involve some rather complicated expressions in the second partial derivatives of F (Appendix H). Note that the number of equations in the system increases with the number of parameters to be identified, and the analytic requirements for differentiability of the error function do as well. This makes it increasingly difficult to construct and solve the system of equations. 9 If differentiability conditions are met, gradient search procedures can be used to find the m i n i m u m value of F. As in the two-parameter case, a standard approach is to search for candidates to update an initial value a 0 in the direction of the gradient OF/Oa_ of the objective function at a 0. A new value for _a is chosen that minimizes the error function along the gradient. At the new value, the partial derivatives for F are computed and a new search is initiated along the resulting gradient. This process continues iteratively until no further change is found in the value of a. Note again that the partial derivatives of the error function must be computed at each point in the iteration. 9 Derivative-free search procedures can be used in the multiparameter case, in which the initial value for a is updated through evaluation of the error function at other points near a. Because of the increased dimensionality of the parameter space, it is necessary to evaluate the error function at a large number of different parameter values in each iteration. The value of a is replaced at each iteration with a new value that most reduces the error function. This process continues iteratively until reductions in the objective function cease. Obviously, the amount of computation with this method increases dramatically as the number of parameters increases.
9.7. Stability of Parameter Estimates
181
ables )~t that were not required for the exponential model. This example displays the key features of model identification with Lagrangian multipliers:
9.6. I D E N T I F Y I N G MODELS USING LAGRANGIAN MULTIPLIERS The approaches described above require a closed form for population size as a function of time. In the more usual situation in which the population trajectory cannot be determined in closed form, model identification must be modified to allow the transition equations to serve as constraints on the minimization of the error function. This is accomplished with Lagrangian multipliers, by means of which the objective function is modified to include the transition equations. The use of Lagrangian multipliers is most easily described by an example. Consider the rodent population of the previous example and assume that, prior to implementation of any management strategy, a population size of 162 is recorded for the fifth year after introduction. Because of the much reduced rate of population growth between years 4 and 5, a logistic model now is thought to be appropriate for the population. However, resource managers remain concerned about the potential for additional population growth and continue to need a prediction of population size. Identification of the model now involves specification of the initial population size N 0, intrinsic growth rate r, and population carryin~ capacity K, based on the observations [N(1), N(2), N(3), N(4), N(5)] = (20, 35, 68, 121, 162). As before, model identification involves the minimization of mean squared error subject to the model transition equations. However, in this case the logistic transition equations can be incorporated into the objective function by means of Lagrangian multipliers:
9 The mathematical form of a model characterizing population dynamics must be assumed, with model identification described as a minimization problem constrained by the model transition equations over the time frame. 9 The transition equations are incorporated into the objective function by means of Lagrangian multipliers, with a distinct multiplier ~'t for the transition equation at each time t in the time frame. This extended objective function, called the Lagrangian function, is influenced not only by the parameters of interest, but also by the Lagrangian multipliers. 9 The Lagrangian function is minimized with respect to both the parameters of interest and the Lagrangian multipliers. In case the Lagrangian function is differentiable, setting its derivatives with respect to )~t equal to zero reproduces the transition equations. These equations, along with analogous equations based on the derivatives of the Lagrangian function with respect to the parameters of interest, constitute a system of equations that can be solved numerically for the parameter estimates. For example, a two-parameter model would involve a gradient search utilizing the gradient (OL/cOa 1, OL/3a 2, 3L/OK_) of the Lagrangian function. Lagrangian procedures for constrained optimization are discussed in some detail in Section 22.3 and Appendix H.
4
L(N 0, r, K, K) = F ( N o, r, K) + ~, ht{N(t + 1) --
t=0
9.7. S T A B I L I T Y O F PARAMETER ESTIMATES
- N ( t ) - r[1 - N ( t ) / K ] } 4
= ~ ([N(t + 1) -/~/(t + 1)]2/5 t=0
+ )~t{N(t + 1) - N ( t ) - r[1 - N ( t ) / K ~ } ) ,
where K t is a "Lagrangian multiplier" for the corresponding transition equation, and the extended objective function L(N 0, r, K, ,~) is the "Lagrangian function." The problem now is to minimize the Lagrangian function by choosing the parameters N 0, r, K, and h. Differentiation with respect to N 0, r, and K results in three equations in the parameters, and differentiation with respect to _h reproduces the transition equations. As before, this reduces to the problem of finding the zeros of functions defined by the derivatives; however, the problem is complicated by the need to consider the additional logistic parameter K and additional varim
An important consideration in model identification is the size of the data set {/~/(ti)" t i ~_ S} relative to the number of model parameters to be identified. In essence, the size of the data set used to define the error function should be substantially larger than the number of parameters; otherwise, variation in the data may lead to identification of parameter values that are unreasonable a n d / o r highly unstable. This can be seen with a simple example involving identification of the exponential model N ( t ) = N0(1 + r) t, based on a data set {N(ti): t i ~ S}of k observations. Here we assume that the exponential model is structurally correct, in that the form of the underlying process for the data is exponential in its mean: E[N(t)] = N0(1 + r) t. Thus the objective is to estimate the parameters r and N O by fitting the model to the data set. The problem can be
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Chapter 9 Models and Time Series Data
simplified greatly by transforming the data and the model with a logarithmic transform and by using logarithms for model fitting. The net effect of the logarithmic transform is to "linearize the model," wherein the exponential model is transformed into log[N(t)] = log(N 0) + [log(1 +
r)]t.
Thus the model can be expressed as
Yt = a + bt, with Yt = log[N(t)], a = log(N0), and b = log(1 + r). The transformed model, which now is linear in the parameters a and b, can be expressed in matrix form as
where 1 is a vector of ones, t is a vector of observation times, and Y consists of transformed model values. Model fitting based on mean square error is equivalent to least-squares estimation from linear regression, with
9 The greater the amount of data with which to identify the model, the easier it is to ensure that the model is structurally correct. With only a few data points, it is difficult to tell whether the model actually characterizes population dynamics. With additional data, the parameter estimates are more precise, and it becomes possible to assess the "goodness-of-fit" of the mathematical form as well as the parameterization of the model. We note that these descriptions apply to the estimates/~/0 and f as well as the estimates l/and/~. Because /~0 and f are obtained from ~ and/~ by an exponential transform that is monotonic and mathematically well behaved, the same patterns of variation hold for/~/0 and f. Indeed, these same patterns apply for a broad range of models and estimation procedures.
D
k
kt
-1
[~]-= [k-[ ~i t2]
[~,itkYiYi]
= ~,i(ti- -t)(yi- ~t) , ~,i(ti - i)2 where -t = ~ i ti/k. Back-transformation of the estimates ~ and/;, then produces the estimates/~0 = exp(~) and f = 1 - exp (b). It can be shown (Graybill, 1969) that the dispersion of the estimates ~ and b is given by ~;~(t~- i)2
9.8. IDENTIFYING SYSTEM PROPERTIES IN THE ABSENCE OF A SPECIFIED MODEL
-
with the diagonal elements representing variances of ~ and/~, respectively, the off-diagonal element representing their covariance, and oa representing the variance associated with the transformed observations Yt = log[/~(t)1. Three points are particularly germane here: 9 The variances of the parameter estimates decrease with increasing^amounts of data. This is easily seen for the estimate b, because the variance of b is inversely related to ~ i ( t i - i) 2 in the dispersion matrix. This sum of squares increases with the number of observations, so that the variance decreases. Similarly, the variance of ~ decreases with increasing amounts of data. 9 The variances of the estimates decrease with the spread of the observations over the time frame. The influence of spread again is seen in the t e r m ~ i ( t i i)2 which increases with increasing dispersion of the data over the time frame.
The preceding methods have dealt with estimation of parameters from time series data, conditional on an underlying model structure. During the past two decades, efforts have been directed at the problem of drawing inferences from time series data about system dynamics in the absence of an assumed model structure (e.g., see Schaffer, 1985; Sugihara and May, 1990; Abarbanel, 1996; Ellner and Turchin, 1996; Kantz and Schreiber, 1999). The absence of a priori knowledge about the underlying system model extends to uncertainty as to the appropriate number of state variables that are needed to describe a system of interest. Remarkably, Takens (1981) has shown that it is possible to draw certain inferences about a multidimensional system based on analysis of time series data for a single state variable of the system. Ecological examples might involve a local food web with predator and prey populations, or a system of competing species, or a system of interacting populations of the same species. In fact, ecologists often are interested in making inferences about a system of interacting species or populations, based on an analysis of a time series of abundance estimates for a single member population of the system (e.g., see Schaffer, 1985; Sugihara and May, 1990; Turchin and Taylor, 1992; Pascual and Levin, 1999). One approach to identification of system properties is based on the concept of system attractors. Strogatz (1994) defined an attractor informally as a closed set of points A in the state space of a system that possess the following properties:
9.8. Identifying System Properties in the Absence of a Specified Model 1. A is invariant, in that any trajectory beginning on A will remain on A from that time on. 2. A attracts an open set of initial conditions (termed the attracting set) such that if a trajectory begins in this open set (i.e., if the initial values of the system state variables are located within the attracting set), then the trajectory will tend toward A with time (as t ~ oo). 3. A is minimal in that there is no subset of A that satisfies properties 1 and 2. Simply put, A "attracts" a system in the sense that a trajectory starting on or near the attractor (within the attracting set) will converge to the attractor [see Milnor (1985) for more detailed definition]. The attractor A of a system may thus be thought of as a phase diagram of the asymptotic trajectory of a system, with the attracting set composed of A along with the set of system initial conditions for which system dynamics converge to A. Ecological examples of an attractor include limit cycles of the Lotka-Volterra predator-prey model and equilibria of the Lotka-Volterra competition model. It is not difficult to find simple models with similar structural features but fundamentally different attractors. Indeed, Caswell (2001) showed that a simple bivariate projection model with density-dependent elements can produce attractors with widely differing geometries, simply by changing the magnitude of one parameter in the projection matrix. Takens (1981; also see Packard et al., 1980) demonstrated that it is possible to identify the geometry of an attractor with data for a single system state variable. That is, we can use data from a time series trajectory of one state variable to produce a "reconstructed attractor" that is topologically equivalent to the true attractor. With this univariate assessment it then is possible to identify certain properties of the complete system that are useful in system analysis and prediction. The numerical methods used in attractor reconstruction from a single time series are fairly involved and will not be described here, but the interested reader is referred to Abarbanel et al. (1993), Abarbanel (1996), Ellner and Turchin (1996), Kantz and Schreiber (1999), and Nichols and Virgin (2001). Applications of these and related methods to biological problems include Schaffer (1985), Sugihara and May (1990), Turchin and Taylor (1992), Ellner et al. (1998), Pascual and Levin (1999), and Nichols and Nichols (2001). A number of measures can be obtained via attractor reconstruction that convey information about the nature of a dynamical system. One such measure is system dimension, which can be viewed in various ways but basically is a metric reflecting the geometry of the attractor. A dimensional metric computed from time
183
series data provides information about the number of state variables or system components that are active determinants of system dynamics and thus are needed to describe system dynamics adequately (also see Schaffer, 1981). If the metric for a natural system is relatively low (e.g., 2-3), then it may be possible to reconstruct attractors accurately based on only a few dimensions. On the other hand, if a system is of high dimension, then attractor reconstruction from observed time series is likely to be impossible (e.g., see Schaffer, 1985). Other system measures that are useful in analysis of system structure and dynamics are the Lyapunov exponents. A Lyapunov exponent ~kn quantifies the behavior of trajectories (e.g., stretching or contracting) with respect to the nth principal axis of the attractor as a system trajectory evolves through time. In simple terms the idea is to track a measure of the difference x 1 (t) - x 2 (t) of neighboring trajectories xl(t) and __x2 (t) as they evolve through time, with the Lyapunov exponents characterizing the rate of trajectory divergence (or convergence) in each dimension. Local Lyapunov exponents are computed using local neighborhoods of the time series data, whereas global Lyapunov exponents are computed as the average of local ~kn computed over the attractor. The signs of the global exponents (positive, negative, 0) provide information on both the shape of the attractor and the dynamics of the system (e.g., characterized as periodic or quasiperiodic, chaotic, or by the absence of posttransient dynamics). In systems subject to exogenous inputs (e.g., relevant environmental fluctuations), the distribution of local Lyapunov exponents characterizes short-term transient dynamics following exogenous perturbations (Ellner and Turchin, 1996; Ellner et al. 1998). When based on an appropriate choice of the dimension and delay parameters, a reconstructed attractor can be useful for prediction. That is, prediction algorithms utilizing reconstructed attractors can be used to project system changes into the future. These predictions can serve as forecasts of system behavior (e.g., Sugihara and May, 1990) and also can be used for other purposes such as identification of the appropriate spatial scale for the aggregation and study of ecological systems (Rand and Wilson, 1995; Keeling et al., 1997; Pascual and Levin, 1999). Methods for attractor reconstruction appear to work well in practice with physical and mechanical systems, for which the time series data are characterized by large numbers (e.g., tens to hundreds of thousands of points) of very precise measurements with little noise. But ecological time series typically include many fewer data and much more noise from sampling variation as well as environmental and other influences. Thus, the
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Chapter 9 Models and Time Series Data
ultimate utility of attractor reconstruction for the investigation of ecological problems is not known at this time. We believe that the investigation of system attractors will be an active area of research over the next decade.
9.9. D I S C U S S I O N There is a strong association between the statistical estimation procedures described in Chapter 4 and model identification as described in this chapter. For example, maximum likelihood estimation relies on maximization of a "likelihood function" as the basis for parameter estimation. If the underlying distribution is normal, this reduces to the minimization of a quadratic form in the distribution parameters, in analogy to the process of model fitting via minimization of a mean squared error function. In fact, the problem of statistical estimation can be seen as an application of (statistical) model fitting. Indeed, the objective is to choose estimates of distribution parameters that best "fit" an assumed statistical distribution, and in the case of maximum likelihood estimation, this means choosing parameter values for which the likelihood function is maximized. The selection of a statistical model from, say, two alternatives is facilitated by choosing the "best" parameter estimates from the two corresponding distributions, determining the "goodness" of fit of the models based on these estimates, and selecting the model with the better fit (see Chapter 4, especially Section 4.2 on parameter estimation and Section 4.4 on model selection). Indeed, the preceding discussion on the effects of observation data on estimator stability is indicative of the strong association between statistical estimation and model identification. Their similarities notwithstanding, we note that in general, dynamic model identification and statistical estimation are not identical. Recall that the process of identification was developed in terms of an error function and a weighting scheme for its components. These attributes are in some sense arbitrary, in that the analyst has very wide flexibility in the choice of both. This flexibility distinguishes model identification from statistical estimation, which is tied to the form of an assumed underlying statistical distribution of the data. This distribution influences the choice of both the metric by which goodness of fit is measured and the weighting scheme of the metric. Recall that the objective of model identification is to represent (time series) data as well as possible with a dynamic model, by appropriate choice of parameter values. The notion of stochastic variation, and the need to account formally
for random variation, is not necessarily a part of the process. On the other hand, the stochastic nature of statistical data, and the need to account for, measure, and model stochastic effects, are at the heart of statistical modeling, i.e., the modeling of components of random variation in a system. Model identification also shares many attributes with dynamic optimization, as described in Chapter 21. Both involve the optimization of an objective function over a range of values for some decision variable. Both incorporate the transition equations of a dynamic system as constraints on the optimization. Both involve (or can involve) initial conditions and possible boundary conditions on the optimal solution. However, there are substantive differences between dynamic optimization and model identification, involving the nature of the objective function, the character of the decision variables, and differences in the models that are used. Whereas model identification seeks with temporally referenced data to identify parameter values in a dynamic model, dynamic optimization seeks to identify a trajectory of controls to optimize an objective function in the control and system state variables (see Chapter 21). The identification process involves an iterative refinement and revision of structural and parametric model features, whereas dynamic optimization typically involves the use of a developed model (or set of models) to guide management a n d / or research. Indeed, one result of model identification is to produce models that can be used for dynamic optimization. It should be noted that it is not uncommon for the effort to identify a model to fail, i.e., for one to fail to construct a model that is adequate for its intended purposes. Several potential reasons for this failure can be recognized (Ljung, 1999), which tie directly to the key elements of identification that were articulated above. For example, the suite of models under consideration may focus inadequately on system features of interest to the investigator or may fail to incorporate structural features (e.g., age or stage structure in a population) that are needed describe system behaviors of particular interest. Another common failure in model identification occurs when the information set is inadequate for identification. A case in point is a mismatch between the extent of the data and the range of biological conditions intended for the model. In this situation one might identify a model that fits the data but nevertheless fails to perform adequately over the biological range of interest. Yet another source of potential failure is a poor choice of the selection criterion by which to compare, contrast, and select the most appropriate model. As mentioned above, the choice of a model fitting criterion influences the weights given
9.9. Discussion to data points entering into the identification process and thereby influences the fitting of models to the data. For example, the least-squares criterion of Section 9.1 allows data at the extremes of the data set to influence heavily the fitting of a model, whereas an absolute difference criterion (Section 9.2) weights the data equally across the data range. Depending on the intended use of the model, the choice of a fitting criterion can potentially result in a model of marginal value. Finally, model identification can fail simply because the numerical procedure used to recognize optimal values of model parameters fails. Finding optima can be quite difficult for complicated models with nonlinear features, discontinuities, complicated constraints, and other features. For such models a search procedure may "home in" on a suboptimal parameterization for
185
the model or may simply fail to recognize any optimum whatsoever (see Appendix H for further discussion). We note in closing that one usually is less than certain about the mathematical structures describing biological process, yet it nonetheless is necessary to make decisions in the face of this uncertainty. One approach is to seek optimal decisions that recognize management objectives, while also accounting explicitly for structural uncertainty in the decision-making process. Such an approach essentially integrates system identification and system control into a single optimization problem, with decision-making pursuant to the dual goals of management and improved system understanding. In Chapter 24 we describe the combination of system identification and optimization under the rubric of adaptive resource management (Waiters, 1986).
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C H A P T E R
10 Stochastic Processes in Population Models
10.1. BERNOULLI COUNTING PROCESSES 10.1.1. Number of Bernoulli Successes 10.1.2. Bernoulli Waiting Times 10.2. POISSON COUNTING PROCESSES 10.2.1. Extensions of the Poisson Process 10.2.2. Poisson Interarrival Times 10.3. DISCRETE MARKOV PROCESSES 10.3.1. Markov Chains 10.3.2. Classification of States in a Markov Chain 10.3.3. Stationary Distributions in Markov Chains 10.4. CONTINUOUS MARKOV PROCESSES 10.4.1. Birth and Death Processes 10.4.2. The Kolmogorov Differential Equations 10.5. SEMI-MARKOV PROCESSES 10.5.1. Stationary Limiting Distributions 10.6. MARKOV DECISION PROCESSES 10.6.1. Discrete-Time Markov Decision Processes 10.6.2. Objective Functionals 10.6.3. Stationary Policies 10.6.4. Semi-Markov Decision Processes 10.7. BROWNIAN MOTION 10.7.1. Extensions of Brownian Motion 10.8. OTHER STOCHASTIC PROCESSES 10.8.1. Branching Processes 10.8.2. Renewal Processes 10.8.3. Martingales 10.8.4. Stationary Time Series 10.9. DISCUSSION
tion according to a sampling or experimental design, could be used to estimate population parameters based on the rules of statistical inference as described in Chapter 4. Stochastic factors arising in the investigation as a result of r a n d o m sampling were included in the corresponding statistical models and accounted for via statistical treatments outlined in these chapters. With some exceptions (e.g., see Sections 6.5-6.6), the patterns of r a n d o m n e s s were a s s u m e d to be absent any covariation across time. An extension to this f r a m e w o r k that is particularly useful in population modeling includes sequences of r a n d o m variables that are temporally indexed. Probability structures for sequences of r a n d o m variables that are temporally indexed constitute the subject matter of stochastic processes. In simplest terms the joint distribution of a set {X(t): t ~ T} of r a n d o m variables over a time frame T describes a stochastic process over T. One elementary example of a stochastic process is the sequence of statistically independent r a n d o m variables produced by r a n d o m sampling of a population over time. The probability structure of a stochastic process typically is defined in terms of the distribution of X(t) at each point in time, as well as the statistical associations of these r a n d o m variables across time. If the potential values for X(t) are countable [e.g., if X(t) takes only integer values] then the process is said to be a discretestate process; otherwise, it is a continuous-state process. Stochastic processes also can be characterized as discrete time or continuous time, d e p e n d i n g on the discrete or continuous nature of the time frame. A particular sequence of observed values of the r a n d o m variables of a stochastic process constitutes a realiza-
In our development of statistical procedures in Chapters 4-6, we focused primarily on the treatment of r a n d o m variables that lack an identifiable reference to time. The idea there was that r a n d o m samples of observations, collected over the course of an investiga-
187
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Chapter 10 Stochastic Processes
tion of the process. A realization is essentially a time trace of the process, a particular manifestation from the collection of all possible time traces defined by it. Example
Consider a sequence of counts {N(t): t e T}, for which N ( t ) is a random variable of nonnegative integers at each time t in a time frame T. If T consists of discrete points in time, the process is a discretestate and discrete-time process; otherwise it is a discrete-state and continuous-time process. Because populations frequently are characterized by counts at discrete points in time, such counting processes often are used to model populations with stochastic components. Example
Consider a continuous-state, continuous-time process with bell-shaped distribution at each point in time, the variance of which increases proportionately with time. Under certain conditions involving temporal correlation in the process, this describes the well-known Brownian motion process (see Section 10.7). Because Brownian motion characterizes continuous change in systems, it sometimes is used in continuous-time population models. Example
Figure 10.1 displays realizations for a discrete-time process for which X(t) --- N ( ~ t , or) and corr[(X(tl), X(t2)] = 0. A process with this statistical structure is known
FIGURE 10.1 Discrete-timewhite noise process with X(t) --- N ( ~ t 1).
,
as Gaussian white noise. For white noise processes the transitions between times are independent of process history; i.e., the state of the system at time t is independent of all previous states. In essence, the future state of the process is not influenced by past or present states, and except for temporal variation in its means, process behaviors in the future look statistically like process behaviors at each time in the past. Because all random variables in a white noise process are statistically independent, this class of processes possesses the simplest possible stochastic structure. Example
Except for white noise processes, the simplest probability structure for a stochastic process is one in which the process state at time t is influenced only by its state immediately prior to t. Figure 10.2 exhibits realizations for a discrete-time process such that the process state at each time is the previous state plus a random component: X ( t + 1) = X(t) + Y(t), where Y(t) --- N(0,1). Processes with the property that future process behavior is influenced only by the present state of the system are known as M a r k o v processes. In the following discussions we describe these and other stochastic processes that arise in the modeling of biological populations. Because they are especially prevalent in the modeling of populations, we focus on Bernoulli and Poisson counting processes, along with some probability distributions that are derived from them. We then describe Markov processes and Markov decision processes, a large class of stochastic processes that play an important role in later chapters on decision-making. Then we deal with Brownian motion for
FIGURE 10.2 Realizationof a discrete-time Markov process, with transfer equation X(t + 1) = X(t) + Y(t) and Y(t) a white noise process with Y(t) ~-- N(0, 1).
10.1. Bernoulli Counting Processes continuous time and finish with brief descriptions of a few other processes that can arise in population biology. In what follows, we use the index t to designate the time at which an event occurs and k to designate the temporal order of events. For discrete-time processes like the Bernoulli, the sequential order of an event and the timing of its occurrence can coincide, d e p e n d i n g on the nature of the event. For most stochastic processes there is no such coincidence, and it therefore is convenient to include indices for both temporal order and time.
10.1. B E R N O U L L I COUNTING PROCESSES We focus here on the class of white noise stochastic processes k n o w n as Bernoulli processes, which are defined by i n d e p e n d e n t binary r a n d o m variables at each point in a discrete time frame. As described in Chapter 4, a binary r a n d o m variable X(t) can be assigned a value of X(t) = 1 if the outcome of the process at time t is a "success" (however defined), and X(t) = 0 if the outcome is a "failure." Success or failure occurs at each point in time with probabilities p and 1 - p, respectively. A formal definition for Bernoulli processes is as follows: The stochastic process {X(t): t = 1, 2, ...} is Bernoulli with probability p if 1. The r a n d o m variables X(1), X(2), ... are independent.
2. P[(X(t) = 1] = p and P[(X(t) = 0] = 1 - p for all t. Realizations of a Bernoulli process consist of sequences of unit-height rectangles, corresponding to the sequence of failures and successes. Figure 10.3 displays a realization with p = 0.5, consisting of a series of
FIGURE
10.3
R e a l i z a t i o n of a Bernoulli process w i t h p = 0.5.
189
unit increases and decreases d e p e n d i n g on outcomes at each point in time.
Example Consider a hunter check station at which the success of waterfowl hunters is determined during the waterfowl hunting season. As hunters come through the station, each harvested bird is checked for species, sex, and age. Let k designate the temporal order of birds that are checked; thus k = 1 corresponds to the first bird checked, k = 2 corresponds to the second bird checked, and so on. Let X(k) = 1 if the kth bird is a mallard and X(k) -- 0 if it is not a mallard. Provided the probabilities for harvesting and reporting mallards and nonmallards are invariant over the time frame (i.e., the probability is p that the kth bird is a mallard for all k), this situation defines a Bernoulli process.
Example Assume that a cartographically correct m a p of the state of Vermont is overlaid with a fine-grained grid system. Assume that grid plots are chosen sequentially by some r a n d o m process and each plot is field checked for forest vegetation. Let k represent the kth plot that is chosen in sequence and assign X(k) = 1 if the kth plot consists of greater than 50% forest cover. Provided the grid sampling is r a n d o m with replacement, this defines a Bernoulli process, for which the Bernoulli parameter p represents the proportion of Vermont that is forested.
10.1.1. N u m b e r of Bernoulli Successes Other processes can be derived from the Bernoulli-for example, the n u m b e r of successes N(t) = X(1) + .-. + X(t) by time t. This is again a discrete-state, discretetime process, but now the state space varies with t. Thus two values are possible at t = 1 [N(1) = 0 or 1], three values are possible at t = 2 IN(2) = 0, 1, or 2], and so on. Realizations of the process consist of unitlength step increases at those times for which X(t) = 1 (Fig. 10.4). It can be s h o w n that N(t) is binomially distributed with parameters p and t:
Because t is a parameter in this density function, the distribution of N(t) varies over time. The r a n d o m variables {N(t)lt = 1, ...} also are correlated over time: because N(t 1) and N(t 2) share r a n d o m elements in their sums, corr[N(tl), N(t2)] ~ 0.
190
Chapter 10 Stochastic Processes hunter successes are independent, then nonoverlapping periods of the record represent sequences of indep e n d e n t Bernoulli r a n d o m variables, and accumulated successes for these periods represent i n d e p e n d e n t binomial r a n d o m variables parameterized by the record lengths and the success rate. This allows one to test via m a x i m u m likelihood procedures the assumption of stationary hunter success over time.
10.1.2. Bernoulli Waiting Times
FIGURE 10.4 Realizationof a process consisting of the number of Bernoulli successes over time.
Example Consider a hunter check station at which the success of deer hunters is determined as they exit a hunting area. A s s u m i n g that hunters are independent and identical in their probability p of hunting success, a sequence of k hunters coming through the check station corresponds to k Bernoulli r a n d o m variables, the sum of which is binomial with parameters k and p. Sequential sampling of hunters provides an opportunity to test the assumption that hunters are identical in their success rates. For instance, the hypothesis that success rate depends on the age of the hunter is easily tested via m a x i m u m likelihood theory (see Chapter 4), based on the binomial distributions for samples of y o u n g and mature hunters coming through the station. It can be s h o w n that interval counts for Bernoulli processes are binomially distributed; i.e., the n u m b e r N(t 2) - N(t 1) of successes in the interval t 2 - t I is binomially distributed with parameters p and t 2 - t 1. Because N(t 2) - N(t 1) includes only the r a n d o m variables X(t) for times between t I and t 2, this count is i n d e p e n d e n t of interval counts for periods prior to t I and after t 2. This property, k n o w n as independent increments, holds for any process for which the elements X(t) are independent. Interval counts also are stationary, in that the distribution of N(t 2) - N(t 1) is independent of the starting time t I of the interval. Again, this property holds for any process with independent, identically distributed elements X(t).
Example Again using the check station, assume that records are kept of hunter success over a period of several weeks during the hunting season. On assumption that
We have a s s u m e d thus far that the time t is fixed and the n u m b e r of successes is random. It is useful to consider a role reversal for these indices, whereby the n u m b e r of successes is assumed given and the time required to achieve that n u m b e r is random. It seems intuitive that if the n u m b e r of successes over a given a m o u n t of time is random, then the time required for a given count also is random. For example, consider the time Z(1) required to record the first success in a Bernoulli process. For the first success to occur at time t, no successes can have occurred by time t - 1, and a success must occur at time t; i.e., the event {Z(1) = t} is equivalent to the joint event N(t 1) = 0 and X(t) = 1, with probability of occurrence -
P{Z(1) = t} = P{EN(t - 1) = O] n [X(t) = 1]}. Because N ( t - 1) = X(1) + ... + X ( t - 1 ) a n d X(t)are independent, their joint probability is given by P{Z(1) = t} = (1 - p)t-lp, which describes a geometric distribution (see Appendix E). Thus, the waiting time for a single occurrence of a Bernoulli process is a geometrically distributed r a n d o m variable, the value of which can be any positive integer. The distribution is parameterized by the probability p of success in any trial, with the average waiting time given by 1/p. This supports one's intuition that the time required for a success ought to increase as the probability of success declines (and vice versa).
Example A team of biologists is investigating the feeding behaviors of an endangered species. Observation stations have been set up in several k n o w n feeding areas; however, individual animals are only rarely observed there and only during the hours just before and just after dawn. In planning the team's field season, it is important to have some idea of the a m o u n t of time required at each observation station in order to observe feeding behaviors there. From previous studies the probability of sighting individuals on a given day is
10.1. Bernoulli Counting Processes about p = 0.1. Based on the geometric distribution for waiting times, the expected time for an observation at any particular station is therefore 1 / p = 10 days. A study design that requires observations at each station thus should anticipate at least 10 mornings of observations per station. By extension, n o w consider a r a n d o m variable Z ( k ) characterizing the time of the kth success in a Bernoulli process. For the kth success to occur at time t, k - 1 successes m u s t have occurred by time t - 1, and a success must occur at time t. Thus, the event {Z(k) = t} is equivalent to the joint event { N ( t - 1) = k - 1} and {X(t) = 1}, with probability of occurrence P { Z ( k ) = t} = P { [ N ( t -
1) = k - 1] A IX(t) = 1]}.
the a m o u n t of time beyond k that is required for k successes. In that case Eq. (10.1) can be written as P [ Z ( k ) = k + x] = ( k + -x - 1
-
1) = k -
= [(;-1) pk-l(ll
1 ] P [ X ( t ) = 1] _p)t-k]p
(10.1)
(;1) -
1 pk(1 - p ) t - k
which describes the negative binomial distribution (see A p p e n d i x E). Note that k in the derived process {Z(k): k = 1, ...} is the n u m b e r of s u c c e s s e s , rather than an index of time. Indeed, k is n o w a distribution parameter, and the time index t actually has become a value that the r a n d o m variable Z(k) can take. The average waiting time for k successes is given by k / p , which indicates that the waiting time increases with the number of required successes and declines with greater success rate. Example
A study of small m a m m a l s involves the capture of individuals with drop traps. Traps are visited twice a day, and individuals found in the traps are tagged and released. From a pilot study the probability of a trap being occupied on a given visit is p = 0.25. A s s u m i n g i n d e p e n d e n t trapping events, the n u m b e r of visits needed to record a specific n u m b e r of captures is given by the negative binomial distribution. A design that targets four captures per trap therefore should anticipate k/p = 16 visits to each trap and so should call for a study period of at least 8 days. Note that at least k units of time are necessary to achieve k successes; thus, the negative binomial distribution is defined for values of t such that t >- k. Sometimes the index t is written as t = k + x to emphasize
pk(1 _ p ) k + x - k
( ,xl t x
pk(1
p)X
a standard form of the negative binomial distribution (see A p p e n d i x E). It can be shown that the process describing time to success is Markovian, in that the distribution of Z(k) depends only on Z(k - 1) but not on the r a n d o m variables prior to k - 1" P i g ( k ) = tlZ(1) = t I . . . . , Z ( k -
By virtue of the independence of N ( t - 1) = X(1) + 9.. + X ( t - 1) and X(t), their joint probability is given by P [ Z ( k ) = tl = P [ N ( t -
191
= P[Z(k)=
tIZ(k-
1) = tk_l]
1 ) = tk_l].
Thus, in order to predict the time of the kth occurrence in the process, one need not keep track of the history of previous occurrences. Once one accounts for the most recent occurrence, all previous occurrences are of no value in predicting the timing of the next occurrence. This property simplifies enormously the task of modeling the time to success. With the Markovian property, one can show that the waiting time Z ( k ) - Z ( k - 1) between successive occurrences is i n d e p e n d e n t of previous waiting times and is geometrically distributed: P[Z(k)-
1) = tIZ(1) = tl, ..., Z ( k -
Z(k-
= P[Z(k)-
= p(1
-
Z(k-
1) = tk_l]
1 ) = t]
p)t-1.
It follows that the waiting times Z(1), Z(2) - Z(1), ..., Z ( k - 1) are all i n d e p e n d e n t and identically distributed r a n d o m variables. Of course, their sum
Z(k) -
k-1
Z(k) = Z ( 1 ) + ~ [ Z ( j + j=l
1)-Z(j)]
is simply the time required for k successes, which from Eq. (10.1) is distributed as a negative binomial: k-1
P{Z(1) + ~ [ Z ( j + j=l
=
(;1 t
1)-Z(j)]
1 pk(1 -- p) t
=t}
,
The independence of the increments Z(k) - Z(k - 1) also guarantees that the times m
Z(k + m) - Z ( k ) = ~ [ Z ( k + j) - Z ( k + j - l ) ] j=l
192
Chapter 10 Stochastic Processes
between multiple successes represent independent increments, and these increments also are distributed as negative binomial, with parameters p and m.
process {N(t): t -> 0} is defined under the following four conditions:
Example
2. The process has stationary and independent increments, i.e., P[N(t + At) - N(t)] is the same for all t and N(t 1 + At 1) - N ( t 1) and N(t 2 + At 2) - N ( t 2) are statistically independent for t2 ~ t I + At 1. 3. For an arbitrary time t, the probability of exactly one occurrence in a "small" interval [t, t + &t] is approximately )~At:
As part of an investigation of waterfowl movements during migration, bait traps are located in a wetland complex and checked daily for waterfowl. Trapped birds are weighed, banded, and released each morning during the course of the study. Trapping is conducted over a limited period during the peak of migration, so that the number of birds exposed to traps is not expected to vary systematically during the study. Let X(t) be a random variable representing daily trapping success: X(t) = 1 if the trap is occupied on day t, and X(t) = 0 if the trap remains unoccupied. On assumption that trapping effort remains constant over the course of the study and trapping success is not influenced by previous trapping success (i.e., there is no behavioral response to being trapped; see Chapter 14), the sequence {X(t): t = 1, 2, ...} constitutes a Bernoulli process, with parameter p representing the probability that a trap is occupied on any given day of the study. The number N(t 2) - N(t 1) of occupied traps during an interval [tl, t2] has the binomial distribution B(t 2 - tl, p), with expected value (t 2 - tl) p. Thus, the average number of captured birds can be increased either by increasing the duration of the study or by increasing the probability of capture (or by increasing both factors). Furthermore, changes in the average number of captures are directly proportional to changes in either the duration of the study or the capture probability. The waiting time between successive captures in the study has a geometric distribution, with expected value 1/p. The waiting time for, say, five captures is distributed as a negative binomial, with expected value 5/p. Thus, the average length of time required for a given number of traps to be occupied can be decreased by increasing the probability of capture, with the required time decreasing from oo to 5 as the probability of capture increases from 0 to 1.
1. N ( 0 ) = 0.
P{~N(t + a t ) - N(t)] = 1} = )~At + o(at),
where o(&t) is some value with a limiting magnitude that is of degree less than &t: lim at--*0
o(At) - 0. At
4. The probability of more than one occurrence in It, t + &t] is negligible when compared to the probability of a single event: P{[N(t + a t ) -
N(t)] > 1} = o(At).
If these four conditions are satisfied, then {N(t): t ~ 0) is a Poisson stochastic process. As illustrated in Fig. 10.5, realizations of a Poisson process exhibit unit increases at random points in time. The rate at which these increases occur is influenced by the parameter )~, as discussed below. Example
An experiment involves the maintenance of minnows in individual aquaria over an extended period of time. When a minnow dies, its aquarium is replaced
10.2. P O I S S O N COUNTING PROCESSES Poisson processes are discrete-state, continuous-time processes that often are applicable to counts over continuous time frames. The idea behind a Poisson process is that events occur at random times over a continuous time frame, subject to certain stationarity and independence conditions. The process records the total number of occurrences throughout the time frame, which typically is assumed to start at t = 0. Formally, a Poisson
FIGURE 10.5 Realizationof a Poisson process with ~ = 0.25.
10.2. Poisson Counting Processes with another containing a live m i n n o w of the same species, age, and genetic stock. Let N ( t ) represent the n u m b e r of m i n n o w s that have died by time t. If the flow-through water system is maintained properly, the death rate of m i n n o w s should remain constant through time and the sequence {N(t): t -> 0} can be modeled by a Poisson process. Clearly, the total number N ( t ) of m i n n o w deaths at any point in time will be greater or less d e p e n d i n g on the experiment-wide rate of mortality. Example
Reproduction for a panmictic endangered species occurs as a result of the r a n d o m encounter of males and females, which is indexed by the parameter ~. The likelihood of one such encounter in a unit of time is directly proportional to K, and the likelihood of k encounters declines as a p o w e r function of K. Because the probability of more than one reproduction event in a unit of time is negligible w h e n compared to that for a single event, one can model total reproduction over time as a Poisson process. Again, reproduction will be greater or smaller d e p e n d i n g on the parameter K. A probability structure for Poisson processes can be obtained through a decomposition of the event { N ( t + &t) = n} into {N(t + At) = n} = {IN(t) = n]
U{[N(t) = n - 1] and
and
[N(t + At) - N(t) = 0]}
[N(t + at) - N ( t ) = 1]}.
(10.2)
Equation (10.2) asserts that there are essentially two ways in which N ( t + At) can assume a value of n: no events are a d d e d to a count of n over At [the lead term of the union in Eq. (10.2)]; or one event is a d d e d to a count of n - 1 over At [the second term of the union in Eq. (10.2)]. On assumption that increments are stationary and independent, the probabilities for these events can be added, to produce the probability
193
where Pn(t) is the derivative of Pn(t) with respect to t. The solution of this differential equation can be s h o w n to be P I N ( t ) = n] = e - ~ t ( K t ) n / n !
(see Appendix C). Thus, the probability structure for the Poisson process {N(t): t ~ 0} is given by a Poisson distribution for N(t), with Poisson parameter Kt. Because this distribution is parameterized by t, it obviously varies as t takes different values. Note that the Poisson parameter is directly proportional to t, so that both the mean and variance of N ( t ) are proportional to the time since initiation of the process (see Appendix E). This is illustrated in Fig. 10.6. Example
In the a q u a r i u m study described above, the total n u m b e r N ( t ) of m i n n o w deaths by time t has a Poisson distribution with Poisson parameter Kt. Because both the mean and variance of a Poisson distribution are given by the Poisson parameter, the average n u m b e r of deaths and the spread in these n u m b e r s increase linearly as the study progresses. For example, if K = 0.1, the average n u m b e r of deaths after the first d a y is 0.1; after the second day it is 0.2; and so on. If an average of Kk mortalities is recorded in the study over k days, we can expect twice that n u m b e r to be recorded in a replicated study lasting twice as long. It should be noted that the Poisson distribution provides an alternative but equivalent definition for Poisson processes. Thus, a counting process {N(t): t -> 0} is Poisson with rate parameter ~ if the following conditions are met:
P[N(t + at) = n] = P[N(t + at) - N(t) = 1]P[(N(t) = n - 1] + P[N(t + at) - N(t) = O]P[N(t) = n].
Using assumptions (3) and (4) listed above for the Poisson process, this equation can be expressed as Pn(t + At) = Pn_l(t)[KAt + o(At)] + Pn(t)[1 - KAt - o(At)]
with Pn(t) = P [ N ( t ) = n], or Pn(t + a t ) - Pn(t) = [Pn_l(t) -- Pn(t)][KAt + o(&t)].
Dividing the equation by At and allowing &t --~ 0, we get the differential equation Pn(t) = [ - P n ( t ) + Pn_l(t)]K,
FIGURE 10.6 Probabilitydistribution for a Poisson process at a particular time t. The rate parameter Kt is a linear function of the process time t, and therefore the distribution evolves over the time frame.
194
Chapter 10 Stochastic Processes
1. N(O) = O. 2. The process has independent increments. 3. The number of events occurring in any interval of length s is Poisson distributed with parameter Ks: P[N(t + s) - N(t) = n] = e
-as ( h s ) n
n!
,
n = 1,....
It is instructive to compare these two definitions for the Poisson process. Both assume a starting value of zero for the process, and both assume independent process increments. One definition leads to the Poisson distribution by considering incremental (single-step) changes in process magnitudes, whereas the other starts with an assumed Poisson distribution for interval counts. However, both lead to the same stochastic framework, with Poisson distributed counts over specified intervals of time. On assumption that {N(t): t -> 0} is a Poisson process, the probability that no event occurs over [0, t] is given by P{N(t) = 0} = e -at. It follows that the probability of at least one occurrence over the interval is P{N(t) > 0} = 1 - e -at. The assumption of stationary increments ensures that this probability holds for any interval of length s in the time frame, irrespective of its starting points: P{N(t + s ) - N ( t )
>0}=
1 -e-aS.
(10.3)
10.2.1. E x t e n s i o n s of the P o i s s o n P r o c e s s
A number of stochastic processes can be derived from the Poisson process. Here we emphasize some of the more common processes that incorporate additional structural features into the Poisson stochastic framework.
10.2.1.1. Poisson Superposition It is possible to combine two Poisson processes into a single process with an identifiable probability structure. Suppose that P1 = {N(t): t -> 0} and P2 = {M(t): t >- 0} are two independent Poisson processes, with rates h I and h 2, respectively. It can be shown that the process P1 + P2 defined by P1 4- P2 = {N(t) + M(t):t >-0} is a Poisson process with parameter ~'1 4- )k2" The combined process P1 4- P2 is called the superposition of P1 and P2. Example
Consider a wildlife population that is subjected to hunting mortality over an extended period of time. Assume that the process P1 records the number of male
deaths that occur through time as a result of hunting, and P2 records the number of female deaths from hunting. If the number of deaths )k 1 and h 2 per unit time for these processes are stationary, then the total mortality for both sexes is tracked by the superposition P1 4P2 with parameter )k1 4- ~k2.
10.2.1.2. Compound Poisson Processes In addition to recording the time at which a Poisson event occurs, it often is useful to record some value associated with the event. For example, both the time of death and the weight at death might be recorded for each mortality event in the aquarium study described above. Such data form the basis of a c o m p o u n d Poisson process. More formally, let {N(t): t -> 0} be a Poisson process with parameter h and {Y(i): i = 1, ...} be a sequence of independent and identically distributed random variables. Assume that {N(t): t -> 0} and {Y(i): i = 1, ...} are statistically independent. Then the process {X(t): t -> 0} with N(t)
X(t) = ~, Y(i) i=1
is a compound Poisson process. In essence, X(t) accumulates values of Y(i) as the process progresses through time. Thus, X(t) assumes a value of 0 until the first Poisson event, at which time X(t) is updated by the value for Y(1). X(t) retains this value until the second Poisson event, at which time the value for Y(2) is added to that for Y(1). X(t) retains the value of this sum until the third Poisson event, at which time the value for Y(3) is added to the sum, and so on as time advances. Basically, the compound Poisson process {X(t): t -> 0} evolves like the Poisson process {N(t): t -> 0}, except that the unit steps of {N(t): t -> 0} are replaced by steps of size Y(t) (Fig. 10.7). It can be shown that the expected value of a compound Poisson process is the product E[X(t)] = (ht)ix
of means for the random variables N(t) and Y(t), and the variance is the product var[X(t)] = (ht)0-2 of their variances, where tx and 0 -2 a r e the expected value and variance of the random variables in {Y(i)" i = 1, ...}. In addition, the compound Poisson process inherits the property of independent increments from the underlying process {N(t)" t >- 0} and the independence of the random variables in {Y(i)" i = 1, ...}.
10.2. Poisson Counting Processes
F I G U R E 10.7 Realizations of a compound Poisson process. The unit step of the Poisson process is replaced by a step of size Y ( t ) when a process event occurs, with Y ( t ) --- N(0, 1) for this particular example.
195
occurrences at time t is distributed as a Poisson random variable with parameter Kt. Assume also that at the time of the nth occurrence there is a Bernoulli trial, with outcomes that are independent of the Poisson process. The stochastic structure thus includes both Bernoulli and Poisson processes, albeit with different temporal indices: the index for the Poisson process represents time, whereas the index for the Bernoulli process characterizes the temporal sequence of Bernoulli trials. N o w let X 1 be a new process that records the accumulated number of Bernoulli successes at each point in time and X 2 be a process that records the number of failures. Then X 1 and X 2 are compound Poisson processes with N(t)
Xl(t) = ~ Y(i) i=1
and N(t)
Example
X2(t) = ~ [1 - Y(t)],
Biologists retain records of the timing and weights of catch for each species of fish in a fishery. For planning purposes it is thought important to predict the size of the bi-catch of an infrequently caught species that exhibits considerable variation in individual sizes. Assuming that catch events are independent, the accumulated catch over time can be modeled by a Poisson process, with Poisson parameter )~t expressing the expected catch in an interval of length Kt. Based on historical records, one can estimate both the rate parameter and the mean ~ and variance 0 -2 o f the size of individual fish. If N ( t ) is the total catch over an interval of length t and Y(i) represents the weight of the ith fish at the time of its capture, then the accumulated weight N(t)
X(t) = ~, Y(i) i=1
of the catch over the interval is distributed as a compound Poisson distribution. Thus, the expected weight of the bi-catch is simply the average number )~t of individuals caught in the interval, times the average weight ~ of an individual fish. Likewise, the variance associated with the bi-catch weight is the variance for the number individuals caught, which for the Poisson distribution is also )~t, times the variance of the weights of individual fish. This information can be used by biologists to adjust the fishing season length appropriately to target an amount of bi-catch to maintain stocks while allowing for fishing opportunities. A useful example of a compound Poisson process results from the combination of Poisson and Bernoulli processes. Assume that events occur according to a Poisson process {N(t): t >- 0}, so that the number of
i=1
respectively. Both can be shown to be Poisson processes, with parameters Kp and M1 - p), respectively. Furthermore, they can be shown to be independent of each other. This particular example of a compound Poisson process is sometimes referred to as Poisson decomposition (Cinlar, 1975).
10.2.1.3. Nonstationary Poisson Processes A key assumption thus far is that the Poisson parameter )~ is constant for all t -> 0. Seasonal, diurnal, and other variations in many biological processes can combine to make this an unrealistic assumption. A generalized Poisson process allows the Poisson parameter to vary with time: ~ = )t(t). The counting process {N(t): t -> 0} is said to be nonstationary (or nonhomogeneous) if all the Poisson assumptions except stationarity remain valid when )~ is replaced with Mt): 1. N ( t ) = O.
2. {N(t): t >- 0} has independent increments. 3. P [ N ( t + & t ) - N ( t ) = 1] = K(t)At + o(&t). 4. P [ N ( t + At) - N ( t ) >- 2] = o(&t). Under these conditions it can be shown that the increments {N(t + s) - N(s)} are Poisson distributed, based on the parameter t
m(t) = f
Ms) ds. o
That is, P{N(t + s) - N(s) = n} = e-lmr162
f o r n ->0.
+ s) - m(s)]n/n!
196
Chapter 10 Stochastic Processes
A useful application of nonstationary Poisson processes involves a stationary process for which the recording of an event is less than certain. Assume that an event at time t is detected with probability )~(t)/)~. Then P[one event is counted in (t, t + At)] = P[one event occurs in (t, t + At)] P[event is detected[event occurs] =
[(),at)
X(t)
o ( a t ) ] ~x
+
o'(at),
= Mt)At +
which demonstrates that a stationary Poisson process, with follow-up sampling to confirm event occurrences, can be described as a nonstationary Poisson process. Example
10.2.2. Poisson Interarrival Times Just as the times between successes can be investigated for Bernoulli processes, so can the times between occurrences for Poisson processes. As before, let the r a n d o m variable Z(k) represent the time of occurrence of the kth event in a Poisson process, with Z(0) = 0. The derived process {Z(k): k = 0, 1, ...} records the waiting times for occurrences of events in a Poisson process. Let Y(k) = Z(k) - Z(k - 1) in turn represent the a m o u n t of time between the k - 1st and kth occurrences. The derived process {Y(k): k = 1, ...} records the interarrival times between occurrences of the Poisson process. Note that the index k is an ordering index for arrival times and not an index of time. A probability structure can be ascribed to realizations of interarrival times by noting that {Y(1) -< t}, {Z(1) -< t}, and {N(t) > 0} all describe the same event, so that P[Z(1) -< t] = P [ N ( t ) > 0]
A s s u m e that in the absence of r a n d o m influences, a continuously growing population can be described by the exponential model d N / d t = r N ( t ) , with solution N ( t ) = No eFt. However, the effect of r a n d o m influences alters this pattern in such a w a y that population dynamics are described by a nonstationary Poisson process, for which Mt) Noert/r. Increments for this process are distributed as Poisson r a n d o m variables, with Poisson parameter =
t )~(s) ds
m ( t + s) - re(s) = f s
= No[er(t+s)-
e rs].
=l-e
-at
from Eq. (10.3). Thus, the waiting time for the first event of a Poisson process has an exponential distribution, with exponential parameter )~ inherited from the Poisson parameter )~t. Because Z(0) = 0 by assumption, this means that the first interarrival time Y(1) = Z(1) - Z(0) is exponentially distributed. By extension, the probability for the second interarrival time can be obtained by conditioning on the first, recognizing that the events {Y(2)- 0} are equivalent: P[Y(2) -< t Z(1) = s] = P[N(t + s) - N(s) > 0IN(s) = 1]
Because = P[N(t + s) - N(s) > 0] t
N ( t ) = ~] [N('r + 1) - N('r)],
= P[Y(2) -< t],
it follows that the average population size at time t is
which holds by virtue of the independence of the increments N ( t + s) - N ( s ) and N ( s ) - N(O). It follows that
"r----1
t
E [ N ( t ) ] = ~,~ E[N('r + 1 ) -
N('r)]
P[Y(2) -< t] = P [ N ( t + s) - N ( s ) > 0]
"r--1
=l-e
-at
t
__ No ~_jEer(r+ 1 ) _
err]
~=1
from Eq. (10.3). A similar a r g u m e n t can be used to show that in general
= N o e rt. P [ Y ( k ) < t] = 1 - e -at
Thus, a nonstationary Poisson process with Poisson parameter Mt) = Noert/r provides a model for population growth with r a n d o m l y timed events, while maintaining exponential growth in the m e a n population size.
i n d e p e n d e n t of previous interarrival times, and therefore the process {Y(k): k = 1, ...} of interarrival times consists of i n d e p e n d e n t exponentially distributed rand o m variables with exponential parameter )~; i.e., {Y(k):
10.3. Discrete Markov Processes k = 1, ...} is an exponential white noise process. Therefore the sum k
Z(k) = ~ [ Z ( j ) - Z ( j j=l k = ~, Y(j)
1)]
j=l
of interarrival times, i.e., the time Z(k) of occurrence of the kth event, is g a m m a distributed with parameters k and k (see Appendix E). Because Z(i) and Z(j) share some of the same interarrival times in their sums, they are not statistically independent r a n d o m variables. Thus the waiting time process is not a white noise process. Example Returning again to the m i n n o w experiment, assume that a stock of 100 m i n n o w s is obtained for the experiment. To reserve the use of laboratory facilities for an appropriate a m o u n t of time, it is useful to predict the time required to exhaust the stock of minnows. If minn o w deaths follow a Poisson process with Poisson parameter k, then the time until 100 deaths is distributed as F(100, k). The mean 100/k of this distribution therefore is the expected length of the experiment. Because the variance of the distribution is 100/h 2, a conservative schedule for laboratory time of 120/k would allow the experiment to continue two standard deviations beyond its expected length.
197
This definition asserts that, conditional on the value of the value of X(t), the sequence {X(s): 0~s
10.3.1. Markov Chains We restrict ourselves here to Markov processes defined over a time frame of equal time steps. Stationary countable Markov processes with equal time steps are k n o w n as Markov chains. Without loss of generality, we assume in what follows that At = 1, so that the process time steps consist of positive integers. The single-step transition probabilities then are denoted by pq = P[X(t + 1 ) = j [ X ( t ) = i],
10.3. DISCRETE M A R K O V PROCESSES
and these probabilities can be arrayed into a Markov matrix
Markov processes constitute a large and important class of stochastic processes that are defined by a lack of process memory. In what follows we restrict our attention to first-order Markov processes, in which the state at time t + 1 depends only on the state at time t (in Chapter 17 we mention second-order Markov processes in which the state at t + 1 depends on states at times t and t - 1). Thus, a Markov process is considered here to have the property that the future state of the process is influenced by its current state, but not by past states. That is, the future of the process is conditionally independent of its past, provided the present state of the process is known. A formal definition is given in terms of the distribution of X(t + At) conditional on the states {X(s)" 0 ~ s _< t} up to time t: the process {X(t)" t ~ 0} is said to be Markovian if P[X(t + &t)
= Xt+At}X(s)
-- Xs; s ~
= P[X(t + at) = xt+atlx(t) = xt].
t]
Pu P21
P12 P22
9
9
P13 P23
... ...
PlN P2N
p
9
PN1 PN2 PN3
~
".
PNN
of transition probabilities, assuming the n u m b e r of possible system states is finite. Thus, row i of the Markov matrix consists of the probabilities of single-step transitions from state i to state j (including the probability Pii of remaining in state i). The sum of probabilities across any row of the Markov matrix is one: s Pij 1 for every row i in the matrix. Example The simplest possible example of a Markov chain is the Bernoulli process. The system state for a Bernoulli
198
Chapter 10 Stochastic Processes
process is either one or zero depending on success or failure of the Bernoulli trial. If the probability of success is p, then the Markov matrix has the form (l-p)
P=
(l-p)
p
p'
with rows representing failure or success at arbitrary time t, and columns representing failure or success at time t + 1. Recall that the probabilities of success and failure for a Bernoulli process at time t + 1 do not depend on the outcome at time t: Pij = PEX(t + 1 ) = jlX(t) = i] = P [ X ( t + 1) = j]
=~, with the result that the same probabilities apply irrespective of the row index i. For that reason the matrix has identical entries for both rows, as appropriate for independent Bernoulli trials. A stationary Markov matrix allows us to express two-step transition probabilities for a Markov chain in terms of its single-step transition probabilities. Thus, the probability of moving from state i to state k in two time steps is simply the sum of the product of probabilities of moving from i to j and then from j to k, s u m m e d over all j: N
P [ X ( t + 2) = klX(t) = i] = ~, PijPjk. j=l
Note that the two-step transition probability from i to k is simply the (i, k) element of the square of the Markov matrix p2. By extension, the probability of moving from state i to state k in m time steps is given by P [ X ( t + m) = k[X(t) = i] N =
N
N
~" ~' "'" ~" PijlPjlj2""Pjm-2jm-lPjm-lk" jl = 1 j2 = 1 j m - 1= 1
If this probability is denoted by p/~, it follows that the probability of transition from i to k in r + s time steps can be expressed in terms of the transition from i to j in r steps, followed by the transition from j to k in s steps: N r+s Pik =
~1
j=
r s PijPjk.
from i to k in m steps is simply the (i, k) element of pro. Note also that the m-step transition probability can be decomposed in terms of the probabilities of transition from i to j in r steps, followed by transition from j to k in m - r = s steps: pm __ pr p m - r . m
Example
Two genetic strains of white mice are used in a controlled behavioral experiment involving a total of 10 mice. The mice are tested for their ability to learn, based on both positive and negative behavioral reinforcements. Each day, two mice are replaced with new mice that are selected at random from a laboratory population, two-thirds of which are of genetic strain S 1. To account for the level of behavioral reinforcement, it is important to keep track of the number of $1 mice in the experiment, and for this situation the "system state" essentially tracks that number on a daily basis. Because the number of states is finite (N = 11) and the time frame consists of equal time steps, system dynamics can be characterized by a Markov chain. Transition probabilities for the system are given in terms of the replacement of mice. Thus, the count of S 1 mice remains unchanged if as many S 1 mice are added as are removed each time; the count increases if more are added than are removed; and the count decreases if more are removed than are added. Probabilities for the counts can be assigned based on the number of $1 mice in the experiment each time. It seems reasonable to expect that, on average, representation of genetic strains in the experiment eventually should reflect their proportionate representation in the pool of available mice.
10.3.2. Classification of States in a Markov Chain The transitions of a Markov chain, and thus its dynamics and statistical properties, are determined by patterns in its transition probabilities. In large part, the long-term behavior of a Markov chain depends on structural linkages among these transition probabilties, which in turn can be described in terms of classes of states in the process. We describe here several structural properties that are important in analyzing Markov chains over time. We focus on patterns in the transitions between states, and groups of states, as a framework for the analysis of stochastic behaviors.
(10.4)
This is known as the Chapman-Kolmogorov equation (Ross, 1996). Note that the m-step probability p~ of transition
10.3.2.1 Communicating States A state j in a Markov chain is said to be accessible from state i if the probability of transfer from i to j is
10.3. Discrete Markov Processes
199
nonzero over some transfer period m: p~ > 0 for some value of m. Two states i and j in a Markov chain are said to communicate with each other if state i is accessible from j, and state j is accessible from i. Communication between states is established w h e n it is possible to get from either state to the other in a finite n u m b e r of transitions. The property allows the states in a Markov chain to be grouped into classes of communicating states. A Markov chain consisting of a single class with communication a m o n g all states is said to be irreducible. By definition, every state in an irreducible Markov chain is accessible to every other state, and any Markov chain for which all states can be reached from each other is irreducible.
is said to be aperiodic. It can be shown that communicating states all have the same period, so that an irreducible Markov chain, which contains only a single class of communicating states, inherits their periodicity. In particular, a Markov chain containing only aperiodic states is said to be aperiodic.
Example
for a process with three states. From the second row of the matrix, it is possible to remain in state 2 (because P22 --- 0.25), and therefore state 2 is aperiodic. From the first row, it is possible to transfer from state 1 to state 2 or 3, but not to remain in state 1. Having entered state 2, the probability P21 = 0.25 in the second row indicates that it is possible to transfer from state 2 back to state 1. This establishes that states 1 and 2 communicate and therefore have the same period. Thus, state 1 shares aperiodicity with state 2. Last, entries from the third row indicate that it is possible to transfer from state 3 to state I or 2, but not to remain in state 3. Because state 3 is directly accessible from state 2, states 2 and 3 communicate, and therefore state 3 is also aperiodic.
A s s u m e that a population can be classified into one of three classes d e p e n d i n g on population size, with states 1, 2, and 3 representing low, m e d i u m , and high population numbers. Population transitions are ass u m e d to be Markovian, with transition probabilities P=
[
0.5 0.2 0.5
0.5 0.3 0.2
]
0 0.5 . 0.3
From row 3, state 1 can be seen to be accessible from state 3, because direct transfers are possible: P31 ~ 0. In addition, state 3 is accessible from state 1, by means of transfer from state I to state 2, followed by transfer from state 2 to state 3:P12 P23 ~: 0. Because there is accessibility in both directions, states I and 3 communicate with each other: it is possible to get from either state to the other in a limited n u m b e r of transfers. In fact, all three states in this example communicate, and therefore the matrix is irreducible. On the other hand, if the transition probability P23 had been 0, i.e., P =
[05 0] 0.3 0.5
0.7 0.2
0 , 0.3
then it no longer w o u l d be possible to transfer from state 1 to state 3 either directly or indirectly. Under these conditions, state I w o u l d be accessible from state 3, but state 3 would not be accessible from state 1. Therefore the two states would not communicate, and the matrix would not be irreducible. A property that is associated with communication describes the pattern of returns to a state that previously was left. Thus, state i is periodic with period d if a return to i is possible in n steps only on condition that n is a multiple of d, i.e., pn = 0 unless d is a divisor of n. A state with period 1 (for example, with Pii > O)
Example Consider the Markov matrix
I0:5~ ~ Lo.5
0.5
o
10.3.2.2. Recurrent and Transient States
It also is useful to recognize recurrent and transient states in a Markov chain. The state i is said to be recurrent if the process is sure to return to i once having left it; otherwise, i is transient. A formal definition of recurrence is based on the probability fi~ that, starting in i, the first transition into j occurs in n steps. If fij represents the probability of ever making a transition from i to j, i.e., fij -- ~1 f~, then a recurrent state i is simply defined by fii = 1 . In words, a process leaving a recurrent state must eventually return to it, that is, the state must "recur" over time. On the other hand, transient states, though certain to be left, are not guaranteed of a return. Thus, the expected n u m b e r of transitions needed for a return to a transient state is infinite. In contrast, the expected n u m b e r of transitions Ei(n) -- ~1 Ylfi n needed for recurrence in state i can be either finite or infinite. If finite, the state is said to be positive recurrent; if infinite, it is said to be null recurrent. Obviously, the finiteness of Ei(n) (and therefore the pattern of recurrence) depends on the sequence {f~: n = 1, ...} of recurrence probabilities. oo
200
Chapter 10 Stochastic Processes
Example Consider transitions among three states in a Markov chain, such that transfers from states 1 and 3 to state 2 are not possible:
p
05 0 1 0.30.50.2.
i
0.4
00.6
Under these conditions, states 1 and 3 are recurrent, because it is possible to transfer from any state to either of them in finitely many transfers. However, state 2 is transient, because it is possible to transfer out of but not into state 2. It can be shown that if state i is recurrent and state j communicates with i, state j also must be recurrent. Thus, a recurrent state only communicates with other recurrent states. As seen below, this has implications as to the structure of Markov chains and their limiting behaviors.
in the absorbing class, the Markov transition matrix has the form
P
=
IP211
2
I
with 0 and P12 representing row and column vectors, respectively. The value P l l - - 1 indicates that state 1, having been entered, cannot be left, i.e., state 1 is absorbing.
Example An easy way to locate absorbing states is to look for rows in a Markov matrix with off-diagonal entries that vanish, i.e., rows with pq = 0 for i =/=j. Alternatively, one can look for rows with unit diagonal elements: Pii = 1. States meeting either of these criteria cannot be exited once they are entered and thus are absorbing. For instance, row 2 in the matrix I0050.210.3]0 P = [_0.30.20.5
10.3.2.3. Absorbing States Certain states in a Markov chain form a closed or absorbing class, such that no state outside the class can be reached from any state in the class. Put simply, an absorbing class, having been entered, cannot be exited. If an absorbing class consists of a single state, the state is said to be an absorbing state. Because an absorbing state can be entered but not left, the process is thereby "absorbed" into it. Clearly, state i is an absorbing state if and only if Pii = 1. Any state not in an absorbing class must be transient; otherwise it would always recur, violating the condition that it eventually must transfer into an absorbing state that cannot be left.
Example A Markov transition matrix can be expressed in partitioned form as
t e l l P12] P = LP21 P22_]' where P/j is a submatrix of transition probabilities among states represented in the rows and columns of the partition. Assume that Pll and P22 are square matrices, each representing transitions among the respective partition states. If P12 -- 0 but P21 =/= 0__rthen the states represented by Pll constitute an absorbing class, in that it is possible to transfer into the class (because P21 :/= 0), but once Pll has been entered, transfers out are not possible (because P12 = 0). On assumption that Pll is a single value, i.e., there is a single state
represents an absorbing state, because the off-diagonal elements of row 2 vanish.
Example An obvious biological example of an absorbing state is population extinction. Consider a population with the potential to become depauperate over time, through the influence of random environmental effects, species interactions, or management. The transition from one population size to another remains possible as long as the population is extant. On the other hand, when the population becomes extinct, no further population change is possible, i.e., N(t) = 0 is an absorbing state. By extension, a critical population level below which population declines are inevitable defines an absorbing class, in that the population, once having attained a size less than the critical level, can never grow beyond it. We note that Markov chains with absorbing states cannot be irreducible, because an absorbing state i cannot communicate with any other state (because Pij = 0 for i =/= j). By extension, Markov chains with more than one absorbing class cannot be irreducible, because pairs of states from different classes cannot communicate (otherwise, they would belong to the same class). Absorbing classes can be identified in terms of recurrent states, based on the fact that transfers from a recurrent state can only be to other recurrent states. This fact allows for a partitioning of a Markov chain into irreducible absorbing classes. With appropriate relabel-
10.3. Discrete Markov Processes ing of states, the Markov matrix for states thus grouped can be put in the form B
p
___
Pl
0
0
...
0
P2
0
...
0
0
P3
...
m
Q1 Q2 Q3
. . .
Qk
Each of the matrices Pi represents an irreducible absorbing class, all states within which are recurrent and accessible to each other. States corresponding to the rows of the Qi matrices are transient, in that they can transfer into one or more of the absorbing classes.
10.3.2.4. Ergodic States Recurrence, periodicity, and communication between states are incorporated in the notion of ergodicity, a key attribute of Markov chains that determines their behaviors over the long term. The term ergodic relates generally to probabilities of state recurrence and sometimes is used to describe sequential samples that are equally representative, i.e., statistically equivalent. A state i that is positive recurrent and aperiodic is said to be ergodic, and an irreducible Markov chain consisting exclusively of ergodic states is said to be an ergodic Markov chain. An ergodic Markov chain allows for the transition from any state i to any other state j, absent any periodicity in the transfer and return.
Example Consider a four-state Markov chain defined by
P =
0 1 0.25 0.25
1 0 0.25 0.25
0 0 0.25 0.25
0 0 0.25 " 0.25
The first row of this chain gives the probabilities of transfer from state I to state j; thus, transfer from state 1 to 2 is certain, and transfer to any other state (including state 1) is excluded. Similarly, rows 2, 3, and 4 correspond to the probabilities of transfer out of states 2, 3, and 4, respectively. Because it is possible to transfer from states 3 and 4 to state 1 or 2, the latter states are accessible from states 3 and 4. On the other hand, states 3 and 4 are not accessible from 1 and 2, because transfers from the latter to the former states are not possible. This defines two classes of communicating states: (1) a class consisting of states 3 and 4 (because each state is accessible from the other) and (2) a class consisting
201
of states 1 and 2. Because there is more than one class of communicating states, this process is not irreducible. States I and 2 are periodic of period d = 2, because two transitions are required before the process can return to the same state. States 3 and 4 are aperiodic, because the process can return to the same state at any time. States 1 and 2 also are recurrent, because it is possible to reach these states from any state in the process. On the other hand, states 3 and 4 are transient, because transfers into these states are not possible from states 1 and 2, no matter h o w m a n y transition periods are allowed. None of the states in this process is ergodic: states 1 and 2, though positive recurrent, are periodic and therefore nonergodic; states 3 and 4 are transient and thus nonergodic. The class consisting of states 1 and 2 is closed, because it is not possible to leave the class once it is entered. The class consisting of states 3 and 4 is transient, because it is possible to transfer out of the class but not into it.
Example If the Markov matrix in the previous example is changed to allow for the transfer to state 3 or 4 from either state 1 or 2, the process becomes ergodic. Consider a change in the transition matrix to [ ~ p
0.25 10.25 L.-
0.9 0 0.25 0.25
0.1 0 0.25 0.25
0 0 0.25 ' 0.25
which allows for transfers from state I to state 3. Under these conditions any state is accessible from any other state, and a single communicating class is defined. Therefore the new process is irreducible. Furthermore, all states are n o w aperiodic, because they communicate with the aperiodic state 3. The process states also are recurrent, because any state can be reached from any other state in finitely m a n y transfers, and indeed are positive recurrent, because the expected n u m b e r of transitions is finite. Thus the process states are all ergodic, and the process is an ergodic Markov chain.
10.3.3. Stationary Distributions in Markov Chains An ergodic Markov chain was defined above as an irreducible (single communicating class) aperiodic chain consisting of positive recurrent states (once left, a state is guaranteed of a return in finitely m a n y steps). For ergodic Markov chains, it can be shown that the probability p~ of transfer from i to j over n steps is asymptotically stationary: 'rrj -- lim p~j > 0. y/---) oo
(10.5)
202
Chapter 10 Stochastic Processes
Thus, successive steps in an ergodic Markov chain will eventually stabilize on a stationary state distribution, irrespective of the initial system state. Furthermore, this stationary distribution can be shown to be unique, a property that offers a simpler method for its identification than finding the limit in Eq. (10.5). To identify the limiting distribution of an ergodic Markov chain, consider a probability distribution defined by {pj'j = 1.... , N}, with pj the probability of initially being in state j and p' = (Pl .... , PN) the vector of these probabilities. For p to be stationary, it must be reproduced after each transition of the process. Thus,
They also have been used to model colony site dynamics (Erwin et al., 1998) and movement probabilities of individual animals (e.g., Hestbeck et al., 1991; Brownie et al., 1993; Nichols, 1996). Hestbeck et al. (1991) assumed a stationary distribution for a Markov movement process for Canada geese in order to compute the stationary distribution of geese over three wintering regions. In addition, stationary distributions for Markov chains factor importantly in applications of Markov chains to conservation and management. We discuss these applications in greater detail below, when we introduce Markov decision processes.
N
Pj = i~-'1 PiPij,
10.4. C O N T I N U O U S M A R K O V PROCESSES
or, in matrix notation, P ' = p'P,
(10.6)
where P is the transition matrix of the Markov chain. Because an ergodic Markov chain possesses a unique stationary distribution, the distribution satisfying Eq. (10.6) also must satisfy Eq. (10.5), i.e., pj = "rrj for every state j. Thus, the stationary probability for state j can be obtained either by repeated transitions of the Markov chain starting at some arbitrary state i or by solving the system of equations represented by Eq. (10.6). Either approach yields the unique stationary distribution guaranteed by the ergodic property of the Markov chain. Example
Consider the Markov matrix P=
0.3 0.6 0
0.5 0 0.4
Up to now we have focused on discrete Markov processes, specifically Markov chains. Recall that the defining characteristics of a Markov chain include Markovian independence, process stationarity, and a discrete state space and time frame, the latter consisting of equal-length time intervals. These conditions give rise to the Markov matrix, which captures the stochastic structure of a Markov chain. The key attributes of a Markov chain are inherited from patterns among the transfer probabilities in the Markov matrix. In this section we continue to focus on processes that have a discrete-state space, and we retain the Markovian assumption that the future state of the process is influenced by its current state but not its history: P[X(t + at) = x t + •
0.2] 0.4 0.6
for a Markov chain with transfers among three states. It is straightforward to show that the states are all recurrent aperiodic, and therefore the chain is ergodic. Its limiting distribution is given by ~' = (6/23, 7/23, 10/23), as shown by (6 7 10)(6 7 10)[00~~ 0"5 0"2] 23,23,2-3 = 23,23,2--3 0 0.4. 0.4 0.6 Starting with any nonzero initial distribution p' = (Pl, P2, P3), repeated application of P ' t + l = P'tP ultimately will produce the limiting distribution w. Discrete Markov processes have been used for a variety of applications in population ecology. For example, in population genetics, discrete Markov processes have been used as a way of modeling gene frequency dynamics under genetic drift (e.g., Roughgarden, 1979).
= P[X(t + at)
= Xs; s ~
= xt+at]X(t)
tl
-- xt].
We also retain the assumption that the process is stationary, i.e., Pij(At[t) = Pij(&t) for all values t >- 0. However, we relax the assumption that &t is fixed over the time frame of the process and instead allow for continuous and random waiting times between process transfers. It can be shown that the length of time in which a memoryless process stays in a particular state is exponentially distributed (Ross, 1996). This property gives us an easy way to model continuous Markov processes; thus, a discrete-state process is Markovian over continuous time if (1) the amount of time the process remains in state i before making a transition to another state is exponentially distributed with rate parameter v i that depends on the current system state, and (2) the transfer from state i to state j occurs with probability Pij, with ~,jPij = 1. Thus, a continuous Markov process is simply a Markov chain in which transfers between states can occur
10.4. Continuous Markov Processes at r a n d o m times. Stated differently, it is a stochastic process with transfers between states in accordance with a discrete Markov chain, except that the a m o u n t of time between transfers is exponentially distributed. Note that the a m o u n t of time between transfers must be i n d e p e n d e n t of the terminal state of the transfer; otherwise, the duration of time prior to the transfer w o u l d inform the transfer probability, in violation of the Markovian assumption. We let Pij(s) = P [ X ( t + s) = jlX(t) = i] represent the probability that the process in state i at time t will be in state j at time t + s. This probability is a function of a discrete distribution (for the transition between states i and j) and a continuous distribution (for the length of time the process resides in state i before the transition). The product qij = viPij, k n o w n as the transition rate from i to j, parameterizes the joint distribution (see below). Note that
qij
viPij
~,jqij
vi~'jPij
203
for intervals of length s. Thus, row 1 of the matrix records the probabilities e -~s()~s)k Plk(S) = k! that a step of size k - 1 _> 0 will be taken after s units of time, starting at state i = 1. Row 2 records the probabilities that a step of size k - 2 - 0 will be taken, starting at i = 2. And so on. The subdiagonal elements of 0 indicate that the Poisson process takes only nonnegative values, so it is not possible to transfer to a smaller state. An equivalent model for the Poisson process focuses on interarrival times rather than the Poisson counts. From Eq. (10.3) the interarrival times of a Poisson process are exponentially distributed. Thus, a model for the dynamics of N(t) allows for a unit increase in N(t) at times given by the exponential distribution. The corresponding transition matrix is simply B
= Pij, so that the transition probabilities Pij reproduce the transition rates, after the latter are scaled to unity. Thus, w h e n a transition occurs, the process transfers from state i to state j with probability Pij, and these probabilities are directly proportional to the process transition rates. Note also the aggregate of transition rates reproduces the exponential parameter vi:
~ , qij = v i ~ J
0
0
0 0
1 0
0 1
m
with the u p p e r off-diagonal elements of unity indicating that w h e n the time for a transition arrives, the transfer from state i to state i + 1 is certain.
10.4.1. Birth and Death Processes
j
A simple example of a continuous Markov process is the Poisson process. If {Nt: t >- 0} is a Poisson process, then
Pij(s) = P ( N ( t + s) = j I N ( t ) = i) 0 e -~'S(Xs)J-i ( j - i)!
1
o 9
Pij = vi.
Example
=
[Pi3 =
i
0 0 0 .
if
j
if
j->i.
An important class of continuous Markov processes in biology represents transition rates in terms of birth and death events. We consider here that the process represents population size, with transfers that allow only for unit changes in state, i.e, Pij = 0 w h e n e v e r Ii - jl > 1. Thus, a population of size i can only increase to size i + 1 or decrease to i - 1. An increase obviously corresponds to a birth event, whereas a decrease represents a death event. Let birth and death rates be represented by ~'i = qi,i+l and ~i - qi,i-1, respectively. Because the two nonzero transition probabilities are related by Pi,i-1 q- Pi,i+l -- 1, w e have
This probability structure can be described at each point in time by the transition matrix
ki if- ~i = qi,i+l q- qi,i-1 --- viPi,i+ 1 q- viPi,i_ 1
-P11(S) 0 0
P(s) =
P12(S)
P13(S)
...-
P22(S) 0
P23(S) P33(S)
... ...
-- vi
and Ki
viPi,i+l
~.i q- ~i
viPi,i+l q- viPi,i-1
m
9
,.,
0
oo
o
o
9
= Pi, i+l"
204
Chapter 10 Stochastic Processes
Thus, the transition probabilities Pi,i+l and Pi,i-1 c a n be expressed in terms of the birth and death rates )k i and ixi. We may think of a birth and death process in terms of two independent Poisson processes, such that whenever there are i individuals in the population, the time until the next birth is exponentially distributed with rate parameter h i and is independent of the time until the next death, which also is exponentially distributed but with rate parameter ixi.
represents the probability that a process in state i at time t will be in state j at time t+s. With the aid of certain limiting relationships involving the probabilities Pij(s), it is possible to derive an equation for the instantaneous rate of change in the probability distribution. Given the continuous-time Chapman-Kolmogorov equation,
Example
[Eq. (10.4)], one may write
Consider a process describing the number of individuals in a population. Individuals are added to or subtracted from the population at times that are exponentially distributed with exponential parameters Xi and ixi, respectively. Then if j = i + 1
~-i
Pij(t + h ) = ~ , Pik(t)Pkj(h) k
Pij(t + h) - Pij(t) = ~ , Pik(t)Pkj(h) -- Pij(t) k = ~
Ix;
if j = i -
1
)ti q- ~i
0
lim
h--,oo
otherwise,
0 ql [Pij] =
.
9
o
0
9
0
Pl
9
9
q2
0
P2
9
~
0 w h e r e Pi = )ki/()ti q- ~l,i) a n d
9
.
o
9
1
~
pij~t)
which, under suitable regularity conditions that allow for the interchange of the limit and summation in this expression, yields the Kolmogorov forward differential equations,
P'ij = ~qkjPik(t) -- vjPij(t) k,j
o
9
[1 -- pjj(h)]Pij(t).
Pij(t + h) - Ply(t) = limfK"lz_~" ~'~Pkj(h) t'ik~'J h " h h~oO l. k.j 1 - pjj(h) ,.}, -
indicating that each event adds or subtracts an individual to the population. The Markov transition matrix for this problem is
--
Therefore
)ki q- ~i
P[N(t + s ) =jIN(t) = i] =
Pik(t)Pkj(h)
k*j
o
0
qi = ~ i / ( ) t i
nt- ~Li). T h e
entries in row 1 indicate that if the process is in state i = 0, then the only possible change is for an individual to be added to the population (necessarily through migration rather than reproduction). Entries in the last row indicate that the only possible change is for an individual to be subtracted from the population. All other states allow for either the addition or subtraction of an individual from the population, as indicated by nonzero entries in the off-diagonal positions. However, the process allows only for an increase or decrease of one individual with each transfer; hence the zero entries are everywhere but in the off-diagonal positions.
10.4.2. T h e K o l m o g o r o v Differential Equations
Recall that
Pij(s) = P[X(s + t ) = jlX(t) = i]
(Ross, 1996). They are called forward equations because the computation of the probability distribution at time t + h is conditioned on the state at time t through the Kolmogorov equation (see Kolmogorov, 1931). On reflection this transition equation makes sense. Thus, the summation term represents the addition to Pij(t) of probability mass from Pik(t), whereas the second term in the equation represents the loss of probability mass from Pij(t). By conditioning on h rather than t, we also can write the Kolmogorov backward equations,
P;j = E qikPkj (t) -- viPij(t)" k.~ i
Again, this equation makes sense; the summation term represents the addition to Pij(t) of probability mass from Pkj(t), whereas the second term again represents the loss of probability mass from Pij(t).
Example Consider a metapopulation of mice in a patchy environment, with local extinctions at a particular patch followed by recolonization from nearby patches. Let
10.5. Semi-Markov Processes
X(t) represent the presence of mice in the patch at time t, with X(t) = 1 if mice are present and X(t) = 0 if they are not. Let Ix be the extinction rate when the patch is occupied and )~be the colonization rate when the patch is unoccupied. Because there are only two states, we have transition probabilities P01 = Pl0 = 1 and P00 = P l l = 0. Furthermore, P01(t) = 1 - P00(t), so that the Kolmogorov forward equations for this system yield d
d~ P~176 = transfer of probability from Pl0(t) to Poo(t)
205
and
pij(t)-
Kj_lPi,j_l(t
) -
hjPij(t)
for j > 0, where h i -- qi.i+l and Ixi,i-1. It is straightforward to show that
Pii(t)
= e-Xit,
which is consistent with the fact that transition times are exponentially distributed. More generally, one can show that t
-
Pij(t) = Kj-le -~jt f o e~lSpi'J-l(s) ds
transfer from Poo(t) to P01(t)
= IxP01(t) - )~Poo(t)
for j > i (Ross, 1996). On assumption that )~j = j)~,
= -()~ + Ix)Poo(t) + Ix. By substituting back into this equation, it can be shown that IX
q_
P~176 = X + tx
~"
for j >-- i >-- 1. These equations provide a simple algorithm for the modeling of a pure birth process.
-(x+,)t
X + IX
with P01(t) = 1 - Poo(t) ~"
~"
---
-
-
-(x + ~)t
e
o
An analogous argument shows that P11(t) --
)k
h+~
IX
+-
)~+IX
e
Pij(t) = (i - 1 )1 e -xti(1 - e xt)j-1
-(x+~)t
The birth and death process models introduced above have been applied in population ecology, epidemiology, actuarial sciences, and evolutionary biology (e.g., see Bartlett, 1960; Bailey, 1964; Chiang, 1968), as well as in current conservation biology (see Chapter 11). Additional applications of the Kolmogorov forward and backward equations to population biology include the modeling of gene frequency dynamics (e.g., Wright, 1945; Kimura, 1957; Crow and Kimura, 1970).
and 10.5. SEMI-MARKOV
Pl0(t) = 1 - Poo(t) Ix K+IX
Ix e - ( x + , ) t . K+IX
Example For the general birth and death process described above, the Kolmogorov forward equations are
p;o(t)
= IxlPil(t)-
)~oPio(t)
and
p;j(t) = Kj_lPi.j_l(t) + Ixj+lPi.j+l(t) - (hj + Ixj)Pij(t) for j :/: 0. The first equation essentially says that the change in probability mass for Pio(t) is given by gains from Pil(t) (via death) minus losses to Pil(t) (via birth). The second equation asserts that for j :/: 0, the change in probability mass for Pij(t) is given by gains from Pi,j+l(t) (via death) and Pi,j_l(t) (via birth) minus losses from Pij(t) (via birth and death). In particular, the forward equations for a pure birth process reduce to
p;i( t) = - )~iPii(t)
PROCESSES
Thus far we have discussed processes satisfying the Markovian assumption that the future state of a process is influenced by its present state but not its past. In particular, we considered continuous Markov processes that are stationary over a discrete-state space and exhibit continuous random intervals of time between transitions. The transition probability structure Pij(s) for such a process is characterized by statistical independence of state transitions and the waiting times between those transitions. In this section we relax the Markovian assumption, but retain several other features of continuous Markov processes, including process stationarity, a discretestate space, and continuous random intervals between transitions. We also retain certain structural features that will allow us to recognize an "imbedded" Markov chain in the process. Thus, a semi-Markovian process is defined by the following characteristics: (1) at any given time the probability of transferring from state i to state j is Pij, and (2) the time until transition from i to j has a distribution that depends on both i and j.
206
Chapter 10 Stochastic Processes
From condition (2) a semi-Markov process fails to satisfy the Markovian assumption, because a prediction about the future state of the process is informed not only by the present state, but also by the length of time one has been there. Essentially, the stochastic prediction of transition times requires one to know the terminal state of the transition as well as its initial state. This adds considerable complexity to the process and stands in contrast to the continuous Markov process, for which the transition waiting times are assumed to be independent of the terminal system state. Let Fij(s) represent the distribution of time required for a transfer from state i to state j. By way of contrast, recall that transfer times for a continuous Markov process are distributed exponentially with rate parameter v i that applies to all transitions from state i, irrespective of the particular terminal state j to which the transfer is made. From condition (1) above, the semi-Markov process "imbeds" a Markov chain within it, in the sense that the stationary matrix _P = [Pij] defined in the semiMarkov process corresponds to a Markov chain. The latter process is called the imbedded Markov chain of the semi-Markov process, and it inherits its properties from patterns in the transfer probabilities in P. In particular, the semi-Markov process is said to be irreducible if the imbedded Markov chain is as well. If the process is irreducible, the expected value id,ii of the time sii between successive transitions into state i is finite.
10.5.1. Stationary Limiting Distributions Under certain conditions the limiting distribution of states for a semi-Markov process is stationary, with probabilities given in terms of the average transfer times. To see why, let Hi(s) be the average time required to transfer out of state i, based on the distributions Fij(s) of transfer times and the transfer probabilities Pij:
Hi(s) = ~_, PijFij(s). J Using the mean ~ii of the return time sii and the mean ~i
-~
f ~ S dHi(s)
of the distribution Hi(s), under rather mild conditions, the semi-Markov process can be shown to have a limiting distribution
pj = lim Pij(s) S ---~oo
= ~j/~jj that is independent of the initial state i (Ross, 1996). In essence, the value pj is the limiting proportion of
time the process spends in state j. Fortunately, these values can be identified without having to determine the mean return times txjj. If {'rri: i=1, ..., n} represents the stationary distribution of the imbedded Markov chain, i.e.,
"rrj = ~ "rriPij , i
then pj can be expressed as
~rj~j PJ = ~,i
"fribl'i"
On reflection this result makes sense. It asserts that the long-term proportion of time spent in a state increases with the stationary probability for the state from the imbedded chain and for the average amount of time the process resides in the state before making a transition. It is intuitive that large values for either of these factors will increase the proportional representation of j over the long term.
Example The movement patterns of a small mammal population are to be investigated by radio tracking. The study involves the periodic capture of individuals and fitting them with radio collars. At irregular intervals a transmitter fails, or the individual suffers mortality or leaves the study area, and another animal must be captured and fitted with a collar. The mean time required for replacement of an individual in the study is la,1 = 2 days. Experience thus far indicates that about twothirds of the individuals available for trapping are juveniles and adults are about one-third. About threefourths of all juveniles exit the study before becoming sexually active, either from mortality, migration out of the study area, or transmitter failure. The mean time to either maturation or death is t.1,2 = 10 days for juveniles. Of course, all adults ultimately exit the study through mortality, migration, or transmitter failure, with a mean time of hi,3 = 2 0 . This situation can be modeled as a semi-Markov process, with probabilities of transfer between states and with state-specific transition times that depend on both the initial and terminal states of the transition. Let the state indices 1, 2, and 3 represent individuals not in the study, and juveniles and adults that are in it, with the transfer from state I to state 2 or 3 representing capture, fitting with a radio collar, and release. Assuming equal trapping probabilities for juveniles and adults, the transfer probabilities from state I are P 1 2 = 2 1 and P 1 3 = 3, with a mean time of la,1 = 2 for trapping, collaring, and release. Because three-fourths of juveniles fail to enter the adult stage in the study, the trans-
10.6. Markov Decision Processes 1
3
fer probabilities for state 2 are P23 = 4 and P21 = 4, with a mean transfer time of ~2 = 10. Finally, the transfer from state 3 to state 1 is certain, so that P31 = 1 with a mean transition time of ~3 20. Under these circumstances the matrix for the imbedded Markov chain is =
P=
[0 3/4 1
1/3 0 0
2 / 3 ]] 14,
and it is easy to show that the corresponding stationary probabilities ~' = [-rr(1), "rr(2), w(3)] are "rr' = -rr'P =[1242-95] 25, 25,
"
It follows that the time spent in each state is in the proportions 'rrl[l,l:'rr2~2:Tr3~ 3 = 6:10:45.
Thus, one can expect to track juveniles about 16% of the time, to track adults about 74% of the time, and to lose about 10% of the study time capturing animals and replacing collars.
207
Let A i be the set of all possible actions available when the process is in state i. The available actions may well vary from state to state, i.e., it is not necessary that A i = Aj for i 4: j, though the set A = U i Ai of all available actions for the process is assumed to be finite. A policy for the process is defined by a mapping ~r that associates with a given state i at any given time t the action -rr(i, t). If ~r(i, t) = -rr(i) the policy is stationary, i.e., time independent. To apply a stationary policy, one need know only the process state and not the time t when it occurs; a particular state has the same action associated with it at every time in the time frame. To indicate the influence of decisions on the Markov transition probabilities, we represent by Pij[~r(i, t)] the probability of transfer from i at time t to j at time t + 1, assuming action ~r(i, t) is taken at time t:
Pij['rr(i, t)] = P~[X(t + 1) = jlX(t) = i], where the subscript ~r in the probability statement denotes a policy with action -rr(i, t) for state i at time t. If p,~(-rr) represents the probability of transfer in n time steps from i to j under policy -rr, then p~j('rr)
. . .in~ . 1
si1{ Piil , ['rr(i, O)]}{Pin_l,j['rr(in_l, 1"1-- 1)]}
n-2
10.6. MARKOV DECISION PROCESSES In this section we consider Markov processes for which the transition probabilities can be influenced by decisions at each point in time. To retain the Markovian assumption, we assume that at each decision point an action is taken based in the current state of the system, but not on previous states (or previous actions). The state space is assumed to be countable and therefore discrete. We also impose the condition that the range of decisions at each point in time is finite. In general, state-specific decisions are allowed to vary with time; thus, the corresponding Markov decision process is potentially a nonstationary process. However, we assume that the only sources of nonstationarity are statespecific actions that vary over time.
X H P,,//+l['rr(q,J )] j=l
by repeated application of the Chapman-Kolmogorov equation (10.4). Under a stationary policy the Markov decision process becomes a Markov chain that is defined by the stationary transition probabilities Pij['rr(i)]. Furthermore, the nth order transition probability matrix of the process is
pn
[P~j('n')] =
' r r !
where P11['rr (1)
...
PlN('rr(1)]
PTI" u 9
LpNiDr(N)
o
...
,
pNN(rr(N)]
10.6.1. Discrete-Time Markov Decision Processes We focus here on Markov decision processes with N states, over a discrete time frame T that is either finite or infinite in length. To simplify notation, we assume that an action is taken at each time in the time frame, with the action taken at time t influencing the probabilities of transition to a new state at time t + 1.
10.6.2. Objective Functionals An investigation of policies with Markov decision processes requires a measure of policy performance, by which different policies can be compared and contrasted and optimal policies can be identified. In what follows we describe an objective functional for measur-
208
Chapter 10 Stochastic Processes
ing policy performance, which aggregates utilities corresponding to time-specific actions and state transfers. Thus, let Rj[rr(i, t)] be the utility (e.g., returns net of costs) associated with the transfer from state i to state j w h e n action "rr(i, t) is taken. Then the average utility
Note that this expression is indexed by the initial state i. Thus, a policy -rr generates N such values, one for each of the possible states X(0) = i. In what follows we restrict our attention to processes with an infinite time horizon.
//
R[~r(i,t)] = ~ Pij[~r(i,t)]Rj[~r(i,t)l
10.6.3. Stationary Policies
j=l is an appropriate optimality index for discrete-time processes, and a corresponding objective functional is the expected sum of (possibly) discounted utilities, w h e n it exists:
IT
]
V~(i) = E ~ odR{'rr[X(t),tl}lX(O) = i . t=0
Additional structure in the values V~(i) can be recognized if the policy -rr is stationary. Let V'rr = [V~(1), ..., V=(N)] be the vector of values generated by a stationary policy ~r. Because the transition probabilities for a stationary process are time independent, average returns are as well:
(10.7)
N
R[~r(j,t)] = ~ Pjk['rr(j)lRk[~r(j)]
The s u m m a t i o n in this expression accumulates stochastic utilities over the time frame of the process, assuming the process begins in state i. The expectation is with respect to the stochastically determined values of process state, and the term oL _< 1 is a single-step discount factor that essentially devalues future utilities as time progresses. The notation Vrr(i) indicates that the value of the objective functional depends on both the initial state i of the process and the policy -rr that is used. We assume here that state-specific utilities are b o u n d e d for all policies, and therefore the value Vrr(i) exists whenever T < oo. If T = oo then Vrr(i) exists for all discount factors 0 < ot < 1. However, when ot = 1 the expectation in Eq. (10.7) can be finite or infinite, depending on the utilities and the pattern of transfer probabilities. If infinite, a different objective functional is required, based on the limit of timeaveraged utilities:
g~,(i) = lim (n + 1) -1E I ~_, n Rl'rr[X(t), t]} X(O) = i 1 . n--+oo t=0
(10.8)
Equation (10.8) can be shown to produce finite values of V=(i) for any stationary policy. For nonstationary policies the limit may be replaced by limit inferior. It is useful to consider an individual element in the expectation in Eq. (10.7) w h e n T = oo. A straightforward inductive argument shows that the expected utility for time t can be expressed as N
k=l = R['rr(j)]. The vector of average returns for a stationary policy is denoted here by R'~ = {R['rr(1)], R['rr(2)], ..., R[w(N)]}.
10.6.3.1. Finite Markov Decision Processes In matrix form, the objective functional in Eq. (10.9) is oo
Err = s oLtptarr. t=0
(10.10)
Because every stationary policy -rr has corresponding to it a stationary Markov matrix P~ and stationary vector Rrr of utilities, from Eq. (10.10) every policy also yields a vector of aggregate utilities Vrr. A simple alternative to Eq. (10.10) for determining Vrr can be derived by rewriting Eq. (10.10) as a recurrence relation: oo
Vrr = ~ ottptRrr t=O
= Rrr + otPrr[Rrr + oLPrrRrr + ""] = Rrr + oLPrrVrr. The vector Vrr therefore can be obtained as
E(R{,rr[X(t),tl}lX(O)=i ) = ~, ottp~j('rr)R['rr(j,t)],
V~ = (I - c~P:) - 1R~.
j=l and therefore the objective functional for finite processes may be written as
V~(i)= ~=o{~=loLtp~j(rr)R['rr(j,t)] }.
(10.9)
Of course, this computing formula requires the existence of ( / - otPrr)-1. The inverse clearly exists for discounted processes (i.e., 0 < oL < 1), because det(/-
N otPrr)= 1-Ill i=1
oLpii] =/h O.
10.6. Markov Decision Processes For essentially the same reason, the inverse for an undiscounted finite process also exists, though it is less obvious to demonstrate. 10.6.3.2. Infinite Markov Decision Processes As mentioned above, the limiting formula
V~,(i) = lim(n + 1)-IE
R['rr(xt, t)llx o = i
H----~oo
of Eq. (10.8) can be used as a measure of aggregate utility in cases in which Eq. (10.9) has no finite solution. It can be shown that the values V~(i) produced by Eq. (10.8) with a stationary policy -rr satisfy the recurrence relation
V~ + h~ = R~ + P~,h~,
(10.11)
where the vector h~, is defined by
209
using disturbances to manage the population in the most cost-effective way. For simplicity, population size is categorized as small, medium, and large populations, with stochastic transitions from year to year that depend on the type of disturbance. Three different actions can be taken, each with its own impact on the population and on its predators and other competitor species. Actions I and 2 can be used when the population is low, actions 1, 2, and 3 are available for midsized populations, and actions 2 and 3 are available when the population is high. The probability of transition from one population size to another depends on which action is taken at the time. At each point in time, management returns (net of costs) for an action depend on the population size at the time, as well as the action that is taken. The transition probabilities and returns are estimated to be -rr(1) = 1" {P11,P12,P13} {1/2, 1/4, 1/4} 9r(1) = 2: {P11,P12,P13} {1/4,1/8,5/8}
R[-rr(1)] = 8 R[-rr(1)] = 4
-rr(2) = 1" {P21,P22,P23} {1/16,3/4,3/16} -rr(2) = 2: {P21,P22,P23} {1/2, 0, 1/2} 'rr(2) -- 3" {P21,P22,Pa3} = {1/16, 7/8, 1/16}
R[-rr(2)] = 5 R[-rr(2)] = 12 R['rr(2)] = 9
"rr(3) = 2" {P31,P32,P33} {1/4, 1/2,1/4} 9r(3) = 3" {PBl,P32,PB3} {1/8, 3/4, 1/8}
R[~r(3)]= 6 R[~r(3)] = 4
- -
- -
z6 ( I - P,, + P*)h~ = ( I - P~)R~, =
with P* given by
=
n
_P* =
lim(n + 1 ) - 1 ~ _Pt. n~oo
t=O
=
Absent additional structure on the process, Eq. (10.11) represents N equations in the 2N unknowns in h~, and V~ and therefore cannot be solved. However, if the matrix P~, is ergodic, then the vector V~, can be shown to be of the form V~ = g~l, and the system of equations now involves N equations in the N + 1 unknowns h~(1), ..., h~(N), and g~. Setting one of the h~(i), say h~,(1), to zero reduces the system to N equations in N unknowns, which is solvable. The resulting h~,(2), ..., h~(N) then represent state-specific values relative to the value for state 1. From V~ = g~l the process gain g~ applies to every state and thus is independent of the initial state i. It can be shown that for an ergodic process the values h~ and g~ asymptotically satisfy
-
These transition probabilities define a total of 12 stationary Markov processes (two sets of transition probabilities for state 1, three sets for state 2, and two sets for state 3, each with state-specific returns). For example, the choice -rr(1) = 1, -rr(2) = 2, and -rr(3) = 2 in a stationary policy results in the Markov matrix [ i / /2 2 /4
Example Biologists are investigating the effect of disturbance on a population of small mammals, with a goal of
11/2 1/4 1/4
with average single-step returns R['rr(1)] = 8, R[-rr(2)] = 12, and R['rr (3)] = 6. State-specific values and system gain are found as solutions of the system of equations
__V~(n) = n[g.~l_] + h~, where V~,(n) is the vector of (asymptotic) cumulative utilities for policy -rr after n time steps. Cumulative returns thus are composed of a component for average long-term process gain and components specific to the initial state of the process. The values h~(i) may be thought of as utilities due to "transient" process behavior, whereas the gain g~ corresponds to "steady-state" utility.
1/4 0 1/2
3
g~, + h~(i) = R[~r(i)] + ~ Pij['rr(i)]h~(j), j=l
i = 1, 2, 3, and h~(3) = 0. It is easy to show that [g~, h~(1), h~(2)] = (160/19, 24/19, 80/19) solves this system of equations. Thus, the policy produces a system gain of 8.42 and transient values for states 1 and 2 of 1.26 and 4.2 (relative to state 3). Of course, a different policy would produce different state values and system gain. For this simple problem, one could determine the policy that produces
210
Chapter 10 Stochastic Processes
the largest system gain simply by enumerating the solutions for all 12 systems of equations. Obviously, such an approach becomes infeasible as the size of the process increases and the policy options multiply. We deal with optimization approaches for problems such as this in Part IV.
10.6.4. S e m i - M a r k o v D e c i s i o n Processes The results above can be generalized to allow for semi-Markov decision processes, involving sequential decision-making in which the times between decisions are random. A decision model for this situation assumes (1) the probability Pij(a) of transition between states i and j is Markovian, and is influenced by decision a, and (2) conditional on the terminal state j, the time s until transfer from i to j is random with probability density function fij(sla). An algorithm for implementing a semi-Markov process consists of choosing an action, determining the transition between states, identifying a random length of time before the transition, and repeating this sequence indefinitely. A policy ~r identifies the action "rr(i,t) = a to be used in the algorithm for every possible state at every possible decision time. As above, let Rj['rr(i,t)] represent the utility associated with transfer from state i to state j when action ~r(i,t) is taken. Assume also that there is a utility rate ri[~r(i,t)] (perhaps expressing delay costs) associated with the waiting time until transfer from i to j. Then a wait of s units of time followed by a transfer from i toj incurs a total utility of Rj['rr(i,t)] + s{r[~r(i,t)]}. Under these conditions the process is referred to as a semiMarkov decision process. Clearly, if the time between transitions is always unity, then the process is simply a Markov decision process. Note that if the policy is stationary, i.e., "rr(i,t) = "rr(i), the process is also. Then the transfer probabilities are Pij[~r(i,t)] = Pij[w(i)],
Rj['rr(i,t)] + s{r[~r(i,t)]} = Rj[~r(i)] + s{r['rr(i)]},
so that the average utility is N
Pij[~r(i)]Rj[w(i)]
+ s{r[~r(i)]}.
j=l
An appropriate objective functional is simply the expected value of the sum of (possibly) discounted utilities, when it exists: e
+ r[Tr(X n, tn)]
tn '
rs,,
e -~s ds
JO
}
-
-
V~(i) = E [ ~ e - ~ ( s ~
]
xo = i ,
1012
(10.13)
H
+(1-e~S')r{~r[X(n)]}/o~} X0= i]. If the process state space includes N transfer states and the time horizon is infinite, then Eq. (10.13) can be expressed explicitly in terms of transfer and waiting time probabilities:
u~(i) = E
e-~(s~
~. p~.(~r)
=0
•
j=l
RI~r(j)l + r [ ~ ( j ) l OL
•
and single-step utilities for the transfer from i to j are
R[w(i)] + s{r[w(i)]} = ~
where s n tn+ 1 -- t n is the waiting time for the nth state transfer and Xn is the state of the process after n transitions. This expression is analogous to Eq. (10.7) for Markov decision processes, with some notable exceptions. As before, the summation accumulates stochastic utilities over the time frame of the process; however, the number of terms in the summation now is random, because the waiting times between transitions are random variables. The expectation is with respect to the stochastically determined values of the states to which transitions are made, as with Markov decision processes; however, it also accounts for distribution of the waiting times between transitions. The sum of terms within parentheses accounts for both the utility R [ ~ ( X n , tn)] , associated with the decision at time t n, and the accumulated value (or cost) over the interval [tn_l, t n] before the nth transition occurs. Finally, the discount term e -st for continuous time has replaced oL-t for discrete time. Assuming that the process is stationary, Eq. (10.12) reduces to
pjk[~r(j)l
k=l
(1 - e~S)~.k[sl~r(j)l ds
xo = i ,
0
where the expectation now refers only to the waiting times s o, Sl, ..., Sn-1 between transitions. It also is possible to define time-averaged objective functionals for undiscounted processes with aggregate utilities that are infinite (Ross, 1970). The mathematics for this situation become rather complicated, and we leave further investigation to the interested reader.
10.7. B R O W N I A N M O T I O N Perhaps the best known continuous-state stochastic process is the Brownian motion or Wiener process. N a m e d after English botanist Robert Brown, who first discovered it while investigating particle movements
10.7. Brownian Motion in fluids, the process was given a concise definition by Norbert Weiner in 1918. It since has been used to describe behaviors of a great m a n y different phenomena, from q u a n t u m mechanics to m o v e m e n t s of stock prices. Brownian motion describes stochastic behaviors over a continuous time frame and continuous state space, on assumption that the process is normally distributed at any given time. Formally, a stochastic process {X(t): t -> 0} over continuous time is said to exhibit Brownian motion if (1) X(0) = 0, (2) {X(t): t - 0} has stationary i n d e p e n d e n t increments, and (3) for every t ~ 0, X(t) is normally distributed with m e a n ~t. From the assumption of stationary i n d e p e n d e n t increments, one can show that the variance of X ( t ) is var[X(t)] = 0.2t, where 0 .2 is linked to the underlying process and must be determined empirically. The probability density function for X(t) is 2
]" (10.14) 1 r 2x/ t W h e n ~ = 0 and 0. = 1, the distribution has the form ft(x) =
exp [ - ~l ( x - ~ tcr)
ft(x) =
1
exp -
(10.15)
and the corresponding process is called standard Brownian motion. Because a normal distribution can always be rescaled and translated so as to have any mean and variance, we assume in what follows below that the Brownian motion is standard.
211
tions of t), but nowhere differentiable. Basically, the random, i n d e p e n d e n t nature of the transitions over infinitesimally small time steps means that change is continuous but abrupt, so that the function X(t) cannot be differentiated. Based on the assumption of stationary i n d e p e n d e n t increments, it is possible to define a joint distribution for Brownian motion. Thus, the probability density function for X(t 1) = x I ~ "'" ~ X ( t n) = x n can be factored into ftl ..... t n ( X l , ...,Xn) = ftl(Xl)ft2_tl(X2
X ft,_t,,_l(Xn
-- Xn_l)
,
and stationary independent increments allow us to recognize the joint probability distribution in Eq. (10.16) as multivariate normal for all values t 1, ..., t n. Processes that meet this condition are said to be Gaussian. Because a multivariate normal is completely determined by its first two m o m e n t s (see Appendix E), one need only identify the covariance terms in the probability density function, Eq. (10.16), which can be shown to be c o v [ X ( t i ) , X(tj)] = min{ti, tj}. Thus, the probability density function of standard Brownian motion is Gaussian with E = 0 and tl tl tl
tl t2 t2
tl t2 t3
... ... ...
t1 t2 t3
.tn
tn
tn
...
tn.
X(tl, ..., tn) =
Example
Consider a population N ( t ) that fluctuates over time according to a combination of n o n r a n d o m and r a n d o m factors. N o n r a n d o m variation can be modeled by the continuous logistic equation, such that the population mean at each point in time is given by
(10.16)
-- X l ) " "
Equation (10.16) also allows us to compute conditional probabilities. For instance, it can be shown that the conditional distribution for X ( t ) given X(t 1) = A and t ~ t I is just the normal with mean E [ X ( t ) I X ( t 1) = A] = A t / t 1
E[N(t)] =
l+e
--Ft"
R a n d o m fluctuations about these average values can be modeled as Brownian motion with ~ = 0. Thus, a stochastic model by which to predict population size at time t -> 0 is described in terms of a normal distribution with logistic mean and variance var[N(t)] = 0.2t. The variance for the model increases linearly in t over the time frame. Though the probability distributions, Eqs. (10.14) and (10.15), have the familiar form of a normal distribution, inclusion of the continuous variable t in the distributions induces the very unusual property that the process is everywhere continuous (as might be expected, because its m e a n and variance are linear func-
and variance v a r [ X ( t ) l X ( t 1) = A] = t(t I - t ) / t 1.
Letting t/tl = e~, we thus have the conditional m e a n oLA, which increases from 0 to A as t increases from 0 to tl, and conditional variance o~(1 - oL)tl, which increases from 0 to a m a x i m u m of t 1/2 w h e n t = t 1/2, followed by a decrease back to 0 as t --> tl. These patterns make intuitive sense, in that the conditioning equation X(t 1) = A means that X(t) must converge to A as t approaches tl, which in turn means that the distribution variance must vanish as t approaches t 1. What is not so intuitive is the remarkable property of Brownian motion that the conditional variance of X ( t ) given X ( t 1) = A is i n d e p e n d e n t of A over 0 K t K t 1.
212
Chapter 10 Stochastic Processes
By extension, the conditional distribution for X(t) given X(t 1) -- A, X(t 2) = B, and t I < t < t2, is just the normal with n,ean E [ X ( t ) [ X ( t 1) = A, X(t 2) = B] = A + ~(t - t 1)
+
[B - A + ~(t I - t2) J t 2 -- t 1
(t-
t 1)
and variance
where Tx is the first time the process attains a value of x -> 0. Then {Z(t): t -> 0} is said to be absorbed, i.e., once having attained a value of x, the process remains at x forever. An example involves the absorbing state of zero for biological populations, in which stochastic p o p u l a t i o n change stops only w h e n the p o p u l a t i o n is extinct. It is easy to see that the first m o m e n t s for standard Brownian motion that is absorbed are
v a r [ X ( t ) l X ( t 1) = A, X(t 2) = B] = (t2 - t ) ( t - tl)" t2 - t 1
The conditional m e a n of the distribution therefore changes in a linear fashion from A w h e n t = t 1, to B w h e n t = t 2. O n the other hand, the conditional variance increases from 0 to a m a x i m u m of (t 2 - t l ) / 4 w h e n t = (t 2 4- t l ) / 2 , followed by a decrease back to 0 as t approaches t 2. As above, the conditional variance is i n d e p e n d e n t of the parameters A and B. Example
Biologists investigating the d y n a m i c s of a population of fruit flies record the n u m b e r of organisms at each of several points in time. Recognizing that the p o p u l a t i o n size is N(t 1) = A at the beginning of the observation period and is N(t 2) = B at its end, the investigators wish to determine w h e t h e r n o n r a n d o m factors have influenced population change over [tl, t2]. One w a y to investigate this issue is to compare the population size at several points in [tl, t 2] against the m e a n p o p u l a t i o n size predicted by Brownian motion. The equations s h o w n above for the m e a n and variance of constrained Brownian m o t i o n can be used to determine h o w well the recorded data fit a Brownian motion process. A reasonable fit suggests that changes in population size over [tl, t 2] are essentially r a n d o m , whereas a lack of fit suggests that population change is being influenced in some systematic w a y over [tl, t2].
10.7.1. Extensions of Brownian Motion A n u m b e r of stochastic processes that are applicable to biological populations can be derived from Brownian motion. Here we mention a few w e l l - k n o w n processes that result from simple process transformations or from restrictions on process values. 10.7.1.1. B r o w n i a n M o t i o n A b s o r b e d a t a Value
One potentially useful derived process assumes that Brownian motion is absorbed once the process attains a specified value: Z(t)={X~t)
if if
t- Tx,
0 x
E[Z(t)] =
if if
t < Tx t -> Tx
and {~ var[Z(t)] =
if if
t Tx.
10.7.1.2. B r o w n i a n M o t i o n R e f l e c t e d a t the O r i g i n
A n o t h e r variation on Brownian motion that is relevant to biology assumes that it can never be negative: Z(t) = IX(t)l
t -> 0.
Such behavior is said to be reflected at the origin. Reflected Brownian motion is especially applicable to processes such as population dynamics, in which process size m u s t remain nonnegative. It is not difficult to s h o w that the m e a n and variance for Z(t) is E[Z(t)] = V ' 2 t / ~ r
and var[Z(t)] = (1 - 2/~r)t. W h e n c o m p a r e d to m o m e n t s of the probability density function [Eq. (10.15)] for the standard Brownian motion, the nonnegativity restriction is seen to increase the process m e a n and decrease process variance. On reflection these results are intuitive; the nonnegative condition restricts the range to positive values, thereby reducing their spread and ensuring that their average m u s t be positive. 10.7.1.3. G e o m e t r i c B r o w n i a n M o t i o n
Yet another derived process is geometric Brownian motion, defined by Y(t) = e X(t). Y(t) is nonnegative, with m e a n ElY(t)] - e t/2
and variance var[Y(t)] = E[e 2X(t)] - [et/2] 2 = r 2t _ e t.
10.8. Other Stochastic Processes Again, these results are intuitive. The exponential transformation is monotone increasing and positive, so the mean and variance of the transformed process should reflect both the sign and the structure of the transformation.
213
by its mean, which is E[Z(t)] = 0, and its covariance structure, which for s ~ t is given by cov[Z(t), Z(s)] = s2(t/2 - s/6). Of course, process variances correspond to t = s: var[Z(t)] = t3/3.
Example
Geometric Brownian motion is especially useful for the modeling of percentage changes [i.e., Y ( n ) / Y ( n 1) rather than Y(n) - Y(n - 1)] that are held to be independent and identically distributed over time. An example involves the modeling of population trends over time. Consider a population that is represented by the exponential model
Example
Consider a population in which the intrinsic rate of growth (rather than the population itself) is assumed to be Brownian. Letting Z(t) represent the population and X(t) represent the population rate of growth, we have d Z(t) = X(t), dt
N(t + 1) = gt N(t),
with growth parameter gt - 1 + r t. We can describe population size in terms of a product of the growth terms:
or t
Z(t) = Z(O) + f
t
N(i) N(t) = N(O) I-I N ( i - 1) i=1 t-1 = X(0) I-I i=0
o
X(s) ds.
Because X(t) is Brownian, Z(t) is integrated Brownian, with E[Z(t)]
= e X(t) +
Z(0)
gi"
and variance as above.
On assumption that the values gt are independently and identically distributed and that the mean of r t is 0, the Central Limit Theorem ensures that the logarithm
10.8. OTHER STOCHASTIC PROCESSES
X(t) = ln[N(t)/N(0)] t-1
)
In this section we briefly mention some other stochastic processes that may arise in the modeling of animal populations.
t--1 = E In(g/) i=0
is approximately normally distributed with mean zero and variance to"2. With appropriate scaling, it follows that N ( t ) / N ( O ) = e X(t)
is geometric Brownian motion.
10.7.1.4. Integrated Brownian M o t i o n Yet another extension is integrated Brownian motion, as expressed by /. t Z(t) = ~ X(s) ds. d0
10.8.1. Branching Processes A useful class of stochastic processes with biological applications consists of branching processes (Harris, 1963; Jagers, 1975). To illustrate, suppose that a semelparous organism produces a random number Z of offspring and then dies (i.e., the generations do not overlap), as is the case with many species of insects, fish, and other taxa. Let {pj: j = 0, 1, ...} describe the probability distribution of Z for individuals in the population, assuming that all organisms reproduce according to the same distribution. Suppose also that offspring act independently of each other and produce their own offspring according to the same probability distribution. If there are, say, N(t) individuals in the population at time t, then N(t)
It can be shown that because {X(t): t - 0} is Gaussian, {Z(t): t -> 0} is as well. Thus, the process is specified
N(t + 1 ) -
~ i=1
Zi
214
Chapter 10 Stochastic Processes
describes the population transition from t to t + 1, where the time step corresponds to a single generation. This equation essentially aggregates the results of random, independent reproduction events across all individuals in the population, and realizations of such behavior over time describe a branching process. Because of the independence of reproduction events, it is easy to see that {N(t): t = 0, 1, ...} is a Markov process. If p~ and 0.2 represent the mean and variance of a random reproduction event, i.e., oo
I~ = ~, j(pj) j=O
and oo
0 .2=
~,(j-
~l,)2pj,
j=O
then the mean and variance of N(t) can be shown to be
E[N(t)]
=
p t
1,1,t -
1
and var[N(t)] =
0.2pt-1
p~- 1 to"2
if
p~ ~ 1
if
~ = 1.
Because each individual in the population produces individuals on average and then dies, it seems reasonable that the population should exhibit geometric growth in its mean for ~ > 1. One also might expect the variance to increase over time, either by tracking the growth of the population mean (if p~ :~ 1) or by increasing linearly with time when the population is stochastically stable (if ~ = 1). It is easy to see that the coefficient of variation for a growing population converges asymptotically to 0.(~2 _ t.i,)-1/2, which, if substantially exceeds unity, is approximately 0./~. Simple branching processes provide a ready model for species that reproduce only once in a lifetime, given that reproductive events are independent and reproduction is only stochastically predictable. In this situation one needs little more than an estimate of the mean and variance of individual reproductive success, to forecast population dynamics and other population attributes over the process time frame. Simple branching processes can also be adapted to organisms with other life histories (e.g., iteroparous) by defining reproduction in a manner that includes survival [i.e., the number of animals at time t + 1 "produced" by an animal at time t includes not only new individuals produced by reproduction but also the survival of the
animal itself (Caswell, 2001)] or by rescaling the time step to correspond to one generation. The original applications of branching processes are usually attributed to the French mathematician I. J. Bienayme (1845; also see Heyde and Seneta, 1972) and to F. Galton (1873) and H. W. Watson (see Watson and Galton, 1874), who used them to study extinction probabilities of family names. They were used in population genetics to study the probability of fixation of a mutant gene (Haldane, 1927; Fisher, 1930; Crow and Kimura, 1970), and they have been recommended for the study of extinction probabilities for animal populations in conservation biology (e.g., Caswell et al., 1999; Gosselin and Lebreton, 2000; Caswell, 2001). Multitype branching processes (Harris, 1963; Ney, 1964; Sevast'yanov, 1964; Pollard, 1966, 1973; Crump and Mode, 1968, 1969; Mode, 1971, 1985; Athreya and Ney, 1972; Jagers, 1975) relax the assumption of simple branching processes that all individuals are similar in their probabilities of survival and reproduction. Multitype branching processes thus can incorporate the more general age and stage structures presented for deterministic models in Chapter 8. For example, Pollard (1966, 1973) focused on stochastic analogs of the age-structured Leslie matrix (also see Mode, 1985), whereas Crump and Mode (1968, 1969) and Mode (1971) developed branching process analogs of agestructured models in continuous time. Note that the variation considered in branching process models as described above concerns the stochasticity of birth and death processes. Thus, an individual either survives until the next time step or it does not, and this process is a simple Bernoulli trial. Similarly, animals may produce 0,1, 2, ... offspring with probabilities described by a multinomial or Poisson distribution. This type of stochasticity typically is referred to as demographic stochasticity (e.g., Chesson, 1978; Shaffer, 1981). One also can envision environmental variation such that the underlying probabilities of death and of producing specific numbers of offspring vary with time and environmental conditions. This variation in the underlying probabilities of the birth and death processes often is called environmental stochasticity. Smith and Wilkinson (1969), Athreya and Karlin (1971a,b), and Keiding and Nielsen (1973) considered branching processes in random environments, thus incorporating both demographic and environmental stochasticity in stochastic process models. Mountford (1973) presented an ecological application, and Mode and Root (1988) applied a generalized branching process with both age-structure and environmental stochasticity to study bird populations. Lebreton (1982, 1990; also see Gosselin and Lebreton, 2000) considered
10.8. Other Stochastic Processes parameter estimation and demographic modeling of bird populations using a branching process model that included environmental variation and density dependence. Gosselin and Lebreton (2000) and Caswell (2001) noted the limited use of branching process models in ecology and conservation biology and provided excellent descriptions and examples of the approach. We suspect that readers of Caswell (2001) and Gosselin and Lebreton (2000) will devote increased attention to this class of models, and we thus expect to see increased use of branching process models to study animal populations over the next decade.
10.8.2. R e n e w a l Processes Renewal processes can be thought of as a generalization of the Poisson process. Recall that Poisson processes accumulate counts over a continuous time frame, with exponentially distributed interarrival times between Poisson events. Because interarrival times for a Poisson process are assumed to be independent, they constitute an exponential white noise process. Renewal processes generalize this situation, by allowing for independent and identically distributed interarrival times with nonexponential distributions. Using an earlier notation, we characterize a renewal process in terms of the interarrival time Y(i) between the i-lst and the ith occurrence in a process, assuming an arbitrary distribution F(Y) for interarrival time. Because process occurrences are independent, the process effectively "starts over" with each occurrence or "renewal." Let Z(k) represent the time until the kth renewal, i.e., k
Z(k) = ~ , Y(i) i=1
accumulates interarrival times for the first k renewals of the process. Letting b~ = E(Y)
it can be shown that average of the first k renewal times Z(k)/k converges to the mean renewal time tx for the process,
k--+oo
215
Furthermore, if N(t) is the number of renewals in the first t units of time, then the renewal rate N(t)/t converges to the inverse of ~, lim IN, t____))] t -*o"
1 -
~.
This same limit also applies to the mean m(t) = E[N(t)] of the number of renewals by time t: lim [mlt__~)] t-+oo
1 -- ~"
None of these results is particularly surprising. As the number of renewals increases, it is reasonable to expect that the finite average of renewal times will converge to the mean renewal time. It also is reasonable to expect the number of renewals per unit time, and the mean number of renewals per unit time, to converge to the reciprocal of the mean time per renewal. Though the expectation re(t) = E[N(t)] can be difficult to compute for certain underlying distributions of interarrival times, the renewal equation f t
m(t)
= F(t) + | d
m(t-
x)dF(x),
0
sometimes can be used to solve for re(t). It also can be useful in recognizing patterns of behavior in renewal processes. Example
Consider a process with alternating renewals between "on" and "off" conditions (e.g., feeding/nonfeeding behaviors), each with its own distribution of renewal times. The renewal equation for this process can be used to show that "on" and "off" conditions occur over the long term in the proportions of the distribution means:
E(X)
lim P(t) = t-+oo E(X) + E(Y)' where P(t) is the proportion of time spent in the "on" condition and E(X) and E ( Y ) a r e the mean renewal times for "on" and "off" renewals, respectively. The Euler-Lotka equation (Section 8.4), expressing population growth rate as a function of the life table birth and death parameters, can be derived as a renewal process (Sharpe and Lotka, 1911; Lotka, 1939; also see Caswell, 2001). Similarly, renewal equations have been applied to stage-based population projection models (Houllier and Lebreton, 1986), multisite projection models (Lebreton, 1996), and nonlinear agestructured models with density dependence (Tuljapurkar, 1987).
216
Chapter 10 Stochastic Processes
10.8.3. Martingales
for supermartingales. It is straightforward to show that
Martingales formalize the concept of a "fair game" over a discrete time frame. Specifically, a martingale is a stochastic process {Zt: t = 1, 2 .... } such that
E[Zt+I] >~ E[Zt]
E[Zt+I] <- E[Zt]
E[lZtll < oo
for all t > 0, and
E[Zt+l ] Zl, Z2, ..., Z t] -- Zt"
(10.17)
Equation (10.17) indicates that the expected process value at time t + 1 is simply the actual process value at time t, irrespective of the process history. For example, if Z t represents a gambler's fortune at time t, then his expected fortune at time t + 1 after his next gamble is simply the current value of his fortune, no matter what has occurred previously. Because the stochastic behavior of a martingale is independent of its past behavior, martingales satisfy the Markovian independence assumption and therefore are special cases of a Markov process. It is easy to show that
E[Zt+I]--
EEZt]
from the martingale condition above, and therefore E[Zt] = E [ Z l l for all t > 0. A derived process of some interest involves the time until some value of a martingale is attained. Thus, a r a n d o m time N for the process {Zt: t = 1, 2, ...} is determined by the r a n d o m variables Z 1.... , Z,, in that knowledge of Z1, ..., Zn is sufficient to k n o w whether N = n. For example, let N = n if n is the first occurrence in which Z t exceeds some value Zmin. If N can only take finite values, then it is said to be a stopping time, and the process -
-
Zt =
{Z t ZN
if if
for submartingales and
t --< N t > N
defines a stopped process. This essentially says that the process continues to vary stochastically over time until a condition on the r a n d o m values is met, and then it retains the last process value from that time forward. It is not difficult to show that a stopped process is also a martingale. It also is possible to define submartingales and supermartingales in a natural way, by replacing condition (10.17) with
E~Zt+l ] Zl, Z2, ..., Zt] >~ Z t for submartingales and
E[Zt+l ] Zl, Z2, ..., Z t] <~ Z t
for supermartingales. A large n u m b e r of derived processes can be recognized as martingales. For example, the sum of independent identically distributed r a n d o m variables with zero mean can be shown to be a martingale, as can the product of independent r a n d o m variables with unit mean. For St = X1 + "'" + Xt with E[X t] = 0 and var[X t] = r the process {Zt = S2t - to2: t = 1, 2, ...} is a martingale. If X(t) is a branching process with mean ~t, then X ( t ) / ~ t is a martingale. If X, Y1, ..-are arbitrary r a n d o m variables, then Z t = E[XIY 1, ..., Yt] for t = 1, 2, ... defines a martingale (known as a Doob-type martingale). It is not difficult to generate an unlimited n u m b e r of martingales based on independent identically distributed r a n d o m variables. Martingales also can be generated easily from the stochastic processes described above and are especially useful in analyzing Markov processes, Brownian motion, and r a n d o m walks. Martingales have been used to model capturerecapture experiments for closed populations in continuous time (see Chapter 14). In particular, m o m e n t estimators of abundance have been derived based on these models (Becker, 1984; Yip, 1989, 1991).
10.8.4. Stationary Time Series Here we mention a large and useful class of timeseries models that satisfy certain stationarity conditions in their behaviors. Stationary (or strongly stationary) time series processes are stochastically invariant, in that their distributions are not parameterized by time. Formally, a process {X(t): t -> 0} is stationary if for any given combination t I .... , t, of times the r a n d o m vectors [X(tl), ..., X(tn)] and [X(t I 4- s), ..., X(t n 4- s)] have identical distributions irrespective of the value s. A less stringent requirement, k n o w n as second-order (or weak) stationarity, requires only that process covariances be time invariant, i.e., cov[X(t), X(t + s)] must be independent of t. It follows that the first two moments of a second-order stationary process are temporally invariant, so that the covariance between X(t 1) and X(t 2) depends only on ]tI - t21. Gaussian processes can be used to illustrate the linkage between strong stationarity and second-order stationarity. Second-order stationarity manifests in the first and second moments, which parameterize a
10.8. Other Stochastic Processes Gaussian process because it is multivariate normal. Because the process is determined by its means and covariances, a second-order stationary Gaussian process is necessarily strongly stationary. Of course, most processes are not determined by their first two moments, so that weak stationarity does not guarantee strong stationarity. The advantage of stationarity in a process is that in order to predict process behaviors, one need know only the relative positions of process values with respect to time and not the actual times of their occurrence in the time frame. Thus, the same temporal sequencing of random variables anywhere in the time frame produces the same stochastic behaviors. In the following discussion we briefly mention two important classes of second-order time series models, the well-known autoregressive and moving average processes, that are applicable over discrete time frames.
10.8.4.1. Autoregressive Processes Let Z(O), Z(1), ... be a sequence of uncorrelated random variables with E[Z(t)] = 0 and
I var[Z(t)] =
0-2 1 -- q)2
0-2
if if
t = 0 t -> 1,
where q)2 < 1. Then the process {X(t): t = 0, 1, ...} defined by X(0) = Z(0) and X(t) = ~ p X ( t - 1) + Z(t)
(10.18)
for t >- 1, is called a first-order autoregressive process. An algorithm for implementing an autoregressive process updates the process state to X(t + 1) simply by multiplying the process state X(t) by q~ and adding a random term. It is straightforward to show that
t X(t) = ~_,
@t-iz(i)
i=0 and that
0-2q)s
cov[X(t), X(t + s)] = 1 -- q)2" Because E[X(t)] = 0 and the covariance is independent of process time t, it follows that {X(t) = 0, 1, ...} is second-order stationary. The covariance formula indicates that the statistical association between process values declines exponentially as the time s between values increases, with ~p controlling the rate of
217
decline. Of course, when s = 0 the covariance formula yields the process variance 0 "2/(1 - q~2). A straightforward generalization of autoregressive processes is obtained by allowing for lags of order greater than one, along with lag-specific weighting parameters. A general autoregressive process of order p is given by
P X(t) = ~ , q~iX(t-i) + Z(t).
i=1 Note that this expression reduces to Eq. (10.18) for the special case of p = 1 and q)i = q)" Autoregressive processes arise naturally in population dynamics through the consideration of density dependence, where vital rates (and hence population growth) for the period t to t + 1 are functions of abundance at time t (see brief discussion in Chapter 1). Thus, abundance at time t + 1 (Nt+ 1 o r log Nt+l), or population growth rate from t to t + 1 (Kt = Nt + 1/ N t or logK t) is modeled as a function of the abundances at time t and in previous periods ( N t, N t _ 1, ..., Nt_d). Royama (1977, 1981, 1992) presented general autoregressive models of population growth, describing them as "density-dependent" and "density-influenced" processes. Autoregressive models of population growth have been used extensively in the modeling of density dependence and as a basis for tests of density dependence (e.g., see Bulmer, 1975; Slade, 1977; Vickery and Nudds, 1984; Pollard et al., 1987; Wolda and Dennis, 1993, Dennis and Taper, 1994). However, it is difficult to obtain unbiased estimates of the parameters of autoregressive models using time series of population estimates. The source of the problem is simple: the sampling variances of abundance estimates reflect sampling and the uncertainty of the estimation process (see Part III of this book). Because abundance estimates/r t appear in the denominator of population growth rate estimates ~t = /Qt+l/fi4t, the sampling variance of/~t leads naturally to a negative sampling covariance between N t and ~t. Although this problem was identified some time ago (e.g., Kuno, 1971; Ito, 1972), it frequently has been ignored. In the simulation study of Shenk et al. (1998), it was concluded that sampling variation invalidated most of the tests for density dependence based on autoregressive models (Shenk et al., 1998). However, Viljugrein et al. (2001) used a Bayesian state-space modeling approach that accommodates sampling variation in an autoregressive population model, based on a time series of estimates of duck population size. The approach appears to work well and should prove useful in fitting such models in the future. In recent years, efforts have been made to fit autore-
218
Chapter 10 Stochastic Processes
gressive population models to time series data for animal populations for purposes other than the investigation of density dependence. One such use, described in Section 9.8, involves inferences about system characteristics based on general nonlinear autoregressive models. In particular, Section 9.8 contained a brief discussion of attractor reconstruction for the purpose of estimating Lyapunov exponents and system dimension. The numerical methods for attractor reconstruction use an autoregressive model of a system state variable (e.g., Takens, 1981; Cheng and Tong, 1992; Nychka et al., 1992), in this case, population size. An estimate of the dominant Lyapunov exponent in the reconstruction can be used to draw inferences about divergence or convergence of nearby trajectories in the attractor based on the behavior of the system (Turchin, 1993; Falck et al., 1995a,b). We note that by means of reconstruction of an attractor, autoregressive models can be used to draw inferences about the number of trophic interactions influencing population dynamics. For example, Stenseth et al. (1996) found evidence that most microtine populations can be characterized as two-dimensional systems and suggested that this dimensionality is consistent with density dependence and the simultaneous influence of rodent-specialist predators. Similar analyses with snowshoe hare (Lepus americanus) data from boreal forest areas of North America provided evidence of a three-dimensional system, indicating influences from density dependence, predation, and food plants (Stenseth et al., 1997). Analyses on Canadian lynx (Lynx canadensis) from the same region suggested two dimensions, indicating density dependence and the influence of prey populations (Stenseth et al., 1997). In both of these autoregressive modeling efforts, Stenseth et al. (1996, 1997) developed mathematical models of the relevant ecological interactions (e.g., density dependence, predation) and then rewrote parameters of the ecological models as functions of the coefficients of the autoregressive model. This work led to general inferences about system dynamics (e.g., about system dimension) from the more phenomenological (see Section 3.4.2) autoregressive modeling and more focused inferences based on a mechanistic, ecological reparameterization of these models. Finally, we note that Dennis et al. (1995, 1997) developed mechanistic models that included autoregressive parameters and fit these to time series data using methods of nonlinear time series analysis (e.g., Tong, 1990). The application of this work to flour beetles (Tribolium sp.) provides a nice example of the interplay between mathematical modeling and laboratory experimentation (Constantino et al., 1995, Dennis et al., 1997; also see Mertz, 1972).
Although the autoregressive modeling described above represents important efforts to investigate animal population dynamics, they still are hindered by reliance on time series of estimated, rather than true, abundance. The existence of sampling variances and covariances remains a problem that has not been dealt with in a completely satisfactory manner. The degree to which the conclusions from the cited analyses are influenced by sampling variation is unknown, but the potential problem is great. Efforts by Viljugrein et al. (2001) and others to develop methods to deal with this problem likely will lead to important contributions in population ecology.
10.8.4.2. Moving-Average Processes Let Z(O), Z(1), ... be a sequence of uncorrelated random variables with E[Z(t)] = ~ and var[Z(t)] = ~2, and consider the average X(t) =
Z(t) + Z(t - 1) + ... + Z(t - k) k + 1
(10.19)
for t --- k. An algorithm for implementing a moving average process simply updates the value X(t) to X(t + 1) by X(t + 1) = X(t) +
Z(t + 1 ) - Z ( t k+l
k)
It can be shown that E[X(t)] = p, and cov[X(t), X(t + s)] =
(k+l-s)~r 2 (k 4- 1) 2 0
I
if if
O<sk.
Because the covariance formula depends on the time s between process values but not on time t, the process is second-order stationary. Process values X(t) and X(t + s) that are less than k units of time apart share some of the same random variables in {Z(t): 0, 1, ...}, so the stochastic association between them is nonzero. However, this association weakens linearly as s increases [because fewer random variables are shared between X(t) and X(t + s)], until the association vanishes for process values farther apart than k units of time. As with autoregressive processes, it is possible to generalize moving-average processes by allowing for lag-specific weighting parameters. A general movingaverage process of order k is given by k
X(t) = ~
O i Z ( t - i).
i=0
Note that this expression reduces to Eq. (10.19) for the special case of 0 i = (k 4- 1) -1. A further generalization combines an autoregressive process and a moving-
10.8. Other Stochastic Processes average process into a mixed autoregressive-movingaverage (ARMA) process of the form p
219
and t-1
k
X(t) = ~ , ~ i X ( t -
i) + Z(t) + ~ , O i Z ( t -
i=1
Nt = No l-I ~'i, i).
i=0
i=1
A process of this form is often referred to as an ARMA process of order (p, k) (Box and Jenkins, 1976). Applications of ARMA models include the modeling of environmental processes and factors influenced by environmental variation. Some uses of such processes in stochastic demographic theory are briefly discussed below.
where N t is abundance at time t and )~t is the finite rate of population increase from t to t + 1. Environmental variation is assumed to be iid, so that kt is drawn from a stationary, nonnegative probability distribution. As noted by Lewontin and Cohen (1969), there are at least two ways to think about computing an average growth rate of the population. One way is to consider the growth of the mean population:
t l)
10.8.4.3. Demographic Stochasticity and Population Projection
With the exception of models for branching processes in random environments (Section 10.8.1), we have focused thus far in this chapter on demographic stochasticity and variation associated with the binomial nature of birth and death processes (a distinction was made between demographic and environmental stochasticity in Section 10.8.1). Because environmental stochasticity often is discussed in the context of stationary time series, in this section we highlight some efforts that have been made to incorporate environmental stochasticity into population projection models. The bulk of this work focuses on environmental stochasticity only, in the absence of demographic stochasticity. We restrict attention here to discrete-time models, although stochastic versions of continuous-time models have also been developed (e.g., see Goel and RichterDyn, 1974). Among the approaches to modeling environmental stochasticity, Caswell (2001) lists three as especially useful. The first approach simply uses independent and identically distributed (iid) sequences of random variables as environmental drivers, whereby the state of the environment at any time t is drawn from a specified distribution. The second approach considers a finite number of environmental states, with changes in state modeled as a Markov process (Section 10.3). Although many applications involve stationary Markov processes, it is possible to consider time-varying processes as well. The third approach uses autoregressive moving-average models as described above. In this case, the environment is modeled as a continuous state variable that is dependent on past states of the system. The simplest discrete-time model of a population in a stochastic environment may be that of Lewontin and Cohen (1969): X t +l --- ~.tX t
Because of the iid nature of the population rates of growth ki, the above expression can be rewritten as
tt-1 E(Nt) = N o ( I [
) E()~i)
= NoE(~ki) t
i=0 or
log E(n t) = t log E(Ki) + log N 0. Caswell (2001) showed with a numerical example that log E(K i) predicts growth of the mean population reasonably well. The other view of average growth rate is to use time-averaged growth of individual realizations of population size trajectories. Population size at time t can be written as: t-1
log N t = log N O + ~
log
~ki,
i=0
so that the time-averaged growth rate for such a trajectory is log N t - log N O lt-1 t = t-/~0 log
)k i.
The right-hand side of the above expression is simply the arithmetic mean of the logarithms of the realized )~i. Because the )ki a r e iid, as t becomes large, this quantity converges to E(log )ti): i.e., lim log(Nt) = E(log t~oo
)ki).
t
Thus, one measure of average growth, log E()~i), represents the growth of the mean population, whereas the other, E(log ~,i), represents the average rate at which
Chapter 10 Stochastic Processes
220
individual realizations will grow. By Jensen's inequality (Mood et al., 1974), E(log hi) -< log E(Ki); that is, the mean population size grows at a higher rate than most individual realizations. As noted by Lewontin and Cohen (1969), there are many cases where E ( N t) may grow infinitely large, yet each population may exhibit a very high probability of going extinct. Thus, the metric E(log h i) associated with individual realizations of population growth is more appropriate for population dynamics. Although the Lewontin-Cohen model development assumes that h i are iid, the inference that E(log h i) provides a good measure of population growth rate applies to more complicated stationary stochastic processes, such as Markov chains and ARMA models (e.g., Tuljapurkar, 1990; Caswell, 2001). The model of Lewontin and Cohen (1969) is relatively simple, yet some of their inferences about population dynamics in a stochastic environment also hold for more complicated models. For example, Cohen (1976, 1977a,b, 1979) and Tuljapurkar and Orzack (1980) considered population dynamics in terms of the product of a stochastic sequence of matrices. Denote as a i the projection matrix for environment i, representing one of k possible sets of environmental conditions. To obtain a realization of the population age-stage structure at some time t (denote as nt), we premultiply an initial vector n 0, by a random sequence of projection matrices: Fit "- [ a t - l A t - 2 " " 6 0 ] / / 0
9
If population size N t is simply the sum of the elements of the age-stage vector n t obtained as above, it can be shown that lim l~ t-,oo
t
-- E(log h s)
where h s is known as the stochastic growth rate and characterizes most realized population trajectories (see Cohen, 1976, 1977a,b; Tuljapurkar and Orzack, 1980; Tuljapurkar, 1990; Caswell, 2001). Computation of h s can be accomplished using results of a long simulation: lo---ghs = log N T - log N O T where log h s is a maximum likelihood estimator of log h s (Heyde and Cohen, 1985; Cohen, 1986). The stochastic growth rate can also be computed analytically in some cases (Cohen, 1977b; Tuljapurkar, 1990) and can be approximated as well (Tuljapurkar, 1982b, 1990).
As was the case for the Lewontin-Cohen model, it is possible to consider other measures (than log h s) of average growth rate for populations with random projection matrices. Cohen (1977a,b, 1979) and Tuljapurkar (1982a, 1990) showed how to compute the growth rate of mean population size (denote this growth rate as ~) lim log E ( N t) = log ~, t--,oo t from population projection matrices. In the case of iid environments, bL can be computed as the dominant eigenvalue of the average projection matrix (A) m
bL = h~a). Variances and confidence intervals for the estimation of average growth rates and population size are available (Heyde and Cohen, 1985; Cohen, 1986; Cohen et al., 1983; Caswell, 2001). In addition, stochastic analogs of deterministic life history quantities (see Chapter 8) such as reproductive value, stable age distribution, sensitivity, and elasticity have been proposed and investigated as well (e.g., see Tuljapurkar, 1990; Caswell, 2001). Bierzychudek (1982) and Cohen et al. (1983) presented two of the first examples of the use of these methods with stochastic matrix models of plant and animal populations, respectively, and Lee and Tuljapurkar (1994) considered the development of shortterm forecasts of human populations based on stochastic demographic models. The past decade has brought a number of examples [e.g., see Doak et al. (1994) and review in Caswell (2001)]. Much of stochastic demographic theory is based on the assumption of stationarity (see above, in this section) or else assumes only demographic stochasticity (see Section 10.8.1). However, short-term forecasts must be able to account for possible changes in vital rates. Lee and Tuljapurkar (1994) used time-series analyses to estimate trends in vital rates with an ARMA model and in turn used these vital rates to project changes in population size and stage structure.
10.9. D I S C U S S I O N From the preceding sections it should be clear that the field of stochastic processes has much to offer biological modeling and investigation. Indeed, stochastic processes can play a natural role in the modeling of animal populations, by providing a convenient way to introduce environmental variation and other stochastic elements into a population model. For example, one might think of a population trajectory as consisting of
10.9. Discussion two parts: (1) a deterministic component that essentially tracks the population mean through time, and (2) a stochastic component that allows for random variation about the mean, as modeled by a stochastic process. For example, a renewal process can be used to model the occurrence of random environmental events over continuous time, whereas Markov chains might be used in combination with dynamic population models to project random population change over discrete time. We illustrate the application of stochastic processes in later chapters, with special emphasis in Part IV on the use of Markov decision processes for dynamic decision-making under uncertainty. As potentially useful as they are, stochastic processes present a real challenge to most biologists, not least because of the mathematical complexities involved, but also because of the scope of the subject matter. We have presented only a few classes of stochastic processes and only a few of the issues that can be addressed for processes in each class. We have restricted attention here to certain well-known processes with obvious biological applications, omitting discussion about queuing processes, nonstationary time series processes, random walks, stochastic gaming, stochastic order relations, and a host of other processes with potential applicability. Nor have we dealt comprehensively with stochastic features and process derivations for the stochastic processes that are included here. For example, we have touched little if at all on such issues as delayed renewals, Markov renewals, stopping times, process approximations, process control (except for Markov processes), asymptotic process properties, and many other interesting and important features. The breadth and variability of the field of stochastic processes arise from the virtually limitless number of ways of defining linkages among random variables
221
across a time frame. Indeed, there are as many stochastic processes as there are relationships among timeindexed random variables. For example, a profusion of mathematical structures could be imposed on the moments of a joint distribution of process variables, with each corresponding to a stochastic process with its own properties and behaviors. Only relatively few of the possible processes that could be considered have been, no doubt in part because of formidable analytic complexities that can arise. As with mathematical models, it is important to limit this complexity to the extent practicable, by including only those stochastic features thought necessary to capture important stochastic associations. As in mathematical modeling, the application of stochastic processes in an investigation requires the identification of an appropriate stochastic structure and the estimation of means, variances, and other process parameters. Basically, model identification and estimation are required for stochastic processes, just as they are for the mathematical models of Chapters 7 and 8. Consider, for example, the sequential sampling of a biological system over some period of time. Assuming no statistical association among samples across time, sequential samples can be thought of as realizations of a white noise stochastic process, the distribution for which must be identified and parameterized via estimation of the distribution moments and other parameters. Repeated measures and other sampling approaches that induce a correlation structure across time periods also can be treated in terms of stationary processes, but with more complicated probability structures and greater numbers of parameters to estimate. We note in closing that there is an extraordinarily large, and often quite complex, literature on the subjects of process identification and estimation, which we leave to the interested reader to explore.
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CHAPTER
11 The Use of Models in Conservation and Management
11.1. DYNAMICS OF HARVESTED POPULATIONS 11.1.1. The Concept of Sustained Yield 11.1.2. Maximum Sustained Yield 11.1.3. Compensatory and Additive Mortality 11.1.4. Methods for Determining the Impacts of Harvest 11.2. CONSERVATION AND EXTINCTION OF POPULATIONS 11.2.1. Population Size and the Risk of Extinction 11.2.2. Extinction and Metapopulations 11.2.3. Models of Patch Dynamics 11.3. DISCUSSION
11.1. D Y N A M I C S O F
HARVESTED POPULATIONS 11.1.1. The Concept of Sustained Yield Management of many renewable resources is based on the idea that a portion of a resource stock (forests, stocks of fish, wildlife populations) can be removed without causing long-term resource depletion. This idea has become enshrined in the concept of a sustained yield that can be taken year after year without causing the resource stock to decline (Beverton and Holt, 1957; Clark, 1976; Errington, 1945; Caughley, 1977). Sustained yield can be expressed in discrete time by the generic harvest model
N(t + 1) = N(t) + f(N, H, t),
In this chapter we revisit some basic concepts and certain of the models covered earlier, as they apply to problems in the management and conservation of populations. In what follows we address two important areas of population management, the first of which involves managing harvest of stocks of animals. In particular, we discuss the sustainability of harvested populations and explore the linkage between harvest rate and population size under sustained yield. Closely related is the concept of compensatory mortality, which we describe in terms of the association between harvest rate and annual mortality. The second issue of concern involves the analysis of animal populations that are vulnerable to extinction, which we describe in terms of stochastic models that explicitly allow for the possibility of extinction.
where f(N, H, t) is the population annual growth increment and H is the annual harvest. On assumption that annual growth can be disaggregated into harvest and population growth, this model can be expressed as
N(t + 1) = N(t) + f(N, t) - H,
(11.1)
from which it is easy to see that population size is sustained by a level of harvest H* that just matches annual growth:
H* = fiN, t).
(11.2)
The growth function f(N, t) in Eq. (11.1) can be expressed as
f(N, t) = r(N, t)N
223
224
Chapter 11 Conservation and Management Models
with r(N, t) the per capita rate of growth, so that the maximum permissible harvest rate sustaining a population of size N is H*/N
= h*(t) = r(N, t).
This equation highlights the rather obvious fact that a population is sustained in equilibrium at size N whenever per capita harvest rate coincides with per capita growth. On further assumption that annual growth is dependent on population size but not on time, the growth function f ( N , t) can be written as f(N), with population dynamics described by N ( t + 1) -- N ( t ) + f i N ) - H,
(11.3)
with equilibrium harvest rate h* = f ( N ) / N
= r(N).
(11.4)
From Eq. (11.4) the sustainable yield h* is dependent on population size, in that the harvest level necessary to sustain a population varies with the size of the population. Conversely, population equilibrium depends on harvest rate, in that a given harvest rate h induces a particular equilibrium population size. If monotonic in N, the function r(N) in Eq. (11.4) possesses an inverse, which allows one to express the equilibrium population size for harvest rate h as N * = r-l(h). The point here is that a particular harvest rate induces an equilibrium in the absence of harvest population size, just as a targeted equilibrium population size requires a particular harvest rate to sustain it. Analogous forms apply for continuous models. Thus, a harvest model in continuous time is dN/dt
= f( N , t) - H
= Jr(N, t) - h(t)]N(t), with f ( N , t) = r(N, t ) N the instantaneous rate of growth of the population in the absence of harvest and h(t) the instantaneous per capita harvest rate at time t. Thus, the equilibrium condition d N / d t = 0 is satisfied by a harvest rate of h*(t) = r(N, t).
(11.5)
As above, the assumption that per capita growth is autonomous but dependent on population size leads to an equilibrium harvest rate of h * = r(N).
There are certain conceptual and practical difficulties with sustained yield as described above, especially as concerns discrete-time models. For example, many populations change over the interval [t, t + 1], and in consequence, the growth function f(N) also varies over the interval. However, the growth function describes change over the interval It, t + 1] in terms of a single population size for that interval. The corresponding biological notion is that growth (through reproduction or migration) occurs at a single point in time in the interval, an assumption that may or may not be appropriate for a particular biological situation. Another potential difficulty is that harvest frequently is assumed in discrete-time models to occur at a single occasion in the interval [t, t + 1]. Two points are noteworthy here. First, the timing of a discrete harvest event during [t, t + 1] can have consequences as to harvest yield and population response (Table 11.1). Second, harvest usually is seasonal, often occurring continuously over some part of the year. Thus, it often is necessary to model the occurrence of harvest during a part of the year, either by changing the definition of the time intervals [t, t + 1] to capture the seasonality of harvest or by replacing harvest rate h with an average harvest rate that accounts for variation over [t, t + 1]. A more fundamental problem is the potential for environmental and stochastic variation in the growth function. Assuming population change is of the form in Eq. (11.3), it is possible to determine a stationary harvest level h* that maintains the population at some specified size N. However, stationarity is sacrificed in the more general case of Eq. (11.1) with nonautonomous growth function f ( N , t). Under these conditions an equilibrium population size can be maintained only if one varies the harvest in accordance with Eq. (11.2), in which case the concept of a sustained yield that is stationary over time ceases to be applicable. Because it can be induced by environmental variation, biological interactions such as predation and competition, and other stochastic influences, temporal variation in the growth of populations is widespread. In this situation, the rather restrictive condition of an autonomous population growth can limit the usefulness of the concept of sustained yield for analysis of harvested populations.
11.1.2. M a x i m u m Sustained Yield Because sustained yield is defined by Eq. (11.5) in terms of population size, wherein different values of h* correspond to different population sizes, it often is useful to identify the maximum stationary harvest (and the corresponding population size) that is possible. Assuming autonomous growth as in Eq. (11.3), the
11.1. Dynamics of Harvested Populations TABLE 11.1
225
Illustration of Effect on Yield H of Differential Timing of Harvest a Population size
Month
Unharvested population
Harvest in month 1
Harvest in month 5
1
1000 b
1000 - 167 = 833
1000
2
976
813
976
3
953
794
953
4
930
775
930
5
908
757
908 - 152 = 756
6
887
739
738
12
768
640
640
138
750
625
625
13 c
750 + 750(0.6) = 1200
625 + 625(0.6) = 100
625 + 625(0.6) = 1000
a For a population with growth rate k = 1.20 and initial size N = 1000 in the spring just after reproduction. Annual birth rate is assumed to be b = 0.6, with birth occurring at the beginning of each year. Annual mortality rate is d = 0.25 in the absence of harvest, with mortality occurring throughout the year. C o l u m n 1 corresponds to m o n t h of the year, column 2 records population size in the absence of harvest, and column 3 corresponds to harvest in the first m o n t h at a rate of h = 0.167, producing a harvest of H = 167 individuals. Column 4 corresponds to harvest in the fifth m o n t h at the same harvest rate, producing a harvest of H = 152 individuals (from Caughley, 1977). Harvest occurs at the beginning of each m o n t h before nonhunting mortality. b After recruitment. c Before recruitment.
maximum sustained yield (MSY) can be determined simply by finding the zeros of the derivative of f(N) when it is differentiable: dH/dN
= f' (N) = O,
from which the optimal sustainable population size N*, and in turn the maximum sustained yield H* = f(N*), can be found. A typical application of MSY is described in terms of the model N ( t + 1) = N ( t ) + rN(t)[1 - N ( t ) / K ]
N = K(1 - h / r )
(11.6)
dH/dN
= r-
2r(N/K)
from Eq. (11.6), which gives N* = K/2
and a value for MSY of H* = rN*(1 - N* / K)
or
= (r/2)N* = rK/4.
h(t) = H ( t ) / N ( t )
(11.7) = r[1 - N ( t ) / K ] . This is the largest harvest rate that can be imposed on a population of size N without causing the population
(11.8)
that can sustain it and thus to a specific level of sustained harvest H = h N . It is clear from Eq. (11.8) that the population can be sustained in equilibrium for any value h that is less than the intrinsic growth rate r, which raises the question of which value corresponds to the largest sustainable harvest. The maximum sustained yield for this model is given by
=0
- H(t)
with logistic growth and postreproduction harvest. Population equilibrium is given by H ( t ) = rN(t)[1 - N ( t ) / K ]
to decline. From Eq. (11.7) a given harvest rate h(t) corresponds to a specific equilibrium population size
Then the optimal per capita harvest rate is h* = H * / N * = r/2.
226
Chapter 11 Conservation and Management Models
Thus, for this particular model, the optimal per capita harvest rate h* is one-half the intrinsic rate of growth and is not influenced by the carrying capacity K. On the other hand, the optimal equilibrium population size N* is one-half the carrying capacity K and is not influenced by the intrinsic rate of growth r. Finally, the m a x i m u m sustainable harvest H* is given by the product h'N*, or H* = rK/4. The m a x i m u m sustainable yield maintains the population in equilibrium at a size that is one-half the carrying capacity, where the population grows as rapidly as possible. These relationships are displayed in Fig. 11.1 for a population with intrinsic rate of growth r = 0.1 and carrying capacity K = 500 individuals.
H
"o
m
It is of course possible to incorporate more realism into the logistic model by including time lags, various forms of stochasticity, age structure, and other features (see Chapters 7 and 8). Nonetheless, the above model captures the biological ideas that frame the concept of MSY: 9 Populations have the potential to produce more offspring than the environment can sustain. 9 At low abundance, populations tend to grow rapidly, with the rate of growth slowing as population size increases. 9 The population asymptotically approaches a level at which it sustains itself.
*
a
8
.m
6 c-
b
'A"
0.100
0.075 (1)
(D
;=, o.o5o c-
0.025
0.000
N 250
~
500
population size
FIGURE 11.1 Yield relationships under logistic model with intrinsic rate of growth r = 0.1 and carrying capacity K = 500. (a) Harvest yield as a function of equilibrium abundance. Maximum sustainable harvest is H* = 12 for a population of size N* = 250. (b) Per capita harvest as a function of equilibrium abundance. Harvest rate for maximum sustainable harvest is h* = 0.05.
11.1. Dynamics of Harvested Populations 9 The population can be maintained below that level by removing the growth increment. For a logistic population, the MSY is achieved by maintaining population size at K / 2 , the point at which population growth is maximum. This result is of course specific to the logistic model, and other models can be expected to exhibit different equilibrium values. We note that an analogous treatment is possible for the continuous model dN/dt
= rN(t)[1 -
N(t)/K]
-
H(t),
with H ( t ) now representing instantaneous harvest. Equilibrium for this model is defined by d N / d t = 0, which requires that H(t) = rN(t)[1 -
N(t)/K]
227
and harvest rates. The additive mortality hypothesis was defined there by an approximately linear relationship between survival and harvest, with strict additivity producing an equivalent decrease in survival rate when harvest rate is increased. On the other hand, compensatory mortality was defined by changes in nonharvest sources of mortality that compensate for changes in harvest mortality, so that within limits, survival rate remains unchanged as harvest rates increase. Here we describe the compensatory and additive mortality hypotheses in terms of the relationship of harvest to mortality (rather than survival). Assume that harvest mortality occurs at the beginning of the year and is followed by nonhunting mortality. Then a simple linear model relating nonharvest mortality m ( t ) in the presence of harvest to mortality from harvest h(t) is given by
or
m(t) = m o + h(t)
= r[1 -
~h(t),
(11.9)
N(t)/K].
But this is the same equilibrium condition as for the discrete model, with the same formulas (properly interpreted in terms of instantaneous rather than discrete rates) for maximum sustainable yield and equilibrium population size.
where m0 is natural mortality in the absence of harvest, with - 1 -< ~ -< - ~ m 0 and oLclose to 1 (e.g., oL = 0.95) (Anderson and Burnham, 1976). The value f~ = - 1 represents complete compensation, with ~ = -oLm0 representing strict additivity. Thus, we have re(t) = m o -
11.1.3. Compensatory and Additive Mortality The concept of compensatory mortality was introduced previously in Chapters 3 and 8. Recall that under the compensatory mortality hypothesis (CMH), increasing harvest rates are compensated by densitydependent changes in nonharvest mortality factors. Thus, when harvest rates are reduced, densities are higher than they otherwise would have been, and natural mortality rates consequently increase. Conversely, as harvest rates increase, densities are lowered and natural mortality operates at a reduced rate. For complete compensation, there is no change in the annual mortality rate with changes in harvest rate, up to a threshold harvest rate, the maximum value of which is the mortality rate in the absence of hunting (Anderson and Burnham, 1976). Opposing the CMH is the additive mortality hypothesis (AMH), which presumably operates in the absence of density-dependent mechanisms that would affect nonharvest mortality. Thus, the AMH implies that as harvest mortality increases, total annual mortality increases proportionately, thereby producing a linear relationship between harvest rate and annual mortality. Recall that these relationships were described in Chapter 8 in terms of the relationship between survival
h(t)
(11.10)
under the CMH, which essentially says that within limits the sum of hunting and nonhunting mortality remains constant: m(t) + h(t)=
m o.
Note that Eq. (11.10) describes a linear relationship between nonhunting mortality and hunting mortality, with a slope of -1. Because the CMH operates through compensatory decreases in nonharvest mortality m(t), the amount of compensation cannot exceed m0, and this value provides an upper bound for the threshold C, beyond which additional harvest mortality becomes additive. On the other hand, Eq. (11.9) with oL = 1 yields m(t) = m o -
moh(t)
(11.11)
= m0[1 - h(t)] under the AMH. This expression is intuitively reasonable, because an animal must survive harvest mortality {with probability [1 - h(t)]} in order to have a chance of dying from nonharvest mortality. As with Eq. (11.10), this equation describes a linear relationship between hunting and nonhunting mortality. However, the slope of the relationship in Eq. (11.11) is - m 0, which is greater than -1. Thus, the reduction in nonhunting mortality attendant to increases in hunting mortality is less than that occurring beyond the threshold C under the CMH.
228
Chapter 11 Conservation and Management Models
The relationship between m ( t ) and h(t) is summarized under both the CMH and the AMH in Fig. 11.2a. Note that m ( t ) declines with increasing h(t) under both hypotheses, as a result of competition between risks associated with harvest mortality and nonharvest mortality (animals killed by harvest cannot be lost to other sources). However, this decline does not represent compensation, which is engendered by densitydependent mortality mechanisms (Anderson and Burnham, 1976; Nichols et al., 1984d). The region be-
m(t)
tween the two curves represents the range in potential compensation, from completely compensatory (lower curve) to completely additive (upper curve). By definition, total annual mortality is the sum of mortality from both harvest and nonharvest sources; that is, 1 -
S(t)
=
h(t)
m(t),
so that S(t) = 1 -
h(t) -
m(t)
= 1 -
h(t) -
[m 0 + f~h(t)].
Substitution of
m(t)
from Eq. (11.10) yields S(t) = 1 -
mo
+
mo
under the CMH [for h(t) < C], whereas substitution of m(t) from Eq. (11.11) yields S(t)
h(t) C
= S011 - h(t)]
(11.12)
under the AMH. Thus, the effect of the additive hypothesis is essentially to add a harvest component to nominal mortality m 0, thereby decreasing the survival rate as in Eq. (11.12) (Fig. 11.2b). On the other hand, the CMH leaves unchanged the survival rate over a range of values for harvest rate up to the compensation limit, with declines thereafter as harvest rate increases:
s(t) S(t)
=
1
So
0.0
(11.13) SO
0.2
0.4
0.6
0.8
1.0
h(t)
C
F I G U R E 11.2 Hypotheses of compensatory (CMH) and additive (AMH) mortality. (a) Relationship between natural mortality m and harvest rate h. Under both hypotheses, m declines from m 0 (natural mortality in absence of harvest, here taken as 0.5) as h increases, because of competition between these sources of mortality. Under the CMH, the decline is steeper and is sufficient to balance h. (b) Relationship between annual survival S and harvest mortality h. Under the AMH, each increment in h is additive to overall mortality, resulting in a linear decline in annual survival. Under the CMH, there is complete compensation up to the threshold C m i n this example C = 0.4; the maximum potential for compensation, Cmax = m0, is 0.5 in this example.
h(t) ~- C.
Thus, compensatory harvest has no effect on population dynamics if the harvest rate is sufficiently small, but reduces survival if the harvest rate is in excess of C. The relationships in Fig. l l.2a represent the results of a mechanism (density-dependent, compensatory mortality) by which a relationship between changes in harvest rates and survival rates (Fig. 11.2b) arises. The contrast between the phenomenological and mechanistic modeling of compensation can be clarified by a slight recasting of the definition of annual survival: S(t)
= 0t[1 - h(t)]
for 0 -< h(t) <- 1, where 0t is survival from nonharvest sources, and may vary according to density or abundance. If this compensating variation in nonharvest survival is thought to occur immediately after the harvest period (e.g., in the wintering period for waterfowl), then a reasonable model for 0 t may be ea+bN(t)[1 - h(t)] O t - - 1 + e a+bN(t)[1 - h ( t ) ] '
11.1. Dynamics of Harvested Populations where N(t)[1 - h(t)] represents the n u m b e r of animals surviving the harvest period and thus influencing nonharvest mortality. From this model one can deduce compensatory relationships that are very similar to that portrayed in Eq. (11.13) and Fig. 11.2b, except now the strength of the compensatory relationship varies according to the initial (preharvest) population size. In the special case for which nonharvest mortality is independent of density, the coefficient b in the expression is zero, and the expression simplifies to
229
m(t) a
m~I
S(t) = 0[1 - h(t)] for 0 -< h(t) <- 1, which is equivalent to the additive model in Eq. (11.11). Values of the coefficient [3 < 0 indicate density dependence and reflect a degree of compensation.
11.1.3.1. Partial Compensation Model with Variable Thresholds The C M H and A M H hypotheses represent extremes, in that the density-dependent mechanism for the C M H is thought to compensate perfectly for changes in harvest rates up to the theoretical threshold C = 1 - S0. On the other hand, it is assumed under the A H M that nonharvest mortality is independent of density, so that harvest is not compensated by changes in nonharvest mortality. These hypotheses provide logical extremes of a gradient of possible responses to harvest. However, it is perhaps more reasonable to expect compensation, if it occurs, to be less than complete, with the strength of compensation depending on both life history attributes and environmental conditions. Under a partial compensation hypothesis (PCH), it is assumed that below a threshold C, compensation occurs, but at a level so as not to compensate completely for harvest mortality. Thus, annual survival declines with increasing harvest, but not as much as if there were no compensation (Fig. 11.3). This hypothesis is a special case of Eq. (11.9) with - 1 < [3 < - m 0. We illustrate the PCH for a case where 13 = -0.75, intermediate between complete compensation ([3 = - 1 ) and complete additivity (~ = - m 0 = -0.50). PCH implies that there is a density-dependent relationship between nonharvest mortality and abundance, but that this relationship is not as strong as under complete compensation, and therefore is inadequate to balance completely changes in harvest mortality (Fig. 11.3b). Both the slope of the compensatory relationship and the threshold beyond which harvest mortality must be additive m a y depend on the life history of the animal (Patterson, 1979; Conroy and Krementz, 1990). For example, long-lived animals have a low natural mortality rate in the absence of harvest, so that the total a m o u n t
h(t)
s(t) So[
b
MH
0
C
1
h(t)
FIGURE 11.3 Hypothesesof partially compensatory (PCH) and additive (AMH) mortality. (a) Relationship between natural mortality m and harvest rate h. Under both hypotheses, m declines from m0 (natural mortality in absence of harvest, here taken as 0.5) as h increases, because of competition between these sources of mortality. Under the PCH, the decline is steeper, but not sufficient to balance h completely. (b) Relationship between annual survival S and harvest mortality h. Under the AMH, each increment in h is additive to overall mortality, resulting in a linear decline in annual survival. Under the PCH, there is partial compensation up to the threshold C.
of possible compensation is lower than for shorter lived animals, where m 0 is higher (recall that m 0 provides an upper b o u n d for C; i.e., C - m0). However, socalled K-strategists, which have a lower threshold with respect to harvest mortality, may also have stronger density-dependent mortality responses (and thus stronger compensation) below that threshold than rstrategists (Conroy and Krementz, 1990). In addition, even for species in which compensation might be theoretically strong, temporally varying environmental conditions can override density dependence in some years, effectively creating variable thresholds to harvest mortality (Conroy and Krementz, 1990).
230
Chapter 11 Conservation and Management Models
It is important to remember that the AMH, CMH, and PCH relate only to the relationship between harvest and mortality rates. Because population growth is determined by reproductive rates as well, it is possible for increases in harvest rates to be balanced (i.e., compensated for) by increasing reproductive rates, even if mortality operates according to AMH (i.e., there is no mortality compensation). Indeed, many arguments for compensation in large herbivores focus on the reproduction side of life histories, often suggesting that increased population levels (e.g., in populations subjected to no or low harvest pressure) result in depression of birth rates because of nutritional stress or other factors (Fowler, 1987; Gaillard et al., 1998, 2000). Of course, similar arguments can be made for mortality rates, e.g., increased population levels result in increased starvation during wintering periods. Although much of the discussion regarding compensation in waterfowl has focused on mortality (Anderson and Burnham, 1976; Nichols et al., 1984d; Conroy and Krementz, 1990), temporary decreases in survival with increasing harvest rates (under additive mortality assumptions) still might not result in decreased population growth, because of density-dependent increases in reproduction rates (e.g., Nichols et al., 1984d). Finally, it is important to keep in mind that nearly all arguments regarding compensation refer to the relationship between per capita rates of mortality from harvest and other causes, and corresponding rates of population growth. Knowledge of the absolute numbers of animals harvested or dying from other causes is inadequate to make inferences about the overall population impacts of harvest, unless (1) these statistics can be used to compute per capita rates, and (2) the relationship between rates of harvest, rates of other mortality, and overall rates of population growth can be established.
11.1.4. Methods for Determining the Impacts of Harvest Much of harvest management is based on an underlying conceptual or mathematical model of the effect of harvest on population vital rates. While we advocate the use of models as a provisional basis for management, the literature is replete with examples in which models have been used in the absence of adequate empirical justification, with potentially serious impacts on resources (e.g., Conroy, 1993; Heppell and Crowder, 1996). Here we briefly examine some methods for evaluating the impacts of harvest on populations and thus for selecting an appropriate harvest model and estimating its parameters. This discussion is general, and a
full development must await the estimation methods developed in Part III and the optimal decision methods in Part IV, where the estimation and assessment of harvest impacts are treated in some detail. 11.1.4.1. Observational Studies Most of the studies dealing with the impact of harvest investigate empirical relationships between harvest rates and population growth rates, birth rates, and death rates based on time series of historical data. We term these observational studies (Section 6.8.2), to distinguish them from experiments in which a deliberate attempt is made to manipulate harvest rates or population densities under design conditions (see Chapter 6). For example, band-recovery methods have been used extensively to investigate the effects of hunting on survival of waterfowl (e.g., Anderson and Burnham, 1976; Nichols and Hines, 1983; Burnham et al., 1984) [see reviews in Nichols et al. (1984d), U.S. Department of the Interior (1988), and Nichols (1991b)]. Though these studies have provided good evidence for compensatory mortality in some species, the evidence for other species is incomplete. Even for the heavily investigated mallard (Anas platyrhynchos), inferences suffer from an overreliance on statistical tests based on attributes of additivity vs. compensation, rather than the testing of mechanistic hypotheses of density dependence (Nichols et al., 1984d; Conroy and Krementz, 1990). 11.1.4.2. Experimental Studies Because of the absence of direct manipulation and experimental controls (see Chapter 6), purely observational studies suffer from an inability to ascribe causation. There have been few studies in which harvest rates or population densities have been directly manipulated in order to observe the population response (e.g., Bartmann et al., 1992). Obviously, a manipulative study is feasible only under special conditions, which are especially difficult to meet with large mobile populations. Nonetheless, we believe that experimental approaches, perhaps under constrained designs, are needed to answer questions about the impacts of harvest (Anderson et al., 1987).
11.1.4.3. Adaptive Resource Management We advocate a particularly powerful method for scientific management, adaptive resource management (ARM) (Waiters, 1986). In ARM, the emphasis is placed on decision-making (e.g., the setting of harvest regulations) to reach a long-term resource goal (e.g., maxi-
11.2. Conservation and Extinction of Populations m u m sustainable harvest over the long term). But a key to ARM is that decision-making accounts not only for current resource goals, but also for the information needed to improve management in the future. Thus, information is gathered continually about system response to management as decisions are being made, and this information is used to revise understanding of the system processes and thus to improve decisionmaking. We are aware of only a few successful examples of the application of ARM, the most notable being its application in the harvest management of North American waterfowl (Johnson et al., 1993,1997; Nichols et al., 1995a; Williams and Johnson 1995; Williams et al., 1996). We consider ARM in detail in Part IV, where we highlight its use for waterfowl harvest management and the design of forest reserves.
231
11.2.1. Population Size and the Risk of Extinction 11.2.1.1. B i r t h Processes
A model for the probability of extinction can be developed from a simple Poisson process of birth. To see how, recall from Section 10.2 that there are essentially two ways in which a Poisson counting process N ( t + At) can assume a value of N: (1) no events are added to a count of N over [t, t + At], or (2) one event is added to a count of N - 1 over [t, t + At]. Thus, the event {N(t + At) = N} can be decomposed into {[N(t)=N-1]
and
[N(t+at)-N(t)=l]}
U {IN(t) = N] and ~N(t + at) - N(t) = 0]}. On assumption that increments are stationary and independent (see Section 10.2), the probabilities for these individual events produce the probability
11.2. C O N S E R V A T I O N A N D E X T I N C T I O N OF P O P U L A T I O N S In this section we focus on the persistence of populations at risk of extinction. We note that population harvest and persistence share common management concerns, in that ill-advised management strategies in either arena can be detrimental to the long-term maintenance of populations at desired levels. For population harvest and persistence alike, a generic problem is to identify management actions that contribute to long-term persistence at desirable levels. Absent a long-term goal that accounts for future consequences of present actions, short-term economic considerations can favor unsustainable exploitation and eventual extinction of a population (Clark, 1976). In the case of rare and endangered species, the threat of extinction is obvious by definition, and conservation goals for these species typically are described in terms of minimizing the extinction threat. A major difference between models of population dynamics for threatened species and those for harvested populations is the focus on small population sizes and the relationship of population size to the probability of extinction. Most harvested populations are large enough that the probability of extinction, at least in the short term, is negligible. Thus, management efforts often are focused on the population growth rate. Of course, all populations, whether large or small, always face the threat of extinction, because factors such as cataclysmic environmental change, demographic stochasticity, inbreeding depression, genetic drift, and other factors may result in increased mortality rates, decreased birth rates, or both.
P[N(t + a t ) = N] = P[N(t + at) - N ( t ) = 1] • P[N(t)= N - 1]
(11.14)
+ P[N(t + at) - N(t) = O]P[N(t) = N].
For a pure birth process with instantaneous birth rate b, the probability of exactly one occurrence in the small interval [t, t+At] is approximately bNAt: N(t)] = 1} = bN&t + o(&t),
P{[N(t + & t ) -
where o(At) is some value with a limiting magnitude that is of degree less than At: lim •
o(&t)
&t
-0.
We invoke the Poisson assumption that the probability of more than one occurrence in [t, t + It] is negligible when compared to the probability of a single event, P{ [N(t + &t) - N(t)]>l} = o(&t),
so that Eq. (11.14) can be expressed as PN(t + At) -- PN(t) = P N - I ( t ) [ b ( N -
1)At + o(At)]
- PN(t)[bN~t + o(At)],
with PN(t) = P [ N ( t ) = N]. Dividing this equation by &t and allowing A t e 0 , we get the differential equation P~(t) = - ( b N ) P N ( t ) + b ( N -
1)PN_I(t),
(11.15)
where P ~ t ) is the derivative of PN(t) with respect to t. On assumption that individuals are independent, a
Chapter 11 Conservation and Management Models
232
solution to Eq. (11.15) is given by the negative binomial distribution PN(t) -- ( N No-
11)-Nobt(1 e - e -bt)N-No,
(11.16)
Under the assumption of independence, the fate of a population of initial abundance N O is equivalent to the separate fates of N O populations, each with initial abundance of one. For each population it is possible to show that abundance at time t is distributed as
for N --- N o (Bailey, 1964), with mean
Po(t) = or(t)
E(N) = No ebt
PN(t) = [1 -- or(t)][1 -- f3(t)][f3(t)] N-1
and variance
for N > 0, where var(N) = Noebt(e bt - 1)
(see Appendix E for a discussion of the negative binomial distribution). Both the expected population size and the dispersion increase linearly with initial population size N O and exponentially with the birth rate b. Because the terms b and t occur together in Eq. (11.16), the net effect on population dynamics of a large birth rate over a short period of time is the same as a small birth rate over an extended period of time. Note that if N O = 1, the negative binomial in Eq. (11.16) reduces to the geometric distribution (11.17)
PN(t) = e-bt(1 - b - b t ) N - 1 .
On reflection this makes sense, in that the sum of independent geometrically distributed random variables has a negative binomial distribution (see Appendix E). Thus, the sum of N Orandom variables, each distributed according to Eq. (11.17), follows a negative binomial distribution as in Eq. (11.16).
oL(t) =
and b(e (b-d)t f3(t) =
The birth model is readily extended to processes that include both birth and death, wherein the state N can be reached from the state N - 1 by birth, from N + 1 by death, or from N in the absence of birth and death:
1) d
_
be(b_d)t_
(Feller, 1939; Bailey, 1964; Renshaw, 1991). Then the mean abundance for a population of initial size N o is given by E[N(t)] - No e(b-d)t,
with variance [b + d-] (b_d)t(e(b_d) t
var[N(t)] = 1,40[b _--C-~]e
~e(b_--d~
so that if death rate exceeds birth rate, [de (b-d)t t ~ ~ 1 7L6" "
: ( )N0
~
d
]
(11.20) (11.18)
+[N(t + M ) - N ( t ) = - 1 ] P [ N ( t ) = N + 1]. Given the same Poisson assumptions as above, Eq. (11.18) can be expressed in terms of the instantaneous birth and death parameters b and d as PN(t + At) - PN(t) = P N _ l ( t ) [ b ( N -
diN~
P0(oo)- lim|h-~_- ~
P[N(t + at)] = N] = P[N(t + at) - N(t) = 1]PIN(t) = N - 1] + P[N(t + A t ) - N ( t ) = O ] P [ N ( t ) = N ]
- 1)
From the geometric model, the probability of extinction by time t is given by (Renshaw, 1991) Po(t) =
11.2.1.2. Birth-Death Processes
d(e (b-a)t _ 1) be(b_d)t_ d
=1. A conclusion from Eq. (11.20) is that eventual extinction is certain if d -> b. On the other hand, if birth rate exceeds death rate, P0(~176= (~)N0.
(11.21)
1)At + o(At)]
- PN(t)[(b + d ) N k t +
o(at)] (11.19)
+ PN+I(t)[d(N + 1)At +
o(at)].
Dividing both sides of Eq. (11.19) by At and allowing At--+0 then yields the differential equation P;v(t) = b ( N - 1)PN_I(t) -- (b + d)NPN(t) + d(N + 1)PN+I(t).
From Eq. (11.21) it follows that the probability of extinction is always nonzero, i.e, no matter how large the population is or how fast it is growing, there is always a risk of eventual extinction. However, from Eq. (11.21) the probability of extinction decreases exponentially as a function of initial population size and growth rate. By rearranging the terms in Eq. (11.21), one can identify
11.2. Conservation and Extinction of Populations values of N 0, b, and d that ensure an extinction probability below some desired threshold probability P*: log(P*) No ~ log(d/b)"
(11.22)
11.2.1.3. Persistence Time A useful parameter is the expected time to extinction or persistence time TE, which can be derived from stochastic birth-death processes (Feller, 1939; RichterDyn and Goel, 1972; Goodman, 1987a). Given initial abundance N o and a maximum abundance (e.g., a carrying capacity) of Nmax, it can be shown that
N~ Nmax TE(N0)- ~ ~
~-'-1= y=x Y
l
[Yl-Il b(Z) Lz=x
'
where d(z) and b(z) are the mean per capita death and birth rates for a population with z individuals. If b and d are assumed constant, this model predicts an approximate relationship between Nmax and TE(Nmax) of 1 b TE(Nmax)-~ bN-max(~)
Nmax
(11.23)
(MacArthur, 1972). From this expression the expected persistence time can be seen to increase as an exponential function of the maximum abundance. Although the theoretical relationship in Eq. (11.23) suggests long and rapidly increasing times to extinction with increasing abundance, other factors may modify this relationship. For example, the model only considers demographic stochasticity, absent environmental variation or other sources of variability in birth and death rates. If the model is modified to incorporate environmental variation, the expected time to extinction increases much more slowly with increases in initial population size (Goodman, 1987a,b). Including irregular catastrophic events along with environmental stochasticity reduces the expected time to extinction even further. These results suggest that persistence cannot be assured simply by increasing the size of a single population (Goodman, 1987a; Shaffer, 1987) (Fig. 11.4). Finally, we note that none of the models presented here incorporates genetic effects such as founder effects, drift, and inbreeding, which may be particularly severe in small populations (Shaffer, 1981).
11.2.1.4. Minimum Viable Populations Population and genetic models have been used to explore abundance levels that are likely to sustain populations over extended periods of time. Recognizing that both the time period over which persistence is
233
evaluated and the definition of "likely" persistence (e.g., a persistence probability of 0.95) are necessarily arbitrary, the predicted abundance for a given model and stated criteria is called the minimum viable population (MVP). MVPs are potentially useful for qualitative comparisons of the effects of abundance on viability and for determining minimum population sizes for managing populations and their habitats. There are several difficulties with the MVP concept. First and perhaps most obvious is the fact that specifications of a desired time horizon and probability of persistence are subjective exercises. Conservation biologists frequently disagree about these criteria among themselves and especially with others who wish to extract goods and services from ecosystems. These disagreements are exacerbated by the adoption of extremely long time horizons (such as 1000 yr) and persistence probabilities of 0.99 and higher (e.g., Shaffer, 1987). It is readily seen in even in the simplest models [e.g., Eq. (11.24)] that moderate changes in persistence probabilities can produce dramatically different results in terms of a minimal abundance. In addition, many estimates of demographic parameters are inadequate, and the functional forms of the relevant biological processes and the sources of variability affecting them are poorly understood. For example, differing assumptions about the form of density dependence lead to very different predictions about viability for grizzly bears (Ursus horribilis) (Mills et al., 1996) and consequently to differing management policies for optimizing viability. In Part IV we consider species conservation as a decision problem containing (at least) three sources of uncertainty: (1) environmental uncertainty, (2) partial observability (sampling error in estimating abundance or demographic parameters), and (3) structural uncertainty (ignorance of the "true" functional form for population dynamics). We argue there that reducing uncertainty through the use of adaptive management leads naturally to better decision-making in the future.
11.2.2. Extinction and Metapopulations Habitat and environmental conditions that influence survival and natality typically are not uniform across landscapes, and environmental conditions (e.g., absence of an essential habitat) sometimes determine both the range and local distribution of animals. Even if environmental conditions are "suitable" at a particular location, animals may not be present there because no animal of the species has ever reached the area from other occupied habitats (e.g., isolation of island habitats). Conversely, habitats may be unsuitable but still occupied (at least temporarily), if nearby suitable habi-
234
Chapter 11 Conservation and Management Models
4000
3000 x
v
b--w 2000
a ...~
1000 .-.-9 ""
0
""
"~
""
"""
""
..~
..=.
_.....-
b .,.--- " ' "
"~"
"~
""
""
C
|
!
!
!
i
i
20
40
60
80
100
120
N max
F I G U R E 11.4 Hypothetical expected times to extinction T E as a function of maximal abundance Nma x. (a) Demographic stochasticity alone. (b) Demographic and environmental stochasticity. (c) Demographic and environmental stochasticity with inclusion of catastrophic events.
tats produce surplus animals, which then move into the unsuitable habitats. Thus, a comprehensive understanding and management of populations in heterogeneous environments requires consideration of the spatial components of a population.
ticity and no migration among populations, from Eq. (11.21) the probability of eventual extinction for population i is
11.2.2.1. Metapopulation Dynamics
i = 1, 2, 3. The probability P0(oo) of eventual extinction for the metapopulation depends on these probabilities, based on assumptions about the independence of demographic rates and the absence of migration between the populations. If there is no interchange among the local populations, then each Poi(OO) is independent of the others and P0(oo) is simply the product
Consider a regional population that is stratified into geographically defined local populations across a heterogeneous range, with the individual populations occupying (relatively) homogeneous subranges. Because the probability of eventual extinction is always greater than zero, the existence of multiple populations increases the likelihood of local extinction of individual populations. However, interest frequently centers not on the fate of isolated local populations, but on the regional "population of populations," or metapopulation (e.g., Levins, 1969, 1970; Hanski and Gilpin, 1997; Hanski, 1999). To illustrate, assume that there are three separate populations with abundances Nl(t), N2(t), and N3(t) at time t, with metapopulation abundance N(t) = Nl(t) + N2(t) + N3(t).
The local populations all have probabilities of persistence over some time horizon, and the metapopulation inherits a persistence probability from them. Let N 1(0), N2(0), and N3(0) be initial population abundances and b i and d i be the population birth and death rates with b i > di, i = 1, 2, 3. Assuming only demographic stochas-
Poi(OO)=(dfii)Ni(~
3
Po(oO) = 11 Poi(OO) i=1
3 (dfii)Ni(O)
-"1 i=
In the special case in which demographic rates are identical for each population, probability of overall extinction is 3 ( d ) Ni(0)
P0(oo) = 111 i=
3 __
i=1 N(0) !
11.2. Conservation and Extinction of Populations that is, overall extinction is simply an exponential function of metapopulation abundance, and the metapopulation is essentially a single population with three biologically identical components. Individual probabilities of extinction are no longer independent if there is migration among populations. For instance, population 1 (with low growth rates) might decline to the threshold of extinction, but be "rescued" from extinction by immigration from populations 2 and 3. The situation is even more complicated if demographic rates are stochastic and nonindependent, i.e., there is a covariance structure among randomly varying parameters of the different populations. This might be expected if the separate populations share common environmental and habitat features, e.g., they all are subject to similar annual variation in climatic conditions that affect birth and growth rates. Though it is possible to incorporate these and other features into extinction models, the models quickly become analytically intractable. In practice, it is usually more straightforward to simulate metapopulation dynamics in terms of a system of interacting populations. By following the "fates" of a large number of simulated metapopulations with common initial conditions and parameters, one can determine how many populations persist and use that information to estimate extinction and persistence probabilities. This approach has the advantage of allowing for the inclusion of other sources of variation (environmental stochasticity, random catastrophic events, genetic effects) in addition to demographic stochasticity, thus providing a more comprehensive assessment of population viability. Its disadvantage is that the biological models and model parameters underlying the approach often must be identified in the absence of adequate field data.
11.2.3. Models of Patch Dynamics Here we consider two analytical and simulation approaches for modeling the dynamics of spatially structured populations: (1) patch-dynamic models, in which the population abundance is defined by the numbers of animals in discrete "patches" (habitats, areas, or other spatially defined regions), and abundance and other statistics are summarized for each patch; and (2) spatially explicit individual models, in which the spatial coordinates of individuals and their fates are simulated. A particularly simple model for patch dynamics considers only the presence (N > 0) or absence (N = 0) of animals in a system of patches, with probabilities of occupancy that are functions of patch-specific growth rates and the probabilities of migration among patches (Levins, 1969, 1970; Hanski 1992, 1994, 1997; Lande and Barrowclough, 1987; Lande, 1988). Our ap-
235
proach here extends this framework, to account for patch-specific abundance (also see Hastings and Wollin, 1989; Gyllenberg and Hanski, 1992; Gyllenberg et al., 1997). Let Ni(t) represent abundance in patch i at time t, with hi(t) the finite rate of population growth from birth and survival in patch i during time interval [t, t + 1] (i.e., excluding immigration into the patch or emigration from the patch). Let ~ri,j(t) represent the probability of movement from patch i to patch j during [t, t + 1]. Then the population dynamics for patch i are given by Ni(t + 1) = Ni(t)hi(t)'rri,i(t) 4- s Nk(t)Kk(t)'rrk,i(t) , k~i
(11.24)
where movement (if any) follows birth or mortality. For example, the dynamics of a system of three populations in a metapopulation are characterized by Nl(t + 1)= Nl(t)hl(t)'rrl,l(t) 4- N2(t))~2(t),rr2,1(t) 4- N3(t)h3(t)~rg,l(t) N2(t 4- 1)= Nl(t)hl(t)Trl,a(t) 4- N2(t)h2(t)'rr2,2(t) 4- N3(t)h3(t)~r3,a(t) N3(t 4- 1)= Nl(t)hl(t)'rrl,3(t) 4- N2(t)h2(t)'rr2,3(t) 4- N3(t)h3(t)~rg,3(t).
By specifying initial populations sizes Ni(O) and functional forms for Ki(t) and ~ri,j(t) (e.g., stationary patch-specifi~ migration rates), one can determine the trajectories of patch-specific population abundances as functions of time. For certain special cases, the population trajectories can be expressed analytically as function of time, but more typically one must use computer simulation. With simulation, the fates of simulated populations can be tracked over a selected time horizon (e.g., 100 yr), and the influence of the rate functions hi(t) and "rri,j(t) can be investigated via repeated simulation. Note that the above model incorporates patch-specific abundance, but no additional structure. The geographically structured projection matrix models of Section 8.6 can be used to develop detailed models of withinand between-patch dynamics for metapopulation systems (e.g., see Rogers, 1966, 1968, 1975, 1985, 1995; Schoen, 1988; Lebreton, 1996). Note also the close connection between the model in Eq. (11.24) and statistical models such as the multistate extensions of the Jolly-Seber model (e.g., Arnason, 1972, 1973; Hestbeck et al., 1991; Brownie et al., 1993; Schwarz et al., 1993a). In Chapters 17-19 we describe methods to estimate the demographic and movement parameters in Eq. (11.24).
11.2.3.1. Source-Sink Models A special case of Eq. (11.24) is the source-sink model described by Pulliam (1988). Suppose there are two "habitat types," one that is "suitable," in that hi(t) =
236
Chapter 11 Conservation and Management Models
)kI ~ 1 (e.g., habitat I provides adequate nest sites, food, and cover for its population component to increase) and one that is "unsuitable" [X2(t) -- X2<1 ]. Suppose further that abundance in the suitable or "source" habitat is limited (e.g., by nest sites) to a maximum of N~ animals and that animals in excess of N~ disperse to the unsuitable or "sink" habitat. Population dynamics for this system are given by
X1NI(t) N~
Nl(t+l) =
XINI(t) < N~ XINI(t) ~ N~
and
N2(t+l) =
N~
X2N2(t)
NI(t)X 1 <
X2N2(t) + ~XINI(t)-N' ~]
hlNl(t) -> N~.
The solution to this system of equations provides an equilibrium population of size N~=X1 -1
1 -- )k2 N~
(11.25)
in the sink habitat. For example, suppose X1 = 1.3, ~'2 = 0.85, and N~ = 1000. Assume there initially are small numbers of animals in each habitat [NI(0), N2(0) <~ 1000]. Until abundance in the source habitat reaches 1000 we have N l ( t + l ) = (1.3)N1(t), so that abundance increases exponentially in the source up to N T, but the number of animals in the sink habitat decreases monotonically until Nl(t) exceeds 1000. Once the abundance in habitat 1 reaches 1000, we have N2(t+l) = (0.85)N2(t) + 100011.3-1] and at equilibrium N~ = 1000 = 1000
(1.3-1)
(1-0.85) 0.3 0.15
= 2000. Several interesting conclusions follow from this simple model: 9 In the absence of source populations, populations in the sink habitat decline to extinction (because X2K1). 9 The combined population can be at equilibrium, even though a majority of it is in unsuitable habitat, as defined by X < 1. 9 It may not be possible to infer habitat suitability (or lack of it) by observing animal abundance (or density) in habitats. In equilibrium, poor habitats can have
higher abundance than "suitable" habitats because of immigration, with the equilibrium abundances maintained through dispersal. 9 Elimination of the source habitat results in eventual extinction in all patches. Thus, management directed at high-density patches could be counterproductive, if at least some of these are not source habitats. 9 Inference about habitat suitability requires a knowledge not just that animals are present, but why they are present (or at least evidence, such as patchspecific vital rates, to indicate that the habitat is a source and is not simply attracting dispersers into a population "sink"). From Eq. (11.25), one only needs )kI and ~'2, along with the capacity of suitable habitat (i.e., the value N~), to predict equilibrium abundance in the sink [in fact, N~ and the ratio (X1-1)/(1-X 2) are sufficient]. The source-sink model implicitly assumes that sinks are (1) available, i.e., near enough to source habitats to allow dispersal, and (2) capable of absorbing dispersers in virtually unlimited numbers (unless a density-dependent growth function for the sink is invoked). Clearly, not all patches of sink habitat are equally available, and useful extensions to the above models could include differential movement of animals among habitats, density limitation in the sink habitats, movement corridors, impediments to movement, etc. Furthermore, there is no need to limit dispersal to a one-way phenomenon or to limit habitats to simple "suitable" and "unsuitable" categories. This additional realism can be incorporated in spatially explicit models, although such models are substantially more complex than the above source-sink model (see Pulliam et al., 1992; Conroy et al., 1995; Dunning et al., 1995).
11.2.3.2. Spatially Explicit and Individual-Based Models
In the models considered above, the landscape (and animals therein) is described in terms of "patches" or other discrete regions of space, with animals in a particular patch (or patch type) possessing common survival and reproduction rates, and common probabilities of movement among patches. The relevant information about animal population dynamics is contained in these parameters. For example, the area or shape of a patch might influence survival, with long, linear habitats possibly creating many opportunities for predators (Gates and Gysel, 1978), and the probability of movement to another patch might be influenced by the quality, proximity, and occupancy of adjacent patches. These and other influences can be summa-
11.3. Discussion rized and encoded into discrete models such as those described above (e.g., Day and Possingham, 1995). An alternate approach is to use models in which both the fates (alive, dead, reproductive, etc.) and the map coordinates of individual animals are simulated. Time-specific location is an individual-level state variable and thus a component of an animal's/-state (see Metz and Diekmann, 1986; Caswell and John, 1992). In an individual-based model (see Huston et al., 1988; DeAngelis and Gross, 1992), N O animals are spatially distributed over the landscape, and additional animals are added over the time frame via birth or immigration events. Similarly, animals are removed via death and emigration. Movement, survival, and reproduction for each animal are modeled as a series of random events based on some underlying mechanistic model. It is necessary to have a digitized cover map of the area with spatially indexed biological attributes. The attributes at a particular location, along with those of surrounding habitats within some radius determined by the animal's movements, determine the subsequent survival, reproduction, and movement rates, recognizing impediments to travel, such as absence of cover, water barriers, and human artifacts. Survival, reproduction, and movement then can be simulated for all animals in the population at each point in time and carried through a specified number of time steps to obtain a realization of the population's "fate" (e.g., abundance and spatial distribution) after t time steps. The simulation can be repeated under identical initial conditions, to obtain estimates of the probabilities of various outcomes (e.g., population or local extinction). Such spatially explicit population models (Dunning et al., 1995) have been used to describe complex problems involving the potential effects of habitat fragmentation, size, shape, and other characteristics, especially in relation to management activities (e.g., Liu, 1993; Lamberson et al. 1994; McKelvey et al., 1992; Pulliam et al., 1992; Turner et al., 1994). However, spatially explicit models can be highly complex and typically contain many parameters and assumptions. Their usefulness is therefore constrained by one's ability to estimate their parameters and validate the resulting models (Conroy et al., 1995). In addition, much work remains to connect these and other models to conservation decision-making (Conroy and Noon, 1996). Nevertheless, when properly developed and used in the context of scientific method (Chapter 2) and statistical design and sampling (Chapter 4), spatially explicit models can be extremely useful tools. We explore the decisiontheoretic use of models (Chapter 3) more thoroughly in Part IV, where we integrate predictive modeling with statistical inference and decision-making.
237 11.3. D I S C U S S I O N
The use of models has been pervasive in the field of population ecology [e.g., see historical review in Hutchinson (1978)]. Fields as disparate as biogeography, species interactions, habitat selection and preference, sociobiology, and conservation biology have benefitted from the use of models in framing the underlying theory and analyzing the dynamics of populations. In this chapter we have focused on two familiar applications of models, recognizing that many others also could have been discussed. The use of models to investigate harvest impacts has a long and often fractious history, with continuing controversy up to the present as to the effects of harvest on mortality and reproduction and thus on abundance. The usefulness of such concepts as sustained yield has been criticized frequently and on many grounds, and the appropriateness of additive vs. compensatory mortality continues to be a subject of active discussion. On the other hand, the modeling by conservation biologists of extinction probabilities, times to extinction, minimum viable populations, and other issues of importance for the conservation of living resources is of more recent vintage and is now an area of active research for biologists (see, e.g., Chapter 20 and references therein). Until recently, much of this work has consisted of simulation gaming with models based on statistically unreliable parameterizations and inadequate validation with field data. Under these circumstances, models can be useful in guiding research and focusing field investigations, but they cannot replace field work as a basis for inference about actual populations. In recent years there has been a worrisome tendency by many to refer to the alteration of model assumptions and parameter values in an evaluation of predicted outcomes as a modeling "experiment." To sharpen the distinction between simulation exercises and experiments, and to distinguish more clearly between what models can and cannot do, in this book we reserve the term experiment for the controlled manipulation and observation of real systems (Chapter 6). As noted in Chapters 2 and 3, models constitute an abstraction of our (possibly faulty, and undoubtedly incomplete) knowledge of how a system works. Models may provide predictions about, e.g., the impacts of management, but these predictions are dependent on the underlying structural assumptions and parameter values in the model. The usefulness of models for making management decisions therefore depends on the degree of empirical support for them and the estimated values (and associated variances) for the parameters in them. In this context, models can be very useful
238
Chapter 11 Conservation and Management Models
tools for summarizing complex systems and for investigating hypothetical responses of these systems to change (e.g., management). As indicated at several points in this book, they can play a key role in the scientific enterprise; however, it is important to remember that they are not substitutes for scientifically based observation and experimentation. In the chapters in Part III, we turn our attention to the estimation of parameters in population models and assessment of reliability for these estimates, including tests of underlying model assumptions. Our focus thus
shifts to the formulation of "statistical models" that are based on (1) structural assumptions about model parameterizations and (2) statistical assumptions for particular sampling situations. These models allow for the estimation of model parameters, as well as the evaluation of alternative model assumptions. We will see in Part III that straightforward field procedures, and relatively simple data recorded in the field, can be highly informative of population status and the biological processes driving population change through time.
P A R T
III E S T I M A T I O N M E T H O D S FOR ANIMAL POPULATIONS
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C H A P T E R
12 Estimating Abundance Based on Counts
12.1. OVERVIEW OF ABUNDANCE ESTIMATION 12.2. A CANONICAL POPULATION ESTIMATOR 12.2.1. Detectability 12.2.2. Spatial Sampling 12.3. POPULATION CENSUSES 12.4. COMPLETE DETECTABILITY OF INDIVIDUALS ON SAMPLE UNITS OF EQUAL AREA 12.4.1. Abundance Estimators 12.4.2. Assumptions 12.4.3. Survey Designs 12.4.4. Accounting for Stratification 12.5. COMPLETE DETECTABILITY OF INDIVIDUALS ON SAMPLE UNITS OF UNEQUAL AREA 12.5.1. Ratio Estimator of Abundance 12.5.2. Weighted Estimators 12.5.3. Estimators Based on Stratified Designs 12.6. PARTIAL DETECTABILITY OF INDIVIDUALS ON SAMPLE UNITS 12.6.1. Estimation of Detectability Based on a Subset of Sample Units 12.6.2. Estimation of Detectability Based on the Set of All Sample Units 12.7. INDICES TO POPULATION ABUNDANCE OR DENSITY 12.7.1. Survey Counts as Population Indices 12.7.2. Relationship of Indices to Abundance 12.7.3. Indices Other Than Counts 12.7.4. Design and Interpretation of Index Surveys 12.8. DISCUSSION
estimation of population parameters such as population size a n d / o r density, annual reproduction, survival/mortality, and other biological attributes relating to spatial distribution, habitat use, and interspecific interactions. In this and the next two chapters we emphasize the estimation of population size and density. There are good reasons for such an emphasis. Population size is the state variable of interest in most of the models described in Part II. On initiating an investigation, the first questions confronting a researcher concern the number of individuals in the population and where they are found. Beyond a first look at size and range, studies often focus on the investigation of biological relationships that are sensitive, or at least potentially sensitive, to the number of organisms participating in them. Without some idea of the size and spatial distribution of a population, it is effectively impossible to investigate such size-dependent or density-dependent relationships. Another reason to focus on abundance estimates is that they can be used to assess the performance of a population model. Though a specific biological process can be verified with data collected for that process, population models essentially integrate all the component processes into a holistic representation of population dynamics. As seen in Chapter 9, population size is an especially useful measure by which to assess the adequacy of this representation. Model behavior that faithfully tracks changes in population status suggests that the model incorporates the key biological factors affecting change. Conversely, the failure of a model to track population changes indicates that further assessment and model development are necessary.
In Parts I and II we have seen how models can be used to describe populations and to assess the influence of environmental conditions, management actions, and other factors on population dynamics. The assessment of influencing factors often begins with the
241
242
Chapter 12 Estimating Abundance Based on Counts
A final reason for an emphasis on abundance concerns the goals of management, which often focus on population size. Managers typically wish to increase populations of species that are rare or are seen as beneficial, such as endangered species and many populations subject to sport hunting, and to decrease populations of nuisance or pest species. Ultimately the measure of success in these efforts is the change in population status through time. Good estimates of population size are essential components of the data needed for evaluation of many conservation, wildlife, fisheries, and pest management programs. There is a long, productive history of methodological development in the area of population estimation. Currently a wide array of methods, including population surveys, radio telemetry, banding and tag-recapture procedures, and other indirect methods, is available for estimating population size and density. A comprehensive review of methods for estimation of animal abundance was given by Seber (1982). However, much work has been done in this fast-growing field since the publication of Seber's text (e.g., Seber, 1986, 1992; Schwarz and Seber, 1999), and we focus on some of these developments in the ensuing chapters. All the methods discussed below share features that distinguish them from informal or ad hoc assessment of abundance. Thus, the methods described below are directed at the estimation of a parameter, typically abundance (N) or density (D). They employ methods with known statistical propertiesmfor example, the degree of bias and variance of the estimator a n d / o r the statistical distribution of estimates from repeated samples. The estimation procedures focus not simply on the point estimation of the parameter, but also on measures of precision of the estimates, typically provided by estimated variances and confidence intervals. Finally, the methods are designed to answer questions about how parameters vary across space or time or among subcategories of the population (e.g., age and sex) and how they relate to factors under management control (e.g., habitat manipulation, harvest regulations). In the sections below (and, indeed, throughout Part III) we follow three general principles. First, we restrict the discussion to methods that meet minimal standards for scientific rigor, and in particular, that satisfy the above methodological criteria. Methods that do not meet these guidelines are of little use in the scientific investigation and management of populations. Second, we emphasize methods that rely on few assumptions and are robust to moderate deviations from them. There is little value in statistically elegant models and estimators, if in practice their critical assumptions remain unmet. Third, we emphasize sampling and esti-
mation schemes that make effective use of the limited resources (time, money) that can be devoted to field sampling. Often this means identifying study objectives and then using preliminary information and a model to estimate the sampling effort required to meet those objectives. In some cases creative sampling designs and models can be used to integrate information from different sources (e.g., ground and aerial counts of waterfowl; the robust design for capture-recapture; see Chapter 19), resulting in estimates that are superior to those obtained from the respective sources separately. A careful preliminary analysis often reveals that study goals cannot be reasonably met, at least within identified time and monetary constraints. Such a determination ideally is made at the planning stages of a study rather than the analysis phase, as it then is possible to determine whether additional resources can be committed or whether they are better utilized elsewhere.
12.1. OVERVIEW OF ABUNDANCE ESTIMATION This and the next two chapters consider several different methods for the estimation of abundance and density. In this chapter we focus on methods involving simple counts. There are two basic approaches: a census, or complete count of a population, and a sample survey. Except for confined populations, true censuses seldom are possible, and almost all wildlife counts are based on surveys, notwithstanding the prolific use of the term census in the literature. Sample surveys fall into two somewhat overlapping categories: (1) counts that include all individuals on the sampling units (e.g., sample quadrats), the units representing a sample of the area or volume occupied by the population; and (2) counts that are incomplete on the sampling units because some individuals are missed by observers. This distinction provides a motivation for the partitioning of two sources of potential error in estimation and gives rise to a canonical form for estimating abundance and other parameters (see Section 12.2). In what follows, we also discuss the use of incomplete counts as indices to abundance or density, and the use of auxiliary information not associated with animal counts but thought to be correlated with abundance or density (indirect indices). We devote Chapter 13 to distance methods, where in addition to counts, auxiliary information is obtained in the form of observer-to-animal (or other object) distances. These data can be used to estimate the fraction of individuals in a sampled area that are actually counted (detectability). There are three basic types of
12.2. A Canonical Population Estimator distance sampling methods and associated models: (1) nearest-neighbor methods, (2) line transect (perpendicular distance) methods, and (3) point-to-object (radial distance) methods. We emphasize the latter two sampling methods, especially line transect, along with flexible and robust modeling approaches developed by Buckland et al. (1993). In Chapter 14 we describe methods based on the capture or removal of animals from a population. Emphasis is given to capture-recapture methods, in which samples of animals are captured and tagged or otherwise marked, and then returned to the population, where they may appear in subsequent samples. The simplest capture-recapture method is the LincolnPetersen procedure, which involves an initial capture, marking, and release of individuals back to the population, and the subsequent capture of both marked and unmarked individuals. The Lincoln-Petersen procedure can be extended to accommodate multiple releases and recaptures of marked animals, thereby enabling one to address heterogeneity of capture probabilities among animals. Also covered in Chapter 14 is the estimation of abundance when sampling is by removal from the population, either by marking or by physical removal (harvest, euthanizing, relocation). Data in the latter situation consist only of captures of animals not previously caught, i.e., there are no recaptures, and two kinds of removal schemes are emphasized. In the first, removal effort is typically under the control of the investigator and is constant (or nearly so) over time. Under this scheme, estimates of abundance are based on the models used for recapture data (Chapter 14) (see Otis et al., 1978). In the second, removal effort is variable over time (and often beyond the control of the investigator), but a measure of removal effort (e.g., hunting or fishing effort) is available. Catch-per-unit effort models then are used with the removal and effort data. The simplest of these depends on an assumption of a constant relationship between effort and capture probability (see Section 14.4). A variation of the "removal" theme occurs when animals can be categorized into two or more classes (ideally, visibly distinct from afar), and known numbers of animals of one or more classes are removed, typically through harvest. The observed shifts in the proportion ("ratio") in each class before and after the removal can be used to estimate abundance via changein-ratio estimators. In Section 14.5 we describe models, estimators, and testing procedures for change-in-ratio methods. All of the above methods depend on the assumption of sampling from a population that is closed over the sampling period (except for known removals in re-
243
moval and change-in-ratio methods), i.e., there is no birth, death, immigration, or emigration during the investigation, and thus population size N remains constant between samples. This assumption is almost certainly violated in any population except for brief periods of time over relatively large areas, but is approximately true under certain circumstances, depending on the species, area, and time of year. For instance, a population of meadow voles (Microtus pennsylvanicus) in a 10-ha meadow might be considered approximately closed to birth, death, and migration over 1 week in November, whereas a population of white-tailed deer (Odocoileus virginianus) could hardly be expected to confine its movements to the same 10 ha in a week, and a migratory population of mallards (Anas platyrhynchos) could move hundreds of kilometers and be subject to harvest mortality during the same period. For each of the methods we describe, we discuss the consequences of violating closure and other assumptions. A number of familiar methods for estimating abundance or density are of limited inferential value, because their statistical properties are unknown or suspect. For example, "territory mapping" frequently is used to estimate abundance of many bird species (see Williamson, 1972), but this method typically is applied in an ad hoc and subjective manner that does not permit statistical inference (e.g., see Oelke, 1981; North, 1977, 1978). In general, we discourage the use of ad hoc estimation techniques with questionable statistical reliability, except as a last resort and only as a mechanism to suggest patterns to be investigated in follow-up studies that utilize statistically reliable methods.
12.2. A C A N O N I C A L POPULATION ESTIMATOR Virtually all of the estimators described in this book are based on count statistics. Count statistics include the number of animals counted in an aerial survey, the number counted from line transects, the number caught in traps in a given night. In general, counts represent some unknown fraction of the target population of animals of interest and by themselves may be of little or no value. We almost always need additional information about this fraction, in order to estimate population parameters and make inferences about abundance over time or space. Indeed, the central issue in this part of the book concerns the design of sampling plans and estimation procedures to estimate the sampling fraction, so as to obtain useful estimates of parameters of interest. Two issues requiring special
244
Chapter 12 Estimating Abundance Based on Counts
consideration are the observability or detectability of organisms and the need for spatial sampling.
12.2.1. Detectability
Often oL is known with reasonable accuracy (e.g., by mapping) and may be treated as a known constant rather than an estimate. Under these assumptions an estimator of N is
F4 = c/~.
Assume that one has a count from an area occupied by a population, with sampling occurring over the entire area (i.e., there is no spatial sampling of the area). All animals in the population potentially can be counted, and the only animals not included in the count are those not detected by the observers. Let C be the resulting count statistic, with 13 the probability of detection for an animal given that it is present in the population. The expected value of the count statistic then is
is approximately unbiased as long as 6~ is unbiased. In general, both observability and the spatial sampling fraction are of concern and therefore should be combined into a single canonical estimator, as in
E(C) = f3N,
lCq = C/(oL~)
where N is the actual population size. We can think of 1 - ~ as the fraction of the population present that remains undetected. If we knew the value for 13, we could obtain an unbiased estimate of N by adjusting for this value, i.e.,
1~= c/~. This estimator clearly is unbiased, because E(/~) = (13N)/13 = N. Thus, if we capture 100 mice in a night of trapping, and somehow we know the capture probability to be 0.25, then 100/0.25 = 400 is an unbiased estimate of abundance. Of course, we usually do not know the value of 13 and instead must estimate it from other data, in this case, recaptures of previously marked animals. An intuitive estimate of N then is based on the count C and an estimate ~, which is approximately unbiased provided ~ is unbiased. The lack of bias follows from E(lXl) ~ E ( C ) I E ( ~ ) = f3N/[3
=N.
12.2.2. Spatial Sampling Typically, time and money limitations prevent the sampling of the entire area of interest to obtain count statistics, and an investigator must select areal units consisting of some fraction e~ of the total area A over which a population is distributed. Assuming the sum C = ~iCi of counts Ci on m randomly selected sampling units records the population in the sampled area (i.e., for the moment, we are assuming the unusual situation of 13 = 1), we then have E(C) = oLN.
If o~ is not known and must be estimated, then the estimator
F~- c / a
(12.1)
or, more generally, =
Where possible, we will present estimators of abundance and other parameters in the above form, to show the relationship of the estimator to these two sources of variability. As noted earlier, point estimates are of limited value in the absence of corresponding measures of reliability. For example, estimates of the sampling variation of an estimator are needed for construction of confidence intervals. The approximate variance of the canonical estimator in Eq. (12.1) can be derived via the delta method (see Appendix F), as var(/qD ~ [var(C) var ([3)] E(C) 2 + 62 N 2.
(12.2)
From Section 5.3.1, var(C) can be expressed in terms of variation S2 in the number of organisms on the areal units in A, by var(C) = m[13S2(1 - c~) + ~2], where ~2 is the mean variance for counts on the areal units in A. Thus, the sampling variance of an estimate of population size depends on both observability and sampling fraction. As the sampling fraction oLincreases, the spatial component of var(N) declines, and if the entire area is sampled, the spatial component vanishes. For o~ < 1, var(N) depends on sampling variation in the count statistic. The component var(C) of this variation typically is estimated from spatial replicates at which the counts are obtained. Likewise, as 13 increases, the component due to observability declines to zero. For 13 < 1, var(/~) depends on the precision with which 13 is estimated. Of course, the variance component due to observability vanishes if [3 is a known constant.
245
12.4. Complete Detectability of Individuals on Sample Units of Equal Area Besides its heuristic value, the canonical formulation in Eqs. (12.1) and (12.2) has practical value in sampling design, in that increasing e~ typically requires more spatial sampling units (e.g., transect lines, trapping grids), whereas increasing ~ and its precision typically requires more sampling effort at each unit (e.g., longer transect lines, more trap nights, or more traps per grid). A knowledge of how the variance of the estimator is influenced by each component, along with the relative cost of each, is critical in decisions about allocating resources to meet study objectives (Cochran, 1977; Skalski and Robson, 1992). Specific variance estimators incorporating components for both spatial sampling and detectability depend on sampling specifics and are presented in Thompson (1992) and Skalski (1994) (also see Secti o n 12.6).
12.3. P O P U L A T I O N C E N S U S E S A census or complete count of the population may be possible under idealized circumstances--for instance, if the population is very small, is highly conspicuous, or is in a confined area (such as a zoological park or small, fenced area) (see Jarman et al., 1989, 1996). If organisms are sessile and the population of interest is confined to a relatively small area, a complete count may be practical. In most situations, however, a complete census is either impossible (because the area is too large or the animals are inconspicuous) or if possible, would be less efficient than a sampling procedure. This can be particularly true for large populations, even if the organisms are all "countable." For instance, the decennial national "census" of the United States relies on sample-based estimation methods for characteristics of the human population in the United States (Cochran, 1977). In the following two sections, we assume that a complete count can be obtained, but only for a sample of the total units on the study area. We defer discussion about the size and shape of the sampling units until Section 12.4.3, where we discuss the relative advantages of different physical configurations.
12.4. C O M P L E T E D E T E C T A B I L I T Y OF I N D I V I D U A L S O N S A M P L E U N I T S OF E Q U A L A R E A In what follows, our objective is to use sampling procedures and estimation methods to estimate population size a n d / o r population density. Here N represents the total size (abundance) of the population of interest, A represents the area occupied by the popula-
tion, and D = N / A represents the population density. The idea is to select randomly a sample of spatial units and to use the counts from them to fashion an estimate of population size. As a practical matter it often is desirable to choose spatial sampling units of equal size. When sampling units are the same size, errors in delineating the units on the ground often are reduced or eliminated, estimation formulas generally are simpler, and sampling variances of the resulting estimators generally are lower. We assume for now that spatial sampling units are of identical size and shape.
12.4.1. Abundance Estimators Assume that the relevant area A for a population is divided into M sampling units, each of area a = A / M . Suppose that a sample of m of these units is selected at random. Then the total area sampled is ma and the sampling fraction is ~ = m / M . On each sampling unit a count of Yi animals is recorded. If m
C=~yi i=1
is the total count across the m units, a natural estimator of N is m
1~--" M ~_j m
i=1
(12.3)
= M~,
with estimated sampling variance v~r(/~) = M2S2(1 - m / M ) , m
(12.4)
where m
$2-- ~_j(Yi-
Y)2/( m
--
1)
i=1
(see Section 5.3). Formula (12.4) for the variance is based on the assumption that sampling is without replacement (see Section 5.3.1). The variance of/~/can be seen as a scaled product of the sample variance s 2 and the finite population correction (1 - re~M). Thus, the variance of N declines linearly to 0 as the sample fraction m / M increases to 1.
12.4.2. Assumptions The assumptions for estimator (12.3) are (1) m sampling units are randomly selected from a population of M units, and (2) all individuals are counted on each sampling unit. Assumption (1) normally is required to assure proper estimation of the variance of/~/, but
Chapter 12 Estimating Abundance Based on Counts
246
is unnecessary if individuals are distributed completely at random. Because a random distribution of individuals over an area is so infrequent in ecology, random sampling is an important safeguard to avoid bias in estimating variance. In the absence of random selection of spatial units,/~/still can be approximately unbiased, but variance estimates and the resulting confidence intervals are incorrect, i.e., confidence interval lengths no longer correspond to their specified probability significance levels. We refer the interested reader to Seber (1982), who discusses tests of randomness for count data and variance estimates under alternative (e.g., negative binomial) spatial processes. For sufficiently large sample sizes, in excess of m :> 30 samples, the distribution of the sample mean is approximately normal, a result of the Central Limit Theorem of statistics (Mood et al., 1974). Approximate confidence limits for abundance then are given by
TABLE 12.1 Example of Muskrat Houses with Complete Counts on Fixed-Area Plots Plot
Houses counted
1
13
2 3
18 10
4
6
5 6 7 8 9 10
16 13 12 13 9 11
Sy2
12.1 11.66
/Q -+- Zi_e,/aV/v~(/Q), where Z1_~/2 is the standard normal deviate corresponding to the oL/2 upper tail of the standard normal distribution. For example, a 95% confidence interval is provided by + 1.96V'v~ (/~).
Example An estimate is desired of the abundance of muskrat
(Ondatra zibethicus) houses in a 100-ha marsh on a wildlife refuge. The marsh is divided into 50 2-ha square plots, with plots at the boundary of the marsh included if >50% of the plot is marsh. Ten plots are selected at random for a complete search by air boat crews for muskrat houses. The resulting counts are presented in Table 12.1. These data yield the estimate
= 50(12.1) = 605
v ~ ( / ~ = M 2s2(1 - m/M) m
(12.5)
The bias in N due to undercounting ((3 < 1) is not remedied by increasing the number of replicate plots. Thus, with undercounts one can obtain very precise estimates, but of an unintended or undesired population attribute. In addition, the estimated variances may underestimate true sampling variance, even if sampling is random, counts are complete, and organisms are distributed randomly. This is particularly problematic for m < 30 sample units and high variability in Yi (as with aerial surveys; see Section 12.6.1) (Cochran, 1977). Under these conditions we recommend the use of alternative procedures for estimating sampling varia n c e - f o r example, based on bootstrapping (see Appendix F).
My
of abundance, with estimated variance
= 50211.66 lO
(1 -
lO/5O)
= 2331. These results can be used to compute an approximate 95% confidence interval on N, by /~ + 1.96V'v~(/~ = 605 ___ 94.63 = (510,700). 12.4.3. S u r v e y D e s i g n s
Thus far we have made no assumptions about the size, shape, number, or placement of sampling units. The following general guidelines assist with survey design.
12.4.3.1. Size and Shape of the Sampling Units The optimal size and shape of sampling units depend on a number of factors, and it is difficult to make recommendations that apply to all circumstances. However, some general guidance can be provided. With respect to shape, the choice usually is between circular and rectangular units. In some situations (e.g., counts of plants in small areas) circular plots can be convenient, because they require only the specification of a central point and a radius. Of all geometric plot shapes, circular plots minimize perimeter length per unit area and therefore minimize the error in deciding
12.5. Complete Detectability of Individuals on Sample Units of Unequal Area whether an individual is in or out of the plot. On the other hand, it is impossible to subdivide an area into M circular plots without overlapping or excluding areas, whereas it is easy to do so with square or rectangular units. In comparing square and rectangular units, the former have smaller perimeter lengths per unit area and therefore less potential for error in including individuals in th e plot. Square plots often are easier to lay out than rectangular plots, though in some circumstances (notably aerial surveys) the opposite is true. In determining the size of the sampling units, two factors tend to operate in opposing directions. First, the smaller the plot size a relative to the survey area A, the larger will be the necessary sample size m to sample the same proportion of the survey area. Under these circumstances, the increased sample size reduces the variance of the estimator, as per Eq. (12.4). However, very small units also have large perimeter-to-area ratios and thus a greater possibility of inclusion error. Also, very small (relative to the density Of animals) units will result in m a n y zero counts, which tend to inflate the variance of the count statistic. A useful "rule of thumb" is that plot size should be sufficient for one to expect more than half the plots to contain individuals (Greig-Smith, 1964).
12.4.3.2. Sample Size If possible, a pilot study should be conducted to obtain preliminary estimates of the mean ~ and variance S2 for the sampling units, which then can be used to determine the numbers of additional units needed to meet specified precision goals. A simple formula for the necessary sample size is based on the coefficient of variation
cv(y)
247
Example Suppose we have a study area of area A = 10,000 ha, divided into M = A / a = 100010-ha plots. A preliminary sample of 10 plots is used to obtain the estimates = 100 and s 2 = 2500. If the desired coefficient of variation is 10% (CV 0 = 0.10), from Eq. (12.6) the necessary sample size is given by 1 1 1002 m = 1000 + (0"10)22500' or 1/m <- 0.041. Thus, m = 24 and m - m 0 = 14 more plots are needed to achieve the desired precision.
12.4.4. Accounting for Stratification The estimator in Eq. (12.3) and its variance in Eq. (12.4) can be extended in the manner of Section 5.4 to incorporate stratification over the sampling area. The need for stratification arises when population densities vary over an area in response to heterogeneity in habitat conditions or other factors. Under these conditions, stratified sampling designs can reduce the variance of an estimator, provided the variation among strata is substantial compared to variation among the sampling units within strata. An estimate of population size is given by combining stratum-specific estimates
mi 1Qi = M---ii~.~ Yij mi j=l
= Miy i to produce the aggregate estimate I
1Q = ~_j M i y i i=1
~/var(y) =
for I strata. The corresponding variance is
SN/1 - m/M
i
VGm
v~(/~ = ~
i=1
s2
M 2 - (1 - m i/Mi),
mi
which can be rewritten as
s
Z if_ CV(y )2~2
m
M
where
S2
Suppose that a pilot survey consisting of m 0 randomly selected units produces the estimates y and s2 for the population mean and variance. By substituting y and s 2 for ~ and S2 in the above equation, we then can determine the additional sampling effort necessary to achieve a desired coefficient of variation CV o. The resulting formula specifies a sample size m such that 1 m
1 -
M
CV2y 2 t
S2
'
so that an additional m - m 0 units are required.
(12.6)
mi $2-- ~_J(Yij- yi)2/(mi--1 ). j=l
12.5. C O M P L E T E D E T E C T A B I L I T Y OF INDIVIDUALS ON SAMPLE UNITS OF UNEQUAL AREA Thus far we have focused on sampling designs and estimators when the sampling units are all of equal size. With variable sizes of the sampling units, an additional source of variability is introduced, and the esti-
248
Chapter 12 Estimating Abundance Based on Counts
mators in Eqs. (12.3) and (12.4) must be modified accordingly. In particular, the estimator of variance now must include variability due to unequal sample unit size.
(Cochran, 1977), where sy2 a n d s 2a are the estimated variances for the counts and the areas, respectively, and m
~ (Yi - Fd)(ai - -d)
12.5.1. Ratio Estimator of A b u n d a n c e
Say --
Under certain assumptions, ratio estimators (see Section 5.4) are appropriate for obtaining estimates of density with sample units of unequal areas. As before, let M be the total number of available quadrats in a sampled population, with m the number of quadrats sampled and Yi the sample count on each sampling unit i, i = 1, ..., m. We introduce a i to represent the area of unit i from a total survey area of M ~ai=A. i=1
The idea is to fashion an aggregate estimate of the number of individuals per unit area, which then can be scaled by A to get an estimate of overall population size. From Section 5.4, a ratio estimator of N is i
I~r = YA a
(12.7)
=/3A, where m
Y = ~ Yil m
i=1
m-1
is the estimated covariance between the counts and areas. This estimator is the best linear unbiased estimator on assumption that (1) the Yi and a i are related by E(y i) = ~a i and (2) var (yi) is proportional to a i (Cochran, 1977). If plots of Yi and a i indicate that the relationship has nonzero intercept, a regression estimator (Cochran, 1977) should be used instead of the ratio estimator. If plots of residuals indicate violation of assumption (2), a weighted ratio (Section 12.5.2) or regression estimator should be used. Evidence of nonlinearity in the relationship between Yi and a i casts doubt on the adequacy of any of these approaches and may suggest the need for stratification to account for nonhomogeneous densities. In the presence of nonlinearity, the above methods will provide unreliable estimates even of trends in abundance. Note that if the areas of the sampling units are equal (a i = a), then the last two terms of Eq. (12.8) vanish. One implication is that the variance of an estimator of abundance based on equal-area sampling typically is smaller than an estimator based on unequal areas (though not necessarilymit is possible to h a v e 2E)Say > ~)s2).
i=1
and
Example m
-d = ~ , a i l m i=1
are the respective sample means of the counts and areas, and D = y/~ is an estimate of population density D = N / A . That this estimator is a particular case of the canonical form [Eq. (12.1)] can be seen by defining
Consider the example in Section 12.3.2 involving counts of muskrat houses, but now consider the sample plots to be variable in area (Table 12.2). An estimate of abundance for these data is now I
F=Y-A_ a 7.5
m
-
C = ~_j yi
i~100
i=1
= 536
and with variance
m
oL = ~
ai/A ,
i=1
V ~ ( / ~ r ) = M2 (1 - m / M ) ( s ~
= 200116.94 + (5.36)2(0.27)- 2(5.36)(1.56)~
/~/= C / o L .
= 1596.
The estimated variance of/~ is provided by V~(/~r ) = M2
+ f)2s2- 2L)Say)
m
so that
(1 - m / M ) ( S m
2 if- ~)2S2 _ Y a
2~)Say )_
(12.8)
From Eq. (12.5) the resulting approximate 95% confidence interval for N is (458, 614).
12.5. Complete Detectability of Individuals on Sample Units of Unequal Area TABLE 12.2 Example of Muskrat Houses with Complete Counts on Variable-Area Plots Plot
Houses counted
Plot area
1
15
2
2
8
1
3
6
1
4
8
1
5
7
2
6
3
1
7
3
1
8
3
1
9
9
2
10
13
2
y, a
7.5
1.4
2 sy, s2a
6.94
0.27
Say
1.56
12.5.2. W e i g h t e d Estimators
If var(yi) is not proportional to ai, estimators other than that shown in Eq. (12.7) are appropriate. It can be shown (Brewer, 1963; Royall, 1970) that if the residual variance of Yi about a i is proportional to var(ai), then the weighted estimator
249
aerial surveys, stratification often can be accomplished by delineating areas of similar habitat types (e.g., wetland types), which can be expected a priori to have similar densities of animals. Ideally, estimates of variance from pilot samples would be available to provide a basis for the optimal allocation of sampling units within strata. In practice, relative densities may be all that are available, but these are adequate if there is a linear relationship between the mean and variance of counts. Absent any other information, allocation should be proportional to the size of the strata. If one fails to stratify when it is appropriate, heterogeneous densities induce heterogeneity in the linear relationship between Yi and a i, resulting in suboptimal estimation of D and N. One solution to this problem is the separate estimation of a subpopulation size for each stratum. Even if densities are similar among strata, there still may be nonhomogeneous variances among strata (e.g., because of differing stratum areas), in which case an estimator based on stratification of the counts, but estimating a common value of D for all strata, may be more efficient. On assumption that a ratio (vs. weighted) estimator is optimal (there are analogous choices for weighted estimators) (see Cochran, 1977), this suggests two choices for estimators in stratified designs. The separate ratio estimator (Cochran, 1977) is
&=
I --
l~s = ~_j Yi a i i=1 ai
--
~,~ wiaiy i
~_j wi a2 A,
i=1
i=1
/
with w i = 1 / v a r ( a i ) , is best linear unbiased. On assumption that var (a i) -- ai,2 the best linear unbiased estimator of population size becomes
]~)iai
i=1
with estimated variance
I
i=1
._ ( ~~ / i ~ l ~ ) a i = l
m m
= ~
v~(/~s ) = ~ M2 (1 - m i / M i) (S2y + D" 2i s2 ia - 2Disiay),
Nw = Dw A
=
(12.9)
I
mi
where Siy , Sia, and Siay a r e the estimated variances and covariance of counts and areas for stratum i. An alternative is the combined ratio estimator
l(lc- 19cA
a "m..~
--(i=~1 Mi~]i/i~= 1 Mi-ai) a (Cochran, 1977), with estimated variance with estimated variance V~r(~w ) = A21 - m/M m
~_~ 1 a~ (Yi -- E)wai )2/(m - 1). i=1
I
var(/~c) = ~ M/2 (1 - m i / M i) (S2y + ~2s2 a _ 2~cSiay). i=1
12.5.3. Estimators B a s e d o n Stratified D e s i g n s
Under conditions described in Section 5.3.2, stratification can reduce variance. In sample counts such as
mi
The separate ratio estimator has lower variance if density D is not constant across strata and is appropriate when there is sufficient replication within strata. However, if stratum samples are "small" and there are many
250
Chapter 12 Estimating Abundance Based on Counts
strata, there may be substantial bias in estimator (12.9) (Cochran, 1977). Because of the potential for bias, the separate ratio estimator should be used only if there is good empirical evidence for between-stratum variation in D (Cochran, 1977).
abundance from the total sample is obtained as in Eq. (5.23), = )~/~,
(12.10)
where f =1/~
12.6. P A R T I A L D E T E C T A B I L I T Y OF I N D I V I D U A L S O N SAMPLE UNITS As noted earlier, incomplete counts on sampling units can result in estimates that are biased low, in that E(/Q) < N. To the extent that detectability 13 varies over time, space, or among individuals, comparative inferences also can be compromised. It therefore is important to estimate 13so as to obtain unbiased estimates of N and also to test for homogeneous detectability so as to ensure comparability across population cohorts. Two general approaches are described below. In the first, detectability is estimated on a subset of sample units that appear in the sample, and this estimate is effectively applied to all sample units (double sampling; see Section 5.4.3). In the second approach, detectability is estimated on all sample units selected in the survey.
12.6.1. Estimation of Detectability Based on a Subset of Sample Units 12.6.1.1. General Approach The adjustment of counts by the detectability 13ordinarily requires collection of auxiliary data in addition to the counts. Sometimes these data can be collected simultaneously with the count data, but often they must be collected via independent or interleaving sampling. We consider here the use of separate but complementary surveys, each survey recording counts on sampling units according to a double sampling scenario as discussed in Section 5.4.3. Thus, an extensive survey records counts x i on m' sampling units, with counts Yi recorded by independent observers on a subset of m units. The Yi counts on the subset of m units are assumed to be obtained with detection probability of 1, and data from this subsample are to be used to estimate 13 for the extensive count data. The bivariate values (x i, Yi), i = 1, ..., m are used to establish a linear relationship between counts from the two surveys, which then can be used to adjust counts for the units sampled in the extensive survey. On assumption that the relationship between the two survey counts has zero intercept, an estimator of
m
m
= i=1 ~ Y i / ~i =1 Xi estimates the reciprocal r = 1/13 of detectability and m'
2=M~xi
.
m r
i=1
The variance of this estimate is v,.d~(/Qr) =
M 2
(1 - m / M ) (s} + f 2 s 2 - 2fSxy). m' x
(12.11)
When an assumption of a zero intercept between the two counts is not warranted, a regression estimator should be used, based on the model E(y) = f3o + [31x rather than E(y) = f3x as above. The parameters of this model can be estimated using ordinary or weighted least-squares methods (Draper and Smith, 1966) as appropriate. In either case, var(/Q) must be estimated using a variant of Eq. (12.2), where in addition to the variance due to incomplete sampling (described in the previous section), a component due to the estimation of [3 ([30 and [~1 for the regression model) is required (see Thompson, 1992). Although the double-sampling approach frequently is described as above for the situation where detection probability is one for the subset of m sample units, this restriction is not necessary. In many cases, complete counts on even a subset of sample units is not possible, and the data needed to estimate detection probability are expensive to collect. In such cases it is reasonable to estimate detection probability (and thus the actual number of organisms) on the sample units in the double sample, and then apply this estimate to all surveyed units. Survey design in double-sampling includes specification of both the number of sample units selected for extensive survey and the number of units selected for intensive survey and detectability estimation. Optimal survey design in such cases is an area of active research, and some initial recommendations have been provided by Thompson (1992) and Pollock et al. (2002). 12.6.1.2. A i r - G r o u n d Comparisons in Aerial Surveys Aerial surveys from fixed-wing aircraft offer an important example of survey-based correction for detect-
12.6. Partial Detectability of Individuals on Sample Units ability. Two sampling features typically are associated with aerial surveys. First, narrow, rectangular sampling units are usually more practical than square or circular sampling units (although the latter are feasible with helicopters). Second, the area of each sampling unit typically is variable. The width of these units is usually fixed, as a function of the flight altitude and an observation angle determined by window or strut marks (Rudran et al., 1996). Typically, observations are obtained from either side of the aircraft, with a "blind spot" directly below the aircraft, although in practice the observations are often aggregated into a single plot (e.g., Conroy et al., 1988). The length of each unit (and thus the area) usually is variable. The use of aerial surveys in wildlife biology has been reviewed by Pollock and Kendall (1987). A recommended procedure for delineating aerial survey units (Seber, 1982) is to (1) stratify the study area into more or less homogeneous strata (e.g., forest or wetland types), (2) establish a baseline in the direction of least environmental change (e.g., parallel to a coastline), and (3) select sampling units at random, perpendicular to this baseline (Fig. 12.1). In practice, the rectangular units (sometime called "transects") will be flown in sequence, with a random starting point. In some circumstances, an aerial survey may use sampling units that are either square or circular and of either fixed or variable area. For example, counts of breeding American black ducks (Anas rubripes) typically are based on square, fixed-area quadrats. Aerial surveys of breeding ducks in North American
/ boundary
/_,
,: "-.../
Aerial quadrat Baseline FIGURE 12.1 Illustration of an aerial survey design. Baseline (dashed line) is oriented perpendicular to environmental gradient (e.g., upland to wetland to pelagic habitats). Aerial quadrats are selected randomly along baseline, with central lines perpendicular to baseline and with endpoints determined by the study area boundaries (e.g., marsh habitat for ducks). Quadrat width is determined by aircraft elevation and angles of detection (see text).
251
prairies (Pospahala et al., 1974; Smith, 1995) employ a double-sampling scheme, in which ground counts of a subsample of sampling units (i.e., transects) are used to correct for visibility bias, on assumption that the ground counts provide nearly unbiased estimates of abundance on the sampled units. In this scenario, m' aerial units are flown, and from this sample, a subsample of m units is selected from which accurate ground counts are recorded. The bivariate values (x i, Yi), i = 1, ..., m are used to establish a linear relationship between the ground and aerial counts, which then is used to adjust aerial counts for the units sampled only by air as in Eqs. (12.10) and (12.11). Stratified (separate) estimates of [3 are warranted when surveys combine data from different habitats or species having differing detectabilities (e.g., Pospahala et al., 1974; Smith, 1995). For instance, Srnith et al. (1995b) found a nearly threefold difference in visibility rates between forested and nonforested habitats in aerial surveys of wintering mallards. Under these circumstances, combining count statistics from different habitats without habitat-specific correction leads to biased and misleading estimates of abundance. Even the use of the counts as indices to trend might be misleading, if the relative numbers in the different habitats change from year to year, as they do for wintering ducks (Smith et al., 1995b). The double-sampling approach assumes that ground surveys are complete (detection probability of one), but this assumption may not be always be appropriate (e.g., Jarman et al., 1996; Short and Hone, 1988). Additionally, it must be assumed that the complete and incomplete counts are of the same, closed population and are independent of one another. Obviously this assumption is violated if immigration or mortality on the sampling units occurs between the times at which complete and incomplete surveys are recorded or if the sampling units are imperfectly matched, as could easily happen with air-ground comparisons. At the same time, simultaneous surveys often are infeasible for logistic reasons and, even if feasible, would be difficult to conduct without violating the independence assumption (e.g., different observers "cueing" on each other; disturbance of animals by observers). Despite these difficulties, double-sampling remains a valuable, if imperfect, means of dealing with incomplete detectability.
Example We use an example of an aerial survey of moose (Alces alces) abundance described by Thompson (1992) to illustrate visibility adjustment via double-sampling. Sample counts of moose were taken on m' = 20 aerial survey plots from a study area of M = 100 plots of equal
252
Chapter 12 Estimating Abundance Based on Counts
area, and 240 moose were counted. For a subsample of m = 5 of these plots, 70 moose were counted on the ground, whereas 56 moose had been seen from the air. The resulting estimate of detection is ~ = 1/~ = 56/70 = 0.80. Based on the aerial survey alone the estimated count for the total population is
or objects are seen by one method but not the other, and which are seen by both; and (4) the population is closed between the two samples. The idea here is to consider as a marked sample the organisms or objects observed in the ground survey and use the proportion of these that are detected in the aerial survey to estimate detectability and hence abundance (see Section 14.1). The resulting estimator for the number of organisms or objects present in the subset of sample units surveyed by air and ground is /~/= (n I + 1)(n 2 + 1 ) _ 1, m+l
m'
x=M~_jx i m'
l=1
- 12~(240 ) = 1200, resulting in an estimate = 1.25(1200) = 1500
where n I is the number of objects seen by the aerial observer, n 2 is the number of objects seen by the ground observer, and m is the number of objects seen by both observers (see Section 14.1). It can be shown that lZ(/~) -- (nl + 1)(n2 + 1)(nl - m ) ( n 2 - m) (m + 1)2(m + 2)
The assumption that all individuals are counted in the sample ground plots frequently is unrealistic. In such cases, detection probability can be estimated from intensive efforts on a subset of sample units using any of a number of approaches (e.g., see Section 12.6.2 and Chapters 13 and 14), and double-sampling still can be used. For example, an aerial survey approach described by Magnusson et al. (1978) allows both aerial and ground counts to be incomplete and uses capture-recapture models (Chapter 14) to estimate abundance. The approach was developed for sessile organisms or objects associated with animal presence and activity (e.g., nests). Thus, some sample units are surveyed both by aerial survey and by a ground crew. In order to match detections from the two surveys, the organisms or objects typically are mapped by personnel conducting both the aerial survey and the ground survey. Following completion of the two counts on the sample units, the maps are compared and numbers of organisms or objects detected by aerial survey only, by ground survey only, and by both surveys are recorded. The assumptions of this approach are (1) that the sightings by aerial and ground observers are independent; (2) the detection probabilities are homogeneous, i.e., the same detection probabilities apply for all organisms or objects; (3) one can determine which organisms
(12.13)
is an essentially unbiased estimate of the variance (see Section 14.1). The detection rate for the survey is thus estimated as ~ 1 --
of abundance after adjustment for detection.
12.6.1.3. Incomplete Ground Counts
(12.12)
nl//~.
This estimate of detection probability then can be applied to all the sample units surveyed from the air using the double-sampling estimator of Eq. (12.10). The estimator in Eq. (12.12) is the well-known LincolnPetersen estimator and is discussed more fully in Section 14.1.
Example This approach to estimating detectability was used with aerial and ground surveys to estimate the abundance of osprey (Pandion haliaetus) nests (Henny and Anderson, 1979; cited in Pollock and Kendall, 1987). A total of n I = 51 nests were seen from the air, n 2 = 63 from the ground, and m = 41 from both locations. In this particular example, the entire area of the study was surveyed by air and ground, so a double sampling approach was not needed. Application of Eqs. (12.12) and (12.13) provides estimates of N = 78.24 and v ~ (/~) - 9.67. The estimated detection rate for the aerial method was ~ = 51/78.24 = 0.65. Unfortunately this method is unlikely to be useful for mobile populations, because of the difficulty of determining which animals are seen by either or both methods (Pollock and Kendall, 1987). Though the method may prove useful for fixed objects (such as nests, roosting sites, biologically relevant terrain features), we do not recommend its application to mobile populations.
12.6. Partial Detectability of Individuals on Sample Units
12.6.2. Estimation of Detectability Based on the Set of All Sample Units Here we again consider a survey with the objective of estimating the total number of animals in some large area of interest. However, instead of using a doublesampling approach, the detection probability is estimated on all sampling units that are surveyed. We consider both simple random sampling and sampling proportional to size of the sample unit. In both cases, we assume that the survey method includes a means of obtaining counts and estimating the associated detection probability. Thus, survey efforts yield an estimate N i of abundance and its conditional sampling variance va"'r(Ni[Ni), for each sample unit i. Additional details on estimation under such survey designs can be found in Skalski and Robson (1992), Thompson (1992), and Skalski (1994)
12.6.2.1. Simple Random Sampling Assume that there are M sample units in the area of interest, from which m are randomly selected with equal probability, yielding c~ = m/M. An unbiased estimate of the total abundance for the area of interest (denote as N T) is given by m NT -- M E m i=1
w h e r e / ~ i is based on counts adjusted as necessary for
detectability. The estimated variance is
253
12.6.2.2. Sampling Proportional to Size Because the variance of the total population estimate under simple random sampling is a function of the variation in abundance among sample units, it may be inflated by variation in the size of the sample units. Thus, if sample units are of unequal size, it may be reasonable to consider sampling the different units with probability proportional to their size. If units are sampled without replacement, then total abundance can be estimated using the Horwitz and Thompson (1952) estimator m /~i
=/El / where Pi is the selection probability for sample unit i; i.e., the probability that unit i appears in the sample of m units from the total of M possible units. Note that when selection probabilities are equal for all units (Pi = P), the above expression equals the previously presented estimator under simple random sampling. In many ecological surveys, sampling costs are fixed and depend on the sizes of the selected sample units. In that situation, sample size m is a random variable, so that the variance for N T is estimated as (see Skalski, 1994) m m m v,.d,r(/~/T) = ~ (1 - Pi)fil2 i=1 p2 + 2 ~i= 1 j>i ~ m
(Pij-
Pi Pj)l~il~j ~gii ~9/~j.
~-
+ E var(NilNi) va"r(/~/T) =
M2[ ( 1 - m/M)d2im
+
E[~r(l~i[Ni)]]M
(see Skalski, 1994), where m "2 __ E i = l
sNi
"
N
~
(l~i _ ~])2
(m - 1)
"
E rn 1~i i=1 ! m
~
and m
E(~r(1Cqi[Ni) =
Ei=l
V~r(l~i[Ni ) m
The first term of the sum in brackets reflects spatial variation in abundance among the different sample units and hence variation associated with selection of the m sample units. If the entire area is surveyed, m = M and this term vanishes. The second term of the sum is the average sampling variance or measurement error associated with the fact that [3<1 and hence that detection probability must be estimated.
i=1
~
r
where Pq is the probability that sample units i and j are both in the sample of m selected units. The probabilities Pi and Pij must be computed on a case-by-case basis. Skalski (1994) presents an estimator for the variance of/~/T for the simpler case in which the sample size m is predetermined. The above expressions are written in terms of 1~i and ~r(l~ilN i) and are very general. Thompson (1992) provides estimators and associated variances in terms of count statistics Ci and the e s t i m a t e s ~i of detection probability, for some specific sampling designs (e.g., simple random sampling with estimated detection probability the same for each sample unit).
12.6.2.3. Other Methods for Estimating Detection Probability The above estimators are based o n / ~ i and v ~ (1CqilNi) and thus on various types of counts and their associated estimates of detection probability [Eq. (12.1)]. Such estimators form the basis for several chapters in Part
254
Chapter 12 Estimating Abundance Based on Counts
III. Distance sampling is discussed in Chapter 13, and estimation of abundance using capture-recapture models for closed and open populations is discussed in Chapters 14, 18, and 19. In the remainder of this section, we introduce methods other than those based on distance sampling or the capture of animals for estimating detection probabilities and abundance with count statistics.
12.6.2.3.1. Multiple Independent Observers Even in the absence of actual capture at multiple times, capture-recapture modeling is applicable when two observers obtain independent counts on the same sampling units with the same counting method, in such a way that animals detected by one or both observers can be identified. This approach has been used in aerial surveys, but also has been applied to avian point counts (T. Simons and J. Sauer, unpublished) and to estimation of numbers of bird nests from ground counts (Nichols et al., 1986b). Assuming that independent counts can be obtained (Magnusson et al., 1978; but see Smith, 1995), one can estimate the number of animals not seen by any observer and thus the total number present, using capture-recapture estimators as above [e.g., Eqs. (12.12) and (12.13); also see Section 14.1]. The method can be extended to include multiple observers (see Nichols et al., 1986b), so that the models described in Chapter 14 can account for heterogeneity in detection probabilities (e.g., certain animals are more wary of detection than others). Magnusson et al. (1978) and Marsh and Sinclair (1989) have applied the independent multiple observer approach with apparent success. However, we note that the method seems most likely to succeed when individuals or clusters of animals can be distinguished easily and therefore can be readily assigned as seen or not seen by each observer (e.g., dugongs and kangaroos). With more abundant and uniformly distributed animals (e.g., waterfowl) (Smith et al., 1995b), it can be quite difficult to make such a determination (indeed, the definition of "group" can be problematic and arbitrary). Nevertheless, when logistics and the characteristics of the population permit, this method can be useful.
12.6.2.3.2. Multiple Dependent Observers A variation on the use of multiple independent observers is the method of multiple dependent observers, which was developed by Cook and Jacobson (1979) for use with aerial surveys, but also is applicable to ground-based surveys (Nichols et al., 2000b). In this case two observers are present during the survey, with one observer designated as "primary" and the other as "secondary." The primary observer notifies the sec-
ondary observer of each animal detected, either pointing out the animal directly or noting the direction and general distance of the detection. The secondary observer then records the animals detected by the primary observer and also surveys the area himself/ herself. Those animals detected by the secondary observer but not by the primary observer also are recorded by the secondary observer. On completion of the count, the data consist of the number of animals (1) detected by the primary observer and (2) missed by the primary observer but detected by the secondary observer. The observers then switch roles as primary and secondary observers for the next count (the switching of roles can occur when observations are initiated on a new sample unit, or it can occur halfway through the sampling for the same sampling unit). This situation can be described more formally using the notation of Cook and Jacobson (1979). Define xij as the number of individuals counted by observer i (i = 1, 2) on sample units when observer j (j = 1, 2) was the primary observer. The counts for the primary observer include all animals detected, whereas the counts for the secondary observer include only animals detected by this observer that were missed by the primary observer. Define Pi as the detection probability for observer i, assumed to be the same whether observer i is serving as primary or secondary observer. Further, let N 1 denote the true number of animals in the sample units for which observer I served as primary observer. The count Xll can be viewed as a binomial random variable with parameters N 1 and Pl and distribution B(N 1, Pl). Conditional on x11, the count x21 also can be viewed as a binomial random variable, withx21"--B(N1 - Xll , P2). The joint distribution of (x11, X21) c a n thus be written as the product B(N1, pl)B(N1 - x11, P2)-Similarly, the distribution of (X22 , X12) can be written as B(N2, pa)B(N2 X22, Pl)" Finally, assuming that the pairs (Xll , X21) and (X22 , X12) are independent, the joint distribution for all four random variables is simply the product B(N 1, pl)B(N1 - - X 1 1 , p2)B(N2, p2)B(N2 X 2 2 , Pl)" Because the N i are unknown, it is difficult to use the above distribution directly for estimation. Cook and Jacobson (1979) thus conditioned on the total number of animals detected in the sample units for which each observer served as primary observer. The probability of detection by at least one of the observers is given by p = 1 - (1 - pl)(1 - P2), and this probability applies to each count. Thus, the distribution of Xll qx21 is B(N 1, p) and that of x22 if- X12 is B(N 2, p). Further, the probability that an animal was a member of Xll, given that it was a member of the sum Xll + x21, is given by Pl/P. The complement of this probability, the probability that an animal was missed by observer I and detected only by observer 2, is given by (1 - Pl)P2/P. -
-
12.6. Partial Detectability of Individuals on Sample Units For estimation purposes, we thus rewrite the joint distribution of the four random variables as B(N1, p)B(x11 if- X21'
Pl/p)B(N2, P)B(x22 +
P2/P)"
X12'
With this notation in mind, the approach to estimation is first to use the conditional (on detections) distributions e(x11 q- x21 , Pl/P) and B(x22 q- x12 , P2/P) to estimate detection probabilities. Cook and Jacobson (1979) present the following maximum-likelihood estimators for the general model in which detection probability differs for the two observers: X11X22 -- X12X21
255
mals detected by observer 2 when observer I is primary is the appropriate abundance, times the probability that an animal is missed by observer 1, times the probability that an animal is detected by observer 2. Once estimates of these detection probabilities are obtained, the natural estimator [following (Eq. 12.1)] for population size over the surveyed area is = x../#
where x.. = Xll + x12 + x21 + x22. An associated variance estimator is va"~(/Q) : (x..)2v~r(p) + (x..)(1 - fi)
X11X22 if- X22X21
]~4
]~2
"
X11X22 -- X12X21 P2 --
Confidence intervals for /Q can be approximated using the approach of Chao (1989). The estimation is based on the estimated number f0 =/Q - x.. of animals not detected, with ln(f0) treated as an approximately normal random variable. This results in a 95% confidence interval of
X11X22 if- X11X12
and /~ = 1
X12X2~1.
-
X22X11
An asymptotic variance estimator for the overall detection probability estimator ]~ is given by var(#lx..)
(1 - p ) 2 p [ X..
1
+ 1_~+
1
p202
p2(1 -- pl)01
[p101
(x + 0/c, x + 0c), where C = exp {1.96[ln(1 + var(1Q)lf2)]l/2}.
12.6.2.3.3. Marked Subpopulation pl(1 - p2)02 ' where 0i = x.i/x., and x.i = Xli + x2i (Cook and Jacobson, 1979). Detection probabilities also can be modeled as, e.g., constant for the two observers or as functions of habitat characteristics of the different sample units, using software developed by J.E. Hines (see Nichols et al., 2000b). Note that the point estimates for detection probability also can be obtained simply by equating the four sufficient statistics with their expectations: E(x11 ) =
NIP1,
E(x21 ) -
Nl(1-pl)P2,
E(x22) --
N2P2"
and E(x12) :
N2( 1 - P2)Pl.
The resulting equations then can be solved to yield the estimators for detection probability. Examination of the above expectations provides an intuitive basis for the dependent-observer approach. For example, the expected number of animals detected by observer 1 as the primary observer is simply the product of the appropriate abundance and the detection probability for that observer. The expected number of additional ani-
In some situations, a marked subpopulation of individuals is available for use in estimating detection probability in observation-based surveys. The general study design involves efforts to capture and individually mark individuals in an initial sampling effort, followed by subsequent sampling of the population via observation rather than capture. Specifically, resightings of marked animals along with counts of unmarked animals during subsequent sampling periods provide the data needed to estimate detection probability. The observation sampling may be carried out for a single sampling occasion or on multiple occasions. This approach has been used for, e.g., aerial surveys of ungulates that have been tagged with marks that are remotely visible (e.g., Rice and Harder, 1977; Bartmann et al., 1987) and for boat surveys of immature eagles marked with patagial tags (Arnason et al., 1991). In many applications, the total number of marked animals present in the sampled area at the time of the observational sample (denote as M) is assumed to be known. This assumption is most likely to be true when marked animals also are radiocollared (so that their presence in the sampled area at the time of the sighting survey can be confirmed) or when the marking is done only a short period before the sighting survey. The observational or sighting survey yields two statistics, the number of marked animals detected (m) and the
256
Chapter 12 Estimating Abundance Based on Counts
total number of marked and unmarked animals detected (n). A natural estimator for detection probability is simply the ratio, = m/M,
of the number of marked animals observed to the number that are known to be marked. If an abundance estimate for the sampled area is desired, it can be computed as in the canonical estimator (Eq. 12.1)
The situation with a known marked subpopulation and a single resighting survey illustrates the approach of utilizing marked subpopulations, which can be extended easily to handle the situation wherein the marked subpopulation is known and multiple sighting surveys are conducted (Rice and Harder, 1977; Bartmann et al., 1987; Minta and Mangel, 1989; White and Garrott, 1990). Investigations of estimator performance have shown that the joint hypergeometric maximum likelihood estimator of Bartmann et al. (1987) performs well (White and Garrott, 1990; White, 1993; Neal et al., 1993). Though the estimator cannot be written in closed form, it is implemented (along with others) in program NOREMARK (e.g., White, 1993). In cases where marking is not conducted immediately before the sighting survey(s), the number of marked animals in the survey area will not be known but must be estimated. Arnason et al. (1991) developed an estimation approach for this situation, which allows for multiple sighting surveys. The estimator cannot be computed in closed form, but implementation software is available from Arnason et al. (1991). 12.6.2.3.4. Sighting Probability Models
Still another approach for estimating detection probability in observational surveys involves development of sighting probability models (e.g., Caughley et al., 1976; Samuel et al., 1987). The basic approach is to develop models that predict detection probability as a function of factors (e.g., aircraft speed and altitude, weather and daylight conditions, habitat in which observation is made, animal group size) that can be measured and recorded during the observational survey. For example, Samuel et al. (1987) developed a model for detection probability of elk (Cervus elaphus) in aerial surveys conducted in Idaho. Elk were radiocollared, experimental surveys were conducted, and some radiocollared elk were visually detected. Radiocollared animals that were not detected during the survey then were located using a radio receiver. Potentially useful covariates such as group size and vegetation cover were obtained for all radiocollared animals (whether detected or not during the survey), and the resulting
data were used with logistic regression to develop a model of sighting probability as a function of the measured covariates. The detectability model of Samuel et al. (1987) has been applied in operational aerial surveys for elk (Samuel et al., 1987). Thus, each time an animal is detected from the air, a vector of covariates (those used in the sighting probability model) also is recorded. Detection probability for animal i then is predicted as a function of the covariates x i for the animal, [3* = f(xi), where the function is the sighting probability model developed during the experimental surveys. Abundance for a surveyed area then is estimated using the HorwitzThompson estimator (see Horwitz and Thompson, 1952; Steinhorst and Samuel, 1989): /(/=~
C
1
i=1 ~ '
where C is the number of animals that are counted. Note that if the predicted detection probability is the same for all animals (i.e., if [3* - [3*), then the Horwitz-Thompson estimator simply becomes the canonical estimator, N = C / ~ . It should be emphasized that the utility of a sighting probability model depends heavily on the correspondence between conditions under which the model was developed and those under which it is applied. 12.6.2.4. Bounded Counts
If all animals in a sampling unit potentially can be counted once and only once, and if repeated sampling of units is possible, then the method of bounded counts can be used to provide estimates of abundance and approximate confidence intervals (Robson and Whitlock, 1964; Regier and Robson, 1967; Seber, 1982). Let N be the true abundance and N(k ) and N(k_l) be the largest and second largest sample counts obtained on k successive sampling occasions. Then an estimate of N having bias of order 1/k a is 1~ = N(k ) + (N(k) -- N ( k _ l ) )
= 2N(k) -- N(k-1)
with approximate 100(1 - oL)% confidence interval N(k ) < N < [N(k ) -- (1 - oL)N(k_l)]/(x.
Though we do not advocate sampling designs with this estimator, it does provide a means for obtaining estimates and some measure of confidence, if the only available data are a series of incomplete counts and estimation of detection rates using other approaches is not possible. However, there is no assurance that the upper limit of a bounded, incomplete count ap-
12.7. Indices to Population Abundance or Density proaches the number of animals in the study area and no assurance that estimated confidence intervals ever achieve coverage of the true parameter value. In situations where detection rates are low, bounded counts typically underestimate abundance. The difficulty is in knowing how low detection rates actually are, given that the method provides no information on them. Because it usually is possible to collect auxiliary data with which to estimate detection rates directly, sampling schemes that incorporate auxiliary information generally are preferred to the method of bounded counts.
257
that if ~1 5h ~2 then differences in the counts can be attributable to either a change in population size or a change in detectability, and without additional information about detectability it is not possible to determine which. Often, the ratio C2//C1 of count statistics obtained at different times or places is used to estimate rates of change over time or relative spatial differences in abundance. However, a ratio estimate of change also can be badly biased, unless the detection probability is equal for the times or places being compared (see discussion in Section 15.1.1). As discussed earlier, a natural estimator of Ni is 1~i Ci/~i , and if an estimator of ~i w e r e available there would be no need to use an uncorrected index. In practice, users of direct indices assume that =
12.7. I N D I C E S T O P O P U L A T I O N ABUNDANCE OR DENSITY An index of abundance or density is "any measurable correlative of density" (Caughley, 1977). That is, an index is a field measure (e.g., a count statistic) that contains information about the relative size or density of the population. Indices typically are used for species that are difficult to capture or observe directly (e.g., because of nocturnal or secretive habits). Sometimes, however, indices are used when other more appropriate methods (e.g., line transect, capture-recapture) are available. Before using an index, we urge readers to consider carefully the objectives of the study and assess whether other methods may be more appropriate. Surveys based on indices frequently are less expensive and require less effort than those based on formal estimation methods. However, indices also yield weaker inferences. The decision of whether to use an index or a formal estimation approach should be based on the relative importance of costs versus inferential strength.
12.7.1. Survey Counts as Population Indices Population counts can be useful as indices to population size, even when they are severely biased. Assuming detectability ~ is constant over time or some other dimension across which comparisons are made, population counts represent patterns in population size irrespective of bias. For example, the estimate C2 - C1 of change in population size has expected value E(C 2 -
C1 ) -- N 2 / ~ 2 -
N1/~1
,
which reduces to E(C 2 -
C1 ) -- ( X 2 -
N1)/~
if ~1 = ~2 = ~" Under these conditions, systematic differences in the counts C1 and C2 are attributable to changes in population size over time. Note, however,
f3 = E ( C i ) / N i ,
i.e., that [3 is constant over i (time, space, or other dimensions of interest). Examples of direct indices are capture or harvest indices and incomplete counts on plots or from line transects, including singing counts of territorial birds. For these types of data there are estimation procedures that, under an appropriate sampling design, allow for unbiased estimation of abundance or density. In lieu of these, the assumption of a homogeneous, proportional relationship for the index is critical for an uncorrected index. It is possible to collect most count data in a manner that allows for a test of homogeneity in detection probability (e.g., see Skalski and Robson, 1992). Where possible, homogeneity of [3 should be tested and the index only used in situations where heterogeneity does not occur or cannot confound comparisons of interest (Yoccoz et al., 2001, Pollock et al., 2002).
12.7.2. Relationship of Indices to Abundance As with incomplete counts, the association of an index to abundance or density typically is positive; that is, as N or D increases, the index increases. Occasionally an index is negatively associated with N or D (e.g., amount of habitat per animal; nutritional indicators). In general, the index should be monotonic over reasonable values of N (Fig. 12.2), because indices that are monotonically related to abundance permit inferences about ordinal or relative changes in abundances. Some indices, however, are nonmonotonic, increasing over one range of N or D but decreasing over others (Fig. 12.3), because of some density-dependent inhibition to the behavior producing the index. In such instances it is not possible to make even ordinal inferences about abundance or density, unless the true relationship between the index and population is known
258
Chapter 12 Estimating Abundance Based on Counts Note that this model allows for a nonzero intercept (Fig. 12.4a), as when there is a threshold abundance N Obelow which individuals become essentially undetectable in the field. In this case, C = 0 for 0 < N < No. If it can be assumed that a zero value for the index is associated with zero abundance (Fig. 12.4b), then a proportionality model
20
15
8r-. "1o r-t'~
10
<
0 X "0
E(C) = f3N
5
0
,
0
200
i
i
400
600
Absolute Abundance
,
,
800
1000
(N)
FIGURE 12.2
Example of a monotonic relationship between an index to abundance (C) and absolute abundance (N).
a priori and is unchanging. Because this is never true in practice, nonmonotonic indices usually are of little practical use. Beyond simple, ordinal comparisons, one typically desires information about the proportional relationship of the index to abundance or density. A typical inquiry might focus on whether the doubling of an index implies the doubling of abundance (see Section 15.1.1). We describe below three categories of relationships of general interest.
is appropriate. Note the similarity between these relationships and the ratio and regression models for adjusting incomplete counts. Most interpretations of indices in the ecological literature implicitly assume that the index is of this form. To perform well at predicting changes in relative abundance, the relationship of an index, in addition to being linear, also must be precise. In some cases, the index-parameter relationship is monotonic, even proportional, but the variance of the observed index is high for this relationship. For example, Diefenbach et
a
~"
6O
(D O r "10 r"
12.7.2.1 Linear Relationship with Constant Slope
40
..Q
<
O
Biologists often think of indices as involving a linear relationship between the index and a relevant biological attribute. For example, a count C might be related to the population size N by
2o
x r"
0
-20 0
200
E(C) = f3o + f31N,
400
600
800
1000
800
1000
Absolute A b u n d a n c e (N)
b 100 20 rv 15
or
O r
"10 r
"10 r
<
10 < @ x (D ~9
O .6..., x (D "10 tin
5
.E
0 0
i
|
|
J
,
200
400
600
800
1000
Absolute Abundance
FIGURE 12.3
(N)
Example of a nonmonotonic relationship between an index to abundance (C) and absolute abundance (N).
0
200
400
600
Absolute Abundance (N)
FIGURE 12.4
Example of a linear relationship between an index to abundance (C) and absolute abundance (N) with (a) detection threshold and (b) the absence of a detection threshold.
12.7. Indices to Population Abundance or Density
al. (1994), in an experimental evaluation of scent station indices for bobcats (Felis rufus), found a large range of values of the index for similar levels of density. In this situation, use of a single index survey by itself provides low precision for temporal or geographic comparisons.
12.7.2.2. Linear Relationship with Nonhomogeneous Slope Figure 12.5 illustrates the situation encountered when detectability differs, for example, among habitats, for an index that is otherwise well behaved (e.g., linear proportional). In the example, the same value of the count statistic represents two different levels of actual abundance, depending on the habitat from which the index was obtained. This condition induces bias in comparisons between the two habitats and affects the validity of aggregate indices if not corrected on a habitat-specific basis. An example involves aerial counts of wintering waterfowl (Smith et al., 1995b). Visibility of mallards in forested habitats is approximately 0.3, but is 0.7 or higher in agricultural and other nonforested habitats. Thus, counts of mallards obtained from aerial surveys represent differing proportions of the actual population, depending on the distribution of birds over the habitats. Consider two surveys separated by a period of a decade, each with actual abundances of 100,000 ducks, but with 25% in forested habitats and 75% in nonforested habitats in the first survey, and the reverse proportions in the second. The expected values of the respective count statistics would be
E(C)
= ~ I N 1 q- ~2N2
= 0.3(25,000) + 0.7(75,000) = 60,000
100
8O O o r-
E
60
.s
<
O X (D "{3
40
t
0
in the first survey, and E(C) =0.3(75,000) + 0.7(25,000) = 40,000 in the second. If the count statistics were used as an index to population change, an incorrect inference of a population decline (K = 40,000/60,000 = 0.67) might result when in fact population size is stationary (K = 1.00). Without stratification by habitats, and correction for differential visibility rates by habitat, the count statistics are meaningless as indices of population size. The assumption that detectability is constant over time a n d / o r space is infrequently tested, and when tested (e.g., Smith et al., 1995b), is often refuted. Thus, reliance on counts as indices can be seriously misleading, notwithstanding the fact that many wildlife surveys fall into this category and data from them often are used for management. We advocate the use where possible of techniques that produce more reliable estimates of abundance, through the collection (perhaps as part of double or multistage sampling designs) of auxiliary data that allow for the calibration of counts via the estimation of detection probability. At a minimum, one should collect the necessary information to test critical assumptions about detectability (Skalski and Robson, 1992).
12.7.2.3. Nonlinear
Relationships
As suggested earlier, some indices can be expected to exhibit a nonlinear (or even nonmonotonic) association with abundance, if there are density-dependent behavioral changes operating to change per-animal manifestation of the index. For example, carnivores may visit scent stations less frequently at high densities than at low densities, because of behavioral inhibitions (Diefenbach et al., 1994). Also, some indicesnfor instance, those based on frequencies or p r o p o r t i o n s n a r e intrinsically nonlinear. If an index is based on the proportion p of positive values for the index (e.g., a visit by an animal to a tracking station) in n sampling units, then p and the number x of positive values are bounded above by I and n, respectively, and the index is theoretically related to density D = N/A by
p=l-e
20
,..
,
,
,
,
w
0
200
400
600
800
1000
Absolute Abundance
F I G U R E 12.5
(IV)
Example of a linear relationship between an index
to abundance (C) and absolute abundance (N) for two habitats in which detectability is different.
259
-D
(Caughley, 1977). Likewise, ordinal classes of abundance are not indicative of a proportional or other ratio relationship for abundance: abundance ranks of {2, 4, 5, 1, 3} could have arisen from the counts {200, 400, 500, 100, 300} just as easily as from the counts {201, 204, 205, 201, 203}.
260
Chapter 12 Estimating Abundance Based on Counts
12.7.3. Indices Other Than Counts Indirect indices are based on evidence of the animal's presence, other than direct observation of the animal. Examples include track counts for terrestrial vertebrates, scent station surveys for carnivores, scat surveys, counts of structures (nests, lodges, food caches, etc.), and auditory or other cues. In most cases, the index is assumed to take a value of 0 when there are no animals present and is assumed to increase (ideally, proportionally) as abundance increases. Under these conditions the relationship
theless, the use of an index may be warranted even if other methods are available, provided (1) an investigation focuses on relative (vs. absolute) abundance and (2) the assumptions of a homogeneous, proportional relationship between the index and abundance (or density) are met. In general, relative differences in abundance (e.g., between two management treatments) can be addressed via hypotheses of proportional abundance (Skalksi and Robson, 1992), e.g., H0: N 1 / N 2 -1. Thus, the ratio T = C1//C2 of counts may provide an estimate of N 1/N 2, the mean and variance of which inform a test of proportional abundance. We note that
E(C) = f3N
may be appropriate, except that the index C no longer represents the number of animals detected, and thus 13need not range between 0 and 1. Otherwise, the same principles that apply to direct indices also apply to indirect indices.
12.7.4. Design and Interpretation of Index Surveys 12.7.4.1. Sampling Design Considerations With proper design and analysis, index surveys can provide important information about relative abundance of the population and in some cases may be the only practical alternative. However, index surveys often are conducted in an ad hoc fashion, with little attention to proper design and analysis. Such surveys are of limited value and may well provide misleading information about the population. The design of an index survey should follow the same principles as discussed in Chapter 5 for sample surveys. Thus, attention must focus on defining the target and sampled populations, establishing objectives (e.g., desired precision levels, cost constraints), and selecting sample units so as to meet survey assumptions and achieve its objectives. In addition to these requirements, attention must be focused on the relationship of the index to the quantity of interest (abundance or other parameter). If the index is to be used to estimate a parameter, the index must be calibrated so that an unbiased estimate can be produced. If the index is to be used for comparative purposes, homogeneity of the associated detection probability across comparative categories (e.g., time, space) must be assured.
12.7.4.2. Use of Indices vs. Other Estimation Methods If used without validation of basic assumptions (e.g., proportional relationship to abundance, homogeneity), indices can provide misleading comparisons. Never-
E(T) = E ( C 1 / C 2) E(C1)/E(C2)
because C 1 and Ca are based on individual samples, with E(C1)
=
~1N1
E(C2)
=
~2N2
and
so that E(T) ~ ~1N1/~2N2 . From this it follows that T is approximately unbiased (but see Barker and Sauer, 1992) when ~1 = ~ 2 = ~ , i.e., E(T) ~ ~N 1/~N 2 = N 1 / N 2.
If instead the indices are adjusted individually by a sample estimate of [3, for each count we now have an estimate of abundance (Ni = Ci/~i) with approximate variance var(~i) ~ [var(Ci)c 2 + var(~i) ] ~ / 2 N2 from the delta method. The components of variance correspond to sampling variation in the count statistic Ci and variation due to estimating 13i. The index, of course, only involves variation in the first component, with var(Ci) ~- var(/~/i). Thus, hypothesis tests based on the former will be more powerful than those based on the latter [see discussion in Skalski and Robson (1992)]. The difficulty with this approach is that most indices have neither been calibrated (i.e., 13 is not estimated) nor validated (tested under a range^of conditions to determine if Ci predicts N i, given 13), but are used as if they have been. This situation is especially serious if 13 is variable over time or with respect to different
12.8. Discussion habitat conditions, treatments, or other comparisons of interest. The dilemma is that potential bias and confounding usually cannot be assessed unless ~i has been estimated, ideally as an adjunct to the study (e.g., in a double sample). If homogeneity is supported, then the index may be used for comparative inferences. For some data structures (e.g., capture-recapture) it may be possible to construct homogeneity tests to assess whether bias is confounded over treatments or other comparisons, without actually estimating ~i- If heterogeneity exists, and particularly if it is confounded with the hypotheses of interest, estimation procedures must be used to estimate f~i, which can then be used to construct the appropriate test statistics. Alternatively, the hypotheses (along with ~i) can be incorporated as part of a model structure (e.g., Lebreton et al., 1992) and examined as part of the model assessment. Although we have provided an approach to index use (e.g., calibrate the index, test for heterogeneity of detection probabilities, and use the index cautiously when detection probability is concluded to be a constant) that is conceptually sound, we confess substantial pessimism about the use of indices. This pessimism stems from experience with formal estimation methods, which involve both count statistics and estimated detection probabilities. When such methods are used in real-world sampling situations, there typically is evidence of variation in detection probability over most dimensions (time, space, habitat). Even in the rare case where evidence of heterogeneous detection probability is not found during a calibration study, this result cannot be applied safely to other times and places, so that calibration and testing for homogeneity are required there as well. Hence the recommendation is to always estimate detection probabilities on at least a subset (double-sampling approach) of sample units. More generically, we advocate the use of formal estimation methods for most surveys designed for abundance estimation (also see Thompson et al., 1998; Nichols et al., 2000b; Yoccoz et al., 2001; Pollock et al., 2002).
12.8. D I S C U S S I O N The methods in this chapter apply basic principles of sampling design and estimation to the problem of estimating abundance or density for closed populations. Some of the methods make strong assumptions
261
about the detectability of organisms on the individual sampling units (study plots, quadrats, transects, etc.), either assuming that all individuals are counted or that a fixed relationship exists between the counts and the parameter of interest. In some cases, these assumptions can be relaxed, as when auxiliary data (e.g., ground comparisons of aerial surveys) are available to estimate this relationship and thereby obtain unbiased estimates, or when only a relative measure is needed and the relationship is shown to be homogeneous over the dimensions of comparison (e.g., time, space). In some instances, simple counts or indices may be all that is possible or practical, but in most cases it is possible to collect auxiliary data under more elaborate sampling designs, to allow for robust modeling of detectability. We urge readers to consider more elaborate sampling schemes like those described in the remaining chapters of Part III, as alternatives to simple counts or indices. We particularly discourage the practice of (1) the collecting and use of count data under ad hoc designs, in which inadequate attention has been paid to sampling principles; (2) the use of ad hoc estimators with poor or unknown statistical properties; and (3) the use of index data that have neither been calibrated nor tested for homogeneity of detection. In Chapter 13 we consider sampling designs in which incomplete count data are collected from sample lines or points, together with line-to-individual and point-to-individual distances. These auxiliary data are used to estimate detectability, which in turn can be used to produce unbiased estimates of density and abundance. In Chapter 14 we consider designs in which the counts are from captures of animals from the population, and recaptures are used to model and estimate rates of detection. The methods of Chapters 13 and 14, together with the methods in this chapter, adhere to the principles outlined at the beginning of this chapter, under population closure. We will revisit abundance estimation under open population conditions in Chapters 18 and 19. Because abundance and density estimation are amenable to different sampling schemes and assumptions, multiple methods are potentially available to estimate these parameters. In Appendix G we provide a guide to computer software for estimating abundance. The appendix can be used as an adjunct to the material in Part III and serves as a convenient guide and reference to the methods for abundance estimation for situations that arise in the field.
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CHAPTER
13 Estimating Abundance with Distance-Based Methods
13.1. POINT-TO-OBJECT METHODS 13.1.1. Sampling Scheme and Data Structure 13.1.2. Models and Estimators 13.2. LINE TRANSECT SAMPLING 13.2.1. Sampling Scheme and Modeling Approach 13.2.2. Assumptions 13.2.3. Statistical Models 13.2.4. Estimating the Distribution of Distances 13.2.5. Maximum Likelihood Estimation 13.2.6. Estimating the Variance of/~ 13.2.7. Density Estimation with Clusters 13.2.8. Model Selection and Evaluation 13.2.9. Interval Estimation 13.3. POINT SAMPLING 13.3.1. Sampling Scheme and Data Structure 13.3.2. Models and Estimators 13.3.3. Assumptions 13.4. DESIGN OF LINE TRANSECT AND POINT SAMPLING STUDIES 13.4.1. Field Procedures 13.4.2. Sample Size Determination 13.4.3. Stratified Sampling 13.4.4. Experimental Design and Replication of Study Populations 13.5. OTHER ISSUES 13.5.1. Estimation When g(0) ~ 1 13.5.2. Cue Counting 13.5.3. Trapping Webs 13.6. DISCUSSION
of detection, which in turn can be used to adjust the incomplete counts for detectability. Distance-based approaches can be seen as a special case of the canonical estimator described in Section 12.2, in which a count statistic is adjusted with an estimate of detectability. A number of distance-based methods have been used by ecologists; however, many require unrealistic assumptions about the nature of the sampled populations a n d / o r the sampling methods, and many are nonrobust to violations of these assumptions. In this chapter we emphasize more recent robust methods for distance-based estimation, and in particular we discuss methods based on distances from randomly located points and lines. For each approach we describe assumptions, statistical models, estimation procedures, and field sampling protocols.
13.1. POINT-TO-OBJECT
METHODS Point-to-object estimators are mainly applicable to the study of density and spatial pattern of sessile organisms such as plants. In animal ecology the principal applications involve estimation of the density of nests or animal signs (e.g., fecal pellets), for use in determining other population parameters (reproduction rate, abundance). When used to estimate density, these methods typically assume that individuals are randomly distributed (i.e., by a Poisson point processmsee Section 10.2) over an area, so that the expected number of animals in a specific subarea is described by a Poisson distribution. One approach is to select points in the area at random and measure the distances from
The methods described in this chapter use incomplete count data along with distances between an observer and individual organisms to estimate density. Distance data allow one to estimate the probability
263
264
Chapter 13 Estimating Abundance with Distance-Based Methods
each point to the nearest individual. Alternatively, one can select individuals at random and record distances from them to their nearest neighbors in the study area. Applications of point-to-object methods in plant ecology often focus on the reverse problem of estimating a spatial distribution of individuals based on a known density. For example, all the plants on a given study plot of known area might be counted, so that the density is known with certainty. Random points then are chosen in the plot, and the distance to the nearest individual from each point is measured. These distances are used with moment or m a x i m u m likelihood methods to estimate the parameters of a statistical distribution and to test hypotheses about spatial pattern (Pielou, 1977). Again, the density of the population is assumed known, and inference centers on the spatial distribution of organisms. With animal populations, interest typically focuses on estimating abundance or density, and a priori information is not available about the spatial distribution of individuals. Thus, estimation methods that are robust to a variety of potential spatial distributions are especially useful.
13.1.1. Sampling Scheme and Data Structure A typical sampling scheme for point-to-object methods involves two steps. First, n sample points from a study site of area A are selected at random. In practice this might be accomplished by gridding the area at some arbitrary level of resolution and selecting grid points by a pair of random draws, one for each coordinate axis. For each of the n points selected, the distance is measured to the nearest individual (e.g., animal). Variations on this method include measuring the distance to the 2nd, 3rd, ... kth nearest individual. We limit discussion here to first-order methods.
13.1.2. Models and Estimators Estimators of density D for the sampling scheme above are based on an assumed random spatial distribution. Consider a population of size N distributed over an area A, and let xi represent the distance between a randomly located point in A and individual i, i = 1, ..., N. On assumption that individuals are randomly distributed over A, the probability that individual i is in the circular area a = ~rx2 about the point is P(i ~ a) = P ( x i < x) = a/A.
More generally, area a is occupied if this condition is met for at least one individual: N
P(occupancy [ x) = 1 - l-I P(xi > x) i=1
= 1 - (1 - a / A ) N -Da
-~1 -e = 1
-
(13.1)
e -D~x2,
where D = N / A . The exponential approximation in this formula is based on lim(1 + x) 1/x - e, x--*0
which applies whenever a is small relative to the sampling area A. Relationship (13.1) assigns a probability to the distance between a randomly located point and the nearest individual to it. Simple differentiation of Eq. (13.1) yields the corresponding density function f ( x ) = 2(D~rx) exp(-D'rrx2).
(13.2)
This function is parameterized by D, so that a sample of distances x 1, ..., Xn between n random points and the nearest individuals to them can be used to derive a m a x i m u m likelihood estimate of D by /~)=
F/ H
x2
~
A bias-adjusted estimator is /)1 =
n-n 1
'IT ~ i = 1 x2
(13.3)
(Seber, 1982), with estimated variance v~r(/)l)
= (n -- 2)"
(13.4)
More general forms of the above estimator can be developed with samples to the rth closest individual (Morisita, 1957; Seber, 1982). Of course, higher order estimators are based on density functions more complicated than Eq. (13.2). Unfortunately, this estimator has been found to be particularly nonrobust to n o n r a n d o m distributions. Batcheler (1971, 1975) suggested the modified estimator /~)2 =
[
nl
'IT ~ n l I X 2 q-
(n - n l ) a 2
]'
(13.5)
where the distance x i is used in the estimation only if it is less than some value R that is chosen to reduce
13.2. Line Transect Sampling
uals along transects located over the range of a population of interest. It is a specific type of distance sampling in which the sample consists of one or more lines, which are traversed by observers on foot, by vehicle, or other means. Observers count individuals detected from the line and measure the distance from each to the line. The distance data then are used in conjunction with the number of individuals detected to estimate detection rates and thus to adjust the count to obtain an estimate of density.
the resulting sample size n by approximately 50% to a new sample size n 1. Even the modified estimator in Eq. (13.5) is nonrobust to three assumptions not likely to be met for most real populations of animals. First, the estimator requires an assumption that sampling is from a population distributed over a (theoretically) infinite area with constant density D. Second, and following from the first assumption, the number of counted individuals in a randomly chosen subarea must be distributed as a Poisson random variable. Finally, animals must be detected with probability one, i.e., the counts and respective distances must correspond to a population of completely detectable individuals. Unfortunately, these assumptions, especially perfect detection, seldom are met in reality, even for sessile objects (e.g., bird nests). For animal ecologists, these methods are of historical interest only, and they have been supplanted by the robust line transect and point estimation procedures described below.
13.2.1. S a m p l i n g S c h e m e a n d Modeling Approach
In line transect sampling, one records the locations of individuals on either side of one or more transects, as a basis for estimating the effective area that is sampled and hence the population density. Assume for now that line transects of specified length L are located within the range of a population of interest. Multiple transects may be involved, each (possibly) randomly located with (possibly) random orientation over the population range. If individuals are randomly distributed over the area, then systematic positioning of transects is acceptable. Otherwise, random transect placement is necessary to ensure accurate statistical inferences. Sampling designs are discussed in greater detail in Section 13.4. Each transect is traversed systematically, starting at one end of the transect line and proceeding along the line at a constant pace. The locations of all individuals seen within a predetermined (possibly indefinite) distance from the transect are recorded. Locations are described either as distance x i perpendicular to the transect or as radial distance r i from the observer, along with the angle 0 i of incidence to the transect. Because robust estimation depends on perpendicular distances, it is essential that either xi is recorded for every individual or that both r i and 0 i a r e recorded to allow for computation of x i. It is especially important that all individuals located directly on the transect line are observed and recorded. Figure 13.1 illustrates the basic layout of a line transect sample, together with data recorded in such a scheme.
Example
Batcheler (1971) described an effort to estimate density of trees in a pine plantation, involving the measurement of distance from each of 100 points to the nearest tree. In this instance, the assumption of perfect detection likely was met, so that the estimation procedure above may be applicable. Batcheler (1971) provided evidence of n o n r a n d o m distribution of the pine trees from an analysis of mean-variance ratios and thus advocated use of D 2 in Eq. (13.5). We computed/~1 and its asymptotic 95% confidence interval from Eqs. (13.3) and (13.4), using all 100 data points (Table 13.1). Comparison o f D1 a n d L)2 reveals little difference between them, with the estimate/~2 under 50% sampling (truncatingdistances > 7.8 ft) close to the untruncated estimate D1; truncated and untruncated density estimates were well within the 95% confidence interval of each other.
13.2. L I N E T R A N S E C T S A M P L I N G The method described in this section, known as line transect sampling, involves the observation of individ-
TABLE 13.1
265
Summary Statistics and Estimates of Density of Trees in a Pine Plantation a
n
n1
R
~ x2i (ft2)
100 100
100 50
7.8
9648.2 1418.7
a After Batcheler (1971). See text for
further explanation.
/)1 (acre-l) 351.6 .
95% CI
SE(L)I)
.
35.5 .
281.9 .
/)2 421.2
384.1 355.1
266
Chapter 13 Estimating Abundance with Distance-Based Methods
FIGURE 13.1 Exampleof line transect sampling and measurements. The arrow indicates direction of travel. A is the point at which the observer makes the observation; r is the observer-to-object distance; x is the perpendicular distance to the transect line; L is the transect length. On completion of the fieldwork, the distances of individuals from the transects are used to determine density. Assuming complete observability, an estimate of density is = n/A
(13.6)
= n/2wL,
where n is the number of organisms observed, L is the transect length, and w is the distance from the transect line to its edge. Typically, however, not every individual in a transect area is observed, and the probability of seeing an individual decreases with distance from the transect line. It thus is not sufficient to simply divide the number of individuals seen by the area surveyed; some adjustment must be made to account for the fact that not all individuals in the surveyed area are actually observed. A strategy is to determine an "effective transect width" ~ that can be used to calculate an effective transect area by ,~ = 2L~, with the adjusted area used in turn to estimate population density by /5 = n/A.
of density given as the ratio of the number of observations to the effective area. Clearly, the crucial step in this procedure is the estimation of effective strip width ff~, which is necessary to determine the effective area ,4. A number of historically important estimation procedures (e.g., Hayne, 1949b) are based only on the measurement of sighting distances (r in Fig. 13.1), though they depend on restrictive assumptions about the form of the sampling distribution of sighting angles. In principle, models can be developed to allow estimation based on the distribution of sighting angles and distances. However, attempts to develop a general, robust estimation procedure based on these data have been unsuccessful (Burnham et al., 1980). In the remainder of this section we assume that the data consist of perpendicular distances xi, which are either measured directly or are obtained via angular transformation of the sighting angles and distances. To envisage how the general approach to estimation from line transect sampling works, it is useful to begin with the special case in which animals are perfectly detectable out to a distance w on either side of the transect, and the whole transect area is within the observer's field of view. If n animals are counted within distances x i ~ w and animals beyond w are ignored, then a straightforward estimator of density is given by Eq. (13.6). However, for this estimator to be unbiased, one must assume complete detectability of organisms. The modification of transect width adjusts for partial detectability, whereby the estimate of density in Eq. (13.7) increases as the detectability (and thus the effective width) decreases. The parameter ff~ is based on a detectability function that is estimated with the distance data, assuming (1) detection of objects directly on the line is certain and (2) detection probability for an organism away from the line decreases as its distance from the line increases. The crucial step in this procedure is the estimation of effective strip width ~ based on a detection function g ( x ) . Once ~ is determined, estimation of D follows easily by the basic relationship in Eq. (13.7). The details of the theory for developing the detection function, and robust methods for estimating the function, are thoroughly covered by Buckland et al. (1993). We provide a brief introduction to this theory below and refer readers to Buckland et al. (1993) for a more detailed exposition.
(13.7)
Again, the logic of this approach is that the relative positions of individuals from the transects can be used to estimate the effective transect area, with the estimate
13.2.2. Assumptions To develop a statistical model for line transect sampling, it is necessary to make assumptions about
13.2. Line Transect Sampling random sampling and et al., 1993).
field sampling
(Buckland
13.2.2.1. Random Sampling Transect lines are assumed to be randomly positioned with respect to the distribution of objects. This assumption is automatically met if individuals are randomly located over their range, irrespective of placement of the transect lines. If individuals are not randomly distributed, then the transect lines must be randomly located over the population range. In particular, the common practice of orienting transects along roadways should be avoided. It is assumed that sampling is from a defined area, though the size of this area need only be measured if abundance, as well as density, is of interest. Finally, the survey must follow "good survey practices" in its design and conduct, and in the collection of measurements. Readers are referred to Buckland et al. (1993) for more detail on sampling designs.
13.2.2.2. Field Sampling Three assumptions pertaining to field sampling are critical to reliable density estimation. The first is that individuals directly on a transect line are certain to be observed. That is, the probability of detection for individuals on a transect is one. The importance of this assumption is discussed below, and procedures for dealing with its violation are discussed in Section 13.5.1. Second, objects are detected at their initial locations, and the locations of individuals are not influenced by observation. Thus, individuals do not move prior to detection in response to sampling disturbance. Otherwise, large biases may occur in the resulting estimates, with negative biases if animals avoid observers and positive biases if animals are attracted to observers. Also, any movement after initial detection does not result in individuals being counted more than once. Under these circumstances, reliable estimation of density is not possible unless auxiliary information (e.g., on movement patterns) can be gathered and used in a statistical model (Buckland et al., 1993). Third, distances and angles are measured accurately. Thus, neither measurement nor rounding errors occur. This assumption can only be met by accurate field methods and careful data recording. Errors typically occur when data consist of ocular or other crude estimates of distances and angles. These errors introduce difficulties in fitting models to the data, and in some instances (e.g., "heaping," or a tendency of observers to round to convenient values), can result in serious bias. Systematic errors are also possi-
267
ble, e.g., the tendency to underestimate distances at sea, which require calibrations to correct for the bias (Buckland et al., 1993). In addition to these assumptions, the statistical models for line transect sampling require assumptions about the nature of the data collection and the underlying detection model. Of particular importance is the assumption that individual sightings are independent events. This assumption clearly is violated when animals occur in clusters (e.g., coveys of quail). A possible solution is to treat the cluster as the sampling unit, recording as ancillary data the number of elements (e.g., individual birds) as the "size" of the cluster (Section 13.2.7). If independence cannot be assured (e.g., clusters are loosely defined), then point estimates of density and abundance will be relatively unaffected, but the model-based variances are potentially underestimated. Empirical estimates of sampling variance (see Section 13.2.6) alleviate the need for the independence assumption and thus perform better under these circumstances (Buckland et al., 1993). More critical to estimation are assumptions about the shape of the detectability function, especially on or near the transect line. Distance estimators tend to perform better when there is a "shoulder," i.e., detectablity is nearly perfect in a region near the line. On the other hand, when detectablity drops sharply near the line, estimation tends to be poor. The occurrence of a "shoulder" in the detectability function is known as the shape criterion (see Section 13.2.4), and it is an important feature of model selection procedures with distance estimation under distance sampling.
13.2.3. Statistical M o d e l s As noted in Section 13.2.1, the intuitively appealing but naive estimator in Eq. (13.6) does not take into account partial detection of animals in the vicinity of the transect line. To estimate density accurately, it is necessary to introduce a detection function that expresses the probability of detecting an individual as a function of distance x from a transect line:
g(x) = P(individual is detectedlx). It is reasonable to assume that the probability of detection decreases with distance. We also require that the probability of detection be unity for individuals directly on a transect line, i.e., g(0) = 1. To illustrate the effect of the detection function, assume that N individuals within the transect area are distributed such that N x are at distance x from the transect line. On average, Nxg(X) of these individuals are actually observed, where g(x) again is the probability of detecting individuals x units from the transect
Chapter 13 Estimating Abundance with Distance-Based Methods
268
line. The expected n u m b e r of observations over the whole width of the transect is E(n) = N P a
fw
=
Nxg(X) dx,
0
where N is the total n u m b e r of individuals in the transect area and Pa is an "average" detection probability (averaged over the frequency distribution Nx/N)" Pa =
fw o
( N x / N ) g ( x ) dx.
It follows that E n
_
Pa
o --~ g(x) dx
=N, so t h a t / ~ = n / P a is an unbiased estimate of N. On assumption that individuals are r a n d o m l y located with respect to transect position, the average detection probability simplifies to Pa =
fw
g(x) d x / w .
o
(13.8)
As mentioned earlier, this assumption is assured by the r a n d o m positioning of transects in the study area. Then an unbiased estimator for actual density is b = F~IA = (n/Pa)(1/2Lw) = n/2L(WPa).
Substituting the expression if; = w P a =
f
w
g(x) dx 0
from Eq. (13.8) into this formula, we obtain 19 = n / 2 L w P a = n/2CoL
(13.9)
= n/A
as an estimate of density, with ~ replacing w in the expression for area. It is in this sense that ~ = w P a is said to represent "effective" (as opposed to actual) transect width. If, for example, the probability of detec-
tion is I over the whole width of the transect strip {g(x) = I for all x over the interval [0, w]}, then all individuals in the strip are actually observed and the effective strip width is just 2w. On the other hand, if the probability 1 of detection is i everywhere except immediately beside the transect line, then only about 50% of the individuals in the strip are observed, and the effective transect area must be reduced to account for the presence of more individuals than are seen. The use of ~ effectively reduces the strip width as detection decreases, thereby increasing the estimate of density. Note that this density estimator is in the form of the canonical estimator in Eq. (12.1), wherein a total n u m b e r of individuals counted on a sampling area is adjusted for detectability (in this case 13 = Pa)" A useful generalization follows from this development. Though we began with an assumption that observations are recorded over some specified transect width w, the term w no longer occurs in the formula for D except as a limit in the integral term. Assuming this integral exists, the transect width w can assume an arbitrary value of indefinite size. Thus, we no longer need limit ourselves to observations within lateral transect bounds. This has important consequences for field procedures, because a traditional source of difficulty has been to recognize whether an observation near a b o u n d a r y is within or outside the transect strip. The approach described here no longer requires this distinction, so long as distance from the transect line can be accurately ascertained. The relationship between strip width and detectability can be formalized in terms of the probability density function of observed distances. As above, let x represent the distance of individuals from a transect line, with fl (x) the distribution of distances for all individuals in the population. Let y be a binary variable representing whether an individual is observed, with y = 1 if the individual is observed and y = 0 otherwise. Let f2(Y) describe the distribution of values for y over the population. Each individual in the population has associated with it values for both x and y, so that a joint distribution f(x, y) is defined for the population, along with the marginal distributions fl(x) for distance and f2(Y) for observation status. Note that the probability of y = 1 given x, f ( y - 1Ix), is simply the probability of detection given that an individual is x units from a transect. This was referred to earlier as the detection function, denoted by g(x). Applying Bayes' Theorem (see Section 4.1.6), we can express the distribution of distances for observed individuals by f(xly = 1 ) = f ( y = 1Ix)f1 (x) f2(Y = 1) "
13.2. Line Transect Sampling On condition that transects are r a n d o m l y positioned, fl(x) = l / w , and this expression simplifies to
sumption of complete observability on the transect line. If g(0) = go is less than 1, then
f(y = 1 Ix) fl(x)
f(xly = 1 ) =
269
f(0) =
f
rw f(Y = 1Ix) fl(x) dx d
g(0) g(x) dx
0
0
= go~if2.
f(y = 1Ix)/w
-[w f(Y = 1 Ix) dx/ d
w
Then f(O)/go = l/if; and we have the estimator
W
0
b = n/2Lff2
f(y = llx )
fw0 f(y
= 1Ix) dx
= nf(O) / 2Lg o.
.
Replacing f(y = 1 Ix) with g(x) and denoting f(xly = 1) by f(x), we get
f(x) = f
w g(x) g(x) dx
(13.10)
0
as the distribution of distances for observed individuals. In particular, the probability that an observed individual is directly on the transect line is
r(o =
f[
= 1 / [ w g(x)dx
dx (13.11)
=1/~. Substituting this expression into the estimate for density, we have (13.12)
= nf(O) / (2L). This shows that the density estimator, Eq. (13.9), is directly proportional to the probability that observed individuals are directly on the transect line; i.e., the estimate of density increases as observations are clustered near the transect line and decreases as observations are distant from the transect line. Because f(0) must be estimated from data, the density estimator is more properly expressed as
15 = nf(O)/2L,
Thus, the effect of partial observability of organisms on the transect line is to increase the estimate of density by the factor 1/go. For example, the estimate of density for g(0) = ~1 is twice what it w o u l d be for g(0) = 1 (assuming the same count n in both cases). The density estimators shown in Eqs. (13.13) and (13.14) are designed to account for reductions in detectability as individuals are more distant from the transect line. The estimators clearly are dependent on the distribution f(x) of observation distances, and f(x) is dependent on the detection function g(x) through Eq. (13.10).
Example
do
D = n/(2L~)
(13.14)
(13.13)
where f(0) indicates that f(0) is estimated with field data. Thus, the key statistical problem in estimating density with transect data is the derivation of an estimate f(0) of f(0). In the following development, we will focus on this problem as it pertains to the statistical properties of/5. One can generalize Eq. (13.12) by relaxing the as-
To illustrate the effect of g(x), consider two extremes in detectability over a transect 50 m in length, with observations out to 10 m on either side of the transect line. In the first case, assume that detectability is certain over the whole width of the transect, i.e., g(x) = 1 for all values of x between 0 and 10 m. This means that every animal within the boundaries of the transect is seen with probability 1, irrespective of distance from the transect line. Therefore
f(x)
g(x)
fl0~ g(x) ax
s
1 ax
= 1/10; in particular, f(0) = 1-!6,so that the estimate of density is simply the ratio of observations to transect area:
D = n/2L(lO) = n/1000. Thus, an estimate of the average n u m b e r of organisms in an area of 1000 m 2 is n individuals. In contrast, n o w consider a situation in which the same n u m b e r of organisms is observed but detectability declines sharply as individuals are distant from the
270
Chapter 13 Estimating Abundance with Distance-Based Methods
transect line. Assume, for example, that detectability is certain over a strip width of 1 m and is 0 elsewhere: g(x) = 1 for x between 0 and 1 m, and g(x) = 0 elsewhere. In this scenario, the distribution of sightings is 1
f(x) = w ~o dx =1 for x < 1. In particular, f(0) = 1, with a corresponding estimate
D = n/(2L) = n/100 for density. Here an estimate of the average number of organisms in an area of 100 m 2 is n individuals, a density 10 times that corresponding to uniform detectability over the transect area. From this example, it is clear that for a given number of observed individuals, the estimate of density increases as detectability drops off from the transect line. For a given number of observations, one can anticipate larger estimates of D if observations are clustered near x = 0, and smaller estimates of D if they are more evenly distributed over the range of detectability.
13.2.4. Estimating the Distribution of Distances With this background we now consider ways of estimating the distribution function f(x) and in particular the value of the function at x = 0. Because the distribution function f(x) factors directly into the estimation of D, it is necessary to find a statistically reliable procedure for its estimation.
13.2.4.1. Statistical Properties of f(x) The number of individuals directly on the transect line is assumed to be known with certainty, with accurate distance measurements for all individuals observed within allowable transect boundaries. These data form the basis of an estimate of f(x), from which is derived the estimate f(0) of f(0). The approach is to estimate f(x) based on certain distribution assumptions, i.e., to assume a general model for f(x) and then estimate its parameters. The estimator should possess the following properties (Burnham et al., 1980; Buckland et al., 1993). First, it should be applicable to a wide range of distributions, i.e., the form of the assumed model for f(x) should be flexible enough to fit a wide range of distance functions. This property protects against un-
warranted assumptions about the specific structure of
f(x) and thereby avoids potentially severe bias in the estimator. Such a property is called model robustness. Second, the estimator should be robust to the pooling of data across strata. There are two reasons for requiring this property. First, stratifications within the population, arising out of genetic, physiological, behavioral, or other differences among individuals, often go unrecognized. In like manner, habitat stratification over the study area can be overlooked or unrecognized. Second, even if stratification is recognized, to account for it by estimating a separate density for each stratum often is infeasible because of inadequate sample sizes. Thus it is important that the density estimator be robust to the pooling of data across strata, so that an estimate based on data pooled across strata is essentially the same as that based on an average of estimates for each stratum. This property is known as pooling robustness. Third, the estimator should be based on a model for which the probability of detection near the transect line is approximately 1. Because the probability of detection on the transect line is 1 by assumption, this means that the model for f(x) is essentially flat over some region near x = 0, that is, (df/dx)(O) = 0. The effect of this requirement is to limit the potential models for f(x) to those for which detection probabilities decline gradually from 1 as animals are farther from the transect line, at least over some small distance from the line. The requirement is referred to as the shape criterion for f(x). Fourth, in case a m a x i m u m transect width is not part of the field procedures, the estimator should accommodate the truncation of data at extreme distances from the transect lines. This allows for post hoc truncation of data based on some m a x i m u m allowable distance value. Because extreme distance values often occur in transect studies, a procedure that allows for elimination of such values is to be preferred. This is referred to as the truncation requirement. Fifth, the estimator should be useful for handling data that are grouped into a relatively small number of distance intervals. Grouping of data into appropriately defined distance intervals can accommodate a tendency for data to be recorded as certain convenient values, notably as multiples of 5 or 10 units of distance. Grouping of data effectively "smooths out" this tendency to aggregation and improves the estimate of f(0). This condition is known as the grouping requirement. Finally, the estimator should be efficient, in that the bias and variance of the estimator should compare favorably to other estimators of density. This requirement refers to the usual standards of accuracy and precision of a statistical estimator.
13.2. Line Transect Sampling Depending on the nature of the data, three distinct approaches to estimation are possible. If perpendicular distances from the transect lines are recorded accurately and if the tendency to cluster distances about certain units of measurement is avoided, then the data are properly analyzed as continuous, ungrouped measurements. On the other hand, if the field situation allows data to be recorded accurately only within certain categories of distance, then assessment procedures for grouped distance measurements are appropriate. Finally, if the only available data consist of sighting distances between observers and organisms along with the corresponding angles of incidence between sighting radius and transect line, then estimation methods must be used that differ from the methods for perpendicular distances. In general, transect surveys should be designed so that ungrouped perpendicular distances can be analyzed, because the density estimators for this situation are the most precise. In any case one should always collect perpendicular distance measurements, even if they must be grouped for analysis. This is because density estimators based solely on sighting distances and angles are quite sensitive to observations near the transect lines and therefore tend to be imprecise and potentially inaccurate. 13.2.4.2. Estimation Based on Fourier
Series Approximations The estimation of density from line transect observations has a long history (Burnham et al., 1980; Buckland et al., 1993). Early approaches typically were based on parametric models (e.g., negative exponential, half normal) and in many cases required restrictive assumptions about animal behavior, sampling conditions, or both. An important advance came with the application of an approximation technique from engineering mathematics known as Fourier series approximation (Burnham et al., 1980). A Fourier series approximation can be used to represent virtually any function, subject to mild conditions on the analytic properties of the function. In this case, a Fourier series of the form m
f(x) ~ 1/w* + ~ aj cos(jwx/w*) j=l is used to represent f(x), where w* is the maximum distance allowed by the truncation requirement. The Fourier coefficients aj are estimated by t/
~j = (2/nw*) ~ , Cos(j~Xk/W*) , k=l
271
which are used in the Fourier series approximation to estimate f(0). Because cos(0) = 1, the estimate of f(0) is simply m
f(O) = 1/w* + ~
ak,
k=l
which in turn can be used to estimate density. Theoretically based estimates for the variance and covariances of the Fourier coefficients are given by vaAr(~k) = (n -- 1) -1
[ ( a 2 k nt-
2/w*)/w* - ~]
and co'v(~k, dj) = (n - 1) -1 [(t~k+j + dlk_j)/W* -- dtkdj], respectively, and these terms can be combined into a straightforward expression for the variance of f(0): m
va'r[f(0)] = ~ j=l
m
2
c"0"v(dj, ak),
k=l
where cov(t/j, t~j) = var(dj). Note that no structural or parametric assumptions about the distribution function f(x) are involved in its Fourier approximation. Thus, the approach is nonparametric, in contrast to the parametric approaches of more familiar procedures such as maximum likelihood estimation. A rule of thumb is required for determining the number m of cosine terms to include in the Fourier approximation. The theory of Fourier series approximation indicates that the inclusion of additional terms in a Fourier series representation increases the accuracy of that representation. However, inclusion of additional terms also increases the sampling variances of f(0) a n d / ~ . A procedure that recognizes the tradeoff between bias and variance in the estimation procedure is to add terms sequentially starting with dl, until
(1/w*)[2/(n
+
1)] 1/2
--> ] a m + l ] "
This rule requires an estimate of one more coefficient than ultimately will be included in the Fourier expansion. As a practical matter, the number of Fourier terms should rarely if ever exceed six (Burnham et al., 1980).
13.2.4.3. The Key Function Approach to Estimation The Fourier series approach remains a powerful tool for analysis of line transect data and distance data. However, it does not always result in estimators that fulfill the requirements established earlier, especially model robustness, shape criterion, and estimator efficiency. Thus, for a particular data set, other models and their estimators may be superior with respect to
272
Chapter 13 Estimating Abundance with Distance-Based Methods
these requirements. This in turn poses two additional requirements for an estimation procedure. First, one must start with a sufficiently general and robust procedure, such that at least one approximation model for f(x) meeting the specified requirements is included in the investigation. Second, objective methods must be in place for evaluating model adequacy and for selecting an optimal model when more than one is deemed "adequate." We consider the first issue in this section and the second issue in the next. Because the recording of distance is conditional on individuals being observed, sample data are modeled in terms of f(x) and not g(x), though the two functions are fundamentally related through Eq. (13.10). Thus, robust modeling of the detectability function g(x) enables robust estimation of the probability density function f(x) for the observed distances. Two basic approaches have been used for modeling g(x), one based on nonparametric modeling (e.g, the Fourier series method), the other on parametric modeling. Parametric methods have the advantage in that they require the estimation of fewer parameters and typically produce more consistent estimates from study to study than do nonparametric methods. However, parametric models frequently exhibit lack of fit to field data, suggesting the need for additional model terms to reduce bias and improve model fit. The approach described here is essentially a hybrid of parametric and nonparametric methods. The approach involves selection of a key function as a starting point for estimating the detection function g(x), possibly after visual inspection of the data (e.g., via histograms) and removal of obvious outliers. There are
Key functions Uniform
1/w
Half-normal
exp(-y 2/2o-2)
Hazard-rate
1-exp [-(y/o)-b]
Exponential
exp(-y/Z)
Adjustment functions Simple polynomial
y~)
Hermite polynomial
H2j(G)
Fourier series
eosqwys)
F I G U R E 13.2 Key functions and adjustment factors. See Stuart and Ord (1987) for a discussion of Hermite polynomials.
several candidates for key functions. Among the most important of these are the uniform, the half normal, and the negative exponential. The second step involves use of a flexible method (a "series expansion") for adjusting the key function to improve fit of the model to the distance data. The combined approach to estimation gives rise to a generalized form for the probability density function f(x), as the product of a key function and adjustment term:
f(x) = key(x)[1 + series(x)]. More formally,
f(x)
-7
1 + ~ ajpj(x s) ,
(13.15)
j=l
where o~(x) is a parametric key function containing k parameters (typically k = 0, 1, or 2), pj(x s) is a series adjustment function, [3 is a normalizing function of the parameters that scales the product as required for f(x) to be a probability density function, and x s is a standardized value for x (e.g., x s = x/or), the form of which depends on the parameters in oL(x). Useful key functions and adjustment functions are listed in Fig. 13.2. For example, the Fourier series model (Burnham et al., 1980; Laake et al., 1979) is obtained by combining the uniform key function (containing no parameters) with the Fourier series adjustment. The uniform key function combined with simple polynomial adjustments produces earlier models developed by Anderson and Posphala (1970), Anderson et al. (1980), and Gates and Smith (1980). Likewise, the hazard rate model of Buckland (1985) can be produced using the hazard rate key, combined with adjustment factors (Fourier series or polynomial) as needed to improve fit. The key function approach can be seen as a sequential approximation of f(x), with the identification of an approximating function followed by the refinement of the approximation using terms from an adjustment series. Thus, the general form of f(x) is approximated by e~(x), which then is "adjusted" with terms from {pj(x): j = 1, ...} to improve the fit. Put in this context, the key function approach is amenable to maximum likelihood estimation, for which the parameters aj in Eq. (13.15) as well as those in the key function oL(x)can be estimated with standard likelihood methods.
13.2.5. M a x i m u m Likelihood Estimation Maximum likelihood estimation of the parameters in f(x) is performed using Newton-Raphson methods (see Appendix H), the details of which depend on
13.2. Line Transect Sampling whether data are ungrouped or grouped distance measurements. The likelihood function for detection distances (in general, either perpendicular or radial) is developed by a conditional argument, based on the probability
P(n, X 1..... Xn) , of a realization of data from a line transect sample, where {Xl, ..., x n} are the distances associated with n observations (Buckland et al., 1993). This probability can be reexpressed in terms of conditional probabilities, as
P(n)P(x 1, ..., Xnln), whereby the estimation of density is represented by two separate likelihoods. We focus on P(x 1, ..., x, ln) , which we assume can be factored as
P(Xl, "", Xnln) = H f(xi)
i=1
by virtue of the independence of observations.
13.2.5.1. Maximum Likelihood Estimate for Ungrouped Measurements of Distance Define L(0_) = I-[n=1f(xi) as the likelihood of the observed distances xi, where 01, ..., Ok are parameters of the key function o~(x) in Eq. (13.15) and Ok+j = aj, j = 1, ..., m are the coefficients of the adjustment series. Taking logarithms and introducing the normalizing function [3 from Eq. (13.15), we have
n
= ~_, ln[f(xi)f3] - n ln(f3)
i=1
(Buckland et al., 1993). The first derivative of the loglikelihood with respect to the parameters is
i=1 n {
-- i=IE
{ln[f3f(xi) ] - n ln(f3)}
1 0[~ f(xi)]~ ~ f (x i ) O--Ojj J
ti 3[3 ~ c]Oj
for j = 1..... k + m. From Eq. (13.15) we have O[~fix)]_
OOj
[
m aj,pj,(x s) 1} ,
0 oL(x) 1 + ~
j'=l
3[~ f(xi)] OOj
I
, ,[ ~ OPj'(Xis)]OXis oaxi, [j2.,__1 aj,--Oxi--]-- ] 00--7. + [1 +
j'~-I a..p.,(Xis) ]3~ ] ]
[~(xi)Pj_k(Xis )
c)Oj
lGj~k j>k
and
aj_k:kO
The terms OXis/OOj involve the factors used to scale x i to xis. For example, scaling x by x s = x / r leads to OXis/ OOj = - x i / r 2. The terms Opj,(Xis)/OXis are given by
Opj'(Xis) OXis
q
l(Xis)
(simple and Hermite polynomials) (Fourier series)
(Buckland et al., 1993). This system of k + m equations in the parameters 0i, i = 1, ..., k + m, can be solved using numerical optimization procedures to provide maximum likelihood estimates of the parameters. Variances and covariances of the estimates follow from the Fisher information matrix, which is obtained from the Hessian matrix evaluated at the maximum likelihood estimates (see Appendix F). Note that changing the key function oL(x) involves the specification of OoL(x)/ 00j and cOXis/OOj, whereas specifying new adjustment factors requires redefining pj(Xs). The normalizing factor ~ and 013/O0j can be evaluated by numerical integration.
13.2.5.2. M a x i m u m Likelihood Estimation for Grouped Distance Measurements
In [/=I~1{f3 f(xi)}][3
O{ln[L(O)]} " 00/ = ~2
so that
J-P~'~rsin(j-rrx s)
1"/
ln[L(O)] =_
273
It generally is preferable to record distances as continuous measurements, so that the above procedure can be used to estimate density. However, it sometimes is either inefficient or impossible to record distances accurately, and in some instances there is a tendency for measurements to be clustered at certain values irrespective of the care with which they are recorded. In such cases one is required to estimate density with data that are grouped into a limited number of distance categories. Fortunately the robust estimation methods described above are readily adapted to this situation. We assume here that the range of potential distance values is partitioned into a fixed number of distance categories by "cutpoints" {c0, c1, ..., c k} that define k distance categories, where category i includes distances between ci_ 1 and c i, with co = 0 and c k = w. The distance of an observed individual lies in one (and only one)
274
Chapter 13 Estimating Abundance with Distance-Based Methods
of these categories. Thus, grouped survey data consist of the numbers {n1, n2, ..., nk} of individuals with distances in each of the categories. Given the assumptions for transect estimation as listed earlier, the counts for a total of n observed individuals are distributed according to a multinomial distribution, with the multinomial probabilities dependent on the distribution f(x) of observed individuals. Recall that the probability density function for a multinomial distribution is
(
f(nl~r) =
)k
n
H "]T~/i~'
H1, ...r Hk
i=1
where "rrI + "'" + "rrk = 1 and n I + ... + nk = n. In this case, ~ri is the probability that the perpendicular distance of an observed individual is in the ith category. As with ungrouped data, the key to estimating density with grouped data is to estimate the distribution f(x), so that f(0) can be used i n / ) = nf(O)/(2L). This estimation is facilitated by recognizing that the probability '1ii corresponding to the distance between ci-~ and ci is simply the area under the curve f(x): "rri =
k
n i ln(-rri) + C,
i=1
where C = log[n!/II~=l(ni!)] is a constant given the data. Differentiation with respect to the model parameters yields 0{In[L(_0)]}
k ni O~i
c90j
.= gri OOj'
a0j
3f3
]
-~ L ooj aofr~
.
(13.17)
Numerical integration can be used to determine
Pi =
fci ci-1
f(x)f3 dx
and
3P i = OOj ,J
fc, Ci-1
and
0__~ = ~ 3Pi. OOj i - 1 OOj These forms can then be used in Eq. (13.17) to determine MLEs for the values 0j. Note that the same implications and requirements hold for changing the key function and series adjustments as for ungrouped data (Buckland et al., 1993). A computer is necessary to compute the iterative maximum likelihood computations and numerical integrations and to calculate the parameter estimates and the estimated variances and covariances. The program DISTANCE performs these calculations, computes likelihood ratio tests for model comparisons (e.g., to test effects of adding adjustment terms), computes AIC for model comparison and selection, and tests the resulting model for goodness of the fit to the distance data.
[f(x)[~] dx,
The statistical properties of the estimator/) are inherited from f(0) and n, the two components of /) that are subject to random variation. The estimator is sensitive to statistical behaviors of both components and in particular to the behavior of f(0). Of special concern is the variance o f / ) . On condition that f(0) is asymptotically unbiased {i.e., if f(0) converges to E[f(0)] as n increases}, the asymptotic sampling variance o f / ) is var(/))
and the values of 0j for which these expressions vanish are the MLEs. From Eqs. (13.15) and (13.16) the probabilities 7ii are parameterized by the parameters of the key function and the series adjustment that define f(x). A reparameterization by Pi = "rri~ allows us to write 3"rri= 1 [OPi
i=1
13.2.6. Estimating the Variance o f / )
The log-likelihood for grouped data is In[L(_0)] = ~
k
=EPi
(13.16)
f(x) dx.
ci-1
which in turn can be used to produce ~ and 0[3/00j by
= D 2 { [ c v ( F / ) ] 2 q-
cv[f(0)]2]}
(13.18)
(Burnham et al., 1980). An estimated variance is obtained by using estimates of the coefficients of variation: V~(/~)
=
/~)2{[C"V(H)]2 +
C"v[f(0)]2]}.
(13.19)
If var(n) = aE(n) (as is the case with the Poisson and certain other distributions of distance), it can be shown (Burnham et al., 1980) that the variance of/5 is of the form var(/)) = (1/L)[D.f(O)/2][a
+ b/f(O)2],
(13.20)
suggesting that a combined estimate of density based on replicate transect lines of varying length should weight the replicate estimates by transect line length (Burnham et al., 1980). Note the relationship between Eq. (13.18) and the canonical variance estimator, Eq.
13.2. Line Transect Sampling (12.2), both of which emphasize variation in the count statistic n and variation due to the estimation of detectability. There are several ways of estimating the variance of a density estimator (Burnham et al., 1980). For example, if multiple transects are run, one could use the empirical estimator k va'~(b) = ~ Li(]~ i - D ) 2 / [ L ( k i=1
k LiDi/L
13.2.7. Density Estimation with Clusters
i=1
with L - ~ i Li (see Appendix F). In essence, data from transect i are used to develop an estimate of D i, and transect-specific estimates then are treated as estimate replicates. The resulting estimator of variance has the advantage that no assumptions about the distribution of D are required, but it has the disadvantage that minimum data requirements must be met for each replicate transect. Because no distribution assumptions are necessary, it is the estimator of choice when data requirements can be met. Unfortunately, they can be met only infrequently. In the event that minimum data requirements for each transect cannot be met, a second approach involves the use of a "jackknife" estimator (see Appendix F). Here the data from all transects but one are pooled, and an estimate of density is derived. This is repeated for all transects, leaving each transect out and computing a corresponding estimate of density. This results in k such estimates, designated by D_i, i - 1, ..., k, where the negative subscript is used to indicate that transect i is omitted from the computations. These values then are used to define the jackknife "pseudovalues," defined by Di--
combine them according to Eq. (13.19). An estimator of var[f(0)] can be derived from the procedure for estimating f(x). An estimator for var (n) can be obtained from k ff~r(n) = L ~ , Ci[rli/C i - n / C ] 2 / ( k 1) i=1 if replicate lines are available (Burnham et al., 1980). If not, one can either assume some spatial distribution for individuals in the study area, from which is derived a value for w'r (n), or one can simply assume an expression for var(n) as a function of n.
1)1,
where ]~ = ~
275
[LE) - (L - L i ) D _ i ] / L i ,
i = 1,..., k, which in turn are used to calculate
Animals often are detected in clusters, such as coveys, flocks, and schools. In this situation, interest may focus on the density D s of clusters, the total density D of individuals, the average cluster size E(s), or any combination of the above. Clearly, these three parameters are related to one another. Statistical estimation depends on assumptions about the relationship between detectability g(x) and cluster size s, with the possibility that observed cluster size depends on the distance from the transect. 13.2.7.1. O b s e r v e d Cluster Size Is Independent
of Distance Under this situation, estimation of density and its variance is straightforward, with the estimator of overall density simply the product of estimated cluster density [cf. Eq. (13.9)] and estimated mean cluster size, i.e., /~ =/~s g
(13.21)
= [nf(O)/2r]g,
where ~ = ~ 7=1 si/n, si is the observed size of the ith cluster, and n is the number of observed clusters. A large-sample estimate of variance is provided by va'~r(/~) =/~2([cv(n)] 2 + {cv[f(O)]} 2 + [CV(S)]2),
k /~)jackknife-- ~ LiDi/L i=1
(13.22)
where cv(n) and cv[f(O)] from Eq. (13.18) are applied to the observed clusters, and cv(g) = N/~v~(g)/g with H
~i=l(Si -- ~)2 var(g) = n(n - 1)
and k V~(/~)jackknif e) -- ~ Li(D i - L)jackknife)2/C(ki=1
1).
(Burnham et al., 1980; Buckland et al., 1993). Yet a third approach to the estimation of variance for the estimator of density is to estimate the components of variance in Eq. (13.18) separately and then
(Buckland et al., 1993). 13.2.7.2. O b s e r v e d Cluster Size Is Dependent
on Distance This situation typically arises when cluster size influences the detection probability g(x), which naturally
276
Chapter 13 Estimating Abundance with Distance-Based Methods
complicates estimation. If not adequately addressed, this influence can result in positively biased estimates of density, be~.duse of the tendency to overrepresent large clusters and underrepresent small clusters in the sample. There are several alternatives to account for the nonindependence of group size and distance. One approach involves the estimation of the detection function g(x) using robust methods that do not depend on cluster size. A method for this approach uses the observed clusters to estimate E(s), though including clusters only within some maximum distance x0 over which detection is close to 1, so that detection is not an issue. Another is to use regression methods to estimate E(slx). Other approaches that avoid the influence of cluster size are (1) to treat individuals as the observations (thus avoiding the issue of estimating cluster size) but violating the assumption of independent detections, or (2) poststratify by cluster size, fit detection models for each stratum, and compute a weighted average of the stratum counts rlis i. In each of the above, once E(s) is estimated, it is used along with the unconditional estimate of D s to estimate D as in Eq. (13.21). An alternative approach, described by Drummer and McDonald (1987) and Drummer et al., (1990), uses a data transformation and bivariate parametric detection models to estimate detection, average group size, and density, corrected for size bias. Drummer (1991) documented the use of computer program SIZETRAN for implementation of these procedures. Yet another approach is a regression of si or ln(si) on d(x i) to estimate E(s) where ~(x i) ~ 1, i.e., where detectability is certain and size bias thus should not occur. Buckland et al. (1993) particularly discourage replacing the observed clusters by the individual objects, although they concede that this procedure may be useful for "loosely aggregated clusters." If this approach is used, it is most effective if distances to each individual can be measured. Of the methods described above, the regression approaches seem to offer the greatest robustness and efficiency (Buckland et al., 1993). 13.2.7.3. Full L i k e l i h o o d E s t i m a t i o n
The likelihood approaches described above are based on a conditional likelihood argument, in which parametric models are applied to the distance portion of the data x, but not to the observed sample counts n or the cluster sizes s. Parametric models are avoided by using empirical variance estimates for n and by computing confidence intervals o n / ~ under assumptions of log normality. Likewise, E(s) and var(s) are obtained in a least-squares regression framework, thus
avoiding the specification of a probability model for the number and sizes of the clusters (Buckland et al., 1993). In contrast, the full likelihood approach requires that probability modeling be extended to the sample counts and cluster sizes. The full likelihood for cluster data that include both distances and cluster sizes is given in terms of the joint probability density function P(n,
X1, ...,
Xn, $1,
...,
Sn),
where {X1, ..., X n} are the distances and {$1, ..., Sn} are the cluster sizes associated with n observations (clusters). This probability can be expressed in terms of conditional probabilities, as P(n)P(x I ..... Xnln)P(Sl, ..., shin,
X 1, ...,
Xn),
whereby the estimation of density is represented as a series of separate likelihoods. Buckland et al. (1993) note the difficulties of developing such an approach but point to several advantages, including (1) improved estimator efficiency, (2) availability of a welldeveloped likelihood theory for computing profile likelihoods (Section 4.2.3) and model comparison by AIC (Section 4.5), and (3) the possible extension of Bayesian approaches (Section 4.5) to distance estimation. Presently there is no general, full likelihood approach for distance estimation, and the remainder of this chapter is confined to the conditional approach described above.
13.2.8. M o d e l Selection and Evaluation The approach of combining key functions with series adjustment functions can result in a large number of potential models. On the one hand, this provides users with a great deal of flexibility in fitting detection functions to sample data. On the other hand, there is the problem of how to choose an appropriate model from among the large number of possible models that may be constructed. As indicated earlier, a detection model should meet estimation criteria such as model robustness, pooling robustness, shape criterion, and estimator efficiency. For a given data set, these criteria can be achieved with a combination of methods such as data screening, including the use of histograms to identify general patterns of detection and obvious outliers. This step may be helpful in identifying one or more key functions with which to start the analysis. For a given key function, the issue becomes how many terms to include in the adjustment series. The alternatives form a hierarchy, with simpler models (fewer adjustment terms) forming nested subsets within more complex models. Likelihood ratio and
13.2. Line Transect Sampling similar procedures thus are appropriate for model comparisons. However, frequently more than one key function, or type of adjustment series, may be plausible, so that the models do not form a nested hierarchy as required for likelihood ratio testing. For example, consider a model with normal key function plus the lead term of a cosine series and an alternative model consisting of a hazard function and no adjustment. Both models contain two parameters, and they do not form a nested hierarchy and cannot be compared by likelihood ratio. Akaike's Information Criterion (AIC) (Akaike, 1973; Burnham and Anderson, 1998) provides an alternative method for model selection that views model selection as an optimization rather than a hypothesis-testing procedure (see Section 4.4). The computing formula AIC = - 2 In(L) + 2q includes ln(L), the natural logarithm of the maximum of the likelihood function, and the number q of model parameters. Essentially this expression represents the tradeoff between bias reduction through improved model fit [achieved by minimizing the deviance - 2 ln(L)] and a penalty for increased variance as additional parameters are added (the 2q term) (see Section 4.4). For a given data set (AIC comparisons among data sets are meaningless), the procedure is to compute AIC for each candidate model and select the model providing the lowest AIC statistic, recognizing that models with AIC values less than two units apart are essentially equivalent. We note that for the special case where nested models differ by one parameter, model selection based on AIC is equivalent to a likelihood ratio test with X2 = 2.0 (oL = 0.157) (Buckland et al., 1993). AIC thus can be used for ranking models that are either nested or non-nested. Occasionally, the AICs for more than one model are essentially tied (i.e., differ by ~2). In these cases, the models all are seen as acceptable competitors and should be further evaluated based on other criteria, such as prior biological knowledge. Alternatively, model-averaged estimates (Burnham and Anderson, 1998) can be computed. Once the estimated detection function and the corresponding densities are produced, goodness of fit statistics and graphical analysis of residuals are useful in determining model adequacy. Goodness of fit can be tested by a Pearson chi-square statistic (see Section 4.3.3), provided the n distances are first split into, say, k groups with sample sizes n 1, ..., n k. A model fitted to the (original) data then can be used to estimate the cumulative probability "rri under the probability density function between the "cutpoints" ci_ 1 and c i. Finally, these estimated probabilities can be used to compute a test statistic as
277
k (n i _ n~ri)2, X2-- E i~-1 tllTi which follows a chi-square distribution with k - q - 1 degrees of freedom under the null hypothesis that the candidate model appropriately represents the data.
13.2.9. Interval Estimation Variance estimates f o r / ) are obtained from application of Eq. (13.19) or (13.22), with the estimate v'~[f(0)] obtained from the conditional maximum likelihood methods described in Section 13.2.5. As noted in Section 13.2.6, empirical estimates of var(n) can be used in lieu of likelihood approaches. However, empirical estimates are not available if lines are not replicated, and one then is forced to rely on a distribution-based relationship such as var (n) = n for the Poisson distribution, possibly adjusted by a constant (Burnham et al., 1980). An approximate (1 - 2c~)100% confidence interval may be computed by invoking asymptotic normality of D as /~ _ z~X/v~r(/~) where z~/2 is the upper a point of the standard normal distribution. However, Buckland et al. (1993) note that the distribution of /~ is skewed and suggest that a confidence interval based on assumed log normality o f / ) provides superior coverage. This interval is computed as ( ~ / c , [) . c)
where C = exp [ G V ' v ~ (ln/~i ] and v~r(ln D ) = ln[1 + v~(D)//~2]. The above approach is used in program DISTANCE (Buckland et al., 1993) to calculate confidence intervals, except that the normal deviate is replaced by a t statistic with degrees of freedom computed by a Satterthwaite (1946) adjustment. Example
Burnham et al. (1980) describe an experiment in which a known number of wooden stakes were placed in a sagebrush meadow, with a density of 37.5 stakes/ ha. Teams of students walked transect lines and recorded perpendicular distances from the lines to the stakes that were detected. Here we report the results for
Chapter 13 Estimating Abundance with Distance-Based Methods
278
one transect line, from which 68 stakes were detected. Program TRANSECT (Burnham et al., 1980) was used to compute estimates based on Fourier series (equivalent to the uniform key function with a cosine adjustment term), and a model with two adjustment terms was selected, providing an estimate of density /~ = 39.3 s t a k e s / h a (~~ = 0.15). These same data were reanalyzed with p r o g r a m DISTANCE, using (1) the uniform key function with 0, 1, 2, and 3 cosine adjustment terms and (2) the half-normal key function with 0, 1, and 2 adjustment terms (Table 13.2). The seven models formed by these combinations of key functions and adjustment series were ranked by descending AIC, and the top two models were indistinguishable based on AIC (AAIC < 2). The second ranked model is based on fewer parameters, with a resulting higher precision in the density estimate (~'v = 0.13 vs. 0.16); both models evidenced adequate fit (P > 0.20). The second-ranked model yielded an estimated density of 33.08 stakes/ ha with a log-based 95% confidence interval of (25.38, 43.12).
13.3. P O I N T
SAMPLING
In the previous section the sampling units were line transects of fixed length and (possibly) indefinite width. However, in some applications the sampling unit is a point (or "point transect") with observation distances recorded in terms of radial distance from the point. We have already seen some examples of this approach, in the point-to-object methods considered in Section 13.1. Point sampling often is used in surveys of singing birds, whereby observers stop at predetermined stations and attempt to identify all birds in the vicinity, sometimes visually but often by detecting their songs. Point sampling also occurs in the context of cue
TABLE 13.2 Key function
Uniform Uniform Uniform Half normal Half normal Half normal Uniform
O~ r3~
FIGURE 13.3 Example of point sampling and measurements. Open circles represent detected individuals. For detected individuals, r is the observer-to-individual distance.
counting and trapping webs (Buckland et al., 1993) (see Section 13.5).
13.3.1. S a m p l i n g S c h e m e Data Structure
and
The sampling units in point sampling are k replicate points at each of which individuals are detected and the radial distances r i to each individual are measured (Fig. 13.3). Field ornithologists using point sampling ("point counts") have tended to emphasize sampling over an area of fixed radius w about the point, within which detection is assumed to be perfect, or at least uniform. Though sometimes justified, this assumption, which is analogous to perfect detectability near the transect line, is unnecessarily restrictive. We advocate recording distances to all objects detected in point sampling, along with the use of robust methods to estimate empirically detection functions and density. A modification of point count sampling for birds,
Example of Line Transect Estimation Using Laake's Wooden Stake Data a
Adjustment
Number of adjustment terms
Goodness of fit AIC
~AIC
X2
df
Cosine Cosine Cosine Hermitepolynomial -Hermitepolynomial m
2 1 3 2 0 1 0
382.14 384.11 384.14 384.16 385.78 387.73 409.24
0 1.97 2.00 2.02 3.64 5.59 27.1
8.87 13.42 4.65 8.88 16.21 16.13 39.37
9 10 7 8 10 9 11
P
0.45 0.20 0.70 0.35 0.09 0.06 <0.01
95% CI /)
40.577 33.079 40.416 40.793 34.561 34.589 18.817
C"L
C"U
29.213 25.379 27.363 28.574 25.942 23.668 14.839
56.361 43.116 59.695 58.236 46.044 50.549 23.862
CV
0.1658 0.1334 0.1973 0.1798 0.1445 0.1919 0.1195
After Burnham et al. (1980). The analysis is based on a sample of 68 wooden stakes of known density (D = 37.5 stakes/m), utilizing the complete data set (no right censoring).
13.3. Point Sampling called variable circular plots, allows for the modeling of declining detection rates with increasing distance from the central sampling point. This method involves the recording of birds within strata defined by k nested circular plots or annuli and the fitting of parametric or nonparametric estimation models to the resulting data. This approach is a special case of point sampling, whereby the data are collected in discrete distance intervals. Ideally, one should record the exact distances r i from the central point to each individual that is detected, although for practical reasons (e.g., ability of observers to determine accurately distances to birds in the field), grouped distance data also can be recorded. In either case, previous estimation methods (Buckland, 1987; Ramsey and Scott, 1981; Roeder et al., 1987) have been extended to include a number of flexible and robust methods (Buckland et al., 1993). Point sampling methods offer several logistical advantages over line transect sampling, especially for bird surveys (Buckland et al., 1993). Most notably, once at a sample point, observers are free to concentrate on bird detection rather than on traversing a line, which is especially advantageous in difficult terrain. Point sample surveys also can be easier to design than line transect samples. For example, it often is easier to locate points than to locate and traverse transect lines. Additionally, radial distances to animals can be easier to measure than perpendicular distances from line transects. However, point samples may be unsuitable if, in approaching points, the observer disturbs the animals. Furthermore, point sampling may be inefficient compared to line transects, particularly when densities are low and a substantial portion of the time is spent traveling between points (during which detected animals are not being counted).
13.3.2. M o d e l s and Estimators The logic of density estimation with point samples is analogous to the situation with transect sampling, with detectability decreasing monotonically as objects are farther from the sample point. The sample point can be viewed as the center of a circular plot with an unknown "effective radius," which defines an "effective area" as the basis for adjusting counts for partial detectablity. If k replicate points are used and n animals are counted in areas of radius w around the points, a natural estimator of D under complete detectability is
E) = n / A = n/k,rrw 2. Typically, however, not all the animals in the areas are detected, and the estimator needs to be adjusted for
279
detectability to eliminate bias. If Pa is the average probability of detection for an organism in the sampling area, the estimator
=
(
n ) k~rw ~ /Pa
(13.23)
accounts for partial detectability. The challenge is to estimate Pa with the radial distances r from the sampling point. The detectability of organisms typically declines with radial distance according to some probability density function g(r), and one seeks to use the sampling distances to estimate g(r) as a basis for determining Pa" The analysis for point sampling proceeds as with transect sampling, wherein organisms within some sampling area around a sample point can be characterized by the bivariate pair (r, y), with r the radial distance from the sample point and y = 0 or 1 depending on whether the organism is observed. For now we consider a single point and let f(r, y) represent the joint distribution of (r, y) under random sampling, along with the marginal distributions fl(r) for radial distance and f2(Y) for observation status over the sampling area. Then the probability density function of recorded distances is given by Bayes' Theorem as
f(y = llr)fl(r) f2(Y = 1)
f(rly = 1 ) =
g(r)fl(r) w g(r)fl(r) dr
f0
'
where f(y = l lr) = g(r) is the detection function. On assumption that individuals are randomly located with respect to the location of the point sample, the distribution fl(r) of radial distances is given by fl(r) = 2,rrr/,rrw 2
= 2 r / w 2. Denoting f(rIy = 1) by f(r), we thus have the distribution
f(r) = f
w rg(r) rg(r) dr 0
of observations within a distance w of a sample point [note the similarity between this distribution and Eq. (13.10) for line transect distances]. Now consider the number of organisms in an annulus of width dr about a randomly located point, which is expressed by Nfl(r) dr. This number must be adjusted to account for partial detectability; thus, the expected number of organisms actually observed in
280
Chapter 13 Estimating Abundance with Distance-Based Methods
the annulus is Ng(r)fl(r) dr. The total number of observations over the sampling area then is given by integrating over the sampling area" W
ments these procedures by allowing users to specify whether the data are based on line or point samples, which in turn specifies either Eq. (13.13) or (13.27) as the functional form of the density estimator.
E(n) = f o Ng(r)f l(r) dr
13.3.3. Assumptions
-- X~)al
where Pa is the "average" detection probability [averaged over the frequency distribution fl(r)]" W
Pa -- f o
g(r)f l(r) dr.
(13.24)
Substituting fl(r) = 2 r / w 2 into Eq. (13.24) gives
Pa =
fw
g(r) 2r
dr
o _
2
w
w 2 fo rg(r) dr, and the density estimator in Eq. (13.23) becomes
b = (A) 1 (13.25) n
2k'rrf o rg(r) dr" Letting
v = f~
rg(r) dr, it is easy to see that f' (O) = (df /dr)(O) _
rg'(r)
+ 8(r)
m
v
r=O
= g(O)/v = 1/v when g(0) = 1. Substituting this expression into Eq. (13.25) then produces the simplified estimator
= nf'(O)/2k'rr
(13.26)
of density. Because f' (0) must be estimated from data, the density estimator is more properly expressed as
= nf'(O)/2k~r.
(13.27)
A comparison of Eqs. (13.12) and (13.26) shows that line transect and point estimators are similar in form, the primary difference being that line transect estimation utilizes the probability density function for observed distances, whereas point estimation utilizes its derivative. Estimation from point sampling is thus closely related to that from line transect sampling, and the same general maximum likelihood procedures for estimation of the detection function, described in Section 13.2.5, are applicable. Program DISTANCE imple-
The assumptions of point sampling are essentially the same as those for line transect sampling (Section 13.2.2), substituting "point" for "line" as appropriate. The assumption of perfect detectablity at zero distance [g(0) = 1] may be reasonable for bird surveys because of the length of time (typically up to 10 min) spent at each point. As noted by Buckland et al. (1993), point sampling for birds should be conducted so that detection probabilities are highest (i.e., early morning), both to avoid bias from g(0) < 1 and to provide maximal precision (increased overall detection). As with line transects, estimation in point sampling may be unreliable unless a shoulder exists for the detection function. This suggests a design for point sampling such that g(r) ~ 1 for some predetermined radius r < r 0. Violations of the assumption of no movement of animals in relation to the observer appear to be more serious for point sampling than for line transect sampling. This can be particularly problematic for bird surveys whereby observers either attract bird movement (out of curiosity or "scolding" behavior) or cause birds to flee from around the point. On the other hand, increased detection rates from scolding or other behavior, so long as these do not involve movement into or out of a detection area, can be beneficial in increasing detection rates (Buckland et al., 1993). Effects of observer-induced movement in bird surveys on estimation have been modeled by Wildman and Ramsey (1985), Bibby and Buckland (1987), and Roeder et al. (1987).
Example Buckland et al. (1993) described a study of house wrens (Troglodytes aedon) involving data collected from 155 points (14-16 in each of 10 16-ha study blocks). Initially all the data were used, providing a maximum observed distance of w = 92.5 m. The data were fit to four models representing combinations of the halfnormal, hazard, and uniform key functions, and hermite polynomial, simple polynomial, and cosine adjustment factors (Table 13.3). Because all the models exhibited lack of fit (P ~0.05), they were fit to the data a second time after truncating the observed distances at 42.5 m. The results were ranked by ascending values of AIC and were used to select a model that is based on the hazard key function with a simple, one-term polynomial adjustment factor. Estimated density from
13.4. Design of Line Transect and Point Sampling Studies TABLE 13.3
Example of Point Data from Surveys of House Wrens (Troglodytes aedon) Surveyed along the South Platte River, Colorado a
Adjustment
Number of adjustment terms
Half normal Half normal Uniform Hazard Data truncated (w = 42.5 m)
Hermite polynomial Cosine Cosine Simple polynomial
3 3 4 1
Hazard Half normal Uniform Half normal
Simple polynomial Cosine Cosine Hermite polynomial
1 1 3 1
Key function
281
Goodness of fit AIC
AAIC
X2
6624.8
6665.5
0 5.1 8.6 40.8
5523.8 5524.8 5526.0 5528.0
0 1 2.2 4.2
95% CI
df
P
/~
C-'L
C'U
10.8 10.7 18.8 39.3
4 4 3 4
0.03 0.03 <0.001 <0.001
8.28 8.47 6.72 6.05
6.98
9.82
7.24 5.95 5.28
9.91
7.1 7.6 7.0 12.1
3 4 3 4
0.07 0.11 0.07 0.02
8.14 9.01 9.05 7.84
6.44 7.43 7.48
10.30 10.92 10.95 9.07
Data untruncated (w = 92.5 m) 6629.9
6633.4
6.77
7.58 6.93
a From Buckland, S.T., Anderson, D.R., Burnham, K.P., and Laake, J.L. (1993). "Distance Sampling: Estimation of Biological Populations." Chapman and Hall, New York, with kind permission from Kluwer Academic Publishers.
this model was 8.14 b i r d s / k m 2, with a log-based 95% confidence interval of (6.44, 10.30).
13.4. D E S I G N O F L I N E TRANSECT AND POINT SAMPLING STUDIES As with any sample survey, survey design is critical to assure that assumptions are met (at least reasonably well), the resulting estimates are reliable and survey resources are used efficiently. No a m o u n t of statistical "magic" can produce reliable results if the survey design or data collection procedures are fatally flawed. As a first step in producing accurate estimators, an investigator should determine whether line transect or point sampling is an appropriate method for sampling and estimation. If the study is long term, or if the population is subject to significant mortality, birth, or m o v e m e n t over the sampling time frame, open population estimation methods (Chapters 15-19) m a y be more appropriate. Assuming that animals can be detected (e.g., visually) without capture, investigators should consider whether conventional finite sampling procedures based on complete detectability on sample units (see Sections 12.4 and 12.5) might be appropriate. At the very least, such procedures can provide a first approximation of the degree of variability likely to be encountered in counts and can be very useful for a pilot evaluation, even if assumptions of complete detection turn out not to be tenable.
A pilot study is highly r e c o m m e n d e d as a means of obtaining preliminary estimates of encounter rates on line transects (n/L) or point samples (n/k). It also can be informative as to the likelihood that critical assumptions can be met and h o w best to meet them (Buckland et al., 1993). Estimates of encounter rates often can be used directly to identify needed sample sizes, even if pilot data are insufficient for formal estimates of density and variance.
13.4.1. F i e l d P r o c e d u r e s
Density estimates from line transect and point surveys are only as good as the field data incorporated in them. Whether these data are representative of the population d e p e n d s in large measure on the design of the investigation, the field procedures used in their collection, and the skill and dedication of personnel conducting the study. Several points should be emphasized in designing and implementing a transect study. 1. Prior to the actual survey, a pilot study should be conducted. A pilot study increases one's familiarity with the organisms of interest and with the on-site field conditions facing the field crew. It also provides valuable information about the required sampling intensity, placement of transects in the study area, and other aspects of the design and field procedures. 2. The survey should be designed to avoid systematic sampling effects, such as sometimes arise w h e n transects or points are placed along hilltops, stream
282
Chapter 13 Estimating Abundance with Distance-Based Methods
beds, roadsides, etc. Information from the pilot study, maps of the area, field notes, and other information sources should help in avoiding this problem. 3. Because the key to the transect methodology is accurate measurement of distances, it is important to ensure that transect lines are visible and straight. Otherwise, the observer cannot determine the position of a line and thus cannot make accurate distance measurements. This is key to meeting the assumption that distances and angles are measured accurately. Likewise, sample points must be clearly marked or otherwise delineated. 4. Care must be taken to ensure that all individuals directly on the transect line or at the point are observed. One can meet this requirement by carefully traversing the transect lines and being alert to individuals on them or by expending appropriate effort (e.g., duration of listening period for birds) at the sample points. 5. Perpendicular distances of individuals from the transects or radial distances from points must be accurately recorded. Again, because the accuracy and precision of the density estimate are dependent on these measurements, they should be carefully recorded with tape measures and other measuring devices as appropriate. 6. Transect length or equivalent sampling effort should be chosen to meet requirements for precision in the density estimators. Burnham et al., (1980) suggest that, at a minimum, length should be chosen to assure that 40 individuals are observed, with 60 to 80 individuals preferred. Estimator precision is discussed in some detail below.
The term cv[f(0)l is mainly a function of the number of individuals detected and cv(n) represents the variability in counts among replicate lines or points. For a single transect line or point estimate, specification of cv(n) requires assumptions about the mean-variance relationship in counts. In that case one can either assume some spatial distribution of individuals and derive a value for var(n) from it or assume an expression for var(n) as a function of n. 13.4.2.1. Line Transects
Buckland et al. (1993) recommend at least 60-80 observations to provide adequate estimation for line transect surveys. Obviously, the number of observations cannot be determined with certainty prior to the start of a study, but the sample size is clearly related to sampling effort. For line transects, sampling effort is expressed as L, the total length of transect line sampled. The length L may consist of replicate lines, in which case the additional design issue concerns the choice of line length versus the number of replicate lines. Given a pilot sample of one or more lines with total length L0, a sample count of n o observations, and an estimate of the coefficient of variation cv0(/5) from the pilot study, one can obtain a preliminary estimate of the sampling effort needed to obtain a desired coefficient of variation of density. The idea is that sampling effort should be based on the pilot coefficient of variation, the pilot level of effort, and the desired coefficient of variation. On assumption that precision is inversely related to line length, these elements are related by cv(D) = Cvo(E))X/Lo/L,
13.4.2. Sample Size Determination If sample sizes in a study design are inadequate, the resulting estimates of density will be imprecise and thus will provide little information about the population. However, if sample sizes are too large, the information provided may be of high quality (assuming that other design considerations are met), but resources that could have been applied elsewhere (e.g., surveying more replicate sites, monitoring another population) will have been used unnecessarily. In line transect and point sampling, the variance of the density estimate potentially contains several components [Eq. (13.22)]. For nonclustered individuals, the estimated variance is given by V"~(/5) = /52([c"v(n)] 2 + {cv[)~(O)]}2),
so that the coefficient of variation is estimated as c v ( D ) 2 "-
var(/5) / ~ 2
= [~'v(n)] 2 + {cv[f(0)l} 2.
(13.28)
which indicates that a 50% reduction in the coefficient of variation can be obtained by increasing the sampling effort by L = 4L0. The problem of course is that this formula requires an estimate of var(/~), which in turn requires a substantial pilot effort involving multiple transect lines and large numbers of observations. An alternative approach expresses Eq. (13.28) in terms of sampling effort by n = [cv(/~)] 2
,
(13.29)
where b"
{ v a r ( n ) + n var[]~(O)] [f(O)]2 }"
Estimation of b is problematic for small surveys, but appears to be fairly stable in magnitude, in the range of 2 < b < 4. Burnham et al. (1980) argued that the value of b should be in the range 1.5 < b < 3 and
13.4. Design of Line Transect and Point Sampling Studies recommended b = 3 for planning purposes (Burnham et al., 1980). For example, suppose that cv(D) = 0.10 is desired. A pilot survey of L0 = 1 km is conducted, and n o = 25 animals are detected. Then Eq. (13.29) provides
L =
283
Lo[b + (sd(s)/s) 2] no[cv(~)]2
(Buckland et al., 1993), where s"d(s) = ~/
= 12km. By equating no/L o from the pilot survey and n / L for the planned survey, we then can obtain an estimate of the number of detections in the planned survey: n = L(no/L o) = 25(12/1) = 300. For a more extensive survey (n o > 60), b can be estimated directly as/~ - y/0[cv0(D)] 2 (Burnham et al., 1980; Buckland et al., 1993) and substituted into Eq. (13.29) to provide
L-
Lcv0(/~)
L0"
Practical limits to sampling effort Lma x c a n be imposed by money, time, and other constraints. To determine whether a survey is worth conducting, it is desirable to compute the coefficient of variation CVmin(D) that is achievable given resource constraints: 1/2
CVmin ' (Buckland et al., 1993). For the previous example, if at most 5 km can be surveyed, then the achievable precision is 3 CVmin(D) =
]1/2
5(25/1)
= 0.15. This approach can be generalized to provide coefficients of variation over a range of prospective sampling efforts. Using either graphical or formal optimization methods (Cochran, 1977; see also Appendix H), one then can select a sampling effort to achieve the greatest marginal increase in precision. For populations occurring in clusters, precision of the density estimate depends also on the precision with which mean cluster size is estimated. If an estimate of this precision is available from a pilot sample of n o animals, then the necessary sampling effort to achieve desired precision is
i_l!S i __ ~)2 no - 1 "
The previous material has dealt with sample size decisions in terms of the total amount of sampling effort (L) to allocate for estimating density. However, it is usually desirable to allocate this effort among replicate lines, both to assure adequate (ideally, random) sampling of the population and to enable proper estimation of variances. From Eq. (13.18) the components to the variance of/) include variance due to the estimation of g(x) and variation in the rate of encounter of individuals, which in turn is a function of heterogeneous detectability among individuals as well as the spatial dispersion of individuals. Replicate lines allow separate estimation of these components and permit evaluation of alternative schemes for determining the number of replicate lines and the line lengths. Additional considerations in sampling design involve the relative costs of laying out replicate lines, traveling between lines, and surveying individual lines. There is yet to be a complete treatment of this problem in the context of line transect estimation, but we note that Skalski and Robson (1992) developed formulas for sample allocation for a similar problem in the context of mark-recapture (i.e., replicate trap grids, numbers of traps per grid). Given cost functions for each component of the survey, and estimates relating sampling effort to each component of variation in density, similar functions could be developed for line transect problems.
13.4.2.2. Point Samples Determination of sample size for point sampling proceeds in essentially the same manner as for line transect sampling, where now the quantity of interest is the number of replicate points. If a pilot survey is conducted using k0 points, and n o objects are detected, then a desired coefficient of variation [cv(D)] can be achieved by
[CV(-D)]2 J~O0 (Buckland et al., 1993), where b = 3 again is a reasonable value when sample sizes are inadequate for estimation of b. Given a determination of k for this desired precision, an estimate of the number of objects detected in the planned survey is provided by n = kno/k o. Of
Chapter 13 Estimating Abundance with Distance-Based Methods
284
course, this assumes that points are distributed randomly throughout the sampled area. If points are restricted (e.g., to lines), this expression tends to underestimate the sample size needed. In the extreme case in which points are placed very closely (almost continuously) along lines, the sampling unit essentially reduces to a line and line transect sampling and estimation procedures should be used.
13.4.3. Stratified Sampling For sample estimates of density to apply to an appropriate target population, the sampling units (lines transects or point samples) must be distributed so as to represent adequately the population. Ideally, this means a completely random assignment of the units to the area to be sampled (e.g., Fig. 13.4a), although for practical reasons this goal often must be compromised. For example, completely random lines might overlap extensively and therefore create a situation in which the samples are not truly independent. Certain design restrictions (e.g., stratification) can actually improve the statistical behavior of density estimates. On the other hand, one should avoid sampling that is restricted to only a portion of the study area such as roadsides (e.g., Fig. 13.4b).
13.4.3.1. Stratified Estimates of Density As with other sample survey procedures, stratification can be useful in line transect sampling, both to permit separate estimation of density for each stratum and to enable more precise estimation of overall density. In addition, stratification is appropriate when logistical factors (e.g., fewer, longer lines are more efficient) preclude complete randomization (Fig. 13.4c). It can be used to ensure homogeneous detectability, a condition on which density estimation depends. For a line transect survey, stratum-specific estimates of density are obtained by
Oj = njfj(O) 2Lj
(13.30)
with j = 1,..., J, where the subscripts denote stratumspecific values for n, L, and f(0). The corresponding expression for point sampling is =
2wkj "
(13.31)
Average density is then obtained as a weighted estimate across strata,
j=l
FIGURE 13.4 Possible layouts of line transects in a study area. (a) Random placement of transects. (b) Nonrandom placement of transects, leading to nonrepresentative sampling of the target population. (c) Stratified random placement of transects.
where A = ~ = 1 Aj (Buckland et al., 1993). Often data are sparse or there is empirical justification for estimating f(0) by~pooling across strata, in which case a pooled estimate f(0) is used in place of ~(0) [Eq. (13.30)] or f;(0) [Eq. (13.31)]. An appropriate likelihood ratio test for pooled estimation of detectability is provided by X2 - -
2 [ l n ( ~ j ) - ln(~p)],
where ~j a n d ~p here represent the maxima of the likelihoods for the detection models in the stratified
285
13.4. Design of Line Transect and Point Sampling Studies and the pooled models, respectively. The test has degrees of freedom equal to the difference in the total number of parameters estimated under stratification and that estimated under pooling, and rejection indicates that detection should be estimated separately by strata (see Section 4.3). 13.4.3.2. Abundance Estimation
In the preceding sections of this chapter we have emphasized estimation of density (D = N/A) rather than abundance (N). The approach presented here has the advantage that estimation does not depend on the presence of a defined, finite sampling area. However, if the samples are taken from an area of known size A, then abundance can be estimated b y / ~ / = /gA. In particular, if sampling is stratified, each stratum having area Aj, then the estimate of overall abundance is obtained as
l j=l l
j=l where/gj is based on either separate or pooled estimation of detectability, as above.
parameter (density or abundance, possibly stratified by areas). Interest has focused on the variance of the estimate var (/9[D), conditional on a fixed value of density. With nonclustered data there are two components to this variance: (1) variation in the count statistic ("count variance") and (2) variation in the estimated detection function ("model variance") [Eq. (13.18)]. Both components are estimable under certain conditions (e.g, replicate transect lines per area). Another source of variation is important when there is interest not only in estimation, but also in hypothesis testing in an experimental context. In the ideal case, experimental study populations are assigned at random to two or more treatments, and inference centers on testing the effect of the treatment on a parameter of interest (e.g., density). Following the experimental intervention, each population j = 1, ..., J has an unknown density Dj, and it is of interest to distinguish "background" variation in/9 from variation induced by the treatment. One approach, advocated by Burnham et al. (1987) in the context of mark-release studies, is to treat each sample estimate/gj of density as an observation (e.g, in an ANOVA) and to proceed as usual with appropriate hypothesis tests. That is (ignoring the treatment effects),
J (~j_ D)2 v~(/9)
13.4.3.3. Sample
Allocation under
Stratification
Optimal allocation of sampling effort is obtained from minimizing variance expressions and solving for the allocation of the proportion of total sampling effort (L or k) to strata (Lj, kj, j = 1..... J). If detectability is homogeneous, then Buckland et al. (1993) show that the optimal allocation proportions ~rj = Lj/~lm=l L m for line transect sampling are in the range of values between
- Z'
l AmVGm
and
=
EJ=I
l-
1
'
(13.32)
where D = ~=1 L)i/J and v~(/~) is taken as an estimate of interpopulation variation in density. Clearly this empirical variance includes variation among the true densities Dj as well as sampling variation [var(Dj/ Dj)], and both components are relevant to distinguishing experimental treatments from sampling variation. Sometimes, however, it is useful to estimate geographic or spatial variance. Under certain circumstances (e.g., conditional variances constant among populations), this variance component can be estimated by subtraction, va~r(D) = var(/9) - var(/9[D),
~rj=
1
~m=l AmDm
where Aj, Dj are the area and density of the jth stratum, respectively, and L is estimated using methods as described in the previous section.
13.4.4. Experimental Design and Replication of Study Populations Thus far we have considered sampling design and estimation from the standpoint of estimation of a single
where var(D) is the "true" interpopulational variance in density and var(/9]D) is the mean of the sampling variance estimates (an estimate of the constant conditional sampling variance). Alternative assumptions regarding variation among conditional sampling variances for the populations require more complicated iterative weighted estimators (Burnham et al., 1987). Experimental designs based on replicated populations require three levels of allocation of sampling effort: (1) amount of effort per line, (2) numbers of replicate lines per population, and (3) numbers of repli-
286
Chapter 13 Estimating Abundance with Distance-Based Methods
cate populations. We are unaware of procedures for these allocations that are specific to line transect or point sampling, but the general approach is well developed and has been applied in a similar context to mark-recapture experimental design (Skalski and Robson, 1992). In general, within-population (sampling) variation is decreased by either increasing line lengths, increasing the encounter rate, or increasing the number of lines per replicate population. The between-population variation is decreased by increasing the number of replicate populations, and all three components determine the power of hypothesis tests. With information on the relative costs of each factor under control, and how each enters into the variance of the test statistics, functions can be developed relating power to cost or other considerations.
13.5. OTHER ISSUES
Estimation of go ordinarily requires auxiliary data or experiments in conjunction with ordinary line or point sampling. Empirical detection models based on sampling conditions (e.g., weather, vessel speed, observer factors) and animal behavior (e.g., dive response, aggregation patterns) have been used to model detection as described by Buckland et al. (1993). These detection models are similar to modeling approaches that have been used to model sightability in aerial transect surveys (Samuel et al., 1987). In principle, any method could be used for independently estimating density or abundance at the line or point and the results incorporated as adjustment factors. For example, sightings by independent observers could be used in conjunction with mark-recapture models (Chapter 14) to estimate detection probabilities at the line (or in a region very near the line). Some applications of this general approach have appeared in recent years (Alpizar-Jara and Pollock, 1996, 1999; Berchers et al., 1998a,b).
13.5.1. Estimation When g(0) < 1 As we have seen, the assumption of complete detectability [g(0) = 1] is critical to unbiased estimation of density and abundance using distance methods. In some practical situations this assumption may be violated. Most of the work directed at this problem has been motivated by surveys of whales, because these animals may pass directly beneath survey vessels without detection (Buckland et al., 1993). The density formula, Eq. (13.21), can be extended naturally to include less than perfect observability. If the probability of detection on the transect line is go ~ 1, one can rescale g(x) by g(x)/g o, so that the (scaled) detection probability g(x) becomes 1 on the transect line. Then the "average" detection probability can be factored into goPa, and the appropriate formula for density is
D-
nE(s) APag o"
where E(s) is the expected value of the cluster size. Use of data-based estimates of E(s), Pa, and go in this formula produces /~ = n/~ (s) APago ^
^
!
the variance for which is estimated by va'~r(D) = D2([~V(n)~2 + {c'v[E(s)]}2 + ~c"V(A/3a)]2 + {~[d(0)]} 2) on assumption that n and s are statistically independent.
13.5.2. Cue Counting Cue counting is a method for estimating detection rates and densities of animals that exhibit behavioral "cues" indicating their presence. Buckland et al. (1993) describe cue counting as a special case of distance estimation, closely related to point sampling, in which radial distances are measured (often by visual estimation or using rangefinders) from lines or points to the objects detected by cues. A cue counting estimator for sampling from points (Buckland et al., 1993) is expressed as a straightforward extension of Eq. (13.27), with the count now referring to the number of cues per unit time in the study area.
13.5.3. Trapping Webs Usually the estimation of abundance from captures is considered as a mark-recapture problem, and the methods of Chapter 14 (closed populations) or Chapters 17-19 (open populations) are appropriate. Density estimation based on these abundance estimates requires information about the dimensions of the area from which the marked sample was obtained, i.e., an "effective trapping area." Serious biases can occur if this area is inappropriately estimated, for instance by using only the area actually covered by a trapping grid or other array. The problem is well known (e.g., Dice, 1938) and various approaches for estimating this area have been used, such as estimation of abundance on nested trapping grids and development of densityarea relationships (Otis et al., 1978). An alternative proposed by Anderson et al. (1983) is based on an extension
13.6. Discussion of point sampling and distance estimation to captures in an array of traps placed in concentric circles about a central point (resembling a spider web, hence the name "trapping web"). Estimation is based on total numbers of first captures of animals grouped by distances from the central point and directly corresponds to estimation from point sampling (with the data grouped), where n is now the cumulative number of first captures (over some reasonably short trapping period, e.g., ~ 5 days), and the grouped distances and frequencies of first captures are used to estimate f' (0). We discuss trapping webs in more detail in Chapter 14.
13.6. D I S C U S S I O N In observation studies, variable detectability is a recurring problem that, left unaddressed, can induce severe bias and undermine ecological inferences. In this chapter we have described methods to account for variable detectability, based on the probability of detection as a function of the distance between an organism and an observation point or transect line. Line transect and point sampling methods both utilize observation distances to adjust the count of individuals over a study area as a function of a detectability. The statistical treatment of detection combines parameteric and nonparametric approaches, through the inclusion of a key function to establish the basic shape of the detection function and an adjustment series to improve its fit to the observation data. A robust estimation procedure utilizes maximum likelihood methods for estimation of parameters in the key function and adjustment series. Likelihood ratio testing can be used for model comparison and selection with nested models, and Akaike's Information Criterion is available for nonnested models. The approaches described here summarize important advances in the methodology for estimating animal density with point and transect sampling (Buckland et al., 1993).
287
The computational burden associated with distance estimation has been eased considerably by the availability of several high-quality software packages. We recommend the program DISTANCE, developed by Laake et al. (1993) and available at no cost (Appendix G). In addition to estimation based on Fourier series, DISTANCE provides a wide array of models and adjustment combinations, using the key function approach described in Section 13.2.4. DISTANCE thus supplants TRANSECT (Laake et al., 1979). For size bias estimation based on bivariate detection functions, we suggest program SIZETRAN (Drummer, 1991); however, most applications can be handled adequately by DISTANCE, including those involving size bias. In Chapter 14 we introduce an alternative and substantially different approach to the estimation of density and abundance, based on repeated sampling of a population via the capture of individuals. Under certain conditions the record of captures and recaptures for individuals can be used to make inferences about population size and density. Though it shares with distance-based methods the general objective of estimating population size and density, a capturerecapture approach differs from distance-based methods in its assumptions, models, field procedures, and data requirements. In Chapter 14 we explore the use of capture-recapture and removal methods for closed populations. We emphasize in closing that the line transect and point count methods described here are restricted to closed populations. Thus, application of these methods requires that population size remain stationary during the investigation, i.e., there is effectively no mortality, recruitment, or migration into and out of the population over the time frame of the study. When these rather restrictive conditions are not met, methods that are appropriate for open populations must be used. We reserve for later chapters a more comprehensive treatment of capture-recapture methods for open populations.
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C H A P T E R
14 Estimating Abundance for Closed Populations with Mark-Recapture Methods 14.1. TWO-SAMPLE LINCOLN-PETERSEN ESTIMATOR 14.1.1. Sampling Scheme and Data Structure 14.1.2. Models and Estimators 14.1.3. Violation of Model Assumptions 14.1.4. Study Design 14.1.5. Example 14.2. K-SAMPLE CAPTURE-RECAPTURE MODELS 14.2.1. Sampling Scheme and Data Structure 14.2.2. Modeling Approach 14.2.3. Estimating Population Size 14.2.4. Confidence Interval Estimation 14.2.5. Testing Model Assumptions 14.2.6. Model Selection 14.2.7. Estimator Robustness 14.2.8. Study Design 14.2.9. Example 14.3. DENSITY ESTIMATION WITH CAPTURE-RECAPTURE 14.3.1. Uniform Sampling Effort (Grid Trapping) 14.3.2. Gradient Designs (Trapping Webs) 14.4. REMOVAL METHODS 14.4.1. Sampling Scheme and Data Structure 14.4.2. Models and Estimators 14.4.3. Violation of Model Assumptions 14.4.4. Study Design 14.4.5. Example 14.5. CHANGE-IN-RATIO METHODS 14.5.1. Sampling Scheme and Data Structure 14.5.2. Models and Estimators 14.5.3. Violation of Model Assumptions 14.5.4. Study Design 14.5.5. Example 14.6. DISCUSSION
This chapter again addresses the estimation of animal abundance, a state variable of key interest in studies of population dynamics. Chapters 12 and 13 dealt with approaches to estimation based on observations of animals, using methods for which some (though not necessarily all) animals are readily observable without capture. However, many animals are not easily observed in the wild, either because of their size, preferred habitat, behavior, or other aspects of their ecology. For these animals, abundance estimation can be accomplished using methods based on captures of individual animals. In some cases the captured animals are marked and returned to the population (capture-recapture methods), whereas in others the animals are removed from the population. This chapter covers both capture-recapture and removal methods for populations that are closed. A closed population is one that experiences neither losses nor gains between sampling periods. The closure assumption is relaxed for removal methods, and instead the population is assumed to be closed except for known removals. In Chapters 18 and 19 we generalize this situation to include populations that are open to gains and losses between sampling periods. The general approach in this chapter involves some number K > 1 of discrete sampling occasions at which animals are caught, with unmarked animals in a sample given unique marks and released back into the population. Identification codes are recorded for previously marked animals in the sample, after which they also are released back into the population. Repeated sampling results in a "capture history" for each animal caught at any time in the study. A capture history is
289
290
Chapter 14 Mark-Recapture Methods for Closed Populations
simply a vector of K ones and zeros indicating the sampling occasions when the animal was caught (denoted by 1) and not caught (denoted by 0). For example, a study with K = 5 sampling periods might record a capture history of 01101 for a particular animal, indicating that it was captured on occasions 2, 3, and 5, and not caught on occasions 1 and 4. Data from a capture-recapture study can be summarized in matrix form as X -
Xl I X21
X12 X22
X13 X23
... ...
XIKX2K (14.1)
B
ever, this sampling situation is sufficiently rare (although we expect its use to increase) that it will not be discussed in this book. Methods requiring removals of animals from the population frequently are applied in situations in which animals are harvested for recreational or commercial purposes. The catch-effort models of Section 14.4 are based on known, or at least estimated, numbers of animals caught and removed from the population at each sampling occasion, where the effort expended at each occasion also is known or estimated. The change-in-ratio methods of Section 14.5 are based on a combination of sampling approaches, with observations of animals sampled before and after harvest periods when known numbers of animals are removed from the population.
XMK+I,1 XMK+I,2 XMK+I,3 ... XMK+I,K ..
where the subscript i for matrix element Xij denotes an individual animal (i = 1, ..., M K + 1, with M K + 1 the total number of individuals caught during the study) and j denotes the sampling occasion (j = 1, ..., K). Each element of the matrix assumes a value of either I (if the ith individual is caught on the jth sampling occasion) or 0 (if the ith individual is not caught on the jth sampling occasion). Capture-recapture estimators are based on probabilistic models of events giving rise to the data in _X. In models of closed populations, the relevant model parameters include capture probabilities at each of the sampling occasions. Thus, each individual i in the population has a probability Pij of being captured at each sampling occasion j. The challenge is to recognize and model the patterns in these probabilities and to use the corresponding probability models to develop estimates of population size. Some modeling approaches include abundance as a model parameter, whereas others treat abundance as an unknown random variable to be estimated. This chapter describes estimation methods when there are just two samples (Section 14.1) as well as methods for the general K-sample situation (Section 14.2). The special problem of density estimation is covered in Section 14.3. The sampling approach and data structure for capture-recapture correspond to a relatively small number of discrete sampling periods. There may be sampling situations in which animals are sampled continuously in time, such that the exact time of capture is known for each animal (e.g., birds removed from mist nets; photographic sampling in which time is recorded). Capture-recapture models and estimators have been developed for such situations (Craig, 1953; Darroch, 1958; Becker, 1984; Yip, 1989; Wilson and Anderson, 1995; Yip et al., 1996; Lin and Yip, 1999). How-
14.1. T W O - S A M P L E LINCOLN-PETERSEN ESTIMATOR Though the Lincoln-Petersen estimator of population size was developed for a specific sampling situation, it forms the basis for understanding virtually all capture-recapture estimators. Lincoln-Petersen estimation was used by Laplace (1786) to estimate the human population of France (Seber, 1982) and later by Lincoln (1930) to estimate the number of waterfowl in North America. Norwegian biologist C. J. G. Petersen pioneered the study of marked fish, and although his name is attached to the estimator, it appears that he did not use capture-recapture data to estimate abundance (LeCran, 1965).
14.1.1. Sampling Scheme and Data Structure The Lincoln-Petersen estimator is appropriate when there are just two sampling periods and the interval between sampling occasions is relatively short. An example might involve the trapping of a small mammal population on two consecutive days. In this case the animals caught and marked on the first day are released back into the population, and animals caught on the second day are examined for marks. Because there are only two sampling periods, capture histories are known with certainty even if the animals are not given individually distinguishing marks. This is not true when there are more than two trapping occasions. The X matrix for a two-sample capture-recapture study contains only three observable capture histories, namely 11, 10, and 01, leaving unrecorded the number of animals not captured on either sampling occasion
14.1. Two-Sample Lincoln-Petersen Estimator (unobservable capture history 00). We use Xl0, x01, and Xll t o represent, respectively, the number of individuals captured the first time only, the second time only, and both times. Capture history data for two-sample studies often are summarized using the following statistics: 1. n I = Xll if- Xl0; the number of animals caught and marked in the first period. 2. n 2 = Xll q- X01; the number of animals caught in the second period. 3. m 2 = Xll; the number of animals caught in both periods. 4. r = n I + n 2 - m2; the number of distinct animals captured during the study. Although the Lincoln-Petersen estimator typically is discussed in the context of traditional capturerecapture studies, it has been applied in a variety of studies of animal populations. In many situations the initial sample is obtained with traditional capture methods, with the second sample obtained using a different method [e.g., resighting, harvest (removal sampling)]. In other sampling situations, no capture is involved, even for the initial sample. For example, two independent observers can record the geographic location of inanimate objects such as animal nests or other structures, and these data can be used in the LincolnPetersen estimator [see reviews of such studies in Lancia et al. (1994) and Southwell (1996)]. Then Xll denotes the structures seen by both observers, and Xl0 and x01 denote the numbers of structures seen exclusively by either observer but not both. This sampling approach has been used with territorial animals under the assumption that an animal seen on the same territory on successive survey visits must be the same individual (Hewitt, 1967; Thompson and Gidden, 1972). Three key assumptions underlie Lincoln-Petersen estimation: (1) the population is closed to additions (via birth and immigration) and losses (via death and emigration) over the course of the investigation (between sampling periods), (2) marks are neither lost nor overlooked by the investigator, and (3) all animals are equally likely to be captured in each sample. We discuss the consequences of violating these assumptions in Section 14.1.3.
14.1.2. Models and Estimators
14.1.2.1. Estimation of Abundance The Lincoln-Petersen estimator can be derived in a number of ways and under several different probabilistic models. We begin with an intuitive derivation, noting that the proportion of marked animals in a population after the first sample is simply n l / N , where
291
N denotes population size. If all animals have equal capture probabilities, this proportion should approximate the proportion of marked animals in the second sample, i.e., nl/N
= m 2 / n 2.
Rearrangement of terms in this equality leads to the Lincoln-Petersen estimator for population size: lgq = n l n 2 / m 2 .
(14.2)
The Lincoln-Petersen estimator also can be derived using the canonical estimation approach of Eq. (12.1). If we view n I as the count statistic and estimate the capture probability associated with this count a s fil = m 2 / n 2, then the ratio of the count statistic to the detection probability is the Lincoln-Petersen estimator: /Q =
nl/]91
= nlr/2/m
2.
More formally, the estimator can be derived under a number of probabilistic models, including the multinomial. Thus, the probability distribution for the two-sample study can be written as P(nl, n2,m2N, pl, P2) =
N! m2!(nI - m2)!(n2 - m2)!(N- r)! X (plP2)m2(plq2)nl-ma(qlP2)n2-m2(qlq2)N-r
where qi = 1 - Pi. This distribution contains a probability for each capture history, with the corresponding exponent given by the number of animals exhibiting that history. The cell probabilities sum to unity (PIP2 + Plq2 + qlP2 + qlq2 = 1), and the numbers of animals exhibiting each capture history sum to N [m2 + (n I m 2) + ( n 2 -- m 2) + ( N - r) -- N ] . The MLEs under this model are fil = m2/r12 = n 1/1Q, fi2 = m2/rll = na/1Q ,
and 1 ~ = n l n 2 / m 2.
It sometimes is useful (e.g., Section 14.2.2 and Chapters 18 and 19) to consider a conditional approach to estimation, in which N is not considered a parameter of the probability distribution used in estimation. This approach conditions on the numbers of animals that are caught during the study (r) and then estimates capture probabilities using these data. Applying cap-
292
Chapter 14 Mark-Recapture Methods for Closed Populations
ture probability estimates [e.g., Eq. (12.1)] to the appropriate count statistics permits estimation of N. To illustrate, consider the probability that a member of N is captured at least once during a two-period study: p* = 1 -- (1 -- pl)(1 -- P2)" Then a conditional multinomial model for the capture histories {x01, Xl0, Xll} is
and Robson (1992) recommended confidence intervals for the estimator based on the transformation ln(N). The method of Chao (1989) and Rexstad and Burnham (1991) that is based on the estimated number of animals not captured (see Section 14.2.4) also should be useful for Lincoln-Petersen estimates. Finally, confidence intervals can be estimated by the method of profile likelihood (see Sections 4.2.3 and 14.2.4, below).
14.1.2.2. Estimation of Relative Abundance P({xij}l r, Pl, P2) =
X11!X10!X01! (plP2~ TM (plq2~ xl~ (qlP2~TM
In many cases, one is interested in inferences about the relative abundance of animals at two sampling locations. Let K = Nb/Na
One can show that the maximum likelihood estimates for this model are
Pl -- X11/(X11 4- X01) and
for populations at sampling locations a and b (see Skalski et al., 1983; Skalski and Robson, 1992). LincolnPetersen estimators can be used to estimate K when the capture probabilities in a and b differ. An estimator of K is
P2 = Xll/(Xll 4- X10),
= [(nbl + 1)(rib2 + 1) _ 1]/11a111a2,/ma2 mb2 + 1 k
so that ~*
=
1
-
(1
-
]91)(1
-
]92)
with
--rx11/(x11 4- X1o)(X11 4- Xol). The latter formula can be applied in the canonical estimator, Eq. (12.1), which again produces the LincolnPeterson estimator for N: lC4 = rift* = (Xll 4- XIO)(Xll 4- XO1)/Xll = r/1112/m 2.
It can be shown that the Lincoln-Petersen estimator [Eq. (14.2)] is biased and that the magnitude of the bias is inversely related to sample size. The bias-adjusted estimator /Q = (n I 4- 1)(n 2 4- 1 ) _ 1 m2+1
(14.3)
V"~(/~) = ma2rlbl~------~nb2r"
m~2n31n32U'nb2
+ ( n a 2 - ma2)(11a1 - ma2)mbarlblnb2], where again the a and b subscripts specify location for the usual Lincoln-Petersen summary statistics (Skalski et al., 1983; Skalski and Robson, 1992). Estimation of relative abundance simplifies considerably when capture probabilities are the same for the two populations. In that case the estimator uses the total numbers r a and r b of individuals caught from the two populations in
t(1 = rb/ra with estimated variance
is conditionally unbiased when n I + n 2 -> N (Chapman, 1951). The variance for N can be estimated as va,--~(/qr) = (nl + 1)(n2 + 1)(nl - m 2 ) ( n 2 - m2) (m 2 + 1)2(m2 + 2)
v~r (/(1) =
(Seber, 1970a). Confidence intervals for Lincoln-Petersen estimates of population size can be constructed in various ways. One approach (Seber, 1982) is to rely on the asymptotic normality of N and construct the approximate 95% confidence interval for N as/Q + 1.96V'v~(/~). Skalski
(n.2_
m.2)(n.1 _ m.2)(/~1 + /~2)
11.111.2ra
n.1 -- nal + nbl, n.2 -- na2 + 11,b2, and m.2 -ma2 4- rob2. In general, var(/(1) < var(K). This makes intuitive sense, because/( requires separate estimation of capture probabilities for the two populations, so that v~(K) includes variance components for both capture probability estimates. On the other hand, /~1 ass u m e s that capture probabilities are identical for the two populations, so that the resulting variance is smaller. The use of /(1 requires some method for testing
where
(14.4)
- mb2)(11b1 -- mb2)ma211a111a2
14.1. Two-Sample Lincoln-Petersen Estimator whether the capture probabilities for two populations actually are equal. Skalski et al. (1983) and Skalski and Robson (1992) recommended the use of a 2 • 3 contingency table to test for homogeneous capture probabilities. A contingency table for capture history data from populations a and b is given as follows: Capture history
Population a
Population b
Sum
1 1
Xal I -- ma2
Xbl I = mb2
X.11 -- m . 2
10
Xal o = Ylal -- ma2
Xbl 0 = Ylbl ~ mb2
X.10 = 1"/.1 -- m . 2
01
Xa01 -- Yla2 m ma 2
Xb01 -- Ylb2 ~ mb 2
X.01 = r/.2 _ m . 2
ra
rb
r
Sum
The dot notation in this contingency table denotes summation over the two populations, for example r = r a + r b and x.11 = Xal I 4- Xbl 1. Expected cell frequencies under the null hypothesis of equal capture probabilities for the two populations are given by Expected frequency Capture history
Population a
Population b
Sum
11
Na PiP2
Nb PlP2
(Na + Nb) Pl P2
10
Na Plq2
NbPlq2
(Na + Nb) Pl q2
01
Na ql P2
Nb ql P2
(Na + Nb) ql P2
Na(Pl + P 2 - Pl P2)
Nb (Pl + P 2 - PlP2)
(Na + Nb)
Sum
X (Pl + P2 - PIP2)
Under the null hypothesis of equal capture probabilities for the two populations, the test statistic associated with this contingency table follows a chi-square distribution with two degrees of freedom. The test can be extended readily to the situation of more than two sampling periods and more than two populations, by considering expected numbers of animals for each population exhibiting all possible capture histories (Skalski et al., 1983; Skalski and Robson, 1992). 14.1.3. V i o l a t i o n of M o d e l A s s u m p t i o n s 14.1.3.1. C l o s u r e
The closure assumption can be violated in several ways, including mortality during sampling (i.e., during capture in sample 1). One way to deal with sampling mortality is simply not to include the number of sampling deaths (denote this number as d) in the n 1 statistic. Then the Chapman (1951) estimator/~ in Eq. (14.3) estimates the population size after the sampling deaths, whereas/9 + d is an estimate of the presampling population size.
293
The closure assumption also can be violated by deaths between sampling occasions. To see the effect of mortality on the Lincoln-Petersen estimator, define q0 as the probability that an animal alive at the time of the first sampling occasion is still alive and present in the population at the time of the second sampling occasion. Here we assume that r applies to all individuals in the population, whether captured or not. Because population size differs between the two sampling periods, it is necessary to designate by N 1 the population size at sampling occasion 1. Though E(n 1) = N i p 1, the expected values of the other two summary statistics are influenced by q0, with E(n 2) = NlCpp 2 and E(m 2) = NlPlCpp 2. Substituting these expectations into the standard Lincoln-Petersen estimator, Eq. (14.2), we obtain E(Iq) ~ E ( n l ) E ( n 2 ) / E ( m 2 ) = (NlPl)(Nlq~p2)/NlPlCpP2 -- N1"
Thus, the Lincoln-Petersen estimator provides an estimate of the population size at the time of the first sampling period (Robson, 1969; Seber, 1982). N o w consider mortality associated with handling or marking, which is imposed only on members of n 1. In this case, the expected values of the summary statistics can be written as E(n 1) = N i p 1, E(m 2) = NlPlq~p2, and E(n 2) = Nl[PlCpp 2 4- (1 - Pl)P2]. Substituting these expectations into Eq. (14.2), we obtain E(lxl) ~- E ( n l ) E ( n 2 ) / E ( m 2 ) = ( N l P l ) N l [ P l @ p 2 4- (1 - p l ) P 2 ] / N l P l @ P 2
= NI[Pl + (1 - pl)/q~]. Thus, the Lincoln-Petersen estimator is positively biased in the presence of deaths associated with handling or marking. Intuitively, the estimator/31 = m 2 / n 2 is too small because some of the animals marked in sample 1 die and therefore are not available to be caught in sample 2. A negative bias in ~31 then leads to positive bias in/9. The closure assumption also can be violated because of immigration of new animals between the two sampling periods. If we denote the number of animals entering the sampled population between the first and second periods as B, the expectations of the summary statistics become E(n 1) = N i p 1, E(n 2) = (N 1 + B)p2, E(m2) = NlPlP2. Substitution of these expectations into Eq. (14.2) yields E(lCq) ~, E ( n l ) E ( n 2 ) / E ( m 2) = ( N l P l ) ( N 1 4- B ) p 2 / N l P l P 2
=NI+B.
294
Chapter 14 Mark-Recapture Methods for Closed Populations
Thus, the Lincoln-Petersen estimator provides an estimate of the population size at the time of the second sample (see Seber, 1982). Finally, consider the case where both mortality/emigration and immigration occur between the two samples. The expected values for the summary statistics are now E(n 1) = Nip 1, E(n 2) = (Nlq~ 4- B)p2, and E(m 2) = Nlq~plP2. Substitution of these expectations into Eq. (14.2) yields
E(I~I) ~ E(nl)E(n2)/E(m 2) = (Nlpl)(Nlq~ + B)p2/NlPl~P2
(14.5)
same probability of being caught. The assumption of equal catchability can be violated in two ways. First, members of the sampled population can be heterogeneous with respect to capture probability, such that some animals have a higher probability of being caught than other animals. Consider the Lincoln-Petersen estimator in Eq. (14.2) as an example of the canonical estimator, where n I is the count statistic and ]91 = m2/n2 is the estimate of the corresponding sampling probability. Animals with higher capture probability than average have a greater chance of being caught in both samples, so that
= N 1 4- B/q~.
E(]~I)
Because q~ < 1, the expectation in Eq. (14.5) is larger than N 1 + B, and the Lincoln-Petersen estimator is positively biased for population size at either sampling time (also see Robson and Regier, 1968). A special case of a population that is open to both gains and losses considers the animals in the population at time j to represent a subset of animals in a superpopulation of size N ~ with animals in the superpopulation moving freely in and out of the sampled area. Assume that the animals in the sampled area at either time represent a random sample from the superpopulation with probability ,rj, i.e., E(NjlN ~ ,rj) = N%j. If the superpopulation is closed and the capture probability pj is redefined to be conditional on being in the sampled area at time j, Lincoln-Petersen estimation produces an estimator of the capture probability ,rjpj for animals in the superpopulation, and the Lincoln-Petersen estimator for population size now estimates the number of animals in the superpopulation (see Kendall, 1999). Time specificity can exist in "rj or in pj (or both), and the Lincoln-Petersen parameterization and estimators are still appropriate for the superpopulation. This result is consistent with the more general situation expressed in Eq. (14.5), as seen by writing the expected values for the quantities in Eq. (14.5) in terms of the random movement model:
Pl
-
T1)T2,
b,
4-
where b is a bias factor (b > 0). Because the estimated capture probability is too large, the population size estimate is too small, that is
E(l(4) ,~ E(nl)/E(~I) = N[pl/(Pl
4-
b)].
A second form of unequal capture probability is known as trap response, referring to a tendency for animals caught in the first sample to have a different capture probability in the second sample compared to animals not caught in the first sample. Denote the capture probabilities for sample period 2 as Pc for captured, previously uncaught animals and Pr for recaptured animals that were captured in the first sample. The expected values of the summary statistics under this scenario can be written as E(nl)
=
Xpl ,
E(n2) = N[plp r 4- (1 - Pl)Pc], and E(m2)
=
XplPr.
Substitution of these values into Eq. (14.2) yields
E(l(4) ~ E(nl)E(n2)/E(m2) -- ( N F 1 ) N [ F I F r 4-
E(N1) = N~ E(B) = N~
=
-- X p l
4-
(1
-
pl)Pc]/XplPr
N(1 - Pl)Pc/PF-
Substitution of these expectations into Eq. (14.5) yields the approximate expected value of the LincolnPetersen estimator as E(N)-~N ~
The approximate expectation in this expression equals N if Pc = Pr, that is, if there is no trap response. When Pr > Pc (trap-happy response), the Lincoln-Petersen estimator is negatively biased. When Pr < Pc (trap-shy response), the estimator is positively biased.
14.1.3.2. Equal Capture Probability
14.1.3.3. Tag Loss
Capture probabilities for the Lincoln-Petersen estimator need not be the same for the two samples, but within each sample, all animals are assumed to have the
The third assumption underlying the LincolnPetersen estimator is that marks are neither lost nor overlooked. Consider the situation where a mark is
E(q~) = "r2.
14.1. Two-Sample Lincoln-Petersen Estimator lost between the first and second samples with probability 1 - 0, where 0 < 0 < 1. Then expected values of the summary statistics can be written as E(n 1) = Npl, E(n 2) = Np2, and E(m 2) = NplOp2. Substitution of these expectations into Eq. (14.2) yields
E(1Q) ~ E(nl)E(n2)/E(m2) = (Xpl)(Xp2)/XplOp2
= N/O. Because 0 < 0 < 1, tag loss (or failure to recognize tags) produces positive bias in the Lincoln-Petersen estimator. If the probability of tag loss can be estimated [e.g., via a double-tagging study as described in Seber (1982)], an improved estimate of population size is given as the product/Q0.
14.1.4. Study Design A design for a two-sample capture-recapture study should produce precise and unbiased estimates of abundance when the underlying model assumptions are met. Of particular concern is the closure assumption, which is influenced by the time period separating the two capture occasions. Deaths, recruitment, and movement in and out of the population are much more likely to occur over long time periods. Thus, there should be only a short time period separating the two sampling occasions for most populations. To avoid trap mortality at the first sampling occasion, also a violation of the closure assumption, traps should include sufficient bait to keep animals alive while they are in the traps. In areas experiencing high temperatures, trap covers should be used to shield traps from direct sunlight, and traps should not be set during the hot periods of the day. If trap mortality does occur on the first sampling occasion, then the animals experiencing mortality should be removed from the initial computations and not be included in the n I statistic. The number of trap deaths can be added to the estimated population size subsequently, to obtain an estimate of the pretrapping population size. The variance of the adjusted population estimate is unchanged by the addition of a known number of trap deaths. Also important is the assumption of equal capture probabilities, recognizing that in reality this assumption is seldom if ever met exactly. If capture probabilities are likely to vary with visible characteristics of captured animals ( e.g., age, sex, weight), then samples can be stratified and stratum-specific estimates computed. The distribution of sampling devices (e.g., traps, nets) relative to the distribution of animals can be an important influence on the heterogeneity of capture probabilities, and one should avoid leaving some ani-
295
mals with small probabilities of capture and other animals with high probabilities. Even or uniform trap spacing is often desirable, with multiple traps per average home range size of the studied species. Of course, there sometimes are not enough traps to allocate multiple traps per home range over the entire area of interest. In such cases, division of the sampled area into quadrats, with random allocation of traps to quadrats at each of the two sampling occasions, should help equalize the underlying capture probabilities. As noted above, a special case of unequal capture probability involves a behavioral trap response. In this situation, animals caught in the first period have either a lower (trap-shy) or higher (trap-happy) probability of being caught in the second period compared to unmarked animals. Prebaiting before the first sampling period can reduce a trap-happy response, and minimization of handling time in the first period may reduce trap-shyness. We note that although the Lincoln-Petersen estimator is held to require equal capture probabilities of all animals within each sample, certain kinds of heterogeneity are allowed. If the animals exhibit heterogeneous Capture probabilities, yet the capture probabilities for an individual in the two sampling periods are completely independent (so animals with a relatively high capture probability in the first period do not necessarily have a high capture probability, again in the second period), the Lincoln-Petersen estimator still provides an unbiased estimate of population size (e.g., Seber, 1982). This observation has led to designs involving different capture methods for the two sampling occasions. If the initial marked animals are obtained as a random sample, then the second sample can be highly selective and still yield an unbiased estimate of abundance (Robson, 1969; Seber, 1982). For example, sometimes it is possible to use hunting or fishing as a way of obtaining the recapture sample for the LincolnPetersen estimator. In addition to closure and homogeneous capture rates, the assumption of no tag loss is required for Lincoln-Petersen estimation. This assumption is likely to be met for short-term studies for which the LincolnPetersen estimator typically is used. Tag losses can be investigated by the double-tagging of individuals with two standard tags or with a single standard tag and a more durable "permanent" tag. In this way, the loss of standard tags can be recognized, estimated, and accounted for in estimating population size (e.g., see Seber, 1982). While attempting to meet underlying model assumptions, study designs also should focus on obtaining precise abundance estimates. Precision increases with increasing capture probabilities, so design efforts
296
Chapter 14 Mark-Recapture Methods for Closed Populations
should be directed at catching a large proportion of animals in the sampled area. There are many ways of influencing capture probability, depending on the capture methods used. Robson and Regier [1964; reprinted in Seber (1982)] provided plots of sample sizes (n 1, n 2) needed to achieve Lincoln-Petersen abundance estimates with specified levels of accuracy for different population sizes. Robson and Regier (1964) presented an approach to optimal allocation of effort to the first and second samples as a function of the relative costs of the two types of sampling. Because two-sample studies are typically of short duration, it often is possible to conduct a pilot study to obtain an idea of capture probability and population size. Information about these parameters then can be used to design a more comprehensive study with the desired precision.
of animals exhibiting each possible capture history o~. For example, Xl01 denotes the number of animals caught at the first and third sampling occasions of a three-sample study. The counts x~ can be collapsed further into summary statistics for estimating parameters in specific capture--recapture models. There are assumptions underlying capture-recapture models for closed populations: (1) the population is closed to additions (via birth and immigration) and losses (via death and emigration) during the course of the study, (2) marks are neither lost nor overlooked by the investigator, and (3) capture probabilities are appropriately modeled. The first two assumptions are identical to the assumptions for Lincoln-Petersen estimation. The third assumption generalizes the Lincoln-Petersen assumption of equal capture probability.
14.1.5. Example
14.2.2. Modeling Approach
Skalski et al. (1983) reported results from a study of Nuttall's cottontail rabbits (Sylvilagus nuttallii) in central Oregon, in which 87 cottontails were captured and then released after their tails and hind legs were marked with picric acid dye. A follow-up sample yielded 14 animals counted on a drive count, and 7 of these were marked. Numbers of animals exhibiting each possible capture history were X l l -- 7, X01 = 8 0 , and x01 = 7. Thus, the summary statistics were n I = 8 7 , n 2 = 14, and m 2 = 7. The Chapman estimator, Eq. (14.3), for these data is/q = 164, with estimated variance from Eq. (14.4) of v~(/~ r) = 1283.33 and standard error SE(/~/) = V ' v ~ ( / ~ = 35.82.
We consider here a number of models that make different assumptions about the sources of variation in the capture probabilities, the primary parameters needed to model capture-recapture data for closed populations. To illustrate, consider the sampling of a closed population on three occasions, with unique marking of individuals so that individual capture histories can be recorded. For this situation there are 23 = 8 possible capture histories {i,j,k}, with the binary indices i, j, and k indicating capture outcome for the three sampling occasions:
14.2. K-SAMPLE CAPTURE-RECAPTURE MODELS 14.2.1. Sampling Scheme and Data Structure Here we consider capture-recapture models for sampling situations with K > 2 sampling occasions. An example might involve the trapping of a small mammal population for five consecutive nights. At each sampling occasion, previously uncaptured animals are marked with individually identifiable tags, and the identification codes of previously marked animals are recorded. Individual marks or some other scheme permitting reconstruction of the individual capture histories is required, so that the complete capture history of each animal encountered can be known unambiguously. The data from a K-sample capture-recapture study can be organized in an X matrix as shown in Eq. (14.1) and summarized in statistics x~ denoting the number
{1, 1, {1, 0, {0, 1, {0, 0, {1, 1, {0, 1, {1, 0, {0, 0,
1}, capture 0}, capture 0}, capture 1}, capture 0}, capture 1}, capture 1}, capture 0}, capture
all three times first time only second time only third time only first two times only last two times only first and third times only at no time
Let Xijk be the number of individuals with capture history {i,j,k}, where ~i,j,k Xijk = N. If probabilities for these capture histories are the same for all individuals in the population, the appropriate statistical model is a multinomial distribution
P(xijkIN, ,rijk) =
N! x~ I I ~ij~, IIi,j,k Xijk! i,j,k
with eight cell probabilities, "rrijk, where the subscripts representing sampling period take a value of I (indicating capture) or 0 (indicating noncapture). Thus the fully parameterized model includes eight parameters: the population size N and seven of the eight probabilities for capture histories [the eighth probability is given by 1 - (sum of the other seven)].
14.2. K-Sample Capture--Recapture Models Additional assumptions about the capture history probabilities can lead to model simplification. With the assumption of independence of capture events (i.e., no trap response) the probabilities associated with the different capture histories can be expressed as functions of time-specific capture probabilities. For example, the probability of catching an animal on all three occasions can be written as 11"111 = PlP2P3, where Pl, P2, and P3 are the probabilities of capture on occasions 1, 2, and 3, respectively (Table 14.1). This results in a model with four independent parameters (N, Pl, P2, P3), down from the original eight. A further assumption about equiprobable capture across periods (Pl = P2 = P3 = P) leads to a model with only two parameters (N and p). Reductions in model complexity also are possible under an assumption of differences between capture probabilities for marked and unmarked individuals. Under this scenario the probability structure of the model is written in terms of the probabilities Pc for first capture and Pr for recapture (Table 14.1). This assumption allows the number of model parameters to be reduced to three (N, Pc, Pr)" With the additional assumption of independence of capture events (Pr = Pc = P) the model again reduces to one containing only two parameters (N and p). In general, a fully parameterized model for K sampling periods requires 2 K parameters (population size N and 2 K - 1 of the probabilities corresponding to 2K possible capture histories). The corresponding model with an additional assumption of independent captures, allowing for temporal variation in capture probabilities (pj), requires only K + 1 parameters. On the
TABLE 14.1 Possible Capture Histories and Associated Probabilities a Capture
Probability
history
Mo b
Mb a
Mt c
111
p3
PlP2P3
Pcp2r
110
P2( 1 -- P)
PlP2( 1 -- P3)
PcPr( 1 --Pr)
101
P 2(1 -- P)
Pl( 1 -- P2)P3
PcPr( 1 -- Pr)
100
p(1 -- p)2
p1(1 -- p2)(1 -- P3)
Pc( 1 -- PF)2
011
p2(1 -- p)
(1 -- Pl)P2P3
(1 -- Pc)PcPr
010
p(1 -- p)2
(1 -- p1)P2(1 -- P3)
(1 -- pc)pc(1 -- Pr)
001
p(1 -- p)2
(1 -- pl)(1 -- P2)P3
(1 -- pc)2pc
000
(1 -- p)3
(1 -- pl)(1 -- p2)(1 -- P3)
(1 -- pc )3
other hand, the assumption of equiprobable captures across time, allowing for different probabilities of marked and unmarked individuals, results in a model requiring only three parameters. Finally, the addition of an assumption of both independent and equiprobable captures always requires just two parameters. For example, a model for four sampling periods requires either 16, 5, 3, or 2 parameters, depending on the assumptions of the model, whereas a model for five sampiing periods requires either 32, 6, 3, or 2 parameters. Thus, the impact of additional simplifying assumptions is exponentially greater as the number of sampling periods increases. The broadest possible class of models allows for separate probabilities Pij for each individual i and each capture period j. The models below allow for behavioral responses to trapping, differences in capture probabilities over time, and even heterogeneity in capture probabilities among individuals. For example, one can model capture probabilities so that capture events are independent (no trapping response) and equiprobable across trapping periods (no temporal variation in trapping probabilities), but specific to each individual in the population. The assumption of distinct capture probabilities for each individual is referred to as heterogeneity of capture probability. This source of variation is distinct from time-specific variation in capture probabilities, referred to as temporal variation. It also is distinct from a response to trapping, for which the probabilities of capture are the same for all marked individuals and the same for all unmarked individuals, but differ between the two groups. The latter effect is referred to as behavioral response. These three potential sources of variation in capture probability represent key elements in the modeling and estimation of closed populations (Pollock, 1974; Otis et al., 1978; White et al., 1982). Statistical modeling of closed populations based on multiple-recapture data is essentially an exercise in the comparison of models incorporating the various combinations of these three assumptions. Each combination of assumptions results in a distinct parameterization of the capture probabilities, and the challenge is to sift through the associated models to find one that best represents the sample data while minimizing model complexity. Conceptually, eight models can be defined:
M0 a U n d e r d i f f e r e n t m o d e l s in a t h r e e - s a m p l e c a p t u r e - r e c a p t u r e s t u d y of a c l o s e d p o p u l a t i o n . b p = c a p t u r e probability. Cpj = c a p t u r e p r o b a b i l i t y for s a m p l i n g p e r i o d j. a Pc = c a p t u r e p r o b a b i l i t y for u n m a r k e d a n i m a l s ; Pr = c a p t u r e p r o b a b i l i t y for m a r k e d ( r e c a p t u r e d ) a n i m a l s .
297
Mb Mt
Neither behavioral nor temporal variation nor capture heterogeneity (model parameters: N, p). Behavioral response only (model parameters: N, Pc, Pr)" Temporal variation only (model parameters: N, pj, j = 1.... ,K).
298
Chapter 14 Mark-Recapture Methods for Closed Populations
Individual capture heterogeneity only (model parameters: N, Pi, i = 1, ..., N). Mtb Behavioral and temporal variation only (model parameters: N, Pcj, Pry, j = 1, ..., K). Mbh Behavioral response and capture heterogeneity only (model parameters: N, Pci, Pri, i = 1, ..., N). Mth Temporal variation and capture heterogeneity only (model parameters: N, Pij, i = 1, ..., N, j = 1 ..... K). Mtbh Behavioral response, temporal variation, and capture heterogeneity (model parameters: N, P cij, Prij, i = 1.... , N, j = 1, ..., K).
Mh
The models M 0, Mb, and M t all possess MLEs, but additional assumptions or alternative approaches are required for estimation with models Mh, Mbh, Mth, Mtb, and Mtb h. MLEs for model Mtb can be obtained by assuming a relationship between the time-specific initial capture probabilities (Pcj) and recapture probabilities (Pry) (see Otis et al., 1978; Rexstad and Burnham, 1991). Estimates for models Mh, Mbh, and Mth can be obtained by assuming that capture probabilities for individuals are random samples of size N from an underlying distribution of probabilities (Pollock, 1974; Burnham and Overton, 1978, 1979; Chao, 1987) or by using an approach based on the concept of sample coverage (Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994). If time effects on capture probabilities are known (as when temporal variation is associated with relative trapping effort) it is even possible to obtain coverage-based estimates under model Mtb h (Lee and Chao, 1994). Pledger (2000) has used a finite mixture approach to obtain estimates under all four heterogeneity models (also see Norris and Pollock, 1996). Thus, with adequate data and appropriate parametric restrictions the parameters of all eight models can be estimated, and the models can be tested for goodness of fit. In addition, M b, M h, and M t can be compared to M 0, and Mbh can be compared to M h as an aid in model selection. Operationally, the idea is to do as follows:
6. Compare different models to identify the "best" model based on between-model tests, goodness-offit, and parameter parsimony. For the models for which MLEs can be determined, maximization of the likelihood function can be thought of as a two-step process. Let p represent the vector of capture probabilities defining a capture-recapture model, with N again denoting population size. The likelihood function can be expressed as L(N, p]{x~}), where the set {x~}denotes the number of animals exhibiting each capture history. Maximization of L(N, pl{x~}) is accomplished in program CAPTURE (Otis et al., 1978; Rexstad and Burnham, 1991) using the following general approach: 9 Consider N to be fixed, and choose the value ]3 maximizing L(N, p]{x~}) conditional on N. Let ~ denote the (conditional) KILE of p. In all cases for which the MLE can be described in closed form, a mathematical expression can be derived for fi in terms of N. 9 Incorporate ~ into the likelihood function, and choose N maximizing L(N, fi(N)]{xJ). Because fi is a function of N, the likelihood function now involves only the single variable N. With a single exception,/~ must be determined numerically, and because only the single parameter N is involved, this is a relatively easy numerical problem. m
14.2.3. E s t i m a t i n g P o p u l a t i o n S i z e
14.2.3.1. Constant Capture Probability~Model M 0 The simplest K-sample model assumes no variation in capture probability among animals or sampling occasions, i.e., Pij = P for all i and j (Darroch, 1958; Otis et al., 1978). Model M 0 contains only the two parameters, p and N. It is straightforward to show that the joint probability distribution for the data under model M 0 can be written as =
1. "Model" the capture probabilities by incorporating capture heterogeneity, behavioral response, a n d / o r temporal variation into the parametric structure. 2. Identify the probability density function that incorporates this parametric structure. 3. Incorporate the capture-recapture data {x~} in the probability density function, thereby identifying the likelihood function. 4. Maximize the likelihood function by choosing the appropriate estimates for the parameter values. 5. Calculate standard errors and confidence intervals for the estimates of population size and other parameters.
P({x~~
N!
p n.(1 - p)KN-n.,
P) [1-[0xo~!](N_ MK+I)!
(14.6)
where K /'/ -- ~ Y/j j=l is the total number of captures, and MK+ 1 is the total number of unmarked individuals caught during the study. The MLE ]~ for the capture probability under model M 0 can be derived by differentiation of the likelihood function in Eq. (14.6). In this case,/3 is just the number
14.2. K-SampleCapture-Recapture Models of captures divided by the number of opportunities for capture, or p = n/KN.
be missed (not caught) on each sampling occasion of the study in order to be missed for the entire study. 14.2.3.3. Behavioral Response--Model M b
Substituting this expression into the likelihood function and maximizing with respect to N yields the MLE /~ for population size. 14.2.3.2. Temporal VariationmModel M t
This model has a long history (Schnabel, 1938; Darroch, 1958) and is usefully viewed as the K-sample analog of the model underlying the Lincoln-Petersen estimator. Under model Mt, each animal has the same capture probability on any given sampling occasion (Pij = Pj for all i), but capture probabilities can vary from one occasion to the next. The model has K + 1 parameters, N and Pl, ..., PK (Table 14.1). The joint probability distribution for the data under model M t can be written as N~
P({x~o}]N, p)
299
= [l-I~o x 0 0 [ ] ( N - MK+I)[
(14.7)
The behavioral response model (Pollock, 1974; Otis et al., 1978) incorporates change in capture probability as a result of previous capture. Thus, captured animals not previously captured exhibit capture probability Pc, whereas marked (recaptured) animals exhibit capture probability Pr" The response may be either trap-happy (increased probability of capture after initial capture, P r > Pc) or trap-shy (decreased probability of capture after initial capture, Pr < Pc)" The model includes only the three parameters N, Pc, and Pr (Table 14.1). To describe the likelihood function for this model, let mj be the total number of marked animals caught on sampling occasion j, with Mj the number of marked animals in the population at the time of sampling occasion j. The probability distribution for model M b can be described in terms of the total number of recaptures K m =~mj j=2
K
• l-I pTJ(1 - pj)N-nj, j=l
where p is the vector of capture probabilities, Pl, ..., PK. Thus, the statistics needed for estimation are simply the number of animals caught on each sampling occasion (nj) and the total number of individuals captured m
during the study, the total number of marked individuals at the completion of the study (MK+I), and the sum (over all occasions) of the number of marked animals available for capture at each capture occasion K
M = 7_, Mj. j=2
(MK+I).
The MLEs/~j are determined by differentiating the likelihood function, Eq. (14.7). It is easy to show that /~j is just the number of animals captured in each period divided by the number in the population:
The corresponding probability density function is N! cMK+1 P({x~o}lN, Pc, Pr) = [l-I00 x~o!](N - MK+I)[ p X (1 --
~j = n j / N for j = 1, ..., K. The MLE of N is determined by substituting these expressions into the likelihood function and maximizing with respect to N. We note that in the special case of K = 2, the estimator/~/is simply the Lincoln-Petersen estimator [Eq. (14.2)]. Darroch (1958) showed that N could be estimated under model M t by solving the equation
pc) KN-MK+I-M
• prm'(1
-
(14.9)
pr)M.-m..
Under model M b, the MLE of the probability of first capture Pc is determined from Eq. (14.9) as the total number of first captures over the course of the experiment divided by the number of first capture opportunities: Pc = MK+I/(KN -- M.).
1
MK+I -- ~ N
1 --
(14.8)
for N. The left side of Eq. (14.8) estimates the probability that an animal is not caught during the study. The right side of Eq. (14.8) is the product of estimates of not being caught on each sampling occasion of the study (i.e., products of 1 - i0j). Thus, an animal must
The MLE for Pr is the total number of recaptures divided by the total number of potential recaptures: Pr = m . / M . Substituting these expressions into the likelihood function and maximizing with respect to N produces the MLE of N.
300
Chapter 14 Mark-Recapture Methods for Closed Populations
Under likelihood Eq. (14.9), the estimation of N depends only on initial captures, and recaptures are used only~ for estimation of Pr" Because of the dependence of N only on first captures, estimation of population size under the behavioral response model is equivalent to estimation under a removal model (e.g., Zippin, 1956, 1958), in which animals are removed from the population on initial capture (e.g., as in snap-trap surveys of small mammals).
14.2.3.4. Heterogeneity among Individuals--Model
Mh
Under model Mh, there is no temporal variation in capture probabilities and no behavioral response associated with initial capture. However, every individual animal in the population is permitted to have its own capture probability independent of that of every other individual, i.e., Pij = Pi for all j. The model is thus parameterized with N capture probabilities Pl, ..., PN as well as population size N, for a total of N + 1 parameters. The large number of model parameters led Burnham and Overton (1978) to consider alternatives to maximum likelihood estimation for this model. Their approach was to conceptualize the vector of capture probabilities {Pi} as a random sample of size N from some probability distribution F(p) defined on the interval [0,1] (Burnham and Overton, 1978, 1979; Otis et al., 1978). The corresponding statistical model can be described in terms of the number fj of animals caught on exactly j occasions: N!
P(fl .... , &IF)
= [I_[K=I fj!](N
K
-
MK+I)I'ITN-MK+Ij=lI-I Try, 9
where "rrj
= fo
)!j!pJ(1 - p)K-j dF(p).
(14.10)
where k denotes the "order" of the jackknife estimator and the ajk are constants generated by the jackknife procedure (see Appendix F). Each order k of the jackknife generates a different set of constants ajk and thus a different estimator /qk (see Burnham and Overton, 1978). Burnham and Overton (1979) recommended a series of statistical tests for selecting the appropriate order jackknife estimator for any data set. If the appropriate order jackknife lies between the values k and k - 1, an interpolation algorithm is then used to compute an estimate of N lying between/Qk and/~k- 1 (Burnham and Overton, 1979). Usually, k is chosen to be no greater than 5. Although the Burnham and Overton (1978, 1979) jackknife estimator is the most commonly used approach in animal abundance estimation under model M h, other estimators also have been proposed. For example, Pollock and Otto (1983) proposed a momentbased bias-corrected estimator, Smith and van Belle (1984) used a bootstrap estimator, and Chao (1987, 1988, 1989) introduced a moment-based estimator for use with sparse data. In what follows we describe in somewhat greater detail two additional approaches to estimation under M h. Chao et al. (1992), Chao and Lee (1992), and Lee and Chao (1994) have proposed estimators based on the idea of sample coverage C, defined as the sum of the individual capture probabilities for animals that are caught as a proportion of the total of individual capture probabilities for all N animals in the population. If all individuals in the population have the same constant or time-specific capture probabilities (as in models M 0 or Mt), then the sample coverage effectively estimates the probability that an animal is caught during the study. Thus, an estimate of the sample coverage can be used to estimate population size as = MK+I/C
The cell probability "rrj in Eq. (14.10) can be viewed as the average probability that an individual is caught exactly j times. Burnham and Overton (1978) considered estimation in the case where F(p) is the class of beta distributions, but this approach was found not to be satisfactory. Instead, they used an estimation approach based on the generalized jackknife statistic (Quenouille, 1949, 1956; Gray and Shucany, 1972), in which MK+ 1 is viewed as a naive estimator of N, and bias reduction is accomplished using a linear function of the capture frequencies fi. This approach leads to estimators of the form K
~I k = ~ ajkfj, j=l
(14.11)
(see Darroch and Ratcliff, 1980; Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994). Note that Eq. (14.11) is an example of the canonical estimator (Eq. 12.1). Estimators for C can be constructed from capture frequency data (Good, 1953; Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994). For example, a widely used estimator is K
= 1 - fl/j
14.12
and bias-corrected versions are available (Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994). The estimator in Eq. (14.11) is negatively biased when there is heterogeneity in capture probability among the members of the population, with the magnitude of the bias a function of the coefficient of variation
301
14.2. K-Sample Capture-Recapture Models of the capture probabilities. This coefficient of variation can be estimated as a function of the capture frequencies fj and used in turn to estimate population size in the face of heterogeneity (Chao and Lee, 1992; Chao et al., 1992; Lee and Chao, 1994) (also see model Mth below). A disadvantage of the jackknife and sample coverage estimators is that they are not maximum likelihood estimators and thus are not easily evaluated using familiar likelihood-based approaches. For example, likelihood ratio tests between models and model selection criteria such as Akaike's Information Criterion (AIC) are not available for testing sources of variation or selecting parsimonious models. Thus, nonparametric maximum likelihood estimators recently proposed by Norris and Pollock (1995, 1996; also see Agresti, 1994) are promising. This approach considers the joint estimation of N and a generating distribution F for the capture probabilities. The generating distribution is based on a finite mixture model in which the population is viewed as being composed of some finite (hopefully small) number of groups of animals having similar capture probabilities. The number of groups, the proportions of animals in each group, and the capture probabilities for the different groups are unknown and must be estimated. The approach to estimation involves cycling through each integer n between MK+I and some predetermined upper bound on population size, and, for each n, using the EM algorithm (Dempster et al., 1977) to obtain the MLE of F. The nonparametric MLE is then the (n, F) pair that yields the largest value of the likelihood function. This approach is computationally intensive, but is very general, yields MLEs, and thus has considerable promise. Pledger (2000) recently considered a somewhat different approach that utitlizes finite mixture models to deal with heterogeneous capture probabilities. Instead of estimating the number of groups in the mixture distribution directly within a single model, she proposed using multiple models defined by specific numbers of groups. Simulations and work with actual data sets indicate that two-group distributions frequently provide parsimonious models and estimators with good properties (Pledger, 2000). Pledger (2000) has derived estimators for all eight models of Otis et al. (1978). This estimation and modeling approach holds great promise, and we anticipate its becoming a standard approach for dealing with models for closed populations. Many of the competitor estimators under model Mh, including the jackknife estimator of Burnham and Overton (1978, 1979), the moment-based bias-corrected estimator of Pollock and Otto (1983), the bootstrap estimator of Smith and van Belle (1984), the moment-
based estimator of Chao (1987, 1988, 1989), the sample coverage estimators of Chao et al., (1992), Chao and Lee (1992), and Lee and Chao (1994), and the nonparametric MLE of Norris and Pollock (1996), have been investigated via simulation studies. These investigations have been documented in Otis et al. (1978), Norris and Pollock (1996), and the papers cited above. Some of these estimators perform especially well in specific sampling situations and poorly in other situations. On the other hand, the jackknife estimator of Burnham and Overton (1978, 1979), the oldest widely used estimator for this model, performs reasonably well in a variety of situations based on various simulation results. If the investigator can identify covariates (e.g., a size variable) associated with variation in capture probability among individuals, it is possible to use this additional information in estimation under a special case of model M h. Pollock et al. (1984), Huggins (1989, 1991), and Alho (1990) all considered modeling capture probability as a linear-logistic function of individual covariates, e.g., as e f30+ f31xi
Pi = 1 +
e ~O+f31xi'
where 130 and ~1 are parameters to be estimated and x i is the covariate value for individual i. The unconditional approach of Pollock et al. (1984) that includes N in the likelihood requires the grouping of covariates into a finite number of discrete classes and can be implemented using the program LINLOGN (Hines et al., 1984). The conditional approaches (conditional on MK+ 1 animals being captured) of Huggins (1989, 1991) and Alho (1990) do not include N in the likelihood and permit the estimation of individual capture probabilities Pi using continuous covariates. Estimation of abundance following the conditional approach is based on the estimator MK+I
19= i~1 1 . p';' where ~3~is the estimated probability that individual i was caught at least once during the study: K
/~* = 1 - ]-I(1 -19i) j=l
= 1 - (1 -/~i) K. The above abundance estimator is of the general form described by Horwitz and Thompson (1952) and, in the case of equal capture probabilities for all individuals, Pi -- P, is identical to the canonical estimator of Eq. (12.1). The conditional approach of Huggins (1989,
302
Chapter 14 Mark-Recapture Methods for Closed Populations
1991) and Alho (1990) is implemented in program MARK (White and Burnham, 1999). In situations in which the variation in capture probability among individuals is closely associated with easily measured covariates, the models of Pollock et al. (1984), Huggins (1989, 1991), and Alho (1990) should be useful in estimating population size. As with the finite mixture models of Norris and Pollock (1995, 1996) and Pledger (2000), these covariate models have the advantage of yielding MLEs and permitting likelihood-based inference and model selection.
14.2.3.5. Behavioral Response and Individual
Heterogeneity--Model Mbh Thus far we have considered temporal variation, behavioral response, and heterogeneity singly. However, it is also useful to consider sampling situations in which capture probabilities incorporate multiple sources of variation. For example, model Mbh includes both behavioral response and heterogeneity among individual animals. Thus, every animal in the population is assumed to have a specific pair of capture probabilities: Pci, the capture probability if individual i has not been captured previously, and Pri, the capture probability if individual i has been caught at least once. We assume that the pairs (Pci, Pri) are a random sample from a bivariate distribution F(pc, Pr)" Under the most general formulation, this model includes capture probabilities Pci and Pri for each of the N animals, along with population size N, for a total of 2N+1 parameters. Assuming independence of initial and subsequent capture probabilities, the probability density function can be factored as F(pc, Pr) = FI(Pc)F2(Pr). In this instance, all of the information needed to estimate N is provided by initial captures (as was the case for model Mb). To estimate parameters for this model, define uj, j = 1, ..., K, as the number of unmarked animals caught on sampling occasion j. If F l(p) is the unknown distribution of initial capture probabilities (the subscript c is dropped for this development), we can write the probability distribution of the unmarked captures as P ( u 1, ...,
N~
UKIF 1) -~
~I-I~.= 1 uj!~(N - M K+I) !
ta )( / j=l
where
j=l
~rj = El(1 - p)j-lp-] 1
= f (1 - p)j-lp dF l(p). 0
Estimation under this model can be accomplished by first transforming the K parameters -rrj into a new set of parameters pj via the relationship -try = p j j-1 1-Is=1(1 -- Ps), where pj is the average capture probability of individuals that have not been captured prior to the jth sampling occasion. Otis et al. (1978) based estimation on the assumptions that Pl ~ P2 ~ "" ~ PKand (Pl -- P2) > (P2 -- p3) > "" > ( F K - 1 -- FK)" The first assumption captures the idea that individuals with the high initial capture probabilities tend to be caught in the first sample, animals with slightly lower capture probabilities tend to be caught next, and so on until primarily animals with relatively low initial capture probabilities remain uncaught in the later samples. The second assumption is that differences between the average capture probabilities of animals caught in adjacent samples tend to be largest in the initial sampling periods and decline over time. Estimation involves sequential testing for differences among the pj. The first test is for equality of all the ~j. If this hypothesis is not rejected, one concludes that heterogeneity is not important and model Mb is appropriate for the data. If the hypothesis is rejected, then one next allows Pl to differ and tests for equality of the remaining pjs, P2 = P3 . . . . . FK" Sequential testing continues until it is concluded that the final K - r capture probabilities do not differ significantly, where r is the number of initial pjs that are modeled separately (r -~ K). Estimation of N is based on the resulting model. Several other estimators for model Mbh were considered by Pollock and Otto (1983). One of these is K-1
1~ = ~
u i + KUK,
(14.13)
j=l
based on the generalized jackknife statistic of Gray and Shucany (1972). Estimator (14.13) has performed well in simulation studies, especially with relatively small numbers of sampling occasions (e.g., K = 5). Lee and Chao (1994) presented an estimator for model Mbh based on sample coverage, and simulation results (Lee and Chao, 1994) indicated that it performed better than the generalized removal estimator of Otis et al. (1978) in terms of root mean squared error, but not as well as the jackknife estimator of Pollock and Otto (1983). Norris and Pollock (1995, 1996) and Pledger (2000) developed MLEs for model Mbh using the finite mixture model approach outlined above for model Mh, which simulations suggest is competitive with the other estimators referenced here (Norris and Pollock, 1996). As with the finite mixture MLE for model M h, the estimator has the advantage of placing model Mbh in the likelihood framework that is so useful for model evaluation. If capture probabilities can be modeled
14.2. K-Sample Capture-Recapture Models using individual covariates, then the logistic models of Pollock et al. (1984), Huggins (1989, 1991), and Alho (1990) can be used with Mbh (see previous discussion under model Mh).
14.2.3.6. Temporal Variation and Individual HeterogeneitymModel Mth This model permits variation in capture probabilities Pij both over time, j = 1.... , K, and for individual animals, i = 1, ..., N. The likelihood under the model was described by Otis et al. (1978), but associated estimators were not developed until later (Chao et al. 1992; Lee and Chao, 1994; Pledger, 2000). Otis et al. (1978) viewed the set of capture histories {xij} as mutually independent random variables, with Pij described by Pij -- Pi ej, where 0 -< piej G 1. They viewed Pi as a random sample from some probability distribution F(p) and described the probability distribution of the observed sample {xij} as
303
model, Chao et al. (1992) provided guidelines for which estimators work best, depending on the magnitude of the sample coverage and the coefficient of variation of the capture probabilities. The preferred estimator of Chao et al. (1992) is implemented in program CAPTURE (Rexstad and Burnham, 1991). Pledger (2000) also has developed estimators under Mth , using the finite mixture model approach. If capture probability can be modeled using individual covariates, then the logistic modeling approach of Pollock et al. (1984), Huggins (1989, 1991), and Alho (1990) can be implemented for this special case of Mth. In particular, capture probability Pij for individual i at time j can be modeled as e ~oj4- f3lXi
Pij = 1 +
e ~~
Again, the advantage of the mixture model and covariate approaches is that they permit likelihood-based inference and model selection.
P[xij ] = P[xij J MK+I]PEMK+I] ,
14.2.3.7. Temporal Variation and Behavioral ResponsemModel Mtb
with
P[{xq} J MK+I] =
h e;J
j=l
1 1
L i=1
fo
K pyi[I-I(1
-- pej)l-xij] d E ( p ) ,
j=l
This model assumes a behavioral response to capture and also permits temporal variation in both initial capture and recapture probabilities. The model contains 2K parameters: population size N, a vector Pc = {Pc1.... , PcK} of initial capture probabilities, and a vector Pr = {Pr2, "", PrK} of recapture probabilities. The corresponding joint probability distribution for the data can be written in several ways (Otis et al., 1978), including m
where Yi is the number of times animal i is captured and P[MK+ 1] is the probability distribution of the number of animals caught in the study, depending on the parameters N, el, ..., eK, and the distribution F(p). Chao et al. (1992) utilized coverage estimators for this model with the general form 1Q
-
-
MK+I
4
+
f1~]2
4'
n/f
11
N!
I~,T
'-tlXod l l~, Pc, Vrl = [l-]~ Xo~]](N -
MK+I)!
(14.14)
Xihpc~(l_Pcj)N_Mj+l](14.1B)j=l where ,~2 is an estimate of the coefficient of variation of the individual capture probabilities. The latter quantity can be estimated (Chao et al., 1992) as r~J(1 -
x
.~2 = max
K ~,k=l k(k--1)fk (IVIK~+I~ ~R-~ ..... \c ! 2~j=1 ~k=j+l njnk
O) M j - m j
[
j=2 1, 0 .
(14.15)
Chao et al. (1992) presented three estimators for C [including Eq. (14.12)] for use in Eqs. (14.14) and (14.15). In particular, they found that the estimator
4 = 1 -h-2f2/(K1) K Ek=l kfk performed well in simulation studies. Based on their simulation work with model Mth as the underlying
where Uj and mj are the numbers of unmarked and marked animals, respectively, that are caught at time j, and My is the number of marked animals present in the population at the time of sample j. Because the probability distribution, Eq. (14.16), contains only 2K-1 statistics, the 2K parameters of the model are not identifiable. On the other hand, estimation of N is possible if a relationship is specified between Pcj and Prj" Otis et al. (1978) considered the multiplicative relationship PFj = 0pcj, j = 2, ..., K, but concluded that a constant relationship between initial capture and re-
Chapter 14 Mark-Recapture Methods for Closed Populations
304
capture probabilities is not realistic (Otis et al., 1978). Rexstad and Burnham (1991) considered the relation.~1/o ; ..., ship Prj = Vcj , J = 2, K, where 0 -< Pcj <- 1, j = 1, ..1/0 ___1, j = 2, ..., K. This parameterization ..., K and 0 < - Vcj is the most widely used for estimation under model Mtb and is implemented in program CAPTURE (Rexstad and Burnham, 1991).
14.2.3.8. Including All Three Factors--Model Mtb h In their initial description of model Mtbh, Otis et al. (1978) emphasized its conceptual utility as the most general of the models for closed populations. The model contains an initial capture probability for each individual in each sampling occasion (Pcq, i = 1, ..., N; j = 1, ..., K) as well as a recapture probability for each individual in each sampling occasion after the first (Prij, i = 1, ..., N; j = 2, ..., K). This yields (2K-1)N+1 parameters, which clearly are not estimable with the available data. Lee and Chao (1994) considered estimation under this model for Pcij = Pciej, noting that the model can be viewed as a variable catch-effort model (see Section 14.4) with removal probabilities Pcij = Pci for known relative efforts el, ..., e K. Utilizing the idea of sample coverage as discussed for model Mh, Lee and Chao (1994) recommended an estimator that incorporates Ck, the sample coverage of the first k samples, correcting for the negative bias associated with heterogeneous capture probabilities. The resulting estimator is IQ(k) = M k + l
Ak~
Ck +
Ck '
finite mixtures approach. She considered time, behavioral response, and heterogeneity as main effects and incorporated interaction terms as well. Though her "fully interactive" model with all interactions (including the three-way interaction) requires some constraints on parameters, she also considered partially interactive models incorporating most or all of the twoway interactions. Pledger's (2000) models are new and have seen little use to date, but we expect them to be widely used as they become better known to biologists. 14.2.4. C o n f i d e n c e
Interval Estimation
We focus here on variance and confidence interval estimation for population size. As noted in Chapter 4, asymptotic variances, covariances, and confidence intervals can be generated from the information matrix for those models with MLEs (most of the models that do not incorporate heterogeneity, and some of the newer models that do). However, this approach requires asymptotic normality of point estimates N, and this assumption frequently is not met because of small numbers of captures and recaptures. Under these conditions, the information matrix can lead to biased estimates of variances and poor coverage of the resulting confidence intervals. An alternative approach to interval estimation focuses on the number fo = N - M K + 1 of animals not captured, on assumption that this quantity follows a log-normal distribution (Chao, 1989; Rexstad and Burnham, 1991; also see Burnham et al., 1987). Lower and upper confidence interval bounds for N are given by ^
(14.17)
where
+ f0/c, M > I +
ak= Uk+-------~ 1
ej+lUj, \ ek+l / j-1 Uj+I
where
fo = 1 ~ - MK+ 1 Ck = 1 --
/'/k+l/ek+l
and
ul/el
and
2:maxlI,lllUluael,e2, u2
] 10}
for k = 1.... , K - 1, with M k + l the number of distinct animals captured in the first k samples; Simulations by Lee and Chao (1994) suggested that N ( K - 1) from Eq. (14.17) is the most appropriate estimator for population size when the coefficient of variation of the capture probability distribution is greater than 0.4 (i.e., in the face of substantial heterogeneity). Pledger (2000) considered estimation for model Mtbh using linear-logistic modeling in conjunction with her
c exp{19611nlI+vi2111'2} The lower bound of this confidence interval cannot be smaller than M K + 1, but the upper bound frequently is larger than upper bounds computed with the information matrix under the assumption of normality. Another approach to interval estimation makes direct use of the likelihood function, and the resulting intervals are frequently termed "profile likelihood intervals" [for general applications see Hudson (1971) and Venzon and Moolgavkar (1988); for capture-recapture see Otis et al. (1978) and Rexstad and Burnham (1991)]. The profile likelihood approach is based on
14.2. K-SampleCapture-Recapture Models lnL(0_), where 0 is a vector of parameters consisting of N and the capture probability parameters p (see Section 4.2.3 for general discussion). It reduces lnL(0) to a function of a single parameter (N) by treating the capture probability parameters as nuisance parameters and maximizing over them. The profile likelihood confidence interval then consists of all values of N for which the log-likelihood function evaluated at N is no more than 1.92 units from the maximum value of the log-likelihood function (the log-likelihood function evaluated at the MLEs, including N). The value 1.92 comes from the 0.95 quantile of the chi-square distribution, based on the generalized likelihood ratio test (Venzon and Moolgavkar, 1988; Rexstad and Burnham, 1991). Thus, profile confidence intervals include values of N that correspond to values of the likelihood function that are "close" to its maximum (Otis et al., 1978). D
14.2.5. Testing Model Assumptions A discussion of model assumptions is more involved with K-sample models than with the twosample Lincoln-Petersen estimator, because K-sample studies permit tests of underlying assumptions. Here we address both the testing of assumptions and the assessment of estimator performance when assumptions are violated. We focus on population closure and the absence of tag losses during the investigation. The third assumption (Section 14.2.1) of proper modeling of variation in capture probabilities is dealt with in Section 14.2.6 on model selection. 14.2.5.1. C l o s u r e
All of the models described in this section were developed under the assumption that the sampled population does not change during the course of sampling. We first consider tests of the closure assumption and then discuss consequences of its violation. 14.2.5.1.1. Tests for C l o s u r e
The most commonly used closure test (Otis et al., 1978) uses the null hypothesis H0: Pij = Pi, j = 1 . . . . , K, for all animals captured two or more times. The alternative hypothesis is that some capture probabilities were zero prior to initial capture or subsequent to final capture, because the animals arrived after the study began or departed before the study was completed. Thus, the alternative hypothesis is Ha: Pil -- Pi2 ..... Pir = 0 a n d / o r Pis -- Pi,s+l . . . . . PiK = 0, with r and s the first and last times of capture, respectively. Under H a we would expect the time between first and last capture to be less than under H 0. Otis et al. (1978)
305
developed a closure test based on the observed times between first and last capture for all animals caught at least twice, which is computed by program CAPTURE (Rexstad and Burnham, 1991). The test is sensitive to behavioral and temporal variation in capture probabilities (e.g., low capture probabilities at the beginning or end of a study can confound assessment). In addition, the test is not suitable for detecting situations in which animals emigrate temporarily during the middle of the study. Pollock et al. (1974) considered the testing of four hypotheses about time-specific variation in capture probabilities that are relevant to the closure assumption: (1) no mortality and no recruitment (complete population closure), (2) mortality but no recruitment, (3) recruitment but no mortality, and (4) both recruitment and mortality. Burnham (1997) considered the probability distributions under hypotheses 2, 3, and 4 above, and Stanley and Burnham (1999) have used these results to develop an overall test for population closure using time-specific capture-recapture data. The resulting chi-square test essentially tests the null hypothesis of complete closure (hypothesis 1 above, which corresponds to model M t) against the alternative hypothesis of a completely open population with both mortality and recruitment (hypothesis 4 above, which is the Jolly-Seber model to be described in Chapter 17). The overall test statistic of Stanley and Burnham (1999) can be decomposed into components that provide information about the nature of the closure violations. Under the first decomposition, one component represents a test of the null hypothesis of no recruitment (hypothesis 2 above) versus the alternative of the Jolly-Seber model (hypothesis 4). Another component tests null hypothesis M t (hypothesis 1) against the alternative of mortality but no recruitment (hypothesis 2). The chi-square test statistics for these two tests are independent, and their sum (also distributed as chisquare under the null hypothesis of closure) provides a test of null hypothesis 1 (M t) against alternative hypothesis 4 (Jolly-Seber model). Under the second decomposition of the test statistic (Stanley and Burnham, 1999), one component provides a test of the null hypothesis of no mortality (hypothesis 3) versus the alternative hypothesis of the Jolly-Seber model (hypothesis 4). The other component tests null hypothesis M t against the alternative of recruitment but no mortality (hypothesis 3). The chi-square test statistics for these two components also can be summed to obtain the overall closure test of Stanley and Burnham (1999). Thus the two decompositions have the same overall null (M t) and alternative (Jolly-Seber) hypotheses but involve different intermediate hypotheses.
306
Chapter 14 Mark-Recapture Methods for Closed Populations
Stanley and Burnham (1999) provided information about the power of these test components to the alternatives of permanent and temporary emigration and immigration. Thus, behavioral response in the absence of migration can lead to false indications of closure violations, but some violations are still detectable even in the presence of trap response. Stanley and Burnham (1999) recommend that their closure tests be used in conjunction with, rather than instead of, the test of Otis et al. (1978). The null model of the Otis et al. (1978) test permits heterogeneity of capture probabilities, but is sensitive to time and behavioral variation. On the other hand, the null model of the Stanley and Burnham (1999) test permits temporal variation, but not heterogeneity or behavioral response. The closure tests of Stanley and Burnham (1999) are implemented in software CLOSTEST written for that purpose, whereas the test of Otis et al. (1978) is implemented in CAPTURE (Rexstad and Burnham, 1991).
14.2.5.1.2. Consequences of Closure Violation The consequences of violations of the closure assumption for estimates based on closed population models were reviewed by Kendall (1999). An interesting form of closure violation considers animals in a population to be a subset of animals in a superpopulation of size N ~ Members of the superpopulation move freely in and out of the sampled area, and animals in the sampled area at time j are essentially random samples with probability Cj from the superpopulation. Under these conditions the expected size of the population in the sampled area is a function of the size of the superpopulation and the probability ~'j: E(NjlN ~ = N~ To illustrate, consider model Mt, with pj now reflecting the conditional (on being in the sampled area at time j) capture probability. On assumption that the superpopulation is closed during the study, the M t estimator for capture probability now estimates the product r the capture probability for an animal in the superpopulation. Thus, the population size estimator under M t estimates the number of animals in the superpopulation. Time specificity can exist in r or p (or both), and the M t parameterization is still appropriate. If neither 9 nor p varies over time, then estimation for the superpopulation should be based on M 0. A different scenario for the violation of closure allows for the entire population to be available for capture at the first sampling occasion, but permanent emigration by some individuals can occur before the study is completed (emigration only). Alternatively, some animals could enter the population during the study period (immigration only). Under these scenarios, the K-sample model estimators are biased, and the partially open models of Darroch (1959; also see Jolly,
1965; Burnham, 1997) can be used for estimation (see Chapter 18). Yet another scenario corresponds to a migration stopover site, with animals entering the population during the study and then (potentially) emigrating before the study is completed. Again, estimates obtained under closed models are biased in this situation, and models for open populations should be used for estimation. One approach utilizes the idea of a superpopulation (Crosbie and Manly, 1985; also see Schwarz and Arnason, 1996), which permits direct estimation of the number of animals that were members of the population at some time between the first and last sampling occasions (see Chapter 18).
14.2.5.2. Tag Loss As with the Lincoln-Petersen two2sample estimator, tag loss induces a positive bias in N because capture probability following initial capture tends to be underestimated. In certain cases, it may be possible to recognize recaptures as animals that have been caught before (e.g., in small mammal studies, animals losing ear tags can be identified by torn ears), even though individual identification is not possible. It may be possible to reconstruct capture histories fairly reasonably in such cases. Otherwise, the behavioral response models M b and Mbh , which do not rely on recapture information, can be used to provide unbiased estimates if other assumptions hold true. Tag loss can be investigated with double-tagging studies, in which some animals are marked with two tags, either of the same or different types. Recaptures of double-tagged animals with only one tag provide evidence of tag loss, and numbers of recaptures with one and two tags provide the data needed to estimate tag loss (e.g., see Seber, 1982).
14.2.6. M o d e l Selection One strategy to guard against the failure to incorporate important sources of variation in a model would be to select the most general of available models. However, the sample data allow one to estimate a few parameters with greater statistical precision (at a potential cost in bias) or a greater number of parameters with less precision (but potentially less bias). One therefore faces a tradeoff between greater complexity, with the advantages to accuracy and realism it confers, against greater precision with the potential for informative inference that it confers. A useful approach is to select parsimonious models that achieve an acceptable tradeoff between bias and precision (Otis et al., 1978; Burnham and Anderson, 1992, 1998; Lebreton et al., 1992). In this sense the "appropriate model" can be viewed
14.2. K-Sample Capture-Recapture Models as "the simplest model that fits the data" (Otis et al., 1978). If all of the above models and their estimators were based on likelihood theory, we could use likelihood ratio tests and optimization criteria such as Akaike's Information Criterion (AIC) and its relatives (e.g., see Anderson et al., 1994; Burnham and Anderson, 1998) as tools to aid in model selection. If the finite mixture models of Pledger (2000) prove to be as useful as we suspect, then it may soon be possible to use AIC in model selection for the full set of closed models. However, the models of Otis et al., (1978) that include heterogeneity of capture probabilities (models Mh, Mbh , Mth , and Mtbh) do not fit easily into the standard likelihood framework, and model selection strategies must rely on other approaches than maximum likelihood. Here, we follow the approach of Otis et al. (1978) and Rexstad and Burnham (1991) for model selection, based on model goodness-of-fit tests and between-model tests.
14.2.6.1. Goodness of Fit The multinomial distributions in capture-recapture modeling can be used as a basis for assessment of model goodness of fit (see Section 4.3). For example, assume that one wants to test the fit of model M t to data from a capture-recapture study and that maximum likelihood estimation yields the estimates/~ = 200, ]~1 -- 0 . 2 5 , ]92 = 0 . 4 0 , and ]93 -- 0 . 3 0 for a study with three sampling occasions. The expected number of animals exhibiting the capture histories can be estimated with these values [e.g., E(Xll 1) = /~]911921~3 -- 6; see Table 14.1]. The difference between the observed numbers of animals with each capture history and the numbers expected under model M t then provides information about the likelihood that the data were actually generated by this underlying model. Program CAPTURE (Rexstad and Burnham, 1991) computes goodness-offit tests for models Mb, Mt, Mh, and Mtb. The computation of these statistics is described for all models except Mtb by Otis et al. (1978).
14.2.6.2. Between-Model Tests When MLEs can be computed for two nested models (i.e., one model is a special case of a second, more general model), then a likelihood ratio test can be used for comparative testing (Section 4.3.4). The null hypothesis of such a test is represented by the more restrictive model, and the alternative hypothesis is the more general model. The test is conditional on the more general model fitting the data and essentially addresses the question of whether the more restrictive model is adequate to represent the data (see Section 4.3.4). Program CAPTURE computes tests to compare models
307
M 0 vs. M b and M 0 vs. M t based on MLEs, although they are not computed as standard likelihood ratio tests (Otis et al., 1978). Program CAPTURE also computes tests to compare models M 0 vs. M h and M h vs. Mbh , though they are not based on MLEs.
14.2.6.3. Use of Discriminant Analysis for Model Selection In the absence of an optimization criterion such as AIC, it seems reasonable to base model selection on an examination of the results of the described goodness-of-fit and between-model tests. Otis et al. (1978) developed such a model selection procedure, which is included in program CAPTURE. The procedure utilizes data that were simulated under all eight general models for closed populations, with various test statistics and associated probability levels computed for each simulated data set. Discriminant function analysis (e.g., Cooley and Lohnes, 1971) then was used to develop a model classification function based on the test statistics and probabilities. The procedure subjects actual data sets to the various tests of program CAPTURE, with test results used as input data for the classification function to compute a score that is treated as a model selection criterion (McDonald et al., 1981). In a simulation study assessing the performance of their model selection algorithm, Otis et al. (1978) found that the algorithm performs well when capture probabilities are high, but performance declines rapidly as capture probability declines. Menkens and Anderson (1988) also assessed the performance of the CAPTURE model selection algorithm via simulation and noted that when the population and sample sizes are small, the underlying model generating the data is selected relatively infrequently. They concluded that when sample sizes are not large, it may be wise to pool data from multiple periods into two periods and use the Lincoln-Petersen estimator to estimate population size (Menkens and Anderson, 1988). Stanley and Burnham (1998) investigated possible improvements to the model selection procedure of program CAPTURE. Although they followed the same general approach as in the CAPTURE procedure, their methods differed in some important details. For example, they used not only linear discriminant function analysis but also multinomial logistic regression to develop the classification function. They also used a different vector of predictor variables, specifically the probabilities corresponding to between-model and other tests, as well as coefficients of variation of some of the capture-recapture statistics. Finally, they based their classification function not on the ability to select the underlying generating model but instead on the
Chapter 14 Mark-Recapture Methods for Closed Populations
308
root mean squared error of the resulting estimators. The resulting classifiers performed marginally better than that of plogram CAPTURE. In addition to exploring model selection, Stanley and Burnham (1998) investigated a model-averaging approach to estimation (Buckland et al., 1997), in which they estimated population size as 1~I = ~ , W kl~ k,
k where/~k is the abundance estimate from model k, and w k is the predicted probability associated with model k based on the multinomial logistic regression classifier. The associated variance estimator is va"~(/~) = [ ~
WkV'v~r(l~lk[~k)+~12,
k
where
By incorporating estimators and probability significance levels for multiple models, this estimator incorporates model uncertainty. Stanley and Burnham (1998) recommended considering implementation of the above model-averaging procedure in CAPTURE. The mixture models of Pledger (2000) place all eight basic closed-population models and several variants in the likelihood framework. One of the most important advantages of the likelihood framework is the ability to use AIC as a model selection criterion. Use of AIC also permits model averaging and allows for the incorporation of model uncertainty in variance estimates (Buckland et al., 1997; Burnham and Anderson, 1998; Stanley and Burnham, 1998). We thus expect the Pledger (2000) model set to become widely used in closed population estimation. 14.2.6.4. D i a g n o s t i c S t a t i s t i c s f o r Capture-Recapture Models
As mentioned above, model overparameterization leads to declining precision in all model estimators, with extreme overparameterization leading to parameter estimates containing so little information that they are essentially useless. For this reason it is important to select a model that includes the fewest parameters necessary to fit the data (see Burnham and Anderson, 1992, 1998). Overfitting of a model should be evident on investigation of the goodness-of-fit and model comparison tests, along with other diagnostic statistics. An overfitted model typically has quite wide confidence intervals for the model parameters, corresponding to a lack of precision in parameter estimates. The goodness-of-fit
statistic for an overfitted model indicates a good fit between model and data, but typically one or more reduced models also indicate a good fit. This suggests that a reduced model is adequate to represent the data, i.e., that the full model includes more parameters than necessary. Finally, the test statistics comparing an overfitted and a reduced model typically indicate that a reduced model compares favorably to the overfitted alternative, again suggesting that the reduced model does about as well as the overfitted model in representing the data. Of course, underparameterization of a model also carries risks. A model that fails to account for key sources of parameter variation may result in very precise but very biased results. For example, if model M 0 with constant capture probability is incorrectly used when capture probabilities are in fact highly heterogeneous (i.e., model M h is the "true" model), population size is precisely estimated but the estimate can be severely biased downward. Again, such a situation should be evident in the standard diagnostic statistics. Goodness-of-fit statistics typically indicate a poor fit for the model, and model comparisons indicate that a more fully parameterized model compares favorably to the reduced model, i.e., the more fully parameterized model does a better job in representing the data. Beyond the issues of overfitting and underfitting, certain patterns in the data are useful as diagnostics of particular models. Thus, the expected number of captures for model M0 is the same for all sampling occasions [E(nj) = Np], so the actual number of captures should be similar and show no trends over capture periods. The number of captures of unmarked animals should decline through the study according to E(uj) = Np(1 - p ) J - 1, whereas the captures of marked animals should increase according to E(mj) = Np[1 (1 - p ) J - 1]. These patterns are illustrated in Table 14.2, which shows results of a simulation in which 120 animals were subjected to a capture probability of p = 0.30 for each of seven sampling periods. Note that
TABLE 14.2
Summary of Simulated Capture Histories under M o d e l M0 a
Population data Occasion (j)
Measure 1
2
3
4
5
6
7
Animals caught
32
40
35
42
23
41
31
Newly caught
32
30
17
12
8
8
6
0
10
18
30
15
33
25
36
39
24
12
2
0
0
(nj) (uj) Recaptures (mj) Frequencies (fj)
a For a population consisting of 120 individuals, with capture probability p = 0.3.
14.2. K-Sample Capture-Recapture Models TABLE 14.3
Summary of Simulated Capture Histories under Model Mh a
Population data
Measure
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies 0~)
1 38 38 0 34
2 31 20 11 21
3 32 11 20 23
4 27 7 25 10
5 31 5 22 5
6 32 6 25 2
7 34 8 24 0
the n u m b e r nj of captures s h o w s no a p p a r e n t trend t h r o u g h time, w h e r e a s the n u m b e r uj of first captures decreases a n d the n u m b e r mj of recaptures increases as the s t u d y proceeds. U n d e r m o d e l M h the n u m b e r of captures again should be relatively constant over time. Animals with higher capture probabilities tend to be captured early in the study, so that the captures of u n m a r k e d animals should decline m o r e rapidly than u n d e r m o d e l M 0. A high p r o p o r t i o n of captured animals exhibits capture frequencies that are very low (e.g., fl) or very high (e.g., fK-1, fK) relative to expectations u n d e r m o d e l M 0. These general patterns are illustrated in Table 14.3, w i t h a simulated p o p u l a t i o n consisting of 60 animals w i t h p = 0.15 and 60 animals with p = 0.40. U n d e r m o d e l Mb, the n u m b e r s of u n m a r k e d animals in samples should decline over the study, as u n d e r M 0. U n d e r a t r a p - h a p p y response, m a r k e d animals have higher capture probabilities than u n m a r k e d individuals. Thus, the total n u m b e r of captures should increase with time according to E(nj) = N p r - N(1 - pc) j-1 (Pr - Pc), as increasing n u m b e r s of m a r k e d animals are exposed to traps and are recaptured (Table 14.4). U n d e r a trap-shy response, m a r k e d animals have lower capture probabilities than u n m a r k e d animals, so that the
Summary of Simulated Capture Histories under Model Mb a
Population data
Summary of Simulated Capture Histories under Model Mb a
Population data
a For a population consisting of 120 individuals. Sixty individuals have capture probability p = 0.15 and 60 individuals have p = 0.40.
TABLE 14.4
TABLE 14.5
309
Measure
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies (~)
1 42 42 0
51
2 41 32 9 38
3 33 18 15 20
2 48 30 18 19
3 54 17 37 25
4 67 12 55 33
5 57 8 49 19
6 63 8 55 4
7 62 6 56 1
a For a population consisting of 120 individuals. Animals show trap-happy response with initial capture probability Pc = 0.30 and recapture probability Pr = 0.60.
6 29 5 24 0
7 22 1 21 0
a
total n u m b e r of captures should decline w i t h time as m a r k e d animals accumulate in the p o p u l a t i o n (Table 14.5). There typically are relatively more animals captured only once (fl) u n d e r a trap-shy response than u n d e r a t r a p - h a p p y response, given similar initial capture probabilities Pc. The p r i m a r y diagnostic for m o d e l M t is t e m p o r a l variation in the n u m b e r nj of animals caught per trapping occasion, reflecting t e m p o r a l variability in capture probabilities according to E(nj) = Npj. It often is possible to tell from a quick look at a data set w h e t h e r substantial t e m p o r a l variation is present, simply by examining the n u m b e r s caught. It s h o u l d be clear that different patterns of t e m p o r a l variation p r o d u c e different patterns in the capture histories and associated statistics. Table 14.6 s h o w s results of a simulation based on capture probabilities that increase until the m i d d l e s a m p l i n g occasion and then decline, w i t h the n u m b e r of captures reflecting this pattern. The patterns expected u n d e r the models with multiple sources of variation are more complicated and difficult to recognize. For m o d e l M b h , a t r a p - h a p p y response in the presence of heterogeneity still s h o u l d p r o d u c e an increase in the n u m b e r of captures (nj) t h r o u g h time, as animals in the p o p u l a t i o n become
TABLE 14.6
Measure
1 32 32 0 12
5 20 5 15 1
For a population consisting of 120 individuals. Animals show trap-shy response with initial capture probability Pc = 0.40 and recapture probability Pr = 0.20.
Summary of Simulated Capture Histories under Model Mt a
Population data
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies (fj)
4 37 15 22 8
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies (~)
Measure
1 22
22 0 35
2 37 29 8 41
3 35 18 17 19
4 54 20 34 10
5 23 8 15 3
6 37 8 29 0
7 21 3 18 0
a F o r a population consisting of 120 individuals with capture probabilities Pt = 0.20, 0.25, 0.30, 0.35, 0.30, 0.25, and 0.20.
Chapter 14 Mark-Recapture Methods for Closed Populations
310 TABLE 14.7
Summary of Simulated Capture Histories under Model Mbh a
Population data
Measure
Occasion (j)
1
2
3
4
5
6
7
Animals caught
28
34
39
47
39
41
40
Newly caught
28
20
17
16
9
5
2
0
14
22
31
30
36
38
28
19
17
19
10
3
1
Recaptures
(nj) (uj) (mj)
Frequencies (fj)
a For a population consisting of 120 individuals showing a traphappy response. Sixty individuals have initial and recapture probabilities of 0.15 and 0.20, respectively, and 60 individuals have initial and recapture probabilities of 0.40 and 0.53.
marked and thus have increased capture probabilities. Patterns in the data are more difficult to predict under a trap-shy response, but numbers of captures should decrease through time, or at least not increase. Tables 14.7 and 14.8 show results of simulations under model M b h with trap-happy and trap-shy response, respectively. Temporal variation in capture probabilities can be a dominant feature producing patterns in capture history data. Any of the multiple-factor models containing time as one of the sources of variation in capture probability (Mth , Mtb , and Mtb h) c a n produce temporal variation in the number nj of animals caught each sampling occasion. However, general patterns are difficult to predict, because they depend on the magnitude and specific pattern of temporal variation. 14.2.7. E s t i m a t o r R o b u s t n e s s
Even with testing, model diagnostics, and model selection algorithms, selection of an appropriate model is not guaranteed. It thus is important to determine
TABLE 14.8
Summary of Simulated Capture Histories under Model Mbh a
Population data Occasion (j)
Measure 1
2
3
4
5
6
7
(nj) Newly caught (uj) Recaptures (mj)
33
31
35
24
34
30
25
33
24
17
10
6
6
0
0
7
18
14
28
24
25
Frequencies (~)
38
41
22
4
2
0
0
Animals caught
a For a population consisting of 120 individuals showing trap-shy response. Sixty individuals have initial and recapture probabilities of 0.20 and 0.15, respectively, and 60 individuals have initial and recapture probabilities of 0.40 and 0.30.
how well a selected estimator performs when the underlying model on which it is based is not appropriate for the data. Estimator robustness has been investigated primarily by computer simulation, whereby data are generated under a particular model with specified parameters, and the estimates from different capture-recapture models are compared against the known parameter values. In a few instances it has been possible to examine estimator performance with field data for a population of known size (e.g., Greenwood et al., 1985; Manning et al., 1995). The results of these investigations indicate that the MLE for model M 0 is generally not robust to variation in capture probability. Heterogeneity of capture probability among individuals produces negative bias, as does a trap-happy behavioral response, whereas a trap-shy response yields positive bias (Otis et al., 1978). The estimator based on M 0 is somewhat robust to temporal variation in capture probability (Otis et al., 1978). Performance of the estimator for model M t is similar in many respects to that of model M 0. Heterogeneity and a trap-happy response produce negative bias in estimates of population size, whereas a trap-shy response produces positive bias (Otis et al., 1978). The magnitude of bias depends on the degree of heterogeneity a n d / o r the magnitude of the behavioral response. With model M b, valid estimates can be obtained only when K
~ (K + 1 - 2 j ) ( n j - mj) > 0 j=l
(Seber and Whale, 1970). This condition essentially reflects a requirement for "depletion" of the unmarked population through the marking of previously unmarked animals. Temporal variation in capture probabilities can produce pattern in the captures of unmarked animals that is unrelated to change in the number of unmarked animals available for capture, resulting in large biases in estimates of population size (Otis et al., 1978). Heterogeneity of capture probability results in a negative bias in abundance estimates, with the magnitude of the bias strongly dependent on the number of individuals with low capture probabilities (e.g., p < 0.10). As mentioned above, several different estimators have been proposed for model M h. The jackknife estimator of Burnham and Overton (1978, 1979) has been the most frequently used, and it consistently performs well with respect to model robustness. However, simulation studies by Burnham and Overton (1979) and Otis et al. (1978) indicate that the jackknife estimator can exhibit negative bias when some members of the
14.2. K-SampleCapture-Recapture Models population are essentially untrappable. Simulation results indicate relative robustness of the jackknife estimator to temporal variation and to behavioral response under some scenarios (e.g., Otis et al., 1978) although not all (e.g., Chao, 1989). Simulation studies based on small populations with low (p ~ 0.10) and heterogeneous capture probabilities led Rosenberg et al. (1995) to favor first-order and second-order jackknife estimators, but to question the use of higher order jackknife estimators even in cases where they are selected by the algorithm in program CAPTURE. The jackknife estimator performed the best among all estimators tested by Greenwood et al. (1985) on known populations of striped skunks (Mephitis mephitis). The jackknife estimator and the moment-based estimator of Chao (1989) outperformed other estimators for graytailed vole (Microtus canicaudus) population sizes of 60 and 90 animals (Manning et al., 1995). In contrast to the jackknife estimator, the momentbased estimator of Chao (1987, 1988, 1989) has performed well in simulations of heterogeneous populations with sparse data, the situation for which it was developed. The coverage estimators of Chao et al. (1992) and Lee and Chao (1994) also have been found to perform well, especially when sample coverage is relatively high (e.g., >50%). Finally, the nonparametric MLE of Norris and Pollock (1996) did not perform as well in simulation studies as some of the other estimators (including jackknife and coverage). Because the generalized removal estimator for model Mbh requires a substantial drop in numbers of new animals captured over the course of the study (Otis et al. 1978), certain patterns of temporal variation can cause the estimator to perform poorly. In simulations to evaluate the various estimators proposed for model Mbh, the jackknife estimator of Pollock and Otto (1983) performed fairly well [also see simulation results of Lee and Chao (1994) and Norris and Pollock (1996)], as did the coverage estimator of Lee and Chao [1994; also see Norris and Pollock (1996) simulation results] and the MLE of Norris and Pollock (1996). In a simulation study of their coverage estimator for model Mth , Chao et al. (1992) and Lee and Chao (1994) found that the estimator for model M t performed well when heterogeneity was relatively small (coefficient of variation of capture probability distribution ~ 0.4), but the estimator for model Mth performed better in the presence of substantial heterogeneity (Chao et al. 1992; Lee and Chao, 1994). The only simulation work of which we are aware on estimators for models Mtb and Mtbh involves special cases of these models where the pattern of temporal variation in capture probability is known (Lee and Chao, 1994).
311
14.2.8. Study Design The design of studies to estimate population size using K-sample capture-recapture should involve two general considerations. First, the study should be designed in such a way as to minimize violation of underlying model assumptions to the degree possible. Second, study design should focus on producing precise estimates. A key assumption of the capture-recapture approach in this chapter is that populations are closed to gains and losses over the course of a study. It thus is important to design studies with short duration, because shorter studies reduce the possibility of death, recruitment, and movement in and out of the population. Closed models frequently are used with daily sampling (e.g., small mammal trapping; mist-netting of songbirds) over 5- to 10-day study periods. Study timing also is relevant, because it is useful to avoid sampling during migration and during periods of recruitment or high mortality. Trap mortality is a violation of the closure assumption and should be reduced to the extent possible. When trap mortality does occur, there are at least two ways to deal with it (see Flickinger and Nichols, 1990). In studies with relatively small numbers of trap deaths, the capture histories of animals that die prior to the final day of capture can be removed from the data set, with estimation based on the reduced data set. Trap deaths can be added to the resulting population estimate in order to estimate the pretrapping population size. The variance of the adjusted estimate is the same as that for the estimate obtained from the capture-recapture model, because the number of trap deaths is known and does not add additional variance to the estimate. If trap deaths substantially reduce the number of recaptures, it may be necessary to use one of the removal models M b or Mbh. AS indicated above, the estimators for these models are based on initial captures only, so that trap deaths do not reduce the data used for estimation of pretrapping population size. However, elimination of trap mortality is preferable to the use of removal models, which restrict the analysis in not allowing for temporal variation in capture probabilities. In addition, the restriction with removal models to initial captures clearly reduces the data available for estimation, because recapture data are not used to estimate N, resulting in reduced precision. The assumption of no tag loss typically is not a problem with closed population estimators because of the short duration of such studies. It should be noted that all the models described above were developed for use with individual marks. The use of a single
312
Chapter 14 Mark-Recapture Methods for Closed Populations
"batch mark" yields data that cannot be analyzed fully, though the relevant statistics for some of the models described above can be obtained from batch-marking studies. However, batch marking does not permit adequate testing of model assumptions and precludes the use of many of the models. On the other hand, occasion-specific batch marks sometimes are applied so that at each capture, the previous capture history can be ascertained (e.g., see White et al., 1982). Finally, it should be noted that animals of some species are individually recognizable, so that observations and reobservations can be used with closed-population capture-recapture models in the absence of physical captures [see example with camera-trapping of Indian tigers, Panthera tigris (Karanth, 1995; Karanth and Nichols, 1998)]. High capture probabilities are nearly always desirable, regardless of whether one focuses on model selection, estimator precision, or bias reduction. However, the addition of trapping occasions to increase capture probability reflects a tradeoff between competing objectives. Large numbers of trapping occasions usually increase estimator precision and increase the performance of the model selection algorithm. On the other hand, multiple trapping occasions over an extended time also increase the probability that the closure assumption will be violated and increase the probability that time will be an important source of variation in capture probabilities. Design considerations for estimator precision include those that influence both sample size and the sources of variation in capture probabilities. Because estimator precision is a function of the number of model parameters, one should eliminate nuisance parameters associated with capture probability to the extent possible. Time variation is likely to be most easily influenced by the investigator, via standardization of trapping procedures. For example, the investigator should expend the same effort at each sampling occasion and use the same bait, number of traps, and daily trapping schedule throughout a trapping study. Of course, some factors cannot be dealt with adequately via standardization. For example, weather has the potential to influence capture probabilities. In situations where a study includes a single day of anomalous weather (e.g., hard rain; very cold temperatures reducing animal activity) among "normal" weather days, it sometimes is worthwhile to extend the study an extra day. The investigator then can examine both the full data set and the reduced data set (omitting the day of anomalous weather). It may be that the cost in estimator precision of using a model with time variation for the full data set may exceed the cost of discarding the data associated with the single day of bad weather. If
a likelihood-based model (e.g., M t, M b) appears to be appropriate for the data, then it will be possible to build models that are tailored to specific data sets. For example, it would be possible to develop a special version of M t in which capture probabilities were constant for days 1, 2, 3, and 5 of trapping, but different for day 4. Sometimes nontarget animal species can disturb traps and produce temporal variation in capture probabilities. For example, in small mammal studies, traps can be disturbed and "tripped" by both predators (raccoons, Procyon lotor) and large herbivores (white-tailed deer, Odocoileus virginianus). As is the case with anomalous weather patterns, it may be reasonable to either discard data from days on which substantial disturbance occurs (e.g., see Nichols et al., 1984a) or develop special models that have separate parameters for anomalous days. Behavioral response can be a troublesome source of variation in many sampling situations. The use of bait, an important component of many trapping studies, can induce a trap-happy response, yet in most trapping studies it would be foolish to recommend that bait not be used in order to eliminate the response. Prebaiting (placing baits at traps or trap stations several days before the traps are actually set) is useful not only for increasing capture probabilities, but also for reducing trap-happy responses. On the other hand, one can reduce problems with trap-shy responses by minimizing handling time. Trap deaths can be viewed as an extreme trap response, which can be reduced by minimizing the time animals spend in traps, reducing trap temperatures through the use of trap covers, leaving traps open during hot periods of the day, and other common-sense precautions. Some degree of heterogeneity of capture probabilities among individuals is likely to characterize all populations. In some cases, variation in capture probabilities is associated with visible characteristics of captured animals (e.g., age, sex, weight). Heterogeneity of this kind can be accommodated at the analysis stage, either by stratification or by covariate modeling of capture probabilities (Pollock et al. 1984; Huggins, 1989, 1991; Alho, 1990). The source of heterogeneity most likely to be associated with study design involves spatial distribution of capture devices a n d / o r animals. Investigators should try to avoid a situation in which some animals in the sampled area have very small probabilities of appearing in the captured sample, whereas other animals have high probabilities of appearance. When possible, one should include multiple traps per animal home range [e.g., Otis et al. (1978) recommend four traps per home range], though this may not be possible because of inadequate numbers of traps relative to the size of the sampled area. In
14.2. K-Sample Capture-Recapture Models these situations, one can divide the sampled area into quadrats smaller than the average home range size (e.g., four quadrats per home range) and then randomly select quadrats for trap placement at each sampling occasion. Capture probabilities for such a design are likely to be lower than if traps were in all quadrats during all sampling occasions, but because each quadrat has an equal probability of receiving a trap at each occasion, heterogeneity associated with trap placement is reduced. Along with the distribution and density of traps and animals, one also should consider the spatial configuration of groups of traps. In general, placement of multiple traps per home range can be achieved in the interior of a trapping grid, but not on the periphery of the grid. Animals with home ranges overlapping the outer row of grid traps use unsampled areas and therefore tend to have lower capture probabilities than animals with ranges entirely within the grid interior. Heterogeneity associated with grid edges is unavoidable unless the trapping is conducted on a habitat island or other discrete area that can be sampled completely by traps. To alleviate this problem, trap configurations should be used that minimize the ratio of the periphery to area covered by the traps. Thus, a linear transect of traps represents the worst possible configuration with respect to edge problems, and circular arrangement of traps represents the best configuration. For a fixed population size and known sources of variation in capture probability, the magnitudes of the capture probabilities are the primary determinants of estimator precision. Otis et al. (1978) and White et al. (1982) present some computations involving assumed densities and capture probabilities that are useful in determining grid size. Because closed population studies require a relatively small time commitment (e.g., several days; only 2 days with a Lincoln-Petersen study), pilot studies offer an inexpensive way to obtain some idea of the abundance/density and the capture probabilities to be expected. Simulations based on these preliminary estimates then can be used to investigate estimator precision under various designs (grid sizes, numbers of sampling occasions, etc.). Five trapping occasions can be viewed as a minim u m number to estimate population size, and 7-10 often is better (Otis et al., 1978). For grid trapping, trap stations in 10 • 10 grids probably represent a minimum, with larger grids preferable. The necessary capture probability for precise estimation depends on the actual size of the target population. Otis et al. (1978) suggested that a population of size 50 might require an average capture probability as high as 0.40 or 0.50 to produce useful estimates and tests, whereas a popu-
313
lation size of 200 or so might require only an average capture probability of about 0.20. 14.2.9. E x a m p l e
Nichols et al. (1984) trapped meadow voles, Microtus pennsylvanicus, in old field habitat at Patuxent Wildlife Research Center for five consecutive days, 29 August-2 September, 1981. The trapping grid was a 10 • 10 square of trapping stations with 7.6-m trap spacing. A single Fitch trap (Rose, 1973) baited with whole corn and containing hay was placed at each station. The trapping schedule consisted of setting traps for one evening, checking them for animals and closing them the following morning, setting them again in the late afternoon, checking them the following morning, etc. Newly captured animals were marked with individually coded monel fingerling tags placed in their ears. If tags of previously marked animals showed signs of pulling out, a new tag was applied on the opposite ear a n d / o r toes were clipped. Animals were sexed and weighed on each occasion and external reproductive characteristics were recorded. Adults were defined as voles >22 g. The capture-recapture data for adult females (Table 14.9) were analyzed with program CAPTURE (Rexstad and Burnham, 1991). The closure test of program CAPTURE yielded a test statistic of z = 0.43, P = 0.33, thus providing no evidence that the closure assumption was inappropriate. The relatively constant numbers of captures over the 5 days provided little reason to suspect temporal variation in capture probabilities. The discriminant function model selection criteria highlighted M h as a reasonable model for these data (Table 14.10). The test of M 0 versus M h provided strong evidence of heterogeneous capture probabilities, with X2 = 10.0, P < 0.01. The goodness-of-fit test for model M h was X2 = 1.61, P = 0.84, suggesting that the model provides an adequate description of the data. The jackknife estimate (Burnham and Overton, 1978,
TABLE 14.9
Summary of Capture Histories for Adult Female Meadow Voles a
Population data
Occasion (j) Animals caught (nj) Newly caught (uj) Recaptures (mj) Frequencies (fj)
Measure
1 27 27 0
18
2 23 8 15 15
3 26 9 17 8
4 22 4 18 6
5 23 4 19 5
aCaptured at Patuxent Wildlife Research Center, 29 August-2 September, 1981.
314
Chapter 14 Mark-Recapture Methods for Closed Populations
TABLE 14.10 Closed Model Selection Criteria of Program CAPTURE for the Meadow Vole Data of Table 14.9 Model
M0
Mh
Mb
Mbh
Mt
Mth
Mtb
Mtb h
Criteria a
0.65
1.00
0.23
0.37
0.00
0.29
0.24
0.44
a Model selection criteria are based on the linear discriminant function described by Otis et al. (1978) and Rexstad and Burnham (1991).
1979) of abundance for these data was/~/= 65, SE(/Q) = 5.70, with an approximate confidence interval of (58, 81). The estimated average capture probability was = 0.37. The data in Table 14.9 are not sparse, so we prefer the jackknife estimator to the M h estimator of Chao (1988), although in this case the Chao estimate (N = 63) was very close to the jackknife.
14.3. D E N S I T Y E S T I M A T I O N WITH CAPTURE-RECAPTURE Density is defined as the number of animals per unit area (D = N / A , where D denotes density, N is abundance, and A is area). In attempting to estimate D with trapping data, one typically encounters the problem of not recognizing the area actually used by animals that are subject to trapping. Consider, for example, a small mammal trapping grid that is located in a large area of old field habitat. The use of the area enclosed by the outermost traps of the grid as an estimate of A likely results in an overestimate of density, because the grid traps sample animals whose ranges lie partially outside the grid. This is termed "edge effect," and the estimated abundance/Q actually applies to a larger area of unknown size. Edge effect is more pronounced when home range size is large relative to grid size (White et al., 1982). Recognition of the potential problem in estimating the "effective area" sampled by a trapping grid led Dice (1938, 1941) to recommend the expansion of the sampled area by a boundary strip equal to half the average width of an animal's home range. Called the "extra-grid-effect area line method" by Tanaka (1980), this approach provides a conceptual basis for density estimation, though the problem then becomes one of estimating the width of the boundary strip surrounding the grid. The general approach of using a boundary strip is not restricted to grid trapping, but applies to any situation where sampled animals may come from areas larger than the area in which the sampling actually occurs (e.g., see Karanth and Nichols, 1998). An alternative approach involves direct esti-
mation of density (rather than separate estimation of population and the effective area sampled) using distance sampling or other alternatives (Anderson et al., 1983; Link and Barker, 1994). 14.3.1. U n i f o r m S a m p l i n g Effort (Grid Trapping)
At least three general approaches (Otis et al., 1978) are available to estimate the width W of a boundary strip surrounding a sampled area. One approach uses data on capture locations of recaptured animals to estimate home range size (e.g., Mohr, 1947; Hayne, 1949a; Stickel, 1954; Calhoun and Casby, 1958; Jennrich and Turner, 1969; Van Winkle, 1975; Ford and Krumme, 1979; Dixon and Chapman, 1980; Tanaka, 1980; Anderson, 1982). Half of the average width or radius of the home range estimate is then used to estimate the boundary strip width W. A second approach is to estimate W directly using data from selected subsets of the sampled area (e.g., subgrids), as described by MacLulich (1951), Hansson (1969), Seber (1982), and Smith et al. (1975). Based on this idea, Otis et al. (1978) developed their "nested grid" approach for joint estimation of density D and boundary strip width W. A third approach is based on "assessment lines" designed specifically to estimate both the effective area sampled and the corresponding population size (Smith et al., 1971, 1975; Swift and Steinhorst, 1976; O'Farrell et al., 1977). In what follows we focus on the first two of these approaches. 14.3.1.1. M o v e m e n t D i s t a n c e s B a s e d on T r a p p i n g D a t a
Distances between captures of individual animals have been long used to index the extent of home range (see Stickel, 1954; Brant, 1962). Wilson and Anderson (1985c) investigated a potentially useful approach to estimation of boundary strip width, based on the maximum distance d i between capture locations for each individual i that is captured at least twice. The mean --
1
d=
di
m
of these distances is computed across all m individuals (or all individuals in the age-sex class of interest) caught at least twice, with associated variance m
va'~r(d) --- ~ i = l ( d i _ ~)2
m ( m - 1)
"
Following the suggestion of Dice (1938) that W should be computed as half the average home range width, Wilson and Anderson (1985c) added a boundary strip of width 1~ = d/2 to the perimeter of their simulated
14.3. Density Estimation with Capture-Recapture trapping grids to estimate the effective area sampled. For square trapping grids with sides of length L (see Fig. 14.1), the effective area ~i(l/~ is A(I~V)
proaches to estimation of maximum distances. For example, Jett and Nichols (1987) used an estimator recommended by K. Burnham (personal communication),
-- n 2 if- 4 L I N + -rrl/V2
E(-dj)
with variance v~[A(l/~] = (4L + 2"rrl/~2 var(l/~
(14.18)
(Wilson and Anderson, 1985c), with sampling variance /Q2var[A(l~V)]
va"}(D) =
[A(I/~]4
=
[_1 -
e-(J-SJb]d *,
var(/~r)
-}- [A(I/~]~
given by a delta method approximation (Seber, 1982). A concern with this approach is that the maximum distance moved for an individual animal increases (at least initially) with the number of captures (e.g., Brant, 1962). This has led some to suggest alternative ap-
f
;(
X
)(
X
)(
X
X
;(
X
X
x ....
x
.... x
...... x
;'
x
x
x
x
x
;:
X
X
X
X
lc
x
x
I
'
X
X-
',~
X
X
X
X
)[
X
X
X
X
X
X
X
" X ..... X
...... X
X
!I~X
X
X
X
X ..... X ...... X . . . . . X
;'
X
;'~
x
x
:'
x
X
X
"
X
X
X
)C
X
X
)(
X . . . . X . . . . . X...... X
x
)c
"X
X
....
X ....... X
9 X
X
-
X
X
t ~W 2
(14.19)
where dj is the mean maximum distance moved by animals caught exactly j times, d* is the expected maxim u m movement for animals observed a large number of times, and b is a model parameter. Weighted nonlinear least squares can be used to estimate d*, which in turn can be used in place of d in the computations of effective area. Although density estimation frequently is based on data from regular trapping grids, the boundary strip approach applies generally to discrete areas of sampled habitat that are located in the midst of a larger expanse of similar habitat. In particular, the approach is appropriate for irregularly shaped sample areas (Karanth and Nichols, 1998), with estimation differing from that outlined above only in the computation of A(W) and its variance. In using observed movement distances to estimate
based on a delta method approximation (Seber, 1982). Utilizing one of the closed population estimators in Section 14.2 for population size, an estimator of density then is
D = 1cq/A(17V)
315
X
"
X X
X
X
..... X
X
)(
.. t ~
/
LxW
F I G U R E 14.1 Square trapping grid with boundary width W indicating "effective t r a p area." T h e c o m e r s of the effective area are q u a r t e r - c i r c l e s of r a d i u s W.
316
Chapter 14 Mark-Recapture Methods for Closed Populations
boundary strip and density, it is important to try to meet the assumptions underlying the estimation of N in Eq. (14.18), as well as the additional assumptions required for the estimation of A(I~V). One assumption is that the trapping grid does not induce immigration into the study area. The estimator in Eq. (14.18) is biased by movement into the sampled area that is induced by the sampling devices (e.g., baited traps). Efforts to minimize such immigration might include use of capture devices without bait. Another assumption is that one-half the mean of the maximum distances moved is a reasonable estimate of W for the purpose of estimating effective sampling area. We know of no strong theoretical justification for use of this ad hoc estimator and can only note that it seems to have performed reasonably well both in simulations (Wilson and Anderson, 1985c) and in comparisons with estimates obtained using the nested grid approach described below (Jett and Nichols, 1987). Regarding study design, it seems clear that use of observed movement distances are likely to be most useful in situations in which animal ranges are small relative to sampling grid area (also see White et al., 1982). In addition, movement distances are more effectively estimated when most animals are captured multiple times (see Brant, 1962; Tanaka, 1980; Wilson and Anderson, 1985c). For the purpose of estimating movement distance, it thus would be desirable to use at least 10 trapping occasions, but this must be balanced against the need for population closure, which requires shortduration trapping studies (e.g., five occasions). In cases when only a relatively small number of trapping occasions can be used, it may be wise to use a ^quantity such as d* from Eq. (14.19) for estimation of W, rather than d. Radiotelemetry also can provide information about movements for computing W and /~ with the above approach. 14.3.1.2. Nested Grid Approach
Direct estimation of W based on subgrids has been discussed by MacLulich (1951), Hansson (1969), Smith et al. (1975), and Seber (1982). The nested grid method proposed by Otis et al. (1978) and White et al. (1982) and implemented in program CAPTURE is the most widely used of these methods. It utilizes the fact that a large trapping grid can be subdivided to delineate smaller subgrids nested within the original grid. A 10 • 10 grid of trapping stations, for example, can be viewed as containing subgrids of dimension 8 • 8, 6 x 6, and 4 • 4 (Fig. 14.2). Denote the different subgrids by i (i = 1, ..., k), with subgrid i = 1 representing the smallest subgrid and subgrid i = k the largest. Capture-recapture models
in A
II A
ii A
ii A
ii A
~i, A
1. A
A
,~
^
^
^
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^
^
Z
~/
li
l#
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,~
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v
x
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x
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.. A
i i
I .
A
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,, A
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F I G U R E 14.2 Nested trapping grids: (a) 4 x 4 grid; (b) 6 x 6 grid; (c) 8 x 8 grid; (d) 10 x 10 grid. After Otis et al. (1978).
such as those described in Sections 14.1 and 14.2 can be used to estimate abundance for each subgrid, where /~i is the abundance estimate obtained using data only from subgrid i. If A i denotes the area of subgrid i (the area covered by the traps), then a naive density estimate for each subgrid is b i -
Ni/Ai
(14.20)
with associated standard error S-E(/~i)- S~E(1Cqi)/ Ai for i = 1.... , k. Animals with ranges lying partially outside each subgrid are included in the estimated subgrid abundance, leading to positive bias in the density estimates. The idea underlying estimation with the nested subgrid approach is that biases in the naive density estimates [Eq. (14.20)] should be ordered from largest for the innermost subgrid (i = 1) to smallest for the entire grid (i = k). To see how, assume constant population density over the entire grid. For this sampling situation it is reasonable to consider a boundary strip of width W to be added to each subgrid to compute the effective area sampled. If W is known and Pi denotes the perimeter of subgrid i, then we can write the effective area sampled as Ai(W) = A i + P i W / c + -n-W2/c,
14.3. Density Estimation with Capture-Recapture where c is a conversion factor to express Pi W or W2 in the same units as A i (e.g., Fig. 14.2). Given the assumption of constant animal density D over the entire grid, and hence over the subgrids, the expected number of animals at risk of capture on each subgrid is E(Ni) = [ A i ( W ) ] D = [A i + P i W / c + ,rrW2/c]D,
so that the naive densities associated with each subgrid are
F)~ = N i l & = D[1 + a i W + bi W2] for i = 1, ..., k, where a i = P i / A i c and b i = ~r/Aic. Substituting an estimate of N from Section 14.1 or 14.2, we then can write E)i = Fqi/Ai
(14.21)
317
Tests for closure have been described in Section 14.2. One approach to testing for induced immigration and for density gradients involves contingency tests for uniform density by rows, columns, and rings of the trapping grid (Otis et al., 1978). These tests are based on the total captures at each grid point and are computed by program CAPTURE when grid location data are included in the input data. Induced immigration often is accompanied by increased numbers of captures in the outermost ring(s) of traps. Tests for uniform density by rows and columns provide evidence of density gradients. The nested grid approach carries substantial data requirements. Although Wilson and Anderson (1985a) concluded that the approach was theoretically sound, their simulation results indicated that it can be unreliable unless sample sizes are large. The field comparison of Jett and Nichols (1987) involved good sample sizes, and the approach appeared to perform well.
= D[1 + a i W + bi W2] + 8i, i = 1, ..., k, where 8i is a random error term with expectation E(~) = 0 and covariance matrix E(88') = ~. Because the subgrids are nested, any pair of density estimates /~i and Dj exhibits nonzero correlation p(/~i,/~j), which Otis et al. (1978) assume to be equal to the proportion of overlapping area between the two subgrids (including the boundary strips). Generalized nonlinear least squares can be used with the estimates /~/ and the covariance matrix with elements r = SE(/~i)S"E(/~j) P(/~i,/~j) to estimate directly the density and strip width in Eq. (14.21). As with the previous approach to density estimation, the nested grid approach requires population closure, which limits the study duration to, e.g., perhaps 5-10 days with small mammals. In addition, one should guard against inducing immigration during the study period. Removal trapping is known to create ecological vacuums and induce movement into trapped areas, and is thus not recommended for density estimation with the nested grid approach. The use of nested grids also assumes that population density is uniform in the sampled area (Otis et al., 1978), i.e., there is no density gradient over the trapping grid. When true densities are similar over the grid, differences among naive density estimates reflect only the differences in bias associated with a biased measure of the sampled area. One thus should select areas of homogeneous habitat for trapping. This general approach need not be restricted to a single grid, and Dooley and Bowers (1998) used multiple grids of different sizes within the same landscape. Uniform density and homogeneous habitat are especially important when multiple grids are used with this approach.
14.3.2. Gradient Designs (Trapping Webs) Distance sampling methods such as line transect and point sampling (see Chapter 13) were developed to estimate density in the presence of spatial variation in detection probability. However, Anderson et al. (1983) developed a distance sampling approach using capture-recapture data. Their idea was to distribute traps (or other sampling devices) so as to induce a spatial gradient in sampling effort and detection probability, which then can be exploited to estimate density.
14.3.2.1. Trapping Web and Distance Sampling The trapping web of Anderson et al. (1983; also see Buckland et al., 1993) consists of rings of increasing radius from the web center. Traps are placed at equal distances along the m lines of equal length, radiating from a randomly chosen central point (Fig. 14.3). Each line contains T traps, usually (though not necessarily) located at a fixed distance interval 0, starting at distance oL1 = 0/2 from the web center. The distance from the web center for any trap i is given by OLi - - 0 ( i - - 0 . 5 ) , i = 1.... , T, with points b i along each line midway between consecutive traps. Thus, point b0 is the web center, and point b T represents the boundary of the web beyond the last trap. All captures in ring i of the web occur at distance o~i from the web center and are treated as grouped data from the distance interval (bi_l, bi). The total area of the web out to interval i is given by c i = 'rr(bi) 2, and the area associated with the ring i of traps is A i - - Ci -Ci_ 1. This design yields a gradient in capture probability corresponding to the gradient in trap density,
318
Chapter 14 Mark-Recapture Methods for Closed Populations
F I G U R E 14.3 Schematic diagram of a trapping web with 16 lines, each of total length A T with T = 20 traps per line (after Anderson et al., 1983). Traps are equally spaced along each line. Points equidistant between traps are denoted by bi, with b0 representing the center of the web and b T located just beyond the last trap. Captures in the eleventh ring of traps are assigned to the annulus All, which has area -rr(b121 - b120). After Anderson et al. (1983).
with probability the highest in the first ring at the web center and lowest in the outer ring T. The typical field procedure for use of a trapping web involves setting out the traps in the web design, prebaiting and leaving the traps open for several days (this step is unnecessary for unbaited capture devices such as pitfall traps) and then setting and running the traps for several consecutive days. The trapping web typically utilizes only initial captures, so captured animals can be removed or marked with either batch or individual marks. The data resulting from a single web are the number of initial captures uij occurring in traps from ring i of the web on day j of trapping, i = 1, ..., T and j = 1, ..., K. These data are pooled over the days of trapping to yield the total number K Ui = ~ ldq j=l
of first captures in each ring of traps throughout the study, as well as the total number
of individuals caught in the study. The values u i are used in conjunction with standard point transect estimation methods (Buckland et al., 1993) to determine f' (0), the slope of the estimated density of capture distances evaluated at zero (see Section 13.3.2). Estimation of f' (0) from the capture data u i is carried out by program DISTANCE (Buckland et al., 1993). If the population is distributed randomly, then Wilson and Anderson (1985b) recommend using [cv(u)] 2 = 1 / u , whereas under situations with spatial aggregation, [cv(u)] 2 = 2 / u or 3 / u may be more appropriate. The necessary assumptions for analysis of trapping web data using distance sampling methods are (after Buckland et al., 1993): 1. All animals at the center of the web are captured during the study. 2. Distances moved by animals during the study are small relative to the size of the web, and migration through the web does not occur. 3. Distances from the web center to each trap are measured accurately.
T
U--~U i=1
i
In addition to these assumptions, the proportion of captures in a given ring is assumed to be the same as
14.3. Density Estimation with Capture-Recapture the proportion of captured animals whose locations were closest to that ring (the "closest trap assumption") (Link and Barker, 1994). Assumption (1) is analogous to the assumption in point counts that all animals located on the sampling point are detected. When it is not true, density estimates will be negatively biased. The number of new captures near the web center provides information about this assumption. If one captures no new animals in the innermost ring(s) of traps for 2 or 3 days in a row, it is reasonable to assume that most or all animals near the center have been caught. However, if the study lasts too long, then the possibility arises that animals initially located away from the web center move to the center and are trapped there. This possibility relates to assumption (2) that movements are relatively small. Thus, directional movement toward the web center (or any particular location) can produce biased estimates. Designing the trapping web relative to animal movements is important in determining whether the trapping data mimic the assumptions of point transect sampling (Buckland et al., 1993). The trapping web is likely to perform well when home ranges are small relative to web trap spacing; however, there are few guidelines for the desired relationship between trap spacing and animal home range size.
14.3.2.2. Trapping Web and Geometric Analysis Link and Barker (1994) considered a different approach to density estimation with a trapping web by focusing on the geometry of the web as a determinant of the degree of competition among traps They noted that the "closest trap assumption" implies that the number of captures at a particular trap in the web should be determined by the size of the region closest to that trap (i.e., an animal within this region would be closer to the trap in question than to any other trap on the web). They refer to this region as the "maximum locus" of the trap. Link and Barker (1994) also define a trap's "locus of radius y" as the collection of points within distance y from the trap that are closer to the given trap than to any other. This locus of radius y is the intersection of a circle of radius y and the maximum locus of the trap. The shape and area of this intersection are determined by web geometry and, for a given web, by the ring in which the trap is found. Link and Barker (1994) enumerated 17 different forms for the locus of radius y and computed the area associated with each form. The area associated with the locus of radius y is denoted by Ai(y), where i denotes the trap ring. Link and Barker (1994) then focused on the location of each individual trap, rather than on the web center as in the distance sampling approach of Anderson et
319
al. (1983). Let trap location be denoted as t 0, and the distance from a randomly selected animal to t o be denoted by X, with Y = ~rX2. Finally, let
g(y) = Pr {captured at
to l Y = y,
no competition between traps}. They modeled g(y) as a step function, taking the value 1, (k - 1)/k, (k - 2)/k, ..., 2/k, 1/k, 0 for distance intervals (measured from the trap) of [P0 = 0, Pl), [Pl, P2), "', [Pk-1, Dk), [[3k, Pk + 1 "- OO). The number of steps used to approximate g(y) is thus specified by k. As with the distance sampling approach, the data used for estimation are the numbers of animals caught for the first time in each ring, ul, u2, ..., UT. Link and Barker (1994) modeled these data as a multinomial random variable conditional on the total captures u with associated cell probabilities defined by k
Zj---1 Ai(f3j) o
~k
Ai(Pj)
Estimation of the parameters pj can be accomplished using maximum likelihood. The expected values for number of animals caught at a given trap [E(uis), where i denotes a trapping ring and s denotes a trapping radius], ring of traps [E(ui)], and the entire web [E(u)l are given by
D k E(uis) = -ff Z Ai(Pj)' 1=1
E(ui )
=
m D k k ~ Ai(PJ)' j=l
and
E(u) - m D T
k
k Z Z Ai(Dj)" i=1 j = l
where m is the number of spokes of the web and D is the (unknown) density of animals. Based on the above expectations, D is estimated as (Link and Barker, 1994)
D=
ku Z T Z k Ai(~)j)" m i=1 j--1
Link and Barker (1994) recommend using the delta method to compute v~(/)lu), the estimated variance o f / ) , using the estimated information matrix for the 6j (conditional on u). This method has seen little use but seems to hold promise. The geometric approach also lends itself to considerations about spatial configurations of traps
Chapter 14 Mark-Recapture Methods for Closed Populations
320
other than the web and permits consideration of optimal configurations.
14.4. REMOVAL METHODS As with capture-recapture methods, removal methods for closed populations involve multiple samples in which animals in the population of interest are captured. As the term implies, however, captured animals are not returned to the population but are removed, thus distinguishing removal sampling from capturerecapture. We include these models in a chapter on closed-population models because the removals are under the control of the investigator and are assumed to be known. In this sense the population can be viewed as open with respect to investigator removals, but closed with respect to natural processes. The relevant population model is Xi+l
in capture probability, thus defining models in which sampling intensity is not equal for all sampling periods. The overparameterization of model Mtb w a s handled in Section 14.2.3 by taking advantage of a presumed mathematical relationship between initial and recapture probabilities. In this section the approach is to utilize auxiliary information, namely, timespecific measures of sampling effort that are assumed to be directly related to the capture probabilities. An assumed direct relationship between effort and capture probability has led to the wide use of "catch-per-uniteffort" statistics as indices to abundance (e.g., Schnute et al., 1989; Richards and Schnute, 1992; Schnute and Hilborn, 1993). For reasons presented in Chapter 12, we do not discuss these indices here and instead focus on statistically reliable procedures for estimation of population parameters. The models in this class are typically referred to as "catch-effort" models.
"- X i ~ Flit
with time-specific removals rl i reducing the population monotonically over the course of the study. Removal methods are most commonly used to estimate abundance for exploited populations; for example, fisheries applications are common (e.g., Hilborn and Walters, 1992). Removal models can be conveniently placed into two categories, the first of which imposes equal sampling intensity at every sampling period. Models for this situation have been described in Section 14.2.2 as behavioral response models, with the idea that removal is an extreme "behavioral response" for which the probability of recapture vanishes. Sufficient statistics for abundance estimation under two of the behavioral response models (Mb, Mbh) described in Section 14.2.2 are the number of animals caught for the first time in each sampling period. Thus, estimation under any behavioral model proceeds as with a removal model, with initial captures essentially "removed" from the population (recaptures are not used to estimate population size). Constant-effort removal models were introduced by Zippen (1956, 1958), but the estimators described in Section 14.2.3 and computed by program CAPTURE are now the preferred means of analyzing such data. Because the removal (behavioral response) models for equal sampling intensity have been described in Section 14.2, we will not discuss them further here. The other class of removal models permits variation in sampling intensity over time and requires additional structure in order to estimate parameters. Note that the other behavioral response models discussed in Section 14.2, Mtb and Mtbh, include time as a source of variation
14.4.1. Sampling Scheme and Data Structure We again assume that animals are captured on K different sampling occasions, and captured animals are removed from the population. The focus of estimation is on initial population size. We denote this as N 1, where the subscript 1 serves as a reminder that the population is changing throughout the sampling as a result of removals. Define the following statistics: ni
i-1
Xi -- ~ j = l nj
fi
i-1
Fi -- ~ j = l
)~
The number of animals removed from the population at sample period i. The cumulative catch prior to sampling period i (i = 2, ..., K + 1, Xl = 0). The units of effort expended on sample i. The cumulative effort prior to sampling period i (i = 2, ..., K + 1, F 1 -- 0).
The basic model parameter is k, the catchability coefficient or capture rate for a particular animal for one unit of effort. As in previous sections of this chapter, let Pi denote the capture probability for period i and define qi = 1 - Pi. Under the assumption of a Poisson sampling process (see Appendix E), the relationship between capture probability and effort can be written as qi = e - k f i
and Pi = 1 -- e -kfi.
(14.22)
14.4. Removal Methods In the development below, we assume that both catch (n i) and effort (fi) are known. However, when this approach is used with harvest data, neither catch nor effort is likely to be known with certainty. The consequences of measurement error for catch-effort estimation have been investigated by Gould et al. (1997), who suggested a simulation-extrapolation method of inference (Cook and Stefanski, 1994) as a means of adjusting for resulting bias. The following assumptions often are specified for this approach: (1) sampling is a Poisson process, with all animals having the same probability of capture per unit of sampling effort; (2) units of sampling effort are assumed to be independent and additive in their effect on catchability; (3) all removals from the population and the level of effort expended in each sample are known; and (4) the population is closed both to gains and to losses other than known removals.
As background, we begin by presenting three different least-squares approaches that follow the historical development of catch-effort estimation. We then describe the general development of Seber (1982) and Gould and Pollock (1997b) for an approach using maxim u m likelihood estimation. For a general treatment of catch-effort estimation via least squares, we recommend the work of Bishir and Lancia (1996). The idea in all approaches is to characterize captures at each point in time in terms of sampling effort and the size of the population exposed to capture.
14.4.2.1. Approach of Leslie and Davis (1939) Under the "Leslie" method, removals from the population are viewed as conditionally binomial, with probability of capture given by Eq. (14.22). The joint distribution of removals thus is modeled as K
pni qNl-Xi+l
(14.23)
1)!
Under this model, the conditional expectation of the catch at time i can be written as E(ni[xi)
=
(N 1 -
xi)Pi.
(14.24)
Define a catch-per-unit-effort statistic as Yi = ni/fi (this statistic frequently is used as an index to abundance). If Pi is small then Pi "~ kfi, and substituting this expression into Eq. (14.24) and dividing each side by fi we obtain the regression model E(YiIxi) ~, k N 1 -- kxi,
where k N 1 is the intercept and - k is the slope. The parameters of Eq. (14.25) then are estimated using leastsquares methods based on Eq. (14.25). 14.4.2.2. A p p r o a c h of D e L u r y (1947)
DeLury (1947) considered the expected catch, E(ni) = Nlqlq2 "'" qi-lPi
(14.26) = Nle-kFipi ' i-1 for sample i, where F i = ~j=l ~. Thus, in order to be caught in sample i, an animal must be missed (not caught) in the previous i - 1 samples. The Pi are again assumed to be small, permitting the approximation Pi kfi. In addition, DeLury (1947) used the approximation E[ln(ni/fi) ] ~ ln[E(ni/fi) ].
Taking logs of both sides of Eq. (14.26), dividing by fi, and substituting the approximation yields the regression model
14.4.2. Models and Estimators
g({ni}) = I-[ (Xl Xi)! i=1 ni! (N1 - x~ +
321
(14.25)
E(yi]Fi) ~, ln(kN 1) - kFi,
(14.27)
where Yi = ln(Yi) = ln(ni/fi). Under this approach, the catch-per-unit-effort (actually its natural log, Yi) is related to cumulative effort rather than cumulative catch. The slope of the regression line, Eq. (14.27), is again -k.
14.4.2.3. Approach of Ricker (1958a) Ricker (1958a) viewed the entire study as consisting of FK+I samples, each of which represented a single unit of effort. Thus, the expected population size at the time of each sample can be written as E(N 1 - xi) -~ N1(1
-
k) Fi.
(14.28)
Given equality (14.28) and the approximation in Eq. (14.25), Ricker (1958a) derived the model E(y i) ~ ln(kN 1) + Fi[ln(1 - k ) ] ,
(14.29)
where Yi is again defined as Yi = ln(Yi) = ln(ni/fi). Expression (14.29) also can be obtained directly from Eq. (14.27) by utilizing the approximation ln(1 - k) ~ -k.
14.4.2.4. Comments on the Three Least-Squares Approaches All of the above three approaches are based on regression models for which least-squares estimation typically is recommended. As discussed by Gould and Pollock (1997b), the approaches all rely on the approximation Pi ~" kfi, which is reasonable only when Pi is small. However, it is also true that the reliability of
322
Chapter 14 Mark-Recapture Methods for Closed Populations
these catch-effort methods depends on a substantial proportion (usually >30%) (Gould and Pollock, 1997b) of the population being removed during sampling. A reliance on large catches is not consistent with approximations that assume small Pi. An additional objection to the DeLury (1947) and Ricker (1958a) approaches involves the approximation in which the expected value of a logarithm is equated with the logarithm of the expectation. Finally, the regression assumption of constant variance structure (of Yi or Yi) is unlikely to be reasonable, because the catch-per-unit-effort (Yi) should decrease as the population is reduced. For these and other reasons presented by Gould and Pollock (1997b), we favor their recommendation to focus on maximum likelihood methods for modeling and estimation in catch-effort problems.
14.4.2.5. M a x i m u m Likelihood Approach Seber (1982) and Gould and Pollock (1997b) wrote the joint distribution of the catch statistics r/i as the multinomial distribution
NI!
P({ni}]k, {fi}) =
P~'(qlP2)n2""(qlq2" "qK - lP K)nK
(l-I/K= 1 ni!)(Nl--XK+l)' x (1 - Pl -
(14.30)
qlP2 . . . . . qlq2""qK-lPK )N'-xK+'
Rather than using this distribution directly in estimation, Gould and Pollock (1997b) recommended rewriting Eq. (14.30) as the product of two distributions:
P({rli}]k, {fi})
=
Pl(XK+l[k,
{fi})
(14.31)
X P2({ni}]XK+l, k, {fi}). Expression (14.31) decomposes the distribution (14.30) of the catch statistics into two components. Component P1 models the total catch for the entire study, XK+1, as a binomial random variable:
Pl(XK+l]k, 0ci}) =
NI! (1 XK+I!(NI--XK+I)!
-
Q ) XK+I Q N1 -XK+l I
where Q is the probability of not catching a member of N 1 during the entire study. Q is written as the product of the probabilities of not catching an animal at each of the K sampling periods:
Q = 1 - Pl - qlP2 . . . . .
qlq2 . . . .
qK-IPK"
The second component of expression (14.31) then conditions on the total number of animals caught throughout the study and models their distribution over the K sampling periods:
p2({ni}lXK+l,k,{fi})__
XK+I! ( P l ) n l l-Ii K, hi! 1 - Q
(14.32)
X(qlP__b)na...(qlqai"q-~-lPK)nK The actual modeling of capture probability as a function of effort can use any reasonable function. For example, Gould and Pollock (1997b) selected the linear logistic form
ef3O+f31fi Pi = 1 + e ~~ for their examples, which has the advantage of being sufficiently flexible to incorporate other covariates in addition to effort in the modeling of capture probability (see example in Section 14.4.5) (see Pollock et al., 1984; Gould and Pollock, 1997b). Estimation proceeds by conditional maximum likelihood (Sanathan, 1972), using P2 to estimate catchability. The resulting estimate of catchability then is used with P1 to estimate N 1 using the familiar form of the canonical estimator [Eq. (12.1)]:
1~1 "- XK+ 1//9,
(14.33)
where/3 = (1-(~), and (~ is estimated using the catchability from P2 [Eq. (14.32)]. One advantage of this two-step approach is that it avoids the difficulty in numerical maximization with a discrete-valued parameter (N1), the magnitude of which is very different from that of the catchability coefficient. Gould and Pollock (1997b) provide further discussion motivating this approach. Variances can be estimated using Taylor series approximations, although Gould and Pollock (1997b) recommend use of the parametric bootstrap (see Appendix F). Pollock et al. (1984) applied maximum likelihood methods to catch-effort problems as well, but they used an unconditional approach. They included N 1 in the likelihood and used a two-step iterative process to obtain estimates numerically (Hines et al., 1984). Because the conditional approach of Gould and Pollock (1997b) is easier to implement numerically, we recommend it for most uses.
14.4.3. Violation of Model Assumptions The assumption that all animals have the same probability of capture per unit of sampling effort throughout the entire study can be violated in numerous ways. In fisheries, for example, different size or age or sex classes of fish may have different susceptibilities to particular fishing methods, causing heterogeneity in catchability coefficients among individuals. In the face
14.4. Removal Methods of such heterogeneity, the more catchable animals are likely to be caught early on, leaving the less catchable animals to comprise larger and larger portions of the remaining population (N1 - xi). Such a pattern should lead to decreases in average catchability over time. Trends in catchability because of environmental conditions also can result in violation of the equal-catchability assumption. In this case, the time trend could be either positive or negative, depending on the trend in the environmental parameter and its effect on catchability. Intuition suggests that a negative trend in catchability over time should lead to a more negative slope in the relationship between catch-per-unit-effort and cumulative effort. This should produce an estimate of the catchability coefficient that is positively biased and an abundance estimate that therefore is negatively biased [e.g., see Eq. (14.33)]. Conversely, a positive trend in catchability should lead to a less negative slope in the relationship between catch-per-unit-effort and cumulative effort. This should produce negative bias in the catchability estimate and positive bias in the estimate of abundance. Simulation results of Gould and Pollock (1997b) confirm these expectations and indicate that the biases can be substantial. Failure of the closure assumption should affect catch-effort estimates in a manner similar to that of temporal trends in the catchability coefficient. For example, consider a population exposed to losses between sampling occasions but no gains (or a completely open population with losses exceeding gains). The number of animals exposed to sampling efforts at each occasion will be smaller than N 1 - xi, because of the losses in addition to the known removals. This should yield a more negative slope of the relationship between catch-per-unit-effort and cumulative effort than if there were no losses, producing negative bias in the estimate of N 1. If the population is exposed only to gains, or if gains exceed losses, then we speculate that the slope of the relationship between catch-per-unit-effort and cumulative effort should be less negative, producing positive bias in the abundance estimate N 1. Populations experiencing fluctuations in the relative magnitudes of gains and losses between the different sampling occasions should lead to biased estimates, although the directions and magnitudes of bias will depend on the pattern of population change. A lack of influence of sampling effort on the resuiting capture probability is speculated to be a common reason underlying the lack of fit of catch-effort models to actual catch data. This assumption violation causes model-based variance estimates to be too small, necessitating use of variance inflation factors (see below).
323
The assumption that removals and units of effort are known with certainty is likely to be violated when catch-effort models are applied to data for harvested populations (e.g., fisheries). Gould et al. (1997) explored the consequences of measurement error for both catch and effort via computer simulation. They investigated the performance of their maximum likelihood approach as well as that of the approaches of DeLury and Leslie. The maximum likelihood approach performed the best, but estimates nevertheless were positively biased by measurement error, with biases becoming substantial with large measurement error variances. Gould et al. (1997) thus recommended a simulation-extrapolation inference method (Cook and Stefanski, 1994) for reducing bias of estimates in the presence of measurement error. Because assumption violations can lead to substantial bias, efforts to assess model fit to the data are important. Pearson chi-square goodness-of-fit tests based on a comparison of observed catches against their expectations under the model frequently are used to assess model fit. When there is evidence of lack of fit, and when it is believed that lack of independence may be responsible, it is reasonable to use a quasilikelihood approach (Pollock et al., 1984; Burnham et al., 1987; Lebreton et al., 1992; Burnham and Anderson, 1998). Estimators for model parameters frequently remain unbiased in the face of overdispersion caused by lack of independence, but model-based variances tend to be too small and should be inflated (McCullagh and Nelder, 1989). If a Pearson chi-square goodness-of-fit test is used to assess fit, and if it provides evidence that the most general model in the model set does not fit the data adequately, then the fit statistic can be used to compute a variance inflation factor (e.g., Burnham et al., 1987) by =
x21df,
where X2 and df correspond to the goodness-of-fit test of the global model (Cox and Snell, 1989) or the most general model in the model set. Model-based variance and covariance estimates then are multiplied by ~ to obtain estimates that properly account for overdispersion. The variance inflation factor also can be used to adjust likelihood ratio test statistics and Akaike's Information Criterion for the purpose of selecting from among competing models (see Section 17.1.8).
14.4.4. Study Design As was the case for closed-population capturerecapture models, study design in catch-effort studies should include efforts to minimize assumption violations and maximize estimator precision. Regarding the
324
Chapter 14 Mark-Recapture Methods for Closed Populations
assumption of equal catchability for all animals in the population, we noted in Section 14.4.3 that different size, age, or sex classes of animals may exhibit different susceptibilities to capture. Stratification is an obvious way of dealing with this problem, so study design should include recording of auxiliary data that can be used to classify animals to strata. Variables that might be associated with capture probability and are measurable must be selected before the initial capture sample is taken. In the data-analytic stage, models permitting different catchability coefficients for different classes then can be compared against models that do not include such variation. If the models incorporating variation among animal classes are selected, then separate estimates of abundance can be obtained for each class and summed to obtain an overall estimate. In studies of limited size, it may be possible to ensure through investigator effort that numbers of removals and units of sampling effort are known. However, in large studies involving harvest situations, it may not be possible to enumerate directly removals or units of effort. In such cases it is important to use an estimation method that provides unbiased estimates of removals and effort. Many survey methods have been developed for estimation under these conditions. These methods are beyond the scope of this book, but we recommend Pollock et al. (1994) for an introduction to the angler survey methods that are commonly used in fisheries investigations. Methods for removals and effort that permit estimation of associated sampling variances are desirable, because variance estimates can be adjusted to deal with measurement error (Gould et al., 1997). The primary aspects of study design that are relevant to the closure assumption involve time and space. Relatively short studies provide the greatest likelihood that the studied populations are closed to gains and losses other than known removals. It is wise to restrict catch-effort studies to times of the year when population processes such as migration, reproductive recruitment, and mortality are likely to be minimal. Similarly, the closure assumption is more likely to be met in spatially restricted study areas (e.g., small to moderately sized ponds or lakes) than in areas lacking spatial restrictions (rivers and oceans). When the population is found to be open despite study design, special catcheffort models for open populations can be used (Section 19.5.3) (also see Seber, 1982; DuPont, 1983; Bishir and Lancia, 1996; Gould and Pollock, 1997a). Precision and bias of abundance estimates resulting from catch-effort studies are heavily dependent on the fraction of the population that is removed, with larger proportional removals yielding more precise and less biased estimates. The number of sample occasions is one element of study design that determines the re-
moval fraction. Gould and Pollock (1997b) presented simulations with catchability coefficient k = 0.01, yielding the following proportions of the population removed: 19% for K = 3 sampling occasions, 30% for K - 5, and 51% for K = 10. Median negative bias of the abundance estimate for these three scenarios was 51%, 20%, and 2%, respectively. As in capture-recapture studies for closed populations, the selection of the number of occasions represents a tradeoff between efforts to approximate the closure assumption (emphasis on closure will lead to fewer occasions) and efforts to obtain precise estimates by removing more animals (leading to more sampling occasions). The magnitude of the catchability coefficient is very important, with higher catchability leading to more precise abundance estimates. Catchability should be a direct function of sampling effort and is thus an important aspect of study design. Finally, for fixed catchability and number of sampling occasions, measures of relative precision are smaller for large population sizes and larger for small populations (Gould and Pollock, 1997b). Choice of study area boundaries may partially determine the size of the studied population.
14.4.5. Example We present the analyses of Pollock et al. (1984) and Gould and Pollock (1997b) for the classic catch-effort data set of Paloheimo (1963). The data correspond to 2-week periods and include estimated catch (number of legal-sized lobsters removed), effort (in number of trap hauls), and a potential covariate [average water (bottom) temperature in ~ for a Canadian lobster fishery at Port Maitland, Nova Scotia, 1950-1951 (Table 14.11). In the most general model, capture probability is modeled as a linear-logistic function of effort fi and the environmental temperature t i for sampling occasion i: e ~o+ f31fi+ B2ti Pi =
1
+ e f~~
We also consider reduced parameter models in which capture probability is modeled as a constant (131 = ~2 = 0) and a function only of effort (132 = 0). The models were fit using the unconditional approach of Pollock et al. (1984), and Pearson goodnessof-fit tests provided strong evidence that none of the models fit the data well (Table 14.12). This was expected because of the extremely large sample sizes, the likely nonindependence of lobster captures, and the likely influences of factors other than effort and temperature on capture (see Pollock et al., 1984). As indicated by the magnitudes of the residuals (deviations between observed and predicted values), the models
325
14.5. Change-in-Ratio Methods
TABLE 14.11 Catch, Effort, and Temperature Data for a Commercially Harvested Lobster Population a Period
Catch (n i)
Effort (fi)b
Temperature
1
60,400
33.664
2
49,500
27.743
7.7
3
28,200
17.254
6.3
4
20,700
14.764
3.5
5
11,900
11.190
3.1
6
15,600
16.263
2.9 3.1
( t i)
7.9
7
13,200
14.757
8
25,400
32.922
3.25c
9
29,900
45.519
3.4
10
32,500
43.523
3.6
11
24,700
37.478
4.0
12
27,600
43.367
5.9
13
22,200
37.960
6.1
a At Port Maitland, Nova Scotia, Canada, 1950-1951; reanalysis of data after Paloheimo (1963), cited in Gould and Paloheimo (1997b). bEffort is in thousands of trap hauls. CThis value was missing, so we used the average of the two adjoining periods.
TABLE 14.12 Comparison of Residuals for Three Models for a Commercially Harvested Lobster Population a Residuals b
Period
Observed catch
1
60,400
2 3
Effort plus temperature Constant (p.) Effort only (Pt)
(Pf+t)
+116.8
+27.6
- 10.8
49,500
+72.8
+51.5
+13.2
28,200
- 30.7
+20.9
+5.7
4
20,700
- 63.5
- 4.3
+22.1
5
11,900
- 106.0
- 43.6
- 22.9
6
15,600
- 77.7
- 31.1
- 3.5
7
13,200
- 85.2
- 34.3
- 14.9
8
25,400
- 2.7
- 31.5
+4.8
a c t u a l l y s e e m e d to p e r f o r m r e a s o n a b l y well. T h e variance inflation factor w a s c o m p u t e d u s i n g the X2GOF a n d a s s o c i a t e d df f r o m the m o s t g e n e r a l m o d e l (Pf+t) as ~ = 300.44, a n d this v a l u e w a s u s e d to a d j u s t v a r i a n c e e s t i m a t e s a n d to c o m p u t e AQAICc v a l u e s ( B u r n h a m a n d A n d e r s o n , 1998) (also see Section 17.1.8 a n d Table 14.13). T h e m o s t g e n e r a l m o d e l h a d the l o w e s t QAICc v a l u e a n d w a s j u d g e d m o s t a p p r o p r i a t e for the d a t a (also see Pollock et al., 1984; G o u l d a n d Pollock, 1997b). T h e e s t i m a t e d linear-logistic coefficient ~1 a s s o c i a t e d w i t h effort w a s p o s i t i v e as p r e d i c t e d , as w a s the coefficient ~2 a s s o c i a t e d w i t h t e m p e r a t u r e . H i g h e r t e m p e r a t u r e s w e r e p r e d i c t e d to p r o d u c e g r e a t e r lobster activity a n d t h u s g r e a t e r p r o b a b i l i t y of b e i n g c a u g h t . T h e coefficient of v a r i a t i o n for a b u n d a n c e u n d e r the g e n e r a l m o d e l w a s small [CV(/~) = 0.087].
14.5. C H A N G E - I N - R A T I O METHODS .
.
.
.
.
.
.
Change-in-ratio m e t h o d s originally were d e v e l o p e d for u s e w i t h h a r v e s t e d species, b a s e d o n the o b s e r v a tion t h a t differential h a r v e s t a m o n g g r o u p s of a n i m a l s in a p o p u l a t i o n can p r o d u c e c h a n g e s in p r o p o r t i o n a t e r e p r e s e n t a t i o n of the g r o u p s . For e x a m p l e , a d e e r harv e s t d i r e c t e d at m a l e s s h o u l d lead to a r e d u c t i o n in the p r o p o r t i o n of m a l e s in the p o s t h a r v e s t p o p u l a t i o n . Kelker (1940, 1944) r e c o g n i z e d t h a t i n f o r m a t i o n o n the ratios of different t y p e s of a n i m a l s (e.g., sex ratio) before a n d after h a r v e s t , c o m b i n e d w i t h i n f o r m a t i o n o n the n u m b e r of a n i m a l s of e a c h type, c o u l d be u s e d to e s t i m a t e a b u n d a n c e . C h a p m a n (1954, 1955) d e v e l o p e d the first stochastic m o d e l s for a b u n d a n c e e s t i m a t i o n ,
TABLE 14.13 Comparison of Parameter Estimates (Standard Errors) for Three Models for a Commercially Harvested Lobster Population a
9
29,900
+34.7
- 55.2
- 17.1
10
32,500
+61.2
+1.8
+22.8
549,974
- 3.94
0.030
0.11
11
24,700
+18.4
+15.4
+12.9
(47,780)
(0.182)
(0.004)
(0.02)
12
27,600
+47.0
+26.7
- 10.8
Effort model (pf)
13
22,200
+17.7
+42.4
- 3.9
472,270
- 3.29
0.037
(21,840)
(0.108)
(0.0036)
X2GOF
df AQAICc
56,282 11 28,600
14,706 10 6200
2704 9 0
a At Port Maitland, Nova Scotia, Canada, 1950-1951; reanalysis of data after Paloheimo (1963), cited in Gould and Paloheimo (1997b). bResiduals computed as (0 i - Ei)/V~i , with 0 i the observed catch and E i the expected catch under the model.
Effort-plus-temperature model (Pf+t)
Constant probability model (p.) 716,860
- 2.89
(84,200)
(0.172)
a At Port Maitland, Nova Scotia, Canada, 1950-1951; reanalysis of data after Paloheimo (1963), cited in Gould and Paloheimo (1997b).
326
Chapter 14 Mark-Recapture Methods for Closed Populations
and the general approach has since been reviewed by Paulik and Robson (1969) and Seber (1982). The approach has been extended to incorporate more than two types of animals and more than one removal period (e.g., Otis, 1980; Pollock et al., 1985b; Udevitz and Pollock, 1991, 1995). The usual implementation of change-in-ratio methods involves a combination of observation data and removals from managed hunting areas. Returning to the example of deer sex ratio, observation-based methods (e.g., spot lighting) frequently are used to estimate the sex ratio before and after harvest, whereas managers at hunter check stations record the number of deer of each sex removed by hunting. The methods used to estimate the ratio of types of animals in the population before and after harvest need not involve direct observation but may involve trapped samples or any other means of assessment. The utility of the approach (as with that of the methods of Section 14.4) is based on its exploitation of data that are collected routinely in local management programs. The methods to be discussed here assume that, with the exception of known removals, the population is closed to gains and losses. It is important to note that the method is useful only when removals are selective with respect to the different types of animals in the population, because selectivity is the basis for the change in ratio that is exploited in estimation. For situations in which removals are not selective, the investigator should use constant-effort removal models (Mb and Mbh) of Section 14.2 or the catch-effort models of Section 14.4.
14.5.1. Sampling Scheme and Data Structure Although the original use of change-in-ratio methods involved two types of animals and a single removal period bracketed by two observation periods, we present here the notation for the more general case. We distinguish between sampling periods, in which the ratios of different types of animals in the population are assessed, and removal periods, during which the removals occur. Let K be the number of sampling periods in the study. Then there will be K - 1 removal periods, one following each sampling period except the final one. Define the following notation, adapted from that of Seber (1982) and especially Udevitz and Pollock (1991 ):
N/,
The number of individuals in the population of type i in sampling period j.
k
Xj
k
= ~i=1 Nij
rij
k /'j = ~i=1
rij
nij g
nj = ~i=1 nij
The number of types of animals in the population. The total number of animals in the population at sampling period j. The number of individuals of type i removed from the population in sampling period j between sample periods j and j+l. The total number of animals (all types) removed from the population between sampling periods j and j +1. The number of individuals of type i encountered in sampling period j. The total number of individuals (all types) encountered in sample j.
The Nj are the quantities of interest to be estimated. As with the removal approaches of Section 14.4, the relevant model here is
for j = 1, 2, ..., K - 1. However, change-in-ratio models are distinguished from the removal models of Section 14.4 in that change-in-ratio models recognize different animal types in the population and include additional (nonremoval) observations over the course of the study. Several assumptions underlie most change-in-ratio estimation methods (e.g., see Conner et al., 1986; Udevitz and Pollock, 1991). The population is assumed to be closed except for the removals, and the numbers of removals for animals in each type are assumed known. Sampling is with replacement, or else the sampling fractions are negligible. Encounters of animals during sampling periods are independent, with a probability Pij of encountering an individual of type i in sampling period j. Some approaches are based on the assumption that during any sampling period j, the probability of being encountered is the same for individuals in all types, i.e., Pij = Pi'j. Other approaches are based on the less restrictive assumption that the ratio Pij/Pi'j of encounter probabilities for individuals of type i and i' is constant over all sampling periods. The necessary assumptions for encounter probabilities are specified with the models and estimators described in Section 14.5.2. The change-in-ratio approach provides useful estimates only in the case in which animals of different types are not removed in proportion to their original abundance in the population. If removals are not selective with respect to animal type, then the ratio of types in the population is not expected to change, and the sample estimates of these ratios will provide no information with which to estimate abundance.
327
14.5. Change-in-Ratio Methods 14.5.2. M o d e l s and Estimators
/~1 = /~11 -}- /~21
We begin this section with a description of the standard (or at least original) application of two sampling periods separated by one removal period, with two types of animals. We then present the more general cases considered by Udevitz and Pollock (1991).
14.5.2.1. Two Sampling Periods, One Removal Period, Two Types of Animals We begin with an intuitive derivation of an estimator for the sampling situation originally considered with the change-in-ratio method. Assume that nll and n21 animals of two types are counted during an initial sampling period, with/'11 and/'21 known removals. In sampling period 2, n12 and 11122animals of the two types are counted again. If animals of both types are counted in proportion to their abundance in the population (i.e., if underlying detection probabilities are equal for the two types), then the following approximate expectations hold:
(14.38) /'1111122 -- /'21//12
//1 11111//22 m 1112111112
for sampling period 1, and /~2 ~" /~12 q- /~22 -- (/~11 -- /'11) if- (/~21 -- /'21) =
NI
-
(14.39)
ri
for sampling period 2. The abundance estimator for sampling period 1 also can be derived by writing the proportion of type I animals in the population at sampling period 2, N12/N2, as a function of the removals and the proportions of type I animals in the population at sampling period 1, Nil~N1" N12
N l l - rll
N2
N1 - r 1 (N11/N1)N1
-
rll
N 1 - r1
E(n111 ~ \11121/
Pll Nll P21N21
(14.34)
Nll N21
Substitution in the above expression of the estimators nil~n1 and n12/n2 for the proportions Nil~N1 and N12/N2 of type 1 animals in the population at the two sampling periods [see Eqs. (14.34) and (14.35)] yields the intuitive estimator
and
E[n12~l l ~" P l 2 ( N l l \/'/22./
/~1 = /'11 -nil/n1
r11)
P22(N21 - r21 )
(14.35)
Nll - rll N21 - r21"
Note that expressions (14.34) and (14.35) require equal encounter probabilities (Ply = P2j) for the two types within each sample, but allow for different encounter probabilities for the two sampling periods. The expressions can be combined to yield the following estimators for the number of animals in each type in the population before removal: *'/~111 /'1111122- /'2111112 -11111 1111111122 -- 1112111112
(14.36)
/~/21
(14.37)
and =
/ ' 1 1 n 2 2 - /'21n12 n21. 1111111122- //21//12
Expressions (14.36) and (14.37) are equivalent to the intuitive estimators of Kelker (1940; see Udevitz and Pollock, 1992). Based on Eqs. (14.36) and (14.37), the estimators/~1 and/~2 of abundance are simply
/'1(n12/n2 )
(14.40)
-- //12///2
(e.g., see Paulik and Robson, 1969; Seber, 1982; Pollock
et al., 1985b). It is not difficult to show that the estimators in Eqs. (14.38) and (14.40) are mathematically equivalent. The estimators in Eqs. (14.38) and (14.40) are dependent on reasonable estimators for the proportions of type 1 animals in the population at each sampling period. If the sample counts nj represent random samples of predetermined size taken with replacement (this corresponds to the usual case in which the counts are based on observations of unmarked animals), then the type-specific counts nij can be modeled as binomial random variables conditional on the sample counts and the true numbers of animals in the population of each type at each sampling period:
f({nij}l{Nij'nj}) = ~l (\nlj,n2j,) (Nlj~Nj lj(N2j~]2j\ \ Nj / "
(14.41)
Under this product-binomial model, the proportion nlj/n2j of type I animals in the observed sample is the maximum likelihood estimator for the true proportion in the population (e.g., see Chapman, 1954; Seber, 1982; Pollock et al., 1985b).
Chapter 14 Mark-Recapture Methods for Closed Populations
328
If the sample sizes nj are not fixed, then the counts nij can be modeled using the Poisson distribution. Under this model, expression (14.41) becomes the conditional distribution for the nij (Seber, 1982; Pollock et al., 1985b), and estimation proceeds in the same manner as above. The above estimators are based on an assumption that individuals of the two types are detected in proportion to their true abundance in the population (i.e., have equal detection probabilities), although these probabilities are permitted to differ between the two sampling occasions (i.e., we might encounter 15% of the population in one sample and only 10% in another). Now consider the case of equal detection probabilities for the two sampling periods, but different detection probabilities for animals in the two types. In this situation, the ratio r/il/ni2 provides an unbiased estimate of the ratio of type i individuals in the population in sampling periods 1 and 2 (see Udevitz and Pollock, 1992). Equating these sample ratios with the corresponding true values for the population yields E[n11] ~ Ln12J
Nll Nll
-
rll
and
E[H211 ~ Ln223
N21
N21
Y/11r11 /'/11
--
The basic change-in-ratio method was extended by Otis (1980) to include the identification and possible removal of three types of animals (e.g., adult males, adult females, young). Later, Pollock et al. (1985b) focused on two types of animals that are sampled at three sampling occasions separated by two removals. This additional sampling permits robust estimation of abundance in the face of unequal sampling probabilities and provides the data required to test assumptions about the sampling probabilities. Here we present the general formulation of Udevitz and Pollock (1991) for K -> 2 sampling periods, K - 1 intervening removal periods, and k >-- 2 types of animals. Define cij as the probability that a given encounter in sample j will be with a particular animal from type i. The probability that a given encounter in sampling period j will be with any individual from type i is cijNij. These probabilities are conditional on an encounter and hence sum to 1" k
(14.42)
/'/12
and /~21 =
14.5.2.2. Generalization to Multiple Samples, Removals, and Animal Types
-- r21
This system of two equations with two unknowns can then be solved to yield the intuitive estimators /~11 --
We obtain the estimated number/~il of animals in type i at sampling period 1 by dividing the number ril of animals removed by this probability. This procedure again yields the estimators in expressions (14.42) and (14.43).
n21F21 /'/21 -- /'/22
s cijNij-i=1
1
(14.44)
for all j. As an illustration of this general expression, return to the initial example of two sampling periods and two types of animals. The assumption of equal encounter probabilities for the two types in the estimators of Eqs. (14.36)-(14.40) can be written as
(14.43)
Cll -- C21'
(14.45)
C12 = C22.
(e.g., Udevitz and Pollock, 1992). Abundances at sampling periods 1 and 2 then can be estimated in the general manner illustrated in Eqs. (14.38) and (14.39). Note that the estimators of expressions (14.42) and (14.43) can also be viewed in the context of the canonical estimator (12.1). Under this perspective, we view ril as a sample from the type i animals in the population at sampling period 1. Under the assumption of equal detection probabilities in the two sampling periods, the estimated probability of an animal appearing in the removal sample is given by Hi1 -- Yli2 Hi1
Combining the constraints of Eqs. (14.44) and (14.45), we can write the probability of encountering any particular individual in a study with two types of animals as
Clj = c2j = 1/(Nlj 4- N2j) forj = 1, 2. Udevitz and Pollock (1991) also recommend rewriting the constraints of expression (14.45) as C21/Cll -- C22/C12, C21/Cll ":
(14.46)
1,
in order to emphasize the two components of the equal encounter probability assumption underlying the esti-
14.5. Change-in-Ratio Methods mators of Eqs. (14.36)-(14.40). The first equality in Eq. (14.46) expresses the assumption that the ratio of encounter probabilities for individuals of the two types remains constant over time. The second component specifies the actual value of this ratio (1 in this case). The assumption of equal encounter probabilities, expressed by both equalities in Eq. (14.46), is stronger than the "constant probability ratio" assumption of the first component of Eq. (14.46). The general approach developed by Udevitz and Pollock (1991) for multiple types and sampling periods also uses the constant probability ratio assumption, expressed generally as Cil/Cll "-- Cij/Clj
= hi, i = 2, ..., k, j = 2, ..., K, where the parameters h i are defined as the ratio of encounter probability for individuals in type i to that of individuals in type 1, with K1 = 1. Then the probability distribution of the {nij} under the general model of Udevitz and Pollock (1991) can be written as
f({nij}l{Nij, hi, nj}) - .il~~
---
X ./I~1~2k= 1 hi Xij
(14.47)
.
This general formulation can be used to derive estimates under the special cases described thus far. If there are only K = 2 sample occasions (the original change-in-ratio design), then the parameters of expression (14.47) are not identifiable without an additional constraint. For example, if we assume that K2 -- a2 (some positive constant), then maximum likelihood estimates of remaining parameters can be obtained. If a 2 -- 1 for the sampling design with K = 2 periods and k - 2 types, then the maximum likelihood estimates based on expression (14.47) are the intuitive estimators of expressions (14.36) and (14.37). Similarly, the special case of K = 3 periods and k = 2 animal types yields the estimators first derived by Pollock et al. (1985b). Other models can be developed and tested by imposing constraints on the h i. Udevitz and Pollock (1991) provide computer code for obtaining estimates using iteratively reweighted nonlinear least squares. The key assumption underlying the general model in Eq. (14.47) is that the ratios h i of encounter probabilities remain constant over the different sampling periods. Udevitz and Pollock (1995) developed an approach that uses additional information in order to relax this assumption (also see Chapman and Murphy,
329
1965). The additional information is the effort expended to obtain each set of sample counts (the nij for each sampling period j). A catch-effort modeling approach (Section 14.4) then can be used to generalize the model of expression (14.47) in a manner that permits various forms of temporal variation in the relative encounter probabilities of the different types.
14.5.3. Violation of Model Assumptions The assumption of population closure except for known removals can be violated by any gains to the population via reproductive recruitment or immigration and by losses from deaths or emigration. If the studied population is not closed, then the sample counts for periods following the initial sample will be improperly modeled. Consider the situation in which there is mortality between the first sampling period and the period of removal for the standard two-sample, two-type, change-in-ratio study. If mortality rates are the same for both types of animals, then the abundance e s t i m a t o r / ~ 1 of Eq. (14.39) now estimates abundance after mortality and just before the removals (see Paulik and Robson, 1969; Seber, 1982). The closure assumption also can be violated by unknown removals associated with illegal or otherwise unreported harvest, or with crippling loss of animals that are not retrieved. For the standard case of two samples and two types of animals, the bias of the estimates for abundance at the time of each sampling period can be evaluated using expressions provided by Paulik and Robson (1969; also see Chapman, 1955). If the proportions of unreported kills are the same for both types of animals, then abundance estimates for times 1 and 2 will be negatively biased. For example, if the reported kills of animals of both types is 15% too low, then the true abundance at time 1 will be approximately 15% larger than the estimated value (see Paulik and Robson, 1969; Conner et al., 1986). Most of the estimators discussed in this section were derived assuming that either sampling is with replacement, or else the sampling fractions are negligible. However, when sampling is carried out without replacement for the two-sample, two-type, change-inratio method, the resulting hypergeometric model (see Appendix E) yields the same maximum likelihood estimates for abundance as does the binomial model described above (Eq. 14.41). However, the asymptotic variances do differ for the two modeling approaches (Seber, 1982; Pollock et al., 1985). Encounters of animals during sampling periods are assumed to be independent. Although we are aware of no work on effects of violation of this assumption (e.g., when animals travel as pairs or family groups
330
Chapter 14 Mark-Recapture Methods for Closed Populations
such that encounters are not independent), we suspect that it will not lead to biased estimates of abundance but will instead produce negatively biased variance estimates. As noted above, the two-sample, two-type, changein-ratio method was developed initially assuming equal encounter probabilities for animals of the two types (Kelker, 1940; Chapman, 1954, 1955; Seber, 1982). When this assumption is not true (i.e., when )~ 4: 1), then n11/n I and n12/n 2 will be too small or too large when viewed as estimators for N 1 1 / N 1 and N12/N2, yielding biased estimates of abundance [see Eq. (14.39)]. On the other hand, when all removals are of a single type, the abundance estimate for the type removed and its estimated variance are unbiased, even in the face of different encounter probabilities for animals of different types (e.g., see Seber, 1982; Conner et al., 1986).
14.5.4. Study D e s i g n The design of studies utilizing change-in-ratio methods should include efforts to minimize the probability of violating model assumptions and to maximize estimator precision. As with other estimation approaches, the assumption of population closure except for known removals is best met by restricting the temporal extent of the study. The longer the study, the more likely that numbers of animals will be influenced by movement, deaths, and recruitment. Similarly, it is desirable to restrict studies to seasons of the year when migration, mortality, and reproductive recruitment are minimal. With respect to geographic closure, studies carried out on areas with clear boundaries over which movement is rare are most likely to be successful. In the two-sample, two-type situation, animals of the two types must be encountered in samples in proportion to their abundance in the population, so sampiing methods should be selected with that assumption in mind. If it cannot be met, then the ratio of encounter probabilities sometimes can be estimated independently with a separate experiment, e.g., based on a marked subsample or the use of a double-observer approach (Section 12.6). These independent estimates then can be used in the estimation of abundance (see Chapman, 1955; Seber, 1982). Perhaps the best way to deal with the assumption of equal encounter probabilities for the different animal types in the population is to implement a study design that does not require it. The designs based on more than two samples permit differences in detection probabilities of the different types and require only that the ratios of detection probabilities for the different types remain constant over all sampling periods (Pollock et
al., 1985b; Udevitz and Pollock, 1991, 1992). Even this assumption can be relaxed when the sampling design includes the recording of the amount of effort expended on the different samples (Udevitz and Pollock, 1995). In many cases, the number of removal periods will be dictated by the management program (e.g., when the removals are via sport or commercial harvest). However, whenever there is design flexibility regarding the numbers of sampling and removal periods, studies can be designed in ways that require minimal assumptions about type-specific and temporal variations in detection probabilities. Expressions and associated figures relating sample sizes (e.g., n I and n 2) to accuracy of resulting abundance estimates are presented for the two-sample, twotype situation by Paulik and Robson (1969) and Conner et al. (1986). The graphs in these papers are especially useful in planning a change-in-ratio study under the traditional design. A quantity of critical importance to estimation is the difference in the proportional composition of the population between the first and second sampling periods, i.e., the magnitude of the change in ratio of the types:
Nll
N12
ml
N2
- AP.
Paulik and Robson (1969) declared &P < 0.05 to be "almost worthless as a means of determining population abundance." They questioned the use of the change-in-ratio method for situations in which AP < 0.10, although Conner et al. (1986) obtained reasonable results with an estimated change in ratio of &/5 ~ 0.07. For a given sample size and change in proportions, the precision of estimates is higher when the total proportion of the population removed is higher and when the initial type proportions are more dissimilar (Paulik and Robson, 1969; Udevitz and Pollock, 1992).
14.5.5. Example We report a hypothetical example used by Udevitz and Pollock (1991) to illustrate their general approach. They assumed a population with three animal types, sampled with replacement at sampling period 1 to obtain n I -- 500 encounters consisting of nll = 128 animals of type 1, n21 119 animals of type 2, and n31 = 253 animals of type 3. The first sampling period was followed by the type-specific removals of rll = 140, r21 = 280, and r31 = 560. A second sample then yielded n12 = 227, n22 = 167, and n32 = 106. Using the constraint that the ratio of individual encounter probabilities for types 1 and 2 are equal, )k2 = a 2 = 1 (see Section 14.5.2), the estimated ratio of =
14.6. Discussion encounter probabilities of type 3 to type I individuals is K3 = 2.58 (SE = 1.55). Udevitz and Pollock (1991) presented the following type-specific abundance estimates for sampling period 1: /~/1~ = 912 ^
(S'E = 632), A
N21 = 848
(SE = 495),
and ^
A
N31 = 700
(SE = 43).
Despite the removal of fairly large numbers of animals, the above abundance estimates are very imprecise, illustrating a feature of nearly all change-in-ratio estimates.
14.6. D I S C U S S I O N In this chapter we have described methods for estimating abundance based on captures of animals. Sections 14.1 and 14.2 concerned capture-recapture methods in which animals are caught, given individual marks, and then recaptured, all over relatively short time periods. A short period for the investigation increases the likelihood that the population remains closed to gains and losses over the period of sampling. The resulting data can be written as individual capture histories, vectors of ls and 0s indicating the sequence of captures for each individual during the study. Closed models do not require parameters for gains and losses, so the modeling of capture history data involves only capture probability parameters. These parameters can be defined in terms of three potential sources of variation (time, heterogeneity, and behavioral response), and models were developed to include one or more of these sources. The model underlying the two-sample LincolnPetersen estimator of Section 14.1 permits only temporal variation in capture probability. This model is useful in many field situations (e.g., Seber, 1982; Menkens and Anderson, 1988) but also provides an intuitive foundation for the use of capture-recapture models to estimate population parameters. This foundation underlies all of the more complicated capture-recapture models for both closed (Section 14.2) and open (Chapters 17-19) populations. The consequences of the violation of model assumptions for Lincoln-Petersen estimation were discussed thoroughly, as a basis for deducing consequences to estimators in more complicated models. The K-sample closed-population models of Section 14.2 form a complete set of models for estimating abundance in the face of the three sources of variation in
331
capture probability listed above. However, the model testing and selection tools developed for likelihoodbased models (Chapter 4) are not uniformly available because the models incorporating individual heterogeneity have too many parameters, and estimation utilizes ad hoc approaches such as the jackknife, bootstrap, and sample coverage. As noted throughout Section 14.2, the finite mixture models of Norris and Pollock (1995, 1996) and especially Pledger (2000) provide a solution to this problem. Once software becomes widely available for them, we expect these heterogeneity models to see substantial use. Section 14.3 describes some approaches to the difficult problem of estimating density from capturerecapture data. One approach involves first estimating abundance (e.g., as in Sections 14.1 and 14.2) and then estimating the area from which captured animals are sampled. This approach usually involves grid sampling, in which a boundary strip of estimated width W is added to the perimeter of the study area in order to compute the area sampled by the grid. Another approach involves the use of a gradient in trap density, via distance sampling (Chapter 13) or the geometric approach of Link and Barker (1994). Because the latter approaches to density estimation have been infrequently used, we have limited experience with their performance. The capture-recapture methods presented in Sections 14.1, 14.2, and 14.3 are likely to be useful for animals that are secretive, nocturnal, or simply difficult to observe. In general these methods should not be considered for animals that are easily observed, because the observation-based methods presented in Chapters 12 and 13 should be preferable. On the other hand, the utility of the removal methods presented in Sections 14.4 and 14.5 is tied less to the observability of the target organisms and more to the existence of harvesting operations (e.g., hunting, fishing, trapping). Catch-effort models and change-in-ratio methods are designed to use catch information (e.g., the timespecific numbers of animals harvested) as a means of estimating population size. Estimators based on both approaches tend to be relatively imprecise unless the harvest represents a substantial proportion of the population. Nevertheless, in the absence of independent monitoring programs, such efforts to estimate population size using information from the harvest may be essential to the success of harvest management programs. Chapters 15 through 20 use variations of the capturerecapture models introduced in this chapter. Unlike population size, which can be estimated with direct counts as well as capture-recapture methods, estimation of demographic rate parameters such as move-
332
Chapter 14 Mark-Recapture Methods for Closed Populations
ment and survival rates typically requires the use of marked individuals. Chapters 17 and 18 thus use capture-recapture modeling for open populations to estimate abundance, survival, and movement. Notwithstanding the need to incorporate additional parameters in models for open populations, the underlying approach with these models is similar to that introduced here for closed models. Chapter 19 de-
scribes a "robust design" in which both open and closed models are used in a single study design, wherein the closed models in this chapter are components of larger, more inclusive models. Finally, in Chapter 20, the closed models of this chapter and the robust design of Chapter 19 are used to estimate parameters at the community level of biological organization.
C H A P T E R
15 Estimation of Demographic Parameters
15.1. DETECTABILITY AND DEMOGRAPHIC RATE PARAMETERS 15.1.1. Population Growth Rates 15.1.2. Survival Rates 15.1.3. Movement Probabilities 15.1.4. Reproductive Rates 15.1.5. Summary 15.2. ANALYSIS OF AGE FREQUENCIES 15.2.1. Life Tables 15.2.2. Survival Estimation from Sample Age-Structure Data 15.2.3. Population "Reconstruction" 15.3. ANALYSIS OF DISCRETE SURVIVAL AND NEST SUCCESS DATA 15.3.1. Binomial Survival Model 15.3.2. Models for Estimating Nest Success 15.3.3. Radiotelemetry Survival and Movement Studies 15.4. ANALYSIS OF FAILURE TIMES 15.4.1. Statistical Models for Failure Time, Survival Time, and Hazard Rate 15.4.2. Parametric Survival Estimation 15.4.3. Nonparametric Survival Estimation: Kaplan-Meier 15.4.4. Incorporating Explanatory Variables: The Proportional Hazards Model 15.4.5. Assumptions of Failure Time Models 15.4.6. Design of Radiotelemetry Studies 15.5. RANDOM EFFECTS AND KNOWN-FATE DATA 15.6. DISCUSSION
gued in Part II and elsewhere that a focus on abundance and density is both useful and natural, in that these quantities are often the state variables of interest in models of population dynamics. However, the investigation of population dynamics frequently is not restricted to an assessment of population size alone. Depending on study objectives it is useful, and often essential, to include information about the biological processes that influence population dynamics. In Chapters 15-19 we turn our attention to the rates of survival, reproduction, and movement that ultimately are responsible for changes in abundance. An emphasis on estimation of demographic parameters, and on quantifying variability in these parameters, is important for several reasons. First, estimates of abundance at a single point in time obviously provide no information about population dynamics, though a series of estimates of abundance may provide insights about the trajectory of the population. However, even a time series of abundance estimates provides only limited information about which demographic processes contribute to the observed dynamics and thus about why the population behaves as it does. Second, demographic rates provide a more detailed picture of the "health" of the population and, when used with population models (Chapter 8), may be useful in forecasting future population growth. Third, most of wildlife management seeks to control populations at desirable levels, which in turn requires an understanding of the factors that influence survival and reproduction rates. Although management objectives frequently are expressed in terms of population size, management actions often focus on the control of demographic pa-
In Chapters 12-14 we described methods for estimating abundance or density of a population. We ar-
333
334
Chapter 15 Estimation of Demographic Parameters
rameters associated with survival, reproduction, and movement in order to bring about desired changes in abundance. An understanding of how these demographic parameters vary in space and time, and in relation to environmental and management factors, is fundamental to the understanding and proper management of animal populations. With respect to estimation methodology, the methods of Chapters 12-14 were based on the assumption that the population is both geographically and demographically closed. By geographic closure is meant that (1) the population is immobile or (2) the geographic area or time scale of the study is such that movements into and out of the population need not be considered. By demographic closure is meant that neither births nor deaths occur over the period of study (or the numbers of births and deaths are negligible). Taken together, these assumptions imply that abundance is constant and can be represented by a single parameter N over the course of the investigation. We note that the creative use of, e.g., stratification in space and time sometimes allows one to use the methods of Chapters 12-14 even if the assumption of closure is somewhat relaxed. In this and the next several chapters, we remove the assumption of closure altogether and explicitly estimate the demographic rate parameters associated with population dynamics, i.e., rates of survival, reproduction, and movement. In this chapter we begin a general exposition of methods for estimating both abundance and demographic parameters for open populations. The methods described here for investigation of demographic parameters follow the same principles that guide the development of estimation methods for closed populations. Thus, the methods (1) are based on sound statistical estimation and sampling procedures, (2) rely on as few assumptions as possible, with sampling and estimation schemes that are robust to assumption violations, and (3) make effective use of limited resources for sampling and estimation. We start with a discussion of general principles, emphasizing the importance of detectability in the estimation of demographic rate parameters. We then discuss several methods that require assumptions of perfect detection. In particular, age frequency analyses are described under the rubric of "life table analysis." We then cover methods of analysis of discrete nest success and survival data, with sampling methodologies that are designed to meet the assumption of perfect detection (for example, by means of radiotelemetry or the monitoring of sessile objects such as nests). Finally, we describe methods for analysis and modeling of failure times, the principal applications of which involve the analysis of data from radiotelemetry studies.
15.1. DETECTABILITY
AND DEMOGRAPHIC RATE PARAMETERS In Section 12.2 we described a canonical estimator for abundance that incorporated two main sources of variation in animal count data, namely, spatial variation and detectability. Both sources are relevant to the estimation of demographic rate parameters, though detailed studies incorporating estimation of rate parameters frequently concern populations at single locations and thus do not involve spatial sampling. Exceptions include programs such as Monitoring Avian Productivity and Survival (MAPS) (DeSante et al., 1995) and the North American waterfowl banding program (Anderson and Henny, 1972; Nichols, 1991a). These large-scale monitoring programs involve estimation of demographic rate parameters at regional and national scales, though point estimates of the rate parameters are obtained at the level of the local sampling unit. If the selection of sampling units is based on an appropriate sampling design (see Chapter 5), these point estimates can be combined to form an estimate that corresponds to the entire area of interest using the approaches of classical sampling theory (e.g., Cochran, 1977; Thompson, 1992) (see also Chapter 5). In the remainder of this section we defer further discussion on spatial variability and focus instead on the much more frequently encountered problem of detectability. Demographic rate parameters include descriptors of overall population change (e.g., population growth rates such as the finite rate of population increase), as well as fundamental demographic parameters, such as rates of survival, reproduction and recruitment, and movement, that are responsible for population change. Estimates of these rate parameters, like those of abundance, nearly always are based on some sort of count statistic and thus require one to account for detectability. Before proceeding to detailed estimation methods for estimating these parameters, we provide a brief motivation for the need to consider detectability in their estimation.
15.1.1. Population Growth Rates Because population growth rate is a function of abundance at two or more points in time, it should be clear from Chapters 12-14 that detectability is an important consideration in its estimation. To see how, define the finite rate of population increase for a population of interest as the ratio of abundances in successive time periods:
Xi-- Ni+l/Ni.
15.1. Detectability and Demographic Rate Parameters As in Section 12.2.1, define C i a s the count statistic (number of animals detected by the survey method, e.g., capture, visual observation, and auditory detection) and ~i a s the associated detection probability (probability that a member of N i is detected and thus appears in Ci). The count can be viewed as a random variable, with expectation given by (15.1)
E(Ci) = N i ~ i.
One approach advocated by many biologists is to view Ci as an index (Section 12.7) and thus to use the ratio (15.2)
~i = C i + l / C i
of counts as an estimate of k i. The expectation of this estimator can be approximated as
periods, then Eq. (15.3) is recommended. A related discussion is presented in Section 14.1.2 on estimating relative abundance under partial detectability (also see Skalski and Robson, 1992).
15.1.2. Survival Rates Consider a study in which R i animals are caught, marked, and released at time i, with a goal of estimating the probability that a member of R i survives until i + 1 (denote this survival probability as q~i). Denote as M i+1 the number of marked animals (members of R i) that are still alive and in the population of interest at time i + 1. This number can be modeled as a binomial random variable [i.e., Mi+ 1 "" Bin(R/, q~i)], SO that the proportion of survivors estimates q~i: ~Pi -- M i + I / R i .
E(~i) ~ E(Ci+I)
E(Ci) Ni+l~3i+l Ni~i
9
From this approximate expectation it can be seen that the ratio of count statistics provides a reasonable estimator for ~ki only if detection probability does not change over time, i.e., only if f~i+l ~ ~i. The bias in estimator (15.2) is a function of the difference between the two detection probabilities, with larger differences leading to more biased estimates. Even if detection probability is viewed as a random variable, the equality E(~i+I) = E(~3 i) still is necessary for the index-based estimator for Xi to be approximately unbiased. Two reasonable approaches for estimating Ki require the collection of data needed to estimate the detection probability ~i associated with count statistic Ci. The first approach is simply to estimate abundance as advocated in Chapter 12, i.e., 1~i -- Ci/~ir
Although some sampling designs permit direct knowledge of Mi+ 1 (see Sections 15.3 and 15.4), this situation is relatively rare. A more typical situation is that a sample of the population at time i + 1 detects mi+ 1 members of Mi+ 1. In this situation mi+ 1 is simply another count statistic that follows the usual relationship described in Eq. (15.1): E(mi+l) = Mi+lPi+ 1
(here we use Pi rather than ~i to characterize detectability, in keeping with the common use of Pi in capturerecapture literature). Because of the inequalities Pi+l ~ 1 and mi+ 1 ~ M i + l , the naive estimator mi+l/ai
~ ~i
based on the count statistic nearly always is biased low (unless mi+ 1 -- Mi+I). However, if the detection probability associated with m i+ 1 can be estimated, then reasonable estimators for Mi+ 1 and survival can be constructed as ]~Ii+l = mi+l/fii+l
and then to estimate ~-i a s Ki -- /~/i+1/1~i"
335
(15.3)
This approach is conservative in the sense that the estimator requires no restrictive assumptions about the detection probabilities f~i. The other approach is first to test for differences between the detection probabilities for the two time periods (H0: f~i+l = [3i) and, if no evidence of a difference is found, then estimate ~ki as in Eq. (15.2). Because of the assumption of equal detection probabilities underlying the latter approach, it tends to be more precise than Eq. (15.3) (see Skalski and Robson, 1992). However, if test results provide evidence of a difference in detectability ~i for the two time
and ~i -- ] ~ i + l / R i .
Detection probability is thus an important consideration in the estimation of survival probability in field studies.
15.1.3. M o v e m e n t Probabilities Consider a study in which Rli animals are marked and released in sampling period i at location 1 in a system of two habitat patches. Assume that with probability S li these animals survive and remain in the
336
Chapter 15 Estimation of Demographic Parameters
study system until sampling period i + 1. Denote the total number of survivors as M]~_1 [as above, this is a binomial random variable, M~_ 1 --- Bin(R~, S~)], with 11 M i + 1 located in patch I at time i + 1 and M ] 2 1 i n patch 2. Thus, 11
1.
two patches. If P~+I and p2+1 can be estimated, then movement probability can be estimated as "12 = M" i1+2 1 / M ~ + I , t~i+l
where
12
M i + 1 = M i+ 1 + M i+ 1 .
Denote as t~ 2 the probability that a surviving member of R~ moved from patch I to patch 2 during the interval i to i + 1 and is thus present in patch 2 at i + 1 [M12i+1 is a conditional binomial random variable, with M~21 " Bin( MIi+I, ~]2)]. If the numbers Mi+111 and 12 M i + 1 of surviving members of R~ at period i + 1 are known, then the movement probability can be estimated as ~ 2 __ Mi+112/MJ~_I 9
(15.4)
Equation (15.4) is reasonable when all surviving animals can be detected at i + 1 (i.e., if detection probabilities are equal to 1), as in some radiotelemetry studies (Nichols, 1996; Nichols and Kaiser, 1999; Bennetts et al., 2001). However, the more frequently encountered situation involves sampling that records mi+ 112 animals 111 animals reto have moved from patch 1 and mi+ maining in patch 1. These animals are detected with probabilities p2+1 and Pi+l 1 for the two patches. Once again, the numbers of detected animals are count statistics (see Section 15.1.1), and their expectations can be written as functions of the numbers of animals in the two patches and the associated detection probabilities: E(mll 11 1 i+1) = Mi+1P1+1
" 11
11
and
15.1.4. Reproductive Rates Reproductive rate frequently is defined as the number of young animals at time i + & that are produced by an adult at time i, with A typically a relatively small time step. For example, reproductive rate for mallard ducks might be defined as the number of young fledged female mallards in August of year i per adult female mallard in the breeding population in May of year i (at the approximate time of breeding). Age ratio at a particular time of the year often is used to approximate or index reproductive rate. Thus, the number of young mallards per adult in August of year i is used to index the reproductive rate of year i (e.g., see Anderson, 1975a; Martin et al., 1979; Johnson et al., 1997). If N (~ i and NI 1) are the true numbers of young (age = 0) and adult (age = 1) animals in the population at time i, then we can define the age ratio as Ai
9~12 2 -- IVIi+lPi+I.
A naive estimator for t~ 2 frequently is constructed
N ~-i! ~
E(n!%, , = NI~
(1))
12
12
2
Mi+lPi+I
11
i
9
(~
and
12 / m 1. mi+l i+1,
where mi+ 11" = mi+111 + mi+1,12 with expectation approximated by E(t~ 2)~
1
Mi+I = mi+l/Pi+l.
as ~2=
12
Of course, we seldom know the numbers of animals in any age class in the population; instead, the population must be sampled to obtain the numbers n~~ and n! 1) of young and adult animals detected at time i. The expectations for these random variables are
and
E(m i+1) 12
" 12
M i + 1 = mi +l / ~2 +l
1
Mi+lp2+I + Mi+1P1+1
9
E(n i
(1)p(1)
i ,
where P i(0) and pl 1) 9 are the age-specific detection probabilities associated with the count statistics. A naive estimator for age ratio is constructed as ai
It is clear from this approximate expectation that the naive estimator is biased if Pi+I 1 ~ p2+1. Thus, the move1 < p2+1 and ment probability is overestimated for Pi+l underestimated f o r P~+I > p2+1. As in Sections 15.1.1 and 15.1.2, the ability to estimate movement probability thus depends on the estimation of detection probabilities associated with the
= Ni
=
(0)-
ni
/ni
(1)
9
However, the approximate expectation of this estimator can be written as E(ai) ~ Xl~176
,~. i
9
and thus is a function of not only the actual age ratio a i but also the ratio of age-specific detection probabili-
15.2. Analysis of Age Frequencies ties. As with the previous sections, the naive estimator m a y not perform well if pl 1) :~ Pi(2) 9 Estimation of the detection probabilities permits unbiased estimation of the true n u m b e r s of animals in each age class. Thus, one can use /~!o) ,
=
n
I0)
/fii
(o)
and (1)/ (1) /~I 1) = ni -fii , to estimate the age ratio as Ai
/~/!0)//~(1)
15.1.5. Summary
Like the estimators of abundance, estimators of demographic rate parameters typically are based on count statistics and thus are functions of both n u m b e r s of animals and the detection probabilities associated with sampling. Naive estimators of demographic rate parameters typically are constructed as ratios of count statistics and therefore are biased unless detection probabilities are either equal to 1 (as with survival rate estimators) or are equal for different groups of animals (as with rates of increase, m o v e m e n t probabilities, and reproductive rates). Like abundance estimation, a key to estimation of demographic rate parameters is to collect the data needed to estimate detection probabilities associated with the count statistics. These data permit the testing of critical assumptions that underlie the naive estimators. If testing provides evidence that the assumptions are indeed true, then the estimators based solely on count statistics m a y perform well (see Skalski and Robson, 1992). If the tests fail to provide such evidence, the investigator should use estimators that directly incorporate detection probabilities. In either case, the key to successful estimation of rate parameters is to obtain the data needed to make inferences about detection probability.
337
a set of parameter values, these models can be used to project the numbers of animals in each age class through time. In this section we address essentially the reverse problem, i.e., to make inferences about demographic parameters, particularly survival rates, given the observed fates of cohorts of individuals, patterns in age structure, or combinations of both. The types of data used for these inferences are organized in a format generically k n o w n as a life table. As seen below, u n d e r certain circumstances, life tables can be used to obtain valid estimates of survival or other parameters. Using the notation of Section 8.4 we consider a population consisting of k age classes, with population growth according to a birth-pulse model. Start with an assumed cohort of birth-class individuals at N0(0). The n u m b e r of individuals in this cohort that survive to subsequent ages can be obtained by repeated application of
Ni+l(t + 1 ) = Si(t)Ni(t),
where Si(t) is the survival rate from time t to time t + 1 of individuals in age cohort i at time t. Over the cohort's full life cycle, age-specific survival is given by
Si(t) = Ni+l(t + 1)/Ni(t).
15.2.1. Life Tables
In Chapter 8 we considered population models that incorporate age structure, whereby the projection of population growth is a function of age-specific survival and reproduction rates (see Section 8.4). In these models the transition of age cohorts through time is a function of fixed survival and reproduction parameters. A s s u m i n g an initial population age structure and
(15.6)
These calculations are illustrated in an artificial example from Seber (1982) and presented in Table 15.1. In the example, N0(0) = 1000 animals are followed from birth at t = 0 until all have died. Thus, survival from birth to age 1 over (0, 1) is S0(0) = NI(1)/No(O) = 250 / 1000 = 0.25. On the other hand, a different cohort of 1200 animals, born in the next year (i.e., at age i = 0 in year t = 1),
TABLE 15.1
Example Cohort Life Table a
Cohort t = 0 15.2. A N A L Y S I S O F AGE FREQUENCIES
(15.5)
Cohort t = 1
Year (t)
Age (i)
[Ni(t)]
Si(t)
Age (i)
[Ni(t)]
Si(t)
0 1 2 3 4 5 6
0 1 2 3 4 5 6
1000 250 40 10 3 1 0
0.25 0.16 0.25 0.30 0.33 0.00 m
m 0 1 2 3 4 5
__ 1200 400 125 50 40 30
m 0.33 0.31 0.40 0.80 0.75
aAlso known as age-specific (horizontal) life table (Seber, 1982).
Chapter 15 Estimation of Demographic Parameters
338
has age-specific survival calculated over the interval (1,2) as
It follows that
= 400/1200 = 0.33. Age-specific survival m a y or m a y not be the same for different time intervals. The example in Table 15.1 illustrates a case in which it is not the same, i.e., survival is both age specific and cohort specific. If the n u m b e r s surviving in each of a series of cohorts are available, one can determine survival over each interval (t, t + 1) for each age class i and thus separate temporal (cohort-specific) variation in survival from age effects. In practice, multiple-cohort data are seldom available, and assumptions must be m a d e about the nature of age or cohort specificity in order to estimate parameters uniquely (Udevitz and Ballachey, 1998). These assumptions have serious implications as to the generality of life table approaches, as illustrated below and more fully in Section 15.2.2. Information on fates from a series of cohorts is sometimes called an age-specific or horizontal life table and is obtained in one of two ways: either by recording the ages of all the individuals at death (d x series) or by recording the numbers still alive at each time (age) x (l x series) (Seber, 1982). Both types of data m a y be difficult to collect and in practice both are obtained via sampling procedures that m a y lead to serious bias, as discussed further in Section 15.2.2. On assumption that the population is (1) at a stable age distribution and (2) stationary (i.e., K = 1), it m a y be possible to use the standing age distribution, also k n o w n as a time-specific or vertical life table, to calculate age-specific survival rates. To see w h y these assumptions are needed, consider the age distribution Ni(t), i = 1, ..., k for a single year t. To obtain the vertical life table estimate of age-specific survival one calculates the ratio of successive age frequencies at the same time t: S*(t) = N i + l ( t ) / N i ( t ) .
(15.7)
The numerator of Eq. (15.7) can be expressed via Eq. (15.5) as Ni+l(t) = S i ( t -
1)ci(t-
1 ) N ( t - 1),
(15.8)
where ci(t) = N i ( t ) / N ( t ) is the proportion of the entire population at t in age class i. Substitution of Eq. (15.8) into Eq. (15.7) then produces S*(t) =
Si(t-
1 ) N ( t - 1) ci(t)N(t)
1)ci(t-
1)
S*(t) = S i ( t -
S0(1) = N 1 ( 2 ) / N o ( 1 )
only w h e n the population is both stable [i.e., ci(t) = ci(t - 1) = c i for all i] and stationary [N(t) = N ( t - 1)], which in turn requires stationary age-specific survival: Si(t) = S i ( t - 1 ) = S i. Example
Assume that a population is both stable and stationary, with 1000 individuals entering the population each year (Table 15.2). Under conditions of stationarity and stable age distribution, the same n u m b e r of individuals is in each age class each year, and the vertical age structure is constant over time. By year 5 the n u m b e r of individuals from an initial cohort of 1000 that are still alive each year is fully described, and the vertical and horizontal life tables have converged. Thus, the standing age distribution is an accurate representation of survival rates from each of the original cohorts, and age-specific survival rates from Eq. (15.7) are accurate (Table 15.2a). N o w relax the assumption of stationarity to allow a stable age distribution but nonstationary growth. With an increasing population (Table 15.2b), the calculations from Eq. (15.7) no longer faithfully represent survival, but instead are distorted by the increasing population size [N(t) = N ( t - 1)M. As a result, Eq.
TABLE 15.2 Relationship between Cohort (Horizontal) and Time-Specific (Vertical) Life Tables Yea~
Age
1
2
3
4
5
1000 250 40 10 3
1000 250 40 10 3 0
1728 360 48 10 3
2074 432 58 12 3 0
Stable, stationary~ 0 1 2 3 4 5
1000
1000 250
1000 250 40
Stable, nonstationaryb 0 1 2 3 4 5
1000
1200 250
1440 300 40
aStable age distribution with stationary population. bStable age distribution with nonstationary population (~ -- 1.2).
15.2. Analysis of Age Frequencies (15.7) underestimates survival by 1/)t. For example, in year 4 S~(4) = 360/1728 = 0.21, whereas actual survival (constant for all years, because this population is at stable age distribution) is S0(4) = SO = 0.25. In cases where the population is not at a stable age distribution, there is no guarantee that the survival rates based on Eq. (15.7) will be reliable even as indices. In general it is not possible to use age frequency data alone to both estimate age-specific survival and to test the assumptions of age stability and stationarity (Seber, 1982). The assumption of stationarity may be relaxed if independent data on population growth rate ()~) or age-specific reproduction rates (F x) are available to allow estimation of age-specific survival rates (Caughley, 1966; Michod and Anderson, 1980); however, the assumption of age stability is still required. If age distributions are recorded for a number of years and do not appear to be temporally varying, it may be possible to infer age stability and to compute survival estimates from the standing age distributions, again provided that estimates of )~ or F x are available. Finally, we note that Eq. (15.7) assumes known age distributions, even though information on age distributions typically is obtained via sampling methods with agespecific detection probabilities (Section 15.1.4), leading to additional problems. Unfortunately, age distribution methods are in common use, with little heed paid to these critical assumptions. In keeping with the general philosophy of this book, we strongly recommend the use of methods such as radiotelemetry (Sections 15.3 and 15.4) and capture-recapture (Chapters 16-18), which do not require assumptions such as age stability or stationarity that are unlikely to be met in practice, particularly for populations that are harvested or are subject to environmental variation.
15.2.2. Survival Estimation from Sample Age-Structure Data Though the development above is strictly deterministic, it can be extended to allow for the stochastic nature of birth-death processes, still under the assumption that either a complete accounting of the fates of all cohorts (horizontal approach) or of the entire age profile (vertical approach) is available. The usual situation involves data that arise from both a stochastic demographic process and a sampling process. Seber (1982; also see Seber, 1986, 1992) reviewed models that deal with one or both processes and provided estima-
339
tors based on either horizontal or vertical data structures. Most of these models, though of historic interest, are not considered in a general framework such as maximum likelihood estimation and depend to varying degrees on assumptions that often cannot be evaluated. Udevitz and Ballachey (1998) provided a unified framework for survival estimation from agestructured data, which allows for maximum likelihood estimation, model selection, and model evaluation, utilizing sample data from standing age distributions and ages at death. The development below is based on their framework, with modifications for notational consistency. In what follows, likelihoods are developed separately for each type of data structure, under the very general assumptions that the age structure may not be stable and population growth rates are unknown. From Eq. (15.8) the number of individuals in age class i - 1 at time t - 1 that survive to t is given by Ni(t) = N ( t -
1)Ci_l(t-
1)Si_l(t-
1).
(15.9)
By subtraction, the number of individuals in age class 0 at time t is k
No(t) = N ( t ) - N ( t -
1) ~ C i _ l ( t -
1)Si_l(t-
= N(t-
[
1) (15.10)
i=1
k
1) M t - 1) - ~ C i _ l ( t - 1)$i_1(t- 1)
1
i=1
[because )~(t - 1) --- N ( t ) / N ( t - 1)]. From Eq. (15.9) it is easy to see that the number of individuals in age class i that die between times t - 1 and t is N(t-
1 ) c i ( t - 1)[1 - S i ( t -
1)].
(15.11)
15.2.2.1. Model Likelihoods
Assume that we have a random sample xi(t), i = O, ..., k, from the age frequencies in the population at time t. Then by Eqs. (15.9) and (15.10)
I
k
]
E[xo(t)] = oLN(t - 1) )t(t - 1) - ~ Ci_l(t -- 1)Si_l(t - 1) i=1
and E[xi(t)] = e ~ N ( t -
1)Ci_l(t-
1)Si_l(t-
1)
for 0 ~ i -< k, where oL is the probability of sampling any individual from the population age distribution, assumed to be independent of age. Conditional on
340
Chapter 15 Estimation of Demographic Parameters
the total sample size n(t), a multinomial likelihood for these data is given by
P[{xi(t)}ln(t)l =
n(t)!
I-Iik xi(t) ! X(t
1/
k(Ci_l(t- 1)Si_l(t- )xi(t) X ( t - 1)
where n(t) = ~ i xi(t). Note that the conditioning on n(t) removes the need to consider o~ in the likelihood. Assuming a r a n d o m sample of natural deaths between t - 1 and t, from Eq. (15.11) we have
E[yi(t)] = f 3 N ( t - 1 ) c i ( t - 1)[1 - S i ( t -
Si -~ Yi+l(t))k xi(t ) ,
(15.12)
1)
x ~.=
Under the assumption of age stability, the time index for the parameters is eliminated from both likelihoods. Thus simplified, the MLE for age-specific survival can then be obtained from the sample standing age data
as
k I)k(t -1) - ~,i=l Ci_l(t -1)Si_l(t-1)l x~ x
15.2.2.3. Known Stable Age Distribution
1)],
where yi(t) is the n u m b e r of animals of age i at time t - I that die between t - I and t and [3 is the probability of sampling any individual from the population of ages at death, assumed to be independent of age. The conditional likelihood for the ages at death is
(15.14)
i = 0, ..., k - 1, where ~ is known. Under the special case of X = 1, Eq. (15.14) is the naive estimator of survival from age distribution data; the more general estimator has been described by Caughley (1977), among others. On condition that X has a k n o w n value that is not unity, Udevitz and Ballachey (1998) provide an estimate of variance for this estimator using the delta method (see Appendix F): va'~r(Si) = [C32/n(t)] [1/c,i(t) + 1/c,i+l(t)],
(15.15)
i = 0, ..., k - 1, where
k n(t) = ~, xi(t) i=0 and
p[{yi(t)}lm(t) ] =
k I-[
m(t)! i=0 yi(t)! i=0
IF
?,i(t) = xi(t)/n(t). (15.13)
[ c i ( t - 1)[1 - S i ( t - 1)] I y;(t) x L~ki=oCi( t _ 1)[1 - S i ( t - 1)] with m(t) = ~ i yi(t), where again the sampling probability [3 disappears because of this conditioning. If independent data are available from both a standing age distribution and ages at death, a joint likelihood is formulated as the product of Eqs. (15.12) and (15.13) (see Udevitz and Ballachey, 1998).
15.2.2.2. Parameter Estimation The parameters for either of the above likelihoods are the population growth rates X(t), age-specific survival rates Si(t), and age class proportions ci(t). In the usual case where both the survival rates and growth rates are assumed to be time independent [i.e., Si(t) = Si; X(t) = ~] there are still 2k+1 parameters to estimate under either data structure. These parameters are not identifiable without additional assumptions. The usual assumptions are (1) that the age distribution is stable and (2) that ~ is known. Udevitz and Ballachey (1998) show that if both data structures are used with the joint likelihood [product of Eqs. (15.12) and (15.13)], then these assumptions can be relaxed one at a time.
For )~ = 1 this expression simplifies to v~r(Si) = (C32/n) (1/ci
-}- 1/ci+1),
(Seber, 1982), because the assumption of stationarity allows one to estimate age-specific survival from a single age frequency distribution. If ~ is independently estimated rather than known, Eq. (15.15) must be modified by adding the term
(Si/~)2 var(~). Similarly, the MLEs for age-specific survival from the ages-at-death data are
Si- 1 -
yi(t))k i
k
~'j=i yj(t) xj
,
(15.16)
i = 0, ..., k - 1, and are the usual estimators (e.g., Caughley, 1977) w h e n X is known. A variance expression for this estimate is provided by Udevitz and Ballachey (1998):
v~r(Si) = ~/2m(t)di(t)2~ j=i+l
[ ~tj(t)[1-~lj(t)]xai+2j~]2
+ 2~tj(t))ki+J1
2~i(t)2 k-1 ~ clj(t)~tl(t))k2i+j+! ~/4m(t)j=i+l l=j+l + 82di(t)[1 - di(t)] m(t) '
(15.17)
15.2. Analysis of Age Frequencies i = 0, ..., k - 1, where
Section 4.3.4). Other comparisons (e.g., between a model assuming known K but not stable age distribution, against a model with stability alone; models in which survival rates are not age specific) cannot always be tested by likelihood ratio because nonnested models are involved. For these situations, AIC or other criteria can be used to discriminate among models.
k
m(t) = ~,. yj(t), ]=1 k
y = s
341
dj(t)aJ,
j=i
and ~.i y
15.2.2.5. Assumptions about Sampling Effort
Cli( t ) ~ 2i y2 "
Again, if the value of a is independently estimated rather than known, then Eq. (15.17) must be adjusted by adding the term 2
[1-Si
k
i)]
var (~).
When both data structures are available and the joint likelihood (simplified by assuming stability) is used, both )t and the k - 1 survival rates can be uniquely estimated via numerical optimization of the joint likelihood. The maximized likelihood obtained by this approach then can be compared via likelihood ratio tests to a product likelihood based on specific values for growth rates ()~ = ~0; e.g., )~0 = 1).
15.2.2.4. Age Stability Unknown If both data structures are available the joint likelihood formed by the product of Eqs. (15.12) and (15.13) is maximized by
S i ( t - 1)= Ci+l(t)h(t- 1)
(15.18)
~i+l(t)Mt - 1) + di(t){1 - Mt - 1)[1 - ~0(t)]}' i = 0, ..., k - 1, where
~,i(t) = xi(t)/n(t ) and
cli(t) = yi(t)/m(t), i = 0.... , k, provided the K(t) values are known. The variance of this estimator is quite complicated and is not presented here [the interested reader is referred to Udevitz and Ballachey (1998), Appendix B]. The model is saturated, that is, there are no degrees of freedom for a goodness-of-fit test. Stability in the age distribution can be tested by constraining the parameters of the product likelihood from Eqs. (15.12) and (15.13) to be equal over time and by comparing this maximized likelihood to the unconstrained product likelihood (see
The above expressions for the likelihoods of the age frequency data, ages-at-death data, and combined data structures make it clear that strong assumptions are invoked regarding the sampling process. The principal assumption is that sampling probabilities (c~, ~) in the likelihoods are constant over time and among age classes, which is required to allow parameter identifiability under any of the data structures. In practice this assumption is likely to be violated, particularly in cases where the standing age structure is obtained from a harvested sample. Heterogeneity in the rate of harvest should be expected a priori. For example, younger age classes of gamebirds typically are more vulnerable to harvest, and fishing gear often is configured so as to exclude fish below or above certain size limits. Under these conditions the sample age frequencies likely do not reflect the population age structure. Sometimes auxiliary data are available to provide independent estimates of age-specific sampling rates (e.g., relative vulnerability to harvest) (Martin et al., 1979; Miller, 2000a), and these data can be used to adjust the sample frequencies accordingly. Of course, it then is necessary to incorporate the additional component of sampling error in the estimated sampling rates used to estimate survival and age distributions [as in the case of incorporating estimates of ~ in Eqs. (15.15) and (15.17)]. All too often, unadjusted standing age frequencies or ages at death are used without critical evaluation of the underlying assumptions, including that of homogeneous sampling from the population. Age data are relatively easy to collect, and a multitude of estimators and models are available that will produce apparently reasonable estimates. It is likely that many, perhaps most, uses of these estimators are based on unverified assumptions and thus are of dubious reliability.
Example This example is from a sample of ages for moose
(Alces alces) harvested in N e w Brunswick during 1980-1984 (Boer, 1988). The authors used auxiliary data from aerial surveys and analysis of a sequence of harvest age ratios to support their claim that the assump-
342
Chapter 15 Estimation of Demographic Parameters
tions of a stable and stationary population are warranted. We have analyzed their data according to Eqs. (15.14) and (15.15), setting )~ = 1. The results are reported in Table 15.3 and are similar to those reported by Boer (1988) (which were reported as l x rather than Sx estimates). Note, however, that the precision of the estimates is poor, with reasonably narrow confidence intervals only for the first few age classes. A reanalysis of this problem under a constrained model involving fewer age-specific estimates, or a parametric form for patterns in age-specific survival, might improve these results. We note that the data in Table 15.3 are not "pure" age frequencies (as suggested by their noninteger values) but are in fact adjusted frequencies based on estimates of age-specific vulnerability to harvest (Boer, 1988). Because they likely are sample-based estimates, an additional component of sampling variability should be included in the variance terms for the survival estimates. Boer (1988) alludes to harvestand survey-based estimates for determining age stability and stationarity, and these components of variability contributed to the sampling variances of the estimates, but were unaccounted for in the variance computations. These remarks are not made as a criticism of the study, but rather to point out the difficulty of using age frequency methods, even under
T A B L E 15.3 Age class
the best of circumstances, to provide reliable estimates of survival rates.
15.2.3. Population "Reconstruction" Population reconstruction is a technique for calculating the size and age composition of a cohort at some initial time from subsequent mortalities of the population. A rationale for the method is "if an animal was killed in a given year at four years of age, then it was a three-year-old the previous year, a two-year-old two years earlier .... and a fawn four years earlier" (McCullough et al., 1990). In theory, if all the mortalities in the population can be observed (e.g., all deer are killed by hunters, and all hunter kills are reported), then an accurate picture of the population can be reconstructed and sometimes is referred to as a "virtual population." The "data" thus reconstructed then are used in population models and statistical procedures, for example, to calculate estimates of age-specific survival and population growth rates. The method rests on several assumptions that we believe are unlikely to be tenable in practice, with potentially grave consequences in terms of the reliability of the "estimates." The claim sometimes is made that reconstructed populations correspond to a "minimum known alive" population, a
Estimation of Age-Specific Survival Rates for M o o s e in N e w Brunswick a
Frequency
ci
Si
S'E(Si)
Li
lIi
0
128.82
0.281
0.676
0.094
0.492
0.860
1
87.10
0.190
0.697
0.116
0.468
0.925
2
60.67
0.133
0.711
0.142
0.434
0.989
3
43.15
0.094
0.728
0.171
0.393
1.063
4
31.41
0.069
0.746
0.204
0.347
1.145
5
23.44
0.051
0.771
0.241
0.298
1.244
6
18.07
0.039
0.776
0.276
0.235
1.318
7
14.03
0.031
0.802
0.321
0.173
1.431
8
11.25
0.025
0.820
0.364
0.106
1.535
9
9.23
0.020
0.841
0.409
0.038
1.643
10
7.76
0.017
0.870
0.458
-0.027
1.767
11
6.75
0.015
0.874
0.493
-0.091
1.840
12
5.90
0.013
0.903
0.540
-0.155
1.962
13
5.33
0.012
0.925
0.578
-0.208
2.058
14
4.93
0.011
457.84
1.000
Total
a Based on age distribution in the harvest (Boer, 1988). Parameter ci is the proportion of the population in age class i, and Si is the survival probability for age class i; ~/i and /~i are u p p e r and lower confidence limits, respectively.
15.3. Analysis of Discrete Survival and Nest Success Data notion that is similar to one invoked (and rebutted) in the case of capture-recapture sampling (Jolly and Dickson, 1983; Nichols and Pollock, 1983b; Pollock et al., 1990; Efford, 1992). In most cases the basis of the reconstruction is the harvest of known-age animals. There are at least two serious difficulties with the use of these data. First, even if all harvest mortality (legal and illegal) can be accounted for, the method will exclude deaths due to other mortality sources. To the extent that these constitute a significant fraction of mortality (which is usually unknown), the harvest-based reconstruction will produce an increasingly skewed picture of the population through time. Second, harvest is almost certainly biased toward certain age and sex components of the population, further distorting the relationship between the data and the actual population structure. The only scenario in which one might expect population reconstruction to provide an accurate picture of population structure involves a random sample of the population that has been destructively sampled (deliberately, or as the result of a catastrophe), thus providing an accurate ages-at-death sample. However, this would not allow reconstruction of even one cohort, unless it could be repeated through time. Even with this sort of sampling we advocate the use of estimation methods [e.g., Eq. (15.14)] that utilize statistical likelihoods and are based on clear (and testable) assumptions. We strongly discourage the use of "virtual data" from reconstruction as if they are actual data, for the purpose of statistical estimation of demographic parameters. For instance, harvest data could be used to reconstruct the number of animals alive in each age class in each of several previous cohorts, and these data in turn used to calculate age-specific survival rates via, e.g., Eqs. (15.14) and (15.16). This approach is flawed on three grounds: (1) the quantities used in the "estimates" of survival were never observed; they were inferred from a model of the population; (2) all biases inherent in the reconstruction will propagate in the estimates; and (3) even if the assumptions of reconstruction can be met, estimates of sampling variation in these estimates will not take into account sampling error in the harvest or other data on which the reconstructed "data" are based. It is a common practice to impose assumptions about mortality (especially natural mortality) and other demographic rates, in order to make reconstruction provide "reasonable" virtual populations (e.g., account for nonharvest losses). It also is unfortunately common for the resulting "virtual data" to be used to "estimate" these same population parameters or to test the assumptions (e.g., of age sta-
343
bility) thus imposed, an exercise in circular reasoning whose futility should be obvious to the reader.
15.3. A N A L Y S I S OF D I S C R E T E SURVIVAL AND NEST SUCCESS DATA In contrast to some of the approaches described in the previous section, designed studies that include maximum likelihood estimation methods and statistical models that account for random variation in the data offer a statistically reliable alternative for inference about the demographic parameters of a population. Here we describe methods that are appropriate when subjects can be visited repeatedly in the course of an investigation, as in the monitoring of nests at known locations or animals that are radio marked. We assume initially that the fates of individual subjects in the study can be determined with certainty during the study, i.e., their probabilities of detection are 1. We later will include features that accommodate the censoring of individuals, and in subsequent chapters the assumption of perfect detectability will be dropped altogether.
15.3.1. Binomial Survival M o d e l The binomial model (see Chapter 4) is appropriate for processes that have two mutually exclusive outcomes, such as occur in simple capture-recapture studies. In this section we use the binomial model for estimating survival from data structures arising in, e.g., radiotelemetry and nesting studies, in which the investigator is able to classify unambiguously the fates of individual subjects (individual animals, nests, etc.). For a sample of n subjects, the binomial probability function can be used to describe the number of these individuals that survive (x) or die (n - x), where survival is ordinarily defined as occurring over a fixed interval of time. If S is the probability that an individual subject survives, then the binomial distribution of the number of survivors is
S =tntSX
x
(15.19)
and a maximum likelihood estimate of S is (15.20)
= x/n
with estimated variance v~(<_,4)
=
<,4(1 -
e3)/n.
(15.21)
344
Chapter 15 Estimation of Demographic Parameters
An approximate (1 - 2oL) • 100% confidence interval is given by + z~V'v~ (S),
(15.22)
where z~ is the upper (1 - oL) deviate of the standard normal distribution. One also could compute confidence intervals based directly on the binomial likelihood [Eq. (15.19)]; however, for large n, this approach is cumbersome and unnecessary, in that the Central Limit Theorem (Mood et al., 1974) (see also Section 4.1.2) allows for the use of the normal approximation as in expression (15.22). Frequently subjects are assigned to, or otherwise occur in, two or more groups, and a test of equality of survival between the groups is of interest. If there are two groups (e.g., a treatment and a control), then a simple approach is to calculate confidence intervals for estimated survival for each group using Eqs. (15.20)-(15.22) and determine whether these intervals overlap. The statistic Z =
(15.23)
can be used to test the hypothesis of equality of survival rates. This statistic is approximately distributed as N(0, 1) under the null hypothesis of no difference between survival in treatments A and B. An approximately equivalent test treats the groups (e.g., treatments) and outcomes (survived, died) as the rows and columns of a 2 • 2 contingency table. A test of the independence of rows and columns for these data is equivalent to a test of homogeneity of survival for the two groups. An advantage of the latter approach is that it can be extended readily to k > 2 groups, whereby a k • 2 test of contingency would be used. The assumptions of the binomial model are (1) the fates of all n subjects are known and (2) the fates are independent events that are identically distributed according to the above model. Violation of the first assumption may occur when some of the subjects cannot be detected or are censored before their fates can be determined. If there is potential nondetection of subjects, we recommend the use of capture-recapture methods (Chapters 17-19) that fully account for this source of variation. On the other hand, censoring can be handled by adapting the binomial model as shown below and by using failure time methods (Section 15.4). Violation of the second assumption regarding nonindependence of events and heterogeneity calls into question the appropriateness of the binomial model. However, simple forms of heterogeneity can be accommodated by stratification and by other variations on the binomial model (e.g., the multinomial models in
Chapters 16-19). Nonindependence can be dealt with via a quasilikelihood approach (e.g., Burnham et al., 1987).
Example We use an example from White and Garrott (1990) in which 120 mule deer fawns in Colorado were equipped with radiotransmitters and followed through winter. Sixty-one fawns were on a study area near an oil shale development ("treatment") and 59 were from areas removed from h u m a n activity. The following data were collected:
Number released
Alive
Dead
Other
Treatment
61
19
38
4
Control
59
21
38
0
The "other" deer in the above table were four individuals in the treatment areas whose radios failed and whose fates thus could not be determined. The treatment data were used with Eq. (15.19) to estimate survival rates for the 61 deer on treatment areas, excluding the four whose radio failed, as = 19/57 = 0.3333 with estimated variance va~r(S) =
0.333 (1 - 0.333) 57
= 0.003899. An approximate 95% confidence interval for S is 0.3333 + 1.96V'0.003899, or (0.211, 0.456). Similarly, the estimated survival of the control group was = 21/59 = 0.356 with 95% confidence interval (0.234, 0.478). White and Garrott (1990) performed a chi-square test of equality between treatment and control, assuming survival of the animals whose radios failed"
Number released
Alive
Dead
Treatment
61
23
38
Control
59
21
38
15.3. Analysis of Discrete Survival and Nest Success Data This test resulted in a computed chi-square statistic with X~I) = 0.058 with P = 0.81, indicating little evidence to support rejection of the null hypothesis of equal survival in the two areas. Note, however, that this assessment is complicated by two issues: (1) the nonexperimental nature of the study and (2) the fates of the four animals whose transmitters failed. In a future study, the first issue could be dealt with by random assignment of animals to "treatments" (areas), whereas the second issue requires an analytical procedure for dealing with "censoring" (a subject of Section 15.4) and a means of assessing whether censoring and fate are independent.
15.3.2. Models for Estimating Nest Success 15.3.2.1. Historical Development: The Mayfield Method Nest success, an important component of reproductive rate for many groups of animals (e.g., birds, many reptiles), can be defined as the probability that a new nest survives predation and other forms of nest destruction to produce at least one fledgling young that leaves the nest. A common way to estimate nest success is based on the binomial model of Section 15.3.1. For this approach nests are located during searches and then revisited until they either fail or produce fledglings. An intuitive estimate of nest success is simply the proportion of nests that are successful, as in Eq. (15.20). If all nests are located just following egg deposition, this approach yields unbiased estimates of nest success. However, in most nesting studies, nests are found at various ages (where age is defined as days since initial deposition) and thus are expected to exhibit heterogeneous probabilities of surviving until the end of the nesting cycle. Nests found near the time of fledging may have to survive only a few days to achieve success, whereas new nests must survive much longer. Assume for example that all nests in the sample have the same probability S = 0.99 of surviving each day of the nesting cycle. If the nesting cycle is 30 days between egg deposition and fledging, then nest success is S3~ ~ 0.74. However, if a nest is found at the beginning of day 29 of the nesting cycle, it must survive only 2 days in order to be successful, and it does so with probability S2 ~ 0.98. Mayfield (1961, 1975) was the first to recognize that estimation of nest success as the proportion successful among a sample of nests of various ages yields estimates that are positively biased (also see Miller and Johnson, 1978). Mayfield (1961) proposed an intuitive estimator that should indeed yield unbiased estimates. He focused on the estimation of daily survival probability S as the
345
proportion of exposure days on observed nests that did not fail (an exposure day is defined by a nest that is active one day and is observed the next day to determine its fate). The daily survival probability estimate S is then raised to the power corresponding to the number of days in the entire nesting cycle to produce an estimate S! of nest success, where J is the number of days between egg deposition and fledging (Mayfield, 1961, 1975). Before proceeding to the statistical development of Mayfield's (1961, 1975) estimation method, we note that not all nesting studies require this approach. For example, wood ducks, Aix sponsa, are cavity nesters and readily nest in manmade boxes (e.g., McLaughlin and Grice, 1952). Although breeding occurs in the spring and summer, these boxes can be checked in the fall and winter to ascertain (1) whether a nest was constructed in the box [indicated by presence of eggs or shell fragments (e.g., Zicus and Hennes, 1987)] the previous breeding season, and (2) whether a nest was successful [indicated by the occurrence of detached shell membranes (Girard, 1939)]. Nest success can be estimated from such postseason checks (e.g., Heusmann, 1984; Haramis and Thompson, 1985; Zicus and Hennes, 1987) using the simple binomial model of Section 15.3.1. Because the sample is not based on active nests, and because there is no difference in detectability between successful and unsuccessful nests, there is no need to implement the approach recommended by Mayfield (1961, 1975).
15.3.2.2. Models for the Mayfield Method Statistical models for nest success data were developed independently by Johnson (1979), Hensler and Nichols (1981), and Bart and Robson (1982). The models of Johnson (1979) and Bart and Robson (1982) were binomial models based on daily survival probability S, which was assumed to be equal for all nests and observed time periods. Their model permits nest visits to be conducted at intervals l of different lengths, where l = 1.... , L days. Note that the maximum interval L between nest visits should be bounded by the number of days J in the nesting cycle (i.e., L -< J). Denote by nt. the total number of observed intervals of length l for which nest fate was determined. This number is the sum of the numbers of intervals of length l for which fate was determined to be success, nls, and failure, F/If: F/I. -~ HIs if- F/if.
The data for the nest success study consist of the set {nl.} of observed nest intervals of all the different lengths.
346
Chapter 15 Estimation of Demographic Parameters
The probability model for such data can be written as
L ( nl! ~ flnlslnl., S) = I-[ (sl)n's( 1 / l = 1 \nls!nlf!
--
sl) nlf 9
(15.24)
Maximum likelihood estimates of daily survival probability S under this model can be obtained iteratively using software such as SURVIV (White, 1983) or MARK (White and Burnham, 1999). Bart and Robson (1982) also provide expressions that can be used to obtain estimates by hand calculator with only a small number (two or three) of iterations. Mayfield's (1961,1975) original estimator is the maximum likelihood estimator for the situation in which all nests are visited daily (all l = 1):
?:3 = ns/n., where the l = 1 subscripts are omitted. Its variance can be estimated by va'-~(~;) = ~;(1 - '~)/n.. Hensler (1985) provided approximations via the delta method (see Appendix F) for the variance of nest success, assuming either a constant daily survival rate or allowing for nest success to be viewed as a product of the probabilities of surviving two or more stages (e.g., incubation and nestling stage). Construction of confidence intervals for S and nest success ~;! is discussed by Hensler and Nichols (1981) and Bart and Robson (1982) for the case of constant S, and by Hensler (1985) for the case of multiple stages with differing survival. The test statistic in Eq. (15.23) can be used to test hypotheses about differences between daily survival rates (Hensler and Nichols, 1981; Bart and Robson, 1982), and Hensler and Nichols (1981) presented simulation results of an investigation of the power of this test for specified sample sizes and differences of interest. Program CONTRAST (Hines and Sauer, 1989; Sauer and Williams, 1989) can be used to test more general hypotheses involving more than two survival rates.
15.3.2.3. Assumptions and Alternative Binomial Modeling The Mayfield method uses a binomial modeling approach to estimate nest success or survival from periodic visits to nests. As with the binomial model in Section 15.3.1, the Mayfield method requires the assumption that fates of all subjects are known at each visit; thus, all subjects must be detected at each visit following initial detection. The method also requires
additional assumptions that follow from the underlying binomial model: (1) survival rates are constant over the study and over the nests included in the sample, (2) all "visits" are recorded, (3) survival probability is not influenced by the observer, and (4) the probability of a visit is independent of the probability of survival. Of these, assumptions (2) and (4) can be met by simply using sound field methods (recording all visits and visiting nests regardless of suspected fate). Assumption (1) of constant survival over time and nests must be relaxed in many situations. Johnson (1979) suggested some useful approaches to detecting heterogeneity of survival among nests, which often can be dealt with via stratification. In some cases stratification may involve different stages of the nesting cycle (e.g., egg stage vs. nestling stage) for observation days on the same nest. In other cases, nests may be stratified by such factors as time of nest initiation within the nesting season (early vs. late nesters), by variables associated with habitat or aspects of nest placement (e.g., concealed vs. open), and even by clutch size. When exposure days are stratified in this manner, point estimates of daily survival probability for each stratum can be used in conjunction with analysis of variance and covariance techniques to investigate sources of variation (see Klett and Johnson, 1982; Flint and Grand, 1996). The point estimates may be weighted by the number of exposure days (Klett and Johnson, 1982) or by the inverse of the estimated variances. Another approach to analysis based on stratification is to build competing nested models that incorporate different levels of stratification. Assume, for example, that a data set contains stratified observations of nests that were either concealed or placed in the open. The most general model would have separate parameters for daily survival probability of concealed and open nests, whereas a reduced-parameter model would incorporate the same survival parameter for exposure days of both groups. A likelihood ratio test between these competing models would provide a formal test of the hypothesis that nest concealment is associated with variation in nest survival. AIC (Burnham and Anderson, 1998) (see Section 17.1.8) also can be used as an aid in model selection. Miller (1999) used this approach with data from an experiment on artificial nests to investigate the influence of plumage coloration and vegetation density on nest success. Still another approach to analysis with stratified nest exposure data is to develop ultrastructural models of daily survival probability as functions of the hypothesized explanatory variables (see Section 17.1.4 for more detailed discussion of ultrastructural modeling). For example, the daily survival probability in the above
15.3. Analysis of Discrete Survival and Nest Success Data models could be rewritten as a linear-logistic function of explanatory variables associated with each stratum: exp([30 + ~,j {3jxj) S
__.
1 + exp(60 + ~,j f3jxj)"
where xj is the value for explanatory variable j and [3j is the corresponding slope parameter. This approach can be implemented in programs SURVIV (White, 1983) and MARK (White and Burnham, 1999) and yields direct estimates of the model parameters 6j. Likelihood ratio tests can be used to test hypotheses about the importance of specific variables, and AIC (see Burnham and Anderson, 1998) can be used for model selection. One problem that sometimes arises with stratification concerns the situation in which a nest may make the transition from one stratum to another and the time of the transition is not known exactly. Consider the daily survival rates of an egg-laying stage, incubation stage, and nestling stage. If successive observations of a nest indicate a transition has occurred from one stage to the next (e.g., egg laying to incubation) and the time of the transition is unknown, it is not clear how many exposure days to allocate to each stage. Stanley (1999) extended the modeling approach of Johnson (1979) and Bart and Robson (1982) to handle this situation. His approach does require information about what stage to expect the nest to be in at the next visit (given the time interval between visits). Stanley (1999) developed a computer program to compute estimates under this model using iteratively reweighted least squares. Assumption (3) of the Mayfield method, that the fate of the nest is not influenced by observer visits, has received substantial study (see review by Gotmark, 1992). Results are mixed, with roughly half of the 68 studies reviewed by Gotmark (1992) providing evidence of reduced nest success associated with disturbance. Susceptibility to disturbance appears to vary with species and with characteristics such as coloniality and life history parameters (Gotmark, 1992). Commonsense recommendations are to try to minimize disturbance, and when it is suspected, to test for effects on nest success (e.g., Nichols et al., 1984c).
15.3.2.4. Models Including Nest Encounter Parameters The model developed by Hensler and Nichols (1981) for nest visit data differs from those of Johnson (1979) and Bart and Robson (1982) in that it focuses directly on nests rather than nest exposure days, and it includes parameters reflecting encounter probabilities for different days of the nesting cycle. Thus, the likelihood function of Hensler and Nichols (1981) incorporates age-
347
specific encounter parameters, where "age" refers to the age of a newly encountered nest. For the case of daily nest visits with constant daily survival probability, their maximum likelihood estimate of daily survival probability is identical to that of the binomial models of Johnson (1979) and Bart and Robson (1982). Pollock and Cornelius (1988) recognized that all the information in nest visit data was not exploited by earlier models. If nests can be "aged" when first encountered [aging is sometimes possible by candling or floating eggs (see below) and by use of photographs or keys of feather development for nestlings], the Hensler and Nichols (1981) model can be stratified by age, or covariates (Section 17.1.7) can be used to model the age effect. Commonly, nests cannot be aged at first capture, but the Pollock and Cornelius (1988) generalization of Hensler and Nichols (1981) still can be used when information on age-specific nest encounter probabilities is provided by nests that succeed. Because success is assumed always to occur on day J of the nesting cycle, the age of a successful nest when it was encountered can be determined as the difference between J and the number of days that the nest was observed. Given information on age-specific nest encounter probabilities, it also is possible to use the information on nest age at encounter to provide information about the survival probabilities prior to encounter. This information is not used in the binomial modeling approaches of Johnson (1979) or Bart and Robson (1982) and was not included as a likelihood component in the model of Hensler and Nichols (1981). The model of Pollock and Cornelius (1988) was developed as a means of permitting specific survival probabilities for different ages or stages of nest development. Under this model, survival is estimated from the time of nest initiation, not the time of discovery. The data for the Pollock-Cornelius model consist of two components: (1) nests that eventually succeed, and thus can be backdated to obtain the age when they were found, and (2) nests that fail. Define F/jH as the number of nests found at age j that later succeed and njF a s the number of nests of unknown age that are observed j units of time (usually days) and then fail. Define the total number of encountered nests that are successful as l
n H - E njH j=l and the total number of encountered nests that fail as
l nF = E njF" j=l
348
Chapter 15 Estimation of Demographic Parameters
Let 8j [notation of Pollock and Cornelius (1988)] be the probability that a detected, intact nest is first encountered at age j (i.e., on day j of the nesting cycle, j = 1, ..., J). Finally, define qj as the probability that a nest is found failed at day j although it was successful to age j - 1. Then the probability of a new nest succeeding is
l 1-Eq
j 9
j=l
The Pollock and Cornelius (1988) likelihood is formed as the product of three conditional multinomial distributions involving these statistics and parameters: P r ( n l H , niH ..... nlH[nH) = (
X [ ~ ~=11 ]nlH~
~2
nH t F/1H, F/2H, ..., F/jHJ
n2H
pr nl ,n2F .....
(15.25)
njH
nF
1
15.26)
F/1F, H2F, ..., HJF/
X
•
x
[ [
81q~ + 82q2 + "'" + gJql 81ql + (81 7 ~2)-q2 -~ "'" ; (~-1 + 82 + "'" 8I)q J
81q2 + 82q3 + "'" + gl-lqJ 81ql + (81 + 82)q2 + - + (81 + g2 + "'" 81)ql
81ql + (81 + 82)q2
....
(81 + 82 + "'" 81)q!
• ...
'
and
(15.27)
Pr(nHnH+nF)=(nH+nF I
\ nH ,/
X
. . . . . . . .I. . .
(,),
I
J ""
1 - ~'j=lqj ~j=l 8j + 81ql + (gl + 82)q2 + "'" + (81 + 82 +'"81)ql
15.3.2.5. R a n d o m Effects Model ] nF
X
(
For example, a nest observed to fail after only 1 day (the first term for multinomial cell n l F ) could have been any age at encounter; hence the sum of encounter probabilities in the numerator ranging from (~1 to ~j. On the other hand, a nest observed to fail after J days is known to have been found at age 1; hence the single term n/F in the numerator of the final cell. The third likelihood component [Eq. (15.27)] is simply a conditional binomial for the number of successful nests, conditional on all encountered nests. Pollock and Cornelius (1988) developed a computer program to perform the maximum likelihood estimation of these parameters using program SURVIV (White, 1983) and provided example data and analyses from a study of mourning doves. If adequate data are available, the approach of Pollock and Cornelius (1988) should be useful in estimating daily encounter probabilities and age-specific failure rates. Unfortunately, the rather heavy data requirements of the approach often mean that adequate data will not be available. In such cases it will be necessary to pool data and specify age intervals (e.g., 1-8 days, 9-16 days, 17-26 days). Heisey and Nordheim (1990) noted that the Pollock and Cornelius (1988) approach can yield biased estimates in the face of such a discretization of the time frame, and they provided a bivariate contingency table approach to estimation (Heisey and Nordheim, 1995) that eliminates this source of bias. Finally, Bromaghin and McDonald (1993) presented a general formulation of nest survival models using the framework provided by weighted distribution theory. They showed that the model of Pollock and Cornelius (1988) is a special case of this general formulation and suggested directions for further developments (Bromaghin and McDonald, 1993).
81q1+ (81 + 82)q2+"" + (81 + 82 -Jr-'" -Jr-~l)ql
)'
1 - ~=lqj ~j=lgj + 81ql + (gl + 82)q2 +"" + (81 + 82 + "'" 8Pql
The first component of the likelihood [Eq. (15.25)] provides the conditional distribution of the numbers of successful nests for each age at encounter given the total number of successful nests. This component provides the information needed to estimate the agespecific encounter probabilities. The second component [Eq. (15.26)] specifies the conditional distribution of the number of nests of unknown age that were observed to fail after j days of observation. The denominator of each term is the total probability that an encountered nest fails, and the numerator specifies the probability of failing after exactly j days, given failure.
Natarjan and McCulloch (1999) developed a random effects modeling approach for nest survival data and showed that it can be implemented for both simple binomial models (e.g., Bart and Robson, 1982) and models including encounter probabilities (Pollock and Cornelius, 1988; Bromaghin and McDonald, 1993). Under the random effects approach, daily survival parameters for different nests (e.g., Si for nest i) are viewed as random variables arising from some underlying distribution. For example, Natarjan and McCulloch (1999) used a model similar to that of Eq. (15.24), but multiplied the distribution by a beta distribution for the Si. This approach induces a nonnegative correlation between the survival status of a nest at repeated visits. Natarjan and McCulloch (1999) also included a likelihood ratio test for the presence of heterogeneity
15.3. Analysis of Discrete Survival and Nest Success Data and developed linear-logistic models to incorporate the effects of covariates (either time specific or at the level of individual nests) on nest success in a random effects framework. At present the complexity of the computations may limit the ability of many biologists to apply this approach. However, random-effects modeling is a reasonable and natural way to view nest survival (or indeed, any sort of survival), and we believe that the approach will see increasing use, especially when computations are simplified or made more accessible with, for example, Markov chain Monte Carlo methods (Gilks et al., 1996; Gelman et al., 1997; Link et al., 2002).
15.3.2.6. Study Design The design of nest success studies should focus on meeting model assumptions and attaining sample sizes needed to meet study objectives. Regarding sample sizes, Bart and Robson (1982) documented the number of observation intervals needed to attain a specified confidence interval width for S and to test a null hypothesis of no difference between a pair of daily survival rates with specified power. Hensler and Nichols (1981) considered the sample sizes needed to obtain a specified coefficient of variation
cv(~) = SE(~) $ for the estimate S. As noted above, the likelihood of Hensler and Nichols (1981) differs from that of Johnson (1979) and Bart and Robson (1982) in assuming daily nest visits and in being based on statistics associated with individual nests (rather than interval lengths). They derived an approximate expression for the number of nests needed in a nesting success study to achieve a specified coefficient of variation of S as (1 -
S*) 2
IT*(1 - S*)2 + ( Y * - 1)(1 - 2S*)]CV 2' where n* is the recommended number of nests, S* is the predicted daily survival probability, T* is the predicted average number of days that a nest is observed, and Y* is the predicted fraction of observed nests that succeed (note that this is not the same as predicted nest success because nests are found at various ages). Note that this expression differs slightly from that presented by Hensler and Nichols (1981), because the original paper contained a typographical error. Finally, Hensler and Nichols (1981) provided recommendations for predicting the average number of days observed and the fraction of nests that succeed based on assumed values for S. B
349
Bart and Robson (1982) considered the question of whether to visit nests more frequently or to devote more effort to finding additional nests. They concluded that, under the constant survival model of Eq. (15.24), precision is increased by visiting less frequently and including more nests in the sample. This conclusion is consistent with the objective of reducing the possibility of disturbing nests and seems especially appropriate for the constant-survival case. However, we note that frequent nest visits can be useful when different stages of the nesting cycle have different survival rates. As noted above, a problem arises when the exact times of transition from one stage to another, and hence the number of exposure days associated with each stage, are not known. A study by Stanley (1999) addressed estimation in this situation, and his simulation studies showed that increasing interval length produces an increase in absolute bias in the daily survival estimates for the two stages associated with the transition. Thus, more frequent visits lead to stronger inferences about stage-specific survival rates. A problem related to that of unknown time of transition between different nest stages is that of unknown time at which success is attained (e.g., unknown time of fledging). For example, in work on nest success of mourning doves at the Patuxent Wildlife Research Center (Nichols et al., 1984c), success was defined as nestling birds reaching day 10 following hatch, the earliest age at which fledging was thought to be possible. Mourning dove nestlings usually were found at nests after age 10 days, with some nestlings remaining at the nest until day 19 following hatch (J. D. Nichols and M. J. Conroy, unpublished). Consider nest visits at day 12 and day 15, in which nestlings are still found in the nest. It would not be appropriate to add these days to the total number of exposure days, because if the nestlings had been present on day 12 but absent on day 15, we would have concluded that they fledged. Thus, visits following day 10 can lead to exposure days with a fate of success, but not exposure days with a fate of failure, and inclusion of such successful exposure days will lead to a positive bias in daily survival rates. This brings up an important point of analysis in nest success studies, namely, that the exposure days to be considered in estimation of daily survival probability should be restricted to those on which either fate (success or failure) can be observed. In the cited mourning dove studies, either the day of hatch was known or else nestlings were aged using the key and photographs of Hanson and Kossack (1963). In addition, each nest was visited on day 10 following hatch. Nest visits occurring after day 10 provided natural history information but were not used in nest success estimation (Nichols et al., 1984c). If a
350
Chapter 15 Estimation of Demographic Parameters
nest had not been visited on day 10, but instead on, say, day 8 (nestlings present) and then day 12 (nestlings absent), we would not have known the fate and would have used exposure days through day 8 only. If we encountered nestlings on days 8 and 12, we would know that the nest survived until day 10. However, we would not want to include the 2 days survived in the exposure days because nest visits at days 8 and 12 would not have permitted identification of a fate of failure. Thus, we recommend visits to each nest on day J, the first possible day on which nests could succeed (e.g., nestlings fledge). When this is not possible, then exposure days should not be accumulated for intervals that include J, unless special models are developed that incorporate the dual possibilities of failure and fledging for such nests that are found empty at the first visit following day J. A simple design recommendation that follows from the preceding discussion is that it is useful to obtain information on the age of the nest in order to predict the days of transition, either from one nest stage to another or from the final nest stage to a successful nest. As noted, keys and photographs may be useful in aging nestling birds (Hanson and Kossack, 1963), and techniques such as candling (Weller, 1956) and floating (Westerskov, 1950) eggs may be useful in aging eggs. Precise information on egg and nestling age can be incorporated directly into likelihoods permitting agespecific variation in survival probability, leading to a simplification of the likelihood of, e.g., Pollock and Cornelius (1988). Even if age cannot be determined precisely, it may be possible to use auxiliary information on age as a covariate in nest survival models (Heisey and Nordheim, 1990). Design recommendations to minimize violations of model assumptions include attempts to disturb nesting birds and surrounding habitat as little as possible during nest visits. Recommended field methods for studies of nesting prairie ducks have been provided by Klett et al. (1986). Field recommendations for other nesting species, including small passerines, were provided by Martin and Geupel (1993). Example
Our example is from a study of nesting success of mourning doves (Nichols et al., 1984c) in which 48 nests found at various ages were visited daily until either success (fledging at day 10 following hatch) or failure was observed. These data were used to estimate the daily survival rate based on the closed-form constant-survival Mayfield estimator: = ns/n
= 411/430 = 0.9558.
As noted above, an estimate of the standard error for is
S"E(S) = X/S(1 - S)/n. X/(0.9558) (0.0442)/430 = 0.0099, providing an asymptotic 95% confidence interval of + Z0.025V'SE(S) = 0.9558 + 1.96 • 0.0099, or (0.9364, 0.9752). The total nesting period (egg laying through fledging) was 26 days. Under the constantsurvival model, overall nest success is estimated to be
sl ~ 0.955826 0.31. An approximate 95% confidence interval for nest success can be obtained by raising the endpoints of the daily survival rate confidence interval estimates to the appropriate power:
CI(S/) ~ (0.936426, 0.975226) (0.18, 0.52).
15.3.3. Radiotelemetry Survival and Movement Studies Mortality studies of radiotagged animals typically are viewed as similar to nest studies, in the sense that the probability of encountering a "radioed" animal is usually assumed to be 1 (though see Pollock et al., 1995). However, studies of radioed animals differ from nest studies in that there is often no natural endpoint to the study of the mortality process (e.g., no time of fledging) in telemetry studies. Trent and Rongstad (1974) developed intuitive estimators of daily survival probability from radiotelemetry data using a binomial approach that is similar in many respects to Mayfield estimation with nest success data. Heisey and Fuller (1985) further elaborated a formal statistical framework for the binomial model. Under the Heisey-Fuller approach, daily survival is allowed to vary among subintervals of differing lengths Li, i = 1, ..., k. These intervals are not based on observation frequency, as in the binomial nest success models, but are simply periods over which survival is suspected a priori to vary. In the case in which relocations are made on a daily basis, tlLe estimator of daily survival for each interval is Si = ( X i -
di)/xi,
(15.28)
where x i is the total number of transmitter days (analogous to exposure days in nest success estimation) in interval i, and d i is the number of deaths during interval
351
15.4. Analysis of Failure Times i. An estimate of survival over all intervals is obtained as k
= l-[ dLi.
(15.29)
i=1
Note that estimator [Eqs. (15.28) and (15.29)] permits "censoring," i.e., the removal of an animal from further consideration for estimation purposes. For example, assume that period i covers 10 days, so that L i = 10. An animal that survives and remains in the treatment area or area of interest for the 10-day period contributes 10 transmitter days to x i and hence to the estimator. However, an animal that dies on day 5 or departs the study area following day 5 contributes only 5 transmitter days to x i. This approach can be used to partition multiple causes of death, when cause of death can be determined by the investigator (Heisey and Fuller, 1985). Denote as mij the probability that an animal alive at the beginning of a day in interval i dies during the day as a result of mortality source j. If dq denotes the number of deaths in interval i resulting from mortality source j, then source-specific mortality is estimated as
Like the Mayfield method, the Heisey-Fuller method relies on the assumption that there exist intervals of time over which daily survival rates are constant. In theory this assumption can be met by selecting very small time intervals; however, in practice the intervals must be of sufficient length so that some deaths occur over the interval. Finally, we note that these models can be used to estimate movement probabilities from radiotelemetry data (Conroy et al., 1996; Nichols, 1996; Nichols and Kaiser, 1999; Bennetts et al., 2001). When relocation data are collected daily, the Heisey-Fuller approach can be used as described above, with the added feature that the event of interest is not mortality, but movement away from a defined area. When relocations are made at greater intervals the treatment will differ, as movement does not necessarily terminate the data from an animal (an animal may move out of an area and then return at a later time), whereas mortality precludes additional relocation data. Movement to specific locations also can be handled with the model in a manner analogous to the treatment of different mortality sources by Heisey and Fuller (1985; see Nichols, 1996; Bennetts et al., 2001).
rhij -- dq/xi. The probability that an animal dies as a result of source j during interval i (denote this probability as Mq) is given by (Heisey and Fuller, 1985)
l~ij = l~lij q- SiYhij -ff s2yhij q- ... q_ ~Li-llhij = [the/(1 - di)](1 - ~L,).
(15.30)
On reflection, estimator (15.30) is intuitive. During the first day of interval i, an animal will die of source j with probability mij. An animal can be exposed to mortality on day 2 of the interval only if it survives day 1, and this survival occurs with probability si. Similarly, the remaining terms in the sum include the probability of surviving all mortality sources to be alive at the beginning of the day and the probability of dying from source j, given survival. The reduced expression following the summation of Eq. (15.30) is simply the probability of dying during the interval multiplied by the relative risk of dying from source j. The Heisey-Fuller approach described above assumes daily relocations of animals and is the simplest case to explain. However, they extend the approach to the general case of relocations at intervals of greater than 1 day (Heisey and Fuller, 1985; also see Bart and Robson, 1982). Heisey and Fuller (1985) developed computer program MICROMORT, and their models also can be implemented readily in the general programs SURVIV (White, 1983) and MARK (White and Burnham, 1999).
15.4. A N A L Y S I S O F FAILURE TIMES The Mayfield, Heisey-Fuller, and other methods described in the previous section depend on assumptions of constant survival rates among individual animals and, to varying degrees, over time. These assumptions may not be reasonable for many telemetry studies, over the course of which survival probabilities can change due to weather patterns, hunting pressure, and other events. A flexible alternative to the enumeration of events over discrete periods is based on the measurement of times to "failure." The data for such an analysis are the times from the onset of the study, to some event called a "failure." Examples of failure time analysis can be found in many fields, including engineering (estimating the lifetimes of machine components), business (duration of strikes, times taken to complete labor tasks), and the medical sciences (survival times of patients in clinical trials). In recent years these methods been extended to the realm of wildlife biotelemetry studies (Pollock et al., 1989a,b). Three components are needed for such a "failure" study: (1) a defined time origin, (2) a scale for measuring time (e.g., days), and (3) a definition of "failure" (e.g., an animal dies from mortality source j before time T). It is useful to recognize an analogy between the "event-based" approach of the previous section and
352
Chapter 15 Estimation of Demographic Parameters
the "duration-based" approach here. Both approaches deal with the same phenomena, namely, the potential occurrence of a binary response (death, emigration, etc.) for individuals over some recognized time frame. Both require the three components listed above, both utilize the same sorts of data for analysis, and both incorporate the same parameter structures. Indeed, the two approaches share roughly the same relationship as Bernoulli counting processes and Bernoulli waiting times (see Section 10.1), in that one focuses on the n u m b e r of occurrences over time, whereas the other focuses on the length of time until those occurrences. Thus, the previous section highlighted models of the n u m b e r of animals surviving over fixed time periods, based on variants of the binomial model. In contrast, the survival distributions described here are based on models of time until failure (e.g., death) or censoring (e.g., loss of a telemetry signal). In what follows we describe models that allow for (1) varying survival rates over time, (2) censoring, and (3) modeling of survival distributions based on grouping (age, sex, etc.) or continuous (e.g., weight) variables. We discuss both parametric and nonparametric models for use in modeling survival distributions, with an emphasis on nonparametric or semiparametric approaches. The data for these models include the time an organism enters a study, the time it leaves (through mortality, migration, radio failure, or survival to the end of the study), the fate of the organism (survival, mortality, censoring), and the values of any relevant covariates. The idea is to describe survival as a function of the time to mortality and to use these data to estimate parameters in that relationship. Censoring, heterogeneity in survival, and staggered entry of organisms into the study are easily accommodated in an analysis of failure times (see Sections 15.3.2 and 15.3.3 for treatments of these issues with discrete survival data).
and before t days is F(t) = q + (1 - q)2q 4- ... 4- (1 - q ) t - l q .
The latter expression can be rewritten as F(t) - 1 - (1 - q)t = 1
F(t) = Prob(T < t).
Conversely, S(t) is the probability of survival until t or later, so that S(t) = Pr(T -> t).
Clearly, survival and failure times S(t) and F(t) are related by F(t) = 1 - S(t).
The probability distribution function f ( t ) for failure time describes the probability of dying "near t" (in a short interval around time t): fit) = lim
at-,0
h(t) = lim
Prob(t <- T <- t + A t l T > t) At
This conditional mortality risk is here denoted as the hazard. In words, the hazard is the instantaneous mor-
tality rate for organisms alive at time t. It should be clear that F(t), S(t), and h(t) are mathematically related. Thus, instantaneous mortality at time t is equivalent to survival to t, followed by death:
F(1) = q.
F(2) = q + (1 - q)q,
Prob(t -< T _< t + At) At
It follows that f(t) = F' (t), which captures the rather intuitive idea that aggregate mortality F(t) -- 1 - S(t) can be expressed as the accumulation of instantaneous mortality risk f(t). Instantaneous mortality f ( t ) in turn can be expressed in terms of an instantaneous mortality risk conditional on having survived up to time t:
at--*0
Similarly, the probability of death before 2 days is
e -ht,
where h = -ln(1 - q). In the above expression, h can be interpreted as the constant instantaneous risk of mortality over (t, t + At), conditioned on survival to time t. More formally, let T represent the observed time of death. Then T is a r a n d o m variable that is modeled with the cumulative distribution function F(t), i.e.,
15.4.1. Statistical Models for Failure Time, Survival Time, and Hazard Rate It often is useful to address survival via its assessment over a discrete, but short, time interval. To motivate the approach, let 1 - q be the probability of success (survival) over a short interval (e.g., a day), with q the probability of failure (mortality). Then the probability that mortality occurs before 1 day is
-
f(t)-
S(t)h(t)
or
h(t) - f(t) / S(t).
15.4. Analysis of Failure Times From F(t) = 1 - S(t) we get
353
it is to describe the function S(t) over the course of a study. A number of useful parameteric forms for the hazard and survival functions are explored in the next section. We also describe the nonparametric KaplanMeier method and various semiparametric approaches (notably Cox proportional hazards) to allow more general modeling and inference, particularly in situations in which the hazard cannot be expected to conform to a parametric form.
fit) = F' (t) d
= ~[1 - S(t)] = - S ' (t), and thus h(t) = - S ' ( t ) / S ( t ) .
Using the mathematical identity d(ln u)
15.4.2. Parametric Survival Estimation
I du m ~
z
dx
u dx
we therefore have h(t) = - S ' ( t ) / S ( t )
= - 41n S(t), C/t
which establishes the relationship between hazard and survival as - f t o h(x) dx = In S(t)
and S(t) = e -
H(t)
l
(15.31)
where H(t) is the cumulative hazard with H(t) = y~ h(x)dx. Note that survival, mortality, and hazard are mathematically equivalent expressions, in that specification of one function is sufficient to determine the forms of the others. Returning to the simple case in which h(t) = h, these relationships become H(t)
=
f t 0 h dx
= ht, S(t) = e-lo h ax -ht --
e
I
and - I n S(t) h
This is the exponential model of survival with constant hazard h and is the continuous time equivalent of the Mayfield and other discrete models wherein daily survival is assumed constant. The mathematical association between S(t) and h(t) allows one to specify survival over time by identifying the corresponding hazard function. This can be advantageous, because it often is easier to specify a priori a functional form for instantaneous mortality risk than
A number of parametric forms for the hazard and survival functions in Eq. (15.31) have proved usefulm for example, the exponential model in the previous section. The hazard h(t) = h for the exponential model identifies a constant instantaneous risk of mortality, and in situations where hazard is thought a priori to be constant over time (and homogeneous among subjects), this model is very efficient. Estimates under the exponential model can be obtained via maximization of the likelihood function based on a sample of observed survival times. Alternatively, if the sample size is large, the proportions surviving to various times t can be plotted against time and used to estimate h. A simple estimation approach based on Eq. (15.31) is to convert to natural logarithms, thus obtaining the relationship In S(t) = ln(e -ht) = -ht.
From this expression a zero-intercept least-squares regression of the log-transformed proportion surviving versus time yields the hazard rate h as a slope estimate. In practice this procedure is more useful as a diagnostic tool than for estimation. For the exponential model, a plot of log-transformed survival frequencies versus time should be a straight line, with systematic deviations suggesting that the model may be inadequate. If it is expected a priori that mortality risk is temporally variable, it is straightforward to incorporate time effects into the model. Several parametric models include time effects, an example being the two-parameter Weibull distribution (Table 15.4). The Weibull distribution is especially appropriate if hazard is a monotonic function of time. Values of the Weibull parameter K less than unity result in monotonically decreasing hazard, whereas values greater than unity (e.g., Fig. 15.1b) result in increasing hazard. Because the Weibull distribution is equivalent to the exponential for K = 1 (Fig. 15.1a), the likelihood of the exponential is nested in that of the Weibull, and either likelihood ratio or AIC can be used to evaluate whether the more complex Weibull model is justified for a given data set.
354
Chapter 15 Estimation of Demographic Parameters TABLE 15.4 Parametric Models for Failure Time Analysis a Survival function Model
Density function
S(t) = 1 - F(t)
f(t)
Hazard h(t)
Number parameters
Exponential
e - ht
he- ht
h
1
Weibull
8 -(ht)K
Kh(ht)~- l e-(ht)~
Kh(ht)~- 1
2
Log logistic
[1 + (ht)K]-1
KhKtK-l[1 + (ht)~]-2
Kt~-]h~/[1 + (ht)q
Proportional hazard
[L(t)]*
O~[L(t)]*-ll(t)
O~h(t)(t)
2 1 extra for proportionality
aAfter Cox and Oakes, 1984.
A potentially useful model for nonmonotonic survival is the log-logistic model (Table 15.4; Fig. 15.1c). Again, this model has two parameters, but its functional form allows the hazard to be a nonmonotonic function of time. Various combinations of the parameters provide a great deal of flexibility in modeling survival data, and the distribution can be compared to the exponential, Weibull, and other survival models using AIC (but not likelihood ratio, because the models are not nested in one another). Estimation for these and
other parametric models can be performed in many standard statistical software packages, such as PROC LIFEREG in SAS (SAS Institute, 1989). Many of these (e.g., SAS) allow for right and left censoring of survival times, thus accommodating situations often encountered in wildlife studies. An important model that can be used to estimate survival rates when the hazard is time varying is the proportional hazards model. This family of models utilizes a "semiparametric" approach, in that the parametric form of the hazard is not specified, but one or more parameters are used to specify a proportional relationship between a "baseline" hazard and hazards at various levels of individual and time-specific covariates. We return to the proportional hazards model in Section 15.4.4, when we consider methods for incorporating the effects of predictor variables on survival. 15.4.3. N o n p a r a m e t r i c S u r v i v a l Estimation: Kaplan-Meier
b
0.08 0.04
0.05
0.03 0.02 0.01
0
20
40
60
80
I oo
t F I G U R E 15.1 Representative parametric hazard functions: (a) exponential (h = 0.05), (b) Weibull (h = 0.05, K = 1.50), and (c) loglogistic (h = 0.05, K = 1.50). See Table 15.3 for general forms and corresponding survival and probability density functions.
The parametric models in the previous section have proved valuable in many studies of mortality phenomena and produce efficient inferences under suitable conditions. However, many wildlife telemetry studies occur under situations in which mortality risk cannot be anticipated to be a smooth function of time. For instance, severe and unpredictable weather events may result in a substantial increase in the number of observed mortalities. In addition, many wildlife studies are characterized by small sample sizes, gradual releases (i.e., staggered entry) of animals into the sample at risk, and right censoring due to radio failure or migration of the study animals. The Kaplan-Meier method (Kaplan and Meier, 1958; Pollock et al., 1989a,b) provides (1) empirical estimation of survival rates that can be graphically examined for temporal patterns (possibly motivating estimation under a parametric model) and (2) hypothesis testing based on experimental or other stratification of individuals. The Kaplan-Meier method is based on an expression
355
15.4. Analysis of Failure Times of the survival function [Eq. (15.31)] in terms of a discrete hazard function hj:
S(t)
=
rj = rj_ 1 - dj_ 1 - cj_ 1 + rlj_l;
hj),
I-I (i -
jlaj
where a I .... , a k are k points in time that are not necessarily equally spaced, and the hazard hj now represents the probability of failure over the interval (aj, aj + 1) (Cox and Oakes, 1984). The notation j[aj < t indicates that the product is over the points in time (the aj) that occur before time t. A nonparametric estimator of the survival function is then
S(t)
=
1-I (1
jlaj
be censored when they disappear and included again when they reappear. In general,
-/~j),
where/~j values are maximum likelihood estimates of the discrete hazard. Each of the hj values is estimated as a conditionally independent binomial
that is, the number at risk at the current period will equal those at risk at the previous period, minus any deaths (dj-1) or censored animals (cj-1) last period, plus new releases (r/j_ 1) during the previous period. Because the "staggered entry" situation occurs so commonly in wildlife telemetry problems, the productlimit estimator provides a very useful procedure for analysis. Note that the Kaplan-Meier estimator of survival reduces to a simple binomial estimator of survival over the interval (0, t) for the special case in which all animals are released at an initial time (t = 0) and there is no censoring. For this case, Eq. (15.32) provides
S(t)= ( r~ 70d~
r2 72 d2) ... (rtTtdt).
f,j = 4 / r , where rj is the number of animals "at risk" at aj, and dj is the number that die at aj. The resulting estimator of the survival function is thus
jlaj
(15.32) --
and is known as the product-limit estimator. Attention is restricted to the times at which deaths occur, so the aj are selected as the times of death in the sample. An estimate
Because there is no censoring, the current number of animals at risk is always the previous number at risk, minus the number that died through the previous period.
~(t) = ( r o - do) [ r ~ (d~ + dl)] ro ro - do x [ro- (do+dl+d2)] r 0 - (dO + d 1)
9""
var[S(t)]
= S(t)2 s
jlaj
rj(rj
-- dj)
(15.33)
of variance of the survival function is known as Greenwood's formula (Cox and Oakes, 1984). However, Cox and Oakes (1984) argued that the variance expression var[S(t)] = S(t)211 - S(t)]
r(t)
(15.34)
performs better in the tails of the survival distribution. Pollock et al. (1989a) showed that the product-limit estimator is applicable not only to situations for which the "at-risk" sample decreases due to death and sampling, but also to situations for which the sample increases, as when animals are added at varying points after some initial study time. In addition, animals that temporarily disappear and then reappear in the sample (i.e., cannot be located due to incomplete searching effort or temporary movement off the study area) can
[ r o - (do + dl + d2 + "'" + dt)] tO_ (d o q7 dl q~ : - -~ 6/;--1) "
It is easy to see that this telescoping series simplifies to S(t) = [r 0 - (do + dl + d2 4- .-. +
dt)]/r 0
= X/ro, which is the usual binomial estimator of survival for x successes (in this case survivors from the original sample) in r 0 trials. Finally, we note that the Kaplan-Meier estimator can be used for ecologically interesting events other than death. For example, Bennetts et al. (2001) used the product-limit estimator for data from snail kites, Rostrhamus sociabilis, that were radiotagged at the time of fledging. They defined the event of interest as departure from natal area (natal dispersal). The analog of the survival function for this kind of movement study is a "fidelity function." The hazard function (reflecting probability of departure from the natal area) showed
356
Chapter 15 Estimation of Demographic Parameters
an early peak at about 30 days, indicating high probabilities of dispersal during this period (Bennetts et al., 2001).
where
E(d2j) = djr2j/r j and
15.4.3.1. Tests of Differences between
k
Survival Distributions
var (d2j)=
Often it is of interest to compare survival functions from two or more samples--for example, over two different periods of time or with respect to a treatment and a control. Here we describe a popular log-rank test statistic for two samples (Savage, 1956), generalized by Pollock et al. (1989a) to allow for staggered entry and censoring. More elaborate k-sample tests, and tests involving the influence of continuous covariates, can be handled using approaches such as proportional hazards analysis (Section 15.4.4). Consider a hypothesis of interest that asserts equality between two survival distributions:
1
~k=l(~rlj/rj)
for j = 1, ..., k, where the aj now denote the times of death for animals in both samples. A test of this hypothesis is performed via the statistic k
k
(15.35)
k
TABLE 15.5
Week
(t)
j
]
~,~-l(djr2j/rj)
The example is from a study of northern bobwhite quail (Colinus virginianus) radiomarked in North Carolina from November 1985 to March 1986 (Pollock et al.,
]2
~j=l var (d2j)
1
Example
~j=l d 2 j - ~;j=l E(d2j) X2 =
+
-1
The values rlj, r2j, and 1) in these expressions are the numbers of animals "at risk" at aj in the first, second, and combined samples, and dlj, d2j, and dj are the numbers that die at aj in the first, second, and combined samples. The statistic in Eq. (15.35) is asymptotically distributed as a chi-square variable with one degree of freedom under H 0. Pollock et al. (1989a) provide three alternative expressions for var (d2j), including Eq. (15.36). Of the three, Eq. (15.36) provides a slightly larger estimate of variance and results in a more conservative test of H 0.
H0: Sl(aj) = S2(aj)
[
(15.36)
j=l
Estimation of Survival Rates of Northern Bobwhites Using the Kaplan-Meier Product-Limit Estimator a
aj
No. at risk (1))
1
20
2
No. deaths
(dj)
No. censored (cj)
0
0
New added
Hazard
(nj)
(hj)
Survival [S(t)]
S"E[S(t)]
1
m
1.0000
0.0000
Confidence limits for Si(t) 1.0000
1.0000 1.0000
21
0
0
1
--
1.0000
0.0000
1.0000
3
1
2
22
2
1
0
0.0909
0.9091
0.0584
0.7946
1.0236
4
2
3
19
5
0
0
0.2632
0.6699
0.0883
0.4968
0.8429
5
3
4
14
3
0
0
0.2143
0.5263
0.0968
0.3366
0.7161
11
0
0
0
--
0.5263
0.1092
0.3122
0.7404
11
0
0
0
m
0.5263
0.1092
0.3122
0.7404
11
2
0
0
0.1818
0.4306
0.0980
0.2386
0.6226
6 7 8
4
7
9
5
8
9
1
0
0
0.1111
0.3828
0.1002
0.1863
0.5792
10
8
0
1
0
~
0.3828
0.1063
0.1744
0.5912
11
7
0
0
3
~
0.3828
0.1137
0.1600
0.6056 0.5692
12
10
0
0
6
~
0.3828
0.0951
0.1964
13
6
12
16
4
0
10
0.2500
0.2871
0.0606
0.1683
0.4059
14
7
13
22
4
0
5
0.1818
0.2349
0.0438
0.1490
0.3207
15
8
14
23
4
1
6
0.1739
0.1940
0.0363
0.1228
0.2652
16
9
15
24
4
0
0
0.1667
0.1617
0.0302
0.1025
0.2209
20
2
0
0
0.1455
0.0301
0.0866
0.2045
17
a
From Pollock
et al. (1989a).
357
15.4. Analysis of Failure Times
suggested lower survival rates in 1986 (Fig. 15.4). However, the 1985 estimates were based on a single observed death, and therefore this test result should be viewed with some skepticism. A comparison of the Kaplan-Meier estimates for survival to week 9 resulted in $1(9) = 0.8571 with a 95% confidence interval of (0.7075, 1.0068), and 52(9) = 0.6250 with a 95% confidence interval of (0.4245, 0.8255). These results provide little evidence of an overall difference in survival.
1.2000 1.0000
=
UL
.---ii- LL
0.8000 0.6000 0.4000
0.2000 0.0000 0
5
10
15
20
F I G U R E 15.2 Kaplan-Meier survival estimates and 95% confidence interval for northern bobwhite data in Table 15.5.
1989a). S u m m a r y statistics and Kaplan-Meier survival estimates are presented in Table 15.5 and are plotted, together with 95% confidence interval estimates, in Fig. 15.2. The data in Table 15.5 illustrate the point that only the intervals over which death occurs contribute to survival estimation [although if Eq. (15.34) is used to estimate variances, changes in the at-risk sample due to censoring or staggered entry will influence confidence intervals]. The survival estimates were compared to those predicted under an exponential model, for which the parameter h was estimated by a regression of In S(t) versus t for the observed survival times (Fig. 15.3). The fit of the latter model to the survival data seems sufficiently strong to warrant further exploration of the exponential model as a competitor to the Kaplan-Meier model. The data were stratified into 9 weeks of observations in the fall of 1985, and these were compared to observations from the same 9 weeks 1 year later (Table 15.6). The value o f X2(1) - 4.13 (P = 0.04) indicated that the survival curves differed between the two years. A plot of the two survival distributions and the distribution from the pooled data
2.5000
2.0000 A
1.5000
~ 1.0000
9 Observed
o
- -
Predicted
0.5000 0.0000 -0.5000
~* 5
,
,
,
10
15
20
FIGURE 15.3 Plotof -log eS(t)for survival estimates in Table 15.4 and corresponding predicted -log e S(t) under exponential model.
Pollock et al. (1989a) also discussed the use of the log-rank test to partition causes of mortality, when death can be classified into two or more observed categories. The approach is to estimate survival based on one cause at a time, treating the other causes as censored at the time of death. As noted by Pollock et al. (1989a), this approach may be of limited value if the mortality sources are not independent.
15.4.4. Incorporating Explanatory Variables: The Proportional Hazards Model It may be of interest to estimate the influence on survival of factors that vary among individual animals (such as age or body mass), or vary over time (such as weather conditions), or both. Several approaches are possible. For example, the influence of body mass on survival could be investigated with a sample of marked animals that is divided into two samples, one above and the other below the median mass, using a two-sample binomial test or a log-rank test. One difficulty with this approach is that results from such a study may be highly variable, depending on the range in values of the predictor variable. Another is that it emphasizes the testing of a hypothesis, whereas one often is more interested in developing a quantitative response model. An alternative would be to model the responses using linear logistic regression (on assumption that the outcomes are simply successes or failures and all animals are monitored with equal intensity over the study period). The logistic approach allows one to quantify responses under a specific parametric form for the relationship between mortality and potential covariates. Yet another "semiparametric" approach is the proportional hazards (or Cox proportional hazards) model (Cox, 1972; Cox and Oakes, 1984), which allows for inference under some general assumptions about the nature of the relationships between the hazard function and covariates. The proportional hazards model postulates a hazard function of the form h(t, z) = ~(z; ~)h0(t),
358
Chapter 15 Estimation of Demographic Parameters TABLE 15.6
L o g - R a n k Test of S u r v i v a l D i s t r i b u t i o n s for N o r t h e r n B o b w h i t e s a
Fall 1985
rlj
Week
Total aFrom Pollock
Fall 1986
dlj
Total
r2j
d2j
r/
d/
E(r2j)
var(r2/) 0.250
7
1
7
0
14
1
0.500
6
0
6
0
12
0
0.000
0.000
8
0
11
1
19
1
0.579
0.244
13
0
10
0
23
0
0.000
0.000
18
0
16
1
34
1
0.471
0.249
18
0
15
0
33
0
0.000
0.000
18
0
15
1
33
1
0.455
0.248
18
0
14
0
32
0
0.000
0.000
18
0
14
3
32
3
1.313
0.691
124
1
108
6
232
7
3.317
1.681
et al.,
X2
df
P
4.126
1
0.042
1989a.
where ~(z; 13) is a link function (described below), z is a vector of explanatory variables measured on each of the study animals, ~ is a vector of parameters, and ho(t) is an unspecified function of time known as the baseline hazard. Under this general model the effect of, e.g., an experimental treatment is to scale the hazard with the term ~(z; 13) (Cox and Oakes, 1984). Parametric forms may be specified for ho(t) as in Section 15.4.2; however, the approach described below does not depend on parametric baseline hazards. Linear, logistic, log-linear, and other forms have been proposed for ~(z; ~) (Cox and Oakes, 1984), though the most popular appears to be the log-linear model. A general model that includes these forms and allows discrimination among special cases is ~(z, 13; K) = (1
For example, the linear and log-linear models are obtained from Eq. (15.37) by K = I and K --~ 0, respectively. In the latter case the link function is ~(z, ~) = e -~'z, which results in the proportional hazards model
h(t, z) = ho(t)e ~-'z.
This model is intuitively appealing in that it specifies a proportional relationship between the baseline hazard and the hazard at different levels of the predictors z. In its simplest form, a single predictor variable takes the values zero and one, in which case the hazard relationship in Eq. (15.38) is
h(t, 1) ho(t)
(15.37)
+ K ~ ' Z ) 1/K .
(15.38)
-ef~
or 1.1000 -
Combined
in h(t, 1) = 13,
----m-- 1985
1.0000 -
ho(t)
-~ 1986
with the parameter ~ thus interpreted as a log hazard rate. Equation (15.38) can be generalized to allow for time as well as individual specificity in the predictor variable:
0.9000 mR i
0.8000
-
0.7000
-
i
i
m..
m..,
-.,.
i
h[t, z(t)] = ho(t)e ~'z(t)
0.6000 0.5000
. 0
2
.
. 4
. 6
. 8
10
F I G U R E 15.4 Plot of survival distributions for 1985 and 1986 (data in Table 15.5).
An investigation of a special case of this model by Conroy et al. (1996) involved treating the locations of animals in habitats at each time t as time- and individual-specific predictors of hazard, and estimating the relationship between putative habitat suitability and fitness as measured by resistance to mortality risk.
15.4. Analysis of Failure Times
359
Example
15.4.5.3. Death Times Are Known Exactly
Pollock et al. (1989b) described an application of proportional hazards to investigate the relationship of body condition to overwinter survival of American black ducks (Anas rubripes). Fifty female black ducks were captured in southern New Jersey in 1983-1984, and an index to body condition was recorded for each duck (Conroy et al., 1989). Each duck was equipped with a radiotransmitter, released, and monitored through the following winter. The proportional hazards model
In many radiotelemetry studies, animals cannot be found immediately on death, so that when found, their times of death must be approximated. Obviously, the greater the amount of time elapsed since the last known time alive, the greater the potential for bias in estimation of survival rates, particularly if that time is long relative to the interval over which survival is estimated. Ideally, this issue can be minimized by a diligent search for animals at frequent intervals and by the use of auxiliary information such as the state of decomposition or degree of scavenging on carcasses of animals found dead.
h(t,
Zi) =
ho(t)e ~zi
was fit to the data, with z i the value of the condition index for each duck, i = 1, ..., 50. The proportionality parameter was estimated as ~ = 1.68 with a 95% confidence interval of (0.112, 3.248), suggesting that survival is positively related to body condition.
15.4.5. Assumptions of Failure Time Models As with other methods, failure time models depend on several assumptions that may or may not be met, depending on study design and field conditions.
15.4.5.1. Random Sampling A standard assumption for sampled populations, random sampling can be violated if capture or monitoring techniques are selective with respect to different components of the population. For example, certain kinds of trapping may be selective based on size, dominance status, or physiological condition of individuals. Violation of the assumption may result in the sample of radiotracked animals or other subjects not representing the target population and may lead to estimates of survival that are biased with respect to the population parameter of interest.
15.4.5.2. Independent Survival Times Violation of this assumption can occur if the fates of individuals are nonindependent, as might occur with mates or siblings in certain species. Under nonindependence, the usual assumptions of a multiplicative likelihood do not apply, and in consequence, parameter estimation and other inference can be considerably more complicated. In practice the main outcome of nonindependence may be an underestimation of sampling variation, resulting in estimated confidence limits that cover the true parameter value less frequently than the nominal (e.g., 95%) value.
15.4.5.4. Censoring Is Random and Independent of Survival Violation of this assumption can result in serious bias, particularly if a large proportion of censored animals represent failures. Examples of the violation of this assumption include the destruction of radiotransmitters by a hunter or predator, or the tendency for individuals to migrate from the study area (and thus be censored) because they are either more or less fit than other individuals. There is no solution to the issue of nonrandom censoring, except to try to minimize censoring systematically in the hope that whatever censoring remains will have a minimal impact on estimation.
15.4.5.5. Well-Defined Time Origin This assumption simply requires that a time origin should be specified and adhered to in estimation. Note that the assumption does not require all animals to be released at the same time. Thus, staggered entry designs (Pollock et al., 1989a) are admissible, with the time origin ordinarily set by the release of the first individuals. In fact, staggered entry designs may be preferable to release of all individuals at a single initial time, in that they provide "replacement" for animals that are lost to the sample through death or censoring, and in turn provide more precise estimation of survival rates for the duration of the study [cf. Eqs. (15.33) and (15.34)1.
15.4.6. Design of Radiotelemetry Studies Radiotelemetry has become one of the most important technical advances in the study of animal population dynamics. Though it often is used to monitor movement and behavior of individual animals, with appropriate designs, telemetry also can be applied in the estimation of population parameters, especially
360
Chapter 15 Estimation of Demographic Parameters
survival rates, and in some cases can be used to identify the sources of mortality. An obvious application involves the use of radiotelemetry to collect data for the analysis of failure times. Nevertheless, one should be aware of the technical and logistical challenges involved in using this technology. Here we discuss the design of radiotelemetry studies, for the purpose of estimating survival a n d / o r cause-specific mortality.
15.4.6.1. General Design In the investigation of survival with radiotelemetry, a general procedure is to capture and attach transmitters to n animals. One then periodically surveys the study area using fixed (e.g., towers) or mobile (handheld or mounted in vehicles, boats, or aircraft) radio receivers to locate the marked individuals and determine their fates (alive or dead); and if dead, the cause of mortality. The study is complete after a fixed study interval has elapsed (e.g., a 90-day wintering period), or else all animals have died, left the study area, or their radios have failed (e.g., the batteries are exhausted). The data collected may be simply the numbers of animals in each of two or more outcome classes or "fates," e.g., alive or dead (with further classification as to cause of death), in which case the methods of Section 15.3 may be appropriate. If information is recorded on the specific times over the study area at which fates occurred, the failure time methods of Section 15.4 may be useful.
15.4.6.2. Selection and Size of the Marked Sample By its nature, radiotelemetry provides the opportunity to "revisit" marked animals multiple times. However, each of these "visits" is essentially a repeated sample from a single experimental unit (the animal), and inference to a population requires that a sample be taken from a target population of interest (see Chapter 5). Ideally, the investigator will be able to obtain a sample via random selection from the target population. In practice, capture methods are selective, and this must be taken into account either in the design (e.g., poststratification from a captured sample, and random selection of individuals from strata), or else by incorporating auxiliary information about individuals into the analyses (e.g., see Section 17.1.7). Sample size is also a key consideration in design of a telemetry study. If too few animals are marked, estimates of survival will have poor precision (Chapter 5) and comparisons of experimental groups will have little power to detect differences among groups (Chapter 6). On the other hand, releasing too many marked animals may create logistical difficulties in tracking the fates of each animal, resulting in large numbers of censored individuals. For survival estimation, the bino-
mial distribution (Section 15.3) often can be used to provide an idea of precision and power for given sample sizes of released animals, even if more complicated models ultimately are used to analyze the data. Finally, it commonly is supposed that the ideal radiotelemetry design is to release all marked animals instantaneously and to determine fates over a common study interval. However, mortality and censoring (see below) both effectively remove subjects from the at-risk sample and therefore decrease the precision of estimates over subsequent intervals. In fact, "staggered entry" (Section 15.4.3) designs, in which animals are periodically released over the interval instead of only at an initial time, may provide higher precision over the course of the study.
15.4.6.3. Study Area Size, Study Interval, and Frequency of Monitoring These three study components are closely related and interdependent. The optimal size of the area for a radiotelemetry study of survival depends on the species and its mobility, the length of the study interval, and costs and other logistical constraints. For nonmigratory animals with small home ranges (e.g., a few square kilometers), it may be practical to detect all signals on a daily basis, because the animals occupy a small area. In other situations (highly mobile or migratory birds or mammals) it may take days or even weeks to search an area that is sufficiently large to detect all the signals, even where multiple crews and aircraft are employed (e.g., Conroy et al., 1989). Arbitrarily defining too small a study area means that many animals may go undetected, resulting in much censoring. Conversely, attempting to cover too large an area may mean that individuals are encountered only infrequently, resulting in inexact determination of the times or causes of death or censoring.
15.4.6.4. Determination of Fates and Censoring As suggested above, it is imperative that the fates of as many animals as possible be determined over the course of the study. In practice, radios fail, animals leave a searchable study area, and other events occur that leave the fates of some animals uncertain at the end of the study. Though censoring can be accommodated analytically, (Section 15.4.3), in general, the fewer censoring events the better. This is particularly true when censoring occurs early in the study interval, so that the individual contributes little information to the study. If the study area can be searched frequently, most animals can be found on a regular basis, and mortality or censoring, if they occur, will be recognized and the event times recorded accurately. In larger study areas with mobile animals, many
15.5. Random Effects and Known-Fate Data animals may be "missing" (i.e., not recently found) at any given time. Many of these subsequently may be located and their fates determined, and it is tempting to consider these animals retrospectively as part of the at-risk sample over the preceding time interval for the purpose of estimating time-specific survival rates ( e.g., Section 15.4.3). However, this temptation should be resisted, and the animals should be treated as censored (at the times last previously detected) and reentered under the staggered entry design (Section 15.4.3) at the time of next detection. The analytical methods that account for censoring require the assumption that censoring is random and independent of fate. We believe that this assumption frequently is violated in radiotelemetry studies. Most obvious is the fact that radiotransmitters tend to have accelerating failure rates as the end of battery life approaches, so that censoring times from this cause will tend to be distributed nonrandomly (i.e., near the end of the study interval). More critically, factors that affect the probability of mortality also can affect the probability that animals with an increased risk of mortality cannot be found, i.e., are censored. For example, in a radiotelemetry study by Conroy et al. (1989), at the same time that mortality risk increased due to severe winter weather, radioed ducks greatly increased their movement rates, and relocation of individuals became extraordinarily difficult. The authors suspected that the risk to mortality was greater for farther-ranging ducks than for those remaining closer to the release points, and that their rates of censoring were almost certainly higher. Likewise, Conroy et al. (1989) documented several instances in which transmitters were damaged by hunters (both incidental to harvest and deliberately) and one case in which a fox (Vulpes fulva) buried a duck and transmitter. Whether other transmitters remained undetected and therefore censored is, of course, unknown. Although it may be possible to construct models in which censoring and mortality act nonindependently, such models would require additional data on the censoring-mortality process that is unlikely to be gathered in field studies. Problems with nonrandom and nonindependent censoring are probably inevitable, but should be minimized if the rate of censoring is low relative to hazard over the course of the study. If high rates of censoring cannot be avoided, investigators must be aware that the resulting estimates of survival, and inferences about mortality processes, may be sensitive to violations of randomness and independence of censoring.
15.4.6.5. Effects of Radios Biologists have observed apparent behavioral and other impacts of radiotelemetry on animals, including
361
apparent impacts on survival and other demographic rates [e.g., Burger et al. (1991), but see Powell et al. (1998)]. Clearly, investigators must be aware of these potential impacts and should understand that study interpretation may be compromised if the effects are severe. Ideally, the effects of transmitters should be investigated via experimental studies under comparable conditions, and the results used to adjust estimates from field studies. Investigators should be especially alert to short-term behavioral and survival effects of transmitters that may make observations of animal movement and fates highly suspect for several hours to several days following release. In most cases, it is advisable to establish a postrelease "adjustment period" and ignore or heavily discount events in that period (frequently, up to 48 hr following release). In some cases, effects undoubtedly persist into the study interval and result in survival and other estimates that are nonrepresentative of the target population. Certain comparative inferences (e.g., relative survival rates between sexes or experimental groups) may be possible, if the radio effect does not interact with the factor under study. However, if radio effects are substantial but cannot be estimated or otherwise controlled in the design and analysis, then the use of radiotelemetry probably should be avoided in favor of other methods (Chapters 16-19).
15.5. R A N D O M EFFECTS A N D K N O W N - F A T E DATA With the exception of the nest survival model of Natarjan and McCulloch (1999), the known-fate models presented above assume either constant survival probabilities among groups of individuals or survival probabilities that vary as a function of measured covariates. Animals may be stratified by age, sex, or other characteristics, but within strata survival is assumed to be the same over all animals. Although the above methods represent the state of the art as of the publication of this book, we expect random effects approaches to survival rate estimation to become useful for both telemetered animals and nest data. Link et al. (2002) and Cam et al. (2002) have developed a model for kittiwake (Rissa tridactyla) capture-resighting data, in which the logits of both survival and reproduction rates are modeled as a linear function of age, calendar year, and individual bird. Because it is neither feasible nor practical to incorporate parameters for each individual, Link et al. (2002) chose to treat individual effects as bivariate random effects. An unobservable pair of latent effects was associated with each bird, and these effects were assumed to have a bivariate normal distribution with mean zero
362
Chapter 15 Estimation of Demographic Parameters
and variance-covariance matrix ~. The effects thus were described by two variances [for survival rate and conditional (on survival) reproductive rate, respectively] and a covariance relating the latent effects. Estimation under this model requires a Bayesian approach using Markov chain Monte Carlo methods, as implemented with the software BUGS (Gilks et al., 1996; Gelman et al., 1997). Capture-recapture data seldom can be treated as known-fate data (see Chapters 17-18), and this particular data set is exceptional in that regard, with birds having resighting probability estimates >0.99 (e.g., Cam et al., 1998). The analysis of Cam et al. (2002) and Link et al. (2002) provided strong evidence that the latent factors governing survival and conditional (on survival) reproduction were not independent. Instead, they exhibited a positive correlation (mean and median values of posterior distribution of about 0.69), providing evidence that individuals with high survival probabilities also showed high reproductive rates, conditional on survival. Thus, there was substantial variation in individual "quality," as indicated by these two important fitness components, and there was no evidence of a "tradeoff" between latent survival and reproductive parameters of individuals. This is the first analysis of which we are aware that permits inference about the correlation structure of fitness components at the individual level. The ability to focus on latent parameters at the level of the individual also permitted Cam et al. (2002) and Link et al. (2002) to address questions about age-specific changes in individual survival probabilities. When age-specific survival probabilities are estimated at the population level using groups of animals at different ages (e.g., Nichols et al., 1997), mortality selection can produce patterns that differ substantially from patterns of age-specific change within individuals (e.g., Manton et al., 1981; Hougaard, 1984, 1986; Vaupel and Yashin, 1985; Johnson et al., 1986). For example, assume individual heterogeneity in survival probability that remains constant over the life of the individual. As such a heterogeneous cohort ages, mortality selection results in disproportionate death of the individuals with low survival probabilities, producing an increase in survival with age at the population level, despite the absence of age-specific change within individuals. Regardless of the findings regarding age specificity of survival at the population level, there is always a question about the relevance of such findings to patterns within individuals (e.g., Burnham and Rexstad, 1993; McDonald et al., 1996; Nichols et al., 1997; Service, 2000). Indeed, the kittiwake analysis of Cam et al. (2002) permitted estimation of age-specific changes in survival of individuals and provided strong evidence of
senescent decline in survival probability. However, at the population level, survival was relatively constant with age. When substantial heterogeneity actually exists, certain kinds of demographic and evolutionary questions are very difficult to address with models that assume constancy of parameters over individuals. Strong inference in the face of heterogeneity requires randomeffects models of the sort explored by Cam et al. (2002) and Link et al. (2002). The Markov chain Monte Carlo methods used by Cam et al. (2002) and Link et al. (2002) offer a Bayesian approach to estimation that is very promising. The approach is based on probabilistic models that are similar to those described in this and following chapters, except that some of the parameters of the more usual frequentist approach (e.g., survival probability of animals of a specific age) are treated as random variables in a Bayesian analysis. Thus, the estimation problem is no longer to estimate an agespecific survival probability, but instead to estimate characteristics of the distribution of individual survival probabilities. Interest in this area of investigation is growing, and we anticipate that the professional literature on estimation methods will be dominated increasingly in future years by approaches that permit such random-effects modeling.
15.6. D I S C U S S I O N In this chapter we have described principles for estimation of survival and other parameters, and we have considered some methods that commonly are used to obtain estimates of survival rates. Methods based on the analysis of age frequencies are attractive, in that age data are commonly available, e.g., from samples of harvested animals. However, they depend on assumptions about sampling or demographic processes (or both) that are untested, unlikely to be met in practice, and have serious consequences in terms of parameter bias. If alternatives to these methods are unavailable, estimation should proceed along the lines described in Section 15.2.2, with the exercise of considerable caution in interpreting survival rates and other parameters when assumptions cannot be evaluated. In Sections 15.3 and 15.4 we considered estimation of survival in nesting and radiotelemetry studies, wherein subjects can be visited repeatedly (as in the monitoring of nests at known locations, or the monitoring of radiomarked animals). For designs and data structures such as those described in Sections 15.3 and 15.4, useful approaches are available for the estimation of survival rates and other demographic parameters
15.6. Discussion (e.g., nesting success as a component of reproduction). These methods are both reliable and flexible, and allow for the modeling of features such as group and covariate (e.g., proportional hazards models) effects. In the chapters to follow we focus on estimation in situations in which inference is based on incomplete samples of animals encountered at each of several sampling occasions. The methods to be described involve the tracking of samples of marked animals over time, utilizing estimates of sampling probabilities in the estimation of survival rates (Chapters 16-19) and, for certain designs, recruitment (reproduction) and abundance (Chapters 18-19). The focus in Chapter 16 is on band recovery models, in which the analysis of band recoveries is conditional on known numbers of released animals. This design and the capture-recapture models considered in Chapter 17 permit estimation of survival and movement rates, but do not allow inferences about abundance or
363
recruitment. In both chapters, emphasis is placed on model selection and evaluation, and on the construction of models for examination of temporal, spatial, and other sources of variation in survival and movement rates. In Chapter 18 the focus shifts to open population models for which initial captures as well as recaptures are modeled, thereby permitting inference about abundance and recruitment. Chapter 19 covers the robust design of Pollock (1982), in which the closed models of Chapter 14 are combined with open-population models of Chapters 17 and 18 in order to estimate abundance, survival, movement, and recruitment. Finally, Chapter 20 describes new methods for the estimation of community parameters. These methods extend capture-recapture methodology to the analysis of community-level statistics and provide a statistically rigorous and comprehensive treatment of the data structures used in estimation of species richness, species turnover, and other community-level statistics.
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C H A P T E R
16 Estimation of Survival Rates with Band Recoveries
16.5. POSTSTRATIFICATION OF RECOVERIES AND ANALYSIS OF MOVEMENTS 16.5.1. Banding and Recovery Areas Differ in Time and Space 16.5.2. Banding and Recovery Areas Coincide 16.6. DESIGN OF BANDING STUDIES 16.6.1. General Considerations 16.6.2. Determination of Banding Sample Sizes 16.7. DISCUSSION
16.1. SINGLE-AGE MODELS 16.1.1. Sampling Scheme and Data Structure 16.1.2. Probabilistic Modeling 16.1.3. Reduced-Parameter Models 16.1.4. Temporary Banding Effect 16.1.5. Multiple Groups 16.1.6. Covariates 16.1.7. Banding Multiple Times per Year 16.1.8. Evaluation and Selection of Models 16.1.9. Estimator Robustness and Model Assumptions 16.1.10. Interpretation of Sampling Correlations 16.1.11. Examples 16.2. MULTIPLE-AGE MODELS 16.2.1. Data Structure 16.2.2. Probabilistic Modeling 16.2.3. Reduced-Parameter Models 16.2.4. Temporary Banding Effect 16.2.5. Unrecognizable Subadult Cohorts 16.2.6. Group and Covariate Effects 16.2.7. Banding Multiple Times per Year 16.2.8. Model Comparison and Selection 16.2.9. Cohort Models and Parameter Identifiability 16.3. REWARD STUDIES FOR ESTIMATING REPORTING RATES 16.3.1. Data Structure 16.3.2. Modeling Survival Rates with Indirect and Direct Recoveries 16.3.3. Modeling Spatial Variation in Reporting Rates with Direct Recoveries 16.4. ANALYSIS OF BAND RECOVERIES FOR NONHARVESTED SPECIES 16.4.1. Data Structure 16.4.2. Probabilistic Models 16.4.3. Banding of Juveniles OnlymCohort Band Recovery Models 16.4.4. Estimation When Numbers of Banded Birds Are Unknown
In this chapter we describe procedures for estimating survival, recovery, and harvest rates, and other key parameters based on recoveries of tags. Recoveries can occur in any of a number of ways (subject to certain assumptions discussed below) but typically come from animals that have been tagged, released, and subsequently (1) are found dead and reported or (2) are harvested, retrieved, and reported by hunters or anglers. The data structure and statistical methods involved are similar in m a n y ways to the CormackJolly-Seber (CJS) models considered in Chapter 17. Like them, band recovery models focus on the estimation of survival rates along with probabilities associated with the sampling process, but not on other demographic parameters such as recruitment and abundance. Depending on the specific study design and nature of the recoveries, sampling probabilities may be informative of harvest rates or other sources of mortality. In some cases, the appropriate interpretation requires different parameterizations, depending on the specific data structure and questions being addressed. The same basic statistical framework as with capture-recapture studies applies:
365
366
Chapter 16 Estimation with Band Recoveries
9 A multinomial model is used to describe field data collected over several periods (usually several years). An individual banded in a given period is recovered in a subsequent period (or fails to be recovered) with a probability that is specific to the banding and recovery years. 9 The recovery probabilities are "modeled" with parameters of biological importance, in such a way that simplifying assumptions can be incorporated directly into the statistical model. 9 Maximum likelihood estimates are derived from the multinomial likelihood function, along with variances, covariances, and other statistical information. 9 Goodness-of-fit tests are used to test how well a particular model "fits" the field data, and a model selection procedure is used that incorporates Akaike's Information Criterion (AIC) (Akaike, 1973) a n d / o r likelihood ratio testing to compare models and identify the "best" model. Though the modeling process is similar to that of capture-recapture experiments, there are some key differences. Of most importance is the nature of the data and the manner in which they are collected: 9 In both band-recovery and capture-recapture studies the tagging of unmarked animals occurs in each of several sampling periods. However, in capture-recapture studies, individuals may be recaptured many times subsequent to tagging, whereas in bandrecovery studies, banded individuals typically are recovered only once. Thus the data requirements are different: for capture-recapture experiments it is necessary to record multiple captures for some individuals, whereas band-recovery approaches require the recording of only the period of banding and the period of recovery. 9 In capture-recapture experiments, the tagging of unmarked individuals and recapture of marked individuals occur at the same time during well-defined sampling periods, and the data are collected by the same people. In band-recovery studies, banding and recovery occur at different times, and the data are collected by different people. Banding is conducted during well-defined sampling periods, but recoveries may occur during long periods that sometimes include the entire interval between banding periods. 9 In capture-recapture experiments the numbers of both marked and unmarked individuals in a sample typically are viewed as random variables that may be influenced by such things as sampling intensity and behavioral responses to trapping. In some bandrecovery studies, on the other hand, a target number of individuals to be banded in each period is specified a priori, as part of the study design. Any recaptures
of previously banded individuals during the banding exercise are disregarded (but see Section 17.5), and all subsequent analysis of band recoveries is conditional on the number of individuals banded in each period. 9 In capture--recapture experiments the number of tagging periods typically is the same as the number of recapture periods: There are k - 1 opportunities to tag individuals (not counting the last sampling period) and there are k - 1 opportunities to record recaptures (not counting the first period). In band-recovery studies, on the other hand, there can be more recovery periods than banding periods. In what follows we designate the number of banding periods as k and the number of recovery periods as l.
16.1. SINGLE-AGE
MODELS
In this section we consider models in which every individual can be considered to be of a single age, with survival and sampling (recovery) probabilities identical for all individuals in the sampled population. In Section 16.1.1 we introduce the data structure for a single-age model and discuss in Section 16.1.2 the standard Seber-Robson-Youngs (SRY)(Seber, 1970b; Robson and Youngs, 1971; Brownie et al., 1985) model with time-specific survival and recovery probabilities. In Section 16.1.3 we introduce several reduced-parameter models that can be obtained by constraining parameters to be constant over time, and in Section 16.1.4 we generalize the SRY model to allow for a temporary banding effect. In Section 16.1.5 we further generalize the SRY model to allow for group-specific parameterization of survival and recovery, as might occur when banding data are stratified by sex or geographic locations. In Section 16.1.6 we allow for incorporation of covariates in the modeling of timespecific and individual variation in parameters, and in Section 16.1.7 we describe models for banding at multiple times per year, allowing estimation of survival rates over seasons or other periods of interest. In Section 16.1.8 we present approaches to the selection of a particular model from a class of candidate models, taking into consideration model goodness of fit and parsimony in the model parameterization. In Section 16.1.9 we consider model assumptions and the impacts of assumption violations, and in Section 16.1.10 we address the interpretation of covariance in parameter estimates. Finally, we close the discussion of single-age models in Section 16.1.11 with several examples of the aforementioned data structures and models.
367
16.1. Single-Age Models
16.1.1. Sampling Scheme and Data Structure We begin w i t h the general situation that m o t i v a t e d the w o r k of Brownie et al. (1978, 1985), w h e r e b y a m i g r a t o r y bird p o p u l a t i o n is subjected to a n n u a l sport hunting. We a s s u m e that the b a n d i n g of individuals in the p o p u l a t i o n occurs once a year. For example, b a n d i n g occurs in the period just prior to or just after the h u n t i n g season. Bands of animals released in each year i = 1.... , k subsequently are recovered as a result of hunting, w h e r e i n a b a n d e d animal is harvested and retrieved, a n d the b a n d r e t u r n e d to a central b a n d repository. Recovery occurs either in the same year of
(a)
~
~
S
K
b a n d i n g (j = i), in a s u b s e q u e n t year (j = i + 1 to l), or not at all (denoted as l + 1). The possible fates of an i n d i v i d u a l b a n d are d i s p l a y e d in Fig. 16.1. The data for a b a n d recovery s t u d y are the n u m b e r a i of individuals b a n d e d and released in year i, i = 1.... , k , a n d the n u m b e r mij of animals released in year i and recovered in year j, j = i to l + 1. We note here that Brownie et al. (1985) use a s o m e w h a t different notation, with Ni and Rij d e n o t i n g releases a n d recoveries, respectively. Because of the close relationship of b a n d recovery m o d els to conditional CJS m o d e l s (Chapter 17) w e have chosen to use a c o m m o n notation, to e m p h a s i z e the close correspondence of the t w o data structures a n d m o d e l i n g approaches.
Survives to next banding period
S Tagged and released just before harvest period
"
1
Dies from natural causes
~ Not retrieved
,"v--.. Shot
_ by hunter
Not reported Reported
("band recovery")
(b)
~
~
Survives to next banding period
S Tagged and released just before harvest period
~'"
f
Killed, retrieved, and reported ("band recovery") Killed, but not reported (crippling loss or nonreporting) OR died from other causes
(c)
~
~
Survives to next banding period
S Tagged and released just before harvest period
~ ~ " ~
(1 - S )X
Dies and is found and reported ("band recovery")
(1 Dies but is not found or found and not reported
FIGURE 16.1 Possible fates of animals banded in harvest and nonharvest situations. (a) Fates of harvested animal separated by kill (K), retrieval (c), and reporting (X). (b) Fates of harvested animals identifiable only as recovered (retrieval and reporting not separately estimable). (c) Fates of nonharvested animals, where recovery is finding of dead animals. Bands are recovered in all three scenarios.
Chapter 16 Estimation with Band Recoveries
368
To illustrate, consider a situation in which bands are placed on randomly sampled individuals in three successive years, and bands are recovered during four years. For this example i = 1, 2, or 3 depending on banding year, and j = 1, 2, 3, or 4 depending on recovery year. The structure of the data can be displayed in a triangular array (Table 16.1), which makes it clear that bands can be recovered only in the banding and postbanding years. Thus, first-year bands can be recovered in years 1 through 4, second-year bands can be recovered only in years 2 through 4, and third-year bands are recovered only in years 3 and 4. The sum of recoveries is less (typically much less) than the number banded, because many bands are never recovered: the banded individuals either continue to survive past the termination of the study, they die from other nonhunting causes and are not found, they are killed by a hunter but not retrieved, or they are killed and retrieved but the bands are not reported (Fig. 16.1).
16.1.2. Probabilistic
Modeling
=
RI!
( R l - ~ i - m F) r 'IT 119 1"1"15
TABLE 16.1
Data Structure for Single-Age Band Recovery Problem a Recoveries in period j
Releases in period i
1
2
3
4
Not recovered
R1
mll
m12
m13
m14
R1 - ~,j mlj
m22
m23
m24
R2 - ~,j m2j
m33
m34
R3 - ~j m3j
R2 R3 a With k = 3 banding
=
R2!
H j m2j!(R2 - ~'j m2j)! X
"IT 2j ,.n.(R2-Ejm2j) "" 25
~
r
and likewise for the third-year bandings: R3! Pg(m3]~3) = Hj m3j!(R 3 - ~_,j m3j)! X
"IT 3j ,.rr(R3-Ejm3j) "" 35
-
r
Corresponding to the above data in Table 16.1, the parameters for these three distributions can be displayed in a triangular array, 11" 12 '11"13 qT14
"rr15
'rr22
'rr23
'rr24
"IT25
I1"33
'rr34
'n'35
where the probabilities in each row sum to one and the last column designates the probabilities of not being recovered. The probabilities "rrq can be expressed as products of annual survival probabilities (the annual survival rates) and the product fj of harvest and band reporting rates, known as the recovery rate. We assume throughout that the annual survival rate is not influenced by the year of banding. For example, individuals banded in years i and i + I and alive in year j > i + I experience the same probability of survival from year j to j + 1. Under this assumption, "rrq can be modeled as "rrij = 5 i 5 i + 1 "'" S j _ l f j ,
H/mlj!(R 1 - ~,,j mlj)! )K
P2(m2],rr2)
'rr11
Here we focus on band recovery models involving the banding and recovery of adults only, such that all individuals banded in a particular year i are subject to the same probability of recovery in year j. Consider for example a 4-year study with 3 years of banding (Table 16.1). One of five possible outcomes awaits any individual banded in the first period of the study: it can be recovered in year 1, 2, 3, or 4, or it can fail to be recovered at all. Let "rr11, 'IT12, "IT13, Tr14, and 'rr15 = 1 - 11"11-- Tr12 -- 11"13 -- '1T14be the probabilities for these outcomes. If individual recoveries are independent, the appropriate statistical model for this situation is the multinomial distribution Pl(ml],rrl)
where {m_1} denotes the set of recovery statistics mll , m12, m13, m14, and m15. A similar model is appropriate for the second-year bandings:
years and l = 4 recovery years.
(16.1)
where S m and fm are, respectively, the survival and recovery rates for year m. Then the expected number of band recoveries in year j, for birds banded in year i, is just the product E(mq) = Ri'rrij of the number of individuals banded in year i and this probability. In the example with three banding periods and four recovery periods, the array of expected returns is displayed in Table 16.2. Each row of the table corresponds to a multinomial distribution, with expected cell frequencies given by the table entries. The multinomial probabilities are simple products of annual survival rates and recovery rates, the parameters of biological interest. Note that the parameters $3 and f4 in Table 16.2 always occur as the product Sgf 4. Because of this cooccurrence, it is possible to estimate the product but not the individual parameters $3 and f4. In general, a
16.1. Single-Age Models TABLE 16.2 Expected Recoveries for S i n g l e - A g e Band Recovery Data in Table 16.1 under M o d e l (St, ft) a Recoveries in period j
Releases in period i al R2
1
2
alfl
R3
3
Not recovered
4
TABLE 16.3
369 Data Structure for S i n g l e - A g e Band Recovery Data a Recoveries in period j
Releases in period i
1
2
3
ml I
ml 2 m22
alSlf2
a181S2f3
a151S253f4
al-
~,jE(mlj)
R1
R2f 2
R2S2f 3
R25253f 4
R2 - ~,j E(m2j)
R2
RBf 3
RBS3f 4
a3 - ~,j E(mBj)
R3
""
l
Not recovered
ml 3
...
ml I
R1 m ~,j ml j
m23
...
m21
R2 - ~j m2j
m33
...
m31
R3 - ~j m3j
...
mkl
Rk -- ~j mkj
aWith k = 3 b a n d i n g y e a r s a n d l = 4 r e c o v e r y years. P a r a m e t e r s are a n n u a l s u r v i v a l rate S i, i = 1, 2, 3, a n d a n n u a l r e c o v e r y rate fi, i = 1 ..... 4.
Rk
band recovery model with k banding periods and l recovery periods includes the parameters S i, i = 1, ..., l - 1, and fi, i = 1, ..., I. If l > k the products S k fk+l, S k S k + l f k + 2, ..., and Sk "'" S l - l f l can be estimated, but not the individual parameters S k, ..., S l_ 1 and f k + l , -.., fl. Thus, only the parameters Si, i = 1 . . . . , k - 1, and fi, i = 1, ..., k, can be separately estimated. Estimation of the l - k products mentioned above brings the total to l + k - 1 estimable parameters for each year of banding. We assume that the year-specific distributions of band recoveries are independent, so that the joint distribution for data across all years is simply the product of the three distributions: 3
P ( m l , m__2, m__3ITll, '112, "IT3) -- H
P i ( m i "rri).
i-1
When data are substituted into this expression, we have the likelihood function for a band recovery experiment involving three banding periods and four recovery periods. The model generalizes in an obvious way to include additional banding years a n d / o r additional recovery years. Thus, recoveries for the general case can be put in a rectangular array that includes k rows, l columns (an l + 1st column could be included to account for individuals never recovered), along with a leading colu m n designating the number banded in each year (Table 16.3). The sum of counts for row i (up to the last column) is the total number of recoveries from the ith cohort. The sum of counts for column j is the total number of recoveries from all cohorts in year j. The general likelihood function for this situation is expressed as the product k
C('n'l,
"", 'rl'kIml . . . . , m__k) = I-[ Pi(mi]'rri) i=1
(16.2)
of the multinomial likelihood functions for each banding period. As before, multinomial probabilities for the general model have a rectangular form with k rows and l + 1 columns (the last column is the probability
'~With i = 1 ..... k b a n d i n g y e a r s a n d j = i..... l r e c o v e r y years.
of a band never being recovered). The sum of probabilities for row i (up to the last column) gives the probability of recovery at some time during the study for individuals banded in period i. Expected cell counts for the general model are displayed in Table 16.4. The expected values for single age-class band recoveries illustrate some of the similarities and differences between the modeling of mark-recapture (Chapters 14 and 17) and band recovery data: 9 Both approaches allow for time-specific probabilities -rrq of recapturing or recovering individuals that are released after initial capture and marking. In mark-recapture experiments of closed populations survival rates are not components of these probabilities, because population closure implies a survival rate of 1. In band recovery studies, on the other hand, survival rates are included in the probabilities ~rq, because they are necessarily less than 1 (the population is harvested) and individuals recovered j periods after banding must have survived j - 1 periods prior to harvest and band recovery. 9 Though both situations include individuals that are never seen, the size of this cohort can be important in capture-recapture studies but is irrelevant in band recovery studies. In the case of some capture-recapture models, the objective is to use information in the capture histories of individuals seen, to determine the number of individuals never seen. In band recovery experiments, on the other hand, the goal of the study is to make inferences about survival rates and other sampling parameters, rather than to determine total population size. The number of unbanded individuals in the population is not germane to this goal. However, because not all marked animals are seen again in band recovery studies, the issue of incomplete detectability comes into play just as it does in population estimation, in that only the animals recovered each year are known
Chapter 16 Estimation with Band Recoveries
370 TABLE 16.4
Expected Recoveries for S i n g l e - A g e Band Recovery Data under M o d e l
(St, ft )a
Recoveries in period j
Releases in period i
1
2
3
-"
aI
alf 1
alSlf 2
alSlS2f 3
a2f 2
R2S2f 3 R3f 3
R2 R3
l
Not recovered
...
a l S i S 2.... Sl_lf l
a I - ~j E(mlj)
...
R2S253 ..... S l - l f l
R2 - ~j E(m2j)
...
R3S3S 4.... S l - l f 1
R3 - ~,j E(m3j)
RkSkSk+ 1.... Sl_lf 1
R k - ~,j E(mkj)
Rk
aWith i = 1..... k banding years and j = i..... l recovery years. Parameters are annual survival rate (Si, i = 1..... l - 1) and annual recovery rate (f i, i = 1..... l).
to have been alive u p to the point of recovery (see Section 15.1). As seen below, the s a m p l i n g probability or rate of detection is m o d e l e d by recovery rates, in the s a m e w a y that capture a n d recapture rates are used in c a p t u r e - r e c a p t u r e analysis.
E(mi. ) = E(mii +
16.1.2.1. Estimation The single-age, t i m e - d e p e n d e n t b a n d m o d e l has the identifiable p a r a m e t e r s i = 1,...,k-
Si,
entiation of the log likelihood function, as described in C h a p t e r 4. However, the MLEs for this m o d e l also can be obtained directly from the m o m e n t estimators E(mi.) -- mi. and E(m.j) = m.j of r o w a n d c o l u m n totals for the cell counts, w h e r e
= Ri[fi + Sifi+ 1 -]-...
recovery
1
The p a r a m e t e r s S i for i > k - 1 and fi for i > k are not separately estimable, but the p r o d u c t s S k ... S k + s _ l f k + s ~ ,
q- ( S i S i + 1 i f - . . . - ] -
Sl_lfl) ]
1 ""
Sj_lfj]
if- [ a 2 s 2 " ' " S j _ l f j ]
q - " ' " q- a j f j .
After substitution of mi. a n d m.j for E(mi.) a n d E(m.j), some algebraic m a n i p u l a t i o n leads to a solution of these equations for the MLEs:
',m'[ mi]i.l
s=l,...,l-k
Si = --a-: 1 are estimable a n d therefore are included in the likelihood. Algebraic, closed-form estimators for the p a r a m eters a n d their a s y m p t o t i c variances can be obtained t h r o u g h application of m a x i m u m likelihood m e t h o d s (Chapter 4) or the m e t h o d of m o m e n t s , as s h o w n below. In practice, closed-form solutions are available only for fully p a r a m e t e r i z e d models, a n d estimates can be obtained for r e d u c e d - p a r a m e t e r m o d e l s only by application of n u m e r i c a l algorithms. We p r o v i d e closed-form estimators for the simple e x a m p l e of b a n d recovery for animals of single (adult) age class, w h e r e the data are unstratified by sex or other attributes. Expressions for more complex m o d e l s (e.g., for multiple age classes) are s t r a i g h t f o r w a r d extensions, and references for these are p r o v i d e d for the interested reader. P a r a m e t e r estimates in a fully p a r a m e t e r i z e d b a n d recovery m o d e l can be obtained t h r o u g h partial differ-
rail)
E(m.j) = E ( m l j + m2j + "" + mjj) = [alS
i = 1, ..., k.
+
and
and fi,
m i , i + 1 -}- . . .
,
i - 1 , ..., k - 1
(16.3)
mi+l
and
f
i
m i.m.i -- RiTi ,
i = 1, ..., k
(16.4)
(Seber, 1970b; Robson a n d Youngs, 1971), w h e r e T i is a total (across all b a n d i n g years u p to and including i) of all recoveries in years including and s u b s e q u e n t to i; i.e., Ti = mi. for i = 1, Ti = Ti-1 - m.i-1 + mi. for i = 2 .... , k, a n d Tk+ s = Tk+s_ 1 -- m.k+s_ 1 for s = 1, ..., l -- k if l > k. As m e n t i o n e d above, the p a r a m e t e r s Si for i > k and fi for i > k are not separately estimable. However, estimates of the p r o d u c t s Skfk+ 1, SkSk+lfk+2~, etc. are given by Sk "'" S k + s - l f k + s =
mkmk+s RkTk
(16.5)
16.1. Single-Age Models for s = 1, ..., l - k (Brownie et al., 1985). Though these values usually are not of interest in their own right, they are required for goodness-of-fit testing. Estimation procedures for the fully parameterized band recovery model are discussed in Brownie et al. (1985). Estimation software is noted in Appendix G. To see how band recovery analyses are related to other "incomplete count" estimators discussed in previous chapters, consider a banding study in which animals are banded and released in each of two years in sequence, and recoveries are obtained from postrelease samples over the two years. The data structure for this problem can be described by
al R2
mll
m12 m22
which is represented in terms of the parameters of the time-specific model as al R2
Rlfl
RISlf2 R2f2
Now take M12 as the true (unknown) number of animals from our released sample (R 1) that survive until the beginning of year 2 [by definition E(M12) = alS1]. Consider m12 to be our index or incomplete count statistic representing M12. We need to adjust this statistic by the sampling fraction of M12 that m12 represents, which in this case is f2, the probability of appearing in the second recovery sample, conditional on being alive at the beginning of the sampling period. A natural estimate of f2 is provided by the recoveries of the marked animals released just preceding this period, that is ?2 = m22/R2,
so that
~/I12 = m12/f2 = m12/(m22/R2) ,
which is of the form of the canonical estimator =
in Eq. (12.1), where f~ = f2. Finally, the estimator of $1 derives from the relationship E(M12) -- RIS1, so that
S1 = ~'I12/a1
(16.6)
= m12R2/m22R 1 .
This derivation is recognizable as a special case of the general approach to survival rate estimation outlined in Section 15.1.2. In general the maximum likelihood estimator for survival rate [Eq. (16.3)] is biased. An adjustment that effectively eliminates bias involves the incrementing
371
of Ri+ 1 and mi+l. by one unit in the formula for Si" Thus, an approximately unbiased estimator for S i is given by
mi t m;t ( i'l+l 1 1
Si-- ~
1 -
\mi+l. +
(16.7)
'
i = 1, ..., k - 1 (Brownie et al., 1985). Because the MLE for recovery rate is unbiased (Robson and Youngs, 1971), an analogous adjustment of the computing formula for fi is not required. Variances and covariances for the parameter estimates can be obtained by application of the delta method (Appendix F) to the moment estimators above, with the variances of m.i and mi. obtained from the multinomial distribution (Brownie et al., 1985). Alternatively, variance estimation may be based directly on the likelihood, via the Fisher Information Matrix (Appendix F), the approach used in modern estimation software such as MARK (White and Burnham, 1999) (see also Appendix G). Regardless of the estimation approach, many of the estimators are uncorrelated (i.e., have covariances of 0). However, Si (Si) is negatively correlated with Si+l (5i+1) and fi+l" The magnitude of the covariance is given by the product of the respective parameters, adjusted by a factor that involves the size of the banded cohort and the total number of recoveries. The existence of nonzero sample correlations among these estimators must be kept in mind, especially when making comparisons among estimates, to avoid spurious conclusions resulting from sampling correlation rather than a true association among the underlying parameters. Following general notation of Lebreton et al. (1992) we denote the model described above as model (S t, ft), where the subscipt t denotes variation in the survival and recovery parameters over time, absent any age, sex, or other stratification in parameters. Model (St, ft) is identical to Model I of Brownie et al. (1985). Numerous procedures exist for obtaining maximum likelihood estimates of the parameters of the model and other reduced-parameter models derived from it. For most applications the numerical estimation procedure implemented in MARK (Appendix G) provides a general approach for obtaining MLEs and for investigating sources of variation in survival and recovery rates via goodness-of-fit and model selection procedures (Section 16.1.8).
16.1.3. Reduced-Parameter Models Straightforward restrictions on the parameters in a band recovery model lead to three simplified models:
Chapter 16 Estimation with Band Recoveries
372
9 Constant survival rates across years (Sj = S, j = 1, ..., l - 1) and variable recovery rates, denoted as model (S, ft'J. 9 Constant recovery rates (fj = f, j = 1, ..., l) and variable survival rates, denoted as model (St, f). 9 Constant survival and recovery rates (Sj = S, j = 1, ..., l - 1; ~ = f, j = 1, ..., l), denoted as model (S, f). Models (S, ft) and (S, f) are identical to Model 2 and Model 3, respectively, of Brownie et al. (1985). There is no model in the Brownie et al. (1985) methodology corresponding to (S t, f), because this model is seldom applicable to the waterfowl data sets for which the model set in Brownie et al. (1985) was developed. Each of these reduced-parameter models is specified by a distinct parametric structure for the multinomial cell probabilities. As with capture--recapture models, the standard approach involves an incorporation of parameter assumptions into the model likelihood function, from which m a x i m u m likelihood estimates are derived. The MLEs then are used to examine the goodness of fit of the model and also to compare models as part of a model selection procedure (Section 16.1.8). It is not possible to derive explicit formulas for estimates of the parameters for the models with parameter restrictions on survival and recovery rates, and a numerical procedure is required to solve the likelihood equations. Models (S, ft) and (S, f) include only a single survival parameter that is constant over time periods, with a total of l+ 1 and 2 parameters, respectively. In the usual case of equal time intervals for banding (e.g., every year at the same time), the constant survival parameter has a biological interpretation. However, in general, intervals between banding may be unequal, in which case the assumption Si = S is likely to be violated simply because of variation in the interval over which survival is estimated, irrespective of true temporal variation in survival per unit time. A solution to this problem is to model survival as
the estimation of taxonomic extinction rates from stratigraphic range data, in which the time intervals between sampling "occasions" (i.e., geologic strata) obviously varied. Both of these applications are handled easily in program MARK by specifying the length of intervals between sampling occasions and thus eliminating the need for special models to handle this situation (e.g., Conroy et al., 1989b). 16.1.4. T e m p o r a r y B a n d i n g E f f e c t
A useful generalization of the band recovery model allows for a temporary banding effect, in which newly banded birds experience a probability of being shot and having their bands recovered that differs from that of previously banded birds. This situation m a y apply for bands recovered near banding sites, where the harvest of banded birds consists primarily of newly banded birds and band reporting rates are lower than reporting rates at other locations (perhaps because of the absence of novelty in encountering a banded bird or the absence of curiosity about banding location). To capture this variation in recovery rates, additional recovery rate parameters are necessary. For example, a banding study involving three banding years and four recovery years would have the parametric structure indicated in Table 16.5, where f* is the recovery rate for individuals banded in year i. The parameters f* are k n o w n as direct recovery rates, to emphasize that banding and recovery occur in the same year. On the other hand, indirect recovery rates apply to recoveries in later years, after the year in which banding occurs. A model with different direct and indirect recovery rates for k banding periods and l recovery periods contains 2l + k - 2 parameters: S;,
i = 1.... , l -
fi,
i = 2, ..., 1;
f*,
i = 1, ..., k.
1;
S i = S ti,
where t i is the length of the interval between banding periods i and i+l. The parameter S now refers to survival over a standardized interval of time (e.g., 1 year), which is hypothesized to be constant during periods of variable length t i. The convention of allowing intervals between banding occasions to be of variable length permits ready extension of band recovery models to m a n y "nonstandard" situations. One obvious application involves gaps in an otherwise regular (e.g., annual) banding operation. Another involves sampling intervals that are inherently variable in length. For example, Conroy and Nichols (1984) applied band recovery models to
TABLE 16.5 Expected Recoveries for Single-Age Band Recovery Data with Temporary Banding Effect [Model (St, f~)]a Releases in period i a1 R2 R3
Recoveries in period j 1
2
3
4
Not recovered
a l f ; R1Slf 2 alSIS2f 3 R1515253f 4 R 1 - ~j E(mlj) R2f 2
R252f 3
a25253f 4
R3f 3
R3S3f 4
R2 - ~,j E(m2j) R3 - ~j E(m3j)
With k = 3 banding years and 1 = 4 recovery years. Parameters are annual survival rates (S i, i = 1..... 1-1), annual recovery rates more than 1 year after banding (fi, i = 2..... l), and annual first-year recovery rates (f~, i = 1..... k). a
16.1. Single-Age Models However, only a limited set of these can be separately estimated if l > k:
Si,
i = 1, ..., k - 1;
fi,
i = 2, ..., k;
f*,
i = 1, ..., k.
The generalization of model (St, ft) to include direct recovery rates is denoted as model (St, fD and is equivalent to Model 0 of Brownie et al. (1985). Estimates of parameters in (S t, fD can be computed using program MARK or other numerical procedures (Appendix G).
16.1.5. Multiple Groups As noted earlier, heterogeneity in survival, recovery, or both can be expected when the banded sample consists of animals in different sex or other (e.g., geographic) strata. If groups can be identified on capture and banding, then the recovery samples can be stratified prior to analysis. The only restriction is that stratum membership must be assigned at the time of banding and remain appropriate for the duration of the study. Two basic approaches can be taken in this situation. In the first, the data are stratified and a separate model is fit for each stratum. Goodness of fit tests then can be used to determine whether estimation based on separate groups is appropriate, or whether groups should instead be combined. A contingency table test developed by Brownie et al. (1985) specifically tests the null hypothesis of identical survival and recovery rates for multiple groups (i.e., it tests the appropriateness of pooling data). In the second approach, the stratum identity is incorporated into the model structure and used to estimate stratum- and timespecific parameters. For example, a generalization of model (St, ft) to allow sex- and time-specific variation in both survival and recovery rates, denoted as model (Ss,t, fs,t), would include identifiable survival and recovery parameters
Sij,
i = 1 ..... k - l ,
373
leads to time-specific (but not sex-specific) survival estimates, in which the same parameters Si, i = 1, ..., k - 1, are shared by both sexes. The term fs,t in the above expression indicates that a different recovery parameter is required for each sex-year combination (e.g., different parameters for males in 1998 and females in 1998). The adequacy of this model then can be assessed by goodness of fit, and model selection procedures (Section 16.1.8) can be used to judge the appropriateness of the model relative to competing models. In addition, one can easily describe models allowing for "parallelism," in which a parameter (e.g., survival) varies over time but in a parallel manner for the groups (e.g., sexes). Parallel effects are denoted by " + " rather than "," in the model notation, so that (Ss+t, ft) describes a situation where survival rates are sex specific but vary over time in a parallel manner (see Section 17.1.5 for a more complete description of interactive and parallel effects in the context of mark-recapture models). Program MARK (Appendix G) allows for construction of both types of group effects and provides a more efficient means of estimation and model selection than does separate estimation by groups or comparison of frequency tables.
16.1.6. Covariates The construction of band recovery and mark-recapture models to allow for inclusion of covariate relationships among parameters (typically survival, but also recapture and recovery rates) has seen tremendous progress in recent years, beginning with important work by North and Morgan (1979) for band recovery models, and Pollock et al. (1984) (see Section 17.1.4) and Clobert and Lebreton (1985) for closed and open capture-recapture models. We consider models for both time-specific covariates (e.g., covariates vary with time but not among individuals) and individual covariates (the covariates vary among individuals). The discussion here anticipates a more general treatment of covariates in Sections 17.1.4 and 17.1.7.
j-1,2
16.1.6.1. Time-Specific Covariates and
fij,
i = 1, ..., k, j = l ,
2
for males (j = 1) and females (j = 2). An obvious advantage of this approach is the ability to readily form models involving parameterizations not possible under the "separate models" approach. For instance, the model
(St, fs.t)
Under model (St, ft) both survival and recovery rates vary with time in an unspecified manner. If independent information exists about the time periods under investigation, it may be possible to model parameter variation over time by taking this information into account. Consider a banding study over k = 11 years, for which 10 estimates of survival are possible under model (St, ft). Suppose that we know years 2, 7, and 9 had especially severe winter conditions, whereas years 1, 3, 4, 5, 6, 8, and 10 were years of "normal" or warm
374
Chapter 16 Estimation with Band Recoveries
winters. With this information we could model survivorship by Si = SL,
= SH,
i-- 2,7,9 i = 1 , 3 , 4 , 5 , 6 , 8 , 10,
where SL, SH are parameters in a new (reducedparameter) model under the hypothesis that survival rates differ among years classed as "severe" and "normal" but not otherwise. This model represents an attempt to capture the temporal variation in survival probability in terms of winter conditions. The model could be compared to model (St, ft) and model (S, ft), with the latter comparison equivalent to the hypothesis of no difference associated with weather severity. A slight reparameterization of this model motivates a more general approach to modeling time-specific covariates. Let X i be an indicator variable, with value X i = 0 in "severe" years and X i = 1 in "normal" years. The above model then can be reexpressed as
Si-- ~o q- ~lXi,
(16.8)
where ~0 and ~1 parameterize a linear relationship between the indicator variable and survival. When ~1 -- 0 there is no relationship, i.e., severe and normal years produce equal survival rates, resulting in model (S, ft) (we assume here that the recovery parameter continues to vary in an unspecified manner through time). A somewhat more complex relationship between survival and winter temperature also can be used. Because temperature is measured on a continuous scale a natural extension is to consider values of the covariate X i as continuous, with ~0 and ~1 parameterizing a linear relationship between specific values of winter temperature and annual survival. Thus, estimates of ~0 and ~1 can be used to predict values of survival, i.e.,
Si-~ ~o q- ~lXi given the value X i = x i. One means of obtaining estimates of ~0 and ~1 under the above model is to obtain time-specific survival estimates under, e.g., model (St, ft), and then use these estimates as the response variable in a standard regression analysis (e.g., Nichols et al. 1982b; Sauer and Boyce, 1983). A more efficient alternative is to incorporate relationships like Eq. (16.8) directly into the likelihood function as constraints on the parameters of a more general model and to proceed with maximum likelihood inference (see examples in Conroy et al., 1989b; Dorazio, 1993). This approach has at least four advantages: (1) a single-step estimation is possible, (2) only two survival parameters (~0 and ~1) are estimated, versus an additional k - 1 parameters for the two-step process, (3) the sampling variances
and covariances associated with survival rate estimation are properly accounted for in the direct estimation procedure, and (4) the procedure fits naturally within the model evaluation and selection process described below. In practice, covariate relationships usually are expressed by means of link functions, discussed more thoroughly in Section 17.1.4. For example, the logit function establishes the relationship l~
_Si Si) ~. ~o -}- ~lXi
between survival and the covariate. An advantage of the logit and certain other common link functions is that predicted values for survival are constrained to the unit interval, whereas a linear relationship as above (under an "identity link") may allow predicted survival to take on logically inadmissible values (e.g., Si 0 or Si > 1), depending on the value of the covariate(s). This approach extends to multiple covariates, e.g.,
t Si ) -- ~o q- ~P ~jXij,
log 1 - S i
j=l
where j is an index denoting p time-specific covariates, including possible polynomial and interaction terms and Xij is the value of covariate j in year i. Models with time-specific covariates may be constructed using program MARK (Appendix G) and are evaluated in comparison with a general model a n d / o r other reduced-parameter models as discussed in Section 16.1.8. 16.1.6.2. Individual Covariates
An assumption of band recovery analysis is that survival and recovery rates of individual marked animals within an identified and modeled stratum (e.g., defined by time, age, sex, and location) are identical (see Section 16.1.9). In practice this assumption seldom is justified, and serious violations of it may lead to a lack of model fit and biased estimation. Some variation among individuals in survival, recovery, and other parameters may be explained by measurable covariates such as size, weight, or another characteristic. In addition, the influence of covariates on survival or other parameters may be of inherent interest. In either case, modeling with individual covariates can be accomplished in a manner similar to that used for timespecific covariates as described above, with the qualification that the covariates characterize an individual for the duration of the study and do not vary over time. We describe models incorporating individual covariates in more detail in our coverage of conditional capture-recapture studies (Section 17.1.7).
375
16.1. Single-Age Models
16.1.7. Banding Multiple Times per Year Here we consider a banding study in which bands are placed on individuals at two or more occasions prior to a single recovery period. For example, migratory waterfowl may be captured and marked twice during the year, in the late winter prior to spring migration and again in late summer or early fall just prior to the hunting season. The motivation for such a design is that within-year banding allows one to partition annual survival rates into a component corresponding to the hunting season and a component for the remainder of the year when other sources of mortality besides hunting are operative. In this way it is possible to better isolate the effects of hunting and to investigate possible compensation for hunting by other nonhunting mortality factors. A factor that sometimes complicates such analyses is the dispersal of individuals into different areas during migration. If migratory populations are banded at different times of the year, it is likely that different population cohorts, with different survival and recovery parameters, will be banded. For example, birds from different breeding ground locations may winter in the same area and be indistinguishable. Banding may thus involve a specific group of birds during the late summer, but a mixture of these and other birds during winter. A potential consequence is heterogeneity in survival and recovery rates among the sampled individuals. One way to help reduce this problem is to choose banding periods when animals are most likely to be sedentary [e.g., summer or winter; see Blohm et al. (1987) and LeMaster and Trost (1994)]. We note that these problems with heterogeneity should not arise in resident populations, so that the use of multiple banding periods per year is likely to be most useful for residents. Consider the special case in which resident animals
TABLE 16.6
Period (year) i 1 2 3
are banded twice per year, an early (e.g., spring) banding and a later banding during the period just before the hunting season. A parameterization of annual survival (as measured from preseason of year i to preseason of year i+1) in the one-age band recovery model allows incorporation of data from these two periods: Si ~ "Yi~i+l,
where q)i is the probability that a bird that is alive at the midpoint of the first (e.g., spring) banding period in year i survives to the midpoint of the second banding period, and "~i is the probability that a bird that is alive at the midpoint of the second period [e.g., later summer (preseason)] of banding in year i survives to the midpoint of the first banding period in year i+1. In the second period, all animals (including survivors from the current and previous years' first-period bandings) are assumed to share subsequent survival and recovery probabilities, leading to a matrix of expected recoveries as illustrated for 3 years of bandings and recoveries in Tables 16.6 and 16.7. The model associated with expected values as in Table 16.7 is denoted as model (q~t, ~/t, ft). This model and a number of associated reducedparameter models are provided by program MULT (Conroy et al., 1989b) and may be constructed in MARK. Reduced-parameter models include the following examples: 9 Model (q0t, '~, ft)--Second period survival constant over years. 9 Model (~, ~/t,ft) mFirst period survival constant over years. 9 Model (q~, ~/, ft)--First and second (and annual) survival constant over years. 9 Model (q~t = ~/t,ft) mSecond period survival equal to first period survival (but varying over years for both periods).
D a t a S t r u c t u r e for S i n g l e - A g e B a n d Recovery Data: Banding Two Occasions a
Subperiod h
Releases in period i and subperiod h b
Recoveries in period jc 1
2
3
Not recovered
1
Rll
mll I
mll 2
mll 3
Rll - ~,j m11j
2
R12
m121
m122
m123
R12 - s m12j
1
R21
m212
m213
R21 - ~,j m21j
2
R22
m222
m223
R22 - ~j m22j
1
R31
m313
R31 - ~j m31j
2
R32
m323
R32 - ~j m32j
aWith i = 1, 2, 3 banding years and j = i..... 3 recovery years. The first two bonding periods precede the first recovery period. bRih is the number of animals released during subperiod h of period i. cmihj is the number of recoveries in year j of animals released during subperiod h of period i.
376
Chapter 16 Estimation with Band Recoveries TABLE 16.7 Expected Recoveries for Single-Age Band Recovery Data under a Time-Specific Model with Two Banding Occasions/Year"
Period (year) i 1 2
3
Subperiod h
Releases in period i and subperiod h
Recoveries in period j 1
2
3
Not recovered
1
Rll
a11q~1f 1
a11~plS1f 2
a11q~15152f 3
Rll - ~j E(m11 j)
2
R12
R12 fl
R12S1 f2
R12S1 $2f3
R12 - ~j E(ml2j)
1
R21
R21qo2f2
R21~252f 3
R21 - ~j E(m21 j)
2
R22
R22f2
a22s2f 3
R22 - ~j E(m22 j)
1
R31
Rglq~gf3
R31 - ~j E(m31j)
2
R32
R32f3
R32 - ~j E(m32 j)
~With i = 1, 2, 3, banding years and j = i..... 3 recovery years, aih is the number of animals released during period (e.g., year) i and subperiod h, with i = 1..... k and h = 1, 2; Si = "~iq~i+l is the probability that a bird that is alive at the midpoint of period 2 in year i survives to the midpoint of period 2 in year i + 1; fi is the recovery rate in year i.
Brownie et al. (1985) described a slightly different parameterization for banding twice per year with their models H 7 and H 8, although they did not develop the corresponding reduced-parameter models. Many additional reduced-parameter models are possible--for instance, the above models in combination with time constraints on recovery. These and other models are available in MULT; in addition, MARK provides a more flexible framework for construction and selection of models (Appendix G). The above approach can be extended to situations in which more than two marking periods occur during each year or other interval. In some cases, the intervals between periods are of inherent biological interest (e.g., corresponding to important portions of the animal's life history, such as the reproductive season). The above approach can be extended to cover these situations, by incorporating the appropriate constraints in MARK or SURVIV (Appendix G). 16.1.8. E v a l u a t i o n a n d S e l e c t i o n of M o d e l s
16.1.8.1. Goodness of Fit Once parameter estimates are obtained for a given model, it is possible to evaluate how well the model describes variation in the data set. Goodness-of-fit testing for band recovery models utilizes the multinomial structure of band recoveries in a manner similar to capture-recapture models (Chapter 17). Recall from Chapter 4 that with multinomial data, the observed cell frequencies can be compared with expected counts via the Pearson or log likelihood chi-square statistics (Section 4.3.3), to determine if the model "fits" the data adequately. Large computed values of the goodnessof-fit statistic with correspondingly small probabilities of occurrence provide evidence that one or more of the model assumptions (Section 16.1.9) are violated. Use
of the band recovery model under these conditions may lead to biased estimators of model parameters, with unrealistically low standard errors. Of the many assumptions that potentially can be violated, the most likely candidate is the assumption that all individuals in a banding cohort experience equal survival and recovery rates.
16.1.8.2. Model Selection Assuming one or more models adequately fit the observed data, the issue arises as to which model to select (and thus which parameter estimates to accept). Two basic approaches have been developed, the first of which is based on likelihood ratio tests between models that are "nested," i.e., one model can be formed by constraining the parameter space of another (more general) model. For example, models (St, f~), (St, ft), (S, ft), and (S, f) form a hierarchy, from more general to more reduced parameter structures. Thus, model (S, ft) is a generalization of model (S, f), model (St, ft) is a generalization of model (S, ft), and model (St, f~) is a generalization of model (St, ft). Under these conditions it is reasonable to subject the models to pair-wise comparisons, via the likelihood ratio testing procedure described in Section 4.3.4. Based on test results from the model comparisons (along with information from goodness-of-fit testing), one can identify the most appropriate model for the data. In general the likelihood ratio test statistic is computed as T = -2[ln(L 0) - ln(La)],
(16.9)
where L0, La are the likelihoods (evaluated at their MLEs) under the null and alternative models, respectively. Under the null hypothesis that the simpler model (H 0) describes the data as well as the more complex model (Ha), the statistic T is distributed as
16.1. Single-Age Models 2 X~,ka-kO" where k0, ka are the numbers of estimable parameters for each model (see Section 4.3.4). As a general rule, the model should be used which (1) adequately fits the data (i.e., the goodness-of-fit test does not indicate rejection) and (2) is not rejected when tested as the null model against more complex models [e.g., see Burnham and Anderson (1992, 1998)]. However, this procedure will work only in cases (as above) in which the models under consideration form a nested hierarchy, so that simpler models can be formed by imposing constraints on the parameters of the more complex models. A more general procedure that removes model selection from the framework of hypothesis testing is based on information theory (Akaike, 1973; Burnham and Anderson, 1992, 1998). The procedure is to select the model that minimizes the Akaike Information Criterion (AIC),
AIC = -21n(L) + 2k,
(16.10)
where again L is the maximized likelihood under a candidate model and k is the number of independently estimated parameters. This approach is very general and can be used with any nested or nonnested set of models, as long as they are likelihood based and are evaluated using the same data. Differences in AIC of 2 or less can be expected if models are essentially equivalent. Occasionally AIC values for two or three models will be very close to one another (differences <2) but much lower than other competing models. In these situations, the subset of models with low AIC values should be retained and model selection should be based on other criteria (see Section 4.4), or else estimates can be computed as weighted averages of estimates from all low-AIC models (e.g., Burnham and Anderson, 1998). Although these two approaches to model selection (sequential likelihood ratio testing of nested models versus optimization using a criterion such as AIC) are both reasonable, they are not equivalent and do not always yield the same results. In our experience, the approach based on AIC tends to result in the selection of models with fewer parameters, compared to the approach based on sequential testing. Simulation results with certain classes of models suggest that estimates based on models selected via AIC frequently have better properties (e.g., lower root mean squared error) than do those based on models selected via hypothesis testing procedures (see discussion in Section 17.1.8). It also should be reemphasized that failure to reject a hypothesis (e.g., H0: Si = S) does not by itself confer strong support in favor of that hypothesis. This is especially true of null hypotheses regarding temporal variation in survival rates, because these tests tend to have
377
low power unless sample sizes are quite large. Proper interpretation of "nonsignificant" tests is a key issue in hypothesis testing (see Johnson, 1999), particularly when these interpretations lead to management actions (e.g., the allowance of more liberal harvest regulations).
16.1.9. Estimator Robustness and Model Assumptions 16.1.9.1. Assumptions Key assumptions underlying band recovery models can be grouped into those relating to study design/ field procedure, stochastic variation, and model structure (Brownie et al., 1985).
16.1.9.1.1. Study Design and Field Procedure It is assumed that (la) the sample is representative of the population under investigation, (lb) there is no band loss, (lc) the age and sex of sampled individuals are correctly determined, (ld) the year of band recovery is correctly tabulated, and (le) survival rates are not affected by banding. Assumption (la) is basic to any sample survey method (Chapter 5) and is violated when, for example, trapping is nonrandom (overrepresentation of certain ages, sexes, or other categories). In these instances, the sampled population is not representative of the target population, and additional information is needed about the different classes of animals in order to connect inferences from the former to the latter. Assumption (lb) is common to all capture-recapture and recovery models, and the effects of violation are similar: band loss generally results in underestimation of survival rates because some of what appears as mortality in the sample is actually band loss (Nelson et al., 1980). Assumption (lc) essentially affirms that cohort structure is properly recognized. Failure to meet this assumption may result in unidentified heterogeneity and consequent lack of model fit, because the appropriate parameters of some portion of the population are not represented in the model parameterization. Violation of assumption (ld) can introduce bias in parameter estimates if the errors are systematically biased [so that, for example, the year of recovery is more frequently too high than too low; see Anderson and Burnham (1980)]. Violation of assumption (le) may result in a situation whereby the marked sample no longer represents the target population. For example, if marked animals have lower survival than unmarked animals, inferences (conditional on the marked sample) no longer apply to the population of interest (containing both marked and unmarked individuals). Models can be developed that allow for a one-time effect of marking on survival (e.g., Brownie et al., 1985), which are similar to the temporary trap response mod-
378
Chapter 16 Estimation with Band Recoveries
els of Pollock (Section 17.1.6) and are generalizations of the parameter structure described above.
16.1.9.1.2. Stochastic Effects These effects are related to the stochastic components of the statistical model used to estimate parameters (generally, by the method of maximum likelihood). It is assumed that (2a) the fate of each banded individual is independent of the fate of every other banded individual, and (2b) fate can be modeled as a multinomial random variable, with the multinomial cells defined by prospective periods of band recovery. Assumption (2a) permits the individual cell probabilities for recoveries corresponding to a given release year to be linked together in a multiplicative likelihood, and these likelihoods in turn to be linked together across release years to form an overall probability model to which maximum likelihood procedures can be applied. In addition, independence allows for proper estimation of sampling variances of the parameter estimates, and thus, reliable interval estimation. Independence is violated when, for example, marked animals travel together in groups (such as pairs). In the extreme case, the fates of individuals within each group are totally interdependent, so that if there are n marked animals in k groups, each having m individuals, there now are k rather than n independent outcomes. In this situation the application of maximum likelihood estimation based on n outcomes will result in underestimation of sampling variances and in overly narrow confidence intervals. Assumption (2b) can be viewed as a consequence of the assumption of independence and the fact that the recoveries are by definition mutually exclusive and exhaustive events. Once released, an animal can be recovered in only one year, and all released animals are recovered in years j = 1, ..., l or not recovered (the complementary event). We note that the stochastic framework for band recovery analysis is a special case of the framework for capture-recapture, wherein animals are encountered in more than one subsequent period (Section 17.1.1).
16.1.9.1.3. Model Structure These assumptions determine the complexity necessary for modeling survival, recovery rates, and other parameters and are thus "assumptions" only to the extent that they are thereby linked to corresponding models. It is assumed that (3a) all banded individuals within a banded cohort have the same survival and recovery rates; (3b) depending on the specific model structure, survival and recovery may vary by sampling period and cohort. Assumption (3a) is violated when heterogeneity exists among the animals within a spe-
cific year or cohort. For instance, the same model allowing time-specific survival might be used for both sexes, when in fact each sex has distinct survival a n d / or recovery rates. In this instance the obvious remedy is to stratify the data and to use a two-sex model, assuming that the sexes can be properly identified on capture and release. Other approaches include the measurement of individual covariates (Section 16.1.6) that can be incorporated into a model to account for individual animal heterogeneity. However, there may be attributes that are less readily observed but nonetheless are related to the subsequent probability of survival and recovery. Clearly, mixtures of animals having different values for these attributes in the marked sample introduce heterogeneity. It is difficult to make generalizations as to the effects of the latter type of heterogeneity; however, studies by Pollock and Ravelling (1982) and Nichols et al. (1982b) suggest that (1) survival rates remain unbiased even if the sampled population is heterogeneous with respect to recovery, and (2) survival estimates are "fairly robust" to "moderate" heterogeneity in the survival rates of the sampled population. Finally, goodnessof-fit tests, though of generally low power to detect heterogeneity in survival, can detect severe forms of heterogeneity (e.g., failure to identify age structure in the model) and are an indispensable part of model construction and evaluation. Burnham and Rexstad (1993) considered the problem of heterogeneity in survival probability that could not be attributed to observable characteristics of the animals. They developed a model that accommodates an increase in the average survival of a banded cohort with increasing time since banding, on assumption that the composition of the heterogeneous banded sample becomes more dominated by the individuals with the high underlying survival probabilities (see Vaupel and Yashin, 1985; Johnson et al., 1986; Nichols et al., 1997). Using this approach, Rexstad and Anderson (1992) found widespread evidence of heterogeneous survival rates of mallard ducks banded throughout North America. Pledger and Schwarz (2002) have developed a finite mixture model that shows promise as an alternative approach for dealing with heterogeneous survival rates.
16.1.10. Interpretation of Sampling Correlations From a common set of sample data, band recovery analyses can lead to a large number of parameter estimates. Estimates based on nonoverlapping subsets of these data are statistically independent; however, estimates sharing the same data typically are nonindepen-
16.1. Single-Age Models dent. Sample covariances for the latter are expressions of the degree of statistical independence. Sampling covariances and correlations are important for at least two reasons: first, for proper interpretations of apparent patterns in the parameter estimates, and second, for the unbiased computation of variances, confidence intervals, and test statistics involving functions of the parameter estimates. As to the interpretation of parameter estimates, consider an analysis of band recovery data from a population for which inferences are desired about the effects of harvest mortality on survival. As we have seen previously, one test of the compensatory mortality hypothesis involves testing for association between annual survival and recovery rates. Under the assumption that recovery rates are valid indices of harvest mortality, a negative relationship between these parameters would be interpreted as evidence for rejecting the compensatory hypothesis in favor of additive mortality. If independent sample estimates of each parameter are available, then ordinary correlation methods can be used to test this hypothesis, though evaluation under an experimental design involving control and random assignment to harvest levels clearly would be preferable. However, if estimates of both recovery (fi) and survival (Si) are generated from a common analysis of the same data, then the issue of sampling covariance arises. In particular, it might be tempting [and in fact has been tried; see Anderson and Burnham (1976)] to use the estimates f and S of average recovery and survival taken from various geographic areas, where (f, S) pairs for each study area are obtained from the arithmetic means of model (St, ft) estimates for that area. Brownie et al. (1985) show that the sample correlation between the estimates f and S is negative and can be substantial (e.g., <-0.5). A standard correlation analysis, naively performed on the (f, S) pairs from replicate study areas, could easily reveal a negative correlation between average survival and recovery rates, and thus could be interpreted as "support" for the additive hunting mortality hypothesis (Anderson and Burnham, 1976), even though the correlation might reflect sampling covariation absent any real biological pattern of association. There are at least two remedies for this problem. One approach is to construct a model that directly represents the hypothesized structural relationship between survival and recovery rates and to proceed with estimation and hypothesis testing using MARK, SURVIV, or other procedures (Appendix G). This approach should appropriately incorporate the variance-covariance structure directly into the estimation process (see Burnham et al., 1984; Barker et al., 1991). An alternative
379
approach is to use independent data sets to estimate the different parameters. Independence could be achieved by splitting the data into two portions for each replicate (e.g., geographic area) and using one portion to estimate survival, the other to estimate recovery (Nichols and Hines, 1983). Obviously, in order to estimate either parameter, estimates of the other also would be obtained, but these would be discarded. The latter approach is statistically inefficient, requiring enough data for separate analysis but only partially utilizing the statistical information in the data. In many instances, investigators do not have the luxury of using data in this way; however, independence also can be achieved by drawing multiple bootstrap samples for computing estimates. Of course, any such retrospective analysis is limited in the extent to which causal inferences can be made and is not a substitute for carefully designed experiments involving control, randomization, and replication (Chapter 6). As to the unbiased computation of variances, confidence intervals, and hypothesis tests, we note that many interesting statistics in band recovery models can be expressed as linear combinations of other statistics. A standard form involves one or more linear contrasts, i.e., K -- C101 if- C202 if- "'" if- CkOk_1,
where 0i, i = 1, ..., k - 1 are model parameters (e.g., survival rates). The corresponding statistics for testing such hypotheses must account for sampling covariances, as in Z C101 q- C202 -ff Ck_lOk_ 1 k-1 V%-~k-1"--'i=1c2var(0,) + 2~-'ikZ2 ~j=i+l q- cicj cov(0i, 0j)
Under assumptions of asymptotic normality of the estimates, the statistic Z follows a standard normal distribution under the stated null hypothesis H0: K = 0. Sample covariances obviously have the potential to influence the computation of this test statistic. In the testing of a contrast between two means, the product cic j would be negative; hence, positive covariances would result in a smaller denominator and a larger test statistic (more rejections), whereas negative covariances would lead to a smaller test statistic and fewer rejections. In the specific case of a series of survival rates [e.g., the mean of the first n years of a sample versus the mean of the second m years, based on model (St, ft)], the estimates of Si for adjacent years are negatively correlated. Neither the magnitude nor the sign of sample covariances can be predicted easily, particularly for complex models and models having no explicit estimators. It thus is critical to estimate the
380
Chapter 16 Estimation with Band Recoveries T A B L E 16.8
Recoveries of Adult Male Mallards Banded during January/February in Illinois" Recovered during hunting season
Year
Number banded
1963
1963
2583
91
1964
3075
1965
1195
1966
3418
1967
3100
1968
2400
1969
2601
1970
4433
aFrom Brownie
1964
1965
1966
1967
1969
1968
1970
1971
1972
1973
89
24
18
16
11
8
7
7
2
6
141
45
52
50
17
30
21
16
7
3
27
31
21
8
19
7
9
4
3
92
44
50
49
34
23
5
113
68
57
65
41
23
10
63
52
59
44
30
12
91
80
58
37
25
222
169
95
46
156
et al. (1985).
sample covariances and include these estimates in the computation of variances, confidence intervals, and test statistics. General expressions for K and var(K) are easily encoded and have been included in MULT (Conroy et al., 1989b), CONTRAST (Hines and Sauer, 1989), and other procedures. Following estimation of the parameters of any model within MULT, users are able to specify the coefficients for a series of tests of H0: K = 0. A Z statistic for each contrast is calculated, displayed on the monitor, and saved as output. This procedure may be repeated as many times as desired for each model. Hines and Sauer (1989) developed program CONTRAST, which allows for construction and testing of orthogonal contrasts (after Sauer and Williams, 1989). CONTRAST accepts vectors of parameter estimates as input, along with either estimated variances absent any associated covariances (in which case the contrasts are performed as if the estimates are independent), or estimated variance-covariance matrices when the latter are available. CONTRAST requires users first to obtain the parameter estimates, variances, and covariances from other procedures (e.g., MARK), but is very general in that estimates may be for different parameters (e.g., survival rates, recovery rates, or nest success rates) arising from different data structures (capture-recapture, nest monitoring, band recovery).
16.1.11. Examples In what follows we present several examples from actual field data, utilizing variants of the singleage model.
16.8). Models (St, f~), (St, ft), (S, ft), (St, f), and (S, f) were fit to these data using program MARK (Appendix G). Both models (St, f~) and (St, ft) fit the data, but model (St, ft) was selected for estimation based on AIC ranking (Table 16.9). The parameter estimates for this model are presented in Table 16.10, which shows survival rate estimates varying from 0.585 in 1963 to 0.776 in 1967.
Example Heterogeneity in survival and recovery rates was investigated with data from adult (i.e., after hatchingyear) male and female American black ducks (Anas rubripes) banded in eastern North America (Reference Areas 1-5) (Smith, 1997) during July-September, 1989-1998 (Table 16.11). The modeling of sex and time effects resulted in a large number of candidate models, ranging in complexity from model (S, f), in which both survival and recovery are constant over both time and sex, to model (Ss,t, fs*t), which considers both factors plus a temporary banding effect on recovery rates for
M o d e l Selection Criteria for Adult Male Mallards Banded during January/February in Illinois a
T A B L E 16.9
Goodness of fit b
Model
Number of parameters
X2
df
P
(St, ft)
18
46.39
42
0.28
0.0
(St, f;) (S, ft) (St, f)
25
34.98
35
0.47
2.6
12
62.30
48
0.08
3.9
11
139.75
49
<0.01
79.3
(S, f)
2
226.49
58
<0.01
148.0
~AIC c
Example In an example originally presented by Brownie et al. (1985) adult male mallards were banded during January and February, 1963-1970, in Illinois (Table
aSee Table 16.8. bDeviance-based chi-square test; see Section 4.3.3. CDifference b e t w e e n model AIC (Akaike's Information Criterion; see Section 4.4) and AIC value for the lowest ranked model.
16.1.
TABLE 16.10
(St, ft)
Parameter Estimates for M o d e l
Single-Age M o d e l s
381
for Adult Male Mallards Banded during January/February in Illinois a .
Year (i)
S,
S"E(Si)
C"L
C~U
.
.
.
SE(fi)
f,
C"L
C~U
1963
0.585
0.050
0.487
0.679
0.035
0.004
0.029
0.043
1964
0.686
0.070
0.536
0.805
0.051
0.004
0.044
0.057
1965
0.647
0.063
0.517
0.759
0.022
0.003
0.017
0.028
1966
0.751
0.052
0.636
0.838
0.412
0.003
0.036
0.047
1967
0.776
0.061
0.635
0.873
0.038
0.003
0.033
0.043
1968
0.745
0.061
0.608
0.846
0.025
0.002
0.021
0.029
1969
0.644
0.046
0.549
0.729
0.035
0.003
0.030
0.039
0.050
0.003
0.045
0.056
1970 aSee Tables 16.8 and 16.9. Parameters are annual survival rates
both
sexes. Program
MARK
u s e d to e s t i m a t e p a r a m e t e r s
(see Appendix and compare
G) w a s
models. A
g l o b a l m o d e l (Ss, t, fs*t) fit t h e s e d a t a a n d p r o v i d e d a q u a s i l i k e l i h o o d a d j u s t m e n t f a c t o r ( s e e S e c t i o n 17.1.8) of ~ = • = 1.04, f o r u s e i n c o m p u t i n g Q A I C c f o r m o d e l s e l e c t i o n ( B u r n h a m a n d A n d e r s o n , 1998). M o d e l
(Ss, ft)
was
TABLE 16.11
selected
( s e e T a b l e 16.12), a n d
the sex-
(Si) and
recovery rates
(fi).
specific survival and time-specific recovery rates for t h i s m o d e l a r e r e p o r t e d i n T a b l e 16.13.
Example The modeling
of time-specific covariates
trated by data from adult during August-September,
is i l l u s -
female mallards banded 1966-1978, in Manitoba,
Adult Male and Female American Black Ducks Banded July-September 1989-1998 in Eastern North America a Year recovered during hunting season
Sex
Year
Bands
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
Male
1989
773
21
12
7
2
3
2
3
1
3
2
1990
1463
4
1991
1270
1992
1209
Female
1993
1162
1994
1081
1995
891
1996
1110
1997
816
1998
864
1989
518
1990
836
1991
964
1992
867
1993
885
1994
794
1995
596
1996
592
1997
569
1998
653
51
29
14
13
8
15
4
3
40
20
14
9
8
5
4
3
41
26
14
20
18
8
4 5
31
17
13
10
8
29
27
18
9
9
35
20
12
12
49
18
19
33
14 48
21
4
5
3
3
1
1
0
0
0
27
8
6
3
1
1
2
0
0
6
8
9
1
1
0
0
19
7
10
5
2
0
1
18
14
5
2
7
3
15
5
3
6
24
7
4
6
31
21
19
6
2
20
11
569
653
23 518
a Banding reference areas 1-5 (Smith, 1997).
836
964
867
885
794
596
592
382
C h a p t e r 16
Estimation with Band Recoveries
TABLE 16.12 M o d e l Selection Criteria for American Black D u c k s Banded J u l y - S e p t e m b e r 1989-1998 in Eastern North America a Goodness of fit b Model
Parameters
df
X2
P
~QAICc c
(Ss, ft)
12
119.367
90
0.021
0.00
(Ss, fs.t)
22
104.698
80
0.033
5.93
(Ss.t, ft) (Ss.t, fs.t)
28
97.558
74
0.035
11.10
38
80.596
64
0.079
14.87
4
153.227
98
0.000
16.54
54
49.814
48
0.401
17.44
3
156.313
99
0.000
17.51
(Ss.t, fs)
20
124.964
82
0.002
21.41
(S, fs.t)
21
123.579
81
0.002
22.08
(St, fs.t)
29
107.279
73
0.006
22.46
(Ss. t, f )
19
128.285
83
0.001
22.60
(St, fs)
11
157.225
91
0.000
34.40
(S, fs)
3
178.202
99
0.000
38.56
(S, ft)
11
184.280
91
0.000
60.41
(St, ft)
19
168.550
83
0.000
61.32
(S t, f)
10
201.307
92
0.000
74.78
(S, f )
2
221.667
100
0.000
78.35
(Ss, fs)
(Ss.t, fs.t) (S s, f )
S~x
1 -
Si
-- ~0 nu ~1Xi+1,
where Xi+ 1 is the number of ducks per pond in the subsequent spring surveys. This relationship was added into the model structure of models (St, ft) and (St, f'~) producing models (Scov, ft) and (Scov, f~) in addition to (St, ft) and (St, f'~). Comparison of these and other one-age models (Table 16.15) resulted in the selection of model (Scov, ft), providing evidence for a negative density-dependent survival relationship (estimates are shown in Table 16.16).
Example Kirby et al. (1986) analyzed recoveries of Atlantic brant (Branta bernicla hrota) banded in the Canadian Arctic over a 20-year period, in which substantial releases of banded brant occurred in only 7 years (Table 16.17). Models (St, ft) and (St, f) were fit to these data, along with models (S, ft) and (S, f) under the parameterization described in Section 16.1.3. Based on comparison among these four models (Table 16.18), survival and recovery rates were estimated for each year of banding, with average annual survival estimated when banding intervals were longer than I year (Table 16.19).
Parameter Estimates for M o d e l (Ss, ft) for American Black D u c k Data and Analysis in Tables 16.11 and 16.12 a
v~(s)
~
~'E(~)
c~
c3
Male
1989-1998
0.679
0.013
0.653
0.705
Female
1989-1998
0.529
0.018
0.493
0.565
All
a
[si]
In
a See Table 16.11. Parameters are annual survival (S) and recovery rates (f); indices t and s denote variation by time and sex, respectively. b Deviance-based chi-square test; see Section 4.3.3. c Difference between model QAIC c (Akaike's Information Criterion, corrected for small effective sample size and adjusted by the quasilikelihood factor ~ = 1.04) and QAIC c value for the lowest ranked model.
TABLE 16.13
Saskatchewan, and eastern Alberta (Table 16.14). Previous analyses (Nichols et al., 1982a) suggested that survival rates of adult female mallards could be predicted by a negative density-dependent relationship of the form
~,
~q,)
C'-L
C"~
1989
0.032
0.005
0.024
0.044
1990
0.030
0.003
0.025
0.037
1991
0.029
0.003
0.024
0.034
1992
0.024
0.002
0.020
0.029
1993
0.025
0.002
0.021
0.030
1994
0.027
0.002
0.023
0.032
1995
0.037
0.003
0.032
0.044
1996
0.035
0.003
0.030
0.042
1997
0.032
0.003
0.027
0.038
1998
0.041
0.003
0.034
0.048
Parameters are annual sex-specific survival rate (S) and time-specific recovery rates
(fi).
16.2. Multiple-Age Models
383
A d u l t F e m a l e M a l l a r d s B a n d e d P r e s e a s o n ( A u g u s t / S e p t e m b e r ) 1966-1978
T A B L E 16.14
in M a n i t o b a , S a s k a t c h e w a n , and Eastern Alberta Recovered during hunting season Year
Bands
1966
926
1967
1413
1968
1147
1969
1233
1970
1674
1971
1727
1972
1864
1973
1438
1974
1235
1975
2351
1976
5215
1977
5256
1978
3615
1966
1967
39
20
1968
1969
6
53
1970
3
7
1971
1974
1975
1976
1977
1978
Ducks/pond
2
1
0
0
0
0
0
0.978
18
17
9
8
5
4
0
0
1
0
0
2.245
34
27
23
11
5
3
0
2
1
0
0
1.268
57
33
21
14
5
2
1
2
0
1
1.090
82
35
22
8
2
4
3
1
1
1.336
71
30
13
1
8
6
2
0
1.307
67
32
12
12
4
4
3
2.260
46
23
21
13
5
4
0.665
43
27
15
10
7
0.890
67
56
26
13
1.318
180
91
56
2.116
89
0.871
167
114
It often is useful to band both adults and juveniles in a banding study. Juveniles typically are easier to capture and band, and information about adult survival and recovery rates can be obtained from birds banded as juveniles. However, adults and juveniles typically have different survival and recovery rates, and therefore a statistical model for adults and juveniles must include parameters for both cohorts. FortuM o d e l S e l e c t i o n Criteria for A d u l t F e m a l e M a l l a r d s B a n d e d P r e s e a s o n ( A u g u s t / S e p t e m b e r ) 1966-1978 in M a n i t o b a , S a s k a t c h e w a n , and E a s t e r n A l b e r t a " T A B L E 16.15
Goodness of fit b ~AICc c
Parameters
X2
df
P
(S .... ft) (S, ft)
15
61.79
62
0.48
0.00
14
68.57
63
0.29
4.78
(S t, f )
13
72.33
64
0.22
6.54
(St, ft)
25
48.74
52
0.60
6.98
(S. . . . f ~)
27
47.40
50
0.58
9.64
(S t, f ~)
36
35.90
41
0.70
16.19
2
122.37
75
0.00
34.56
(S, f )
1973
2
16.2. M U L T I P L E - A G E M O D E L S
Model
1972
aSee Table 16.14. Parameters are annual survival rate (S) and recovery rates (f); indices t and cov denote variation by time and as a function of a density covariate, respectively. b Deviance-based chi-square test; see Section 4.3.3. c Difference between model AIC c (Akaike's Information Criterion, corrected for small effective sample size) and AICc value for the lowest ranked model AICc; see Section 4.4.
nately, the single-age models described in Section 16.1 can be extended easily to include two (or more) age cohorts. Besides involving stratification by the ages at which animals initially are marked, modeling of age specificity must reflect the fact that individuals undergo transition from one age "state" to another, and thus become part of a different stratum.
16.2.1. D a ta S tru cture The general data structure of age-specific band reR(v) animals in each of v = 0, m age covery involves ~'i classes that are banded in each of i = 1, ..., k banding occasions. Animals banded in year i as age class v and _ (v) recovered in year j, j = i, ..., l, are denoted by mij , m (v) animals of each banding class with RI v) - ~,J=i ij never recovered. For m a n y organisms (e.g., birds) it is difficult to identify age beyond distinguishing whether animals are members of a birth (hatching) cohort or not. Thus, much of the development for multiage modeling has focused on the situation in which m = 2. In this case there are two strata of marked animals, RIO) juveniles and RI 1) adults, that are banded in each year i, and two corresponding recovery matrices with elements ,...,(0) '"ij and 4.,.,(1) ,,,q that represent, respectively, the recoveries of juveniles and adults released in year i (Table 16.20). "'"
16.2.2. Probabilistic M o d e l i n g We focus here on the situation in which RI 1) adults and RIO)juveniles are banded in year i, leading to recov-
384
Chapter 16 TABLE 16.16
Y~a~(~) 1966-1978 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978
Estimation with Band Recoveries
Estimates of Parameters for M o d e l (Sco v,
ft) for A d u l t Female Mallards Banded Preseason"
~0
~(~0)
C'~
C~U
#,
s'~(O0)
c'~
c~u
0.987
0.253
0.492
1.482
-0.460
0.171
-0.795
-0.125
~,
s'~(f,)
c~
c'~
0.042 0.037 0.027 0.042 0.048 0.041 0.036 0.033 0.024 0.031 0.035 0.033 0.030
0.007 0.004 0.004 0.004 0.004 0.003 0.003 0.003 0.003 0.003 0.002 0.002 0.002
0.031 0.029 0.021 0.034 0.041 0.034 0.030 0.027 0.019 0.026 0.031 0.029 0.027
0.057 0.046 0.036 0.051 0.056 0.048 0.042 0.040 0.030 0.037 0.040 0.037 0.034
aSee data in Tables 16.14 and 16.15. Parameters are annual recovery rates (fi), and the slope (131) and intercept ([3o) of the covariate relationship between survival and density. eries (0) mlJ~ (0)a n d ~...(1) .tij r e s p e c t i v e l y for j > i. P a r a m e t e r s S i a n d fi for j u v e n i l e s u r v i v a l a n d r e c o v e r y rates are i n c o r p o r a t e d into the m o d e l , a l o n g w i t h p a r a m e t e r s SI 1) a n d fi(1) for a d u l t a n n u a l s u r v i v a l a n d r e c o v e r y rates [the latter c o r r e s p o n d to p a r a m e t e r s Si a n d fi f r o m the o n e - a g e class m o d e l (S t, ft) of Section 16.1]. T h u s , a s e c o n d a r r a y of e x p e c t e d j u v e n i l e r e c o v e r i e s is app e n d e d to the a r r a y of e x p e c t e d a d u l t r e c o v e r i e s f r o m Section 16.1. T h e joint a r r a y of e x p e c t e d r e t u r n s is disp l a y e d in Table 16.21 for k b a n d i n g p e r i o d s a n d l recove r y p e r i o d s . N o t e that j u v e n i l e s u r v i v a l rates differ f r o m a d u l t s u r v i v a l rates, a n d j u v e n i l e r e c o v e r y rates differ f r o m a d u l t r e c o v e r y rates. H o w e v e r , j u v e n i l e s that s u r v i v e b e y o n d the b a n d i n g y e a r b e c o m e a d u l t s , w i t h a d u l t s u r v i v a l a n d r e c o v e r y rates. T h u s , the t w o tables of e x p e c t e d r e c o v e r i e s s h a r e m a n y of the a d u l t s u r v i v a l a n d r e c o v e r y p a r a m e t e r s , a n d d a t a o n the r e c o v e r y of j u v e n i l e - b a n d e d i n d i v i d u a l s can be u s e f u l in the e s t i m a t i o n of a d u l t p a r a m e t e r s .
TABLE 16.17
Recoveries of Atlantic Brant
As s h o w n in Table 16.21, the g e n e r a l t w o - a g e b a n d recovery model with k banding years and l recovery y e a r s i n c l u d e s the f o l l o w i n g p a r a m e t e r s :
SI 1),
i = 1 , . . . , l - 1;
fl 1), SI ~
i=
1, ..., l;
i=
1, ..., k;
fl ~
i = 1, ..., k.
As before, o n l y a l i m i t e d set of t h e s e can be s e p a r a t e l y estimated: SI 1), i = 1,...,k1; fl 1),
i=
SI ~
i = 1,...,k-
fl ~
i = 1, ..., k.
1, ..., k; 1;
T h e a b o v e n o t a t i o n r e a d i l y e x t e n d s to the g e n e r a l case of m + 1 a g e classes. F o l l o w i n g L e b r e t o n et al. (1992),
(Branta bernicla hrota) Banded b e t w e e n 1956 and 1975 in Arctic Canada Recoveries during hunting season
Year
Bands
1956
1957
1960
1961
1965
1966
1975
1976
1977
1956 1957 1960 1961 1965 1966 1975
600 1481 327 1172 1057 1227 792
13
6 20
12 40 19
6 22 3 52
3 10 3 12 29
4 17 6 27 47 57
0 0 0 1 2 1 18
0 0 0 2 1 1 17
0 0 0 1 1 0 2
16.2. Multiple-Age Models
H 0, in which survival rates and recovery rates are time specific but not age specific [model (St, ft) of Section 16.1]. Model H 0 is used to compare age-specific models and single-age models to determine if data can be pooled across ages. Many other models also can be formed by constraints on model (Sa,t, fa,t); for example, (Sa,t, ft) has survival rates that are both time specific and age specific, but recovery rates are only time specific. Other constrained models might include, e.g., time constraints in survival for one age class but not the other, a n d / o r additive (versus interactive) effects of age and time. The latter models are explored in more detail w h e n we consider conditional capture-recapture models in Chapter 17. We note that all of these models can be constructed readily in p r o g r a m MARK (Appendix G) for m + 1 age classes. In the above development, an implicit assumption is that the time between sampling periods corresponds to the time needed for animals to transition to the next age class. For m a n y organisms (e.g., birds) the duration of this transition period is 1 year, so that the typical design has annual banding of the y o u n g of the year and "adults" (i.e., in the second or greater calendar year of life). In other situations (e.g., age transition occurs over longer or shorter intervals than sampling) the survival parameters m a y not be uniquely identifiable, and other approaches m a y be needed. It should be emphasized that investigation of multiple-age models imposes greater data requirements than does a one-age model. Because juveniles typically are much easier to trap than adults, one might be tempted to band only juveniles and fit the part of the two-age model that applies to that cohort. T h o u g h such a sampling strategy can result in large numbers of b a n d e d juveniles, it cannot lead to unique, welldefined estimates of survival and recovery rates. To see why, refer to the expected recovery matrices for the two-age case, involving the banding of both adults
TABLE 16.18 M o d e l Selection Criteria for Atlantic Brant Banded b e t w e e n 1956 and 1975 in Arctic Canada" Goodness of fit b Model
Parameters
X2
df
(st, ft) (s, ft ) (st, f)
15 10 9 2
18.27 30.54 75.04 157.93
17 22 23 30
(s, f )
P
AAICc c
0.370 o.106 0.000 0.000
0.000 2.234 44.730 113.592
,,
aSee Table 16.17. Parameters are annual survival (Si) and recovery rates (fi); index t denotes variation by time. bDeviance-based chi-square test; see Section 4.3.3. CDifferencebetween model AICc(Akaike's Information Criterion, corrected for small effective sample size) and AICc value for the lowest-ranked model AICc; see Section 4.4.
we denote the general model (which includes two age classes as a special case) a s (Sa.t, fa.t)~' where t refers to time variation, a refers to age-specific variation, and the symbol "," denotes interaction between age and time effects, i.e., all levels of age occurring in each year. For two age classes, model (Sa.t, fa.t) is identical to model H] of Brownie et al. (1985).
16.2.3. R e d u c e d - P a r a m e t e r
Models
Parameters in the two-age model can be restricted in the same m a n n e r as the model for adults only, through simplifying assumptions about survival and recovery rates. In addition to model (Sa.t, fa.t), p r o g r a m BROWNIE (Brownie et al., 1985) provides estimates for model (Sa, fa,t) (H02 in BROWNIE notation), in which survival rates are age specific but not time specific; and model (S~,fa) (H01 in BROWNIE notation), in which survival and recovery rates are age specific but not time specific. Brownie et al. (1985) also described model
TABLE 16.19
385
Parameter Estimates for M o d e l for Atlantic Brant Banded b e t w e e n 1956 and 1975 in Arctic Canada a
Y,~,,,"
~,
s'~(~.)
c'%
cA.
f.
~(~.)
C-'L
c'-U
1956 1957 1960 1961 1965 1966 1975
0.702 0.858 0.597 0.862 1.000 0.735
0.139 0.057 0.127 0.032 0.000 0.029
0.391 0.708 0.344 0.785 1.000 0.674
0.896 0.938 0.807 0.914 1.000 0.788
0.022 0.014 0.046 0.040 0.026 0.046 0.022
0.006 0.003 0.009 0.005 0.004 0.004 0.005
0.013 0.009 0.032 0.031 0.019 0.039 0.014
0.037 0.020 0.067 0.051 0.035 0.055 0.034
aSee Tables 16.17 and 16.18. Parameters are annual survival (Si) and recovery rates (fi). hear of banding/recovery; survival estimates are annualized and assumed constant over the interval between banding periods.
386
C h a p t e r 16 T A B L E 16.20
Estimation
D a t a S t r u c t u r e for T w o - A g e
Releases in A g e at b a n d i n g Adult
with Band Recoveries
1
2
3
R(1)
"(1) "'11
"(1) '"12 ,,,(1) "'22
"(1) "'13 ..(1) "'23 ,,,(1) "'33
"LI 1
R(1) R(31)
R~O)
(v = 0)
.~.(o)
rUl I
.~.(o)
.._(o)
"*(0)
"*(0)
ru12
R(20)
"'22
*'1 J1
*'1 "1
J2
g(30 )
*'1 "'1 ~
*'1 '-'1 ~
g(20)s(O)f(31)
g(20)S(20)S(31)f(1)
12(0) ~c(0) *~3 J 3
/2(0) c (o),c (1) *~3 ~ J 4
T A B L E 16.21
~
Expected
Recoveries
for Two-Age
period i R~I)
1
y_,j...,.(1),,,kj
R~~
Ej "*(o) "'lj
R~3~
Ej "*(o),,,3j
m(0) kl
R(kO) _
y_,j..t(0),,,kj
"'23
Band Recovery
]~(1)~c(1) *'1 J l
2
3
/~(1)c(1),(1) *'1 ~ J2 /~(1),(1)
/~(1)c~(1)c~(1),(1) *'1 "1 "2 J3 /~(1)~(1) g(1)
*~'2 J2
R(31)
R~o, a(20) R(~
R(k0)
Data under Model
R~o,f~o)
--. .....
*~'2 "'2 J3
. . . . .
R(31)f(31)
.....
R (1)
(v = 0)
R(kl) _
R ( 3 1 ) - Ej "'3j"*(1)
R~20) _ s "*(o),,,2j
(S,,.t,
fa.t )a
Recoveries in period j
a(21)
Juvenile
(1)
R(21)
only in o2r J4 and o2<(~162 9 The co-occurrence of juvenile and adult parameters in model (Sa.t, fa.t) occurs irrespective of the number of banding and recovery periods, and it persists no matter how many samples are taken. For this model, it is not possible to estimate individual model parameters for juveniles and adults without the banding of adults, so as to allow separate estimation of adult survival and recovery rates. Under certain restrictive model assumptions, some information can in fact be obtained from the banding
J4
Releases in
(v=l)
kl (O) 11 (O) 21 (O) 31
"'33
we see that juvenile survival parameters always occur in products that include adult parameters. For example, S(3~ occurs in the product r~ J 4 ~' and S(2~ occurs
Adult
R~I)
and i = 1 ..... k banding years, j = i ..... 1 recovery years.
g(O)f(20)
A g e at b a n d i n g
y_.,j.,(1) "'lj ~j .,,,(1) '"2j
(1) 11 (1) 21 (1) 31
"*(o)
and juveniles for three years and the recovery of bands for both cohorts for four years. If we focus only on the expected recovery matrix for juveniles, R~O) ~(o)r R(o)r 4:(1) /?(0)q(0)<(1)f(1) /-~(0)r162 R(2O)
Not recovered
rn13
R(3o)
a With ages v = 0 , 1
Data"
Recoveries in period j
period i
(v=l)
Juvenile
Band Recovery
R1(o,S1(o, f(21)
~,o,~(o,~<,, ,(,, *"1 ~'1 "'2 J3
/~(0), (0) *'2 J2
/~(0) c~(0),(1) *'2 ~ J3 R(~176
l
Not recovered
/~(1)c~(1)c~(1)...c~(1)11) 1 J1 ~ ~ //~(1)~(1). c(1) ~1)
a(21)
1~(1)c~(1)...c(1)3 "3 ~
R(1) _
2 ~
"'~
c(1) k ~ ""~
f ~1)
I1)
R~I)
(1)
Do)co)
. . . . .
~(o,,(o,~(1,...sG 1 '-'1 "2
.....
l~(0)~(0)...~(1) ~1) 2 ~ "l-lf
a(0)
.....
/2(~176 o3
e(30) -
9""
~(0)c(0) ~'k ~ ""~c(1) f~l)
o,_1 f 11)
R~o,
E(m~}))
-- Ej E ( m 2 j )
.....
f 11)
R(1)
-- Ej
s E(m(3}))
-- ~j E(m(k} )) -
Ej
E(mlj(o)) (0)
-- ~'j E ( m 2 j )
Gj E(m(3~ ))
R(ko) _ Ej E(m(ko))
aWith ages v = 0 , 1 and i = 1 ..... k banding years, j = i ..... l recovery years. Parameters are annual age- and time-specific survival (S~v), i = 1 ..... I - 1 ) and recovery (f~v), i = 2 ..... I); ages v = 0 (birth/hatching year), v = 1 (after birth/hatching year).
16.2. Multiple-Age Models of juveniles only. For example, a model with constant adult survival and recovery rates allows adult survival and juvenile recovery rates to be estimated, but leaves adult recovery rates and juvenile survival rates inestimable. This is seen by investigating the array a~0) R(0)c(0) R 1(0)$ 1(0)f (1) / ~ ( 0 ) q ( 0 ) q ( 1 ) f ( 1 ) R(0) 1 S(0) 1 S(1) S(1))c(1) *'1 J1
g (2o) R(30)
*"1
R ~20)f (2o)
~'1
~"
J
R (20)S (2~)f (1) r~(0)r (0) *x3 J 3
R (20)S (20)S (1)f (1) /2(0) r *~3 ~
Y
(1)
for three banding years and four recovery years. Adult recovery rate always occurs with juvenile survival rates, so that the parameters cannot be estimated separately. However, because adult recovery rate is constant, adult survival rate can be estimated. Nor is the estimability problem solved by additional restrictions on the juvenile parameters. If juvenile survival and recovery rates also are assumed constant, one can see from the array R~~
a (0)
R30,
a~0)f(~
a~O)s(O)f(1) R~o)S(o)s(1)f(1)a~O)s(O)s(1)S(1)f(1)
a(O)f(O)
a(20)s(O)f(1)
R g0 f 0,
a(20)s(O)s(1)f(1)
R g0,S 0 f l,
of expected recoveries that the parameters S(~ and f(1) always occur together. Thus these parameters cannot be separately estimated. Though it is theoretically possible to devise banding models for which valid estimates of survival and recovery rates can be obtained, such models typically are quite restrictive in their assumptions, and the assumptions almost certainly fail to be met for most migratory species. For example, in the unlikely situation for which survival is age specific but recovery rate is constant and independent of age, survival and recovery rates both are estimable. However, such a model clearly is unrealistic: recovery rates are almost always age specific a n d / o r time specific. The bottom line is that at present strong assumptions are required to obtain valid estimates of age-specific survival and recovery rates if banding is restricted only to juveniles. If band recovery models are to be used, one should be prepared to commit the resources necessary to capture and band sufficiently many adults to obtain good estimates of adult survival and recovery rates. We address the issues of study design and sampling intensity below.
387
to year i. This results in an array of expected returns that is identical in its parameter structure to the array of expected returns for the analogous single-age model (St, f';). At first glance it might also seem reasonable to incorporate a temporary banding effect in the probability structure for juveniles, because juveniles can experience a temporary effect from banding as well as adults. However, maturation from the juvenile to adult stage is a temporary effect that cannot be separated from other temporary effects such as banding. For this reason only a single time-specific parameter is needed to account for the transition of juveniles to adulthood. The parameter structure for the overall model (assuming l > k) thus includes S i(1) ,
i = 1,
fl 1),
i = 2, ..., l;
i , f (1)*
i = 1,
...1
.-.I
l-
k;
SI~
i = 1.... , k;
f i(0),
i = 1'
""1
1;
k;
representing adult survival rates, indirect and direct recovery rates for adults, and juvenile survival and recovery rates, leading to the expected values in Table 16.22. Again, only a limited set of these parameters can be separately estimated if l > k: S(1) i i
i = 1,
fl 1),
i = 2, ..., k;
fl 1)*,
i = 1, ..., k;
.--I
k-
1;
S(0) i ,
i = 1'
""1
k - 1;
i , f (0)
i = 1,
..-1
k.
Programs BROWNIE or MARK (Appendix G) can be used to obtain estimates for model (Sa,t, f**t) for the two-age case [denoted H 2 in Brownie et al. (1985)], as well as goodness-of-fit tests and likelihood-ratio test statistics for comparison with other models. More general cases with m age classes can be constructed in program MARK (Appendix G).
16.2.5. Unrecognizable Subadult Cohorts 16.2.4. Temporary Banding Effect A straightforward extension of the m-age class model (Sa.t, fa.t) allows for temporary effects of banding on recovery rates, in a manner analogous to the singleage class model (St, f D described in Section 16.1. This can be illustrated by the two-age case, in which it is assumed that adults banded in year i have a different recovery rate f* from that of adults banded previous
It sometimes is possible to distinguish juveniles (v = 0) from older individuals, but impossible to distinguish subadults (v = 1) from adults (v = 2). In this situation the banded cohort of juveniles at the time of banding consists exclusively of juveniles, but the cohort labeled as "adults" consists of both subadults and adults in unknown proportions. If subadults and adults differ in their survival and recovery rates, these differences
388
Chapter 16 Estimation with Band Recoveries Table 16.22
Expected Recoveries for Two-Age Band Recovery Data under Model (Sa. v fa.t) a
Releases in i Age Adult
Recoveries in p e r i o d j Bands
(v = 1)
R{ 1)
1
2 R~1)r ~'1 J 2
1;}(1)((1)* *'1 Yl
n(21)
R (21)f (21)*
R(3~)
9.-
3 a ~ l ) s ~ 1)~(1)((1),-'2/3
(1)c(1)f(31) 2 ~'2 (1)((1)* 3 ]3
. . .
k . . .
. . .
. . .
". . .
(v=0)
R~o)
(O)r
~(o)((o) *~1 11
1 `'1
(O)((o)
R(2o) R(3o)
2 J2
~c(1) J2
-'-'1D(1)c'(1)c'(1)'"S11-)1f11)~'1 `'2
.
D(1)c(1)...c(1) f}l) *'2 '-'2 `'I-1
R(1)
~j
E(m(2}))
.
D(1)r162 *~3 " 3
R(31)
-- E j
E(m(3}))
a(k1)
__ Ej E(m(k}))
"'"
`'1-1
f}l)
~(1)c(1)...c(1) fll) ~" `'k `'I-1
R (o)c(0)r
`'2 J 3
.
.
.
.
.
.
D(0)C(0)C(1)...r *'1 ~'1 `'2 `'l-1
R(o)r 2 `'2
]3
.
.
.
.
.
.
~(o)r x~2 ~"2
.
1 "1
R(o)((o) 3 J3
R(k0)
.
.
.
.
R(~
.
0)
R~1) - Gj E(m~} ))
.
(1)((1)* k Yk
Juvenile
Not recovered
l
_
R~0) __
~j
~(o)~(o). c(1) f}l) ~ 3 `'3 " "'-'1-1
R(2~ R(o)
Ej E(m(2~)) s E(m(3~))
*~kD(0)C(0)"'C(1)`'k `"-lf~1)
a(kO) -- Ej E(m(k~))
c(1)f}l)
"'~
f}l)
_
_
E(m~?))
aWith a g e s v = 0 , 1 a n d i = 1 . . . . . k b a n d i n g y e a r s , j = i . . . . . l r e c o v e r y y e a r s . P a r a m e t e r i z a t i o n s i n c l u d e t e m p o r a r y effect of b a n d i n g . P a r a m e t e r s are r a t e s of a n n u a l a g e - a n d t i m e - s p e c i f i c s u r v i v a l (.q(O) --i , i = 1 ..... k; S i(1) , i = 1 ..... l - 1 ); a g e s v = 0 ( b i r t h / h a t c h i n g y e a r ) , v = 1 (after b i r t h / h a t c h i n g y e a r ) , j u v e n i l e r e c o v e r y r a t e s f l ~ i = 1 ..... k, a d u l t d i r e c t r e c o v e r y r a t e s f l 1)*, i = 1 . . . . . k, a n d a d u l t i n d i r e c t r e c o v e r y r a t e s f l 1), i = 2 ..... l.
will be reflected in the survival and recovery parameters of the "adult" cohort for the banding year (after the banding year all surviving subadults in the cohort will have matured into adults). To model this situation, we include the survival and recovery parameters as before for juveniles (S f(~ ,1 subadults,_i (~q(1) and ---i(~ and Ji f(1)~ and adults (SI 2) and (2) f i )" T o these parameters must J i ,i be added the additional parameters SI 1-2) a n d Ylf9 -2)i where "1-2" denotes the mixture of subadults and adults in an "adult" cohort in the year the cohort is banded. Thus, the suite of parameters for this model for l > k includes SI~
i = 1, ..., k;
fl ~
i = 1, ..., k;
511) 1
i = 2, ..., k + 1;
fl 1),
i = 2, ..., k + 1;
S(2) i ,
i = 2,
"",
l-
f i(2),
i = 2,
""1
l;
S(1-2) i ,
i = 1,
"-',
k;
I I f !l-2)
i = 1t
""t
1;
k;
where the first set of survival and recovery parameters applies to juveniles, the second set applies to subadults, the third set applies to adult cohorts after the banding year, and the last set of survival and recovery rates applies to "adult" cohorts for the year they are banded (Brownie et al., 1985). The array of expected recoveries
for a study involving unrecognizable subadults is portrayed in Table 16.23. Note that there are only two arrays of data for this model, even though it is applicable to populations with three age classes. The reason for the limitation to two data arrays is that subadults and adults, because they are indistinguishable at the time of banding, essentially are included in the same "adult" cohort. The year they are banded, these "adult" cohorts have survival and recovery rates that reflect the mixture of age classes; hence the need for the parameters fl 1-2) and S I]-2) in the formulas for expected recoveries. In the years after the banding of an "adult" cohort, all individuals in the cohort are certain to be adults rather than subadults, so that the adult survival and recovery rates then apply. Note also that subadult parameters must be included in the juvenile array to account for the fact that juveniles pass through a subadult stage before becoming adults. In essence, the juvenile cohorts are assumed to consist only of juveniles, but they must mature through a subadult age class before reaching adult status. For this reason there are parameters ill!l) and SI1) that apply specifically to subadults. Because it is only for individuals banded as juveniles that the subadult age can be unambiguously recognized, these parameters only occur in the juvenile data array. From the model parameterization, it can be seen that banding-year variation in survival and recovery is introduced in both the juvenile and the adult arrays. Recall that the two-age class model accounted for juveniles by means of parameters for the year of banding
16.2. Multiple-Age Models TABLE 16.23 Age
Releases
v
R! v)
(v = 1-2)
R~1-2)
Expected Recoveries for Two-Age Band Recovery Data with Unrecognizable Subadult Cohorts a Recoveries in period j 1
/~(1-2)~:(1-2)
*'1
Jl
R(21-2)
2
1~(1-2)r
*'1
"1
3
J
*-2/;? (1-2) f(1-2)j2
/(31-2)
/~(1-2)r162
*'1
"1
...
"2 J3
"'"
R~o) e (0) a(30)
~}(0)~(0) ~Xl J1
~(0)r ~c(1) ~'1 "1 J2 12(0) ,r (0) *"2 ]2
Not recovered
l
/~(1-2)~(1-2)~(2) c(2),c(2)
R~1 - 2 ) -
*'1
"1
~
""~
~j L~,"'lj " lvt"(1-2)~
2 ) , . . ,f2 *x2~(1-2)(c:(1-2)
-.-
*-2/2 (1-2)c~(1-2)..,,,2.o1_1~2(2)f~2)
R(21-2) - ~j ~,"'2jlE'(1-2),/ /'*"
*'3/~(1-2)(1-2)j3 ~e
""
"3~(1-2)r . , , , 3 . .oi_1c(2)f12)
R(1-2) - ~j E(m(1-2))3j
~'k~(1-2)c(1-2)~...OlC(2)_1Y~C(2)l
R(k1-2) -- Ej L~,,,~kjlV f,,,,(1-2)~j
/~(0)c(0)r }2) 1 ~ "2 "'3 '-'l-lf //~(0)C (0)r (1)C(2)...C(2) ~2) 2 ~ a3 ~ ~ /~(0)c(0)c(1)c(2) c(2) ~2) BOB o 4 o 5 ""Ol_lf
R(2O)
/)(o)c(o)c(1) c~(2) c(2)f12) ~"k ~ ~176176
R(kO) _ Gj E(m(~))
R(k1-2)
(v = 0)
389
l~(o)c(0)c(1)f(2) ~xI o 1 o 2 /2(0)C(0),e(1) "x2 ~ J3 /~(o)~c(o) x-3 J3
R (0)
..... ..... .....
R~o)
--~'j
E(m~?)) E(m(2~))
-- Ej a(3O) -- ~j E(m(~))
aWith i = 1 ..... k b a n d i n g y e a r s , j - i ..... I r e c o v e r y y e a r s , a n d u n r e c o g n i z a b l e s u b a d u l t c o h o r t s (v = 1-2). A n i m a l s b a n d e d as " a d u l t s " are a m i x t u r e of s u b a d u l t s (v = 1) a n d a d u l t s (v = 2), b o t h of w h i c h are d i s t i n g u i s h a b l e f r o m j u v e n i l e s (v = 0) b u t n o t f r o m e a c h other. P a r a m e t e r s are a g e - a n d t i m e - s p e c i f i c r e c o v e r y r a t e s (f!v)) a n d s u r v i v a l rates (SIv)) for j u v e n i l e s (v = 0), s u b a d u l t s (v -- 1), a d u l t s (v = 2), a n d m i x t u r e of a d u l t s a n d s u b a d u l t s (v = 1-2).
of the juvenile cohorts. In the present case, both juvenile and "adult" banding cohorts have parameters for the year of banding, and juvenile cohorts also have additional parameters to account for subadults. Thus, the model can be seen as a fairly straightforward extension of the two-age class model. However, because of the addition of yet another set of parameters into the already parameter-rich model structure of the two-age class model, data requirements for this model are quite onerous. A total of 6k + 2l - 3 parameters are identified for the model. As above, not all parameters can be separately estimated. In fact, the only separately estif ! 1 - 2 ) . .i .= . 1 mable parameters are Jf(O) i ' ~, . , k, and the parameters of primary interest, i.e., survival rates, cannot be separately estimated (Brownie et al., 1985). Thus this model is of little interest for estimation. However, lack of fit of the ordinary two-age model compared to this one is diagnostic of the need for inclusion of a marked subadult class in the data structure. The program BROWNIE (Brownie et al., 1985) includes estimation procedures for parameters of this model (denoted as H3), as well as goodness-of-fit tests and likelihoodratio test statistics for comparison with other two-age class models. Alternatively, the above model structure can be formulated in program MARK (Appendix G), together with constraints to form appropriate reduced parameter models (e.g., constancy of parameters over time). The problem for two age classes is a special case of a problem that occurs any time one seeks to estimate
parameters for m + 1 age classes using band recovery data, but the field situation allows identification of only m (or fewer) age classes on release. As above, differences in survival or recovery rates among unidentifiable cohorts may be expressed as lack of fit when a simplified (but identifiable) model structure is used to estimate parameters. Programs such as MARK can be used to construct models for this situation based on all m + I age classes, but the majority of the parameters are unidentifiable unless further constraints are imposed. However, such models may be useful for comparative purposes, particularly when it is of interest to identify reasons for lack of fit that may occur in simpler models. 16.2.6. G r o u p a n d C o v a r i a t e E f f e c t s
The effects of sex, geographic location, or other strata can be incorporated in a manner analogous to the one-age case (Section 16.1.5). For example, an ageand sex-specific generalization of model (Sa.t, fa.t) would have all combinations of age and sex classes at each sampling occasion for both the survival and recovery parameters, and might be denoted as model (Sa.s.t, fa.s.t). Unique estimation of parameters for this model requires a data structure in which animals of each identifiable sex and age class are released at each period, i.e., RI ~'s) > 0 for v = 0..... m ages, s = 1, 2 sexes, and i = 1.... , k banding occasions. Reducedparameter models can be formed by constraining pa-
390
Chapter 16 Estimation with Band Recoveries
rameter indices over the relevant dimensions. For instance, model (Sa.s.t, fa,t) describes a situation in which sex- and age-specific survival parameters are modeled for each time, but recovery parameters are modeled as age specific but not sex specific. As described in Section 16.1.5, parallelism in parameter variation also can be modeled. For example, model (Sa.s+t, fa.s+t) describes a situation in which all 2(m + 1) combinations of sex and age are modeled, but the respective age- and sexspecific parameters vary over time in parallel. Similarly, covariate effects can be modeled as in Section 16.1.6, by imposing functional constraints on the parameters of a general model such a s (Sa.t, fa.t). For example, the constraint
[
]= oov,+ov,xi
In 1 - S}v)
describes a relationship between age-specific survival and a time-specific covariate (e.g., temperature). Though this example models the covariate relationship uniquely for each age class; parallel relationships could be specified--for instance, by specifying a common slope but different intercepts:
is,v,]: oov,+olXi"
In 1 - S}v)
16.2.7. Banding Multiple Times per Year As was the case with single-age models, it is sometimes useful to band at two or more times per year with multiple-age models. Rather than attempting to characterize the large number of potential applications for this banding scheme, here we provide an example involving models with two age classes (Hestbeck et al., 1989). In this sampling situation, both fledged young and adult birds are banded just before the hunting season. The matrices of expected numbers of recoveries for a study with k = 3 banding years and l = 4 recovery years are a~l)
R(21
a~l)f~l)R(1)q(1),c(1)/~ (1)q(1)~(1),c(1)/~ (1)~(1)~(1)c(lhe (1) *'1 "1 J2 *'1 "1 ~ J3 *'1 ~ ~ ~ J4
R(i1f21
/(31)
R(i1 s 1 f(31
R(21s1,s(31,f(41,
1~(1)~r "~3 j 3
R(31)~(1)~-(1) ~ j4
for adult birds, and R~O)
R (0, R(3o)
R~O)f { o ) l ~ ( o ) q ( o ) ~ r 1 6 2 *'1 ~'1 J2 *'1 ~'1 " 2 J3
R(O)f(o)
a study of mallard ducks in southwestern Saskatchewan by Hestbeck et al. (1989) included birds banded as "locals" [ducklings in broods aged in classes II and III of Gollop and Marshall (1954)] during July and early August (median banding date in mid-July). In this case the two banding periods per year applied only to young birds, and the objective was to estimate not only annual survival of adults and fledged young, but also the probability of local young surviving to become fledged young in mid-September (median banding date for fledged young). Let QI~ denote the number of local young banded _(0) denote the probability during July of year i, and Wi that a local young in July of year i survives to become a fledged young in September of year i. The matrix of expected values for recoveries of birds banded as local young can be written as
Q~O) Q(20, Q(3o)
,~(0} (o)r(o) kdl q}l f l
,~(0) (o)~(o)r(1) k~l q~ D1 f l
Q(20)q~(20)f(20)
,~(0) (0),~(0),~(1)r(1) ,-~(0) (0),~(0),~(1)c,(1)t(1) kdl qOl 131 D2 f3 k~l q01 D1 ~2 ~}3 f4
Q(20)q~(20)s(20)f(31) Q(20)~(20)s(20)s(31)f(41) ,~(0) (0).(0) k~3 ~p3 J3
.~(0) (0),~(0).0) k~3 ~3 ~3 J4
The model described by this array of expected recoveries permits estimation of the 2-month probabilities of surviving from the local young stage to the fledged young stage ~" i (0),}, as well as the annual survival rates of adults and fledged young and their associated recovery rates. The model used by Hestbeck et al. (1989) was developed for a specific sampling design and estimation problem, but other designs using multiple banding periods per year with multiple ages can be envisioned. For example, Reynolds et al. (1995) developed a model to estimate spring-summer survival probabilities for yearling (first year) and adult (after first year) mallard females banded in both spring and late summer in Manitoba, Saskatchewan, and Alberta. Birds were aged and released as either yearlings or adults in the spring bandings, whereas all birds were treated as adults in the summer bandings (all birds were > 1 year old). The model was developed for a specific sampling situation and biological question and will not be described here, but the point is that designs involving multiple banding periods per year and multiple age classes can be tailored to sampling situations and questions of interest using the general modeling principles in this chapter.
(o,q(o,~(1)c(1)~.(1) "'1 ~"1 "2 ~ J4
R(o)S(O)f(1)
R(o)S(o)s(1)f(1)
~,(o)r (0) *~3 ] 3
i~(0) r *~3 ~
(1) ./4
for young birds (these are the same expectations as in Table 16.21). The new element here is that young birds (not adults) also are banded at a younger age and thus in an earlier period during the same year. For example,
16.2.8. Model Comparison and Selection As seen above, models involving two or more age classes have a large number of parameters. In situations where the appropriate data structure exists, i.e., release of banded animals in each of m +1 recognizable age classes, age- and time-specific estimates of survival and recovery rates can be obtained under model (Sa.t,
16.3. Reward Studies for Estimating Reporting Rates
fa.t)
o r the generalization (Sa,t, f**t). Once the MLEs for a model are obtained, they can be used to compute matrices of expected values, which in turn provide a means of evaluating goodness of fit. Assuming that a general model such a s (Sa.t, f*a,t) fits the data, one can be reasonably confident that the MLEs are unbiased estimates of the underlying parameters of interest. However, as a general rule one should endeavor to estimate as few parameters as possible and at the same time fit the data. In many cases, especially when data are relatively sparse, estimates under highly parameterized models will have poor precision. As with oneage models, it is desirable to seek a compromise between model fit, which is increased by adding parameters, and estimator precision, which is increased by eliminating unnecessary parameters. The same basic approaches can be used for both multiple-age models and models with a single age class (Section 16.1.8), the obvious difference being that many more reduced-parameter models can be obtained with the more general parametric structure of a multiple-age model.
16.2.9. C o h o r t M o d e l s a n d Parameter Identifiability
It is important to keep in mind that proper identification and estimation of the parameters of multiple-age models is possible only if sufficient numbers of animals of each age class are released at each sampling occasion. Thus, for two-age classes (m = 1), both juveniles (v = 0) and adults (v = 1) must be marked and released each year. If the primary interest is in adult survival, and few juvenile birds can be marked, then it is probably best to use the one-age models described in Section 16.2 and ignore juvenile bandings. However, the opposite problem is more common, in which juveniles are readily captured and marked but few or no adults are marked. A similar situation in the context of m a r k recapture is addressed by means of cohort-specific Jolly-Seber analyses (Section 17.2.3), wherein the availability of multiple recaptures allows for the estimation of time-specific survival rates for each age and cohort. However, with band recoveries such an analysis is not possible. One approach (Section 16.4.3) has been to ignore time specificity and construct a completely age-specific band recovery analysis. But even with this approach, arbitrary constraints generally are required to obtain identifiable parameters. In addition, the necessary assumptions of time constancy of parameters, especially of recovery rates, frequently are not justified. For many, perhaps most, species both survival and recovery (harvest) rates can be expected to be highly variable through time, and in fact this variability may be of
391
primary interest as it relates to changes in harvest or habitat management. We recognize that some circumstances may warrant the use of cohort or other modifications of band recovery analysis. In Section 16.4, we discuss issues related to analysis of band recovery data from studies that differ from those typically encountered in the investigation of North American waterfowl (e.g., nonharvested species, whereby recovery is principally through incidental discoveries of dead birds).
Example Multiage band recovery models can be illustrated with a two-age example originally presented by Brownie et al. (1985). Adult and young male mallards were banded during July-September, 1961-1973, in the San Luis Valley, Colorado. Band recoveries in each of the 1963-1971 hunting seasons are presented in Table 16.24. Fifteen models were fit to these data using program MARK (Appendix G), ranging in complexity from model (Sa.t, fa**t), which allows time- and agespecific variation in survival and recovery and includes a temporary banding effect on recovery, to model (S, f), which estimates one survival and one recovery parameter for all ages and sampling occasions. In addition, several models were fit that allow a parallel response over time--for example, model (Sa+t, fa+t)" Model (Sa, fa+t), describing age- but not time-specific variation in survival and age-specific, parallel time variation in recovery rate, was selected on the basis of comparison of QAIC c statistics (Table 16.25). Parameter estimates for this model are reported in Table 16.26 and indicate higher survival rates and lower recovery rates for adults than for juveniles.
16.3. R E W A R D S T U D I E S FOR ESTIMATING REPORTING RATES 16.3.1. D a ta S tru cture
As noted earlier, recovery rates for harvested birds implicitly incorporate the rate of mortality from harvest, as well as the probability of retrieval of harvested birds and the probability that a retrieved bird is reported. If we define the rate of harvest H i as the proportion of (legally) killed animals that are retrieved, recovery rates can be reexpressed as fi = ~iHi,
where H i = c i K i is the probability that an animal alive at the midpoint of the banding period in year i is harvested (K i) and retrieved (ci) during the hunting season of year i, and ~'i is the probability that a har-
392
Chapter 16 Estimation with Band Recoveries TABLE 16.24
Recoveries of Adult and Young Male Mallards Banded Preseason (July-September) in the San Luis Valley, Colorado a Recovered during hunting season
Age class
Year
Bands
1963
Adult
1963 1964 1965 1966 1967 1968 1969 1970 1971 1963 1964 1965 1966 1967 1968 1969 1970 1971
231 649 885 590 943 1077 1250 938 312 962 702 1132 1201 1199 1155 1131 906 353
10
13 58
83
35 103
Young
1964
1965
1966
1967
1968
1969
1970
1971
6 21 54
1 16 39 44
1 15 23 21 55
3 13 18 22 39 66
1 6 11 9 23 46 101
2 1 10 9 11 29 59 97
18 21 82
16 13 36 153
6 11 26 39 109
8 8 24 22 38 113
5 6 15 21 31 64 124
3 6 18 16 15 29 45 95
0 1 6 3 12 18 30 22 21 1 0 4 8 1 22 22 25 38
aFrom Brownie et al. (1985).
vested a n d retrieved bird is reported (Fig. 16.1). Frequently, an estimate of the rate of harvest H i is of intrinsic interest a n d m a y be obtained if the reporting rates are either k n o w n or can be estimated with auxiliary data. One a p p r o a c h is to divide the m a r k e d sample into t w o groups, a "control" sample that is m a r k e d using s t a n d a r d m a r k s (e.g., n o r m a l a l u m i n u m leg b a n d s for birds) a n d a " r e w a r d " sample of animals m a r k e d with b a n d s indicating that the finder will be paid a m o n e t a r y or other r e w a r d for reporting the band. The r e w a r d b a n d s are a s s u m e d to have high rates of reporting (ideally, ~ close to 1), a n d the ratio of recovery rates for the t w o samples can be used to estimate )~. This can be seen by c o m p a r i n g the expected values for the direct (first-year) recoveries from a simple r e w a r d study, w h e r e R control b a n d s and R' r e w a r d b a n d s are applied, a n d recoveries are from a single harvest period i m m e d i a t e l y following release:
Number marked
Expected number recovered
Control
R
RKH
Reward
R'
R'H
A simple m o m e n t estimator for )~ is obtained by equating the n u m b e r of control (m) a n d r e w a r d (m') recoveries to their expected values and solving for the u n k n o w n parameters. This leads to H=
m'/R'
and
m/R m'/R'"
A complication occurs w h e n deliberate efforts are u n d e r t a k e n to obtain b a n d s from hunters in the field, usually in the course of law enforcement or g a m e checking activities. These efforts result in t w o types of reports of s t a n d a r d bands: normal, v o l u n t a r y reports by hunters, a s s u m e d to occur at the rate K, a n d reports t h r o u g h solicitation, w h i c h are a s s u m e d to be perfectly reported (K = 1). In this situation the probability that a b a n d on a harvested animal is reported t h r o u g h solicitation (~/) is usually not of intrinsic interest, but m u s t be taken into account in the estimation of reporting a n d harvest rates. The recoveries for s t a n d a r d b a n d s
16.3. Reward Studies for Estimating Reporting Rates TABLE 16.25 Model Selection of Adult and Young Male Mallards Banded Preseason (July-September) in the San Luis Valley, Colorado a Goodness of fit b Model
Parameters
X2
df
P
(Sa, fa+t) (Sa, fa*t) (Sa*t, fa*t) (Sa*t, f*a*t) (Sa*t, fa+t)
12
110.1
76
0.006
0.00
20
95.43
68
0.016
3.05
34
68.49
54
0.089
7.31
42
53.11
47
0.250
9.79
26
97.06
63
0.004
16.53
(Sa+t, fa*t) (Sa+t, fa+t)
AQAICc r
27
114.3
62
0.000
33.82
19
133.2
70
0.000
34.53
4
172.7
84
0.000
39.45
(St, fa+t)
18
142.3
71
0.000
40.52
(Sa.t, fa)
18
148.3
71
0.000
45.89
(S, f~)
3
186.7
85
0.000
49.83
(Sa, f)
3
253.2
85
0.000
108.64
(Sa, f~)
(Sa+t, ft)
18
232.8
71
0.000
120.64
(S, f)
2
278.0
86
0.000
128.59
(St, ft)
17
248.4
72
0.000
132.42
a From Brownie et al. (1985). Parameters are annual survival (S i) and recovery (fi) rates; subscripts a and t denote variation in parameter over age and time, respectively; " , " denotes interactive effects of age and time, " + " denotes additive effects of age and time. b Deviance-based chi-square test; see Section 4.3.3. c Difference between model QAICc (Akaike's Information Criterion, corrected for small effective sample size and adjusted by the quasilikelihood factor ~ = 1.13; see Section 4.4.), and QAIC c value for the lowest ranked model.
now are poststratified by those obtained from hunters (mh) and by solicitation (ms); reward bands are assumed reported by hunters (m'), with K = 1 as before. The expected values for these observed recoveries are Expected number recovered Band type
Number marked
Hunter reported
Solicited
Control
R
RM1 - ~/)H
Reward
R'
R'(1 - ~/)H
R~H R' ~/H
and moment estimators of K, ~/, and H are
H = m'/R', ~/
ms/R m'/R"
and
mh/R m' /R' - ms/R
393
(Henny and Burnham, 1976; Conroy and Williams, 1981). Variances can be derived for these estimators (e.g., Henny and Burnham, 1976) or obtained numerically from the Fisher information matrix, using SURVIV (White, 1983), for example. As noted by Conroy and Williams (1981), the estimator of reporting rate for standard bands is sensitive to the assumption that ~ = 1 for both the reward bands and the bands obtained through solicitation. Positive biases in K occur whenever either the first condition (100% reporting of reward bands) or both conditions (100% reporting of reward and solicited bands) do not hold. Bias can be reduced (but probably not eliminated) by increasing the reward value; alternatively, bands of differing rewards can be used to allow estimation of a response curve for reporting rate of reward bands, and estimation of a reward band reporting rate ()tr < 1) (Nichols et al., 1991). The above estimators form the basis for a more complete investigation of the variation in reporting, solicitation, and harvest rates, as indicated in the following section. 16.3.2. M o d e l i n g S u r v i v a l R a t e s w i t h Indirect and Direct Recoveries
In this situation both control (standard) and reward bands are applied to animals during each banding period, and the data consist of recoveries of both types of bands in the subsequent harvest periods. The data are stratified by the two types of bands (control and reward), with expected values for the recoveries described in terms of the released sample sizes R or R' and time-specific harvest, reporting, and survival rates. To illustrate, consider a study not involving recoveries by solicitation, in which control and reward bands are applied in each of three years, followed by four years of recoveries (Table 16.27). The expected numbers of recoveries for this data structure are shown in Table 16.28. The model allows for time-specific estimation of reporting, harvest, and survival rates. Parameter estimates and tests of hypotheses are programmed in MULT (Conroy et al., 1989b) and also can be constructed using SURVIV (White, 1983). Notice that survival rates are specified separately for the control and reward groups; this is the fully parameterized model for which closed-form estimates are possible. Under most reasonable study designs (e.g., animals randomly assigned to control or reward marking) the model can be simplified under the hypothesis H0: S; = Si, i.e., common survival rates for the two groups. If band recoveries are solicited, then the data must be poststratified by type of reporting (hunter report vs. solicited) and analyzed by accounting for solicita-
394
Chapter 16 TABLE 16.26
Age
Estimation with Band Recoveries
Parameter Estimates for Model (S a, fa+t) for Data and Analysis in Tables 16.24 and 16.25 a
~(v)
Year
~'-E(~(v))
C'~L
CAU
flv,
S"E(f~v))
C'L
CU
0.006
0.042
0.066
Adult
1963-1971
0.654
0.013
0.629
0.678
Young
1963-1971
0.541
0.027
0.486
0.592
Adult
1963
0.053
Young
a
1964
0.087
0.007
0.075
0.101
1965
0.052
0.004
0.044
0.061
1966
0.072
0.005
0.063
0.082
1967
0.059
0.004
0.051
0.067
1968
0.066
0.004
0.058
0.074
1969
0.075
0.004
0.067
0.084
1970
0.076
0.004
0.068
0.085
1971
0.055
0.004
0.047
0.064
1963
0.083
0.009
0.067
0.102
1964
0.135
0.010
0.117
0.155
1965
0.082
0.006
0.071
0.095
1966
0.112
0.007
0.099
0.126
1967
0.092
0.006
0.081
0.105
1968
0.103
0.006
0.091
0.116
1969
0.116
0.006
0.104
0.130
1970
0.118
0.007
0.106
0.132
1971
0.087
0.007
0.074
0.101
Estimates are annual survival (S (v)) and recovery (fi(v)~, rates for young (v = 0) and adults (v = 1).
Example
tion rates as indicated above. For example, in a 3-year banding study with recoveries of both types for each of the three subsequent harvest periods, the data structure and expected recoveries are displayed in Tables 16.29 and 16.30. As above, program MULT provides estimates for this data structure and provides tests of hypotheses of time specificity in each of the parameters )ki, "~i, Hi, and Si. Alternatively, models can be constructed using SURVIV (White, 1983).
TABLE 16.27
Band type Control
American black ducks (Anas rubripes) were trapped in eastern Canada during 1978-1980 following the hunting season (i.e., during late January-March) and banded with control and reward bands (Conroy and Blandin, 1984). Because of the long interval between banding and the subsequent opportunity for harvest (during September-January), the estimated harvest
Data Structure for a Reward Band Study I n v o l v i n g Indirect Band Recoveries in the A b s e n c e of Band Solicitation a Recoveries in period j
Releases in period i
1
R1
mll
R2
3
4
Not recovered
m12
m13
m14
R1 - ~,j mlj
m22
m23
m24
R2 - Xj m2j
m33
m34
m~3 m~3
m{4 m~4
R3 - ~,j m3j a~ - ~,j m{j R~ -- ~ j m~j
m~3
m~4
R[3 - ~j m~j
R3 Reward
R~
m~l
R~
R~ aWith k = 3 banding years and l = 4 recovery years.
m~2 m~2
16.3.
TABLE 16.28
Expected Recoveries for a Reward Band Study I n v o l v i n g Indirect Band Recoveries in the A b s e n c e of Band Solicitation a
Releases in Band type Control
Recoveries in period j
period i
1
2
3
al
Rlf l
alSlf 2
R15152f 3
RISIS253f 4
al - ~j E(mlj)
R2f l
R2S2f 3
R2S2S3f 4
R2
Rgf 3
RBSBf 4
R3 - ~j E(m3j)
R~S{H 2
R~S~S~H 3
R~S{S~S~H 4
R~ - Xj E(m~j)
a~H 1
R~S~H3
R~S~S~H4
R~ - Y_,jE(m~j)
R~H 3
R~S~H4
R~ - ~,j E(m~j)
R2 R3 Reward
R~
R~H 1
a~ R~
4
aWith k = 3 banding years and l = 4 recovery years (Table 16.27). Parameters are annual survival (h i) rates; annual recovery rate is fi = hiHi.
TABLE 16.29.
Band type
395
R e w a r d S t u d i e s for E s t i m a t i n g R e p o r t i n g Rates
(S i
Not recovered
-- ~ j
E(m2j)
and S;), harvest (Hi), and reporting
Data Structure for a Reward Band Study I n v o l v i n g Both Indirect and Solicited Band Recoveries a
R1
Solicited
Recoveries in period j
Recoveries in period j
1
2
3
1
2
3
mh11
mh12
mh13
ms11
ms12
ms13
mh22
mh23
ms22
ms23
Releases in period i
Control
Hunter reports
R2 R3 Reward
ms33
mh33
R~
m~l
a~
m~2
m{3
m~2
m~3
R~
m~3
a With k = 3 banding years and l = 3 recovery years, with poststratification of recoveries by type of reporting (hunter reported vs. solicited). Excludes final column of animals released in i and never recovered, R i - ~,j mhij -- ]~j msij and R; - ~ j m;j. Because expectations are the same for hunter-reported and solicited reward bands, there is no need to stratify recoveries into hunter-reported and solicited recoveries.
TABLE 16.30.
Band type Control
Expected Recoveries for a Reward Band Study I n v o l v i n g Both Indirect and Solicited Band Recoveries a
Releases in period i R1
1 R l fhl
R2
Hunter reports
Solicited
Recoveries in period j
Recoveries in period j
2
3
1
R1Sl fh2
R I S1S2fh3
alfsl
a2fh2
R2S2fh3
R3 Reward
a~ R~ a~
R3fh3
R~H 1
R~S~H2 R~H2
2
3
alSlfs2
RiS1S2fs3
a2fs2
R2S2fs3 R3fs3
R~S~S~f3 R~S~H3 R;H 3
a With k = 3 banding years and l - 3 recovery years, with poststratification of recoveries by type of reporting (hunter reported vs. solicited). Excludes animals released in i and never recovered: R i - ~,j mhij -- ~,j msq and R~ - "dq m;j. Parameters are annual survival (S i and S;), harvest (Hi), solicitation (~/i), and reporting (Ki) rates; annual recovery rate for hunter reports fhi = Ki(1 -- ~li)Hi, and for solicited reports fsi = "~iHi 9 Because expectations are the same for hunter-reported and solicited reward bands, there is no need to stratify recoveries into hunter-reported and solicited recoveries.
396
Chapter 16 TABLE 16.31
Estimation with Band Recoveries
Reward Study of American Black Ducks (Anas rubripes) Banded in Eastern Canada a Number of recoveries by year Solicited
Hunter reported Band type
Control
Reward
Year banded
Number banded
1979
1980
1981
1979
1980
1978 1979 1980 1978 1979 1980
925 694 758 208 150 150
20
9 9
16
10 7
11
11 11
17 10 23 6 8 13
1981
a Banded during 1978-1980 following the hunting season (i.e., during late January-March), with control and reward bands (Conroy and Blandin, 1984), and recovered during the 1978-1980 hunting seasons.
rates (/-?/i) included mortality during the period J a n u a r y September. H o w e v e r , estimates of reporting (Xi), solicitation (~/i), a n d survival rates w e r e unaffected. Substantial effort at b a n d solicitation w a s exerted (Conroy a n d Blandin, 1984), so recoveries of control b a n d s w e r e poststratified into those obtained by v o l u n t a r y reports from h u n t e r s a n d those obtained t h r o u g h solicitation (Table 16.31). P r o g r a m MULT (Conroy et al., 1989b) w a s u s e d to fit several m o d e l s to these data, w i t h results s u m m a r i z e d in Table 16.32. All the m o d e l s fit the data (P ~ 0.15), a n d several w e r e r a n k e d nearly the s a m e w i t h respect to AIC scores (AAIC < 2). The m i n i m u m AIC w a s p r o d u c e d for m o d e l ()~, "Yt, Ht, S), with time-
TABLE 16.32 Model Selection Criteria for a Reward Study of American Black Ducks Banded in Eastern Canada a Goodness of fit b Model
0~, ~/t, Ht, S) (Kt, "~t,
Ht, S)
(K, ~/t, Ht, St) (~, ~, H, S) (~'t, "~t, Ht, St) 0~, ~/, Ht, S) Okt, "~t"Ht, St, St)
Parameters
8 10 9 4 11 6 13
X2
df
P
10.013 10 0.439 6.228 8 0.622 9.673 9 0.378 19.027 14 0.164 5.892 7 0.552 16.421 12 0.173 5.095 5 0.404
AAICc 0.0 0.3 1.4 1.6 1.7 3.0 4.9
specific variation in solicitation a n d h a r v e s t rates b u t constant survival a n d reporting rate (Table 16.33).
16.3.3. Modeling Spatial Variation in Reporting Rates with Direct Recoveries
M a n y questions, such as that of g e o g r a p h i c variation in reporting rate, can be a d d r e s s e d by considering only direct recoveries, ideally obtained i m m e d i a t e l y following the m a r k i n g period (i.e., not s e p a r a t e d by a l e n g t h y p e r i o d over w h i c h mortality can occur). This design simplifies m o d e l i n g because survival rates no longer n e e d be considered, a n d it also p r o v i d e s a d d e d flexibility w i t h respect to the testing of hypotheses. As above, the s a m p l e s of control a n d r e w a r d b a n d s (Ri, R I) are again g r o u p e d by the time periods of b a n d ing (i = 1, ..., k). Direct recoveries are stratified by the year of b a n d i n g a n d also by g e o g r a p h i c locations of recovery (t = 1 .... , b), w h e r e "location" signifies a n y arbitrary g e o g r a p h i c stratification (e.g., states, flyways, s t r e a m segments). M o d e l structure can be illustrated w i t h a b a n d i n g s t u d y consisting of three years of control a n d r e w a r d b a n d i n g , a n d three possible areas of
TABLE 16.33 Parameter Estimates for Model 0~, Yt, Ht, S) for a Reward Study of American Black Ducks Banded in Eastern Canada a Year
Banded during 1978-1980 following the hunting season (i.e., during late January-March); with control and reward bands (Table 16.31). Parameters are annual survival (S i and S;), harvest (Hi) , solicitation (~/i), and reporting 0~i) rates; subscript denotes variation in respective parameter over time. bPearson chi-square test; see Section 4.3.3. cDifference between model AIC (Akaike's Information Criterion; see Section 4.4.) and AIC value for the lowest ranked model.
~
~(~)
~,,
S'E(~,,)
I-/i
~(I-/i )
S
S'E(S)
a
1978 0.389 0.071 0.250 0.069 0.066 0.012 1979 0.244 0.063 0.056 0.010 1980 0.113 0.037 0.082 0.015
0.697 0.077
a Banded during 1978-1980 following the hunting season with control and reward bands (Tables 16.31 and 16.32).
16.3. Reward Studies for Estimating Reporting Rates TABLE 16.34
Band type
397
Data Structure for a Reward Band Study with Multiple Recovery Locations a Hunter reports
Solicited
Area of recovery (t)
Area of recovery (t)
Releases in period i
1
2
3
1
R1
mlhlb
m21
rrl31
R2
m~2
m22
m32
a3 R~
mlh3 ml 'c
m23 m 2'
m33 m 3'
ms1 1 ms2 1 ms3
R~
m 1'
m 2'
m 3'
R;
m~'
m 2'
m 3'
Control
Reward
1
2 2
ms1 2 ms2 2 ms3
3
ms1 3 ms2 3 ms3
a Poststratification of direct recoveries by type of reporting (hunter reported vs. solicited) and area of recovery, with k = 3 b a n d i n g years t, and b = 3 recovery areas. Excludes animals released in i and not recovered: R i - ~t mthi -- ~t msit and R; -- ~t mi" b mthi and msit denote the n u m b e r of direct recoveries in location t resulting from birds b a n d e d in year i with standard (control) bands, that are reported by hunters (h) and are solicited (s), respectively. c m~' denotes the n u m b e r of direct recoveries in location t resulting from birds b a n d e d in year i with r e w a r d bands. Because expectations are the same for hunter-reported and solicited r e w a r d bands, there is no need to stratify recoveries into hunter-reported and solicited recoveries.
recovery (Table 16.34). The expected numbers of control recoveries from such a study are displayed in Table 16.35, based on area- and time-specific harvest, reporting, and solicitation rates. Program MULT provides estimates for several models under different assumptions about variability in parameters over time and space (Conroy et al., 1989b). Reward band models incorporating both spatial and temporal variation and including both direct and indirect recoveries can be implemented using SURVIV (White, 1983; see Nichols et al., 1995b). Example
This example is from the previously mentioned study of American black ducks, conducted in eastern TABLE 16.35
Canada during 1978-1980 (Conroy and Blandin, 1984; Conroy, 1985). As before, bandings were included for all three years of the study, and direct recoveries of control and reward bands were stratified by categories of distance from banding stations: 0-20, 21-100, and > 100 km. At issue was whether proximity to banding stations had an effect on the reporting rate of band, as suggested by Henny and Burnham (1976). Data for this study are presented in Table 16.36. Program MULT (Conroy et al., 1989b) was used to fit four (of 19 available) models to these data, focusing on variation in X over space and time (Table 16.37). Model (ha,t, '~a.t, Ha, t ) in Table 16.37 is saturated, i.e., the number of parameters is equal to the number of independent observations (27). Of the other three models, only (ha,
Expected Recoveries for a Reward Band Study with Multiple Recovery Locations a'b Hunter reports
Solicited
Area of recovery (t)
Area of recovery (t)
Band type
Releases in period i
1
2
3
Control
aI
alflhl
Rlf21
alf31
a l f lsl
R l f s2
a l f s3
R2
a2f12
R2f22
R2f32
R2f12
R2f s2
R2f s3
a 3 f ls3
R 3 f s2
a 3 f s3
Reward
R3
a3f~3
a3f~3
a3f33
R~ R~
R~H~ R~I41
R~I42 R~H2
R~I-I3 R~H3
R;
R;I4~
R;I4~
R;I43
2
a With k = 3 b a n d i n g years and b = 3 recovery areas, with poststratification of recovery by type of reporting (hunter vs. solicited) and area of recovery. Excludes animals released in i and not recovered: R i - Xt E(mthi) - ~Zt E(mtsi) and R; - ~,t E(m~'). b parameters are reporting rate (h~), b a n d solicitation rate (~I), and harvest rates (/-~i); subscripts and superscripts denote variation in respective p a r a m e t e r over time (i) and recovery area (t); annual area-specific recovery rate fthi = h~(1 - - "Yi)Hi t t for hunter reports and t for solicited reports. ftsi = "~iI-~ii
398
Chapter 16 Estimation with Band Recoveries
TABLE 16.36
Numbers of Bandings and Recoveries of American Black Ducks (Anas rubripes) Banded in Eastern Canada a Number of recoveries by distance from banding stations Hunter reported
Band type Control
Reward
Year banded
Number released
0-20 k m
1978
2719
111
26
1979
2809
83
28
1980
3113
142
1978
374
1979
599
1980
627
21-100 km
Solicited >100 km
0-20 k m
21-100 km
>100 k m
61
65
18
5
64
26
15
15
29
100
34
19
10
41
16
34
46
20
52
47
20
74
a Banded during 1978-1980 following the hunting season (January-March) with control and reward bands, stratified by distance from banding stations.
"~a,t, Ha.t) fit the data (P > 0.25); the latter model also ranked lowest in AIC score. The model allows for areaand time-specific variation in solicitation and harvest rates, with reporting rates constant over time but varying with respect to distance from banding sites. Parameter estimates are shown in Table 16.38. Further tests of linear contrasts among the distance intervals (Section 16.1.10) indicated that reporting rates were higher (P < 0.05) within 20 km from banding stations, contrary to suggestions by Henny and Burnham (1976) that proximity to stations results in depressed reporting rates for mallards (Conroy et al., 1989b; Conroy and Blandin, 1984).
TABLE 16.37 Model Selection Criteria for a Spatially Stratified Reward Study of American Black Ducks (Anas rubripes) Banded in Eastern Canada a Goodness of fit Model b (ka, %*t, Ha*t) (Ka*t, (K,
"~a*t, Ha*t)
~[a*t, Ha*t)
(Kt, "~a*t, Ha*t)
Parameters
X2
df
pc
&AIC a
21
4.706
6
0.582
0.0
27
me
m
m
7.3
19
24.5712
8
0.0018
16.3
21
22.5033
6
0.001
18.0
a Banded during 1978-1980 following the hunting season (January-March) with control and reward bands; recoveries stratified by distance from banding stations (Table 16.36). b Parameters are reporting rates 00, band solicitation rates (y), and harvest rates (H); subscripts denote variation in respective parameter over areas (a), time (t), or both area and time (a't). c Pearson chi-square test; see Section 4.3.3. a Difference between model AIC (Akaike's Information Criterion; see Section 4.4.) and AIC value for the lowest ranked model. e Saturated model.
16.4. A N A L Y S I S OF B A N D RECOVERIES FOR N O N H A R V E S T E D SPECIES In this section we extend the methodology developed in Section 16.1 through Section 16.3, in which recoveries are reported via bands on animals harvested or found dead by sportsmen, to a situation in which the general public reports bands on animals found dead during the entire year. We also deal with two situations that have rendered analysis of band (or ring) recovery data more difficult, particularly in many European bird ringing studies: (1) banding of young of the year only and (2) unknown numbers of banded birds released. 16.4.1. D a t a S t r u c t u r e
The models described above and by Brownie et al. (19~5) emphasize the situation wherein recoveries of marked animals are obtained via reports from hunters or anglers. In this case, the recovery rates fi can be interpreted in terms of harvest pressure, particularly if adjustments can be made for reporting rate or crippling loss (see Fig. 16.1). However, there is no special requirement that the process of band recovery must involve harvest, versus the general finding of bands by the public. This is especially the case for the reporting of bands (rings) from nonharvested birds, and much of the European literature on the subject is oriented toward this type of reporting. 16.4.2. P r o b a b i l i s t i c
Models
In these applications, the basic data structure for band recovery data (Section 16.1.1) still holds, but the
16.4. Analysis of Band Recoveries for Nonharvested Species TABLE 16.38
Parameter Estimates for M o d e l (ha, ~a.t, Ha*t) for A m e r i c a n Black D u c k s B a n d e d in Eastern Canada a
Recovery Year (i)
1978
1979
1980
399
Band solicitation rates
Band reporting rates
stratum (t)
Distance (km)
1 2 3 1 2 3 1 2 3
0-20 21-100 >100 0-20 21-100 >100 0-20 21-100 >100
~t
b
0.549 0.328 0.274 n m m ~ -~
(Anas rubripes)
Harvest rates
~'E(~t)
~/~
~'~(~//t)
/?/~
S'~(/?//t)
0.063 0.064 0.028
0.241 0.178 0.026 0.143 0.151 0.061 0.119 0.181 0.027
0.034 0.049 0.011 0.030 0.045 0.016 0.023 0.048 0.009
0.101 0.038 0.087 0.067 0.035 0.088 0.089 0.033 0.119
0.010 0.007 0.010 0.007 0.006 0.009 0.009 0.006 0.011
a Banded during 1978-1980 following the hunting season (January-March) with control and reward bands; recoveries stratified by distance from banding stations (Tables 16.36 and 16.37). bEstimates vary by geographic stratum (distance interval), but not by year: ~/I = ~/t for i = 1, 2, 3.
parameterization of Seber (1970b) replaces that of Brownie et al. (1985), i.e.,
fi = (1
-
Si)ri,
where r i is equivalent to ~'i in Seber (1970b) and is often termed the "reporting rate." This parameter, which is not to be confused with the reporting rate discussed earlier (Fig. 16.1), refers to the probability that a marked, dead animal is found and its band reported by the finder. The parameterization thus differs from that of Brownie et al. (1985) in that the probability 1 Si of a mortality event leading to the recovery is treated separately from the process of recovery (finding and reporting), whereas u n d e r the Brownie et al. (1985) formulation both processes are s u b s u m e d in fi. This distinction is important because in a typical ringreporting study, m a r k e d animals are found dead t h r o u g h o u t the year, i.e., recoveries are not confined to a well-defined harvest period. The contrast between the two m o d e s of recovery is clarified by contrasting Fig. 16.1(a and b), in which animals are b a n d e d and released shortly before the harvest period in each year and are recovered only during the harvest period, and Figure 16.1c, in which animals are b a n d e d on an anniversary date each year i and recovered t h r o u g h o u t the interval [i, i+1]. In the Brownie et al. (1985) parameterization, recovery is viewed as a destructive sample of a population alive at the time of sampling and is thus conditional on survival to the time of recovery. If survival for the interval following b a n d i n g to the first recovery period is nearly 1, as is reasonable in m a n y
preseason b a n d i n g situations, fi is then interpretable as an index to the harvest mortality process (after appropriate adjustment for reporting rate and crippling loss). On the other hand, recovery of nonharvested species is viewed as a sample of a population that is dead at the time of sampling and is thus d e p e n d e n t on mortality (1 - Si) d u r i n g the interval [i, i+1], followed by sampling with probability equal to the "reporting" rate r i. The above a s s u m p t i o n s and interpretation of parameters lead to expected values of recoveries u n d e r the simple, one-age model that are similar to those for the one-age model considered in Section 16.1. For example, with k = 3 b a n d i n g and l = 4 recovery periods, the expected n u m b e r s of recoveries are given by R1
R2 R3
R1(1
-
Sl)rl R1S1(1 - -
$2)~"2
a2(1 - S2)r 2
R15152(1
-
$3)~-3
R1515253(1
-
$4)/. 4
R2S2(1 - S3)r 3
R2S253(1
R3( 1 - S3)r 3
R3S3(1 - S4)r 4
-
S4)r 4
One statistical and computational a d v a n t a g e of the r i parameterization is that because r i is a conditional probability [conditional on death, which occurs with probability (1 - Si)], it can logically a s s u m e any value on the interval [0,1]. In contrast, fi is a probability that includes both death and reporting. Because an animal cannot experience both recovery and survival d u r i n g the same interval, the p a r a m e t e r s fi and Si are implicitly related as fi <- (1 - Si). However, there is nothing about the structure of the Brownie et al. (1985) models that "enforces" or imposes this relationship, leading to the possibility of estimates of fi and Si that are logically
400
Chapter 16 Estimation with Band Recoveries
impossible. Having noted this possibility, we point out that logically impossible estimates almost never arise in the modeling of North American gamebird data. One can use the Brownie et al. (1985) models in Section 16.1 and associated programs such as ESTIMATE or MULT (Conroy et al., 1989) to estimate fi, and then obtain estimates of r i by Yi --
fi/(1
-
Si )"
However, program MARK (Appendix G) provides the option of parameterizing the band recovery model as above so that estimates of r i are obtained directly. Although analytical models for nonharvested species are straightforward extensions of those for harvested species (again, the approaches only differ in terms of parameterization), additional problems can arise in the analysis and interpretation of these data. One problem is that recovery rates [i.e., fi = (1 - Si)r i] for many species (e.g., passerine birds) are very low (see Section 16.6), resulting in sparse recovery data and estimates of poor reliability (e.g., see Francis, 1995). In many cases the data for these species are virtually unusable for survival analysis, and investigators can obtain far more reliable estimates of survival with focused capture-recapture studies (Chapters 17-19). When the r i parameterization is used for hunted species, the interpretation of the reporting rate parameter r i requires additional consideration. As noted above, band recoveries for hunted species typically are restricted to birds shot or found dead during the hunting season. However, (1 - Si) denotes the probability of death any time during year i. Therefore, for hunted species the "reporting" parameter r i reflects the product of two probabilities: (1) the conditional probability that a bird dying during year i died during the hunting season (this can also be viewed as the proportion of all annual deaths that occur during the hunting season) and (2) the conditional probability that a bird dying during the hunting season is reported.
16.4.3. Banding of Juveniles Only-Cohort Band Recovery Models Frequently juvenile animals are easily captured (e.g., at the nest) and tagged, whereas adults are more difficult or expensive to capture. Therefore, many data sets are composed solely of marked and released juveniles and their recoveries. One approach that has been used in studies of nestling birds (e.g., Haldane, 1955) has been to suppose that survival rates are age dependent but not time dependent. The expected values for a
study with three years of juvenile bandings and three years of both juvenile and adult band recoveries are R~~ R~~ - s(O))rl R~~176 R(20~ R(2~ -R(3~
-
5(1))/, 2
S(0))r2
a~~176
_
S(2))/,3
R(2~176 -- 8(1))/,3 R(3~ - S(0))r3
where S (v) is the annual survival rate from age v to age v + 1 for birds still alive v years following the banding year. Thus, survival rates are associated with age but not calendar year. Various authors including Anderson et al. (1985), Lakhani and Newton (1983), and Lakhani (1985) have discussed difficulties in parameter estimation for this model and its special cases. In addition to the assumption of age (but not time) dependency of survival rates, "reporting" rates (i.e., rj) generally are assumed to be constant (rj = r, j = 1, ..., l - i; i = 1, ..., k), to allow for identifiability and estimation of parameters. However, Seber (1971) and others (e.g., Anderson et al., 1981, 1985; Lakhani, 1985) note that additional constraints [e.g., S (I-1) = S ~l)] are needed for unique estimation. The challenge is to develop constraints that are neither arbitrary nor biologically meaningless. Even given "meaningful" constraints, Anderson et al. (1985) identified numerous other difficulties in the mathematical behavior and interpretation of estimates resulting from the above and similar life table approaches. Brownie et al. (1985) concluded that "there is no valid way to estimate age-specific survival rates from only the banding of young," assuming that only recoveries of the banded birds are available. In our own work, we have followed the recommendation of Brownie et al. (1985) and have avoided the modeling of band recovery data resulting only from banded young (see Section 16.2.3). However, we note that potentially useful approaches to the modeling of such data have been provided by North and Cormack (1981), Morgan and Freeman (1989), Catchpole and Morgan (1991, 1996), and Catchpole et al. (1996, 1998). Finally, we note that if recaptures are available for animals tagged in the year of birth (hatching), then it is possible to estimate both age- and time-specific survival rates, using the cohort extensions of the Cormack-Jolly-Seber model (Section 17.2.3). Whereas the methods described above are based on maximum likelihood principles (albeit under possibly unrealistic assumptions about parameter structure), certain other methods do not provide even this degree of statistical rigor. It still is possible to encounter the application of ad hoc methods such as the "composite dynamic method" (e.g., Hickey, 1952) that lack a rigorous statistical basis. This particular approach rests on three biological assumptions (Burnham and Anderson,
16.4. Analysis of Band Recoveries for Nonharvested Species 1979): (1) annual mortality (survival) varies by age but not time, (2) annual recovery rates are a constant proportion of annual mortality, and (3) virtually none of the banded sample is alive by final recovery year (i.e., essentially all birds are dead at time l + 1). Burnham and Anderson (1979) developed an equivalent probabilistic model, m a x i m u m likelihood estimators, and goodness-of-fit tests for the composite dynamic method, and evaluated the procedure on 45 band recovery data sets from a variety of species of waterfowl. The authors found strong evidence of lack of fit for the majority of these data and determined that parameter estimates are biased, even when the model assumptions appear valid. Because of the ready availability of valid estimation procedures for band recovery data, we strongly discourage the use of ad hoc methods that depend on questionable biological and statistical assumptions.
16.4.4. Estimation When N u m b e r s of Banded Birds Are U n k n o w n Knowledge of the n u m b e r of animals banded and released at each occasion seems an obvious prerequisite for the estimation of parameters using band recoveries. Unfortunately, some banding operations, such as some bird ringing studies in Europe, fail to keep accurate records of this information. Thus, the only information available from such a study is a matrix of recoveries, m~, i = 1, ..., k, j = i, ..., l. Under the agespecific cohort model considered in Section 16.4.3, the u n k n o w n quantities RIo) of releases appear in the expectations of the mi~, 0 so that estimation apt, ears not to be possible, even with constraints such as S (I - 1) = S (I-2) However, Burnham (1990) showed that the " u n k n o w n ring n u m b e r " problem can be addressed by describing a likelihood function for the data in terms of a series of conditional binomial likelihoods. On condition that survival depends on age but not year, it is possible to combine band recoveries into aggregates of recoveries with age-specific survival probabilities. Thus, the number of recoveries of age v birds is given by Hti, i + v ~
i
where the index v of age ranges from 0 to l - 1 and the index i of banding year ranges from 1 to min{l v, k}. For example, a study with l = k + 1 recovery years would aggregate age-specific recoveries into y(0) ~(0) + ~,(0) + ... + ,~(0) - - Irt11
~22
,~.(0) + , , ( 0 ) + =
Irt12
y(2) = m l (0) 3 +
~'~23
,,,(0) "~24 +
and so on. On assumption that reporting is constant ( r I -- r 2 . . . . . rl), the joint probability distribution for these statistics can be expressed as a product of the conditionally independent binomials
(y(V)]T(V)) ,,, bin(T(V), ~.(v)), where v = 1, ..., l - 1. The binomial parameter T(v) is given by Td
j=0
with the proviso that the y(J) in this sum include only recoveries from release cohorts for which age v recoveries are possible. The parameter -dv; is defined by ,r(v) = [1 -- S (v)]IIh=O v-1 s(h) 1 - II~= 0S (h) " with "r(~ = 1. The denominator of "r(v) is the probability that a recovered bird died sometime before age v + 1, and the numerator is the probability that a bird died at age (v). Then 7(1)__
S(~ 1 -
,1.(2)--
_
S(~
S(~
1
-
8(1))
~1)" _
S(2))
8(~
and so on. On further assumption that S (1-1) = S (l), age-specific survival rates S Cv), v = 0, ..., l - 1, n o w can be estimated. Burnham (1990) showed that the use of this modeling approach results in little loss of precision or power of goodness-of-fit tests, compared to age-specific estimation w h e n numbers banded are known. However, assumptions about age or time dependency in reporting rates cannot be tested, because this parameter no longer explicitly appears in the likelihood. Thus, the lack of knowledge about numbers banded further reduces the flexibility of modeling the recovery process, even beyond the limitations attendant to the absence of releases of older age classes.
Example
y(V) = ~_, ~(o)
y(1)
401
~kk
. . . + ~,(0) ~k, k +lr
. . . + re(k0_)1, k + l ,
The models discussed in this section can be illustrated with data from mallards banded in Colorado (Table 16.24), which previously were used to illustrate the two-age models in Section 16.2. Here we analyze the banding and recovery data for birds banded as juveniles only. Several age- and time-specific models were fit to these data using program MARK (Appendix G). Model selection criteria (Section 16.1.8) were used to select model (S~~ 5 (1), rt), which provides for timespecific juvenile survival rates, constant second-year
402
Chapter 16 Estimation with Band Recoveries
T A B L E 16.39
Model Selection Criteria for Male Mallards Banded as Juveniles Preseason (July-September) in the San Luis Valley, Colorado, 1963-1971"
analysis. Although apparently making little difference in the point estimates of parameters, the analysis based on juvenile bandings alone clearly limits the degree to which model assumptions can be tested via goodness of fit and the range of alternative models that can be considered, particularly as concerns time and age specificity. These limitations are compounded when the numbers banded are unknown, forcing an assumption of constant reporting rate (no age or time specificity). Finally, arbitrary assumptions such as S(t) = S (1-1) are required for many models, to enable parameter estimation. For these reasons we encourage the use of sampling designs such as those described in Section 16.2, that allow for robust modeling of survival and recovery probabilities via the release of individuals in each identifiable age class at each banding occasion.
Goodness of fit b Model
Parameters
X2
df
P
AQAIC~
(S~~ S (1), r t)
19
36.377
25
0.066
0.00
(S (~ S (v), r t )
17
48.530
27
0.007
3.44
(S~~ S (v), r)
18
52.453
26
0.002
7.85
(S (~ S (1), r)
3
101.867
41
0.000
8.09
(S~~ S (v), r t )
26
29.385
18
0.044
9.78
a Second portion of Table 16.24. Parameters are S (~ = juvenile survival rates; S (1) = first-year and older survival rates; S (~') = agespecific survival, v = 1..... 7, S (8) = S(7); and r = reporting rates. Subscript t denotes time variation in respective parameter. b Deviance-based chi-square test; see Section 4.3.3. c Difference between model QAIC c (Akaike's Information Criterion, corrected for small effective sample size and adjusted by the quasilikelihood factor g = 1.63; see Section 4.4.) and QAIC c value for the lowest ranked model.
16.5. P O S T S T R A T I F I C A T I O N OF RECOVERIES A N D ANALYSIS OF M O V E M E N T S In the preceding sections, recoveries were grouped according to the years of banding and recovery, without regard to the geographic region in which recovery occurred. The geographically stratified reward band models in Section 16.3.3 were an exception to this, in that they allowed for geographic variation in harvest rates, reporting rates, and other parameters. A natural extension involves banding in one or more study areas, along with the stratification of recoveries by regions so as to allow for area-specific survival and either fidelity or movement among areas. The appropriate data structure ordinarily involves a distinction between
and later survival, and time-specific reporting rates (Tables 16.39 and 16.40). To illustrate the situation that occurs when numbers banded are unknown, we used these same recovery data but eliminated the numbers of birds banded each year. Application of model selection criteria resulted in selection of model (S~~ S (1)) with time-specific juvenile survival rates and constant second-year and later survival rates (Tables 16.41 and 16.42). Note that reporting rate r, although not a parameter in the model, is assumed to be constant over time. The parameter estimates in Table 16.41 are similar to those found by including banding information in the
T A B L E 16.40
Age
Year
Adult Young
Parameter Estimates for Model (S~~ S (1), r t) f o r M a l e Mallards Banded as Juveniles Preseason (July-September) in the San Luis Valley, Colorado, 1 9 6 3 - 1 9 7 1 a Si
S~E( S i)
C~L
ri
S"E ( ri)
1963-1971
0.692
0.025
0.641
0.739
1963
0.441
0.062
0.325
0.564
1964
0.481
0.069
0.154
0.026
0.110
0.211
0.351
0.614
0.280
0.042
0.206
0.368
1965
0.585
1966
0.459
0.061
0.462
0.698
0.181
0.028
0.132
0.243
0.050
0.363
0.558
0.223
0.024
0.180
1967
0.273
0.425
0.059
0.316
0.542
0.173
0.019
0.138
0.214
1968
0.604
0.054
0.496
0.703
0.239
0.028
0.189
0.298
1969
0.589
0.059
0.471
0.698
0.271
0.030
0.217
0.332
1970
0.560
0.069
0.423
0.688
0.236
0.026
0.189
0.292
1971
0.293
0.172
0.075
0.679
0.152
0.022
0.114
0.200
a See Table 16.24. Parameters are SI~ = juvenile survival rates; parameterization; see Section 16.4.2).
C~U
S (1) -
C~L
C"U
first-year and older survival rates; and r i -- reporting rates (Seber
16.5. Poststratification of Recoveries and Analysis of Movements TABLE 16.41 Model Selection Criteria for Models That Do Not Use Numbers of Bandings; Male Mallards Banded as Juveniles Preseason (July-September) in the San Luis Valley, Colorado, 1963-1971 a .
.
.
.
G o o d n e s s of fit c Model b
Parameters
X2
df
P
AQAICc a
(SI0), S (1))
9
46.336
26
0.008
0.0
(S (~ S (1))
2
77.861
33
0.000
3.1
(S~~ S (v))
15
36.614
20
0.013
6.9
(S (~ S (v))
8
68.228
27
0.000
9.9
16.5.1.1. D a t a Structure
b Parameters are S (~ juvenile survival rates; S (1) = first-year and older survival rates; SCv~= age-specific survival, v = 1..... 7, S (8) = S (7). Subscript t denotes time variation in respective parameter. c Deviance-based chi-square test; see Section 4.3.3. a Difference between model QAIC c (Akaike's Information Criterion, corrected for small effective sample size and adjusted by the quasilikelihood factor ~ = 1.83; see Section 4.4.) and QAICc value for the lowest ranked model.
areas in which banding occurs and those in which recoveries occur. A different design involves banding in multiple areas, where recoveries also occur. In what follows, we consider both types of designs. In w h a t follows, we use superscripts on s u m m a r y statistics and parameters to designate geographic area. For example, R k denotes the n u m b e r of releases of individuals in period i from area k. This notation convention was used previously, e.g., in Section 15.13, to characterize m o v e m e n t probabilities, and in Section 16.3.3, to describe spatial variation in direct recoveries. Note that the convention contrasts with the use of
TABLE 16.42 Parameter Estimates for Model (S~~ S(1)) in Which Numbers of Bandings Are Not Used, for Male Mallards Banded as Juveniles Preseason (July-September) in the San Luis Valley, Colorado, 1963-1971 a
Adult Juvenile
parenthetical superscripts in Section 16.2 to designate age (with, e.g., RI k) denoting the n u m b e r of releases of age k individuals in period i). Superscripts on s u m m a r y statistics and parameters will be used throughout the remainder of the book to denote geographic area.
16.5.1. B a n d i n g a n d R e c o v e r y A r e a s D i f f e r in T i m e a n d S p a c e
a See second portion of Table 16.24, excluding numbers banded.
Age
403
Year
Si
S"E(Si)
C'u
C"U
m
0.677
0.027
0.621
0.728
1963
0.537
0.050
0.437
0.634
1964
0.403
0.051
0.307
0.507
1965
0.624
0.045
0.531
0.708
1966
0.447
0.042
0.365
0.532
1967
0.497
0.049
0.400
0.594
1968
0.596
0.044
0.505
0.680
1969
0.499
0.053
0.395
0.604
1970
0.449
0.077
0.305
0.602
a See Table 16.24. Parameters are constant first-year and older survival r a t e s (S (1)) and time-specific juvenile (SI~ survival rates.
Consider a banding study that includes a single banding stratum and b recovery strata. In each of k banding periods (years) R i animals are released, and bands are recovered in each of j = i, ..., 1 subsequent recovery periods. Assume that the location of each recovery is recorded, so that the recoveries can be stratified into t = 1, ..., b geographic strata. For example, the data structure for k = 3, l = 3, b = 2 is
Stratum 2
Year
Number
b a n d e d (i)
banded
j = 1
Stratum 1
2
3
j = 1
2
3
1
aI
m~l
m]2
m13
m21
m22
m23
2
a2
m22
m23
3
R3
m12
m13 m13
m23
In general there m a y be more than one banding stratum, with the potential that animals released in each banding stratum (s = 1, ..., a) m a y be recovered in each recovery stratum. The data structure for a banding s t u d y w i t h k = 3, l = 3, a = 2, b = 2 i s s h o w n in Table 16.43. 16.5.1.2. Probabilistic M o d e l s
Schwarz et al. (1988) and Schwarz and Arnason (1990) developed models for the above data structure u n d e r varying assumptions about the fidelity of animals to migratory routes subsequent to banding and release. A "complete fidelity" model includes the following parameters" t~ t is the probability that an animal in stratum s at time i migrates to stratum t during [i, i + 1]; S st is the probability of survival between i and i + 1 for an animal that migrates from stratum s to stratum t during [i, i + 1] and fst is the probability that an animal migrating from stratum s to stratum t during [i, i + 1] is harvested and reported during the harvest period. Index ranges for these paramenters are s = 1, ..., a banding strata, t = 1 ..., b recovery strata, i = 1, ..., k banding years, and j = 1, ..., l recovery
404
Chapter 16 Estimation with Band Recoveries
T A B L E 16.43
Data Structure for Multiple-Stratum Band Recovery Study with Different Banding and Recovery Strata a
Banding stratum
Year banded
Number banded
1
Recovery stratum 1 2
3
1
Recovery stratum 2 2
3
1
R~
11 Nil
R~
12 m12 12 m22
3
R~
11 m13 11 m23 11
12 Nil
2
11 m12 11 m22
12 m13 12 m23 12
1
r~2 ~
21 mll
R22
22 m12 22 m22
3
R~
21 m13 21 m23 21
22 mll
2
21 m12 21 m22
m33
m33
22 m13 22 m23 22
m33
m33
a With k = 3 b a n d i n g years, l = 3 recovery years, a = 2 b a n d i n g strata, and b = 2 recovery strata.
years. The average survival for banding stratum s over all migratory routes is
A "partial fidelity" model assumes that animals always return to the original banding area, but choose migration routes (i.e., potential recovery areas) independent of previous movements (Schwarz and Arnason, 1990). The cell probabilities for this model are parameterized with the overall survival probability for the banding stratum, S~ and require a movement probability parameter indexed by the year of recovery, e.g.,
b S~
-~ E
ilist.qst T l v l 9
t=l
The "complete fidelity" model assumes that animals always return to the banding and recovery areas to which they migrated during their first migration. This assumption is reflected in the matrices of expected recoveries (Table 16.44), where each cell probability contains a single movement parameter I~/st that is indexed by the banding year i. Unique parameter estimates are not possible for the model without imposing additional constraints. In particular, the recovery rates fst and migration r a t e s ~st are confounded and thus cannot be estimated uniquely without further assumptions. Similarly, stratum-specific survival rates are confounded with functions of movement probabilities, and separate estimation of these parameters requires additional assumptions, e.g., that probabilities of migrating to a particular recovery area do not vary over time, ~st = ~ t __ i[ist (see Schwarz et al. 1988).
T A B L E 16.44
Banding s
i
E(mSlt31RSl) = lps c.s.c,S.~l,St,CSt 9- 1 ~ 1 o 2 , g 3 j 3 9
The "partial fidelity" model allows for estimation of an overall survival rate associated with each banding area, but not separately by recovery areas. Harvest derivation can be estimated if estimates of abundance are available for the banding strata (Schwarz and Arnason, 1990). Finally, a "nonfidelity" model allows free movement between banding and recovery areas. The model parameters are confounded under this model, and parameter estimation requires either auxiliary data or strong assumptions (see Schwarz and Arnason, 1990). A special case of the "complete fidelity" model con-
Expected Recoveries under the "Complete F i d e l i t y " M o d e l for the Multiple-Stratum Band Recovery Study in Table 16.43 a Recovery stratum t = 1
R~
j=l 1 11f11 alt~l
/;~lfl,ll<11f11 *'1"t"1 ~ 1 11fll R2t~2
Riq/1221 f21
alq/1221521 f21
1
RI
2
R1
3
R~
1
R12
2
R22
3
R32
2
2 21 f21 a2t~2
Recovery stratum t = 2 3
//Jllhll
j=l /~l~h12 , c 1 2 *'1'el yl
"'1,'2d'22 1F~ j1~c22
2
3
/~1d,12<12f12 *'1'+'1 ~ j/~ld,lagl2 *'2~2 J2
/1~1,h12<12~12 ~c12 *'1'4"1 ~ '-'2 J3 ]~ld,12~lag12 *'2'+'2 ~ J3 11~1d,12,c12 ~'3~3 J3 1/~2'h22<22~2~~ *'l'e
*'1'el/~ 2d'22~22~ j2~c22 ~2,1,22 ~:22 *'2'q2 J2
~2,1,22~22 ,c22 *'2~2 ~ J3 tP2,h224:22 *'3'+'3 J3
a Parameters are ~ t = the probability that an animal in stratum s at time i migrates to stratum t d u r i n g [i, i + 1]; S~ t = the probability of survival between i and i + 1 for an animal that migrates from stratum s to stratum t d u r i n g [i, i + 1]; and fst _ the probability that an animal migrating from stratum s to stratum t d u r i n g [i, i + 1] is harvested and reported d u r i n g the harvest period.
16.5. Poststratification of Recoveries and Analysis of Movements sidered by Schwarz et al. (1988) occurs w h e n there is a single banding area (a = 1). Schwarz et al. (1988) developed a reparameterization of the model in Table 16.44 to allow unique estimation of survival rates and a "recovery rate" (r~), n o w defined as the probability that an animal alive at the time of banding in year i is harvested and reported during the harvest period in year i in recovery stratum t (Table 16.45). This parameter is influenced not only by harvest rates in the respective strata, but also by the probability of migration from the banding area to each recovery stratum. In the notation of Table 16.4.4, r it = ~stfst, where s is n o w the single banding stratum. Furthermore, variation in the rates of migration a m o n g cohorts can produce heterogeneity in estimates of parameter r ti, because under Schwarz et al. (1988) multiple cohorts may contribute to a single estimate. For example, in Table 16.45 the parameter r~ is included in the expected values for recoveries from both year 1 and year 2, and thus involves migration to the stratum in each of these years. Nevertheless, this parameterization should prove useful for analyzing stratum-specific survival rates, though one must keep in mind that the "recovery rates" produced do not have the usual interpretation as indices to harvest. 16.5.2. B a n d i n g a n d R e c o v e r y Areas Coincide 16.5.2.1. D a t a S t r u c t u r e
In this situation, animals are banded and released from multiple strata, and recovery occurs in these same strata contemporary with releases. Interest typically focuses on survival rates for the respective strata and rates of interchange between strata. The data structure for two banding and recovery strata is the same as
TABLE 16.45 Expected Recoveries for a Single Banding Area with Multiple Recovery Strata a Recovery stratum t = 1
Recovery stratum t = 2
Ri
1
2
3
1
2
3
R1
air1
112 RISI?.
R1S~ 521, 113
Rlr2
R2rl
1 1 R252r3
2 2 RISlr2 R2r2
2 2 2 R1S1S2r3 2 2 R252r3
R2 a3
Rgrd
R3 r2
a The S c h w a r z et al. (1988) p a r a m e t e r i z a t i o n . P a r a m e t e r s are SI = the p r o b a b i l i t y that an a n i m a l alive at the time of b a n d i n g in y e a r i in r e c o v e r y s t r a t u m t s u r v i v e s until the time of b a n d i n g in y e a r i + 1; r t -- the p r o b a b i l i t y that an a n i m a l alive at the time of b a n d i n g in y e a r i, is h a r v e s t e d a n d r e p o r t e d d u r i n g the h a r v e s t p e r i o d in y e a r i in r e c o v e r y s t r a t u m t.
405
that in Table 16.43, except that the recovery strata are identical to the banding strata. Thus, a i is an a • 1 vector whose sth element is the n u m b e r of animals banded and released in stratum s in year i, s = 1, ..., a, i = 1,..., k, and mq is an a • a matrix whose element (s, ;9 is the n u m b e r of animals released in year i in stratum s that are recovered in year j in stratum t. For example, for three banding strata, the releases in each year i are
a i
Fr l =/R2/, Lr3j
and the recoveries in year j of animals banded in year i are mij __
my
m~j2
mq21
mq22
m~3| .
31
32
m~3_]
_mij
mij
13 m/j 1
16.5.2.2. P r o b a b i l i s t i c M o d e l s
Schwarz et al. (1993a) model this data structure via matrix extensions of standard band recovery models (Sections 16.1 and 16.2). Under a Markovian (first-order Markov process; see Chapter 10) assumption, the model includes the parameters {i, an a x a matrix whose element (s, t) is the probability that an animal alive and present in stratum s at the time of banding in year i will be recovered in stratum t during [i, i + 1]; and q~i,an a x a matrix whose element (s, t) is the probability that an animal alive in stratum s at the time of banding in year i will survive and be present in stratum t at the time of banding in year i + 1. On assumption that survival over [i, i + 1] depends only on the location s at time i, and not on the location t at time i + 1, and further assuming that survival and m o v e m e n t are independent events, we can reexpress each element of ~i as the product of annual survival and m o v e m e n t probabilities st qsilsst ~i -- "i'vi ,
where S~ is the probability that an animal alive in stratum s at the time of banding in year i will survive to the time of banding in year i + 1, and t~7 t i s the probability that an animal alive in stratum s at the time of banding in year i migrates to stratum t at the time of banding in year i + 1, given that it has survived to i + 1. With this notation in hand, the expected values for the above data structure can be described as j-1
E(m,) = D(Ri)H x=i
~xf4,
Chapter 16 Estimation with Band Recoveries
406
where D ( A ) is an operator that places the elements of an n • 1 vector A along the diagonal of an n • n matrix; e.g.,
~~
D(R1) =
a 2
0
Year banded .
e3
Thus, the expected recoveries for a = 3, k = 3, and j = 3 can be written as Expected recoveries in year j
Number banded in year i
1
2
al
D(R1)fl
D(R1)~lf2 D(R2)f2
e_2 e_3
TABLE 16.46 Adult Male Mallards Banded in Southern Alberta in Late Summer 1975-1982 a
3
Number banded
1975
453
1976
1337
1977
1380
1978
1079
1979
2253
1980
888
1981
1924
1982
1107
Number recovered in western stratum 1975 1976 1977 1978 1979 1980 1981 1982 6
3
3
1
1
0
0
2
25
12
9
6
3
4
1
22
18
12
7
5
2
17
13
9
8
4
29
25
18
13
12
Number recovered in eastern stratum 1975 1976 1977 1978 1979
Example
We illustrate poststratification of band recoveries with a portion of the data on adult male mallards released in southern Alberta, Canada, and stratified by region of recovery (Table 16.46). Because the data structure involves banding in one area and time (southern Alberta during late summer) and recovery at another (western and eastern flyways during fall and winter), we use the Schwarz et al. (1988) parameterization described in Section 16.5.1. Thus, the only strata requiring specification in the superscripts are the recovery strata, and only stratum-specific survival Ss and recovery r~ rates can be estimated, the latter again involving migration to the respective recovery strata. Several models can be constructed, ranging from a general model that includes time- and stratum-specific survival and recovery, to models that include constant recovery and survival over both strata and time. Model parameters were estimated and compared using program SURVIV (Appendix G). Model (S, ra), specifying constant survival over time and strata and area-specific recovery that is constant over time, was selected based on AAIC (Table 16.47), with model (Sa, r a) a close competitor (AAIC = 1.04). The estimates of overall survival rate S and stratum-specific "recovery rates" r s are pre-
5 23 19
D(R1)q~lcP2f3 D(R2)cP2f3 D(a3)f3
Hilborn (1990) and Schwarz et al. (1993a) have developed multistratum models that allow for the estimation of parameters for multiple stratum designs. Schwarz et al. (1993a) used this modeling approach with band recoveries of North American mallard ducks banded and recovered during the winter. These models fit within the general framework of multistate Cormack-Jolly-Seber modeling and will be discussed in more detail in Section 17.3.
10 28
1975 1976 1977 1978 1979
9
1980 1981 1982
18
4
3
4
1
1
1
26
31
16
8
7
5
4
35
24
18
13
10
2
17
12
16
9
8
37
33
18
19
1980
19
1981 1982
14
9
50
21 23
a Mallards were recovered in either the Pacific f l y w a y (western) or the central, Mississippi, and Atlantic flyways (eastern).
sented in Table 16.48; the latter must be interpreted carefully, as they involve products of the probabilities of birds migrating to the respective strata and the harvest and reporting rates within each stratum. 16.6. D E S I G N OF BANDING STUDIES As with any estimation method, the results of a band recovery analysis are only as good as the data used in the procedure. It is important that a banding study be conducted in such a way that the assumptions of band recovery models will be reasonably met, so as to assure that estimates are unbiased and apply to the target population. Estimates should be sufficiently precise that meaningful statements can be made about the parameters of interest. For instance, estimates of survival with confidence intervals that span most of the unit interval are of little value, even for descriptive purposes. If hypothesis testing is of interest, then the data must provide sufficient test power for meaningful comparisons (e.g., of time-specific patterns in survival).
16.6. Design of Banding Studies TABLE 16.47 M o d e l S e l e c t i o n Criteria for A d u l t M a l e Mallards B a n d e d in S o u t h e r n Alberta in Late S u m m e r 1975-1982 a Goodness of fit c Model b
(S, ra) (Sa, ra) (S a, r) (S a, r'a. t)
(St, ra,t)
(S, r) (Sa. t, r t) (Sa*t, Ya*t) (S t, r t)
Parameters
X2
df
P
•AIC d
3 4 3 18 23 2 22 30 15
58.408 57.456 70.692 46.159 44.64 90.002 53.575 37.613 79.765
69 68 69 54 44 54 50 42 57
0.82 0.82 0.43 0.77 0.65 0.11 0.34 0.66 0.02
0.00 1.05 12.28 17.75 26.23 29.59 33.16 33.20 45.35
a Mallards were recovered either in the western or eastern recovery strata [Table 16.46; Schwarz et al. (1988) parameterization; see text]. bParameters are survival (S i) and recovery (r i) probabilities under the Schwarz et al. (1988)parameterization (see text); subscript a denotes recovery area and t denotes time variation in respective parameter. c Deviance-based chi-square test; see Section 4.3.3. Difference between model AIC (Akaike's Information Criterion; see Section 4.4) and AIC value for the lowest ranked model.
Finally, as with any sampling study, resources should be used both effectively and efficiently. 16.6.1. General
Considerations
As noted earlier (Section 16.1), inference in banding studies is conditional on the sample of animals marked and released, and the subsequent observations (i.e., recoveries) from this sample. It thus is critical that the sample of marked animals be representative of the population of interest. Although statistical modeling can accommodate some types of heterogeneity (e.g., age and sex structure; time- and individual-specific covariates) in the sample, it is incumbent on bioloTABLE 16.48 Parameter Estimates for M o d e l (S, ra) for A d u l t M a l e Mallards B a n d e d in S o u t h e r n Alberta in Late S u m m e r 1975-1982 a
Parameter
Recovery stratum
Estimate
SE
CL
CU
Both Western Eastern
0.706 0.015 0.022
0.017 0.001 0.001
0.673 0.013 0.020
0.738 0.017 0.024
a Mallards were recovered either in the western or eastern recovery strata (Tables 16.46 and 16.47). Parameters are overall survival (S) and stratum-specific recovery (r~) probabilities, with s - 1 for western recovery area, s = 2 for eastern recovery area.
407
gists conducting the field study to assure that, to the extent possible, factors inducing unidentified heterogeneity and nonrepresentativeness of the samples be controlled. 16.6.1.1. C a p t u r e
and Marking
Methods
Under most circumstances, marking animals requires capture, but the capture and handling of animals disrupts their normal patterns of behavior and in some cases can predispose animals to mortality. Methods that minimize these negative effects obviously are to be preferred over more intrusive methods, and in any case the effects should be taken into account in the analysis and interpretation of the data. For example, recoveries (e.g., birds found dead) in the vicinity of the trap site immediately after release could be rem o v e d from the numbers subject to later hunter recovery. However, problems occur with (1) deaths detected following release that were not a result of trapping and (2) birds not found dead, but predisposed (by trapping) to mortality. Because a key assumption of banding models is that the b a n d e d population is representative of the population at large, diligence must be taken to assure that the effects of marking and handling are minimal. Ideally, field sampling should produce a representative sample of the population in the vicinity of the study area. In practice, sampling methods, especially those that involve baits, scents, or other attractants, attract animals in relation to the individual's age, sex, reproductive status, nutritional status, dominance, and other individual characteristics (e.g., Weatherhead and Greenwood, 1981; Weatherhead and Ankney, 1984, 1985; but also see Burnham and Nichols, 1985; Reinecke and Shaiffer, 1988). This m a y create no difficulty if the animals' attributes can be observed and appropriate stratification and age- or sex-specific models used. However, it m a y not be possible to identify certain characteristics in the field (e.g., dominance or nutritional status), and the ages of m a n y animals are difficult to determine beyond simple classes (e.g., birth or hatching vs. older). Finally, stratification and modeling are of little use if trapping methods are selectivemfor instance, capturing only d o m i n a n t adult males. Unless study goals are similarly selective, the results will not generalize to the population at large. Other factors that might lead to nonrepresentative marking should be controlled. Thus, animals that are sick, injured, or have been transported (e.g., following nuisance complaints) should be excluded from the sample for the purposes of parameter estimation, unless these factors are of interest or have been taken into account through appropriate stratification of the data
408
Chapter 16 Estimation with Band Recoveries
and inclusion of the necessary parameters. Certain types of special markers such as radiotransmitters or color marks can be important adjuncts to standard (e.g., leg band) marking, but these marks cannot be intermingled with ordinary marking without introducing the possibility of serious heterogeneity in rates of recovery (Atwood and Geis, 1960; Craven, 1979; Samuel et al., 1990; Reinecke et al., 1992) and even survival (Craven, 1979; Zicus et al., 1983; Small and Rusch, 1985; Marks and Marks, 1987; Marcstrom et al., 1989; Ward and Flint, 1995; Castelli and Trost, 1996). For instance, color-marked gamebirds may be more likely to be singled out and shot by a hunter. Even if the color mark does not increase the probability of being shot, color marks may be more likely to be noticed on a harvested bird by the hunter and thus more likely to be reported. Likewise, radiotransmitters can enhance the probability that a hunter or other finder notices an ordinary band and thus reports both the band and the radio (Reinecke et al., 1992). Animals with these types of "special" marks can be removed from consideration in analysis of the data. Another approach is to develop joint likelihoods that permit different recovery and survival rates for animals with different mark types. There are situations in which the deliberate use of more than one marking method can be advantageous, if proper care is taken with the organization and analysis of the data. We have seen in Section 16.3 that special "reward" bands or tags, thought to enhance reporting rates (ideally, to 100%), can be used in conjunction with appropriate models to estimate the proportion of standard bands that are reported. Some studies may involve the deliberate use of more than one marking a n d / o r encounter method to estimate common parameters of interest. For instance, banded birds may be reencountered both by reports from hunters or by live recaptures at banding sites. If bands or color marks are visible at a distance, additional reencounters may be obtained via visual sightings without recapture or recovery. Models have been developed that take advantage of these sorts of data structures, by incorporating parameters that account for different kinds of observation (e.g., harvest recovery, live recapture, or resighting) (see Szmczak and Rexstad, 1991; Burnham, 1993) (see also Section 17.5.1). Use of multiple bands, either of the same or different types, also can be used to estimate rates of band loss (e.g., see Kremers, 1987, 1988; Nichols et al., 1992a; Nichols and Hines, 1993; Blums et al., 1994; Spendelow et al., 1994; Fabrizio et al., 1999).
16.6.1.2. Age and Sex Determination As noted above, ages and sexes of the captured animals must be determined accurately for those spe-
cies for which age and sex differences in survival and recovery rates are pronounced. For waterfowl, especially ducks, males and females frequently differ in their survival rates and vulnerability to harvest (and thus recovery rates). It therefore is important that these parameters be estimated separately, and that the sample sizes in age- or sex-stratified data are adequate to do so. If ages or sexes cannot be identified, or if this identification is neglected (e.g., if field technicians are inadequately trained to identify ages), one has little choice but to analyze the data as if these strata do not occur and hope that the underlying heterogeneity in parameters is minor. For age-stratified modeling of recoveries, it is essential that adequate cohorts of adults be captured and marked. Marked samples of juveniles and subadults alone require untestable assumptions about survival and recovery rates (see Section 16.2.9). Banding of multiple age cohorts provides information about agespecific survival and recovery rates, and also improves the estimator precision of older age classes with information from younger age classes. If multiple ages for non-adult animals cannot be identified, or adequate samples cannot be obtained, but age-specific heterogeneity is nonetheless important (as it typically is between at least the juvenile and adult age classes), then it may be better to confine analysis to only the adult data. 16.6.1.3. Time of Year Unless animals occupy the same geographic area for an entire year and do not experience seasonal variations in life history, the time of year of banding is an important consideration. To best meet the assumptions of band recovery methods (Section 16.1.9), it usually is desirable to conduct capture and marking during periods in which movement, mortality, or recruitment are absent or negligible. For migratory populations, sampling should be conducted at a time when representative samples can be obtained (e.g., during the breeding season). However, this is not always feasible or practical. The same goal of representative sampling can be accomplished if animals can be captured and marked at specific sites just before migration. Migratory birds frequently congregate at staging areas. Often, birds banded in such areas subsequently migrate over the same paths and are subject to the same hunting pressures. In such cases, it is reasonable to sample birds at such migration staging areas and to view such birds as a relatively homogeneous group with respect to survival and recovery rates. For purposes of survival estimation, it usually is best to select a time of year when animals are relatively sedentary and avoid marking while migration is in
16.6. Design of Banding Studies progress. The duration of banding should be as short as possible and, in particular, should be short relative to the length of the period over which survival is to be estimated. If the banding period extends too long, mortality will occur, leading to heterogeneity in survival rate wherein birds banded at the beginning of the marking period are at risk longer than birds banded at the end. Approximations of the percent relative bias in survival estimates resulting from mortality during the marking period were generally small for the scenarios investigated by Smith and Anderson (1987). In cases in which such mortality is large a n d / o r variable, it should be possible to develop models that correspond to this situation. One possibility is to model the first survival probability following banding as a function of the date of banding (e.g., expressed as number of days following the initial day of the banding period) using program MARK (White and Burnham, 1999). Another is to include extra survival parameters corresponding to the probability of surviving from one release period to the next. This modeling could involve either a daily survival probability, assumed to be constant during the banding period and thus raised to the appropriate power as determined by the day of banding relative to the final day of banding, or a discretization of banding periods with survival parameters corresponding to the probabilities of surviving from one portion of the banding period to the next (Tavecchia et al., 2002). The latter approach can be viewed as a generalization of models developed for banding at multiple times per year (see Sections 16.1.7 and 16.2.7). Because interest in game animals focuses so heavily on the impacts of harvest, releases of marked animals frequently occur immediately before the hunting or fishing season begins. Again, for migratory birds or fish it is best to band at a time when animals are relatively sedentary (pre- and postseason) so that it is possible to determine the reference population. However, it is best to avoid capture during the breeding season to avoid disruption of breeding activities, unless estimates of reproductive success or survival of adults or offspring during this period are of interest. It also is best to avoid marking and release of animals during the harvest season, to enable clearer separation of survival rates and to avoid excessive losses of marked animals during the marking period. However, if marking and release during the harvest period is unavoidable, then variation in survival over the period can be modeled as in Tavecchia et al. (2002). In many instances, survival over portions of the year is of interest, especially when the sources of mortality (e.g., harvest versus nonhunting) vary through the year. If feasible, semiannual banding (e.g., Sections
409
16.1.7 and 16.2.7) will allow estimation of seasonal and other periodic survival rates and in some instances will allow the direct investigation of relationships between harvest and other sources of mortality (see Blohm et. al., 1987).
16.6.1.4. Duration of the Banding Study Although it is theoretically possible to obtain estimates of survival from only two years of banding and recovery, in practice at least three and preferably five or more years are required (1) to obtain sufficient numbers of marked animals in the population for precise estimation and (2) to investigate temporal variation in survival and other parameters. If animals are banded for two years, only one survival rate can be estimated and tests of time specificity are precluded; with three years, two estimates are possible, with five, four estimates, and so on. One should continue to record and analyze data after a banding program has been terminated, because recoveries in later years provide information about survival rates in earlier years and thereby improve the precision of their estimators, even though no further marked cohorts are released. 16.6.1.5. Banding Station/Location
Ideally, captures and releases of marked animals should be distributed at random throughout some geographic area of interest. In practice, animals usually are caught in batches at stations or other locations that have been established with logistic considerations (e.g., proximity to roads) in mind. The bandings at these stations can be envisaged as a type of cluster sampling of the population at large. In practice, banding stations often must be grouped into strata (e.g., banding reference areas) (see Anderson and Henny, 1972; Anderson, 1975a; Nichols and Hines, 1987) to achieve minimum aggregate sample sizes. Often (e.g., with migratory bird banding) these areas are large, and questions naturally arise as to the heterogeneity (in survival, movement, and subsequent recovery rates) of the population from which the banded sample is drawn. If possible, samples from each banding station should be sufficiently large to allow separate analysis for each. Such sample sizes permit development of a general model (e.g., implemented in MARK; White and Burnham, 1999) that includes data from all areas, with separate parameters for each location. Tests of this general model against models with parameters constrained to be equal across areas permit inferences about the appropriate level of pooling. The contingency table tests of Brownie et al. (1985) can also be used to test the null hypothesis of equal survival and recovery rates for multiple samples (e.g., representing multiple geographic areas).
410
Chapter 16 Estimation with Band Recoveries
In cases in which it is not possible to develop a general model that includes all areas, other tests (Section 16.1.5) can be used to determine whether pooling is justified, or whether separate estimation of parameters by station or other geographic strata is warranted. Tests for homogeneity of spatial distribution (e.g., Mardia, 1967) can be useful for migratory populations, to ascertain whether different banding locations share common migratory patterns (see Nichols and Haramis, 1980; Munro and Kimball, 1982; Perdeck and Clason, 1983; Nichols and Hines, 1987; Pendleton and Sauer, 1992). These tests also tend to be more sensitive to heterogeneity than tests based on band recovery or other models. However, they do not directly test for variation in survival or recovery rates, but instead are based on the premise that animals found in the same locations should have similar demographic parameters, whereas animals in different locations have the potential for differences in demography.
16.6.1.6. Efforts to Enhance Recovery Rates Unless a deliberate effort is made to obtain live recaptures of marked animals (in which case the methods of Chapter 17 are applicable), the principal sources of reencounters from marked animals in a recovery study are voluntary reports by the public. These often take the form of reports from sportsmen who shoot or catch banded animals. It is well known that the majority of encounters of banded animals are not reported. For example, as many as 70% or more of banded migratory game birds shot by hunters are not reported (Bellrose, 1945; Henny and Burnham, 1976; Conroy and Blandin, 1984; Nichols et al., 1991, 1995b). This represents a serious loss of information, which if remedied could substantially improve the quality of data in a banding study. The estimation of the rate of nonreporting is itself of interest, in that it is required for the estimation of harvest rates from band recovery rates. As seen in Section 16.3, reward bands can be used in conjunction with standard bands to estimate the rate of reporting, which in turn can be used to estimate harvest rates. Of course, reward bands could be used simply to enhance the reporting rate of bands, a potentially effective approach when banded sample sizes are necessarily limited, harvest or encounter rates are low, or both. Additional approaches could include the use of lotteries (e.g., based on band numbers), advertizing in sport hunting and conservation magazines, and toll-free telephone numbers to facilitate reporting of found bands. The latter approach has been implemented in the United States and Canada, and preliminary estimates show an approximate doubling of reporting rate to about 80% (J. A. Dubovsky, personal communication). In any of these approaches, care must be taken to avoid
introducing geographic and temporal heterogeneity in reporting rates, thereby complicating analysis and interpretation of the data.
16.6.2. Determination of Banding Sample Sizes Sample sizes must be adequate to produce precise estimators. In the words of Brownie et al. (1985), "estimates of population parameters, made after years of expensive field work, that have extremely wide confidence intervals represent wasted time and money." A difficulty in band recovery (and capture-recapture) studies is that precision of estimators and power of tests depend not only on the number of animals marked and released, but also on the numbers surviving to subsequent recovery periods and then recovered. These are, of course, random events that are influenced by, among other things, the very parameters under investigation. Thus it is not possible to prescribe standard levels of banding effort necessary to achieve given levels of precision or power. Nonetheless, intelligent planning, in conjunction with some prior estimates or guesses for parameter values, can be used effectively to design banding studies. 16.6.2.1. Factors Influencing Sample Sizes
The factors that influence the numbers of band recoveries, and thus estimator precision, essentially fall into two categories: those under the direct control of the investigator and those determined by population or sampling phenomena outside the investigator's control. Included in the first category are both the number k of banding periods and the number R i of individuals to be banded in each period. Of course, the latter is not under perfect control, as the number actually captured in a given period varies at random in most situations. However, as a practical matter, realistic goals or quotas that can be met with some consistency can be established for banding stations, and these numbers can be treated as fixed. As noted earlier, banding should continue for a minimum of 5 years for most purposes. A general guideline of a minimum of 300 birds banded each year for gamebirds has been suggested (Brownie et al., 1985), although the desired sample size varies greatly among species and as a function of recovery rates and other factors. 16.6.2.2. Studies w i t h Targeted Precision Levels
A reasonable objective for sample size determination is some measure of estimator precision. For example, it is useful to specify a target value for a measure of relative precision, such as the coefficient of variation
16.6. Design of Banding Studies cv(6)
and
V'var(0) E(0)
=
of the estimator of a parameter 0 of interest. A reasonable approach in band recovery studies is to target equal numbers of bands each year (R 1 = R 2 . . . . . R k = R). It can be shown that the coefficient of variation of the estimate of average adult survival rate S is proportional to 1/R, according to
CV(~)2 =
h ( S , fr k)
R
'
-S = ~, S j / ( k -
1)
(16.11)
where
k-1 j=l
and k
f = j= fj/k, with the function h(S, f, k) specified as below (also see Brownie et al., 1985). If juveniles as well as adults are to be banded, the juvenile sample sizes necessary to meet minimum precision requirements for average juvenile survival rate can be shown to depend on both the adult and juvenile parameters [see Brownie et al., (1985) for computing formulas]. This approach also can be used to identify combinations of the number of years of banding and sample size per year required to estimate survival with a specified level of precision. Thus, from the variance formula for a parameter estimate one can determine the required sample size R i, given specified values for parameters (S i, fi) and the number of years of banding. For the one-age, time-specific model (St, ft) the expression h(S, f, k) in Eq. (16.11) is a function of Ri:
h(S, f, k) =
1 t ~i=1 q'r~ -- 'rr.j -- ~i=1
q - -'rrl. -1q -
(k - 1)2 where
k 'rri. -- E
"rrij
j=l =y(1-
(~)k-i+l)_ 1-S
J "IT.j = E "rr/j i=1 = ~(11__ ( ~ i ) ,
411
'
1
"iTk.
1-S
1 --- ~ J'
i = 1, ..., k (Brownie et al., 1985). These relationships and others (e.g., for two-age models) have been encoded into computer algorithms such as BAND2 (Wilson et al., 1989) to allow users to investigate combinations of sample sizes and years of banding to achieve specified precision goals, for both single-age and agestratified (two-age classes) band recovery models. Program BAND2 allows for user-specified survival and recovery rates, or else it computes them from band recovery matrices provided by the user from a pilot study. In what follows we use this program together with hypothetical examples to illustrate the influence of various factors on banded sample sizes and study duration. In order to identify adequate sample sizes, the following must be known or specified by the investigator: (1) the parameter(s) of interest, (2) the number of banding and recovery years, (3) an expected annual survival rate for each year of the study (usually a guess or an estimate from a previous or pilot study), and (4) an expected recovery rate fi from a guess or previous study. Finally, a quantitative goal is needed, which often is expressed in terms of the coefficient of variation of the parameter estimator [e.g., CV(S) -< 0.05 for mean estimated survival rate]. Because each of these quantities affects sample size, we consider each below.
16.6.2.3. Parameter of Interest Before sample sizes can be determined, the specific parameter or parameters of interest must be identified. Typically these parameters are either annual or average survival rates. Studies are directed at survival rates more frequently than recovery rates, for at least two reasons. First, biological interest often focuses primarily on survival a n d / o r time-specific variation in survival, especially in relation to harvest or other management intervention. Precise estimates of annual survival rates for temporal comparisons and of average survival for comparison among groups (e.g., geographic regions; sexes) are necessary to allow these sorts of inferences. Second, precise estimation of survival requires both direct (recovery period just after banding) and indirect (subsequent recovery periods) recoveries, whereas recovery rates can be estimated from only direct recoveries. Because indirect recoveries rapidly diminish with time after banding, survival rate estimation tends to be considerably more "data hungry" than recovery rate estimation. Consequently, if the study design is adequate to assure precise estimation of
412
Chapter 16 Estimation with Band Recoveries
survival rates, then precise estimation of recovery rates is virtually guaranteed. It may be important to ensure that time-specific survival rates are estimated at some minimum level of precision. For instance, consider a study conducted for k = 6 years, with expected survival and recovery rates assumed to be constant over years (Si = S = 0.6, fi = f = 0.07). If a coefficient of variation of 10% is desired for a given parameter [CV(0) = 0.10], the necessary sample sizes for each year of the study are as follows: Parameter
Ri
needed each year
$1 S2
1500 1400
53
1500
54
1800
$5
2800
600 (Note that the entry for S corresponds to a CV of 0.05 rather than 0.10, in keeping with a requirement of increased precision for an estimate of a mean.) For example, the banding of 1800 animals per period for six consecutive periods will permit the estimation of $4 with CV(S 4) ~ 0.10. Note that the estimation of mean survival rate requires less sampling effort than estimation of annual estimates with the same precision. Note also that survival rates from earlier years in the study tend to be estimated more precisely (require fewer bandings for the same precision) than later estimates. This is because indirect recoveries from throughout the study contribute to the estimation of the former, whereas the latter are estimated only from the recoveries from later periods.
16.6.2.4. Desired Level of Precision The desired level of precision directly influences the required sample sizes. For example, consider a oneage band recovery analysis with k = 6 years of banding, constant annual survival rates of S i = S = 0.6, and constant recovery rates fi = f = 0.07. The required annual banded sample sizes to estimate $2 and S with varying levels of precision are as follows: Required sample sizes CV
S2
0.03
15,300
1700
0.05
5500
600
0.07 0.10 0.20
2800 1400 400
300 200 100
As a general rule, the sample sizes needed to estimate both individual and average survival rates increase rapidly as precision requirements increase. However, the sampling requirements are less onerous for estimation of average survival rates. For this example the precise estimation of $2 (e.g., CV ~ 0.10) requires very large samples (R i ~ 1,000), whereas precise estimation of S can be achieved with considerably less sampling effort ( R i ~ 5 0 0 ) .
16.6.2.5. Number of Years of
Study
The number k of sampling periods in the study can have a major influence on the precision of estimators and the resulting sample size requirements for each year of the study. For example, consider a study with Si = S = 0.6, in which the goal is the estimation of period 2 survival with precision CV(S 2) = 0.10 and the estimation of average survival with a coefficient of variation of 0.05. The required sample size varies with the number of years of the study according to the following tabulation: Required sample sizes k
S2
5
1500
6
1400
900 600
8
1300
300
10
1200
200
15
1200
100
Thus, increasing the number of years of the study reduces the burden of the sample size requirement for each year, especially for estimates of average survival.
16.6.2.6. Expected Survival Rates Except in experimental situations, the survival rate of the studied population is not under the investigator's control. However, survival influences the precision of estimates and thus the sample size requirements for precise estimation. Marked populations with low survival tend to generate fewer recoveries and thus to produce estimates with lower precision than those with higher survival. For example, consider the influence of expected survival rates on the sample size requirements for a 6-year study with expected recovery rates of fi = f = 0.07, for which the objectives are to estimate period-2 survival with CV(S2) = 0.10 and average survival with CV(S) = 0.05. Considerable variation in survival can be expected among species, and
16.6. Design of Banding Studies the resulting impact on sample size requirements is illustrated below (note that substantial variation also is expected in species-specific recovery rates)" Example species
Expected survival
Required sample sizes
Si = S
S2
S
Dove
0.2
5900
3700
Wood cock
0.4
2500
1300
Duck
0.6
1400
600
Goose
0.8
800
400
Clearly, as average survival decreases, sample size requirements can increase dramatically. These results illustrate the importance of considering historical or pilot estimates of survival as part of study design. In studies designed to detect an experimental impact on survival, the anticipated change in precision for the treatment group should be considered in establishing sample size guidelines.
16.6.2.7. Expected Recovery Rates The expected size of the recovered sample is strongly influenced by average recovery rates, which in turn are influenced by harvest and band reporting rates. The latter are at least partially controlled by managers through mechanisms such as harvest regulations and band solicitation efforts. To illustrate, consider the sample sizes needed to achieve to CV(S 2) = 0.10 and CV(S) = 0.05 in a 6-year study of a population with expected annual survival Si = S = 0.6. The necessary sample sizes to achieve the desired levels of precision depend on the annual recovery rate fi = f, as shown in the following table: Required sample sizes
413
high recovery (harvest) rates would have an opposite effect on sample size requirements. Planning is most effective when reasonably good estimates of both average survival and recovery rates are available from a previous study, ideally under experimental circumstances not too different from those under consideration. We note that most nonharvested species have very low recovery rates (typically fi < 0.01), because recoveries mostly depend on chance encounters with banded animals. For this reason, band recovery data have proved much less useful than capture-recapture data (Chapter 17) for estimation of survival with nonharvested species.
16.6.2.8. Two-Age Analyses The previous examples are based on band recovery analyses in which all of the animals are marked at a single age and thus are considered to be "adults." In age-stratified analyses it is important to recognize that the precision of adult estimates of survival depend not only on the numbers of adults that are marked and released each year, but also on the numbers of marked young that survive to adulthood and are recovered. This can be seen by reexamining the matrices of expected recoveries for the two-age analysis (Section 16.2), wherein indirect recoveries of juveniles contribute to the estimation of adult survival and recovery rates. For example, consider a 6-year banding study of animals marked as both juveniles and adults, with expected adult survival rates of S i = S = 0.6, juvenile survival rates of SI = S' = 0.45, expected recovery rates for adults and juvenil~ of fi = f l = 0.07,_and desired precision levels CV(S2) = 0.10 and CV(S) = 0.05 for adult survival. The following combinations (among many others) of adult and juvenile sample sizes fulfill these goals" Parameter Adults banded
Juveniles banded
fi = f
$2
0.02
5300
2200
52
1400
2300
0.05
2000
900
S
600
10,000
0.08
1200
500
1000
3000
300
1500
2500
2000
2100
0.15
500
Clearly, sample size requirements are strongly influenced by expected recovery rates, making prior estimates or good guesses at these rates extremely important in study design. Of course, the above table does not take into account possible negative impacts of high recovery rates on annual survival under additive mortality. Decreases in annual survival attendant to
Again, program BAND2 (Wilson et al., 1989) can be used to compute combinations of adult and juvenile bandings needed to meet precision goals, given values for expected survival and recovery rates and study duration. Note that precision of the survival estimates for young birds also depends strongly on adult banded
414
Chapter 16 Estimation with Band Recoveries
sample sizes. In fact, it often is the size of adult samples that is the limiting factor in study precision; indeed, a desired level of precision may be impossible to obtain if adult sample sizes are below certain levels, regardless of how many juveniles are banded. This emphasizes the point made in Section 16.2.9 that band recovery studies based on samples of marked juveniles without accompanying adult samples are of limited value for survival estimation. 16.6.2.9. Studies Directed at Detecting
Differences
Interest often is less on the precision of estimates per se than on the ability to discriminate among biologically important effects. Ideally, banding data in the latter situation would come from a study in which populations or individuals are assigned at random to experimental groups. For example, a 5-year study might involve 10 populations of bobwhite that are randomly assigned to each of two treatment groups, with one group subject to heavy harvest pressure (T) and the other to no harvest (C) for a 5-year period. Interest might center on whether the mean annual survival rates for the two groups of populations differ. If Sh. is the true annual survival rate in year i for population j in the treatment k group, the hypothesis Ho: ~T = ~C
corresponds to the biological hypothesis of no treatment effect. If each population is banded every year starting in the first treatment year and continuing for 6 years, then annual estimates of survival S~ would be available for use in subsequent tests. One approach to testing would be to compute the mean annual survival for each replicate. These estimates then could be used with replication-based testing such as a t-test or with more complex models in an analysis of variance framework [Chapter 6; also see Skalski and Robson (1992) for general discussion and Coffman et al. (2001) for an example]. Sometimes manipulative experiments are not possible when studying animal populations, and constrained designs such as those discussed in Chapter 6 are required (also see Green, 1979; Burnham et al., 1987; Skalski and Robson, 1992). Point estimates from band recovery models may be useful in such tests, and studies can be designed with such tests in mind [see Nichols and Johnson (1989) for specific examples]. Regardless of the specific design, the objective of the study is no longer the estimation of parameters at (arbitrary) levels of precision, but is now the detection of a hypothesized effect under the experiment, usually with preestablished levels of Type I and Type II error
(Chapter 6). Given the desired Type I and Type II error rates, it then is important to know how many experimental populations to employ, and how many animals to mark for each population, in order to detect a putative effect. Similarly, the researcher may wish to know what effect size is detectable given the practical limitations of the study, or whether the study is likely to be informative about the effect. The answers to these questions in turn are influenced by (1) variation in true survival both over time for the duration of the study and among replicate groups of animals, and (2) sampiing variance of the estimates for each population. The formulas of the previous section can be helpful in assessing the impact of sampling variation on the number of marked animals needed for each population in a replicated study; however, in the absence of spatial replication, precise estimates alone are not sufficient. It is not possible to formulate general rules regarding sample sizes needed to detect experimental effects, because sample sizes depend on the experimental effect in question, the test statistics used, the required Type I and Type II error rates, and a number of design factors. Experimental studies frequently require much more sampling effort (especially in terms of replication of populations) than descriptive studies. It is no surprise that very few true experimental studies have been conducted using band recovery models with marked animals, and those that have been conducted generally failed to find the anticipated effect, or detected only very large effects.
16.7. D I S C U S S I O N The models of this chapter correspond to relatively simple data structures for estimating survival rates, in which animals are encountered only two times: an initial capture occasion and a subsequent (and terminal) band recovery. Nevertheless, data from band recovery studies can be information rich, allowing the estimation of survival rates, recovery rates, and other parameters, and, depending on the sampling design, the examination of variations in these parameters according to age, sex, time, and geographic strata, and with respect to covariate relationships. The challenge frequently lies in a judicious selection of a model that is sufficiently complex to capture biological relationships of interest, yet not so complex as to result in model overparameterization and loss of estimate precision. Model diagnostics such as goodness of fit, likelihood ratio testing for model comparison, and, more recently, information criteria such as AIC provide databased approaches for making these decisions. The pro-
16.7. Discussion cess of model identification is facilitated by a number of sophisticated computer packages for computing estimates and model diagnostics that in some cases allow specification of models tailored to specific ecological questions. In Chapters 17, 18, and 19 we extend these methods to the more general situation in which animals are encountered on more than one occasion subsequent to the initial capture and release. In Chapter 17 we consider models in which the likelihood is conditioned on the total numbers of animals released in a "cohort" (defined variously according to combinations of time of release, age, sex, geographic stratum, or other factors). The data structures and models for these situations allow for estimation of demographic parameters of interest, typically survival over defined time intervals and nuisance sampling probabilities (e.g., capture, recovery, or resighting rates depending upon the mode of sampling). As with band recovery designs, certain
415
sampling schemes lend themselves to the estimation of additional parameters of interest, such as movement or fidelity rates. Conditional capture--recapture models allow for rich parameterizations to describe survival and other parameters in multiple dimensions and thus require objective methods for selecting an optimal model. The models in Chapter 17 also include cases in which auxiliary data, such as recoveries, telemetry data, and covariates, are available. In Chapter 18 we consider unconditional (Jolly-Seber) models, in which the numbers of unmarked animals in the sample enter explicitly into the model likelihood functions, enabling estimation of abundance and recruitment in addition to survival rates. In Chapter 19 Jolly-Seber models are linked to closed population estimation (Chapter 14) under the robust design (Pollock, 1982), enabling estimation of abundance and demographic parameters under relaxed assumptions about heterogeneity in capture probabilities.
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C H A P T E R
17 Estimating Survival, Movement, and Other State Transitions with Mark-Recapture Methods
17.1. SINGLE-AGE MODELS 17.1.1. Data Structure 17.1.2. Probabilistic Modeling: The Cormack-Jolly-Seber Model 17.1.3. Reduced-Parameter Models 17.1.4. Time-Specific Covariates 17.1.5. Multiple Groups 17.1.6. Effects of Capture History 17.1.7. Individual Covariates 17.1.8. Model Selection 17.1.9. Estimator Robustness and Model Assumptions 17.1.10. Example 17.2. MULTIPLE-AGE MODELS 17.2.1. Data Structure and Considerations 17.2.2. Pollock's (1981b) Multiple-Age Model 17.2.3. Age-0 Cohort Models 17.2.4. Age-Specific Breeding Models 17.3. MULTISTATE MODELS 17.3.1. Markovian Models 17.3.2. Memory Models 17.4. REVERSE-TIME MODELS 17.4.1. Single-State Models 17.4.2. Multistate Models 17.5. MARK-RECAPTURE WITH AUXILIARY DATA 17.5.1. Capture-Recapture with Band Recoveries 17.5.2. Capture-Recapture with Resightings between Sampling Occasions 17.5.3. Capture-Recapture with Radiotelemetry 17.6. STUDY DESIGN 17.6.1. Sampling Designs and Model Parameters 17.6.2. Model Assumptions 17.6.3. Estimator Precision 17.7. DISCUSSION
In this chapter we focus on the survival and movement of individuals in a population. Along with reproduction, the deaths of individuals and their movement into and out of a population are directly responsible for changes in population size. Here we explore ways in which capture-recapture methods can be used to estimate the probabilities of survival and movement for open animal populations. Investigations of open populations differ generically from those of closed populations as described in Chapter 14, in that the times between sampling occasions are sufficient to expect losses a n d / o r gains to the sampled population. Thus, the modeling of open populations must include parameters in addition to those for closed populations, to account for population gains and losses. As with the study designs in Chapter 14, the capture-recapture studies described here include K 1 sampling occasions at which animals are caught or observed. On each occasion, new (unmarked) animals are given unique marks and then are released back into the studied population. Previously marked animals also can be included in the samples, and after their identification codes are recorded, they too are released back into the population. The procedures described below utilize these records of recapture a n d / or resighting for estimation of survival and movement parameters. The actual mechanisms for marking vary widely among organisms. Some organisms can be marked with tags that are observable from a distance--for example, the use of neck collars on geese and swans (e.g., Hestbeck and Malecki, 1989a; Nichols et al., 1992b), patagial tags on birds (e.g., Cowardin and Higgens,
417
418
Chapter 17 Estimating Survival, Movement, and Other State Transitions
1967; Stromborg et al., 1988), vinyl collars on ungulates (e.g., Storm et al., 1992), and color bands on birds (e.g., Spendelow et al., 1994). In such cases, sampling primarily involves efforts to resight marked animals, and the capture of new organisms for tagging may be restricted to the beginning of the study or to a limited number of sampling occasions during the study. Individuals of some animal species can be identified from photographs based either on natural markings [e.g., whales (Caswell et al., 1999); tigers (Karanth and Nichols, 1998)] or other markings that are not applied by the investigator [e.g., manatees (Langtimm et al., 1998)]. The repeated sampling associated with all these approaches results in a record of captures a n d / o r sightings for each animal caught or observed during the study. We use the term capture history to mean a row vector of K ls and 0s indicating whether an animal was caught (denoted by 1) or not caught (denoted by 0) at each sampling occasion (see Chapter 14). The task in capture-recapture modeling is to develop probability models for the biological processes giving rise to capture history data. Whereas parameterizations for the closed population models of Chapter 14 require only capture probabilities, the changing nature of open populations requires a considerably larger suite of parameters to accommodate these changes. The most general models for capture-recapture data consider both gains and losses of individuals between sampling periods and permit the estimation of population size (e.g., Jolly, 1965; Seber, 1965; Crosbie and Manly, 1985; Pollock et al., 1990; Pradel, 1996; Schwarz and Arnason, 1996). In this chapter we restrict attention to models that are conditional on the capture, marking, and release of individuals at each of several times over the time frame of a study. Because the likelihoods for these conditional models represent components of the full likelihoods discussed in Chapter 18, it seems natural to discuss the conditional models first. For models discussed in this chapter, the usual approach is to condition on the initial capture of an animal and then model subsequent entries in its capture history as functions of parameters associated with both sampling (capture probabilities) and real population change (survival probabilities). The modeling thus focuses on losses of individuals from the population and permits the modeling and estimation of survival probabilities (e.g., Cormack, 1964; Burnham et al., 1987; Lebreton et al., 1992). Most of this chapter is devoted to the statistical modeling of capture a n d / o r observation data and the estimation of the associated survival and capture probabilities. We note that the band recovery models of Chapter 16 can be viewed as special cases of these conditional models, in which there are no releases after the first recapture of an animal (see Brownie and Pollock, 1985; Brownie et al., 1985).
In this chapter we also describe reverse-time modeling (Section 17.4), which focuses on gains (rather than losses) to the population as one proceeds backward in time. An alternative to the standard conditional modeling with multiple capture-recapture data, this approach involves conditioning on the final capture of each individual, incorporating parameters that reflect whether an animal present at one sampling period was also present at the previous sampling period, recognizing that new recruits may have entered the population after the previous sampling period (Pollock et al., 1974; Nichols et al., 1986a; Pradel, 1996).
17.1. S I N G L E - A G E M O D E L S In this section we consider models for populations in which every individual can be considered to be of a single age, in that the capture and survival probabilities are identical for all individuals in the sampled population. In Section 17.1.1 we describe the data structure corresponding to a single-age model, and in Section 17.1.2 we describe the standard CormackJolly-Seber (CJS) model with time-specific probabilities of capture and survival. We also describe the maxim u m likelihood estimators of these parameters. In Section 17.1.3 we consider several reduced-parameter models that can be obtained by imposing stationarity and other restrictions on the parameters in the CJS model. In Section 17.1.4 we allow for the inclusion of environmental and other covariates in the modeling of capture and survival probabilities. In Section 17.1.5 we relax the assumption of homogeneous model parameters, to allow for group-specific parameterizations that accommodate differences between, e.g., sex cohorts in a population. In Section 17.1.6 we allow for trap response and other forms of influence that are tied to the capture history of individuals. In Section 17.1.7 we relax the assumption that individuals in a group have identical capture and survival probabilities and allow for individual covariates such as organism weight, length, or other measures. In Section 17.1.8 we consider approaches to the selection of a particular model from a class of candidate models, based on the parsimony of a model's parameterization as judged primarily by Akaike's Information Criterion (AIC) and secondarily by model goodness-of-fit tests and likelihood ratio tests. Last, we consider in Section 17.1.9 the robustness of model estimators to violations of the assumptions of the CJS model. 17.1.1. Data Structure
The data for a standard capture-recapture study of an open population are of the same form as those for
17.1. Single-Age Models a capture-recapture study of a closed population. Thus, the data can be summarized in a so-called X matrix (Chapter 14), denoted [Xq], where i represents an individual animal (i = 1.... , MK + 1, where MK+ 1 denotes the total number of individuals caught during the study) and j denotes the sampling occasion (j = 1, ..., K). Element Xij of the matrix assumes a value of 1 if the ith individual is caught on the jth sampling occasion, and 0 if the ith individual is not caught on the jth sampling occasion. As was the case for the closed population models of Chapter 14, estimation for open models is based on the number x~ of animals exhibiting the observable capture history 00. For example, a three-period study can have seven observable capture histories, and the numbers of animals exhibiting each of them are expressed as Xl11, X110, X101, X100, X011, X010, and x001. Modeling can proceed directly from these capture-history data, or it can be based on summary statistics computed from the capture histories that carry all the information needed for parameter estimation under a specified model. Summaries of capture history data typically present each observed capture history as a row vector followed by the number of animals in the data set that exhibit that history (Table 17.1). Note that Table 17.1 contains two rows for some histories, and the number of animals for one row in each pair is negative. Negative numbers indicate the number of animals exhibiting a capture history that are not released back into the population
TABLE 17.1 Capture-Recapture Data Summary for a Three-Period Study Using Capture History Notation Capture history 1 00
100 110 110 101 111 111 010 010 011 011 001 001
Number of animals"
89 -3 41 -2 16 19 -1 75 -4 37 -2 82 -3
aNumbers preceded by a " - " indicate animals that were not released following their last capture.
419
following the final capture in the history. For example, the first row entry in Table 17.1 is "100, 89", indicating that 89 animals were caught in the first capture period, released, and never caught or seen again during the study. The second row entry is "100, - 3 " , indicating that three animals caught in period 1 were not released back into the population. These animals may have died on capture or were otherwise purposely removed from the population by the investigator, so they had no opportunity to be recaptured. For the single-age Cormack-Jolly-Seber model (Cormack, 1964; Jolly, 1965; Seber, 1965), data frequently are summarized in an mq-array, where mij denotes the number of animals released in period i that are next caught (or observed) in period j (j > i). For example, m13 corresponds to animals that were released at period 1, not caught or seen at period 2, but caught or seen at period 3. Let R i denote the number of releases in period i (these may be either physically captured and released or observed alive, depending on the sampling methods). Table 17.2 shows an mijarray for a four-period capture-recapture study. An individual capture history can contribute to multiple statistics in the mij-array. For example, capture history Xll 0 contributes to R 1 and R2, as well as to m12. Animals not released back into the population following capture (e.g., in period j) are simply not incorporated into the corresponding number of releases (e.g., into the Rj). So if m13 includes an animal that dies in the trap and is not released back into the population following capture at period 3, then this animal will not contribute to R 3. 17.1.2. P r o b a b i l i s t i c M o d e l i n g : The Cormack-Jolly-Seber Model 17.1.2.1. Model Structure
As with capture-recapture for closed populations (see Chapter 14), we use a multinomial distribution to model captures and recaptures for single-age open populations. We define two primary parameters for the conditional modeling of capture-recapture data: Pi is the probability that a marked animal in the study population at sampling period i is captured or ob-
TABLE 17.2 The mi--Array Representation for Data from J a Four-Period Capture-Recapture Study Recapture period j Releases in period i
2
3
4
R1 R2
m12
m13 m23
m14 m24
R3
m34
420
Chapter 17 Estimating Survival, Movement, and Other State Transitions
served during period i; ~i is the probability that a marked animal in the study population at sampling period i survives until period i + 1 and remains in the population (does not permanently emigrate). In addition we define Xi as the probability that an animal alive and in the study population at sampling period i is not caught or observed again at any sampling period after period i. For a K-period study, • = 1, and values for sampling period i < K can be computed recursively as:
Xi-~ (1
-
q~i) if- q~i(1 -
Pi+l)Xi+l.
(17.1)
Equation (17.1) expresses the fact that there are two ways an individual can fail to be recaptured after time i: it can fail to remain in the sampling area (with probability 1 - q~i), or it can survive and not be captured thereafter [with probability q~i(1 -- Pi+l)Xi+l]" The probability Xi plays an important role in the probability modeling of capture histories, as seen below. Note that the parameters q0i combine the probability of survival and the probability of not permanently emigrating out of the study area between sampling occasions. It is because both factors are included in the parameterization that we use ~i rather than the more familiar S i to characterize the continued presence of individuals over time. For economy of presentation we characterize q~i as a "survival rate" in what follows, recognizing that the parameter includes both factors. In the capture-recapture literature, q~i sometimes is referred to as "apparent survival" and "local survival," whereas Si is known as "true survival" (true in the sense that its complement includes only mortality, not emigration). Note also that we use the subscript i to denote sampling occasion, rather than the more generic t. It is useful to think of the time frame of a study as extending from an initial time 1 to the final sampling
occasion, partitioned by K sampling occasions that typically (though not necessarilymsee below) are evenly spaced over the time frame. In later sections of this chapter we allow for observations not only during the sampling occasions, but also at any time between occasions. We now consider the modeling of capture history data using these capture and survival probability parameters. The modeling process is illustrated in the tree diagram of Fig. 17.1. An animal alive at any time i either survives until i + 1 (with probability q~i) or not. If it does survive, then it is either captured (with probability Pi + 1) or not. Each survival or capture event has an associated probability, and these probabilities are used to model the observable events as coded in the capture histories. To illustrate, consider the capture history 011010, which indicates capture at periods 2, 3, and 5 of a sixperiod study. We condition on the initial capture at period 2 and then model the remaining events in the capture history by Pr(011010 I release at period 2) = q~2PBq~3(1 - p4)qo4P5X5. Beginning with the initial capture in period 2, an animal with this history survived from periods 2 to 3 (the probability associated with that event is q~2), was caught at period 3 (the probability associated with that event is P3), survived from periods 3 to 4 (associated probability is q~3), was not caught at period 4 (associated probability is 1 - P4), survived from periods 4 to 5 (probability q~4), was caught at period 5 (probability P5), and was not caught following period 5 (probability X5). Note that the model for capture history 011010 would differ if the animal was removed from the population and not released following its last capture in
Capture History
Period 2
Period 1
Animal caught
11
Animal not caught
10
Animal / alive
Animal caught, , , ~ J marked and released
~
Animal dead or emigrated
10
F I G U R E 17.1 Tree diagram of events and associated probabilities for an animal released in period 1 of a two-period study under the Cormack-Jolly-Seber (CJS) model for open populations.
17.1. Single-Age Models period 5. In this case, the capture history is modeled as follows: Pr(011010 ] release at period 2 and removal at period 5) = q~2Pgq~ (1 -- P 4 ) ~ 4 P 5 "
The modeling is identical to the previous case, except that there is no need to model events following period 5; hence, the • term is removed. Table 17.3 shows the probabilities of the four possible capture histories for an animal released in the initial period of a three-period study. These probabilities correspond to a conditional multinomial distribution, as all of the R 1 animals released at period 1 must exhibit one of the four capture histories:
RIX~o! ! (q~lP2q~2pg)X111[q~1P2(1 Pr({x~} ] R1) - II~o
-- q~2P3)] x110
421
enter into this conditional modeling, as we are modeling the portion of each capture history that follows an initial release (at times 1 and 2). A different but statistically equivalent approach utilizes the mij statistics. Thus, the data in each row of an mq-array are modeled via a multinomial distribution, with expected values for the mij statistics shown in Table 17.4. For R i releases at time i, define K ri =
s
mij
j=i+l
as the number of recaptures at any subsequent sampiing period. Then R i - r i is the number of period i releases that are never recaptured during the study. We can write the multinomial distribution corresponding to the first row of Table 17.4 as
(17.2) x [q~1(1 - -
RI!
p2)q~2P3]Xl~ )xl~176
P r ( m 1 2 , m 1 3 , m14 ] R1)
where the term I - q~2P3can also be written as • and the index ~o in II~ x~! ranges over the capture histories 111, 110, 101, and 100. A complete description of the probability model also allows for unmarked animals to be caught at sampling times 2 and 3. We denote as u 2 the number of unmarked animals caught and released in period 2 and note that all of these animals have entries 01 for the first two elements of the capture history. The animals counted in u 2 are either caught in period 3 or they are not, yielding the following conditional binomial distribution: //2! E,~, , ~x011(1_ - ~x010 Pr({x~}]u2) = [I~ x~----~.~,~2F31 q02P31 ,
(17.3)
where the index ~o in IIo~x~! now ranges over 011 and 010. The probability distribution for the six possible capture histories observed for animals first caught in periods 1 and 2 then is written as the product of Eqs. (17.2) and (17.3). Note that the history 001 does not
TABLE 17.3 Possible Capture Histories and Associated Probabilities for Animals Released in Period 1 of a Three-Period Study, under the Cormack-Jolly-Seber Model a
Capture history
Probability
1 1 1
q~lP2q~2P3
1 1 0
q~lP2(1 - q~2P3)
rl)!
(17.4)
x {(q~1P2)m'2[q~1(1 - P2) X q02pg]m'3[q~1(1 -- p2)q~2
x (1 -
,
-
-
-lml4,.(Rl-rl)) X1
P3)q~3P4]
9
The probability distribution for all six of the mij statistics shown in Table 17.4 (m12, m13, m14, m23, m24, and m34) for a four-period study can then be written as the product of three conditional multinomials corresponding to releases R 1, R2, and R 3. The model forms in Eqs. (17.2)-(17.4) and their associated data structures are examples of the Cormack-Jolly-Seber model. In what follows we use the notation (q~t, Pt) to denote the fully parameterized CJS model, to emphasize time-specific values for the survival parameters q~i and capture probabilities Pi. Later in this chapter we discuss model forms for reduced parameterizations of the CJS model that allow for stationary values of ~i, Pi, or both.
E(mijIRi)
TABLE 17.4 Expected Numbers of Recaptures for the Data of Table 17.2 under the Cormack-Jolly-Seber Model Structure
Releases in period i
2 Rlq~lP2
1 0 1
q~1(1 - p2)q~2P3
al
1 0 0
(1 - q~l) + %(1 - p2)(1 - q~2P3)
R2 R3
aprobabilities conditional on releases in p e r i o d 1.
m12!m13!m14!(R 1 -
Recapture period j 3
4
alq~l(1 - p2)q~2P3 R1q~1(1 - p2)q02(1 - p3)q~3P4 R2q~2P3
R2q~2(1 - p3)q~3P4 R3qo3P4
422
Chapter 17 Estimating Survival, Movement, and Other State Transitions
17.1.2.2. Model Assumptions The following assumptions typically are listed for the Cormack-Jolly-Seber model (e.g., Seber, 1982; Pollock et al., 1990): 1. Every marked animal present in the population at sampling period i has the same probability Pi of being recaptured or resighted. 2. Every marked animal present in the population immediately following the sampling in period i has the same probability q~i of survival until sampling period i+1. 3. Marks are neither lost nor overlooked, and are recorded correctly. 4. Sampling periods are instantaneous (in reality they are very short periods) and recaptured animals are released immediately. 5. All emigration from the sampled area is permanent. 6. The fate of each animal with respect to capture and survival probability is independent of the fate of any other animal. Assumptions (1) and (2) concern homogeneity of the rate parameters that underlie the capture history data, recognizing that survival and capture probabilities frequently vary as a function of the attributes of a captured or observed animal. These attributes can be categorized into four functional classes depending on whether they are discrete or continuous and whether they are static or dynamic. Variation of survival a n d / or capture probabilities as a function of discrete, static attributes is best dealt with via stratification. Sex is a static variable (it does not change over the course of a study for an individual) for most vertebrate species, and males and females of many species are thought to exhibit different survival and capture probabilities. In such cases, data for different sexes can be modeled independently, although it may be parsimonious to develop models in which some parameters are sex specific whereas others are common to both sexes (e.g., Burnham et al., 1987; Lebreton et al., 1992). Modeling for discrete groups is discussed in Section 17.15. Body mass at birth or fledging is a continuous variable that may influence survival probability throughout the life of an organism. One way to handle the influence of such a variable is to group animals into discrete mass classes and develop separate models, as in the case of sex. Alternatively, birth or fledging weight can be viewed as a continuous variable and survival rate can be modeled as a function of this variable (Skalski et al., 1993; Smith et al., 1994; Hoffman and Skalski, 1995; White and Burnham, 1999). Development of models in which survival a n d / o r capture prob-
abilities are modeled as functions of static, continuous variables is discussed in Section 17.1.7. Many attributes with the potential to influence survival a n d / o r capture probabilities change over time and can be viewed as discrete. For such discrete, dynamic variables, the predictability of change is an important determinant of the best way to incorporate them into models. For example, age is a dynamic attribute that varies deterministically, in the sense that age of an animal in year i + 1 is perfectly predictable from its age in year i. Age-specific capture-recapture models for multiple age classes (e.g., Pollock, 1981b; Stokes, 1984; Brownie et al., 1986; Pollock et al., 1990; Lebreton et al., 1992) are discussed in Section 17.2. Of course, many dynamic attributes may be only stochastically predictable. For example, assume that capture and survival probabilities vary by body mass; specifically, capture probability at time i and survival probability for the interval i to i + 1 are dependent on body mass at time i. If animals are categorized into discrete mass classes (rather than viewing mass as a continuous variable), then an animal in mass class r at time i may be in that same class at time i + 1, or may instead be in mass class s. Thus, transitions between mass categories are probabilistic, such that an animal in class r at time i is in class r at time i + 1 with probability ~rr, and in class s at time i + I with probability ~s. Multistate models have been developed to deal with this kind of situation (Arnason, 1973; Nichols et al., 1992a, 1994; Brownie et al. 1993; Schwarz et al., 1993a) and are discussed in Section 17.3. Finally, we can envision models in which continuous attributes are dynamic and influence capture a n d / o r survival probabilities. We are unaware of the development of such models for capture-recapture data. One approach could rely on animals that are caught or observed in two (or more) successive periods, to predict an individual's state at time i+1 based on the value of its state at time i. Then, capture and survival probabilities for each animal could be modeled as functions of that state. Such an approach would be highly model dependent, however, in that it would require an explicit mathematical form of the relationship between the attribute and the capture and survival probability. In contrast, the multistate approach for use with discrete, dynamic state variables seems more flexible, in that time- and state-specific survival and capture probabilities can be directly estimated without any a priori knowledge of the form of the relationship(s). In addition to parameter heterogeneity that is not under the control of the investigator, heterogeneity also can be tied to certain aspects of study design. For example, sampling methods used to obtain recapture or resighting data should cover the entire population of
17.1. Single-Age Models interest as homogeneously as possible. Thus, trap locations in a trapping study should be located in such a manner that two to four traps lie within the home range of any animal in the study population. Traps need not be placed in a uniform grid pattern (e.g., see Karanth and Nichols, 1998), but the trap configuration should not contain "holes," or trap-free areas, that are large enough to contain an animal's daily movements. The investigator should place traps so that every animal in the sampled population is expected to encounter 1 trap during the course of daily movements. Study design is also relevant to heterogeneous survival probabilities. For example, many studies based on reobservations of animals [e.g., goose neckband studies such as that of Hestbeck and Malecki (1989a)] involve an initial capture for the application of marks and subsequent encounters via reobservation. It is possible for trapping and handling effects to produce lower survival probabilities for the interval following initial capture and marking, leading to different survival probabilities for releases of captured animals vs. "releases" of reobserved animals. Certainly, efforts to reduce any adverse effects of capture and handling should be incorporated into all capture-recapture study designs. Study design is considered further in Section 17.6, but we emphasize here the importance of reducing heterogeneity of capture and survival probabilities of animals in the target population. Study design is clearly relevant to assumptions (3) and (4) as well. Certainly selection of the appropriate mark is important in ensuring negligible loss of marks. If tag loss is suspected, then the study design should include double tagging (application of two tags to the same individual) for at least a fraction of releases in order to permit estimation of tag loss rates. The assumption of instantaneous, or at least very short, sampling periods is also an important aspect of study design. We recommend that the duration of sampling be short relative to the interval over which survival is to be estimated. The actual length of time of the sampling period is not as relevant as the probability of mortality during the interval, recognizing that long sampling periods create heterogeneity of survival among released animals. If mortality occurs during a long sampling period, then an animal released in the early portion of the sampling period may have a lower probability of surviving to any arbitrary point in the future compared to that of an animal released at the end of the period. Estimated survival probabilities typically are considered as applying to the interval extending from the approximate midpoint of one sampling period to the approximate midpoint of the next. Sometimes, however, investigators observe con-
423
tinuously and then arbitrarily subdivide data (e.g., by month or year). In such cases, the end of one sampling period occurs on one day and the beginning of the next period may occur the very next day. The concept of a discrete survival probability (the probability of surviving between periods i and i+1) begins to lose meaning in such situations. The assumption that all emigration is permanent is required for the interpretation of estimates of capture probability. Basically, interior 0s in a capture history (a "0" preceded by at least one capture in an earlier period and followed by at least one capture in a later period) are assumed to represent an event described by probability (1 - p), the complement of capture probability. If temporary emigration represents another possible explanation for an interior 0, then the interpretation of resulting ]~ must change and biased estimates of capture and survival probability can result (Kendall et al., 1997). Violation of any of the assumptions (1)-(5) can result in biased point estimates of survival a n d / o r capture probability (Section 17.1.9). The nature of these biases and methods for testing model assumptions are discussed in Sections 17.1.8 and 17.1.9. Assumption (6) concerns independence of fates and largely depends on the characteristics of the animals under study. For example, animals that travel in family groups (e.g., many goose species) tend to show some dependence of fates. Violation of this particular assumption typically does not result in bias of the point estimates of survival and capture probabilities. Instead, dependent fates violate the assumptions of the underlying multinomial distribution and lead to biased estimates of variance. Quasilikelihood methods can be used to adjust variances and inference procedures for the extra variation induced by dependence of animal fates (Section 17.1.8) (also see Burnham et al., 1987; Lebreton et al., 1992). 17.1.2.3. E s t i m a t i o n
Armed with a probability model [e.g., Eqs. (17.2)-(17.4)] and associated data [new releases u i and the number x~ of animals exhibiting each capture history 00 for Eqs. (17.2) and (17.3); or releases R i and the number mij of recaptures under Eq. (17.4)], we can estimate model parameters using methods such as maximum likelihood (Chapter 4) (also see Edwards, 1972; Mood et al., 1974; Burnham et al., 1987). Conditional on new releases u i in each period and the cell probabilities -rr~ f({pi},{q~i}) associated with each capture history, the probability distribution of the capture history data {x~} can be written generally as =
P({x~o} l {~i}{Pi}{ui}) = 1-IK----11 Ui! I-Io~ "ITx~~ II~ox~o! ~o,
(17.5)
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Chapter 17 Estimating Survival, Movement, and Other State Transitions
where the index co ranges over all possible capture histories. Examples of the cell probabilities associated with specific capture histories are shown in expressions (17.2) and (17.3) and Table 17.3. The right-hand side of Eq. (17.5) can be viewed as the likelihood function for the parameters of interest, L({~i}{Pi}), and thus can serve as the basis for maximum likelihood estimation of the survival and capture probabilities. Alternatively, the probability distribution of the recapture data, mij, can also be modeled as conditional on releases Ri and the same survival and capture probability parameters:
P({mij}I{ai}, {q~i}, {pi}) = K-1 II
Ri
i=1 (mi, i+i)!(mi,i+2)! "'" (miK)!(R i -- ri)! (q~ipi+l)mi'i+l
(17.6) ( x /[q~i(1 - Pi+l)q~i+lPi+2]mi,,+2 ... [q~i(1 - Pi+I)"'"
X
~K_lPK]mi,KxRi-ri).
The above distribution is simply a more general version of Eq. (17.4) (also see Table 17.4). Equations (17.5) and (17.6) provide alternative (but statistically equivalent) ways of writing the likelihood L({q~i}{Pi}). As discussed in Chapter 4, maximum likelihood estimates (MLEs) of the model parameters Pi and q~i are the values (denoted fii, ~i) that maximize L({q~i}{Pi}) for the observed data (the R i and mij). The above model with fully time-specific survival and capture probabilities was first studied by Cormack (1964), Jolly (1965), and Seber (1965), who derived closed-form estimators for the parameters P2.... , PK-1 and q~l. . . . , q~K-2. The model thus is known as the Cormack-Jolly-Seber model, and it contains 2K - 3 identifiable and estimable parameters (the parameters PK and q~K-1 in the likelihood can be estimated only as the product q~K-lPK, but not separately). Because there sometimes is confusion about estimable parameters in capture-recapture modeling, we provide here an intuitive explanation for the CJS model. The information needed to estimate capture probability for period i in open-population capture-recapture models essentially comes from the marked animals known (because of capture before and after i) to be alive and in the population during sampling period i (see Manly and Parr, 1968). The capture probability Pi can be estimated by conditioning on all animals caught both before and after period i, and hence known to be alive during i, and then asking what proportion of these animals were actually captured at i. However, a subset of animals known to be alive because of previous and subsequent capture does not exist for animals
caught in the first sample period (there are no previous sampling periods to use for such conditioning), so Pl cannot be estimated. A similar problem applies for the final sampling period K, because there are no subsequent sampling periods with which to establish a subset of animals known to have been alive at K. For this reason, only the joint probability of survival and capture (q~K-lPK) c a n be estimated for the final sampling interval. Current software readily permits numerical maximization of likelihoods for the CJS and related models, along with computation of their parameter estimates and associated variances and covariances. For example, program MARK (White and Burnham, 1999) was developed specifically for this sort of problem, and more general programs for multinomial estimation also have been developed (SURVIV) (White, 1983). Although we recommend software such as MARK for estimation, it is worthwhile to consider the closedform estimators derived by Cormack (1964), Jolly (1965), and Seber (1965), because examination of these estimators provides some insight into the manner in which capture history data supply the information needed to estimate capture and survival probabilities. Define the following statistics:
mj -- ~J-1i=1mij ni ri zi
--- ~ j K - i + I
mij
Number of marked animals caught in sample period j. Number of animals (marked and unmarked) caught in period i. Number of animals released at i (R i) that subsequently are recaptured. Number of animals caught before sample period i, not caught in i, and caught at some period after i.
In addition, define the random variable M i as the number of marked animals in the population at the time of sampling period i. With these statistics, closed-form MLEs can be described for survival and capture probability under the CJS model. The estimation approach is analogous to that of Lincoln-Petersen estimation, and on reflection, the estimators of parameters are intuitive. For example, the capture rate Pi is estimated by
fii-- mi/l~ii"
(17.7)
Thus, the estimate of Pi is simply the proportion of marked individuals in the population that also are in the ith sample. To estimate q~;, successive estimates of the marked cohort are used in the formula /~i+1
~i = 1~ i __ mi + Ri.
(17.8)
17.1. Single-Age Models The denominator of this expression is the estimated number of marked individuals in the population after trapping at time i. This can be seen by dividing the number of marked individuals into two groups: a group of M i - m i individuals not captured at time i, all of which survive the trapping event, and a group of n i individuals that are captured at time i, of which R i are released back to the population. The number of marked individuals in the population after trapping at time i is simply^ the sum of these two values, and it is estimated by M i - m i + R i. The ratio of this number to the estimated size of the marked cohort at time i+1 produces the estimator q~i. Note that the estimator applies to the period from the end of one trapping event to the start of the next. Thus, the estimated survival rate ~i encompasses all other sources of mortality except trapping loss. To complete the estimator derivations, an estimator Mi of the number M i of marked individuals in the population at time i is required. An intuitive estimator for M i is obtained by equating the expected proportion E ( r i / R i) of individuals released in period i that are later seen, and the expected proportion of marked individuals in the population that are not seen in period i but are seen later: E
ri
[
= E Mi_
zi mi
]
,
with M 1 = 0. After some algebra this leads to the estimator l~/li = m i + R i z i / r i ,
(17.10)
i = 2, ..., K-1. Jolly (1982) and Seber (1982) have developed bias-adjusted estimators for Eqs. (17.7) and (17.8), respectively. Closed-form variance and covariance estimators for the survival and capture probability estimates are presented in Seber (1982) and Pollock et al. (1990).
17.1.3. Reduced-Parameter M o d e l s The general CJS model includes separate survival and capture probability parameters for each sampling period. This generality carries a cost in terms of estimator precision, because time-specific parameter estimates have larger variances than stationary estimators. To capitalize on improved precision, biologists have developed models with capture a n d / o r survival parameters that are constant over time (Cormack, 1981; Sandland and Kirkwood, 1981; Jolly, 1982; Clobert et al., 1985; Crosbie and Manly, 1985). Here we denote the CJS model as model (q~t, Pt), with the t subscripts on survival and capture parameters
425
indicating time specificity (Lebreton et al. 1992). Let model (q~t, P) denote a model in which survival probability is time specific, but capture probability is constant over time; i.e., Pi = P, i = 2 . . . . , K. This model includes K - 1 survival probabilities and a single capture probability. Because we make the assumption that PK = P, q~K-1 also can be estimated. Note, however, that the data provide no extra information with which to test this assumption. An alternative parameterization, denoted by (q~, Pt), includes a single survival parameter for time periods 1 to K - 1, and time-specific capture probabilities for periods 2 through K. On assumption that q~K-1 = q~, the final capture probability PK can be estimated. Note that for models with variable periods of time separating successive sampling periods, the assumption of q~i = q~ is not biologically reasonable. For example, if 1 month elapses between sample times 1 and 2, and 6 months elapses between sampling times 2 and 3, then we could hardly expect survival probabilities for the two periods to be identical. However, it is reasonable to ask whether monthly survival probability between times 1 and 2 is similar to that between sampling times 2 and 3. This question can be investigated by modeling survival as ~i -- q~ti, where t i is the number of units of time separating sample times i and i+ 1, and q~is the per-time-unit survival probability corresponding to the same units used to express t i (Brownie et al., 1986; Pollock et al., 1990). In the above example, the reduced-parameter survival model (q~,Pt) would model survival between sampling times 1 and 2 using the parameter q~, and it would model survival between sampling times 2 and 3 using q~6,where the 6 denotes the number of months separating sample times 2 and 3. In what follows, the notation q~as a model descriptor refers to equal survival probability per unit time, rather than equal probability of surviving between all pairs of consecutive sampling times. Model (q~,p) denotes the simplest capture-recapture model, with single parameters for survival and capture probability, each of which is constant over time and sampling period. The likelihoods of the reducedparameter models have the same general appearance as the CJS likelihood [see Eqs. (17.5) and (17.6)], the only difference being the substitution of the constant parameters for the time-specific parameters of the CJS likelihood. For example, under model (q~t, P), the Pi in expression (17.6) are simply replaced by p in order to create the new likelihood. Identification of parameter estimates that maximize the likelihoods for these reduced-parameter models involves numerical approximation and therefore can be accomplished only with the aid of a computer. As discussed in Chapter 4, likelihood ratio tests
426
Chapter 17 Estimating Survival, Movement, and Other State Transitions
(LRTs) between models may be used when models are "nested" in the sense that the simpler model can be obtained by constraining the parameters of a more general model. The above three reduced-parameter models are all nested (see Chapter 4) with respect to the more general CJS model. So for a data set for which the CJS model provides an adequate fit, an LRT between model (r Pt) and CJS model (OPt, Pt) provides a test for temporal variation in survival probability. Specifically, model (q~,Pt) serves as the null hypothesis, and model (
17.1.4. Time-Specific Covariates In many situations, survival a n d / o r capture probabilities can be modeled as functions of time-specific external variables. For example, it may be reasonable to model variation in annual survival rates as a function of some variable (e.g., mean daily temperature during specific months) reflecting severity of winter weather. Variation in capture probability can be modeled as a function of investigator effort, based on statistics such as number of trap-nights or number of personhours expended in capture efforts. Models that include such information (e.g., Sauer and Boyce, 1983) can be constructed via a two-step process of first obtaining time-specific estimates (e.g., of q~i) and then using a regression approach with these estimates and the timespecific covariates (denoted here by xi). A preferred approach, known as "ultrastructural modeling," involves the modeling of certain parameters as functions of relevant covariates (North and Morgan, 1979; Clobert and Lebreton, 1985; Clobert et al., 1987; Lebreton et al., 1992) and the use of maximum likelihood estimation to estimate the parameters of the functional relationship directly. Following the development in Lebreton et al. (1992), we consider the approach of generalized linear models (McCullagh and Nelder, 1989), where we write the parameter of interest as a linear function of (possibly) multiple covariates [e.g., ~i = f(f3o + ~j ~jXji)]. The function f is said to "link" the parameters r to a linear function and is frequently called a link function. The link function typically is expressed using the inverse f - l , whereby r -- f(f3o + ~j ~jXji) is written in the
mathematically equivalent form f-l(q~i) = ~o + ~,j ~jxji. Link functions include the following functions: 9 The identity function: f-l(x) -- x. 9 The logit function: f-l(x) = logit(x) = log[x/(1 - x)]. 9 The logarithm function: f-l(x) = log(x). 9 The hazard function: f-l(x) = log[-log(x)]. Each of the above link functions is suitable for specific kinds of modeling (e.g., see Cox and Oakes, 1984; Aitken et al., 1988, Lebreton et al., 1992). The logit link function has been used frequently in capture-recapture modeling (e.g., see North and Morgan, 1979; Pollock et al., 1984; Clobert and Lebreton, 1985; Lebreton et al., 1992) and has the advantages of providing a flexible form and bounded estimates for q0i and Pi in the interval (0, 1). The logit link function is emphasized in software such as SURGE (Pradel et al., 1990; Lebreton et al., 1992) and is available in MARK (White and Burnham, 1999) as well. Additional information on the use of the logit link function in capturerecapture modeling, including confidence interval estimation and back-transformation to modeled parameters and their estimated variances and covariances, is found in Lebreton et al. (1992). For example, under the logit link, the estimate of survival probability is (~i =
exp(~0 + ~,j ~jXji) 1 + exp(~0 + ~,j ~jxji)'
(17.11)
where the ~j are estimated directly as part of the maxim u m likelihood procedure. We note that the covariates Xji used in the modeling of survival and capture probabilities need not be continuous. For example, we might categorize years as having "warm" or "cold" winters and consider modeling survival probability using two levels of the parameter. One approach would use the identity link with two different survival parameters r and
17.1.5. Multiple Groups Capture-recapture data sometimes can be grouped into distinct cohorts that share survival or capture probabilities. For example, males and females of the
17.1. Single-Age Models same species often are captured and marked in sampling efforts of a particular study. One approach to parameter estimation would be to treat the capture histories for the two sexes as completely separate data sets, with each used to estimate sex-specific survival and capture probabilities. Such a model can be denoted as (q~s.t, Ps,t) where superscripts s and t denote sex and time, respectively. Sex and time can be viewed as factors associated with variation in the subscripted parameters. The "," notation follows the convention for generalized linear models and indicates that the model includes parameters for all interaction terms between the different levels of the associated factors s and t. This notation emphasizes the alternative means of writing the model as, for example, a linear-logistic (logit-link) model. In the case of general model (~s.t, Ps,t), no parameters are "shared" by the two sexes. However, there are many situations in which either survival or capture probabilities or both should be similar for the two sexes. Such models can be constructed by imposing various constraints on the general model. For example, a model might permit different survival probabilities for males and females, yet assume common capture probabilities for the sexes. This model is denoted as (%,t, Pt), where the absence of the s subscript for Pt indicates that the same time-specific capture probabilities apply to both sexes. The advantage of such a reduced-parameter model is that it contains fewer parameters than the general model and thus allows for more precise estimation of model parameters than with the general model. Model adequacy can be assessed via an LRT of (q~s.t, Pt) vs. (q~s.t, Ps,t), which compares models with and without sex-specific capture probabilities. A rejection of the reduced-parameter model is taken as evidence of the need to include sexspecific differences in the capture probabilities. Alternatively, the inclusion of these two models in an a priori model set allows for the use of the informationtheoretic measures AIC, AICc, and QAICc to select the most reasonable model (see Sections 4.4 and 17.1.8). An interesting and potentially useful kind of constraint for multiple groups involves the idea of "parallelism" (Lebreton et al., 1992), which typically takes the form of a constraint linking the temporal variation in a parameter for two or more groups. For example, Lebreton et al. (1992) use the notation (q~s+t, Ps,t), to denote a model with time specificity and sex specificity of both survival and capture probability parameters, with the "s + t" notation of the survival parameter indicating that survival varies over time, but does so in a parallel or additive manner for the two sexes. If sex is treated as a d u m m y variable (s = 1 denotes males;
427
s = 0 denotes females), then the logistic link leads to the following parallel parameterization: logit(q~s) = as + ~i
or q~s =
exp(oLs + f~i) 1 + exp(oLs + ~i)'
(17.12)
where q~ denotes survival probability for an animal of sex s for period i, and c~ and ~i are the parameters to be estimated. Other forms of parallelism also can be defined--for example, by expressing a time-specific parameter for one sex as a constant multiple of that for the other sex (this form can be expressed as an additive model using the log link). Unless specified otherwise, we will use the " + " notation to refer to an additive model with a logit link, as in Eq. (17.12). Sex was used as an example of a group factor in the above discussion, but the interest in multiple groups extends to a variety of other questions. In cases with replicate study sites, one often is interested in variation in parameters across sites (spatial variation). In experimental studies, different groups of animals may receive different "treatments" (e.g., Burnham et al., 1987; Stromborg et al., 1988). In this case, possible differences among groups in the parameters representing response variables are of primary interest, and knowledge of these differences can inform ecological inferences (e.g., see Skalski and Robson, 1992). Despite the relevance to community ecology, we have seen few multispecies studies where species are treated as groups. Examination of models incorporating parallelism over time should provide an interesting means of assigning species to "guilds" or other functional groups. For example, species that respond to the same environmental variables would not necessarily be expected to exhibit the same survival rates, say, but would be expected to exhibit the same general pattern of temporal variation in survival. Examination of models with capture/resighting probabilities that are constant over species can be used to test an important assumption commonly used to translate multispecies count statistics into estimates of relative abundance. In any case, numerous interesting questions can be addressed in the context of modeling multiple groups of capture histories.
17.1.6. Effects of Capture History As noted in Section 17.1.2, the CJS model assumes that all marked animals present in the population at the time of sampling period i have equal probabilities of being caught (or resighted) at i and of surviving to
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Chapter 17 Estimating Survival, Movement, and Other State Transitions
any subsequent sampling period. One of the initial efforts to relax this homogeneity assumption was by Robson (1969) and Pollock (1975), who considered models in which individuals alive in the sampled population at period i could exhibit different capture and survival probabilities at period i depending on their previous capture history. Models that include transient individuals, or account for responses to previous trapping, are included in this class of models.
17.1.6.1. Trap Response in Capture Probabilities Several workers have considered models involving trap response in capture probability (Cormack, 1981; Sandland and Kirkwood, 1981; Lebreton et al., 1992; Pradel, 1993). For example, Sandland and Kirkwood (1981) considered a simple model of trap dependence with different capture probabilities for an animal at period i, depending on whether the animal had been captured at period i - 1: Pi is the capture probability at sampling period i for an animal that was caught at i - 1; p; is the capture probability at sampling period i for an animal that was not caught at i - 1. Sandland and Kirkwood (1981) considered models in which the above capture probabilities were constant over time and referred to the parameterization as Markovian, indicating the dependence of capture probability on the capture history of the previous period. These firstorder Markov process models can be denoted as (Pro), so that model (Pm, q~t) indicates trap response in capture probabilities with no time dependence, and time dependence in survival probabilities. To illustrate the parameterization under model (Pm, q~t), consider the following capture history and associated probability: Pr(01101 I release at period 2) = q~2pq~3(1 - P)q~4P'. The capture probability parameters associated with sampling periods 3 and 4 correspond to animals caught the previous time periods (2 and 3, respectively), whereas the capture probability for period 5 corresponds to an animal not caught the previous period.
17.1.6.2. Trap Response in Survival Probabilities The models with capture-history dependence that have seen the most use model survival between periods i and i + 1 as a function of capture history prior to period i. In particular, Brownie and Robson (1983) considered the sampling situation in many captureresighting studies, in which a mark is applied at initial capture, and subsequent encounters with marked animals are resightings. If trapping or handling adversely affects survival, then such an effect most likely occurs during the interval immediately following capture (the
initial encounter). Brownie and Robson (1983) thus parameterized survival as follows: q~i is the probability that a previously marked animal in the sampled population that is caught at time i survives until time i + 1 and remains in the sampled population; q~; is the probability that a previously unmarked animal in the sampled population at time i survives until time i + 1 and remains in the sampled population. Because the model was developed to deal with negative trap response in survival, the expected relationship between the two survival parameters is q~; < q~i. Capture (resighting) probability is not dependent on previous capture history under the Brownie-Robson (1983) model, and we can denote the model as (q~m.t, Pt), with the m subscript denoting dependence of survival probability on mark status (marked or unmarked) and the t denoting time dependence. As an illustration of the parameterization under the Brownie-Robson (1983) model (q~m.t, Pt), consider the following capture history and associated probability: Pr(01101 I release at period 2) = q~pgq~3(1 - p4)q~4P5. The initial survival probability includes the prime notation because it corresponds to an animal that has not been previously marked, whereas the subsequent survival parameters correspond to resightings of marked animals. The Brownie-Robson (1983) model permits estimation of P2, ..., PK-1, q~2, ..., q~K-2, and ~p~, ..., q~K-2. The sampled population contains no marked animals at the initial capture period, so qo1 is not a relevant parameter. Of course, reduced-parameter models (e.g., certain parameters set constant over time) can be constructed. The LRT of model (~t, Pt) vs. (q~m.t, Pt) provides a test of the null hypothesis of no trap response in survival. !
17.1.6.3. Parameterization for Transient Individuals Pradel et al. (1997a) adapted the Brownie-Robson (1983) trap-response model for studies in which unmarked animals are viewed as either "transients" or "residents." Transients are animals passing through the study area with negligible probability of again being in the area and available for capture at a subsequent sampling period. Residents, on the other hand, are animals with home ranges in the study area and typically are the animals of interest in capture-recapture studies. The problem with the use of standard open models such as CJS in the presence of transients involves heterogeneity of survival. Residents survive at some nonzero rate, and transients exhibit survival probability approaching 0 with respect to the study area. Under the CJS model, survival estimates in the presence of transients are negatively biased with re-
17.1. Single-Age Models spect to resident survival. This problem has been recognized by biologists and statisticians working on small mammals (e.g., Andrzejewski and Wierzbowska, 1961; Wierzbowska and Petrusewicz, 1963; Boutin and Krebs, 1986; Paradis et al., 1993) and birds (MacArthur and MacArthur, 1974; Manly, 1977; Snow and Lill, 1974; Buckland and Baillie, 1987; Peach et al., 1990). Following the approach of Brownie and Robson (1983), Pradel et al. (1997a) separated captured animals into two groups based on previous mark status. Their modeling approach can be described using the following parameters: ~pr
~p~
"ri Pi
The probability that a resident in the sampled population at time i survives until time i + 1 and remains in the sampled population. The probability that a transient in the sampled population at time i survives until time i + 1 and remains in the sampled population. The probability that an unmarked animal caught at sampling time i is a transient. The capture probability at sampling occasion i.
Using the above parameters, the survival probabilities for marked (q~i) and unmarked (~p;) animals at period i can be rewritten as: q)i - - q)~,
and q); - - Tiq) ~ q-
(1 - Ti)q~.
(17.13)
The first equality in Eq. (17.13) indicates that all previously marked animals are residents by definition and hence are exposed to resident survival probabilities. The second equality expresses the survival probability of unmarked animals as a mixture model including survival rates of both transients and residents, in proportions given by % The parameterization of Eq. (17.13) is too general to permit identification of all parameters. However, Pradel et al. (1997a) operationally defined transients as animals with no chance of returning to the study area; thus, q~ = 0 and ~p; = (1 - T i ) q ) r. Estimation of both ~pr and T i is possible under this parameterization. Identifiable parameters under the transient model with full time specificity include P2.... , PK- 1, T2. . . . , "rK_1, and q~, ..., q~:-2. Separate estimation of T i and ~prrequires releases at period i of known residents (previously marked animals); hence only the product ~p~ = (1 - "rl)q~~ is estimable the first time period because all releases are unmarked. Again, reduced-parameter models can be constructed based on the transient parameterization, and these have proved useful in work with avian capture-recapture (e.g., De-
429
Sante et al. 1995; Rosenberg et al., 1999). LRTs of models with Ti = 1 against alternative hypothes~s models with no constraint on Ti provide tests for the presence of transients. The transient model of Pradel et al. (1997a) and the Brownie-Robson (1983) trap-response model were developed with very different biological mechanisms in mind, and these mechanisms lead naturally to different parameterizations. However, the models are equivalent in that they lead to identical data structures. A consequence of this is that it is not possible to distinguish between mortality associated with capture and handling vs. existence of transients. Instead, auxiliary data or background information must be used to decide on the appropriate parameterization for a particular study situation. A further complication arises from the fact that the data structure of the models of Brownie and Robson (1983) and Pradel et al. (1997a) can be produced by the presence of unidentified age specificity (see Loery et al., 1997). This situation is believed to occur with annual sampling of small birds (e.g., passerines) and involves an inability to distinguish fledged young-of-the-year from adult (> 1 year of age) birds. Thus, any recapture (previously marked) is known to be an adult bird, but unmarked birds can be a mixture of young and adult. This leads to a mixture model for survival of unmarked birds that is similar to that under the transient parameterization, with the parameter Ti now specifying the probability that an unmarked bird is a young bird with separate survival parameters for adult and young (Loery et al., 1997). Unlike the transient model, whereby one of the survival parameters equals 0 by definition, both young and adult survival parameters are nonzero, and there are too many parameters to estimate. Again, auxiliary information is needed to decide whether a particular data set was likely to have been produced by this type of underlying model. The distinguishing feature of all the models considered above is that survival a n d / o r capture probability for period i is allowed to differ between groups of individuals, with the proviso that survivors from both groups share the same probabilities of capture and survival after time i+1. Thus, a model for temporary trap response has one survival probability over (i, i + 1) for those individuals captured at time i that previously were caught and another survival probability for those individuals not previously caught, with survivors from both groups sharing the same survival probability over (i + 1, i + 2) (because surviving individuals from both groups have been captured previously). A model for the presence of transients allows for a mixture of different survival probabilities for transients and residents, resulting in one survival probability over (i, i + 1) for
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Chapter 17 Estimating Survival, Movement, and Other State Transitions
unmarked (resident + transient) individuals that are captured at time i and another probability for marked (and therefore resident) individuals captured at time i, with survivors from both groups sharing the same survival probability over (i + 1, i + 2) (because surviving individuals from both groups are necessarily resident at time i + 1). Finally, a model for indistinguishable age cohorts has a mixture of different survival probabilities for young and adults, resulting in one survival probability over (i, i + 1) for unmarked (young + adult) individuals that are captured at time i and another probability for marked (and therefore adult) individuals captured at time i, with survivors from both groups sharing the same survival probability over (i + 1, i + 2) (because surviving individuals from both groups are necessarily adult at time i + 1). As indicated above, the same model parameterization can accommodate each of these situations, and auxiliary information is needed to distinguish among the underlying biological situations.
17.1.7. Individual Covariates As described above, the CJS model and the models derived from it assume homogeneity of capture and survival probabilities among individuals at some level of grouping. Under the standard CJS model, all marked animals alive at sample time k have the same probability of being caught at a future time (e.g., k + 1) and, assuming they are not removed from the population, of surviving until k + 1. Under the general models with different groups of animals, homogeneity is assumed to apply within each group. In the case of dependence on previous capture history, homogeneity is assumed to apply to all animals with a particular history. Of course, strict homogeneity (exact equality) of survival and capture probabilities of different individual animals is unlikely ever to be true, regardless of the manner in which animals are grouped or categorized. Often a substantial amount of variation among individuals in survival a n d / o r capture probabilities may be explained by some measurable covariate, for example, size or weight at some critical stage in the early life of an animal [e.g., mass at hatch or fledging; see Perrins (1963, 1965)] or parental size and experience (see Hastings and Testa, 1998). Individual covariates of this type fall in the category of static, continuous state variables as discussed in Section 17.1.2, and their influence violates the assumption that groups of animals have similar characteristics, as in Eqs. (17.1)-(17.3). Instead, individual-based modeling is based on the view that the capture history of each individual animal is a multiple Bernoulli trial [a
multinomial sample with sample size of 1 (Smith et al., 1994)]. In describing models for individual covariates, we follow the development and notation of Skalski et al. (1993) and Smith et al. (1994). In particular, sample times are represented by k and m rather than i and j, and the following variables are used:
Ikj m
tkj
dkj
An indicator variable that has a value of 1 if the jth tagged animal released at sampling period k is detected (recaptured or resighted) at sampling period m, and a value of 0 if the animal is not detected. The number of the sampling period on which the jth tagged animal released at sampling period k was last seen alive. An indicator variable that has a value of 0 if the jth tagged animal released at sampling period k is known to be removed from the marked population at event tkj, and equal to 1 if it is not known to have been removed during the study.
We then define model parameters associated with each individual animal:
q~kjm
Pkjm
Xkjtkj
The probability of surviving the interval between sampling periods m and m + 1 for the jth tagged animal released at sampling period k. The probability of detection at sampling period m for the jth tagged animal released at sampling period k. The probability of never being detected following sampling period tkj for the jth tagged animal released at sampling period k. By definition • = 1, and the other Xjktkj a r e defined recursively as in Eq. (17.1).
The probability associated with an individual capture history [expressed as the set {Ikj m} of indicator variables] can then be written as follows: tkj--1
pr({ikjm}, dkj) = ,;( dkj ]--[ -Ikjm kjtkJ ~n=~k q~kjmP kj,m + l
(17.14)
x (1 - Pkj,m+l) 1--Ikjm (see Smith et al., 1994). If the fate of each individual is independent of that of all others, then the likelihood associated with the entire study is given by the product of the probabilities in Eq. (17.14) over all individuals released during the study. However, such a likelihood is overparameterized and estimation is not possible without modeling individual parameters as functions of individual-level covariates.
17.1. Single-Age Models Modeling with individual covariates is accomplished in a manner similar to that used for timespecific covariates (Section 17.1.4). Again, we focus on the different link functions used to express the parameter of interest in terms of a linear function of covariates. For example, let Xikj be the value of the ith covariate for the jth individual released at period k, let ~im be the regression parameter relating individual covariate i to survival probability for the interval between sampling periods m and m + 1, and let Pm be an "interval effect" parameter associated with the interval between sampling periods m and m + 1 (Smith et al., 1994). Under the hazard link defined in Section 17.1.4, we might model survival probability as
q~kjm =
q~exp(pm+~i
f3imXikj)"
(17.15)
Alternatively, an intercept parameter oLcould be added and the logit link used to model individual survival as
~kjm
exp(oL + f)m q- ~i ~imXikj) 1 + exp(oL + Pm q- ~i ~imXikj)"
(17.16)
Time-specific covariates and associated regression parameters can be added to such models as Eqs. (17.15) and (17.16) if desired. It is also possible to incorporate group effects in the case of multiple groups (Skalski et al., 1993; Smith et al., 1994; Hoffman and Skalski, 1995). Capture probabilities also can be modeled in a similar manner, as functions of individual and possibly timespecific covariates. Analysis of deviance and likelihood ratio tests can be used to test hypotheses about the existence of relationships between the different individual covariates and probabilities of both survival and capture (Skalski et al., 1993; Smith et al. 1994). AIC, AIC c, and QAICc (see Sections 4.4 and 17.1.8) can be used to select the most appropriate models from the model set, in the same manner as for models without individual covariates. Note that with individual covariates, each individual is characterized by a vector of covariates measured at a single time, typically at the time of initial capture and release. The relationship between this vector of covariates and survival (or capture) probability may be characterized by time-specific regression parameters, but the values of the covariates do not change over time. For some covariates this is a reasonable form for the expression of covariate influence. For example, mass at fledging or weaning could be expected to be relevant to survival throughout the life of an organism in some situations. However, for covariates such as mass measured at the time of initial capture for adult animals, the modeling approach described above is
431
likely to be inadequate. The idea of mass at period k influencing survival during the interval k to k + 1 may be reasonable for adult animals, but survival during later intervals is much more likely to be influenced by mass at those later times than by mass at some previous time period. Thus, use of this approach requires careful consideration of the appropriateness of static covariates for modeling nonstationary survival and capture probabilities. Discrete, dynamic state variables influencing survival a n d / o r capture probabilities can be modeled using the multistate approach of Section 17.3.
17.1.8. Model Selection Because a model can be viewed as set of assumptions about processes that generate data, the topics of model selection and model assumptions are closely related. Biologists and statisticians working with capture-recapture models historically have devoted substantial effort evaluating the degree to which data meet model assumptions. Some early work involved specific assumptions. For example, Leslie (1958) developed a test for heterogeneous capture probabilities, which later was extended by Carothers (1971). Robson (1969) developed a test for the influence of initial marking on survival during the subsequent interval prior to the next sampling period (also see Manly, 1971; Brownie and Robson, 1983). An alternative approach to the testing of specific model assumptions is to focus on an overall assessment of the fit of the model to the data. Leslie et al. (1953) developed an ad hoc approach for assessing fit of the CJS model, and formal goodness-of-fit tests were presented by Seber (1982), Jolly (1982), Pollock et al. (1985a), and Burnham et al. (1987). Even when these tests are based on sufficient statistics (e.g., the mij array), data typically are sparse for some cells of the resulting contingency tables. Sparse data and the resulting low expectations for many contingency table cells produce test statistics that do not follow the chisquare distribution. This problem has motivated various approaches to cell pooling. The most general approach for the CJS model is that of Burnham et al. (1987) implemented in program RELEASE. A goodness-of-fit test for a reduced-parameter model usually can be constructed from a goodness-offit test for an associated general model and an LRT of the general model against the reduced-parameter model. For example, Brownie et al. (1986) developed tests for reduced parameter models as the sum of chisquare statistics from (1) the goodness-of-fit test for the general CJS model and (2) the LRT between the reduced-parameter model and the CJS model. The LRT tests the assumption incorporated into the reduced-
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Chapter 17 Estimating Survival, Movement, and Other State Transitions
parameter model (e.g., the absence of temporal variation in a parameter). In like manner, a goodness-of-fit test for the CJS model can be constructed from an LRT involving a model that includes capture history dependence (Section 17.1.6), because the latter is more general than the CJS model. Brownie and Robson (1983) developed a goodness-of-fit test for their model of trap response in survival probability, and this same test can be used for the transient parameterization of Pradel et al. (1997a). Because the chi-square distribution is approximated poorly by contingency table test statistics for sparse data and low expected cell values, other approaches to testing and model selection besides an LRT are needed. One alternative is to use a bootstrapping approach to investigate the distribution of the deviance (see Section 4.4), a metric reflecting model fit. Under this approach, parameter estimates for the model to be assessed are taken to be the true parameter values underlying the data. These values then are used to simulate capture-recapture data sets that resemble the original data, based on the observed numbers u i of new unmarked animals released at each sampling occasion. A large number of such data sets are simulated with the same underlying model and same parameter values (i.e., the parameter estimates from the analysis of the real data set). A model is fit to each simulated capture-recapture data set and a corresponding deviance is computed. The deviance corresponding to the original data set and the original model then is compared to the distribution of deviances generated with the original model. A conclusion that the original deviance could reasonably have been drawn from the distribution provides evidence that the original model fits the data. If the observed deviance is very unusual (e.g., is in the upper 1% of values in the deviance distribution) one concludes that the original model fails to fit the data. Modern capture-recapture analyses typically begin with a series of models representing competing sets of assumptions about the structure of the capture-recapture data and the processes that gave rise to the data. The most general model in the model set is presumed to describe the data adequately, because model selection procedures are conditional on the existence of at least one adequate model in the set. Most approaches to model selection therefore begin with a test of the goodness of fit for the most general model in the model set. For many model sets involving single-age data, the CJS model represents the most general model, and the test implemented in RELEASE can be used to assess fit. An alternative approach involves the bootstrapped deviances described above. Given an acceptable general model, the process of model selection can proceed in at least two ways. One
approach places model selection in a hypothesistesting framework and proceeds with sequential LRTs between nested models. Sequential testing frequently begins with the most general model and proceeds systematically through the comparison of more general models against reduced-parameter models (e.g., Brownie et al., 1985, Lebreton et al., 1992). Of course, likelihood-ratio testing also can proceed in the reverse direction, beginning with simple models and using tests to decide when additional complexity is warranted [see Catchpole and Morgan (1996) for a similar approach using score tests]. We note that capture-recapture analysts seldom limit themselves solely to a hypothesis-testing approach to model selection. However, the use of LRTs to test model assumptions is still a common practice. For example, the LRT between model (q~,Pt) versus CJS model (tpt, Pt) provides a formal test of the hypothesis that survival rates vary over time. Another approach to model selection places the problem in an optimization (rather than hypothesistesting) framework. Akaike's Information Criterion (Akaike, 1973; Burnham and Anderson, 1992, 1998) typically is used as the optimization criterion with this approach. An information-theoretic criterion essentially combines goodness-of-fit and parsimony of model parameterization with a tendency in consequence to select mathematically simpler models. An information-theoretic approach with AIC has performed well in simulation studies with capture-recapture models (Anderson et al., 1994; Burnham et al., 1994, 1995), and it is the recommended approach to model selection in capture-recapture studies (e.g., Lebreton et al., 1992; Burnham and Anderson, 1998). AIC is computed as AIC = - 2 logiC(0 ly)] + 2K,
(17.17)
where log[~(0ly)] denotes the log of the likelihood function evaluated at the MLEs of 0 given the data y, and K denotes the number of estimable parameters. The small-sample correction of Hurvich and Tsai (1989) is applied to AIC to yield AICc, which is recommended unless sample size is large relative to the number of estimated parameters (Burnham and Anderson, 1998). AICc is computed as AICc = - 2 logiC(0 ]y)] + 2K +
2K(K + 1) n-K-1
(17.18)
2K(K + 1) = AIC + n-K-l'
where n is sample size. In the context of capture-recapture models, sample size is usually obtained by summing the releases R i over all time periods.
17.1. Single-Age Models Both the hypothesis-testing and optimization approaches to model selection are conditional on the presence in the model set under consideration of a general model that adequately fits the data. However, it is not unusual to encounter situations in which the most general model does not fit the data well. In this case, a quasilikelihood approach typically is recommended (Burnham et al., 1987; Lebreton et al., 1992; Burnham and Anderson, 1998). This approach is based on the presumption that overdispersion of data is the reason for lack of fit in capture-recapture models. Overdispersion in capture-recapture data can be the result of a lack of independence of capture and survival events, as might be expected for animals that travel together in family groups (e.g., many goose species) or even in groups of unrelated individuals. Estimators for model parameters frequently remain unbiased in the face of overdispersion, but model-based variances tend to be too small (McCullagh and Nelder, 1989). The quasilikelihood approach involves computation of a variance inflation factor ~ as -- x2/dfl
(17.19)
where X2 and df correspond to the goodness-of-fit test of the most general model (Cox and Snell, 1989) in the model set. If the bootstrap approach to assessing goodness of fit is adopted, then ~ can be estimated by dividing the deviance from the original data set by the mean of the deviances from the simulated data sets (this mean estimates the expected value of the deviance, which is the value of df under the null hypothesis that the model fits the data). The quasilikelihood variance inflation factor d can be multiplied by modelbased variance and covariance estimates to compensate for the underestimation of variances and covariances caused by overdispersion. This inflation factor also can be used as a means of dealing with overdispersion in likelihood ratio tests. Lebreton et al. (1992) have suggested that LRT statistics for nested models be modified to compute the corresponding F test statistics as Fdfl,df 2 =
LRT/dfl ~ ,
(17.20)
where dfl denotes the degrees of freedom for the LRT and df2 denotes the degrees of freedom associated with the goodness-of-fit test for the most general model under consideration. The quasilikelihood approach to model selection using AIC leads to computation of QAIC, which is modified to deal with overdispersion: QAIC = - 2 log[~(0 ] y)]/~ + 2K.
(17.21)
433
Finally, overdispersion and small sample size are considered in the computation of QAICc: QAICc = - 2 logiC(6 ]y)]/d + 2K +
2K(K + 1) n-K-1
(17.22)
2K(K + 1) = QAIC + n-K-l" Because many capture-recapture studies are characterized by both sparse data and some indication of lack of fit, QAIC c should be used frequently in model selection. We note that interest often is focused on the relative AIC values for members of a candidate model set. Thus, model selection results can be displayed in terms of AAIC values, where &AIC for model i is given by
&i-- A I C i - AICmi n, with AICi the AIC value for model i and AICmin the minimum AIC value for models in the candidate set. Models with small &AIC (e.g., < 2) are relatively well supported by the data, whereas models with large AAIC (e.g.,> 10) are not well supported. In like manner, &AIC values can be defined for the other forms of AIC, i.e., &AICc, &QAIC, and &QAIC c. As noted in Section 4.4, it is sometimes useful to compute "Akaike weights" (Burnham and Anderson, 1998) for the models in a candidate set. Specifically, e x p ( - A i / 2 ) can be viewed as the relative likelihood of model i, given the data and the other members of the model set (Burnham and Anderson, 1998). These relative values are normalized to unity by e x p ( - ai/2) Wi
--
R
r
exp( - a m/2) m=l
where R denotes the number of models in the candidate model set (Burnham and Anderson, 1998). The resulting Akaike weights, w i, can be interpreted loosely as the weight of evidence in favor of model i being most appropriate, given both the data and the model set. The Akaike weights provide a mechanism to incorporate the uncertainty of model selection into the process of parameter estimation. Let 0 be a parameter of interest that appears in all of the models in the candidate model set. A weighted average estimate of 0 can be computed as R
i=1
(see Buckland et al. 1997, Burnham and Anderson 1998), where 0i is the parameter estimate under model
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Chapter 17 Estimating Survival, Movement, and Other State Transitions
i, and w i is the corresponding model weight. An estimator of the variance for 0 utilizes the sum of two components for each model: the weighted conditional variance var(Oi]Mi), where M i denotes model i (note that this variance component is obtained directly from the model-based estimation procedure), and the squared deviation of each model-based estimate from the weighted average, (0i - 6)2. Each of the terms var(Oi/Mi) + (Oi - ~)2 can be viewed as the mean squared error for model i with respect to the weighted average estimator 0. The resulting variance estimator is
R v'~(0)
~-~
E wi
V/
~r(OilM i)
_ -}- (0 i
]2 6) 2
.
"i=1
Simulations have indicated that this estimator performs acceptably (Burnham and Anderson, 1998). We note in closing that the statistics reflecting model selection and associated uncertainties all are computed by program MARK (White and Burnham, 1999). &AIC values, Ai, and the Akaike weights w i are displayed regularly with model selection results, and modelaveraged estimates and associated variances also are computed when specified by the user.
17.1.9. Estimator Robustness and Model A s s u m p t i o n s When models inadequately represent data and alternative models providing a good fit are not available, there is a potential to be misled by the biased estimates that can result. In the past, when the CJS model was used in capture-recapture studies of open animal populations, the issue of estimator robustness received considerable attention from biologists and statisticians. However, the development of more general models permitting relaxation of CJS model assumptions [e.g., the capture history dependence models of Robson (1969), Pollock (1975), Brownie and Robson (1983), Pradel (1993), and Pradel et al. (1997a)] has reduced substantially the need for such studies. Today the response to a lack of model fit is to consider ways of modifying the CJS model to accommodate additional sources of variation. Nevertheless, no model exactly describes the processes giving rise to a set of capture history data, and estimator robustness in the face of assumption violations continues to be relevant in model estimation and assessment. The investigation of estimator robustness offers some insight into how the information provided by a capture-recapture study is used to estimate demographic parameters. I n w h a t follows we explore the consequences of violating the assumptions from Sec-
tion 17.1.2 that underlie capture-recapture methodology. 17.1.9.1. H o m o g e n e o u s Capture Probabilities
Recall from Section 17.1.2 that the first assumption of the CJS model concerns equal capture probability for every marked animal in the population at a given sampling period i. Heterogeneous capture probabilities typically produce relatively small bias in survival estimates, and the direction of this bias is typically negative (Carothers, 1973, 1979; Nichols and Pollock, 1983b). Cormack (1972), Carothers (1973), and Pollock et al. (1990) discussed the reason for this small bias, which involves the fact that survival estimates are based on marked animals [e.g., Eqs. (17.8) and (17.10)], for which variation typically tends to be much smaller than that for all marked and unmarked animals (see Section 18.2.5). Nevertheless, many different causes, and hence patterns, of heterogeneous capture probabilities can be envisioned, and it sometimes is useful to tailor an examination of bias to the specific type of problem encountered. For example, Carothers (1979) found strong evidence of heterogeneous capture probabilities in a long-term data set for fulmars, Fulmarus glacialis. Using an interesting approach based on simulation and inverse prediction, he approximated the relative bias in the CJS survival estimates and found the bias to be quite small (<0.01) (Carothers, 1979). Stromborg et al. (1988) conducted a capture-resighting study of postfledging starlings, Sturnus vulgaris, in which a group of marked fledglings moved to a location that was not discovered until midway through the study. The resulting heterogeneity was extreme in that one group of birds was nearly invisible (resighting probabilities of 0.05) for the first two resighting periods. However, the fact that the group was later discovered led to high capture probabilities at the end of the study, and the relative bias of survival estimates was <0.02. Pledger and Efford (1998) extended the simulation and inverse prediction approach of Carothers (1979) to the correction of bias resulting from unequal capture probabilities. Similarly, Hwang and Chao (1995) used an approach based on sample coverage to approximate the bias in Jolly-Seber estimators and to propose methods for bias correction. These approaches are discussed in Section 18.2.5, because they assume more importance when dealing with abundance estimators, which can be severely biased by heterogeneous capture probabilities. Permanent trap response refers to the situation wherein capture probability is influenced by initial capture such that within a sampling period, unmarked
17.1. Single-Age Models animals have one capture probability and marked animals have a different capture probability. Trap-happy refers to a higher capture probability for marked animals, and a trap-shy response denotes a lower capture probability for marked animals. Because survival and capture probability estimates are based on marked animals [e.g., Eqs. (17.8) and (17.10)], permanent trap response produces no bias in survival or recapture probabilities (Nichols et al., 1984b), although the capture probability estimates do not apply to unmarked animals (see Chapter 18 for further discussion). 17.1.9.2. Homogeneous Survival Probabilities
The second CJS model assumption involves homogeneity of survival probabilities. The potential effects of heterogeneous survival rates have been investigated for estimators in band recovery models, which represent a special case of conditional CJS models (Pollock and Raveling, 1982; Nichols et al., 1982b) (see Section 16.1.9). The magnitude of estimator bias was found to depend on the covariation between survival and recovery probabilities (i.e., the degree to which animals with high survival probabilities tend to have high or low recovery probabilities). Effects of heterogeneous survival on CJS survival estimates have not been thoroughly investigated. Pollock et al. (1990) considered the case of heterogeneous survival probabilities that are positively correlated within individuals (some individuals tend to have low survival probabilities across time and others tend to have high survival probabilities) and are independent of capture probability. Survival rate estimators tended to be positively biased in simulations, but the bias was not large for the situations considered (Pollock et al., 1990). Use of single-age models in the face of age-specific variation in survival probabilities can result in positive bias in survival estimates, although the bias is not large for small to moderate variation in age-specific survival (Manly, 1970).
435
1987, 1988; Nichols et al., 1992b; Nichols and Hines, 1993). Tag loss rates can be estimated using doublemark studies (animals are marked with two tags) with conditional models. These models condition on animals recaptured with at least one tag and use information on the presence or absence of the other tag to estimate the probability of tag loss (e.g., Seber, 1982; Blums et al., 1994; Spendelow et al., 1994; Fabrizio et al., 1999).
17.1.9.4. Instantaneous Sampling The assumption of instantaneous sampling is related to heterogeneous survival probabilities. If some animals are captured at the end of a long sampling period and others only at the beginning, then we might expect animals in the two groups to exhibit different survival probabilities over the remainder of the study. Animals caught near the end of the period will have a greater probability of surviving compared to animals caught near the beginning, because the latter must survive the entire sampling period. Smith and Anderson (1987) approximated bias in survival estimators for band recovery data produced by lengthy sampling periods. As expected, the magnitude of bias depends on the pattern and magnitude of variation in survival probabilities for individuals banded at different times. One possible approach to dealing with this problem involves using multistratum models, with different strata corresponding to the relative timing of capture within a given sampling period. Another approach involves the use of a robust design approach (Chapter 19) and the modeling of survival using captures and recaptures within each primary sampling period. Tavecchia et al. (2002) has used this approach with band recovery data. Schwarz and Stobo (1997) and Kendall and Bjorkland (2001) have extended the robust design to the fully open model situation. 17.1.9.5. Permanent Emigration
17.1.9.3. Absence of Tag Losses
Loss of tags or marks produces negative bias in survival estimates, because estimation under these circumstances applies only to animals retaining marks rather than to the animals themselves. "Correction" of CJS survival estimates with estimates of tag loss rates is relatively easy when tag loss rates are constant over time and exhibit neither time nor age specificity (Arnason and Mills, 1981; Pollock, 1981b; Nichols and Hines, 1993). If tag loss rates vary in an age-specific manner (where age refers to age of the mark, not the animal), then correction of bias in CJS survival estimators is more difficult and may even require joint modeling of capture-recapture data and tag loss data (Kremers,
Violation of the assumption that all emigration is permanent does not always result in biased estimates. If temporary emigration is a random process such that every individual within an age-sex category has the same probability of being in the area exposed to sampling efforts, then estimates of survival probability remain unbiased and capture probability estimates now estimate the product of the probabilities of (1) being in the sample area and (2) being caught, given that the animal is in the sample area. An alternative model might allow animals that were not temporary emigrants at period i - I to exhibit one probability of being a temporary emigrant at period i, whereas animals that were temporary emigrants at i - 1 exhibit a different
436
Chapter 17 Estimating Survival, Movement, and Other State Transitions
(frequently higher) probability of being an emigrant at i. Under this model, CJS survival and capture probability estimates al,~ biased, with the direction and magnitude of the bias depending on the nature of the emigration process (Kendall et al., 1997). Alternative models for temporary emigration processes are considered in more detail in Chapter 19, and the robust design described there provides extra information to estimate temporary emigration and to distinguish between alternative models of the emigration process (Kendall and Nichols, 1995; Kendall et al., 1997; Schwarz and Stobo, 1997; Kendall and Bjorkland, 2001).
17.1.9.6. Independence of Capture Histories Finally, the CJS model assumes independence of the fates of individual animals. The multinomial distribution used for most capture-recapture modeling [Eqs. (17.2)-(17.4)] is appropriate only when fates are independent. Under dependent fates, the effective sample size is some number smaller than the number of marked individuals, so that the computed variances for the parameter estimates are smaller than are appropriate (see Pollock and Raveling, 1982; Schmutz et al., 1995). However, as noted in Section 17.1.8, the point estimates tend to remain unbiased. Quasilikelihood procedures for variance inflation, model testing, and AIC model selection provide a reasonable approach to drawing inferences in the face of dependent fates. Schmutz et al. (1995) used Monte Carlo simulations to demonstrate the positive relationship between dependency in survival probability among individuals and increased variance in survival estimates. Schmutz et al. (1995) then estimated the magnitude of bias in survival rate variances resulting from dependent fates in resighting data for black brant, Branta bernicla nigricans. They estimated the true variance in survival probabilities for paired birds to be > 2.5 times larger than for the same number of unpaired birds, whereas the model-based variance estimates were nearly identical. Unfortunately, this bias was substantially underestimated by the variance inflation factor computed based on goodness-of-fit testing (Schmutz et al., 1995).
17.1.10. Example We illustrate the concepts and methods for singleage capture-recapture analysis with a live-trapping study of meadow voles, Microtus pennsylvanicus, at Patuxent Wildlife Research Center, Laurel, Maryland (Nichols et al., 1984a). Data for the study were collected on a 10 • 10 grid of trapping stations spaced at 7.6-m intervals in old field habitat. A single, modified Fitch live trap (Rose, 1973) was placed at each station. Hay
and dried grass were placed in the traps and whole corn was used as bait. Sampling occurred for five consecutive days each month, from June 1981 through December 1981. During each 5-day trapping session, traps were opened in the evening of the first day, checked the following morning, locked open during the day, and reset in the evening, with the sequence repeated each day until 5 days had elapsed. A raccoon, Procyon lotor (later captured), visited the traps on the final two nights of the second trapping session, essentially leaving only 3 days of trapping for this session. At each capture, animals were examined for a tag, sexed, weighed, and examined for external reproductive characteristics. Untagged animals were ear-tagged with numbered fingerling tags, and tag numbers of marked animals were recorded at each capture. For a single-age analysis conditional on releases, we focus only on "adult" animals (->22 g) (see Krebs et al., 1969) and collapse the 5 days of sampling each month into a single assessment of presence or absence, leaving six monthly sampling occasions. Thus, an animal caught for five consecutive days in month I would receive a "1" as the first entry of the capture history, as would an animal caught on only 1 of the 5 days. Data can be summarized as capture histories (Table 17.5) or in mq-array format (Table 17.6). A small number of animals was lost on capture (died in the trap). We first applied the CJS model (Section 17.1.2) to the data, because the model is very general and is frequently used as a starting point in capture-recapture modeling (e.g., see Burnham et al., 1987; Lebreton et al., 1992). We assessed model fit to the data using the overall goodness-of-fit test of program RELEASE (Burnham et al., 1987). Fit was judged to be adequate for each sex separately (females, X2 = 11.7, P = 0.17; males, • = 12.9, P = 0.23) and for the combined data set (with a test statistic obtained by adding the sexspecific test statistics and degrees of freedom, yielding • = 24.6, P = 0.14). These goodness-of-fit statistics provided no motivation to investigate more general models such as those incorporating capture-history dependence (Section 17.1.6). The estimates of capture probabilities under the general model (%,t, Ps,t) were very high, reflecting the five consecutive days of trapping at each sampling period (Table 17.7). Point estimates of monthly survival probability showed more variation, ranging from 0.58 to 0.89 (Table 17.7). Several reduced-parameter models also were fit to these data, imposing various constraints on sex and time specificity of the capture and survival parameters (Table 17.8). Model (%+t, P) showed the smallest AICc value, although several other models seemed appropriate for these data as well (Table 17.8). Under model (%+t, P), survival probability varied by
17.1.
TABLE 17.5 Capture-History Data for a Six-Period Study of M e a d o w Voles, (Microtus p e n n s y l v a n i c u s ) at Patuxent Wildlife Research Center, Laurel, Maryland, 1981 Capture history
Number of females a
437
Single-Age Models TABLE 17.6 History Data of M e a d o w Wildlife
The m/j-Array Representation of the Capture Presented in Table 17.5 for a Six-Period Study Voles (Microtus p e n n s y l v a n i c u s ) at Patuxent Research Center, Laurel, Maryland, 1981
Number of males a Release
1OOO0O
7
8
100000
- 1
- 3
110000
10
21
110000
- 4
111000
7
111100
2
3
111100
0
111110
1
111110
-1
Sex
Recapture period j Releases R i
2
3
4
5
6
1
51
40
4
0
0
0
- 2
2
49
3
0
0
5
3
52
34
1
0
Female
-1 1 0
111111
14
10
101111
1
0
110111
1
1
101011
1
0
101110
1
0
101100
0
1
101000
1
0
010000
2
10
010000
0
- 1
011000
2
3
011100
3
3
011110
1
0
011111
3
3
011101
0
3
011011
0
1
010111
1
2
010100
1
1
010001
0
1
001000
7
7
001000
0
- 1
001100
2
3
001110
1
0
001111
4
3
001111
0
- 1
000100
6
13
000110
0
2
000110
0
- 1
000111
2
4
000101
0
1
000010
5
4
000011
18
13
000001
27
34
aNumbers preceded by a " - " indicate animals that were not released following their last capture.
Male
period i
4
45
5
54
1
53
2
69
3
48
4
56
5
45
34
31
0 45
44
1
0
0
33
4
0
1
32
1
0
28
0
4 38
sex a n d time, b u t the t e m p o r a l v a r i a t i o n w a s p a r a l l e l (on a logit scale) for the t w o sexes (Fig. 17.2). S u r v i v a l for f e m a l e s w a s slightly h i g h e r t h a n t h a t for m a l e s , as is c o m m o n in s m a l l m a m m a l s t u d i e s . As n o t e d p r e v i o u s l y , the c o m p l e m e n t of s u r v i v a l in c a p t u r e recapture studies includes both death and permanent e m i g r a t i o n , a n d the h i g h e r m o v e m e n t rates of m a l e s of m o s t s m a l l m a m m a l s r e s u l t in l o w e r s u r v i v a l p r o b a bilities t h a n for females. C a p t u r e p r o b a b i l i t y w a s b e s t m o d e l e d u s i n g a single c a p t u r e p r o b a b i l i t y (]~ = 0.90, S E(~) = 0.020) t h a t w a s c o n s t a n t o v e r t i m e a n d the s a m e for b o t h sexes. M o d e l (q~t, P) also w a s r e a s o n a b l e (AAICc = 1.23). This m o d e l a s s u m e s e q u a l s u r v i v a l rates for b o t h sexes, an a s s u m p t i o n that does not seem unreasonable given the r e l a t i v e l y s m a l l difference b e t w e e n t h e sex-specific s u r v i v a l e s t i m a t e s of Table 17.7 a n d Fig. 17.2. P e r h a p s the s t r o n g e s t i n f e r e n c e e m e r g i n g f r o m the d a t a set inv o l v e s the s t r o n g e v i d e n c e of t e m p o r a l v a r i a t i o n in m o n t h l y s u r v i v a l probability. The m o d e l w i t h t h e s m a l l e s t AAICc a m o n g t h o s e w i t h s t a t i o n a r y s u r v i v a l w a s m o d e l (%, Ps,t), w i t h AAICc = 14.22. H o w e v e r , t h e large AAICc v a l u e i n d i c a t e s little s u p p o r t for this m o d e l a n d p r o v i d e s e v i d e n c e of t h e e x i s t e n c e of t e m p o r a l v a r i a t i o n in s u r v i v a l . T e m p o r a l v a r i a t i o n can be t e s t e d f o r m a l l y b y cont r a s t i n g m o d e l s t h a t are p a r a m e t e r i z e d similarly, except for the p r e s e n c e or a b s e n c e of t e m p o r a l v a r i a t i o n in s u r v i v a l . For e x a m p l e , w e can c o n s t r u c t a l i k e l i h o o d ratio test of m o d e l (%, Ps,t) v e r s u s the m o d e l (q~s+t, Ps,t) as DEV(%,
Ps,t) - DEV(q~s+t, Ps,t) = 23.3,
438 TABLE 17.7
Chapter 17 Estimating Survival, Movement, and Other State Transitions Parameter Estimates under the General Two-Sex CJS model (~s.t, Ps.t) for M e a d o w Voles Studied at Patuxent Wildlife Research Center, Laurel, Maryland, 1981 a
Survival probability
Capture probability Capture period
Sampling dates
Female A
Male
Female
~
/~; (SE[j0;I)
Male
A
A
/3; (SE[/~/I)
q~i (SE[~;I)
q~i (SE[q~i])
1
6/27-7/1
ub
wb
0.89 (0.052)
0.86 (0.052)
2
8/1--8/5
0.88 (0.055)
0.96 (0.039)
0.78 (0.066)
0.58 (0.066)
3
8/29-9/2
0.90 (0.057)
0.82 (0.071)
0.68 (0.066)
0.71 (0.072)
4
10/3-10/7
0.96 (0.037)
0.91 (0.059)
0.69 (0.069)
0.59 (0.069)
5
10/31-11/4
6
12/4-12/8
1.00 c b
0.83 (0.069) b
b
mb
mb
b
See data in Tables 17.5 and 17.6. b Parameter not estimable under CJS model. c Standard error not estimated.
a
where model deviance DEV = - 2 log[~(61y)] is based on the likelihood ~(01Y) for the model (and data) under consideration, and the values of Table 17.8 are used in the computation. Under the null hypothesis of no difference in the abilities of these two models to describe adequately the variation in the data, the above LRT statistic is distributed as a chi-square statistic with five degrees of freedom (the difference in number of parameters between the two models). The value 23.3 is very unlikely (P<0.01) under the null hypothesis, providing strong evidence of the existence for temporal variation in survival probabilities. Analyses of field data sometimes lead to estimates at the boundary of the parameter space (e.g., estimates of survival or capture probability of 1). For example, the P5 for females in Table 17.7 is 1. When such boundary estimates are obtained using numerical estimation algorithms such as MARK (White and Burnham, 1999), it is not possible to estimate an associated standard error (see 1~5for females in Table 17.7). In consequence, it is not obvious that such parameters should be counted among the parameters estimated in the model. Typically, the parameter counts provided by MARK do not include parameters with standard errors that cannot be estimated (these standard error estimates appear as 0 or approximately 0 in MARK output). Thus, the CJS model (~s.t, Ps,t) in Table 17.8 is listed as having 17 parameters, whereas in Section 17.1.2 we specified that the CJS model should have 2K - 3 parameters or, for the data in Tables 17.5 and 17.6, 18 parameters (with K = 6 periods, we should have 9 parameters estimated for each sex). In this case, although ]~s = I is a legitimate estimate, it is not included in MARK's parameter count.
17.2. M U L T I P L E - A G E M O D E L S In the previous section we considered models for populations in which individuals can be considered to be of a single age, in that the capture and survival probabilities are the same for all age classes. In this section we relax that assumption and allow both capture and survival probabilities to vary with age. In Section 17.2.2 we consider Pollock's (1981b) multipleage model, which generalizes the CJS model by allowing for both time- and age-varying capture and survival probabilities. In Section 17.2.3 we consider models for the situation in which newly captured individuals can be distinguished only as young and older animals, even though there is age specificity in capture and survival probabilities among the older animals. Because an accurate accounting of age is possible only for individuals first captured as young, this situation requires a different approach to estimation and testing. Finally, in Section 17.2.4 we account for the possibility that individuals younger than a minimum breeding age temporarily emigrate from the breeding area, returning only when they attain breeding status. The occurrence of such temporary emigration induces a particular form of heterogeneity in the population, which must be accounted for in the modeling and estimation of population parameters. Because of the frequent occurrence of age-specific survival and reproductive rates in animal populations, it is important to be able to accommodate age in the estimation of demographic parameters. As noted in the discussion of CJS model assumptions (Section 17.1.2), age can be viewed as a discrete, dynamic attribute, by means of which animals can be categorized
17.2. Multiple-Age Models TABLE 17.8 Model Selection Statistics for Different Models of Time- and Sex-Specific Variation in Capture and Survival Probabilities of Meadow Voles a
439
1.0
Males
08 Model
Parameters b
Deviance
AAIC c
t~
E
. m
09
(q~s+t, P)
7
74.9
0.00
(q~t, P)
6
78.2
1.23
(q~s+t, Ps)
8
74.2
1.33
(~s+t, Ps,t) (qh, Ps) (~s+t, Pt)
14
61.7
1.40
7
76.9
1.98
10
72.5
3.82
(q~s,t, P)
11
70.5
3.94
(q~s,t, Ps)
12
68.6
4.14
(q)t, Ps*t) (~s+t, Ps+t) (qot, Ps+t) (q~s,t, Ps,t) (qot, Pt) (q0s,t, Pt) (q~s,t, Ps+t) (q~s, Ps,t) (q~, Ps,t) (q~ Ps+t) (q~s, Pt) (qo, Ps+t) (~P, Pt)
14
64.9
4.60
11
71.4
4.77
10
73.8
5.12
17
59.1
5.25
9
76.2
5.40
14
68.4
8.17
15
66.7
8.50
9
85.0
14.22
8
87.9
15.05
7
94.6
19.67
6
97.1
20.13
6
97.6
20.67
5
100.4
21.44
(q~s, P)
3
108.2
25.15
(q~, p~)
4
106.6
25.62
(9o, p~)
3
109.8
26.74
(q~, p)
2
112.0
26.90
a At Patuxent Wildlife Research Center, 1981; see data in Tables 17.5 and 17.6. b Parameter numbers computed in program MARK (White and Burnham, 1999).
into age classes. Of course, the change in age with time is deterministic, so that if the interval between sampling periods coincides with the interval required to graduate to the next age class, then an animal in age class 2 in year i necessarily will be a member of age class 3 in year i+1, given that it survives. Multipleage models can be viewed as special cases of multistate models (see Section 17.3), which also include probabilistic changes in state variables, as with size a n d / o r geographically defined classes. 17.2.1. D a t a S t r u c t u r e a n d C o n s i d e r a t i o n s
We begin with a description of the data needed for a capture-recapture study with age specificity in survival a n d / o r capture probabilities. The structure of
w
0.6
n," ~9
0.4
09
0.2
o.0
,
,
,
,
,
1
2
3
4
5
Time Period F I G U R E 17.2 Estimated monthly survival probabilities and 95% confidence intervals from model (q~s+t, P) for male and female m e a d o w voles at Patuxent Wildlife Research Center, 1981.
these data is similar to those for single-age models (Section 17.1.1). Data again can be summarized in an X-matrix, with the rows corresponding to individuals captured during the study. As with the single-age case, each matrix element Xij assumes a value of I if the ith individual is caught on the jth sampling occasion, or 0 if the ith individual is not caught on the jth sampling occasion. The added feature here is that in addition to its capture history, the age of each animal at initial capture must be included in the data structure. If age is known at initial capture, then the deterministic nature of changes in age ensure knowledge of age class in all subsequent sampling periods. The ability to assign an age to an unmarked animal determines what kind of age-specific modeling is possible. In many vertebrates, it is possible only to distinguish young (first-year) animals from adults (>1 year). In some species, it is possible to distinguish first-year, second-year, and adult animals. If age specificity in a parameter structure is restricted to age classes that can be distinguished at the time of initial capture, then the models of the type developed by Pollock (1981b) and Stokes (1984) can be used (Section 17.2.2). In other situations, the investigator can only distinguish young from older (> 1 year) animals on capture, but nevertheless wishes to consider age specificity for ages 2, 3, 4, etc. In such a situation, only the animals marked as young will be of known age in any subsequent year. Estimation of age-specific parameters therefore relies on animals marked as young. Cohort models (Buckland, 1982; Loery et al., 1987; Pollock et al., 1990) were developed for use with such data (Section 17.2.3). An important consideration for age-specific models involves the need for correspondence between the time separating sample periods and the time required to
440
Chapter 17 Estimating Survival, Movement, and Other State Transitions
make the transition to the next age class. For example, if one is interested in variation in survival or capture probabilities among annual age classes, then sampling should be conducted (at least) annually. Under the most common sampling scheme, the time separating sample periods i and i + 1 should be the time required to move from age class v to class v + 1. The models to be discussed in this section are based on this common design feature. Inferences about age specificity are possible under other sampling designs, but parameterizations of associated models must correspond to the temporal sampling frame and are best considered on a case-by-case basis. A final consideration relevant to modeling agespecific data involves the ages exposed to sampling efforts. In many seabirds and passerines, both young, first-year (<1 year) birds and adult breeders are marked on the breeding grounds. However, prebreeders are not found on the breeding grounds and are not exposed to sampling efforts until their initial year of attempted breeding. In this situation, individuals of certain age classes (sometimes all individuals of particular age classes) cannot be sampled, even if sampling is conducted annually. Approaches to modeling data for such species and estimating prebreeding survival probability (Section 17.2.4) have been investigated by Rothery (1983), Nichols et al. (1990), and Clobert et al. (1994).
17.2.2. P o l l o c k ' s (1981b)
Multiple-Age Model 17.2.2.1. Data and Sampling Design The model developed by Pollock (1981b) (also see Stokes, 1984) assumes the existence of l + 1 age classes (0, 1, ..., l) that can be distinguished for newly caught (unmarked) animals, with age class l denoting all animals of at least age l. Typically, this model is used with a small number of young age classes and an adult class. Its most frequent use involves the case of l = 1, with young (v = 0) and adults (v = 1) as the distinguishable age classes (Pollock, 1981b; Stokes, 1984; Brownie et al., 1986). In the following development we sometimes retain the general situation of l + 1 age classes, though we frequently use the two-age model for illustrative purposes. Estimation under the Pollock (1981b) model is based on the numbers of animals in each age class exhibiting each of the observable capture histories (denoted by x (v) for capture history co and age v). Assuming a threeperiod study with two age classes, the observable capture histories u. l.~._ . "4"111, (0) "4"110, ~,(0) .4.~,(0) ~,(0) ~,(0) ~,(0) ~,(0) ~,,~ 1 0 1 I "r 1 0 0 , "~ 0 1 1 ! "~ 0 1 0 , .4. 0 0 1 i u l young animals and -r~.(1) Xll0, (1) ~.(1) (1) ~ . ( 1 ) ~.(1) -r x100~ -~011, -r
X(1) 001 for adults. Note that the superscript corresponds to the age at initial capture. For example, -r176 is the number of animals released as young in period 2 and recaptured (as adults) in period 3. Summaries of capture history data typically are of the type presented for single-age models in Table 17.1, except that there now are capture history data for multiple age classes. Negative numbers again are used to indicate the number of animals that exhibit a particular capture history but are not released back into the population following the final capture in the history. As with data from only adults, modeling can proceed directly from the capture-history data, or it can be based on summary statistics computed from the capture histories. For example, data can be summarized in a mij-array similar to that used for the singleage CJS model, where-mij{v) denotes the number of animals of age v released in period i that are next caught (or observed) in period j. Thus, ,,(0),,,13denotes animals that were released as young at period 1, not caught or seen at period 2, but caught or seen at period 3. Because of the deterministic nature of aging, we know that under a two-age model, the animal was an adult (v = 1) in periods 2 and 3. Releases in period i are denoted using the superscript notation, --i R(v). Table 17.9 shows a m!V)-array for a four-period capture-recapture study ] under a two-age sampling situation. An individual capture history can contribute to multiple statistics in the array, though animals marked as young, R! ~ can appear in only one ml~ recapture statistic, after which they are recorded as adult releases (and then recaptures) in subsequent periods (e.g., in period j they may become members of R~I)). For example, capture history x(O) 111 contributes to R~~ R(21), and R(31), as well as to mm) 12 and ,4,,(1) "~23 9Animals not released back into the population following capture (e.g., in period j) are not incorT h e m ~ ) - A r r a y Representation for the Data Resulting from a Four-Period Capture-Recapture Study a
T A B L E 17.9 .
.
.
.
.
Age at release v
.
Recapture period j
Releases in
Young
period i
2
3
4
17(0)
,.,(0) "'12
1.,.,(0) "'13
.,(0) "'14
~.(0) '1~23
..(0) "~24
R(201 R(30) Adult
17(1)
R(21) R(31)
.,(01 "'34 ,.,(1) "'12
,4.,(1) "q3
1.1.,(1) "q4
~(1) "~23
.,(1/ "~24 ~.~,(1) "'34
a For a population with two age classes, y o u n g (v = 0) and adults the n u m b e r of animals of age v released at time i, and mljv) denotes the m e m b e r s of R! v) next caught at time j. In the case of two age classes all recaptures -mij(v) are of adult animals. denotes (?d -- 1 ) . . . /~(~') i
17.2. Multiple-Age Models porated into the corresponding number of releases R(v)~,. (e.g., into the ,,j
17.2.2.2. Model Structure Parameters are defined in a manner similar to the single-age case, with the additional notation of a superscript denoting age: p!~)
q;lv)
Xi{v)
The probability that a marked animal of age v in the study population at sampling period i is captured or observed during period i. The probability that a marked animal of age v in the study population at sampling period i survives until period i + 1 (to age v + 1) and remains in the population (does not permanently emigrate). The probability that an animal of age v in the study population at sampling period i is not caught or observed again at any sampling period after period i. As in the case of singleage models, the-xi(~) parameters are written as -(v) parameters. In the functions of P i(v) and '.pi two-age case, for example, X}~ can be written
441
is that of the initial capture, on which the history is conditioned (i.e., the initial capture is not modeled). As with single-age models, if the animal is removed (not released) following the last capture, then the final • (v) term is simply removed from the capturehistory model. In a three-period study with two age classes, the probabilities associated with the different capture histories are given by the conditional multinomial distributions:
[ {
R~~
f (o)_(1)(1)--(1)
"~',.(0) ~llO{
X
q~~
- ~2~(1)" (1)-I~F3 ij
X
X~~
/ H ~ x ~~.(0) I.
.,(0) "~"~101
q:}~0)[1--P2-(1)ljq:}a(1)_p3(1)/
1
"e2 F3 J
r
}xl,
In x{i), f (')- (') (') (')
as X! 0) -- (1 -- q:}~0)) if_ q}10)(1
_
(1) , (1) P i+l}Xi+l.
(17.23)
The modeling of capture-history data proceeds in the same general manner as with the single-age CJS model. Consider capture history 011010 for animals marked in period 2. Modeling of this history is again conditional on the initial capture at period 2 and is dependent on the age of initial marking. In the twoage situation, we have
N
{
()(
,,(1).(1)11 ,.(1),.,(1)1 I, ~l"l /"2 Lx -- ~2 1'3 J l
_ (1)7 (1)- (1) q:}~l)[1 --P2 Jq:}2 P3
}x;1)1 (17.24)
x ; ~ ] ~ U(21)I
x(011}l
~X~o',~o7
~(1).(1)L X 1 -,~2 v3 j
/ j.
Pr(011010 I release at period 2 as young) = q:}(O)~,(1)q:)(31)(1 p (1))qV(41). (1). (1) 2 /"3 -/-'5 A5
and Pr(011010 ] release at period 2 as adult) = {P(21)P1(1)'(1)( ~3 3
-- P 4(1)'}q:}4(1)-P(1)X(1)5 9
Note that these probabilities differ only in the superscript of the initial survival probability. Animals initially released as young survive the interval following initial release (periods 2-3) with a survival parameter associated with young animals. But subsequent capture and survival probabilities correspond to adult animals, because the young animal in period 2 makes the transition to the adult class in the interval between periods 2 and 3. Note also that in the two-age situation, the capture probabilities p}0) for young animals never appear in capture-history models. The only period in which the animal is classified as young
The"Ix~(v)} notation simply refers to the set of capture histories for animals released at age v from any specific group or cohort of release. Note that u! v) denotes the new releases of age v at period i, and _,i/~(v)denotes all releases. For the first time period there are no previously marked animals, so R~v) = u~v). The above conditional multinomials are multiplied to form a product-multinomial distribution corresponding to this three-period, two-age study. As for the single-age CJS model, we also can develop a multinomial model for each row of the .m/j(v) -arra y. The expected values for the elements of the two-age ml~)-array for a four-period study are presented in Table 17.10. Of RIv) animals that are released at period i, let K
r~v)=
~, j=i+l
m,j!v.)
442
Chapter 17 Estimating Survival, Movement, and Other State Transitions TABLE 17.10 Expected N u m b e r s of Recaptures E(mij(v)IRi(v)) for the Data of Table 17.9"
Age at release v
Recapture period j
Releases in period i
2
R{0)
Young
3
~,(0) ,,(0).4(1) *'q ~1 /-'2
n(o) (0),1 /x1 q~l t l -
a(20)
4 (1)x (1)_ (1) P2 )q~2 P3
n(0) (0)_(~) /X2 q32 P3
R(30) R{1)
Adult
/;}(1) .,(1).(1) ""1 ~1 /"2
ln(1)~l(1),~tl /x
R(21)
p(1))~(1)p(1)
~(1) ,.(1)..,(1) *'2 't"2 F3
R(31)
(0) ~(0)1'1 __ .(1)% ,,(1)(1 __ .(1)~ ,,(1).(1) 1 '+'1 ~
/-'2 Jhu2 "~
R{0) (0),~
/-'3 J~3 F4
_(1)x (1)_(1)
2 q~2 kl -- P3 }q~3 P4 (0) ..(0)~.,(1) 3 ~3 V4 R(1) ~(1)(1 .(1)~ ,.(1)(1 .(1)~ ,.(1).(1) 1 't'1 F2 ,~'.1."2 ~,* /-'3 J~3 V4 R(1) (1)z-, .(1)~ ..(1)p(41) 2 q32 t l - F3 Jw3 R(1) ,~(1)_ (1) 3 bY3 P4 _
_
aUnder the structure of Pollock's (1981b) age-specific model for two age classes, v - 0, 1.
represent the number that are recaptured at any subsequent sampling period. Then the conditional multinomial distributions corresponding to rows I and 4 of Table 17.10 can be written as
pr(.,(o)~.(0).,(O)]R{O) ) -q2,
, ~ 1 3 , -,14
,~.{o),..{o),~.{o),[~{o)r~0)l,
L'q2
"'q3 ""q4" ['"1
[{
]
}_0 "'13
(1)1 (1)-
' ~P~~ 1 - P2 ]{P2 P(31)) ..(0)
q0(0)l-1
x
_
_
O)1 (1)rl
,,,(1)-I,,(1)._(1)/
P2 ]q~2 [ 1 --/-'3 -1"3 ]J4 t
r ~,{0} .(0);1 _
/
(1
iX1 )
and pr[.,(1) . ~ , ( 1 ) . ~ ( 1 ) ] R ~ I ) ]
17.2.2.3. M o d e l A s s u m p t i o n s
Model assumptions are very similar to those listed for the single-age CJS model in Section 17.1.2. Assumptions (1) and (2) in the single-age case are modified to restrict homogeneity of capture (p!V)) and survival q~i(v)~J probabilities to members of the same age class (v) at each sampling period (e.g., survival probability must be the same for all animals of age v but not for animals of different age classes). We also assume that the timing of sampling and age class transition are synchronized, such that an individual of age v in sample period i will be at age v + 1 in sample period i + 1. This temporal synchronization can be considered a design restriction, rather than an assumption. The age-specific models of Pollock (1981b), Stokes (1984), and Brownie et al. (1986) also assume that age is correctly assigned to each new animal that is encountered and marked. The discussion of assumptions in Section 17.1.2 for single-age models is relevant to multiple-age models as well.
" H 2 , " H 3 , "'14
17.2.2.4. E s t i m a t i o n |.~,(1)I~,(1)I..(1)--~[;(1)--"H2 " ' q 3 " ' q 4 " L--1 I
r ~1)] t @~I)p (1)
!
(17.25)
.!.,(1)
,,,(1) X
q~1)[1-
(1)~ (1)r~ P2 Jq~2 [ 1 -
- (1)-1 (1)- (1) P3 Jq~3 P4
r l~(1)_,,(1)-~l X~
1)
9
Thus, the probability distribution for the data in Table 17.9 for a four-period study with two age classes would be written as the product of the six conditional multinomials, representing adults and young released at each of the first three sample periods.
Parameter estimation under these multiple-age models is accomplished using maximum likelihood based on multinomial models of the capture history [Eq. (17.24)] or mlg-array [Eq. (17.25)] data. The product-multinomial models have the same general appearance as for the single-age CJS case [Eqs. (17.5) and (17.6)]. The only difference between the probability distributions for the single-age and multiple-age situaR ( V ) . (v)~ tions involves the multiple ages of releases (,~,i , ui , and parameters (q~!v), P i(vh, in the multiple-age models. Closed-form maximum likelihood estimators for q~!v) and p lv) are presented for the general model (q~v), p~v)) by Pollock (1981b), Stokes (1984), and Pollock et al. (1990). MLEs are computed numerically by software such as program MARK (White and Burn-
17.2. Multiple-Age Models ham, 1999). Approximately unbiased estimators are presented by Pollock and Mann (1983; also see Pollock et al., 1990). Capture probability can be estimated for sampling periods i = 2, ..., K - 1 for all ages v > 0, and survival probability can be estimated for periods i = 1..... K - 2 for all ages. As was the case for the single-age CJS model, only the product ,(~)lP(Kv) can be estimated for the final sampling interval. As noted in Section 17.1.2, the information needed to estimate capture probability for period i essentially comes from the marked animals known to be alive and in the population during sampling period i (see Manly and Parr, 1968). The capture probabilities P (v) i can be estimated by conditioning on all animals of age v in period i that were caught both before and after period i and hence known to be alive during i. The proportion of such animals actually captured at i provides an estimate of P (v) i 9 However, there is no subset of age 0 animals known to be alive because of previous and subsequent capture, because there can be no animals in younger age classes in previous time periods. Thus, the capture probability p I~ for the first age class cannot be estimated under this model for any sample period. An analogous explanation applies to the inability to estimate p~V) and p(Kv) for any age class; there are no marked animals of any age available before period 1, and none is caught following period K. Hence, no animals are known to be alive at periods 1 and K other than those actually caught. 17.2.2.5. A l t e r n a t i v e M o d e l i n g
The model of Pollock (1981b) and Stokes (1984) can be viewed as the multiple-age analog of the CJS model, in that both models permit time-specific variation in survival and capture parameters but do not deal with other sources of variation. In Sections 17.1.3 through 17.1.7, we discussed models that were either more general or more specific than the basic CJS model. Alternative models can be formulated with multiple ages as well. As the basic details of these models differ little from the single-age case, these models are mentioned only briefly here. Reduced-parameter models were presented for multiple age classes by Brownie et al. (1986) and Clobert et al. (1987) and are also described in Pollock et al. (1990) and Lebreton et al. (1992). As was the case with the CJS model, some of the initial reduced-parameter models considered for multiple-age data involve stationarity constraints on model parameters. For example, Brownie et al. (1986) considered temporal constraints for both survival (,~v) = ,(2v) . . . . = *~')--1) and capture (p (2v) = p (3v) . . . . p (K v)) probabilities. In the case of multiple ages, it is possible for parameters of
443
some age classes to be modeled as constant over time and those of other ages to be modeled as time varying. For example, model (,~0), ,(1), pll)) denotes a model with time-specific parameters for young survival and adult capture probability, but constant adult survival over time. Such a model is reasonable for many species, because it is not uncommon for young survival to exhibit substantial temporal variation whereas adult survival varies little (e.g., Gaillard et al., 1998). Another class of reduced-parameter model involves absence of age specificity. For example, the single-age CJS model represents a reduced-parameter version of the Pollock (1981b) multiple-age model in which ,~0) = ,!1) ..... ,!') a n d p l 1 ) = p!2) . . . . . p!'). Pollock (1981b), Pollock and Mann (1983), and Pollock et al. (1990) provide contingency table tests of the alternative hypothesis of full age specificity versus the null hypothesis of no age specificity of survival and capture probability parameters. In the two-age case, these tests simply reduce to a series of K - 1 2 • 2 contingency tables, one table for each period of release for which recaptures are possible: Recaptured after i
N o t recaptured after i
R! ~
r! ~
R! ~ - r! 0)
RI 1)
rl ')
R~1) - r! 1)
R e l e a s e d in i
It can be seen from Table 17.10 that the expected proportions .,,(v),~(v) m..l] /t~.l are identical for young and adult releases lI "~ ,i(0) *i(1) 9Intuitively, if the proportions of recaptures of two age classes are identical, then there is no reason to suspect age-specific differences in capture or survival probability parameters. The hypothesis of age specificity also can be tested using models with time-specific constraints. For example, an LRT of model (,(v), Pt) versus model (,, Pt) tests the null hypothesis of no age specificity in stationary survival against an alternative hypothesis of age-specific survival. Both models incorporate the assumption of nonstationary capture probabilities that do not vary by age (hence no superscript v). Time-specific covariates can be used in the modeling of age-related parameters in a manner similar to that for the single-age case (Section 17.1.4.). The use of additive models [e.g., Eq. (17.12)] for time and age are likely to be quite useful with age-specific data. Thus, agespecific probabilities for the different time periods may vary in parallel (on a logit scale) over time. Multiple groups (e.g., locations or sex classes; see Section 17.1.5) also can be modeled in the same manner as for singleage models. Models of capture-history dependence (Section 17.1.6) can be tailored for use with multiple-age data -
-
-
444
Chapter 17 Estimating Survival, Movement, and Other State Transitions
as well, although care must be taken in the process. Consider a model for trap response in survival probabilities (Brownie and Robson, 1983), requiring separate models for marked and unmarked animals at each time period and leading to different survival estimates depending on mark status. In multiple-age modeling, all members of the initial age class (v = 0) are unmarked. Thus, only one survival estimate is possible for this age class, which automatically includes any marking effect. Except for this natural restriction, capture-history dependence can be modeled with multiple-age data in a manner similar to that for the single-age case. Finally, individual covariates also are modeled as in the single-age case, although in multiple-age models it is possible to specify certain parameters for some age classes as functions of the selected covariates, though parameters for other age classes are held to be independent of the covariates.
by the sample period of release. In single-age models, for example, we might have different survival probabilities for sample period i for each cohort or group of releases (e.g., survival for animals released at i - 1 might differ from that of those released at i - 2). In this section we focus on cohorts of animals released at age 0. These models are most useful in situations in which the age of organisms can be distinguished only in terms of young (age 0) and older (age :> 0) individuals. The only way to know the specific age of an adult animal is for the animal to have been released in some previous period at age 0. Thus, the models in this section focus on age-0 or birth class cohorts, recognizing that individuals in a birth class age over time in a deterministic manner. In consequence, the age of any individual in the data base is known with certainty at each point in the time frame.
17.2.2.6. Model Selection, Estimator Robustness, and Model Assumptions
In many situations an investigator can distinguish young (age 0) from older (age > 0) animals only on capture, but nevertheless wishes to account for age specificity for ages 1, 2, 3, 4, etc. Only animals that were marked as young are of known age in any subsequent year, and thus the estimation of age-specific parameters must rely on animals marked as young. In what follows we again assume that the interval between sampling occasions coincides with the time period required for the animals to mature from one age class into the next. As before, capture-history data can be summarized as capture histories or as summary recapture statistics. The notation for capture-history data is the same as that for the Pollock (1981b) model (Section 17.2.2). With the cohort model, however, all animals used for modeling are marked as young (v = 0), so all capture-history statistics are superscripted with "(0)" to indicate this restriction. As an example, ~111~'(~denotes the number of animals released in period 1 at age 0 and caught again in period 2 (at age 1) and period 3 (at age 2). Ages at period 2 and 3 are known only because the animal was known to have been age 0 when marked at period 1. Negative numbers again indicate histories of animals not released back into the population following their last capture. The data also can be summarized in m!~)-array form, where v is the age of last release (Table 17.11). Thus, m(0) 24 indicates the number of animals released at age 0 in period 2 and next caught in period 4 (at age 2). If some of these animals are released again after recapture in period 4, they would be included in R~42). Of the number R~42)in this release group, the number next captured in period 5 is denoted by ...(2),,,459The four-
Goodness-of-fit tests for multiple-age models were developed by Brownie et al. (1986) and Pollock et al. (1990). The discussion of estimator robustness for single-age models presented in Section 17.1.9 is also relevant to multiple-age models. Effects of assumption violations frequently are similar for models with and without age effects. Pollock (1981b) and Nichols et al. (1992a) specifically considered the problem of tag loss in multiple-age models, finding the resulting biases to be predictable based on a knowledge of effects on single-age model estimators. As noted above, the age-specific models of Pollock (1981b), Stokes (1984), and Brownie et al. (1986) assume that age is correctly assigned to each new animal that is encountered and marked. If there really is age-specific variation in either survival or capture probability, then incorrect assignment of age will lead to increased heterogeneity within age classes. In addition to the usual problems caused by heterogeneous rates (see Sections 17.1.2 and 17.1.9), incorrect age determination can increase the apparent similarity between different ages and reduce the power of tests for age specificity.
17.2.3. Age-0 Cohort Models The term "cohort" is used in this book to indicate recognizable classes of individuals in a population-for example, age classes, size categories, geographic location, sex, and combinations of these and other classification factors. For the particular case of capturerecapture models, recognizable cohorts can be defined
17.2.3.1. Data and Sampling Design
17.2. Multiple-Age Models TABLE 17.11 The m~)-Array Representation for the Data Resulting from a Four-Period Capture-Recapture Study a Release cohort (i)
Releases in period i
1
R~O) R(21)
Recapture period j 2
3
..(0) ,,,12
.I(0) ,,.13 .1.,(1/ ,,,23
.~(0) ,,,14 ..,(1) ,,~24 .t.,(2)
.,(0) ,,,23
.I(0) ,,,24 .1.,(1) -s34 .~(0) ,,,34
R(32) 2
R(2~ R(31)
3
R(3~
a O n cohorts of a n i m a l s initially released at age 0. RI v) d e n o t e s the n u m b e r of a n i m a l s of age v released at time i, a n d -mij(v) d e n o t e s the m e m b e r s of RI ~) next c a u g h t at time j.
period study shown in Table 17.11 contains three cohorts of animals released at age 0, R~~ R(2~ and R(3~ . Under the cohort model, release groups of all other ages (e.g., R(21)) are animals from one of these initial cohorts of age 0 releases and thus are always taken from recaptures (there are no releases of unmarked animals of any age > 0). The number of releases thus typically declines monotonically with age, so that estimates of parameters associated with older animals tend to be less precise. Cohort models also have been used for unaged adults (Loery et al., 1987). In such cases, the superscript for both data and parameters corresponds to the number of time periods since initial capture rather than precisely to age. Such analyses are viewed as being relevant to relative, rather than exact, age.
17.2.3.2. Model Structure Models for cohort data were considered by Buckland (1980, 1982), Loery et al. (1987), and Pollock et al. (1990). Parameters are defined as in the Pollock (1981b) model (Section 17.2.2), with probabilities of capture (p!V)) and survival (q~!v)). Modeling is also similar to that for the Pollock (1981b) models, except that age is defined not only for classes recognizable at capture but for animals of all ages, given that they were initially caught at age v = 0. For example, the probability associated with capture history 011010 for individuals first captured as young in year 2 is Pr(011010 I release at period 2 as young) =
(3) q~(20)p(31)q~(31)(1--P4(2),}q~4(2)I,,(3) /"5 X5 "
Note that unlike the Pollock (1981b) model, every increase in sample period (subscript i) is accompanied by an increase in age (superscript v).
445
As suggested in the data summary of Table 17.11, the general cohort model can be viewed as a series of separate CJS models, one model for each cohort of age0 releases. The modeling of m!~)-array data under the cohort model is illustrated in Table 17.12. Each row of Table 17.12 follows a conditional multinomial distribution (conditional on releases, RlV)), and the probability distribution for the entire array is given by the product of these multinomials. The multinomial associated with each cohort of age-0 releases can be viewed as a separate CJS analysis. This is easily seen in Table 17.12, because the modeling for the three different age-0 release cohorts contain no shared parameters. Each parameter is indexed by both time and age, and specific time-age combinations are unique to particular age-0 release cohorts.
17.2.3.3. Model A s s u m p t i o n s The CJS assumptions about homogeneity of survival and capture probabilities are required for animals of a specific age at a specific time. The homogeneity assumption is much more likely to be met in the standard situation in which all releases are of young (age 0) animals. Occasionally cohort models are used with adults of u n k n o w n age, resulting in a situation for which data and parameter superscripts actually correspond to "time since initial marking" rather than to age (Loery et al., 1987). Obviously, homogeneity is less likely in this case. As with the age-specific model of Pollock (1981b), we also assume that the timing of sampling and age class transition are synchronized, such that an individual of age v in sample period i will be at age v + 1 in sample period i+1. As noted above, this temporal synchronization of sampling and aging may be considered a design restriction rather than an assumption.
17.2.3.4. Estimation Parameter estimation under cohort models is accomplished using maximum likelihood based on multinomial models of capture histories or the m(i~f)array (e.g., Tables 17.11 and 17.12). Estimation can be viewed as a series of CJS analyses, with each analysis based on a specific cohort of age-0 releases, R!~ i = 1, .... K - 1. In each analysis, and thus for each cohort, the initial capture probability p !0) cannot be estimated, and the final survival and capture probabilities can be estimated only as the products q~(K v~ v+l~ . The closed -lP (K form estimators of the CJS model can be used with cohort data (Buckland, 1980, 1982; Loery et al., 1987; Pollock et al., 1990). Any software that computes CJS estimates can be used one cohort at a time to produce estimates under
446
Chapter 17 Estimating Survival, Movement, and Other State Transitions TABLE 17.12
Expected N u m b e r s of R e c a p t u r e s L~mij "~" (v) I R (v) i ) for the Data of Table Recapture period j
Release cohort (i)
Releases in period i
2
1
R{~
R(O) ~(0).(1) 1 '+'1 I-'2
2
a(21~ R(32) R(2~ R(31~ R(3~
3 a
17.11a
3
4
(1) ,,(1)p(32) 2 ~-'2
R(O) ( 0 ) , t _(1)~ (1)~-, _ (2)~ (2)_ (3) 1 q01 ~1 -- P2 Jq02 kl -- P3 JqV3 P4 a ( 1 ) ,,(1)(1 _ (2), (2)_ (3) 2 ~ 2 ' ~ - P3 Jq~ P4
R ( 0 (0) ) (1)
R(0) (0),1 _(1), (1)_(2) 2 q02 ~,1 -- P3 )q~3 P4
,~(0) (0),., /~1 q~l t l
-
_ (1), (1)_ (2) P2 /q~ P3
R(2) (2)_ (3) 3 q~ P4 2 q~2 P3
R(1) (1)_ (2) 3 q~ P4
R(0) (0)_(1) 3 q~3 P4
Under the structure of an age-specific cohort model with all initial releases at age v = 0.
cohort models. Software such as SURGE (Lebreton et al., 1992) and MARK (White and Burnham, 1999) can compute estimates under cohort models.
17.2.3.5. Alternative Modeling The cohort model is quite general in that it permits different survival and capture parameters for each age-time combination occurring in a study. It often is of interest to consider reduced-parameter models in which parameters are constrained to be equal over time or age or both factors. The imposition of such constraints was considered by Buckland (1980, 1982) and Loery et al. (1987), and modern software (White, 1983; Lebreton et al., 1992; White a n d Burnham, 1999) permits direct estimation under reduced-parameter assumptions (see Pugesek et al., 1995; Nichols et al., 1997). One particularly interesting application involves the investigation of senescent declines in survival rates (Pugesek et al., 1995; Nichols et al., 1997). For example, Nichols et al. (1997) modeled age-specific survival probability as a linear-logistic function of age for certain age classes over which senescent decreases in survival were expected: q~Iv) =
exp(c~i + ~v) 1 + exp(oti + ~v)'
(17.26)
where v denotes age. The linear-logistic model was used for some age classes, whereas separate timea n d / o r age-specific survival parameters were established for other ages (those hypothesized to be unaffected by senescence). Note that the above survival model [Eq. (17.26)] permits time specificity, with the oLi parameters scaling the survival probability according to calendar year. This model can be viewed as an additive model permitting a form of parallelism of age-specific survival over time. Nichols et al. (1997) also fit models that assumed the same linear-logistic relationship regardless of year [i.e., Eq. (17.26) was modified so that O~i - - O~for all i].
The above description of the cohort model assumes that the release cohorts are of age 0, the sampling situation most frequently encountered. However, the requirement of known age does not necessarily restrict the cohorts to young animals of age 0. For example, data analyzed by Nichols et al. (1997) for the European pochard (Athya ferina) included young birds that could be aged as either age 0 or age 1. These two age classes were discernible in the field, so release cohorts of both age classes were included in the analyses. Releases of birds of age 1 thus included both previously marked and unmarked birds. This extension of the basic cohort model is easily handled and reinforces the general idea that the statistical modeling should be tailored to the details of field sampling methods. The various kinds of alternative modeling described in previous sections can be applied to cohort models as well. Time-specific and individual covariates can be used to model parameters, and forms of capturehistory dependence also can be introduced.
17.2.3.6. Model Selection, Estimator Robustness, and Model A s s u m p t i o n s As noted for multiple-age models, the discussion of model selection in Section 17.1.8 for single-age models is applicable to cohort models, as is the discussion of estimator robustness in Section 17.1.9. However, some important differences should be noted. In particular, age-specific survival estimators of cohort models tend to be much less robust to heterogeneous capture probabilities compared to the standard single-age CJS estimators. In a simulation study, Buckland (1982) found evidence that heterogeneous capture probabilities can produce substantial negative bias in survival estimates for the first survival probability (q~l~ and last few survival probabilities. Buckland (1982) noted that these biases can be misinterpreted as evidence of lower survival for young and old animals. On the other hand, Loery et al. (1987) used simulation to investigate ex-
17.2. Multiple-Age Models treme heterogeneity in capture probabilities and found evidence of substantial bias in the survival estimator for young animals (age 0) but little evidence of bias in the survival estimators for older age classes. That heterogeneous capture probabilities can produce substantial negative bias in the initial survival estimate of cohort models may seem surprising, in view of the relative robustness of standard CJS model estimates. The initial survival estimate under the cohort model can be written as q~!o) =/~,i(21)/ R~O),
(17.27)
where/~4(21) is the estimated number of marked animals (all are age 1) in the population at sampling period 2. As specified in Eq. (17.9), the numbers of marked animals in the CJS and related models are estimated by essentially equating two ratios: the proportion of marked animals caught at time i (R i) that are recaptured at some later time (r i) and the proportion of marked animals not caught (yet in the population) at time i ( M i - m i) that are recaptured at some later time (zi). With heterogeneous capture probabilities, the average capture probability is higher for the m 2 marked animals recaptured at time 2 than for the (M 2 - m 2) marked animals not recaptured at time 2. Under the CJS model, the number of animals released at time 2 consists of both marked and unmarked animals, a 2 = m 2 4- u 2. Because of the m 2 animals with relatively high capture probabilities, the R 2 animals are expected to have a somewhat higher average capture probability compared to the (M 2 - m 2) group. Thus, relatively more of the R 2 animals are likely to be recaptured than the (M 2 - m 2) animals, yielding a small negative bias in/~42, and hence in q~l. Under the cohort model, the releases at time 2 are the marked animals caught in two consecutive sampling periods. These marked animals are "undiluted" by the new unmarked animals that would be present in the CJS treatment. Thus, the larger negative bias of the initial survival estimate under the cohort model with heterogeneous capture probabilities is to be expected. On the other hand, subsequent survival estimates under the cohort model are of the form
~p(v) l~d(v+l)/l~Iv) i -- ~vli+1
(17.28)
Although the ~'~i]~(v) typically is negatively biased, the negative bias appears in both the numerator and denominator of Eq. (17.28), rather than only in the numerator, as with Eq. (17.27). Thus, subsequent survival estimates are affected by heterogeneous capture probabilities considerably less than is the initial survival estimate.
447
17.2.4. Age-Specific Breeding Models Not all ages may be exposed to sampling efforts under some capture-recapture sampling designs. Young of many colonial breeding bird species depart the breeding ground of origin following fledging and do not return to the breeding colony of origin until they are ready to breed. Thus, prebreeders of age > 0 can be viewed as temporary emigrants with 0 probability of being captured or observed prior to their first breeding attempt. There are two basic approaches to dealing with temporary emigration of this sort. One approach involves the use of the robust design, which will be covered in Chapter 19. The other is to use standard open-model capture-history data, but to develop a model structure that accommodates the absence of prebreeders. Here we focus on the latter approach. Rothery (1983) and Nichols et al. (1990) considered estimation in the situation in which all birds begin breeding at the same age. Clobert et al. (1990, 1994) considered the more general situation in which not all animals begin breeding at the same age. The latter approach is described here, recognizing that the models of Rothery (1983) and Nichols et al. (1990) represent a special case of the Clobert et al. (1994) approach. Although the general model has been used primarily for birds, it may be useful for a variety of other groups, including sea turtles, anadromous fish, some amphibians, and perhaps some marine mammals.
17.2.4.1. Data and Sampling Design Sampling can be viewed as a hybrid between the sampling approaches for the Pollock (1981b) and cohort models. Thus, animals are marked at age 0 on the breeding grounds, so that their ages are known throughout the study. However, adults are treated as in the models of Pollock (1981b), in the sense that age is considered no longer relevant once an animal begins breeding. Thus, releases each year can consist of both young animals (age 0) and adult breeders of u n k n o w n age. As with the previous age-specific models, the time separating successive sample periods must equal the time required to make the transition from one age class to the next. The discussion below will use "year" as the unit of time, as this corresponds to the situation most frequently encountered. Capture history data can be summarized using the notation of Pollock (1981b) (also see Section 17.2.2.). Thus, the number of animals exhibiting each capture history again carries a superscript denoting the age at initial capture and release. Young animals are again denoted as age v = 0; however, animals first caught as breeding adults will be indicated as v = k+, where age k is the first age at which animals can become
Chapter 17 Estimating Survival, Movement, and Other State Transitions
448
breeders (we assume that k is known). For example, assume that the first age of breeding is age 3. Then x(0) 100101 denotes the number of animals released as young (age 0) during the first year of the study that are subsequently caught in years 4 and 6. In this instance the capture histories of all animals released as young necessarily have two 0s following the initial release, corresponding to the fact that animals cannot breed until age 3 at the earliest. Animals with the above capture history attempted to breed and were captured in year 4, were not caught in year 5, but were caught again in year 6. The statistic -~010110~'(3+)denotes the number of animals first caught as adult breeders (hence at least 3 years old) in year 2, not caught in year 3, caught again in years 4 and 5, but not caught in year 6. These statistics are compiled as in Table 17.1, with negative numbers again indicating the number of animals not released following capture. (v) rr ay form The data can be summarized in .,,Lij-a (Table 17.13) in a manner similar to that for the Pollock (1981b) model. Note that all m~ ) = 0 for ages j - i, such that j - i < k (i.e., for all ages less than the age of first possible breeding). As was the case for the Pollock (1981b) model, animals released at age 0 can only appear in a single ml~) statistic. They are recaptured only as breeders, and breeders are released following capture as age k+. Of course, animals may appear in a number of releases (RI k+)) and recaptures (m!~+)) as adults.
(v)-Array Representat'on 1 f or the Data TABLE 17.13 The mij Resulting from a Four-Period Capture-Recapture Study on A n i m a l s Released in Two Age Classes a
Age at release Young
Releases in period i R~0)
2
3
4
5
"(0)b "'12
"(0) "'13 "(0)b "'23
1,1,1(0) "'14 "(0) "'24 .,(0)b "'34
"(0) "'15 .,(0) 1"25 "(0) "'35 .,(0)b "'45
(2+) m12
"(2+) -,13 ,..(2+) "'23
"(2+) '"14 .,(2+) "'24 (2+) m34
"(2+) ,,'15 .,(2+) "'25 .,(2+) ','35
R(3o) R(4o) A d u l t (breeder)
R(22+) R(32+) R(42+)
The following material is based loosely on the approach of Clobert et al. (1994). However, we have modified their approach to permit direct estimation and modeling of breeding probability parameters. Clobert et al. (1994) recognized that with a standard capturerecapture model parameterized by survival and capture probabilities, the information about nonbreeding and temporary emigration is incorporated into the capture probability estimates. They estimated age-specific breeding probabilities as functions of these capture probability estimates. We have applied a direct estimation approach to the model of Clobert et al. (1994), which we use here because we believe it is more easily understood and permits more flexible modeling. Define the following threshold ages, which are assumed to be known:
m
.,(2+) "'45
a A g e classes of release, y o u n g (v = 0) a n d a d u l t s (v = 2+). RI ~) d e n o t e s the n u m b e r of a n i m a l s of age v r e l e a s e d at time i, a n d (v) (v) m i. d e n o t e s the m e m b e r s of R i next c a u g h t at time j. All r e c a p t u r e s , t'v) m i j , are of a d u l t animals. P r e b r e e d e r s of age >0 are not e x p o s e d to s a m p l i n g , a n d the first possible age of b r e e d i n g is 2 y e a r s (hence m(0) i,i+1 = 0). b.. (0) = 0, i = 1, "", 4. mi,i + 1
The first age at which a young animal can breed, and thus the first age at which an animal marked as young (R!~ can be exposed to capture efforts and possibly recaptured. The age by which all animals are assumed to be breeding; i.e., the first age at which breeding probability is known to be 1 (or at asymptotic adult rate--see below).
Define the following model parameters:
p!k+~ ~!k+)
Recapture period j
R(20)
R~2+)
17.2.4.2. Model Structure
o•(V) i
The probability that a marked breeder (denoted as age k+) in the study population at sampling period i is captured or observed during period i. The probability that a marked animal of age - k (regardless of breeding status) survives until period i + 1 and remains in the population. The probability that a young animal (age 0) released at sampling period i survives until sampling period i + k (hence, until age k). The probability of breeding for an animal of age v at sampling period i that has not previously bred.
The above parameters differ from those discussed for previous models and therefore require some additional explanation. Capture probability is defined as conditional on being a breeder (hence, exposed to sampling efforts) so a corresponding parameter is needed only for breeders. Prebreeders of age > 0 are assumed to have capture probabilities equal to 0. The adult or breeder survival parameter q~!k+) is equivalent to the q~l) in Pollock's (1981b) model (see Section 17.2.2) in that it applies to all animals above a threshold age. The young survival parameter q~!0)
17.2. Multiple-Age Models differs from previous survival parameters in that it refers to a multiperiod time interval prior to breeding age. No inference can be drawn about time-specific survival probability of prebreeders before age k because the animals cannot be sampled during this interval [though inferences about average annual survival probability of young prebreeders can be obtained v i a (~}0))(1/k)]. Finally, we note that ~ is needed (and estimated) only for ages v = k, k + 1, ..., m - 1. Breeding probability before age k is known to be 0, and breeding probability after age m - 1 is assumed to be 1 (or at least is assumed to be at some asymptotic adult level). In addition, it is assumed that following the initial breeding attempt, an animal breeds with probability 1. To illustrate, consider a situation in which the first possible age of breeding is k = 2 and the age at which all animals breed is m = 4. Consider capture history 10011 for both young (age 0) and adult breeder (age 2 +) releases. The probability associated with this capture history for young animals can be modeled as Pr(10011 I release at period 1 as young) =
~1
IO~3 1
-- p(2+)
'+'3
],%+:,v42+:,d42+:,pF+:, (2
~4
/4
The survival term q4~ corresponds to the survival of the animals from release in year 1 until sampling in year 3. The large term in braces consists of the sum of two different products of probabilities, each product corresponding to a different sequence of events. In the first component of the sum, the animal breeds in the first available year (year 3) and age (age 2) but is not captured during that breeding season. The animal then survives and is captured during each of the next two breeding seasons. The breeding probability parameter is only needed in year 3, because once the animal breeds for the first time, breeding probability is 1 for subsequent years. In the second component of the sum, the animal does not begin breeding in year 3; hence no capture parameter is needed for this year (because prebreeders are not exposed to sampling efforts). The animal survives and then does breed in year 4 and is caught at that time. The animal then survives until year 5 and is caught again. If we dissect the sequence of ls and 0s that comprise the capture history, we see that the "0" in period 2 is required by the restriction that k = 2. The "0" in period 3 corresponds to an uncertain event, because there are two possibilities: the animal bred in period 3 but was not caught, or the
449
animal did not breed. The sum in the above probability statement reflects this uncertainty, with each side of the sum representing a scenario associated with capture history 10011. Given the "1" in period 4, there was no uncertainty associated with the modeling for the final "1." The probability associated with this same capture history for adults is modeled as Pr(10011 ] release at period 1 as adult) --- q~
(2+)1 , . ( 2 + ) r l p ( 3 2 + ) 1 , . . ( 2 + ) . ( 2 + ) ..(2+)p(52+) 1 -- P 2 iv2 L1 -J~3 /4 ~d4 9
This modeling is more straightforward, because there is only one possible sequence of events leading to (10011) and hence no uncertainty requiring a sum of two possibilities. All survival probabilities from period 1 through period 4 are required. Capture probabilities are used for the periods when the animal was captured, and the complements of capture probabilities are used for time periods of no capture. Thus, modeling for adults is identical to that for the standard CJS model. The probabilities associated with different capture histories again are specified by multinomial distributions that are conditional on the releases of previously unmarked animals of both ages [young (0) and breeding adults (2+)]. These product multinomials are of the same basic form as the multiple-age models of Pollock (1981b) and thus are similar to those shown in Eq. (17.24), with the exception that the cell probabilities for capture histories of animals released as young are different (more complicated) in the age-specific breeding model. The probability distribution for this model also can ,(v)-arra y summary statisbe described in terms of the ,lij tics of Table 17.13. Writing out the expected values or cell probabilities for the entire table can be tedious, so we illustrate with two examples. As with the modeling of capture history data, the probabilities for animals released as young are more complicated than those for animals released as adults. Assume the same age thresholds as above (k = 2, m = 4) and consider the animals released in period 1 as young and next seen in period 5 as breeders: Pr{'(~ ~,(0)} " ~ 1 5 *Xl
:
_
450
Chapter 17 Estimating Survival, Movement, and Other State Transitions
This probability includes the sum of three terms inside the braces. The first term corresponds to an animal that began breeding in the first possible year (3) and was simply not captured until year 5. The second term corresponds to the event of first breeding in year 4, and the third term reflects the event of first breeding in year 5. No breeding probability parameter is needed for period 5 even in this last component of the sum, because all animals are assumed to breed at age m = 4. The corresponding probability for adults released in period 1 and not recaptured until period 5 is given by
pr{2+ "~15
• q~22+)[1 - p~32+)]q~32+)
7. Every marked prebreeding animal of age v, where k ~ v < m, in sampling period i has the same probability ~i-(v)of initiating breeding and becoming a breeder in i. 8. Every marked animal that attempts to breed for the first time in period i breeds with probability 1, or with asymptotic adult breeding probability, at all sampling periods after i. 9. Marks are not lost or overlooked and are recorded correctly. 10. Sampling periods are instantaneous (in reality they are very short periods) and recaptured animals are released immediately. 11. Except for the temporary absences of prebreeders, all emigration from the sampled area is permanent. 12. The fate of each animal with respect to capture and survival probability is independent of the fate of other animals.
• [1-p~42+)]q~42+)p~52+). The above probability is again equivalent to the probability under the standard CJS model. The animal survives and is not recaptured for three consecutive sampling periods, survives, and finally is caught at period 5. 17.2.4.3. M o d e l
Assumptions
The age-specific breeding model described above uses standard open-model capture-recapture data and permits estimation of a kind of temporary emigration associated with prebreeding animals. The ability to estimate these temporary emigration probabilities (actually, their complements, the age-specific breeding probabilities) comes at the cost of some fairly restrictive assumptions about the modeled process. The following assumptions are required by the age-specific breeding probability model: 1. The age k of first possible breeding is known. 2. All animals become breeders by age m. 3. Every young animal released at age 0 in sampling period i has the same probability r ~ of survival until sampling p e r i o d / + k. 4. Every marked animal aged ~k in sampling period i, regardless of breeding status, has the same probability q01k+) of survival until sampling period i+1. 5. Every marked breeding animal present in the population at sampling period i has the same probability p~k+) of being recaptured or resighted. 6. Marked prebreeding animals of age > 0 are not exposed to sampling efforts and have a probability of 0 of being captured in any sampling period.
If the age of first breeding in assumption (1) is not known a priori, the investigator may simply set k equal to the first age at which animals are observed to return and breed. Assumption (2) is met when all animals of age m and greater breed with probability 1. As noted above, however, use of this model is appropriate even if all animals are not assumed to breed with probability 1, but instead breed with some asymptotic adult probability. In this case, the age-specific breeding probability estimates are no longer absolute probabilities but instead reflect age-specific breeding proportions expressed relative to those for adults. Although estimation under a particular model is conditional on a priori knowledge of m, it is possible to fit models incorporating different values of m, and to then use LRTs or AIC to select the most reasonable model and therefore the most reasonable value of m. Assumptions (3) and (4) deal with homogeneity of survival probability within an age class. Of particular importance is the assumption that survival probability of animals of age ~k is the same regardless of whether or not the animal has become a breeder. It does not appear that relaxation of this assumption is possible with single-state, open-model data. Assumption (5) of homogeneous capture probabilities is required in most open-population capturerecapture models. However, assumption (6) of capture probability of 0 for prebreeders is specific for this model. If prebreeders are available for sampling on the breeding grounds, then multistate modeling (Section 17.3) can be used, even if prebreeders (or even nonbreeding adults) have different capture probabilities than breeding adults (see Nichols et al., 1994; Cam et al., 1998).
17.2. Multiple-Age Models Assumption (7) deals with homogeneity of agespecific breeding probabilities for animals that have not bred previously. The discussion of heterogeneity of rate parameters for the CJS model is relevant to this parameter as well. Assumption (8) represents another strong hypothesis about the underlying process of accession to reproduction, an alternative to which might involve animals that, having previously bred only once, breed again with a lower probability than older, experienced breeders. 17.2.4.4. Estimation
Clobert et al. (1994) used maximum likelihood estimation to estimate survival and capture probability parameters for this underlying model. Estimates of breeding probabilities then were obtained as functions of capture probabilities of young animals (the complements of their ~Iv) values include the probability of not breeding and therefore of not being exposed to sampling efforts) and adult breeders (the complements of their ]~!v) values include only noncapture when all adults breed, but also include nonbreeding in the more general case of some adult nonbreeding). We have implemented this model using program SURVIV (White, 1983), because it permits flexible modeling of the agespecific breeding probabilities. The model also can be implemented as a multistate model in MARK (White and Burnham, 1999). As with the CJS and other multiple-age models, capture probabilities for the initial sampling period cannot be estimated, and the final capture and survival probabilities can only be estimated as products. Additional information on estimable parameters is provided by Clobert et al. (1994).
17.2.4.5. Alternative Modeling The discussion of modeling under the previously described age-specific models (Sections 17.2.2 and 17.2.3) is relevant to the age-specific breeding models. Time constraints can be placed on capture, survival, or breeding probability parameters. Because breeding probabilities are often difficult to estimate, it can be both useful and reasonable to assume these probabilities are constant over time. Under many reasonable scenarios, breeding probabilities are hypothesized to increase monotonically with age, so it is useful to model them as linear-logistic functions of age; e.g., as o~(V)_ i -
exp(~/i + ~v) 1 + exp(~/i + f~v)'
(17.29)
where "Yi is a parameter associated with year effects and [3 is the linear-logistic slope parameter (expectation
451
is that ~ > 0). Recall that o~ i - (v) is estimable for ages v = k , k + 1.... , m - l and is defined to be 0 for v < k andlforv>(m1). As noted above, it frequently is useful to construct several different models assuming different values of m. AIC or LRTs then can be used to help decide which model, and thus which value of m, is most appropriate. The above model structure is fairly general, and we note that constraints on this model can produce the models considered by Rothery (1983) and Nichols et al. (1990). In particular, they considered the case in which k = m. Animals released as young (age v = 0) in year i do not return to the breeding grounds until year i + k, but breeding probability at age k is 1 (or at least the same as that of adults). So oLlk-l) = 0 and ~Ik) = I by assumption, and a model in which all birds begin breeding at the same age is obtained simply by removing the breeding probability parameters from the general age-specific breeding model.
Example This example is based on a long-term study of roseate terns, Sterna dougallii, on Falkner Island, Connecticut, in Long Island Sound (e.g. Spendelow, 1982; Spendelow and Nichols, 1989). Falkner Island is a breeding colony site for the terns, and banding of both adults and chicks has occurred there every spring and summer since 1978 [for description of trapping methods and other logistical issues, see Spendelow (1982) and Spendelow and Nichols (1989)]. Because of some problems with band losses (Spendelow et al., 1994), color bands were replaced in 1988 with field-readable metal leg bands designed for reobservation. Data from 1988 to 1998 are used in this example. Very few birds return to the breeding colony as breeders until age 3 years, and some are not seen again until ages 6 and 7. Few nonbreeders are seen at the breeding colony, and only known breeders were used in this example analysis. Thus, the estimation problem is equivalent to one in which nonbreeders are completely absent from the colony. The data are summa(v) ~rr ay format in Table 17.14. Having rized in .m/j-~, previously been unavailable for marking, birds banded as chicks (designated as Y for young) were of course unmarked when captured. Note that the first nonzero entries in the array for young are for ,I(0) " q , i + 3 r reflecting the fact that very few birds breed before age 3. On the other hand, releases of adult birds could be divided into unmarked (not captured previously on Falkner Island) and marked birds. This categorization is useful for models that include certain types of capture-history dependence (Section 17.1.6). The estimation problem involved estimating the survival probabilities for young and breeding-age birds,
452
Chapter 17 Estimating Survival, Movement, and Other State Transitions T A B L E 17.14
T h e .,(v) .,/j - -array for Roseate Terns a Year of next e n c o u n t e r
Age
R e l e a s e s R (v) i
R e l e a s e year
M a r k status b
1988
U
206
1989
U
136
1990
U
142
1991
U
158
1992
U
103
1993
U
189
1994
U
186
1995
U
122
1996
U
82
1997
U
97
1989
90
91
92
93
94
95
96
97
98
0
0
17
9
3
0
0
0
0
0
0
0
9
6
3
0
0
0
0
0
0
9
7
3
2
0
0
0
0
3
0
2
0
0
0
0
17
4
4
1
0
0
26
14
7
0
0
15
8
0
0
10
0
0 0
1988
U
160
20
3
0
2
0
0
0
0
1989
U
136
57
78
9
1
0
1
1
0
0
0 0
1989
M
57
37
4
1
0
0
0
0
0
0
1990
U
108
73
7
0
2
0
0
0
0
1990
M
135
100
3
0
2
0
1
0
1
1991
U
72
37
4
3
1
0
0
0
1991
M
206
115
7
0
1
1
0
1
1992
U
31
16
1
0
0
0
0
1992
M
182
158
6
2
0
0
0
1993
U
72
28
1
0
0
0
1993
M
205
177
5
1
0
0
1994
U
29
11
4
1
0
1994
M
233
182
3
1
0
1995
U
21
7
2
2
1995
M
224
175
15
0
1996
U
39
9
1
1996
M
226
173
7
1997
U
23
5
1997
M
234
176
a Captured and released as both first-year young (Y) and adult breeders (A) and then recaptured in subsequent breeding seasons, 1988-1998, Falkner Island, Connecticut. b u denotes previously unmarked, and M denotes previously marked.
as well as age-specific breeding probabilities. We use the modeling approach of Clobert et al. (1994), which was implemented using a specific version of program SURVIV (White, 1983) developed by J. E. Hines for this purpose. The first possible age of breeding was taken as k = 3 years, and the age by which all birds were assumed to be breeding was taken to be m = 6. Two basic models were parameterized as described above, model (~o), ~3+), p~3+), Ot(3,4)) and model (~f0), q~f3+), P t(3+) ' Ot(3,4,5)) 9 Both models contain timespecific survival probabilities for young and adult
birds, as well as time-specific capture probabilities for adults. The superscripts on the c~ parameter indicate which age-specific breeding probabilities are not equal to either 0 or 1, and hence require estimation. For example, Ot(3'4'5) indicates that separate breeding probability parameters are estimated for ages 3, 4, and 5, with the assumptions that k = 3 and m - 6. The model with Ot(3'4) does not include estimation of ~(5), but instead assumes oL(5) -- 1 and thus m = 5. Both forms of trap dependence (transient parameterization and trap response in capture probabilities)
17.2. Multiple-Age Models were needed in the model to deal with permanent and temporary emigration from Falkner Island to other breeding colonies in the Long Island Sound system (see Spendelow et al., 1995). Some emigration is permanent, whereas some can be viewed as Markovian temporary emigration [see Chapter 19 and Kendall et al. (1997)] in that birds emigrate, stay at the new colony site for some time, and then return to Falkner. A transient parameterization of the models was implemented by rewriting survival for unmarked adults allowing for some proportion of transients [Eq. (17.13)]. Model notation for the transient parameterization includes T~3+), indicating time-specific proportion of transients among adults, e.g. model (q~o), q~3+), ,1.13+), pI3+), O~(3,4)).
In addition, a trap-response model (Section 17.1.6) was developed in which animals caught the previous sampling period had different capture probabilities than animals not caught the previous period. The model notation p~3+) and p(3+), for inclusion of trap response indicates that adult capture probability for animals caught the previous period is time specific, whereas the capture probability for animals not caught the previous period is constant over time. The latter constraint is required for parameter identifiability in this model, as for the simpler CJS-type models [see Section 17.1.6 as well as Sandland and Kirkwood (1981) and Pradel (1993)]. Models with age-specific breeding probabilities, transient response in adult survival probability, and trap response in capture probabilities fit the data well and thus could be used as the basis for estimation (Table 17.15). Both the lack of fit of the original models and the need for trap-dependent models could be at-
TABLE 17.15
453
tributed to the movement of birds among the breeding colonies of the study system. Although the best way to deal with movement is via multistate modeling with multiple sampling sites (Spendelow et al., 1995), such models have only recently been extended to deal with age-specific breeding probabilities. The two models with the smallest AICc values are designated as (q~0), ~3+), TI3+), p~3+), p(3+),, o~(V)),the distinction between them being that one contains parameters for age-specific breeding probability for ages 3-5 (denoted as 0~(3'4'5)), whereas the other contains parameters for age-specific breeding probability only for ages 3 and 4 (OL(3'4)), assuming that ~(5) = 1 (Table 17.15). The age-specific breeding probabilities were modeled as constant over time, as AICc values indicated that such models were preferable to models with time-specific c~v). The Pearson X2 goodness-of-fit statistics for both models indicated reasonable fit (Table 17.15). Parameter estimates for the models were consistent with biological knowledge and a priori predictions. Most of the annual survival probabilities for young were in the interval from 0.50 to 0.70, whereas most of the adult estimates were between 0.75 and 0.95 (Table 17.16). A severe hurricane occurred following the breeding season of 1991, so the 1991 survival probabilities were predicted to be low, especially for young birds [see Spendelow et al. (2002)]. This prediction clearly was supported by the estimates, as evidenced by the very low survival estimates for that year. It should be recalled that the survival estimates presented for young are actually estimates corresponding to the 3-year period following release as chicks, expressed as (0) ~1/3 annual rates ,~,q~i,i+3) . Thus, the survival probability
A A I C c V a l u e s a n d P e a r s o n X2 G o o d n e s s - o f - F i t Test S t a t i s t i c s a Goodness of fit b
Model
(q010), q~?+), Tt(3+), Pt~(3+),p(3+)', ~(3,4)) (q010), q0? +), T~3+), Pt'~(3+),p(3+)', 0((3,4,5)) (q0(0), q0~3+), Tt(3+), p~3+), p(3+)', O~(3,4)) (q0~0), q~3+), Tt(3+), /dt ..(3+) , Or (q010), q~(3+), Tt(3+), Pt~(3+)'pC3+)', O~(3.4)) (q0~0), q0~3+)' Pt*'(3+)'p(3+)', O~(3,4)) (q0~0), q0~3+), pl 3+), OL(3'4'5)) (q0~0), q0?+), Vt 4..(3+)' 0((3,4))
Numbers of parameters
j(2
df
0.00
24.8
25
40
2.15
24.8
24
0.41
32
33.14
68.9
33
<0.01
38
82.03
29.8
17
0.03
30
82.74
122.1
35
<0.01
30
159.38
225.3
33
<0.01
30
239.00
237.9
27
<0.01
29
240.23
239.1
28
<0.01
39
&AICc
0.47
a For several age-specific breeding probability models fit to the Falkner Island roseate tern capture--recapture data of Table 17.14. Model notation is specified in the text. b pearson chi-square goodness-of-fit test with cell pooling for low expected cell values computed by program SURVIV (White, 1983).
454
Chapter 17 Estimating Survival, Movement, and Other State Transitions TABLE 17.16
Parameter Estimates a
Year (i)
~j0)
(S~"E)
mi~'(3+)
(SE)'-"
~i~-(3§
(SE)I"
/37 +)
1988 1989 1990 1991 1992 1993 1994 1995 1996 1997
0.57 0.53 0.55 0.33 0.67 0.68 0.56 0.55 -.
(0.033) (0.039) (0.037) (0.049) (0.038) (0.030) (0.037) (0.059) -.
0.59 b
(0.050) (0.063) (0.039) (0.034) (0.022) (0.022) (0.027) (0.028) (0.032)
__b 0.90 0.96 >0.99 0.60 0.45 0.69 0.62 0.32 0.29
b (0.090) (0.068) (0.109) (0.099) (0.065) (0.118) (0.130) (0.088) (0.115)
0.61 0.82 0.86 0.88 0.94 0.96 0.94 0.86 0.91
.
0.78 0.83 0.62 0.92 0.91 0.81 0.89 0.84 .
(SE)"
__
(0.060) (0.038) (0.031) (0.027) (0.019) (0.016) (0.019) (0.029) (0.030)
a For annual survival probability of young (q~!0))and adults ( q 0 1 3 + ) ) , probability that an unmarked adult is a resident (1 - - T (i 3 + ) ~, , and adult capture probability for animals caught the previous period (pl 3+)) for roseate terns at Falkner Island, Connecticut. Time-invariant parameter estimates for breeding probability,,at age v (oL(v)) and adult capture probability for animals not caught the previous period: oL(w3) = 0.00, &(3)=0.77 (S"E=0.081),6L(4)=0.66 (SE=0.251), (x(v>4)=l.00,]~(3+)=0.60(SE=0.043). bInitial adult survival probability represents the product 1 _ hvi-(3+)-(3+)li , because the parameter cannot be separated the first year of marking.
of y o u n g birds b a n d e d in 1989 and 1990 also include the effects of 1991, although these effects are "diluted" by the geometric averaging with two other years. Of course the survival rate for y o u n g birds b a n d e d in 1991 also is "diluted" by survival during 1992-1993 and 1993-1994, so the extremely low estimate for 1991 suggests that first-year birds were affected very strongly by the hurricane. The estimated proportions of transients a m o n g unm a r k e d adults (§ 3+)) varied substantially, but we m a d e no a priori predictions about them. The capture probabilities for animals caught the previous period were predicted to be larger than those for animals not caught the previous period, u n d e r a hypothesis of Markovian t e m p o r a r y emigration (to other breeding colonies), and this prediction was supported by the results of the analysis (Table 17.16). In addition to survival, other parameters of interest included the age-specific breeding probabilities. Several models (not only those in Table 17.15) were fit using different values for the age (m) at which breeding probability for y o u n g nonbreeders was assumed to be 1.0 (or at least is the same as for adult breeders). The most appropriate model from this group was one for which ,m = 5. This model yielded estimates of about 0.77 (SE = 0.081) for the probability that a y o u n g bird of age 3 w o u l d breed at that age (c~(3)) and about 0.66 (SE = 0.251) for the probability that a bird that had not bred by age 4 would breed at that age (c~(4~) (Table 17.16). The estimate for age 4 was very imprecise. The probabilities of breeding for birds older than 4 years that had not previously bred were then 1.0.
17.3. M U L T I S T A T E M O D E L S In the previous section we considered statistical models of populations with age structure, as represented by age-specific parameters for survival and capture probabilities. We saw that the single-age CJS model in Section 17.1 can be modified to accommodate the more general parameterizations of age-structured models. We used the simple expedient of an additional index to characterize age in the s u m m a r y statistics and the survival and capture probabilities. We also can use this index for variables representing the size of the age cohorts in the population, as in Leslie matrix modeling (Leslie, 1945) (Section 8.4). The population then can be described at any point in time by the magnitudes of its age cohorts, with population dynamics given in terms of cohort transitions through time. In this section we generalize the CJS model yet again, to allow individuals in the population to be distributed across multiple sites or a m o n g multiple phenotypic states. Here the term multistate model refers to models that permit stochastic transitions a m o n g states, as distinct from, e.g., the deterministic transitions a m o n g age classes. In the case of multilocation studies, animals move stochastically from one discrete location to another, and a description of population status at any time requires the sampling of all locations. Similarly, some kinds of phenotypic development and change can be thought of as characterized by stochastic transition a m o n g discrete phenotypes. As with multiage models, a multistate model must contain parameters for the capture and survival probabilities of
17.3.
Multistate Models
individuals in different states, i.e., different locations or phenotypic states. However, a new feature for multistate models not shared with multiage models is a suite of transition probabilities. Of special interest to ecologists is the potential to estimate demographic variation with multistate models. For example, assume that survival probability between times i and i + 1 depends on whether an individual attempts to breed in time i. In some instances the relative survival rates of breeding and nonbreeding individuals is biologically interesting in its own right, or is key to the investigation of other biological factors. But even if we are not interested in this variation per se, it sometimes is useful to account for it, as a means of getting better estimates of other model parameters. Multistate generalizations of the CJS model were first considered by Arnason (1972, 1973), who derived moment estimators for the case where sampling was conducted at multiple locations. Hestbeck et al. (1991) used maximum likelihood estimation for a multiple location problem, and Nichols et al. (1992b) computed MLEs for a problem involving multiple phenotypic states (size classes). Brownie et al. (1993) and Schwarz et al. (1993a) considered a more general likelihood treatment of multistate models, which have seen wide use recently. In recent years, there has been increasing interest in problems involving both multiple locations and multiple phenotypic states. Location-specific survival rates and rates of movement among locations are critically important parameters in metapopulation models (Chapter 8). Many important questions in evolutionary ecology require comparisons of the rates of movement among locations of differing quality, or the investigation of movements as stochastic functions of covariates for individual animals (e.g., reproductive condition, social status) or study locations (e.g., distances between sites). Similarly, there is considerable interest in statespecific vital rates, both for testing hypotheses about phenotypic costs and tradeoffs (Nichols et al., 1994; Nichols and Kendall, 1995) and for state-specific population projection models (Nichols et al., 1992b) and optimization models (McNamara and Houston, 1996, 1997). In the following discussions we refer to the "state" of an animal at a particular sampling occasion, recognizing that the term "state" may refer either to phenotypic state or to the location of the animal when captured. Though conceptually different, these two situations are modeled in the same way; hence, no distinction between them will be made. As in Chapter 16, we use superscripts in the development below to denote geograPhic location (or phenotypic state). For example, RJ/might designate the number of individuals of
455
phenotype j that are captured and released on sampiing occasion i, and ~k might characterize the probability of survival for individuals that migrate from location j to location k during the period [i, i + 1]. We continue to restrict the use of parenthetical superscripts for population age structure.
17.3.1. Markovian Models Here we discuss models that represent state transitions as first order Markov processes, in the sense that the state of an animal at time i + 1 is stochastically determined as a function of its state at time i. The Arnason-Schwarz model can be viewed as the basic model in this class (Arnason, 1972, 1973; Brownie et al., 1993; Schwarz et al., 1993a).
17.3.1.1. D a t a Structure
The capture-history data for multistate modeling requires additional notation to identify the state of the animal at each capture. In multilocation studies, the location of each captured animal at each sampling occasion must be recorded. In studies of phenotypic state variables, each captured animal must be examined and recorded as to phenotype. Capture-history data for multistate modeling represent natural extensions of standard single-state capture histories. The capture history for an individual animal still is represented as a row vector, with the number of columns equal to the number of sampling occasions. A "0" in the capture history denotes failure to catch or observe the animal, just as in the single-state case. However, instead of indicating each capture with a "1," we now use a different number (any alphanumeric code can be used) to indicate capture/observation in each of the different states. For example, assume a twostate system with the states denoted as "1" and "2." Then a capture history 022010 describes an animal that was first caught at time 2 in state 2, was caught again in state 2 at time 3, was not seen at sampling time 4, and was caught in state 1 at time 5. The animal was not caught at time 6, the final sampling occasion of the study. As with other capture-recapture models, estimation for multistate models is based on the number of animals exhibiting each of the observable capture histories. For example, consider a three-period study with two states (1, 2). The number of animals exhibiting each capture history associated with releases in state 2 at time 1 are: X222, X221, X211, X212, X220, X210, X202, X201, and x200. Data summaries frequently include the capture history followed by the number of animals exhib-
456
Chapter 17 Estimating Survival, Movement, and Other State Transitions TABLE 17.17
Multistate m~S-Array Representation for Data Resulting from a Four-Period Capture-Recapture Study on a Population with Three States a Period of first recapture j (state of recaptures) 2
Release period i
State of release r
Number released
1
1
R~
1
2
R12
1
3
R3
2
1
R~
3
4
(1
2
3)
(1
2
3)
(1
2
3)
11 m12 21 m12 31 m12
12 m12 22 m12 32 m12
13 m12 23 m12 33 m12
11 m13 21 m13 31 m13 11 m23 21 m23 31 m23
12 m13 22 m13 32 m13 12 m23 22 m23 32 m23
13 m13 23 m13 33 m13 13 m23 23 m23 33 m23
11 m14 21 m14 31 m14 11 m24 21 m24 31 m24 11
12 m14 22 m14 32 m14 12 m24 22 m24 32 m24 12
13 m14 23 m14 33 m14 13 m24 23 m24 33 m24 13
2
2
R2
2
3
R3
3
1
R~
3
2
R32
m34
3
3
R3
m34
m34 21
31
a ~"
m34 22
m34 32
m34
m34 23
m34 33
m34
y
R i denotes the n u m b e r of animals in state r that are released at time i; m~7 denotes the n u m b e r of members of R i that are next captured
at time j and found to be in state s at that time.
iting that history, as with single-state models (Table 17.1). A " - " preceding the n u m b e r specifies the number of animals with a particular capture history that were not released (e.g., because of trap deaths) following the final capture. Capture history data also can be reduced to summary statistics and conveniently expressed as matrices (Brownie et al., 1993). Define the following statistics for a study in which animals are classified into three states:
i = 1, 2 , . . . , K -
1, and
m]j1 mij12 m~j3 mij =
mq21
mq22
mq23 ,
mij31
mq32
mij33
f o r i = 1 , 2 .... , K l a n d j = i + 1,...,K, w h e r e R r denotes the n u m b e r of marked animals in state r (r = 1, 2, 3) that are released at period i (i = 1, ..., K - 1), and m~s is the n u m b e r of marked animals captured in state s at period j that were last captured in state r at p e r i o d i ( i = 1.... , K 1;j = i + l , . . . , K ; r = 1, 2, 3; s = 1, 2, 3). As in our notation for single-state models, the releases R r include both new releases and releases of marked animals that are recaptured or resighted in period i. The data s u m m a r y table for multistate modeling
is similar to the single-state mq-array. Example data s u m m a r y tables are presented in Tables 17.17 (summary statistic notation) and 17.18 (matrix notation) for a study with four sampling occasions and three states.
17.3.1.2. Model Structure Multistate modeling uses parameters similar to those in single-state modeling, except they n o w are state specific, and the survival parameters incorporate the probabilities of transition from one state to another. The basic Arnason-Schwarz model (Arnason, 1972, 1973; Brownie et al., 1993; Schwarz et al., 1993a) can be viewed as the multistate analog of the CJS model and is closely related to certain models for poststratified band recovery data (Section 16.5.2). Define the following parameters: q~s is the probability of being alive and in state s at time i + 1, for a marked animal alive in state r at time i; pr is the probability that an animal alive in state r at time i is captured or observed. The probabilities q~rs reflect both survival and transition
TABLE 17.18 Matrix Representation of the Multistate m~jS-Array Data Summary in Table 17.17 a Recapture period j Release period i
Number released
2
3
m12
m13
m14 m
m23
m24
1
a I
2
a 2
3
a 3
4
m34
aRi and mij are the release and recapture matrices, respectively.
17.3.
Multistate Models
among states and are referred to below as survival-transition probabilities. The parameters ~prSand pr reflect the assumption that survival-transition between i and i + 1 and capture at i depend only on the state at time i and not on the state at i - 1 or previous periods. The model thus can be described as a firstorder Markov process (Chapter 10). The modeling of capture history data proceeds as for the CJS and multiple-age models. For example, assume a five-period study of a system with states 1 and 2, and consider the conditional probability associated with capture history 11020, given a release in state 1 at sampling period 1: Pr(11020 I release at period 1) _ r --
--
pl)12-2 12(-2 ) 22- ] qO3 P4 4- q) 1 P3 qO3 p24
x (1 - ~p22p2_ ~p21p1). This expression indicates that probabilities associated with multistate modeling are more complicated than with single-state modeling, the complications arising when "0s" are encountered in the capture history. For example, the modeling of events associated with the "0"at period 3 reflects uncertainty about the animal's state at that time; an animal exhibiting this capture history could have been either in state 1 or 2 at period 3. Both possibilities are accounted for in the sum in brackets. Thus, the animal could have remained in state 1 at time 3 and then moved from state 1 to state 2 at time 4, as reflected by the first component
Period 1
of the sum. Alternatively, the animal could have moved from state 1 at time 2 to state 2 at time 3 and then remained in state 2 at time 4, as reflected by the second component of the sum. Interpretation of the final "0" indicating that the animal was not captured at time 5 also is ambiguous, but the corresponding probability is simply written as the complement of the probabilities that the animal survived and was caught in either of the two states. The modeling of multistate data is illustrated in the tree diagram of Fig. 17.3. The probability associated with each capture history at the right of the figure is obtained by multiplying the probabilities corresponding to the specific path. In cases in which multiple paths lead to the same capture history, the probabilities associated with the different paths are summed to obtain the overall cell probability for the capture history. As with the other models in this chapter, the probabilities associated with the different capture histories follow multinomial distributions that are conditional on the releases of new (unmarked) animals in each state in each sampling period. The modeling of mij -array data is most easily accomplished by first writing the model parameters as matrices (Brownie et al., 1993). Consider the following parameter matrices for a three-state system:
s i
~ v
Alive in state2
Caught
11
1.~i ~
Not caught
10
~2 ~
Caught
12
Not caught
10
I-~
Dead or emigrated Alive in state 1
Caught and released in state 2 ,~,~,
~ r
Alive in state 2 ~ Dead or emigrated
--
Capture History
Period 2 Alive in state 1
Caught and released in state 1
457
10
1-~1 ~
Caught
21
Not caught
20
P~.,.......--..-'-l~ Caught Not caught
22 20 20
F I G U R E 17.3 Tree diagram of events and associated probabilities for animals released in period 1 from both locations of a two-location, two-period study under the Arnason-Schwarz multistate model.
458
Chapter 17 Estimating Survival, Movement, and Other State Transitions
i= 1,...,K-
1, and 1
p_i-
P2
i = 2, ..., K. Define D(pi) to be a diagonal matrix with diagonal elements equal to the elements p r of the parameter vector Pi. Let qr = 1 - pr, and define D(qi) to be the corresponding diagonal matrix. Using this notation, multinomial cell probabilities for the mq-array data are shown in Table 17.19. As with the other models in this chapter, the data follow a conditional product-multinomial distribution, conditional on the a i vectors of releases at each time period. The q~rsparameters reflect both survival and transition from one state to another. For many interesting biological questions, it is desirable to decompose survival and state transition in order to make separate inferences about these probabilities. Thus, define the following parameters: Sr is the probability that an animal in state r at sampling period i survives and remains in the study population until period i + 1; ~rs is the probability that an animal is in state s at sampling period i + 1, given that the animal was in state r at period i and that it survived until i + 1 and remained in the study population. Note that the definition of Sr requires the assumption that survival between two sampling periods depends only on the state at the first of the two periods. This assumption is appropriate for situations in which movement occurs near the end of the interval over which survival is estimated (e.g., see Hestbeck et al., 1991). Under this assumption, q~rs can be rewritten as q~ rS = .qrilsrs " i "r i 9
(17.30)
Under the parameterization in Eq. (17.30), there is a transition parameter ~rs corresponding to every sur-
TABLE 17.19 Matrix Representation of the Multinominal Cell Probabilities for the Multistate Data Summary in Table 17.18"
Release period
2
1
q~lD(p2)
2
3
Recapture period 3 ~lD(q2)q)2D(p3)
q~2D(p3)
4 ~plD(q2)~p2D(q3)q~3D(p4) ~2D(qB)@_.3D(p4)
q0BD(p4)
a~__i is the matrix of survival-transition parameters. D(pi) is the diagonal matrix with diagonal elements equal to the elements of the parameter vector Pi. D(qi) is the diagonal matrix with diagonal elements equal to the complements of pr (i.e., qr = 1 -- p~).
vival-transition parameter q~s in the original model. It thus may appear from Eq. (17.30) that additional parameters are required, because the parameterization also includes the state-specific survival probability S~. However, the transition probabilities ~rs are conditional on survival (and hence on being present in the study area). Because an animal that survives and does not emigrate must be somewhere in the study system, these transition probabilities must sum to 1: ~ ~rs = 1. S
This constraint permits us to write one transition probability as a function of the others. For example, we might parameterize transitions from state 1 of a twostate system a s ~ 2 and 1 - ~ 2 ( = ~;~1). Thus, the original parameterization and that shown in Eq. (17.30) both yield the same number of model parameters. Even when the assumption that survival between i and i + 1 depends only on the state at i is not true, it still may be useful to define the total survival rate q~r. = ~s q~s for animals in state r at time i (Hestbeck et al., 1991; Brownie et al., 1993; Schwarz et al., 1993a), so that transition probabilities ~rs can be defined by
,rs ,s ri s S rilirs l "t" l ~s
=
"qriljrs --i "r i
rs.
In this case the parameter ~rs is interpreted as a relative survival-transition probability (relative to the total survival rate across all transitions). This parameter ~rs can also be viewed as the probability that an animal in state r at time i and still alive at time i + 1 is in state s a t i + 1. 17.3.1.3. M o d e l
Assumptions
Assumptions underlying the Arnason-Schwarz multistate model are simply multistate analogs of the CJS assumptions. The first two CJS assumptions are modified as follows: (1) every marked animal present in state r at sampling period i has the same probability pr of being recaptured or resighted; and (2) every marked animal in state r immediately following the sampling in period i has the same probability of surviving until period i + 1 and moving to state s by period i + 1. Thus, instead of assuming homogeneity of all marked individuals in the population as in the CJS model, the multistate model permits variation among animals in the different states, but still assumes homo-
17.3.
Multistate Models
geneity within each state. The parameter definitions for this model also carry the assumption that state transition probabilities reflect a first-order Markov process, in the sense that state at time i + 1 is determined only by the state at time i (not states at periods prior to period i). The remaining assumptions (3-6) listed for the CJS model (Section 17.1.2) apply as well in the multistate case.
17.3.1.4. E s t i m a t i o n Maximum likelihood estimation can be carried out using the product multinomials for either (1) capture history data that are conditioned on unmarked releases in each state in each time period (program MARK) (White and Burnham, 1999), or (2) mq-array data that are conditioned on the time-specific release vectors R i (program MSSURVIV) (Hines, 1994). MARK uses the parameterization of Eq. (17.30), with the transition probability for remaining in the same state always written as a difference: ~r=
1 - 2~2 q~s. s#:/.
Program MSSURVIV (Hines, 1994) permits either the q~rs or the (S r, qjrs) parameterization. In addition, the user may specify the transition parameter q~rs to be defined by q~rs= 1
-
~_,
qjrs,.
S'~S
Although the (S r, ~rs) parameterization of Eq. (17.30) is incorporated directly into both MARK and MSSURVIV, it also is possible to estimate the parameters q~s directly in MSSURVIV, so that the estimates q~rs then can be used to estimate the separate survival and transition parameters of Eq. (17.30) by: q~r.= ~ q~s s
(17.31)
and ~S
^ FS ^ ?'. = q~i /q~;.
(17.32)
Of course, q~r. = ~r under the assumption that survival between i and i + 1 depends only on the state at time i (and not on the state at both i and i + 1). As expected, based on knowledge of the CJS model, the state-specific capture probabilities p~ for the initial sampling period cannot be estimated. Similarly, only the products q~-i P~ of the final state-specific capture and transition probabilities can be estimated. Closedform maximum likelihood estimators are not available under the Arnason-Schwarz model, although they are available for the more general JMV model of Brownie
459
et al. (1993) that permits capture probability for time i + 1 to depend on state at periods i and i + 1, p irS.+l . We do not anticipate that this modeling of capture probability will be needed in many situations, so we do not discuss the JMV model in this book. 17.3.1.5. Alternative Modeling Virtually all of the modeling that has been described for the single-age and multiple-age models of this chapter can be applied to multistate models. Reducedparameter models can be used to test hypotheses about state specificity as well as time specificity. For example, if a denotes the number of states in the system, then the hypothesis r = r . . . . . r specifies equal probabilities of transition to state s at time i + 1, for all states at time i. Assuming the survival-transition parameterization r --i'ri'qri[frsof Eq. (17.30), separate hypotheses about state specificity of survival and conditional transition probabilities can be addressed. For example, the dependence of survival probability on location (in multiple-site studies) or on physiological state frequently represents an interesting biological hypothesis (Hestbeck et al., 1991; Nichols et al., 1994; Nichols and Kendall, 1995; Spendelow et al., 1995; Cam et al., 1998). If the model testing and selection process provides no evidence of state-specific variation in parameters, then the Arnason-Schwarz and related models simply reduce to the single-state CJS model. Survival and transition probabilities can be modeled as functions of time-specific covariates (e.g., Nichols and Kendall, 1995), and multiple groups are also possible in the multistate context. Models incorporating capture-history dependence also can be constructed using multistate models (Leirs et al., 1997). State-specific parameters can be modeled as functions of individual covariates using program MARK (White and Burnham, 1999). Age can be incorporated into multistate models in either of two ways. One way is to simply incorporate age into the model structure as another state with probabilistic transitions and then constrain certain transitions to reflect the deterministic nature of the age transitions. For example, consider a model for two age classes, young and adult, with the interval between sampling periods corresponding to the time required for transition from young to adult. Assume that animals are trapped at two study areas, 1 and 2. An approach is simply to define four states, 1 = young in area 1, 2 = young in area 2, 3 = adult in area 1, and 4 = adult in area 2. Using the q~rs parameterization, the survival-transition probabilities to be estimated are ~]3, r q~23, r r opt3, ~33, and ~44, with the parameters q~l, q~22, r q~/21,q~31, ~41, ~p32, and q~42all
460
Chapter 17 Estimating Survival, Movement, and Other State Transitions
constrained to be 0. Under the alternative parameterization of Eq. (17.30), all state-specific survival probabilities (S~, S/2, S3, S 4) are estimated, as are the transition probabilities I]/~3, I]/i14 q/23 I]/24, I]/34, 1]/43 , I]/33, and q,i44. The remaining transition probabilities are constrained to be 0: ~J1 __ 1~i22 __ 1~2 __ ~21 _. ~31 __ ~tl __ 1~32 = i~t2 = 0. ,
,
A different but equivalent approach is to incorporate age in multistate models as with single-state models [e.g., as in Pollock (1981b)]. Under this approach, we simply define age-specific survival-transition probabilities corresponding only to possible transitions (rather than including parameters that must be constrained to 0). For example, in the above situation with two age classes and two locations, we would define the survival-transition parameters q)!0)ll q)10)12 q)~0)21 r r r q)11)21 and q)!1)22 where the superscript in parentheses denotes age (0 = young, 1 = adult) and the other two superscripts reflect the standard multistate notation of state (locations 1 and 2 in the example) at times i and i + 1. Multistate models provide a general computational framework for estimation under models with widely differing structures. For example, Lebreton et al. (1999) have shown that the combined recovery-recapture models of Burnham (1993) (also see Section 17.5) fit within the multistate model framework, with death treated as an absorbing state. Lebreton et al. (in review) also considered the age-specific breeding model of Clobert et al. (1994) (see Section 17.2.4) as a multistate problem, and then extended the single-site problem of Clobert et al. (1994) to a general multisite recruitment problem, again using multistate modeling. Finally, Conroy et al. (1999) applied multistate modeling to modeling sex-specific survival and capture probabilities when the sex of animals, though indeterminate on first capture as juveniles, is unambiguously determined on recapture (e.g, following molt of birds) and can be predicted from body measurements taken at first capture. 17.3.1.6. Model Selection, Estimator Robustness, and Model A s s u m p t i o n s
The general approach to model testing and selection presented in Section 17.1.8 is advocated for multistate models. The discussions of estimator robustness and model assumptions presented in Sections 17.1.9, 17.2.2, and 17.2.3 are relevant to multistate models. We are aware of no in-depth analysis of goodness-of-fit tests for multistate models, and this should be a research priority. The Pearson chi-square goodness-of-fit test based on a comparison of observed and expected multinomial cell frequencies (cell pooling algorithm
of program SURVIV) (White, 1983) sometimes rejects the null hypothesis of reasonable fit too frequently, based on computer simulation studies (J. E. Hines, personal communication). Similarly, the G2 goodnessof-fit statistic in programs SURVW (White, 1983) and MSSURVIV (Hines, 1994) can be used for multistate data, but the test statistic is not assured of following a chi-square distribution with real-world sample sizes and thus cannot be trusted. The use of multistate models is relatively new, and we know of little work exploring the consequences of deviations from underlying model assumptions. In an investigation of violations of assumptions underlying the (S~, ,l,~S)-parameterization of Eq. (17.30), Hestbeck (1995) found only small biases for survival and capture probability estimators (Sr, pr), but relatively large biases for movement or transition probability estimators (t~rs). As with goodness-of-fit statistics, estimator robustness is a topic requiring additional research for multistate models. Example
Here we provide a multistate analysis of capturerecapture data collected on meadow voles, Microtus pennsylvanicus, of mass >_22 g at a single experimental grid at Patuxent Wildlife Research Center, Laurel, Maryland, September 1991 through May 1993. A sampling grid consisted of a 7 • 15 rectangle of trapping stations with adjacent stations within each row or column separated by 7.6 m (25 ft). We divided the rectangular grid into two square strata. Stratum 1 was defined by trapping rows 1-7 and stratum 2 was defined by trapping rows 9-15. Because the overall grid contained seven columns of traps, both strata were squares with 7 • 7 trapping stations. The grid was one of four replicates that received a "fragmentation" treatment. During primary sampling periods 1-4, the grid was continuous, and between periods 4 and 5, it was plowed and disced [see Nichols and Coffman (1999) and Coffman et al. (2001) for illustration of grids]. This "fragmentation" occurred between the two grid halves and around the periphery of the grids, and involved a 7.6-m strip of bare, plowed ground. Sampling periods 5-11 were thus viewed as posttreatment. The two grid halves were the patches of interest, and movement occurred when an animal present on one half in one sampling period was present on the other half in the next sampling period. Primary sampling occurred every 8 weeks, and five secondary periods (consecutive days) were trapped during each primary period (designs with sampling at two different temporal scales are discussed in some detail in Chapter 19). The example reported here uses
17.3.
TABLE 17.20
Capture-Recapture
D a t a (Rir mrS) for A d u l t M e a d o w
Release Release date b
1
9-25-91
Release Mark Sex patch ( r ) s t a t u s M
F
11-20-91
M
F
M
3-11-92
M
5-6-92
M
7-1-92
M
4 2)
(1
5 2)
6 (1 2)
(1
7 2)
(1
8 2)
(1
9 2)
10 (1 2)
(1
11 2)
2
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
1
6
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
13
8
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
U
21
0
9
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
U
7
2
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
M
3
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
U
5
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
M
6
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
U
5
3
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
M
8
5
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
U
6
0
2
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
M
9
0
3
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
1
1
1
1
1
1
1
1
2
F
(1
20
2
6
2)
U
2
F
3
1
2
5
(1
13
2
F
2)
U
2
4
2
U
2
F
(1
1
2
1-15-92
Number released
2
2
3
Voles a
Period of next recapture j (patch of next recapture s)
period (i)
2
461
Multistate Models
1
0
U
1
0
M
5
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
U
6
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
M
7
0
2
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
U
5
2
0
M
9
4
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
U
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
M
7
1
4
1
0
0
0
0
0
0
0
0
0
0
0
0
0
U
10
3
1
1
0
0
0
0
0
0
0
0
0
0
0
M
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
U
13
0
6
0
0
0
0
0
0
0
0
0
0
0
0
M
5
0
2
0
0
0
0
0
0
0
0
0
0
0
0
U
8
1
0
0
0
0
0
0
0
0
0
0
0
0
0
M
8
4
0
0
0
0
0
0
0
0
0
0
0
0
0
U
11
0
4
0
0
0
0
0
0
0
0
0
0
0
0
M
9
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
U
3
2
0
1
M
3
3
0
0
0
0
0
0
0
0
0
0
0
U
8
0
6
0
2
0
0
0
0
0
0
0
0
M
10
0
8
0
0
0
0
0
0
0
0
0
0
U
5
3
0
1
0
0
0
0
0
0
0
0
0
M
7
7
0
0
0
0
0
0
0
0
0
0
0
U
5
0
3
0
0
0
0
0
0
0
0
0
0
M
8
0
3
0
2
0
0
0
0
0
0
0
0
U
10
5
0
1
0
0
0
0
0
0
0
M
6
6
0
0
0
0
0
0
0
0
0
U
12
0
6
0
0
0
0
0
0
0
0
M
14
1
6
0
1
0
0
0
0
0
0
U
15
11
0
0
0
0
0
0
0
0
0
(continues)
462
C h a p t e r 17
E s t i m a t i n g S u r v i v a l , M o v e m e n t , a n d O t h e r State T r a n s i t i o n s
(Continued)
T A B L E 17.20
Release period (i)
Period of next recapture j (patch of next recapture s) Release date b
Release Mark Sex patch ( r ) s t a t u s
2
7
9-2-92
M
1
2
F
1
2
8
10-21-92
M
1
2
F
1
2
9
12-14-92
M
1
2
F
1
2
10
2-10-93
M
1
2
F
1
2
Number released
2 (1
3 2)
(1
4 2)
(1
5 2)
(1
6 2)
(1
7 2)
(1
8 2)
(1
9 2)
(1
10 2)
(1
11 2)
(1
2)
M
10
6
0
0
0
0
0
0
0
0
0
U
18
0
15
1
0
0
0
0
0
0
0
M
6
0
4
0
0
0
0
0
0
0
0
U
5
5
0
0
0
0
0
0
0
M
13
8
0
0
0
0
0
0
0
U
6
0
5
0
0
0
0
0
0
M
14
0
10
0
0
0
0
0
0
U
0
0
0
0
0
0
0
0
0
M
18
12
1
1
0
0
0
0
0
U
5
0
4
0
0
0
0
0
0
M
20
0
11
0
0
0
0
0
0
U
18
8
0
4
0
0
0
M
14
6
0
5
0
0
0
U
8
0
6
0
1
0
0
M
16
0
10
0
2
0
0
0
11
0
0
0 0
U
16
2
M
12
9
0
0
0
0
U
23
0
9
0
2
0
1
M
16
0
12
0
1
0
0
U
8
3
1
0
0
M
14
8
0
0
0
U
6
0
1
0
0
M
16
0
9
0
0
U
5
1
0
0
0
M
12
5
0
0
0
U
4
0
2
0
0
M
21
0
10
0
0
U
9
0
0
M
20
1
0 1
U
4
0
M
14
0
1
U
10
0
0
M
17
2
0
U
11
0
1
M
15
0
3
In a two-patch system on grid 1A at Patuxent Wildlife Research Center, Laurel, Maryland, September 1991-April 1993. b Midpoint of 5-day trapping periods (periods 1-8) and first day of 2-day periods (periods 9-11).
a
only information at the scale of the primary sampling periods. Hence, the analyses are based on capturehistory data (Table 17.20) that simply record whether or not the animal was captured at least once in a primary
period (the number of days captured within a primary period is not used in these analyses). We fit approximately 100 different two-patch multistate models with transient parameterization to
17.3.
Multistate Models
these data. Because the grids were constructed in relatively uniform habitat, we expected no patch specificity of parameters (e.g., we expected ~ 2 = t~21 for patches 1 and 2), and AIC c and LRT results for initial models indicated that patch specificity indeed was not needed. The model with the lowest AICc value (considered the most appropriate model for parameter estimation) is denoted as (St, ~ - 4 , t~5-10, Pt, I" = 1.0). The 8-week survival probability was modeled as time specific, but these values are the same for males and females. The m o v e m e n t probabilities (~) were modeled with three parameters, one for males during the pretreatment periods (1-4), one for females during pretreatment (1-4), and one for both sexes during the posttreatrnent periods (5-10). Thus the notation includes an s superscript (denoting sex) for ~1-4 but not for ~5-10" Capture probability was modeled as time specific. The probability that an u n m a r k e d animal was a resident did not differ from 1.0 and hence was set equal to 1 (no transient parameter needed) for the modeling. Our approach to testing hypotheses about sources of variation in model parameters was to compute LRT statistics between the low-AIC model and neighboring models that incorporated hypotheses of specific interest (Table 17.21). We note that some might view our use of both AIC and LRT as an inappropriate mixture of p a r a d i g m s (model selection versus hypothesis testing). However, this study was designed to test a priori hypotheses, and our use of LRT and replication-based ANOVA testing (see below) is appropriate for this purpose. Our initial candidate models differed in details about sources of variation in nuisance parameters and about variation in parameters of interest that was not relevant to the biological hypotheses of interest. We thus used AIC in an ad hoc fashion to select a model
TABLE 17.21
Competing model
(S~, ~--4, ~5-10, Pt, ~"= 1) (S, t~_4, ~5-10, Pt, "r = 1) (St, ~--4, ~-10, Pt, "r = 1) (St, Us, Pt, I" = 1) (St, ~-4, ~5-10, P~, v = 1) (St, ~-4, ~5-10, P, v = 1) (St, ~--4, ~5-10, Pt, "rS) (St, ~-4, t~5-10, Pt, T1-4,"1"5-10) (St, ~--4, t~5-10, Pt, "r)
24
24 23
that represented a reasonable starting point for our hypothesis tests. We make no claims about the optimality of this approach, but believe it to be reasonable. Estimated capture probabilities varied substantially, ranging from 0.52 to 0.98 (Table 17.22). Estimated m o v e m e n t probability (probability of c h a n g i n g patches) during periods 1-4 was higher for males than for females, as expected based on general knowledge of sex-specific m o v e m e n t patterns (Table 17.22). Movement probability was substantially smaller following fragmentation, providing evidence that the treatment did indeed reduce vole movement. However, the decreased m o v e m e n t probabilities following fragmentation did not differ by sex (Tables 17.21 and 17.22). The average of the time-specific survival estimates for the pretreatment periods 1-4 was 0.57, whereas the average for the posttreatment periods 5-10 was 0.76, indicating substantially higher survival after fragmentation. This result is consistent with the idea that p e r m a n e n t emigration (part of the complement of survival) was reduced by fragmentation (also see Kozakiewicz, 1993). Although model-based tests are useful in d r a w i n g inferences about sources of variation in parameter values within grids, for the purpose of testing research hypotheses, we favor the use of replication-based tests using point estimates from the models in conjunction with procedures such as analysis of variance (ANOVA) (see Coffman, 1997; Nichols and Coffman, 1999; Coffm a n et al., 2001). For such modeling, capture-recapture models m a y contain more parameters than the lowAICc model, if that is required to attain statistical independence of estimates. For example, the selected model in our example did not contain sex-specific survival or capture probabilities, but we w a n t e d to use sex-specific
A A I C c a n d L i k e l i h o o d Ratio Test R e s u l t s for M e a d o w V o l e s a
Number of parameters
32 14 23 21 31 14
463
AAIC c
11.3 36.7 2.0 8.1 9.9 38.0 3.8 2.3 1.6
LR hypothesis H a
Ho Ha Ho Ha
Ho Ha Ha Ha
LRT results
X2
df
P
10.2 53.6 0.1 10.3 9.4 54.9 0.5 1.9 0.5
10 8 1 1 9 8 2 2 1
0.43 <0.01 0.73 <0.01 0.40 <0.01 0.79 0.38 0.50
Hypothesis tested
Sex-specific survival Time-specific survival Sex-specific posttreatment movement Period-specific movement Sex-specific capture probability Time-specific capture probability Sex-specific proportion transients Period-specific proportion transients No transients
a Captured on grid 1A, Patuxent Wildlife Research Center. Likelihood ratio tests (LRTs) correspond to tests between the low-AIC model (St, ~-4, ~5-10, Pt, ~"= 1) and selected neighboring models.
Chapter 17 Estimating Survival, Movement, and Other State Transitions
464
TABLE 17.22
P a r a m e t e r E s t i m a t e s for M e a d o w
Survival probability Dates sampled
S [SE(Si)]
Pi [SE(~i)]
1
09/23/91-09/27/91
0.61 (0.081)
mb
2
11/18/91-11/22/91
0.70 (0.102)
0.66 (0.095)
3
01/13/92-01/17/92
0.58 (0.101)
0.64 (0.099)
4
03/09/92-03/13/92
0.38 (0.059)
0.77 (0.110)
5
05/04/92-05/08/92
0.89 (0.061)
0.96 (0.040)
6
06/29/92-07/03/92
0.70 (0.050)
0.81 (0.070)
7
08/24/92-08/28/92
0.71 (0.051)
0.94 (0.034)
8
10/19/92-10/23/92
0.97 (0.086)
0.98 (0.023)
9
12/14/92-12/15/92
0.54 (0.096)
10
02/10/93-02/11/93
~
Movement probabilities
Capture probability
Sampling period
^
Voles a
^
0.116 (0.043) C
0.040 (0.024) a
0.015 (0.007) e
0.015 (0.007) e
Treatment
a
0.52 (0.064) 0.86 (0.130)
b
C a p t u r e d on grid 1A, Patuxent Wildlife Research Center, based on the model with the smallest value for Akaike's Information Criterion,
AICc (Si, ~ - - 4 , ~ 5 - 1 0 , Pi, ~" = 1 . 0 ) . b Not estimable. c Estimate applies to ~m, i = 1-4. Estimate applies to ~ , i = 1-4. e Estimate applies to ~m and ~ , i = 5-10.
survival estimates with ANOVA in order to d r a w inferences about sex-specific variation in survival. Capture probabilities are not of any particular interest, and hence there is no biological reason to produce separate estimates for the sexes; however, a shared capture probability parameter for the two sexes will likely induce a sampling correlation between survival rates of males and females. We thus used a model with sex-specific capture parameters in order to obtain i n d e p e n d e n t estimates of other parameters to use with ANOVA (Coffm a n et al., 2001).
17.3.2.
Memory
Models
In an initial application of multistate models to a large data set, the A r n a s o n - S c h w a r z multistate model exhibited poor fit to data for Canada geese wintering in the Atlantic Flyway (Hestbeck et al., 1991). Examination of the expected and observed multinomial cell frequencies showed a strong tendency for heterogeneity in the probability of moving from one wintering area to another, d e p e n d i n g on location in past winters (Hestbeck et al., 1991). This observation led Hestbeck et al. (1991) to consider a model in which m o v e m e n t between winters i and i + 1 d e p e n d e d not only on location at time i but also on location at time i - 1. This " m e m o r y model" can be viewed as a secondorder Markov process (Hestbeck et al., 1991 ). The devel-
o p m e n t below follows the p r i m a r y reference on these models (Brownie et al., 1993).
17.3.2.1. Data Structure The capture-history data needed for m e m o r y models are the same as used for the Markov models of Section 17.3.1. For example, x1021 denotes the n u m b e r of animals exhibiting capture-history 1021, indicating capture in state 1 at sampling time 1, no capture (or observation) at time 2, capture in state 2 at time 3, and capture again in state 1 at time 4. In the modeling described below, we follow the approach of Brownie et al. (1993) and focus on sufficient statistics s u m m a r i z e d in mij-array form. Extending the notation of Section 17.3.1, define the following statistics: R rsl,i is the n u m b e r of animals released in state s at time i, having been seen in state r at time i - 1 (i -- 2, ... K -- 1; r = 1,. .., a; s = 1, ..., a);-rrli_l,i, rst j is the n u m b e r of animals in release cohort R rsl,i that next are captured in state t at time j (i = 2, ..., K - 1; j = i + 1 .....
K;r=
1, ..., a; s = 1, ..., a).
The corresponding release and recapture matrices for a = 2 states are given below: 11
Fii-l,il 12
IRi-l,il ai-1, i
--
//221
/'
1~i-1,i1 22
LRi-l,iJ
17.3. i=2,...,K-
Multistate Models
1, and I.
111
. 112
121
211
. 122 m i - 1,i,j . 212 r
221
. 222
m i - 1,i,j
m i - 1,i,j
m i - 1,i,j mi_l,i, j =
rn i - 1,i,j
Imi-l,i,j |.
-
m i - 1,i,j
Lmi-l,i,j
i = 2, ..., K - 1; j = i + 1, .... K. The data s u m m a r y table for m e m o r y m o d e l s (Table 17.23) has the s a m e a p p e a r a n c e as those for M a r k o v i a n multistate m o d e l s (Table 17.18), with the exception that the release and recapture matrices are subscripted b y periods i - I a n d i, reflecting the need in m e m o r y m o d e l s for information about state in two successive periods (see Brownie et al., 1993). Note that the above statistics do not include all available data from a multistate c a p t u r e - r e c a p t u r e study. Instead they pertain only to animals caught in two consecutive s a m p l i n g periods, i - 1 a n d i. Certainly there are animals that are not caught at successive s a m p l i n g times, and these animals do not a p p e a r in the above statistics. For example, all n e w releases fall into this category of u n k n o w n state the previous time period. It is theoretically possible to m o d e l either capture-history data (Hestbeck et al., 1991) or sufficient statistics in a m a n n e r that m a k e s use of all data, but this a p p r o a c h presents c o m p u t a t i o n a l difficulties. In order to take a d v a n t a g e of algorithms similar to those used for M a r k o v i a n multistate models, we follow the conditional likelihood a p p r o a c h of Brownie et al. (1993) in the d e v e l o p m e n t below. 17.3.2.2. M o d e l
465
(1993), w e define the following parameters: q~i-l,i" rst is the probability of being present in state t at time i+1 for an animal that w a s in state r at time i - 1 and is in state s at time i; pS is the probability of capture at time i for an a n i m a l present in state s at time i. The surv i v a l - t r a n s i t i o n p a r a m e t e r n o w reflects the a s s u m p tion that s u r v i v a l - t r a n s i t i o n b e t w e e n i a n d i+1 d e p e n d on state at both i and i - 1. H o w e v e r , the capture probability still is indexed by the state at time i. We also define by q7 = 1 - pS the probability of noncapture. Model fitting is most easily a c c o m p l i s h e d using data s u m m a r i z e d in m i j - a r r a y form with m o d e l p a r a m e t e r s written as matrices (Brownie et al., 1993). C o n s i d e r the following p a r a m e t e r matrices for a = 2 states: I
111
_112
~i--l,i
~i-l,i
0
o
0
_121 q~i- 1,i
211 ~ i 0 1,i
_212 q~i - 1,i 0
[
q)i-l,i
/
0 0 _221
q~i- l,i
/
q~i-l,ij-222 /
0]
f o r / = 2, ..., K - l ,
P i - 1,i =
p2j
for i = 3, ..., K, and 0 D(qi)
Structure
=
0
0
q~3 ~ 0 i q2 0
o
As expected, the p a r a m e t e r i z a t i o n of m e m o r y m o d e l s is quite similar to that used for M a r k o v i a n multistate models, except that extra superscripts are used to indicate that s u r v i v a l - t r a n s i t i o n p a r a m e t e r s d e p e n d on an a n i m a l ' s state at successive times i - 1 a n d i, rather than s i m p l y at one time i. Following the conditional likelihood a p p r o a c h of Brownie et al.
] _122 / qvi- 1,i 0
o
q4
for i = 3 .... , K. These p a r a m e t e r matrices can be used to write the m u l t i n o m i a l cell probabilities (Table 17.24) of the conditional likelihood of Brownie et al. (1993) for the data s u m m a r y of Table 17.23.
TABLE 17.24 Matrix Representation of Multinomial Cell Probabilities a TABLE 17.23 Matrix Representation of the Multistate mi_l,i,j-Array rst Data Summary a Release period i
Number released
Recapture period j 3
4
5
3
4
5
2
q)12P3
~12D(~3)q)23P4
~P12D(q3)q)23D(q4)q)34P5
~23P4
q)23D(q4)q)34P5
3 2
R12
3
R23
4
R34
///123
/?/124
m125
m_234
/?/235
m345
For the conditional likelihood memory models of Brownie et are the release and recapture matrices, respectively. a
al. (1993). ai_l, i and mi_l,i, j
Recapture period j
Release period i
4
q034P5
~Under the memory model of Brownie et al. (1993) for the multistate data summary presented in Table 17.23. q)i-l,i is the matrix of survival-transition parameters, and Pi is the matrix of capture probabilities. The diagonal matrix D(qi) has diagonal elements equal to the complements of p~ (i.e., q~ - 1- pr).
466
Chapter 17 Estimating Survival, Movement, and Other State Transitions
17.3.2.3. Model Assumptions The m e m o r y model relaxes the assumptions of the Markovian multistate models in that all animals in state s at time i do not have the same probabilities of surviving until time i + 1 and moving to state t. Instead, survival-transition probabilities d e p e n d also on the state at time i - 1. This is t a n t a m o u n t to a further stratification of the animals in state s at time i. The assumptions of the m e m o r y model are the same as those for Markovian multistate models with the exception of assumption 2. U n d e r the m e m o r y model this assumption is revised to state that every m a r k e d animal in state s at time i that previously was in state r at time i - 1 has the same probability of surviving until time i + 1 and moving to state t by i + 1. 17.3.2.4. E s t i m a t i o n
Estimates can be obtained u n d e r the conditional likelihood of the m e m o r y model using p r o g r a m MSSURVIV (Brownie et al., 1993; Hines, 1994). The estimates are not MLEs, because they are based on only a limited subset of the available data, rather than the full set of all data. Because the estimation outlined above is conditional on releases of animals seen at both i - 1 and i, capture probabilities p~ for the first two sampling periods cannot be estimated. 17.3.2.5. A l t e r n a t i v e M o d e l i n g
Most of the modeling described previously in this chapter for single-state and Markovian multistate models can be applied to m e m o r y models. Reducedparameter models can be used to test for time specificity and for state specificity. In particular, the nature of the dependence of survival-transition probabilities on previous states can be explored using reduced-parameter models. These models can be used to address a n u m b e r of interesting biological hypotheses [e.g., the possible influence of breeding in one year on survival and reproductive probabilities of subsequent years; see Nichols et al. (1994), and Nichols and Kendall (1995)]. To illustrate the reduction of parameters, consider a model for which there are three possible states. The full m e m o r y model requires different survival-transition parameters for each possible history of states in periods i and i - 1. For example, animals alive in state I at time i require three different parameters associated with survival-transition to state 2 at time i + 1, q~112 _212 _312 i - l , i , ~Pi-l,i, and ~i-l,i, one for each possible state at i - 1. The Markovian A r n a s o n - S c h w a r z model can be obtained by imposing the constraint ~112 = i - 1,i
_212 = q~i- 1,i
_312 ~2 q~i- 1,i - -
9
A reduced-parameter model that is not as restrictive as the constrained Markovian model above can be de-
veloped by using parameter q~i-l,i _tst for the case where state at i - 1 is the same as state at i + 1, and ~ r , t ,1,i s,t where state at i - 1 differs from the state at i + 1 [this is model MV2 of Hestbeck et al. (1991)]. In the context of the above example, the m e m o r y model parameterization for the example transition reduces from three parameters to two: 212
i-l,i
and q)112
_312
i-l,i = q~i-l,i--
_r,2 1 2
~i-l~i"
9
The above parameterization is still viewed as a memory model, but reduces the n u m b e r of parameters relative to those needed for the full m e m o r y model. This particular model was found to be useful with wintering C a n a d a geese that move annually a m o n g different locations. The biological idea is that m o v e m e n t between i and i + 1 is more likely to occur if the location at i + 1 was also the location of the animal at i - 1 (so in some circumstances, animals m a y tend to move to locations with which they are k n o w n to be familiar). M e m o r y models have not seen m u c h use, so the full range of alternative modeling possibilities has not been explored. However, modeling with multiple groups is straightforward, and both survival-transition and capture probabilities can be modeled with time-specific covariates. Some forms of capture-history dependence m a y be difficult to model using the conditional likelihood approach of Brownie et al. (1993) because of the restriction to a subset of capture histories (those with at least some captures in consecutive sampling periods). The incorporation of age should also be possible, although this again increases the n u m b e r of parameters. Consider a two-age model in which y o u n g become adults after a single sampling period. Such a model now requires two parameters for every adult survivaltransition, one for animals that were y o u n g in state r at time i - 1, and another for animals that were adults in state r at i - 1. In this two-age situation, the survival of y o u n g would be treated in a Markovian manner, because there is no state history in i - 1 for y o u n g released at i (such y o u n g were not alive at i - 1). Although we believe it important to note that memory models can be generalized (e.g., by incorporation of age specificity), we recognize that very few data sets merit such modeling. Because of the m a n y parameters to be estimated, m e m o r y models are extremely data h u n g r y and thus will not be useful for most multistate data. However, there are instances in which such models are needed (Hestbeck et al., 1991, Brownie et al., 1993), and we can envision situations for which generalizations to include, e.g., age, could be useful.
17.3. Multistate Models TABLE 17.25
467
Capture-Resighting Data for Wintering Canada Geese a N u m b e r of resightings by year j (and stratum t)
Year of release i
Period (i)
Release cohort
Number released
(1
1985
2
R~
239
75
3
21
4
5
2
1
0
128
2
R~
53
11
5
2
1
0
1
0
1
32
2
R~
84
18
6
4
3
0
2
1
3
47
2
R~
610
15
145
8
68
7
22
1
16
328
1986
1987
1988
3
4 2)
(1
5 2)
(1
6 2)
(1
2)
Not seen again
3
R~
483
159
13
48
7
9
2
245
3
R~
132
10
25
3
6
3
3
82
3
R~
156
30
20
5
17
2
3
79
3
R~
856
15
245
4
80
1
39
469
4
R~
561
183
37
40
15
286
4
R~
197
25
42
11
14
105
4
R~
121
19
16
7
3
76
4
R~
975
19
270
15
75
596
5
R~
400
127
19
254
5
R~
172
21
32
119
5
R~
128
16
22
90
5
R~
817
24
225
568
a In the mid-Atlantic (stratum 1) and Chesapeake (stratum 2) regions of the Atlantic Flyway, 1984-1989 (periods 1-6). Release cohorts are defined by locations in years i and i - 1, and resightings are tabulated by year and location.
17.3.2.6. Model Selection, Estimator Robustness, and Model Assumptions As with Markovian multistate models, little is known about issues of model selection, goodness of fit, and estimator robustness. We advocate the general approach to model testing and selection presented in Section 17.1.8. It is important to keep in mind that the conditional likelihood approach uses only a subset of the data, and that the use of LRTs for model comparisons and AIC for model selection must be based on a single data set. For example, Markovian models can be fit to the entire set of multistate capture histories, whereas the fitting of memory models based on the conditional likelihood approach of Brownie et al. (1993) does not use all histories. In order for AIC and LR statistics involving memory and Markovian models to be meaningful, they must be applied to the same data, so the subset used in fitting the memory models should be used for the Markovian models as well. If, based on AIC or LRTs, it appears that the Markovian models are adequate for the data, these models can be refit using the full complement of data. The discussions of estimator robustness and model assumptions presented in Sections 17.1.9, 17.2.2, and 17.2.3 should all be relevant to memory models. As with other multistate models, the Pearson chi-square goodness-of-fit test encoded in programs SURVIV (White, 1983) and MSSURVIV (Hines, 1994) can be
used for multistate data, but the distribution of the test statistic has not been studied and may not have the assumed chi-square distribution under real-world sample sizes. Example The Canada goose example of Brownie et al. (1993) provides a good illustration of the need for memory models. The data are for geese marked annually with individually coded neckbands during the winter throughout the Atlantic Flyway (see Hestbeck et al., 1991). The wintering grounds were subdivided into three regions for reasons related both to biology and to goose management: from north to south, mid-Atlantic (denoted as state 1), Chesapeake (state 2), and the Carolinas (state 3). For application of the memory model, we focused on only the mid-Atlantic and Chesapeake regions. Data for K = 6 years (1984-1989) are summarized in Table 17.25 in the form needed for the conditional likelihood approach to fitting memory models. Both the memory model (38 parameters) and the Arnason-Schwarz Markovian model (22 parameters) were fit to these data. The logarithms of the conditional likelihoods for the two models were -193.4 and -315.5, respectively. If we compute the standard LRT using these conditional likelihoods, we obtain a statistic (X26 = 244.2, P K 0.01) that strongly suggests the need for the memory model (Brownie et al., 1993). The
468
Chapter 17 Estimating Survival, Movement, and Other State Transitions
Pearson goodness-of-fit statistic output by program MSSURVIV for the memory model was X21 = 45.1, P = 0.002. Thr fit was good except for a single cell in each of two cohorts, leading to reasonable faith in this model despite the fit statistic. Estimates of the survival-transition probabilities (Table 17.26) provide strong support for the parameterization of the memory model. For every year-transition combination, the survival-transition probability was higher when the location at i + 1 was the same as the location at i - 1. The relative difference was largest when location at i + I differed from location at i. Thus, movement from one wintering area in year i to a different wintering area in year i + I was substantially more likely if the wintering area in year i + 1 was also the wintering area used in year i - 1 (Table 17.26) (also see Hestbeck et al., 1991; Brownie et al., 1993). This example thus illustrates the utility of the memory models of Brownie et al. (1993). Because of the large number of parameters, the investigator should hope that these models are not necessary. However, for the Canada goose example, both the a priori biological reasoning and the data point to the need for this type of model structure.
17.4. REVERSE-TIME MODELS The models discussed above are conditioned on numbers of releases at a given sampling period, in that they describe recapture events over the remainder of the capture history. Pollock et al. (1974) noted that if
T A B L E 17.26
Year
Period i
1985
2
1986
1987
3
4
Annual
the capture-history data are considered in reverse time order, conditioning on animals caught in later time periods and observing their captures in earlier occasions, then inference can be made about the recruitment process. Specifically, "a backward process with recruitment and no mortality is statistically equivalent to a forward process with mortality and no recruitment" (Pollock et al., 1974). Additional uses of reverse-time capture-recapture modeling include Nichols et al. (1986a, 1998b), Pradel (1996), Pradel et al. (1997b), and Pradel and Lebreton (1999).
17.4.1. Single-State Models 17.4.1.1. Data Structure
In this section, attention is restricted to single-age, single-state models, the relevant capture-history data for which are the same as presented in Section 17.1.1. For example, under a three-period study the numbers of animals exhibiting each observable capture history are Xll 1, Xll 0, Xl01, Xl00, x011, x010, and x001, and capture history summaries such as those in Table 17.1 provide the data needed for reverse-time modeling. In concept it would be possible to define reverse-time summary statistics based on the numbers of final captures occurring at each occasion, and the numbers of last captures at particular times j that were most recently captured at previous time i (i.e., reverse-time mijarrays). However, we can avoid defining yet more summary statistics by simply developing reverse-time models from standard capture-history data.
Survival-Transition
Probabilities a
Estimated survival-transition probability if
Transition m a d e in i to i+1
Stratum at i + 1 = stratum at i - 1
Stratum at i + 1 ~ stratum at i + 1
11
0.57 (0.05)
0.38 (0.07)
12
0.22 (0.07)
0.04 (0.02)
21
0.34 (0.09)
0.05 (0.01)
22
0.66 (0.04)
0.21 (0.09)
11
0.58 (0.04)
0.31 (0.05)
12
0.37 (0.06)
0.06 (0.02)
21
0.13 (0.04)
0.03 (0.01)
22
0.67 (0.03)
0.42 (0.06)
11
0.54 (0.04)
0.27 (0.05)
12
0.31 (0.06)
0.14 (0.02)
21
0.22 (0.04)
0.03 (0.01)
22
0.59 (0.03)
0.51 (0.06)
a Estimated using the m e m o r y model with the Canada goose resighting data displayed in Table 17.24. Estimates were computed using MSSURVIV (Hines, 1994) and are grouped according to whether location at i+1 is the same as location i - 1.
17.4. Reverse-Time Models
17.4.1.2. Model Structure The modeling is similar to that described in Section 17.1.2, the time direction being the only real difference. We define two primary parameters: "Yiis the probability that an animal present just before time i was present in the sampled population just after sampling at time i - 1; p; is the probability that an animal present just after sampling at time i was captured at i. The parameters "Yi can be viewed as survival probabilities that extend backward in time. Pradel (1996) referred to them as seniority probabilities (probability that an animal present just before i is "old" in the sense that it was present just after i - 1). Note that both the seniority and capture probabilities are defined carefully relative to the time of sampling (just before or after sampling). The reason for this attention to timing concerns losses on capture. In forward-time capture-recapture modeling, losses on capture do not necessarily enter into the modeling (but see Chapter 18), but are handled instead via conditioning. For example, consider the standard mij-array data presented in Table 17.2. An animal that appeared in m12 (released in period I and then recaptured in period 2) that was not released following capture at time 2 would simply not appear in the release statistic R 2 for time 2. However, with a reverse-time model we know that the probability is 0 that an animal present at time i was a member of a group of animals seen at i - 1 but not released following i - 1. In addition to the above parameters, let ~i be the probability of not being seen previous to time i for an animal present immediately before i. The parameter ~i, which is analogous to Xi in standard-time modeling, satisfies the recursive relationship ~i =
(1 - ~ti) nu ~/i(1 -
was an old animal at 5, in the sense that it was a survivor from period 4. The probability associated with an animal at 5 being a survivor from 4 is ~/5. The animal was not captured at time 4 (associated probability is 1 - p ~). It was a survivor from 3 (~/4) and it was caught at 3 (p ~). It was a survivor from period 2 (~/3), when it was again caught (p ~). However, it was not seen before period 2 (~2). Note that unlike the case with standardtime modeling, the capture-history modeling does not differ depending on whether or not the animal was released following the final capture in the history. The reverse-time modeling only involves events occurring prior to this time. As with standard-time modeling, each capture history has an associated reverse-time probability. Conditional multinomial models can be developed by conditioning on the number of animals caught for the last time at each period, i, and then using the numbers of these animals exhibiting each capture history, in conjunction with the probabilities associated with each history. Consider a three-period study and define R; as the number of animals seen for the final time in period i. The probability distribution for the capturehistory data resulting from this small study can be written as the product Of the distributions resulting from each of the two groups of final captures: Pr({x~} I R;) Pr({x~} I R~) where
Pr({x~} ] R;) = II~ x~!
t
t
(3tgp 2"y2p 1 )
X ('Y3[1 --
Xlll
t
(3t3P2~2) x~
P2]~2P1)x101(zxO01%3
and
P;-1)~i-1
for i = 2, ..., K, with ~1 - 1. In order to have not been seen before time i, an animal either must not be a survivor from time i - 1 (this possibility occurs with probability I - ~/i), or it must be a survivor (with probability ~/i) that was not caught in i - 1 (with probability I - p ;-1) and not seen before i - 1 (with probability ~i-
469
1)"
N o w consider the reverse-time modeling of capturehistory data, using the same history that was used to illustrate standard-time modeling in Section 17.1.2. Again consider history 011010, indicating capture in periods 2, 3, and 5 of a six-period study. As noted, for reverse-time modeling we condition on the final capture and model prior events in the capture history: Pr(011010 ]last capture at period 5) = ~/5(1 - p'4)'y4p~/3P~2. Beginning with the final capture in period 5 and working backward, the animal exhibiting this history
Pr({x~ I R~}) = '.~ ----' ' x)-~---~(~2p xo 1 1 ~ 1 7 6 1 7 6
(17.33)
The history 100 is also possible in a three-period study but does not enter the probability modeling, because we condition on the period of final capture and model the events occurring prior to that capture (no events occur prior to period 1).
17.4.1.3. Assumptions The assumptions underlying reverse-time modeling are similar to those underlying standard-time models in Section 17.1.2. The homogeneity assumptions now apply to the seniority and capture probabilities rather than to survival and capture probabilities. All of the various ways of dealing with heterogeneity in the standard-time case (stratification, age-specific modeling, multistate modeling) should be relevant to reverse-
Chapter 17 Estimating Survival, Movement, and Other State Transitions
470
time modeling as well. Despite recognition of the potential utility of reverse-time modeling decades ago (Pollock et al., 1974), the approach has seen little use. In particular, there has been virtually no work on effects of assumption violations. 17.4.1.4. Estimation
Estimation of model parameters is accomplished using the method of maximum likelihood. In fact, estimates of ~i and p; can be obtained by simply reversing the time order of capture-history data and obtaining estimates using software developed for standard-time analyses (Pradel, 1996). Program MARK (White and Burnham, 1999) contains a routine to provide estimates under reverse-time modeling. Estimable parameters for a K-period study under the single-age, time-specific model (the reverse-time equivalent of the CJS model) include ~K, ~K-1 .... , ~3 and PK-1, PK-2 .... , P2. Parameters ~/2 a n d P l cannot be estimated separately under the general time-specific model, but the product "Y2Pl can be estimated. In the case of no losses on capture (all animals released following capture), the capture probabilities estimated from a single data set are the same for the CJS model in standard time as for the CJS analog in reverse time. As with estimation under the CJS model, closedform estimators exist for the capture and seniority parameters. We avoid additional notation here, but simply note that the reverse-time analogs of the summary statistics used for the CJS estimators [Eqs. (17.7)(17.10)] lead to the reverse-time estimators. !
!
17.4.1.5. I n t e r p r e t a t i o n
!
mainder of this section, we view ~-i a s a parameter and define it as
where N i denotes population size at time i. We wish to estimate the relative contributions of two demographic components to population growth between i and i + 1: (1) surviving animals from the population at time i (denote these as L i) and (2) new recruits (denote these as Bi). Recruits are animals not in the population at time i, that enter the population via reproduction a n d / o r immigration between times i and i + 1. We view population size Ni+I, number of survivors Li, and number of new recruits B i as random variables. Population size at time i + 1 can be written as the sum of these two demographic components:
of Estimates
The relevance of survival and movement probabilities to population dynamics should be evident from their use in the various models of Part II of this book. However, the relevance of reverse-time seniority parameters to population dynamics is not so obvious. The following discussion is based on Nichols et al. (2000a), and this reference should be consulted for more detail. The focus here is on the contributions of different demographic components to population growth rate. We consider the estimation of population size and growth rate in Chapters 18 and 19, and restrict attention here to the estimation of relative contributions to these quantities. Consider a single, open (birth, death, emigration, and immigration can occur between sample periods) animal population with no age specificity. It is reasonable to define ~ki a s either the realized population growth rate or a parameter reflecting the expected rate of population growth. For the development in the re-
(17.34)
N i + 1 = L i 4- B i.
P
We can view the components L i and B i of Ni+ 1 as following a binomial distribution conditional on Ni+ 1, which is governed by a parameter ~/i+1 denoting the probability that a member of Ni+ 1 is a survivor from the previous period (i.e., a member of Li). The ~/i are the seniority parameters estimated via reverse-time capture-recapture modeling. Conditional on Ni+l we can write the probability distribution of L i as (Ni+I)! ~/L~_I(1 Pr(Li ] Ni+l) -- (Li)!(Ni+ 1 - Li)!
-
-
,~i+l)Ni+l -Li
'
(17.35)
where B i -- N i + 1 -- Li" Based on Eqs. (17.34) and (17.35), we can decompose the expectation for population growth rate as follows: E(L i) + E(Bi) E(Ki) ~
(17.36)
E(Ni) ~[i+lNi+l 4-
(1
-
"Yi+l)Ni+l
E(Ni )
The ~/i+1 parameters provide useful information about the components of population growth. For example, if ~/i+1 = 0.5, then survivors from N i and new recruits can be regarded as equally important to population growth over the interval i to i + 1. If "~i+1 = 0 . 7 5 , then a member of N i + 1 is three times more likely to be a survivor from time i than to be a new recruit, and survival within the population can be viewed as three times more important to population growth over the interval i to i + 1. The parameters ~/i+1 can be used to draw inferences about the relative effect of hypothetical changes in the two demographic components on the population
17.4. Reverse-TimeModels growth between i and i + 1. For example, assume that recruitment had been reduced by proportion between i and i + 1, such that recruitment during this interval was really (1 - ot)B i. The proportional change in ~ki resulting from proportional change a in recruitment is given by a(1 - ~i+1). Thus, we can compute the new population growth rate resulting from a proportional reduction in recruitment of magnitude a by hi[1 -- c~(1 - ~/i+1)]. The "Yi+I parameters also are closely related to the concept of the elasticity of h i with respect to demographic components and their vital rates (Chapter 8). Consider the above decomposition of expected population growth rate h i into components associated with survivors from the previous period, E(Li), and new recruits, E(B i) [see Eq. (17.36)]. We can compute an analog of elasticity for the survivor component as
E(L i) c] h i h i 3E(L i)
3 log h i
0 log E(L i)
E(Li)E(Ni) 3{[E(Li) + E(Bi)]/E(Ni)} E(Ni+I) 3E(Li) E(Li)E(Ni) 1 E(Ni+I ) E(Ni ) z
(17.37)
E(Li) E(Ni+I) ~/i+1"
If one prefers to focus on vital rates (e.g., ~Pi)associated with demographic components rather than on the components, one can express the numbers of survivors and recruits as functions of the population at time i and an analog of elasticity again computed. Recognizing that survival rate from time i to i + 1 can be expressed as E(Li/N i) = q~i, we can approximate the elasticity of h i with respect to r as 0 log h i (9 log r
"Yi+I"
(17.38)
Although the parameters "Yiare analogous to elasticities derived from population projection matrices, these quantities differ in several respects. Perhaps the most obvious difference involves the asymptotic nature of elasticity measures derived from projection matrices, as contrasted with the applicability of the ~/iparameters to specific time intervals (i - I to i). We cannot comfortably use a specific "Yi t o characterize a population over a long period of time (although a mean of ~/i might be useful for such a purpose); nor can we expect an asymptotic elasticity value necessarily to be a useful
471
descriptor for population change over a specific interval. The asymptotic nature of the sensitivities and elasticities leads to uncertainty about their relevance to situations involving either transient dynamics that precede asymptotic behavior or simple temporal variation in vital rates and population growth. With respect to temporal variation in vital rates, sensitivity and elasticity analyses likely provide reasonable approximations for situations involving relatively small temporal variation, but not necessarily for populations inhabiting highly variable environments. Generally, we would expect the ~/i parameters to be more useful in retrospective analyses and true elasticities to be more useful for prospective analyses (see Caswell, 1997, 2000; Horvitz et al., 1997). Another difference between elasticities and ~/i involves geographic closure and the effect of movement on inferences about population change. Single-location population projection matrices typically reflect an asymmetry with respect to movement. The complements of survival rate estimates in capture-recapture (and several other methods) include both mortality and permanent emigration from the study area. These survival estimates often are combined in projection matrices with fecundity estimates that are based solely on components of reproductive rate (e.g., litter size, clutch size, brood size at fledging). Matrices composed of such estimates thus include movement in the complements of survival rates (the components of loss), but not in the fecundity parameters (the components of gain). One consequence of the movement asymmetry is that asymptotic rates of population increase computed from such matrices frequently are too small. Another consequence is an inability to draw inferences about the relative contribution of immigration to population growth (see Section 17.4.2). Questions involving asymptotic rates of increase could be addressed using multistratum projection models (e.g., Rogers, 1966; Schoen, 1988; Lebreton, 1996) incorporating movement. However, this approach would require that at least one of the modeled strata represent "the rest of the world" or all potential sources of immigrants other than the locations under detailed study. Though very important to asymptotic characteristics of the metapopulation system, the modeling of the dynamics of such "catch-all" strata is likely to be very difficult because of a lack of information. A minor methodological point regarding the comparison of the ~/i and sensitivities/elasticities involves statistical inference. Resampling approaches such as the jackknife and bootstrap (see Appendix F) can be used for inference on any demographic statistics computed from projection matrices (Caswell, 1989), includ-
472
Chapter 17 Estimating Survival, Movement, and Other State Transitions
ing sensitivity or elasticity values. The approach described here permits direct estimation of sampling variances and covariances associated with the '~i parameters.
17.4.1.6. Alternative Modeling As before, reduced-parameter models are of interest for reasons of parsimony and increased precision in parameter estimates. Use of multiple groups (e.g., sex) is certainly reasonable with reverse-time modeling. Use of covariates is also of interest in some situations-for example, when we expect relative contributions of new recruits and old survivors to vary as a function of environmental covariates (e.g., covariates associated with reproductive success and therefore recruitment). With respect to more general models, reverse-time models incorporating age specificity are discussed in Chapter 19. Multiple locations are discussed in the next section. Models with capture-history dependence have not been investigated for reverse-time questions, and this may not be possible; in any case they cannot be implemented by simply reversing standard-time approaches.
17.4.1.7. Model Selection, Estimator Robustness, and Model Assumptions Most of the discussion presented for the CJS model in Sections 17.1.8 and 17.1.9 should be relevant to the single-age, reverse-time modeling presented here. Certainly, the approach to model selection using AIC will be the same. Regarding goodness-of-fit testing, Pearson chi-square tests comparing observed values and their expectations should perform in a manner similar to that for forward-time modeling. We know of no investigations into the applicability of CJS goodnessof-fit tests (Pollock et al., 1985a; Burnham et al., 1987) to reverse-time modeling. In the case of no losses on capture, it would seem that the temporal symmetry of the modeling should result in tests being equally applicable to standard-time and reverse-time modeling. Estimator robustness for reverse-time modeling deserves serious consideration, in part because the robustness of the CJS survival rate estimator (Carothers, 1973, 1979; Nichols and Pollock, 1983b) may not extend to its reverse-time analog, ~i. To illustrate the problem, consider permanent trap response, in which an animal experiences one capture probability before its first capture and a different capture probability after the initial capture. The initial experience with the trap produces either a trap-happy (higher capture probability following first capture) or trap-shy (lower capture probability following first capture) response. Trap response pro-
duces no bias in CJS survival estimates, because these estimates are based only on marked animals and hence only on animals that have been caught at least once (Nichols et al., 1984b). On the other hand, trap response should result in biased estimates of the reverse-time seniority parameters. An intuitive argument for the influence of trap response can be developed by first recognizing that capture probability is estimated with information from marked animals, regardless of whether a reverse-time or standard-time approach is used. An estimation method for capture probability in period i is to condition on animals caught before and after i, and hence known to be alive at i. The number of these marked animals that actually are captured in i provides an estimate of capture probability (e.g., Manly and Parr, 1968; Skalski et al., 1993; Smith et al., 1994). This estimate is clearly based on, and applicable to, marked animals. But problems can arise when one attempts to use this approach in reverse-time estimation. For example, an estimate of the number of marked animals in the population just following period i - 1 includes both previously marked animals and animals just marked at time i - 1. In the case of trap response, these two groups of animals have different capture probabilities, yet our estimation approach applies the capture probability appropriate for previously marked animals to both groups. In the case of trap-happy response, the estimated capture probability is too large for newly marked animals, resulting in seniority estimates that are negatively biased. A trap-shy response in capture probabilities produces a positive bias in seniority estimates. This intuitive argument is no substitute for a rigorous treatment of reverse-time estimator robustness and is simply provided as a warning that robustness arguments based on a standard-time model may not be applicable to reverse-time modeling. Heterogeneous capture probabilities also yield biased estimates of seniority parameters. An in-depth investigation of the robustness of seniority parameter estimates to assumption violations is warranted.
17.4.2. Multistate Models The use of reverse-time modeling in multistate systems should be useful for addressing several kinds of estimation questions. For example, consider a metapopulation with several patches under study. It often is of interest to identify the relative contributions of animals from different patches to growth of a particular patch or of the entire metapopulation system. Use of reverse-time multistate modeling provides one approach to addressing such problems. Reverse-time
17.4. Reverse-Time Models Capture History
473 Time i+1
Time i
Caught in location 2 ~'P~ /
Present at location 2
Not caught~ ' ~ ' J
~.70
Caught in 7,~..,..,,,.,,..~i ~ location 1 _ _ ~'P~~ Not caught,
~' 11 Present at ~ location 1
, Caught in location 1
~ ~~
~ ' J Not p r e s e n t J in study locations
FIGURE 17.4 Tree diagram of events and associated capture probabilities for an animal caught in period i + 1 at location 1 of a two-location study using reverse-time multistate modeling.
modeling with multiple physiological and behavioral states also can be of biological interest. Finally, we note that use of a reverse-time approach with age-specific models requires the use of multistate modeling (Nichols et al., 2000a). If the initial age class is to be included in such analyses (contributions to the population at time i of young at time i - 1 are frequently of interest), then additional information is required (specifically, a capture probability for young animals). This extra information can be supplied by a robust sampling design that uses both closed and open population capture-recapture models and will be presented in Section 19.5.2. The following material deals with single-age multistate modeling. 17.4.2.1. D a t a Structure
The data structure required for reverse-time multistate modeling has been described in Section 17.3.1 on Markovian multistate models. We focus here on the numbers of animals showing various capture histories, where the "state" of an individual is recorded at each capture. For example, assume a study of a twolocation system and consider the capture histories x1020 and x2201. The first statistic reflects the number of animals that were caught in location I at the first sampling occasion, were next caught in location 2 at time 3, and were not caught at times 2 or 4. The second statistic provides the number of animals that were caught in location 2 at sampling times 1 and 2, were not caught at time 3, but were caught in location 1 at time 4. As with single-state reverse-time modeling, it is possible to develop reverse-time mq-array summary statistics,
but here we avoid the introduction of new notation by focusing on capture-history statistics. 17.4.2.2. M o d e l Structure
Multistate modeling in reverse time is very similar to that described in Section 17.3.1. Define the following parameters: ~/rsis the probability that an animal present in state r just before time i was present in state s just yv after sampling in period i - 1; P i is the probability that an animal present in state r just after sampling at time i was captured at i. The yrs are simply the multistate analogs of the seniority parameters % as described in Section 17.4.1. A tree diagram illustrating the events and associated probabilities for reverse-time multistate modeling is presented in Fig. 17.4. The use of the above parameters for modeling multistate capture history data in reverse time can be illustrated with capture history 2201 for a two-state system. As with the single-state reverse-time modeling, we condition on the final capture and model the remaining entries of the capture history: Pr(2201 I last capture at period 4 in state 1) = I y12(1 _
-1
p2,)y22
+
(17.39)
22~2P y11(1 _ P3lr) -3,12] p2 r~/2 /al.
The term in brackets reflects uncertainty about the state at time 3. An animal with this history could have been in state 2 at time 3, or it could have been in state 1 at period 3. In either case, the animal was not caught in period 3, requiring terms for the complement of capture probability in each possible state. The parameters
474
Chapter 17 Estimating Survival, Movement, and Other State Transitions
following the bracketed expression include no additive terms because states, and hence transitions, are k n o w n for periods 2 and 1. Animals showing this history were in state 2 at times I and 2 and were caught there during both sampling periods. The probability distribution for an entire multistate data set using a reverse-time approach involves conditioning on groups of animals last seen in each particular state in each time period. Conditional multinomial models then can be constructed using probabilities such as those of Eq. (17.39). There is a probability corresponding to each possible capture history for the group of animals last seen in state r at period i. The product of these conditional multinomials for all groups of final captures is the distribution for the entire study on which estimation is based.
17.4.2.3. Assumptions The assumptions underlying reverse-time multistate modeling are the same as those presented for the A r n a s o n - S c h w a r z model in standard time (Section 17.3.1). The homogeneity assumption n o w applies to the yrs and pr,. Multistate modeling is one means of dealing with heterogeneity, and the various other approaches discussed previously in this chapter m a y be useful as well. As with single-state reverse-time modeling, further investigation of reverse-time multistate models and their underlying assumptions is needed.
17.4.2.4. Estimation Parameters are estimated using m a x i m u m likelihood. Because software has not been tailored specifically for these analyses, a reasonable approach is to simply reverse the time order of the data and use either MARK (White and Burnham, 1999) or MSSURVIV (Hines, 1994). The multistate seniority parameter yrs is analogous to the survival-transition parameter q~s in standard time (Section 17.3.1). Reverse-time multistate modeling is very new, thus this issue deserves more thought, but we anticipate that the yrs parameters should be relevant to m a n y biological questions, and there should be no need to decompose these parameters in the m a n n e r of standard-time analyses [Eq. (17.30)1.
is in population growth on one of these areas (e.g., area 1). Define L rs as the n u m b e r of animals located on area r at time i that are alive on area s at time i + 1. Using superscripts to denote the study area, we can write the population size on area 1 as the s u m of three components:
N]+ 1 = C~1 + C21 + B~.
(17.40)
The first term on the right-hand side of Eq. (17.40) represents the n u m b e r of animals that survived from i to i + 1 and remained on area 1. The second term denotes animals present on area 2 at time i that survived until i + 1 and moved to area 1. The final term represents the recruitment to area 1 between i and i + 1, including results of reproduction on areas 1 and 2 as well as immigration from locations outside the boundaries of the two study areas. Thus, B] = N~+ 1 L~1 - C21 . We can model the r a n d o m variables contributing to N~+ 1 as a conditional trinomial: Pr(L~ 1, L21, B] IN~+ 1) (X~+l)! = (L]I)!(L21)!B]! x
1
--
11 ~/i+1
11
L] 1
('Yi+I) ('y]21) --
12 "~i+1
(17.41)
L 21
9
The ~/i+1 rs are multistate seniority parameters, defined as the probability that an animal present in area r during period i + I was in area s in period i. In this simple 11 case of two study areas, "Yi+I is the probability that a m e m b e r of Ni+ 1 1 is a m e m b e r of N~ that survived from the previous period (Ell), ~/i+1 12 is the probability that a m e m b e r of Ni+ 1 1 is a m e m b e r of Ni2 that survived from the previous period, and 1 - y]l I - - ~/]21 represents the probability that a m e m b e r of Ni+ 1 1 w a s in neither area 1 nor 2 at time i but was recruited (either via reproduction or immigration) between times i and i + 1. The ~/~+1 parameters quantify the relative contributions of these different demographic components to the population at time i + 1 and thus to population growth between i and i + 1. Based on Eqs. (17.40) and (17.41) we can decompose the expectation for population growth rate of area 1 as follows: E(L] 1) q-
E(L/21) +
E(B])
E(N])
(17.42)
17.4.2.5. Interpretation of Estimates We provide here some explanation about w h y reverse-time multistate modeling should be of interest to biologists. Assume a sampling situation in which animals can move between different study areas (denoted areas 1 and 2), with marking and recapture in both areas. Assume further that our primary interest
,11,,1 12, , +[1_ i+11"~i+1 q- "Yi+ll'~i+l
11
12] gi+l 1
"Yi+I -- "~i+1
E(N]) The ~/i+1 rs parameters can be used to address questions such as " H o w would the rate of increase in area 1 (X~) have differed if the contribution from area 2 had
17.4. Reverse-Time Models been reduced by 25%?" Let oL be the proportional reduction in the contribution from area 2. Then we can predict the proportional reduction in N~+I, and thus in k], as ot'~]21 . As an aside, we note the analogy between the ~rs parameters above and the quantity known as "derivation of the harvest" in studies of harvested animal populations (e.g., Munro and Kimball, 1982; Schwarz and Arnason, 1990). Studies of derivation of the harvest attempt to estimate the relative contributions of various source populations to the animals harvested in a particular area. In such studies, efforts are made to estimate the probabilities that an animal harvested in the area of interest originated in each of a specified number of potential contributing source populations/ areas. Reverse-time, multistate, tag-recovery modeling deserves consideration as a possible means of estimating these probabilities more directly.
17.4.2.6. Alternative Modeling Virtually all of the alternative modeling described for standard-time multistate modeling should be useful with reverse-time modeling. Reduced-parameter models can be used to address interesting biological questions. Consider a metapopulation system with several studied patches or subpopulations. A reducedparameter model with constraint ~/rs= ~/rl = ~1~2= . . . . ~r. expresses the hypothesis that a randomly selected animal at location r at time i has equal probabilities of having been in any other location s ~ r at time i - 1. Rejection of this hypothesis suggests that some patches in the metapopulation contributed disproportionately to the subpopulation at patch r, time i. Use of multiple groups and modeling with covariates should also be useful with these models. As noted above, age-specific models in reverse time require not only a multistate modeling approach but also a method such as the robust design described in Section 19.5.2 to estimate the capture probability for the youngest age class.
17.4.2.7. Model Selection, Estimator Robustness, and
Model Assumptions Most of the discussion presented for the CJS model in Sections 17.1.8 and 17.1.9 and for the ArnasonSchwarz model in Section 17.3.1 should be relevant to the multistate, reverse-time modeling presented here. The use of AIC in model selection is recommended for this modeling. Regarding goodness-of-fit testing, Pearson chi-square tests comparing observed values and their expectations should perform adequately. Estimator robustness may not be similar for standard-time and reverse-time approaches to estima-
475
tion. As was the case for single-state modeling, we expect that the multistate seniority parameter ~/rs will not be as robust to trap response as, for example, the multistate survival-transition parameter q~s. Once again, the entire reverse-time approach is quite new and worthy of much future attention.
17.4.2.8. Example Analysis We use capture-recapture data from a study of meadow voles, Microtus pennsylvanicus, conducted in old field habitat at Patuxent Wildlife Research Center in Laurel, Maryland (also see Nichols et al., 1994). Here we focus on one of eight experimental trapping grids, grid 4A, and use data from the first 11 sampling periods extending from November 1991 through May 1993. The grid was a 7 • 15 rectangle of trapping stations with adjacent stations within each row or column separated by 7.6 m (25 ft). We divided the rectangular grid 4A into two square strata. Stratum 1 was defined by trapping rows 1-7, and stratum 2 was defined by trapping rows 9-15. As the overall grid contained seven columns of traps, both strata are squares with 7 • 7 trapping stations. The robust design (Pollock, 1982) was used, with primary trapping periods occurring at approximately 8-week intervals. For primary periods 1-9, secondary sampling included five consecutive days of trapping, whereas for periods 10-11 only two consecutive days were trapped. A single Sherman live trap containing cotton and baited with rolled oats was placed at each station. The trapping schedule consisted of setting traps one evening, checking them for animals and closing them the following morning, setting them again in the late afternoon, checking them the following morning, etc. Newly captured animals were marked with individually coded monel fingerling tags placed in their ears. If tags of previously marked animals showed signs of pulling out, a new tag was placed on the opposite ear a n d / o r toes were clipped. Animals were sexed and weighed on each occasion, and external reproductive characteristics were recorded. For this analysis, we focused on adult and subadult voles (defined as ___22 g body mass) of both sexes. We used an open-model approach, focusing only on whether an animal had been captured at least once at each of the 11 primary trapping periods (data summarized in Table 17.27). We focus attention on stratum I and on the number of adults (N~) and the rate of increase in number of adults [k~ = E(N~+1/N~)] in this stratum. We first used the model of Pradel (1996) (see Section 18.4), with capture-recapture data only from stratum I (captures from row 8 or stratum 2 were simply entered as 0) to estimate population growth rates for this stratum.
476
Chapter 17
TABLE 17.27 Release
Release
Number
Estimating Survival, Movement, and Other State Transitions
Capture-Recapture Data (R r, mrS) for Adult Meadow Voles a Period of next recapture j (patch of next recapture s)
period
Release
patch
released
(i)
date b
(r)
(R/r)
(1
2)
(1
2)
(1
2)
(1
2)
(1
2)
(1
2)
(1
2)
(1
2)
(1
2)
(1
2)
1
11-13-91
1
18
5
2
0
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
2
12
0
4
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
1-1-92
1
21
11
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
15
3
4
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2-26-92
1
45
34
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
31 83
0
25
4-22-92
2 1
0 40
0 3
0 1
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
28
6-17-92
59 58
1
5
2 1
0 33
1 1
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
48 64
30
8-12-92
2 1
2
6
0 14
1 4
0 1
0 0
0 0
0 0
0 0
0 0
0 1
0 0
60 40
26
10-7-92
2 1
2
7
1 17
4 0
0 0
0 1
0 0
0 0
0 0
0 0
55 39
12
12-1-92
2 1
3
8
0 10
1 0
0 3
0 0
0 2
0 0
27 31
4
1-27-93
2 1
1
9
3 5
3 0
0 5
0 0
18 36
7
3-25-93
2 1
0
10
1 17
2 0
2
20
2
6
3 4
2
3
4
5
6
7
8
9
10
11
a In a two-patch system on grid 4A at Patuxent Wildlife Research Center, Laurel, Maryland, November 1991-May 1993. bMidpoint of 5-day trapping periods (periods 1-8) and first day of 2-day periods (periods 9-11).
We u s e d r e v e r s e - t i m e m u l t i s t a t e m o d e l i n g to estim a t e the relative c o n t r i b u t i o n s to p o p u l a t i o n size a n d g r o w t h o n s t r a t u m 1 f r o m a d u l t s o n s t r a t u m 1 (~]1), a d u l t s o n s t r a t u m 2 ( . ~ 2 ) , a n d n e w recruits (1 - ~ 1 _ q~2). E s t i m a t e d c o n t r i b u t i o n s to s t r a t u m 1 of a d u l t s f r o m s t r a t u m 2 w e r e n o t large, r a n g i n g f r o m ~]2 = 0.00-0.14 (Table 17.28). C o n s i d e r the p o p u l a t i o n c h a n g e in s t r a t u m 1 b e t w e e n s a m p l i n g p e r i o d s 6 a n d 7. T h e e s t i m a t e d c o n t r i b u t i o n s f r o m a d u l t s o n s t r a t u m 1, a d u l t s o n s t r a t u m 2, a n d n e w recruits w e r e 0.57, 0.02, a n d 0.41, r e s p e c t i v e l y (Table 17.28). So a 5% c h a n g e in a d u l t s u r v i v a l o n s t r a t u m 1, or in the p r o b a b i l i t y of an a d u l t r e m a i n i n g o n (not m o v i n g from) s t r a t u m 1, w o u l d h a v e y i e l d e d a 2.85% (SE = 0.325) c h a n g e in a d u l t p o p u l a t i o n g r o w t h o n s t r a t u m 1 d u r i n g the i n t e r v a l b e t w e e n p e r i o d s 6 a n d 7. A 5% c h a n g e in a d u l t s u r v i v a l o n s t r a t u m 2, or in the p r o b a b i l i t y t h a t an a d u l t o n s t r a t u m 2 w o u l d m o v e to s t r a t u m 1, w o u l d h a v e p r o d u c e d a c h a n g e of o n l y 0.10% (SE = 0.085) in ~ . A 5% c h a n g e in the n u m b e r o f r e c r u i t s to s t r a t u m 1 w o u l d h a v e p r o d u c e d a 2.05% (SE = 0.325) c h a n g e in )~1.
17.5. M A R K - R E C A P T U R E AUXILIARY
WITH
DATA
17.5.1. C a p t u r e - R e c a p t u r e Band Recoveries
with
C a p t u r e - r e c a p t u r e s t u d i e s of s o m e species i n v o l v e b o t h r e c a p t u r e s / r e s i g h t i n g s of live a n i m a l s a n d recoveries of d e a d animals. A l t h o u g h d e a d r e c o v e r i e s are possible w i t h v i r t u a l l y a n y species, species that are h a r v e s t e d ( h u n t e d , fished, t r a p p e d ) for c o m m e r c i a l or recreational r e a s o n s are the m o s t likely to y i e l d large n u m b e r s of recoveries. T h u s , the i n v e s t i g a t i o n of species for w h i c h b a n d r e c o v e r y m o d e l s ( C h a p t e r 16) typically are u s e d can benefit m o s t directly f r o m m e t h o d s d e s c r i b e d in this section.
17.5.1.1. D a t a Structure B u r n h a m (1993; also see S z y m c z a k a n d Rexstad, 1991) d e v e l o p e d the initial a p p r o a c h to s i m u l t a n e o u s u s e of live r e c a p t u r e s a n d d e a d recoveries, b a s e d on the sufficient statistics of a m o d i f i e d mij-array. Specifically,
17.5. Mark-Recapture with Auxiliary Data TABLE 17.28
477
Estimates of Population Change (kJ) and Contributions to Population Size and Growth a
Trapping period
Date b
~.jr
~'E(~l)c
2
1-01-92
1.25
0.735
3
2-26-92
0.51
0.360
0.49
0.126
0.00
0.000
0.51
0.126
4
4-22-92
1.03
0.767
0.62
0.166
0.04
0.035
0.34
0.169
5
6-17-92
1.36
0.832
0.49
0.091
0.00
0.000
0.51
0.091
6
8-12-92
0.88
0.249
0.30
0.069
0.08
0.040
0.62
0.076
"Yi11d -"
S'~(,~11)d
.
.
~/i-"12d .
~(,~J2)d .
1 -- "~i"11-- "Yi-"12d .
S"E(1 - ,~11 _ ,~12)d.
.
7
10-07-92
1.54
0.238
0.57
0.065
0.02
0.017
0.41
0.065
8
12-01-92
0.67
0.185
0.51
0.056
0.03
0.020
0.45
0.056
9
1-27-93
0.65
0.472
0.83
0.065
0.00
0.000
0.17
0.065
10
3-25-93
m
--
0.66
0.113
0.04
0.046
0.29
0.111
11
5-20-93
--
B
0.43
0.139
0.14
0.099
0.43
0.141
For patch I of a two-patch system. Relative contributions to population size and growth of adult survival on patch I (~/~1), adult survival and m o v e m e n t from patch 2 to patch 1 (~]2), and recruitment to patch 1 from outside the two-patch system (1 - .~]1 _ .~]2) for m e a d o w voles trapped on grid 4A, Patuxent Wildlife Research Center, Laurel, Maryland, N o v e m b e r 1991-May 1993. bMidpoint of 5-day trapping periods (periods 1-8) and first day of 2-day periods (periods 9-11). CEstimated from patch 1 capture history data using the approach of Pradel (1996); also see Section 18.4). d Estimated using reverse capture-recapture with multistate models.
define the following statistics: R i is the n u m b e r of animals released with marks following capture occasion i; i = 1.... , K - 1; mijl is the n u m b e r of animals released at capture occasion i that were next caught at occasion j; i = 1, ..., K - 1, j = i + 1, ..., K; mijd is the n u m b e r of animals released at capture occasion i that were recovered dead between capture occasions j and j + 1; i = 1.... , K , j = i.... , K. Note that we use the term "capture occasion" rather than "sampling occasion" to emphasize that these are discrete periods at which animals are recaptured or resighted. Recoveries are not restricted to short capture occasions but can occur at any time. An example mij-array representation of data resulting from a study with both live recaptures and dead recoveries is presented in Table 17.29.
TABLE 17.29 The m/j-Array Representation of the Data Resulting from a Three-Period Capture-Recapture Study with Dead Recoveries a Recapture or recovery period j
Release period i
Number released
1 mll d
1
R1
2
R2
3
R3
2 m12 !
3 m12d
m13 !
m13d
m22d
m23 !
m23d m33d
a Ri denotes releases in time i, mqd denotes the members of R; next seen as dead recoveries between capture occasions j and j + 1, and mij I denotes the members of R i next seen as live recaptures at occasion j.
Define the following statistics representing the total numbers of recaptures and recoveries respectively, resulting from each group of releases, ai: K ril =
~
mijl
j=i+l
for i, i + 1, ..., K - 1, and K rid = ~ j=i
mqa
for i, i + 1, ..., K. Burnham (1993) s u m m a r i z e d data for such a study in terms of encounter histories, which are analogous to the capture histories presented previously in this chapter, but reflect the fact that recoveries can be obtained as well as captures. The format for the encounter history is similar to that of the capture history, except now two entries are used for each capture occasion. The first entry denotes possible capture at that occasion and the second entry denotes possible recovery before the next capture occasion. For example, consider the encounter history 100001 corresponding to a three-period study. Each animal with this history was caught and released at capture occasion I and not recovered before the next capture occasion (the first "10" entries), not caught at occasion 2 or subsequently recovered before occasion 3 (the second "00"), and then not caught at occasion 3 but recovered as dead following capture occasion 3 (the final "01"). As with the capture history data presented previously in this chapter, encounter history data can be
478
Chapter 17 Estimating Survival, Movement, and Other State Transitions
summarized using the number x~ of animals exhibiting encounter history o~. The convention of a negative number again can be used to indicate the number of animals exhibiting the encounter history that are not released following their final encounter [other conventions are possible (Burnham, 1993)]. Note that animals are not released following recovery, whereas most recaptures will be followed by release. 17.5.1.2. M o d e l Structure As with the CJS and other conditional models described previously in Chapters 16 and 17, each row of the m/j-array (Table 17.29) can be modeled as a multinomial distribution conditional on releases R i. The probability distribution for the entire mq-array then can be modeled as the product of the multinomials corresponding to the different release cohorts. Let ~r/jt represent the multinomial cell probability associated with mq-array element mifl, with ~rq~ representing the multinomial cell probability associated with m/j-array element mij a, and K
)kil--
E
ITij I
j=i+l
for i = 1, ..., K - 1, and K
kid - E 'lTijd j=i
f o r / - 1, ..., K. Given this general parameter notation, the probability distribution for the recovery-recapture mq-array can be written as
Pr({mija, mijl} l {Ri})
=r l(mii
Ri
)]
, "", miKl, miKd, Ri - l'id -- ?'il
K-1 X {~~~[ (j~_i (ITijd)mija(ITi'j+l'l)mi'j+U )
(17.43)
X(ITiKd)miKd(1--)kid - ~il)Ri--rid--ril]}
responding to live recaptures/resightings are the same as for the CJS model (Table 17.2): (17.44)
7ri,i+l,l = ~PiPi+l,
7rij I = [q~i(1 -- P i + I ) ] "'" [q~j-2 (1 -- Pj-1)](q)j-lPj )
for j > i + 1. Notation for Eq. (17.44) is the same as that used for the CJS model, with q~i denoting the probability of surviving and not permanently emigrating from the study area between periods i and i + 1, and Pi denoting the recapture-resighting probability for sampling period i. The probabilities associated with the band recovery cells of the multinomial (17.43) are similar to those used in Chapter 16: Tfiid = fi, ITijd = S i ... Sj_l~/ijf j,
for j > i, where Si denotes the probability of surviving between sampling periods i and i + 1 and fi denotes the recovery rate for year i. However, ~/q is a new parameter not used in the band recovery models of Chapter 16, denoting the probability that an animal released following capture occasion i and still alive at occasion j has not been recaptured following release in period i. The need for this parameter stems from the need to associate each band recovery with a specific release cohort, and the fact that at each recapture, an animal moves to a new release cohort. For example, an animal released at period i and recaptured in period k usually (e.g., in the absence of trap death) becomes a member of the release cohort R k. Subsequent recovery of the animal in some period j > k is associated with the release cohort of period k, provided the animal was not recaptured again in some capture occasion between j and k. Because the cell probabilities associated with live recaptures/resightings [Eq. (17.44)] and dead recoveries [Eq. (17.45)] share no parameters, it is difficult to see the utility of simultaneously using both kinds of encounters. The link between the two kinds of cell probabilities is established by considering a model in which animals may emigrate permanently from the capture-recapture study area, so that the survival probability associated with live recaptures is q~i-- SiFi,
mKKd
Model (17.43) is not particularly interesting or useful until its cell probabilities are specified as functions of parameters reflecting the biological events giving rise to recoveries and recaptures. The cell probabilities c o r -
(17.45)
(17.46)
where F i is the probability that an animal in the sampled area at capture occasion i remains in the population (exhibits fidelity) and does not permanently emigrate between i and i + 1. Stated differently, F i is the probability that an animal in the sampled area at
17.5. Mark-Recapture with Auxiliary Data time i is also in the sampled area at time i + 1, given that it is alive at i + 1. This parameterization [Eq. (17.46)] corresponds to the permanent emigration model of Burnham (1993) and was anticipated in principle by Anderson and Sterling (1974) and Hepp et al. (1987). Rewriting q~i as in Eq. (17.46) is appropriate for the usual situation in which dead recoveries come from the entire range of the sampled population--for example, with North American waterfowl that are hunted throughout their range. Given that there are no locations to which members of the hunted population can permanently emigrate to escape hunting, the Si estimated from dead recoveries reflect true survival (the complement is mortality rate; see Chapter 16). As noted previously in this chapter, however, the ~i estimated from capture-recapture or resighting data reflect local survival, because the complement includes both mortality and permanent emigration from the area exposed to sampling efforts. The final step in the modeling involves rewriting the parameter ~/q as a function of other model parameters. Recall that ~/q denotes the probability that an animal released following capture occasion i and still alive at occasion j has not been recaptured following release in period i. The parameter is conditional on the animal being alive at times i and j, so no survival probabilities are needed to rewrite ~/q. Instead, the parameter is written recursively as ~/jj = 1, (17.47) ~/ij = 1 - FiE1 - qi+l~/i+l,j~,
for j > i, where qi = 1 - Pi. Thus, an animal can avoid capture between periods i and j [Eq. (17.47)] by either permanently emigrating between i and i + 1 (probability given by 1 - F i) or by remaining in the area yet not being caught (Fiqi+ 1) at i + 1 and then not being caught between i + 1 and j (~i+ld). The model cell probabilities [Eqs. (17.44) and (17.45)]
TABLE 17.30
479
correspond to the multinomial distribution in Eq. (17.43) for the mq-array data. For example, cell probabilities for a three-period study with both live recaptures and dead recoveries are shown in Table 17.30. It is also possible to develop the permanent emigration probability model using encounter-history data (Burnham, 1993). For example, consider the following encounter histories and their associated probabilities for a three-period study: Pr(100011 [release at period 1) =
51F1q2S2F2P3~/33f3
-- 51F1q252F2pgf3,
Pr(101000 [ release at period 1)= SIF1P2(1
- ~ . 2 l - )k2d),
and Pr(100100 [ release at period 1)
=
SiS2"Y12f 2.
The probability distribution for the entire data set is given as the product of the conditional (on first capture) probabilities of the different encounter histories raised to powers corresponding to the numbers of animals exhibiting each history: II[Pr(co [ first capture)] x-. co
The parameterizations for both the mq-array and encounter-history data correspond to Burnham's (1993) permanent emigration model, in which there is some probability (1 - F i) that an animal permanently departs the area in which it is exposed to capture efforts. Burnham (1993) also described an alternative "random emigration" model (also see Kendall et al., 1997). Under this model each animal has some probability of not being in the area exposed to capture efforts at each capture occasion. If this probability is the same for animals regardless of location (in or out of the sampled area) in the previous period, then the capture-recapture survival parameter is the same as the
Expected N u m b e r s of D e a d Recoveries and Live Recaptures for the Data of Table 17.29 a Recapture or recovery period j 1
2
3
Release period i
Number released
Live
Dead
Live
Dead
Live
Dead
~
R 1fl
R 1S1Flp 2
R 151~ 12f2
R 151F1q2S2F2P3
R 1$152'Y13f3
Raf2
R252F2P3
R252~23 f3
1
R1
2
R2
3
R3
aUnder the structure of Burnham's (1993) permanent emigration model.
Rgf3
480
Chapter 17 Estimating Survival, Movement, and Other State Transitions
"true" survival used to model dead recoveries (q~i --
Si), and the capture probability parameter Pi now reflects the product Pr(animal is in the area exposed to sampling efforts) x Pr(animal is caught I animal is in the area exposed to sampling efforts). Decomposition of estimates ]9i into probabilities associated with temporary emigration and conditional capture is discussed in Section 19.5.1.
17.5.1.3. Model Assumptions Assumptions of Burnham's (1993) permanent emigration model simply represent a combination of the assumptions underlying the CJS capture-recapture model (Section 17.1.2) and the Seber-Robson-Youngs band recovery model (Section 16.2.1). The single exception is that permanent emigration from areas exposed to sampling is possible for capture-recapture sampling, but not for sampling for dead recoveries. 17.5.1.4. Estimation
Estimation with the permanent emigration model can be based on either mq-array or encounter-history data. Szymczak and Rexstad (1991) used program SURVIV (White, 1983), in conjunction with the mq-array approach, to provide estimates for a study of gadwalls (Anas strepera). Program MARK implements the model using the encounter-history approach (White and Burnham, 1999). Regarding identifiability, the parameters that can be estimated are similar to those that are estimable under the corresponding models for the single data sources (band recovery models of Chapter 16; capture-recapture models of Section 17.1). For example, under the permanent emigration model with full time specificity of parameters, the estimable parameters are $1, ..., SK- 1, P2, ..., PK-1, fl, ..., fK, F1, ..., FK-2, and the product
FK-1PK. 17.5.1.5. Alternative Modeling The joint use of band recovery and capture-recapture data in general, and of the permanent emigration model of Burnham (1993) in particular, has been infrequently implemented, so there are few examples of alternative modeling (see Szymczak and Rexstad, 1991). One reduced-parameter model of special interest involves the constraint Fi = 1, for all i = 1,... K - 1. This constraint corresponds to a model in which fidelity is perfect, and there is no permanent emigration. It is equivalent to the constraint ~ i -- S i [see Eq. (17.46)]. Other reduced-parameter models with stationary pa-
rameters also are possible, as are models with multiple groups or with parameters modeled as functions of time-specific covariates. An ecologically interesting model would allow permanent emigration probabilities as functions of environmental variables under the hypothesis that emigration is more likely during periods of unfavorable conditions. Age can be incorporated into the permanent emigration models to allow for the testing of hypotheses about age-specific emigration, a matter of some relevance given that emigration often is greater for young animals in many vertebrate species. Use of multistate modeling with joint recovery and recapture data is certainly possible, but requires careful attention to model specifications because many possibilities exist. However, the ability to estimate probabilities of moving among sampled locations within a study system, as well as the probability of moving out of the study system, should be very useful. The above development is based primarily on designs in which band recoveries are obtained by some harvest process (e.g., hunting) that covers the entire range of the sampled population. Burnham (1993) discusses possible model modifications that may be useful in alternative sampling designs, such as when recoveries are obtained as animals found dead rather than harvested, and when recoveries are obtained from constrained areas rather than from throughout the range of the sampled population.
17.5.1.6. Model Selection, Estimator Robustness, and Model Assumptions Burnham's (1993) modeling has seen relatively little use other than that by Szymczak and Rexstad (1991) and the incorporation into program MARK (White and Burnham, 1999). Thus, there has been relatively little work on special issues of model selection and goodness-of-fit associated with these models. The approach to model selection described in Section 17.1.8 is applicable to these models. To our knowledge, specific goodness-of-fit tests have not been developed, but Pearson chi-square tests should be applicable, especially given the development based on mq-arrays. Discussions of estimator robustness and model assumptions presented in Section 17.1.9 should be relevant to these models. It would be interesting to investigate special problems or advantages that result from the two distinct sampling methods. Individual animals may react differently to the two "sampling" methods, in which case both the degree of heterogeneity and the covariance between sampling probabilities (capture probabilities and recovery rates) within individuals should be important determinants of estimator
17.5. Mark-Recapture with Auxiliary Data performance. The problem of heterogeneous capture probabilities has been investigated in the capture-recapture context (Carothers, 1973; Gilbert, 1973), and heterogeneous recovery rates have been studied for band recovery models (e.g., Pollock and Raveling, 1982; Nichols et al., 1982b), but simultaneous heterogeneity in both sampling probabilities has not been investigated.
17.5.2. Capture-Recapture with Resightings between Sampling Occasions The models considered here utilize auxiliary observations of marked animals that can occur at any time between sampling periods. Whereas observations between capture occasions were all of dead recoveries in the previous section, here we consider observations of live animals as well, wherein animals observed between capture occasions are released back into the population. Jolly (1965) considered the use of auxiliary observations by noting that recaptures in his original model "enter into the estimates in two distinct ways, first...as the proportion of previously marked animals in ni, ..., and secondly as the ratio of future recaptures," where n i is the total number of marked and unmarked animals caught at capture occasion i. The ratio of future recaptures could include observations of animals from a much larger area than that at which capture efforts are carried out. These observations then can be used to augment the zi and r i statistics used, for example, in Eq. (17.10) to estimate the number of marked animals in the population, M i (see Jolly, 1965; Pollock et al., 1990). Barker (1995, 1997) considered this ad hoc approach and developed models that explicitly incorporate auxiliary ("ancillary," using Barker's terminology) information. In some studies of migratory birds that are marked with neckbands, virtually continuous sampling for portions of each year produces large numbers of observations that do not occur during the capture occasions, and these observations have the potential to make substantial contributions to estimation of demographic parameters (e.g., Raveling et al., 1992). The use of such observations obtained by sampling methods other than those used for actual capture of animals also may reduce problems associated with heterogeneous recapture probabilities. Band recoveries (Section 17.5.1) also can be viewed as auxiliary observations, and the models of Barker (1995, 1997) reduce to those of Burnham (1993) in the case of no releases on capture (dead recoveries). The development presented here follows that of Barker (1995, 1997). Barker's (1995) Ph.D. thesis in-
481
cludes a thorough examination of auxiliary data and their incorporation into capture-recapture models. 17.5.2.1. Data
Structure
The sampling situation includes two different kinds of observations of marked animals. Live captures and recaptures can occur at capture occasions (i = 1, ..., K), recognizing that at any capture occasion, the sampled population may include animals that are not exposed to sampling efforts (temporary or permanent emigrants, depending on the model considered). The other type of observation involves resightings of marked animals at any time in the interval (i, i + 1), designated as resightings in interval i. There are v resighting intervals, where v -> K. In his modeling, Barker (1995, 1997) also assumed that resighting occurs throughout the range of the animals, such that all animals are exposed to resighting efforts (including those not at risk of capture during some capture occasions). Barker (1995, 1997) notes that the term "resighting" is intended to be general in that it could actually involve capture or dead recovery. The key feature is that the process associated with resighting covers the entire range of the population of interest. As usual, losses are permitted on capture and are handled by simply conditioning on releases. Losses on resighting (e.g., recoveries of dead animals) are included in the model and incorporated into the likelihood. Barker (1997) defined the following statistics: Ri, c
Ri, r mi,j,c, c
mi,j,c, r mi,j,r, c
mi,j,r, r
ri, c ri, r
mi oi
The number of animals released following capture at occasion i. The number of animals released following resighting in (i, i + 1). The number of Ri, c that are next encountered by capture at occasion j. The number of Ri, c that are next encountered by resighting in (j, j + 1). The number of marked animals last encountered by resighting in (i, i + 1) that are encountered next by recapture at occasion j. The number of marked animals last encountered by resighting in (i, i + 1) that are encountered next by resighting in (j, j+l). The number of Ri, c that subsequently are encountered by either method. The number of ai, r that subsequently are encountered by either method. The total number of marked animals captured at time i. The total number of marked animals resighted in (i, i + 1).
Chapter 17 Estimating Survival, Movement, and Other State Transitions
482
The number of animals removed from the population (not released) following resighting in (i, i + 1). The number of animals marked before i, not captured at i, but subsequently caught or resighted [includes animals observed in ( i , i + 1)]. The number of marked animals in the population immediately before i that are resighted or recaptured at or after i (Ti = zi q- m/). The number of animals in the population immediately after i that are subsequently encountered after sampling time i [includes animals resighted in (i, i + 1) ( V i = z i + ri,c)].
Ii
zi
Ti
Vi
in that the probability of capture at i does not depend on whether the animal was at risk of capture at time i - 1. Barker (1997) defined the following parameters: Si Pi
Oi
Fi
Data also can be summarized in encounter history form, but here we follow Barker (1997) and develop the model using summary m i j - a r r a y data. An example m/j-array representation of data is presented in Table 17.31 for a study with four periods of capture and resightings following each capture occasion and occurring up to hypothetical capture occasion 5. Note that all animals enter the study as members of a release cohort of captured animals, Ri,c, but that they may later become members of a release cohort of resighted animals, Ri, r. Multiple resightings of animals in an interval are ignored. The relevant information is whether an animal was seen at least once during an interval.
Vi
Thus, the random emigration assumption is given by the constraint F i = 1 - F;. As is the case in the CJS model (Burnham, 1993; Kendall et al., 1997), F i is confounded with Pi+l so that only the product P~+I -FiPi+ 1 can be estimated. Note that the parameter f differs in its meaning here from previous uses in this book, in particular in Chapter 14 where it is used to denote capture frequency, and in Chapter 16 where it denotes recovery probability. We have retained the use of f for different attributes in order to facilitate cross-
17.5.2.2. Model Structure
Barker (1997) presented a model that corresponds to the random emigration model of Burnham (1993), TABLE 17.31
The probability that an animal alive at time i is alive at i + 1. The probability that an animal is captured at time i, given that it is at risk of capture at time i. The probability that an animal is resighted in the interval (i, i + 1), given that it is alive at time i. The probability that an animal alive at time i is not resighted in the interval (i, i + 1), given that it is alive at i + 1. The probability that an animal alive and at risk of capture at i and alive at i + 1 is at risk of capture at i + 1. The probability that an animal alive and not at risk of capture at i and alive at i + 1 is not at risk of capture at i + 1. The probability that an animal is released, given that it is resighted in (i, i + 1).
The m/j-Array Representation for Data Resulting from a Study with Four Capture Periods and Ancillary Observations Occurring after Period 4 a Period of next encounter j
Release
Number
cohort i
released
Recapture 2
Resighting
3
4
1
2
3
4
ml,3,c, c
ml,4,c, c
ml,l,c, r
ml,2,c, r
ml,3,c, r
ml,4,c, r
m2,3,c, c
m2,4,c, c
m2,2,c, r
m2,3,c, r
m2,4,c, r
m3,3,c, r
m3,4,c, r
Released following capture 1
R1, c
2
R2, c
3
a3, c
4
a4, c
ml,2,c, c
m3,4,c, c
m4,4,c, r
Released following resighting 1
al, r
2
a2, r
3
a3, r
ml,2,~c
aUp to hypothetical capture period 5.
ml,3,~c
ml,4,~c
m2,3,~c
m2,4,~c m3,4,~c
ml,2,~r
ml,3,~r
ml,4,~r
m2,3,~r
m2,4,~r m3,4,~r
17.5. Mark-Recapture with Auxiliary Data referencing b e t w e e n material in this book and the biological literature, where f is similarly used. In constructing the probability model it is i m p o r t a n t to recognize that the survival probability over (i, i + 1) for a m e m b e r of ai, r is not Si, but should be larger than Si because these animals have been seen after i. Here we follow Barker's (1995) d e v e l o p m e n t of an expression for
483
s u m m a r y of Table 17.31 are presented in Tables 17.32 and 17.33. Table 17.32 includes expectations for animals that are released following capture. Some of these releases are next encountered as captures and others are encountered as resightings. As an example, consider the expected value for the entry ml,3,c,c. The expectation begins with the n u m b e r al, c of releases following capture in period 1. Animals associated with this statistic then survive until period 2 (probability associated with this event is $1), are neither seen b e t w e e n 1 and 2 (01) nor captured at 2 (q~), survive until 3 ($2), are not seen b e t w e e n 2 and 3 (02), but are caught at 3 (p~). On the other hand, some animals released following capture are next encountered as resightings, as with m3,4,c, r. The expectation for this statistic begins with the n u m ber R3,c of releases following capture at period 3. In order to a p p e a r as a m e m b e r of mg,4,c,r, a n animal m u s t survive from occasion 3 to 4 ($3), not be seen d u r i n g that interval (03), not be caught at 4 (q~), but then be resighted following capture occasion 4 (f4). Expectations for animals released following resighting are presented in Table 17.33. Thus, the entry m2,4,r, c represents animals released following resighting b e t w e e n periods 2 and 3 and next encountered by capture at period 4. The expectation begins with the n u m ber of releases following resighting b e t w e e n 2 and 3, R2,r. Animals associated with this statistic m u s t then survive until period 3, given resighting b e t w e e n 2 and 3 and release, and the probability for this event is
Pr[individual survives from i to i + 1 ] it was seen in (i, i + 1) and released]. Begin by noting that an animal seen in (i, i + 1) m u s t be released in order to survive until i + 1. Thus, Pr[(survives from i to i + 1 and seen in (i, i + 1)] = Pr[survives from i to i + I and seen in (i, i + 1) ] released] Pr(released). Using the above expression, we obtain Pr[survives from i to i + 1 ]seen in (i, i + 1) and released] Pr[survives from i to i + 1 and seen in (i, i + 1) and released] Pr[seen in (i, i + 1) and released] Pr[survives from i to i + 1 and seen in (17.48) (i, i + 1) I released] Pr(released) Pr[released ] seen in (i, i + 1)] Pr[seen in (i, i + 1)] Pr[survives from i to i + 1 and seen in [i, i + 1)] Pr[released ]seen in (i, i + 1)] Pr[seen in (i, i + 1)]
(1
"
Animals in m2,4,r, c a r e not caught at occasion 3 (q~), survive until occasion 4 ($3), are not resighted b e t w e e n 3 and 4 (03), but are then captured at 4 (p~). The analo-
Using the above notation, and also defining q* = 1 - p*, the expected values for the mq-array data
TABLE 17.32 Release cohort i
02)S 2
f2v2
(1 -- Oi)S i
ivi
-
Expected N u m b e r s of Recaptures and R e s i g h t i n g s for A n i m a l s Released F o l l o w i n g Capture (Ri, c) a Number
Period of next encounter by recapture j
released
1
Rl,c
2
R2, c
3
a3, c
2
3
Rl,cS101p 2
4
Rl,cS101q2S202p3
a1,cS101925202q35303P4
R2,cS202P3
a2,cS202q2SgOgP4 R3,cS303P4
Period of next encounter by resighting j 1 1
a I ,c
2 3 4
R2, c R3, c
Rl,cfl
2
3
4
Rl,cSlOlq2f2
Rl,c51019252q3 f3
R2,cf2
a2,cS20293 f3 RB,cf3
Rl,c510192520293530394f4 a2,cS20293530394f4 Rg,cSgOBq4f4
R4, c
a Under the random emigration model of Barker (1997) (see upper half of Table 17.31).
R4,cf4
484
Chapter 17 Estimating Survival, Movement, and Other State Transitions
TABLE 17.33
Expected Numbers of Recaptures and Resightings for Animals Released Following Resighting (Ri, r) a Period of next encounter by recapture j
Release cohort i
Number released
al,r 2
R2,r
3
R3,r
2
Rl,r
I(1
3
- 01)S11 flVl
P~
4
R1r[ ( 1 - 01)S1] ' flY1 j q~S202p~ Rare(1 - 02)$2~ ,
'[
72q
JP3
al,rl (1 - 01)S1]
71Vl Jq~S202q~S303p~
[(1 ~ 0_2)$2]
R2"r
f2v2 j q~S303p~
R3,r[( 1 - 0__3)$3]
f3v3 JP~
Period of next encounter by resighting j 2
1
R1,r
2
R2,r
3
R3,r
R 1 rl (1- 01)$1]
,
71Vl jq'~f2
3
4
al,r[(1- O1)Sllq'~S202q'~f3 flY1
R2,r{(l[ - 02)$2 ] , 72~
Jq3f3
Rl,rl (1 --flVl01)51]q,~S202q~S303q~f4 - 02)5 2 R2,rl (1 72~2 ] q~S303q~f4 R3r[( 1 - 03)$3-] ,t , [ 73V~ ]94./4
aUnder the random emigration model of Barker (1997) (see lower half of Table 17.31).
gous statistic for animals released following resighting between periods 2 and 3 and next encountered by resighting following period 4 (rather than capture at 4), m2,4,r,r, has a similar expectation, differing from that of m2,4,r,c in that the final capture probability of E(m2,4,r,c) is replaced by q~,f4 (these animals are not caught at occasion 4, but are instead resighted following 4). As under previous models of this chapter, each row of the mij -array can be modeled as a multinomial distribution, and the likelihood for the entire study is given by the product of these multinomials. 17.5.2.3. M o d e l A s s u m p t i o n s
In addition to the usual CJS assumptions listed in Section 17.1.2, Barker (1997) lists the following assumptions underlying his general approach: 1. All animals have the same resighting probabilities 0i and fi at time i. 2. All animals at risk of capture at i and alive at i + 1 have the same probability F i of being at risk of capture in i + 1, and all animals not at risk of capture at i and alive at i + 1 have the same probability F; of not being at risk of capture at i + 1 (this assumption can be modified depending on model specifics; e.g., recall that F i = 1 - F; under the random emigration model). 3. Resightings occur throughout the animals' range, but capture only occurs at a specific location within
the range, so study animals may or may not be at risk of capture at any time. 4. Survival probability does not depend on location within the range, so that all animals alive at i survive with probability Si, regardless of whether they are at risk of capture. Assumptions 1, 2, and 4 listed above are additions to the homogeneity assumptions of the CJS model. In addition to capture and survival probabilities, resighting and emigration probabilities must also be the same for all animals in the study population. As noted in the discussion of CJS assumptions, these probabilities may vary as a function of state variables associated with individual animals. Stratification and the incorporation of age and multiple states provide possible means of dealing with homogeneity assumptions. Assumption 3 is as much a statement about the sampling design as it is an assumption. 17.5.2.4. E s t i m a t i o n
As with previous models, estimation is accomplished by using the mij-array data in conjunction with the product-multinomial likelihood function to obtain maximum likelihood estimates. Barker's random emigration model is an option of program MARK (White and Burnham, 1999), and estimates can be easily obtained in this manner. For the random emigration
17.5. Mark-Recapture with Auxiliary Data model with full time specificity, Barker (1997) presents the following closed-form estimators:
fi
--
oiri'c
i = 1, ..., K;
Ri,cVi, mi
i = 2, ..., K;
I
Riczi , ri,c Vi -
+ m i
oi
Zi+l
Yi,c ai,cWi
r
Ri+l,cZi+l
i = 1 , . . . , K - 1; + mi+l
ri+l,c
,
i = 1 .... , K - l ;
and ~;i -
Ri'r,
i = 1, ..., K.
(17.49)
oi
Additional confounded parameters representing functions of the above parameters can be estimated as well (Barker, 1995). Asymptotic variances and covariances are provided by Barker (1995, 1997). 17.5.2.5. A l t e r n a t i v e
Modeling
Barker (1995) considered several alternative models, all of which make use of auxiliary observation data. One such model is analogous to the permanent emigration model of Burnham (1993), in that animals can depart the location where they are at risk of capture (with probability 1 - Fi), but this departure must be permanent. Barker's model is a generalization of Burnham's (1993) model because animals can be released following resighting. The probability structure looks similar to that of Burnham (1993), and the parameters Fi, i = 1 . . . . , K - 2, can be estimated. Closed-form estimators do not appear to exist (Barker, 1995), and estimates must be computed numerically using MARK (White and Burnham, 1999) or perhaps SURVIV (White, 1983). Another alternative discussed by Barker (1995) involves stationary Markov movement. Under this model, animals may move between the locations where they are at risk of capture and locations where they are not at risk, according to a first-order Markov process (movement of an animal between i and i + 1 depends only on its location at i). If these movement probabilities are assumed to remain constant over time (a stationary Markov movement model), then parameters of interest appear to be estimable numerically (Barker, 1995). Barker (1995) also described the use of auxiliary data with multiple-age models of the form described
485
in Section 17.2.2. He presented a detailed structure for the random emigration model with age specificity and derived closed-form maximum likelihood estimators with asymptotic variances and covariances. Numerical estimates are possible under the age-specific analogs of the permanent emigration and stationary Markov movement models. Barker (1995) also considered models for auxiliary observations with capture-history dependence for both single- and multiple-age models. Finally, Barker (1995) outlined an approach for the use of auxiliary observations in multistate models, although additional work is required in this area. Modeling with multiple groups and time-specific covariates should represent straightforward extensions of the models presented by Barker (1995, 1997) for auxiliary data. Certainly, reduced-parameter models will be useful as well, and the additional parameters of the auxiliary-observation models present many opportunities for potentially useful and interesting constraints. 17.5.2.6. M o d e l
Selection, Estimator
Robustness, and
Model Assumptions The approach to model selection described in Section 17.1.8 is applicable to Barker's models using auxiliary observations. Specific goodness-of-fit tests were developed by Barker (1995, 1997) for the random emigration model as well as for many of the alternative models considered above. Discussions of estimator robustness and model assumptions presented in Section 17.1.9 should be relevant to these models, because the modeling and estimation are similar. The use of two distinct sampling methods (e.g., capture and resighting) should reduce problems associated with heterogeneous capture probabilities. Higher capture probabilities also tend to result in less bias in parameter estimates in the face of heterogeneous capture probabilities (Carothers, 1973; Gilbert, 1973), and the additional information provided by auxiliary observations should similarly lead to reduced bias. In general, estimator robustness has not been addressed with these models, though the topic is worthy of future investigation. Barker (1995) specifically discusses tag loss and notes the potential for using double tagging to model tag loss and estimate parameters in the face of this problem.
17.5.3. Capture-Recapture with Radiotelemetry Radiotelemetry has proved to be useful in studies of animal populations, and biostatisticians have devel-
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Chapter 17 Estimating Survival, Movement, and Other State Transitions
oped useful approaches for estimating survival and movement probabilities from radio-marked (radioed) animals (see Sections 15.4 and 15.5). Telemetry frequently is used in conjunction with other forms of marking and sampling animals. For example, radios are expensive relative to other kinds of tags, so it has become fairly common to conduct a standard capturerecapture study, but also to release a small group of animals marked with radios. In some of these studies, observations of radioed animals are simply used to interpret the estimates of demographic parameters obtained from the capture-recapture data. For example, low survival rates from capture-recapture studies can result from either high mortality or high permanent emigration, and telemetry with even a few animals can be used to judge the likely importance of permanent emigration. In studies using both radiotelemetry and standard capture-recapture with reasonable samples of marked animals having each type of mark, it is common to compute different estimates (e.g., survival probability) using each set of data separately. When resulting estimates are similar, it is generally concluded that both estimation approaches are performing reasonably. However, when they differ, a posteriori stories are developed to explain why and to infer which estimate is "right." Because of the greater sampling intensity that is possible with radioed animals (detection probability approaches 1 in many studies), estimates resulting from telemetry data frequently are assumed to be more accurate (or at least reasonable). However, some researchers have cited possible radio effects (e.g., reduction in survival probability associated with the attached radio) and possible associations between censoring (Sections 15.4 and 15.5) and animal fate as reasons to distrust telemetry-based estimates. For example, Bennetts et al. (1999) found evidence of strong year-to-year variation in survival probabilities of juvenile snail kites (Rostrhamus sociabilis), as estimated from radiotelemetry. Survival estimates were low (and consistent with capture-recapture estimates) for one year in which search effort for dead radioed birds was especially intensive, but high (and different from capture-recapture estimates) for two years of reduced search effort for dead birds. Bennetts et al. (1999) concluded that many of the "censored" birds in the two years of low search effort were actually dead, leading to telemetry-based survival estimates that were biased high. Our purpose in this section is to suggest an alternative to the simple comparison of estimates resulting from two different groups of marked birds (radioed and otherwise marked). Thus, it is possible to combine data from both groups into a single likelihood and
utilize both data sources for estimation simultaneously. In most cases the modeling of such a situation is likely to be tailored specifically to the sampling design of interest [see Powell et al. (2000a) for an application with wood thrushes (Hylocichla mustelina)]. This section is more abbreviated than previous sections in this chapter, primarily because there has been little development of models that combine capture-recapture and radiotelemetry. 17.5.3.1. D a t a Structure
In the previous models in this section, a single type of mark is used on all animals, but the mark can be detected in two different ways, with different detection probabilities and different sampling periods. The situation is somewhat different with radiotelemetry, which involves the release of animals with two different types of marks that then are detected with different kinds of sampling and with different detection probabilities. Even if sampling for radioed animals and animals marked with other sorts of tags (the latter will be referred to as "tagged") is conducted at the same time periods, radioed animals will be detected with much higher probabilities compared to tagged animals. The detection probability typically is assumed to be 1 for radioed animals. There may be cases in which this assumption is not justified, and Pollock et al. (1995) developed a capture-recapture modeling approach for this situation. Data for tagged animals can be summarized in either capture-history form or mq-array form. These data summaries are identical to those presented in the sections above. Data for radioed animals require different summary statistics, because simple detection or nondetection (or in multistate models, state-specific detection or nondetection) is not the only possible fate for radioed animals. Even in a single-state, single-location study, it usually is possible to detect radioed animals that die on the study location. Depending on the area searched for radioed animals, it may be possible to locate animals that have emigrated from the area over which capture efforts occur. As an example of a data structure, we consider the sampling situation of Powell et al. (2000a). They worked at a study location that could be subdivided into two sections, a core section (denoted as area 1) in which all capture (mist-netting) efforts occurred, and a peripheral area (denoted as area 2) that could be searched for radioed animals but in which tagged and radioed animals had zero probability of being captured. Denoting location with superscripts, Powell et al. (2000a) defined the following summary statistics:
17.5. Mark-Recapture with Auxiliary Data
A r
rl ai,i+l birS+l ar
The n u m b e r of tagged animals released in the core area following capture occasion i. The n u m b e r of tagged animals released in the core area following capture occasion i that are next captured in the core area at occasion j. The n u m b e r of radioed animals released in area r (r = 1, 2) following capture or radiolocation at capture occasion i. The n u m b e r of members of A r captured at time i + 1 on the core study area. The n u m b e r of individuals of A r radiolocated (but not caught) at time i + 1 on area s. The n u m b e r of individuals of A r that die between i and i + 1.
The above quantities are defined for r = 1, 2 and s = 1, 2. The first two statistics listed above are used in standard capture-recapture modeling for a multistate (in this case, two locations) system. However, note that the only m~js statistic listed is m~j1. Animals are caught only in the core area (location 1), so releases and recaptures of tagged animals can only occur in area 1. Radioed animals can be captured only in the core area, but can be radiolocated and released in either area, and can be found dead following release in either area. It should be clear that m a n y different designs for such a study are possible.
17.5.3.2. Model Structure Parameterization of models for combined capture-recapture and telemetry data depends heavily on the sampling design. Here, we focus on the model of Powell et al. (2000a) for illustrative purposes. Powell et al. (2000a) defined the following parameters" ~ s is the probability that an animal in location r at time i is in location s at time i + 1, given that the animal is alive at time i + 1; S r(b) is the probability that an animal with m a r k type b (b = 1 indicates a standard tag, b = 2 indicates a radio) that is alive on area r at time i is still alive at time i + 1; p ](b) is the probability that an animal with m a r k type b in the core study area (location 1) at time i is captured at i. Note that this parameterization is quite general, in that it permits different capture and survival probabilities for animals with standard tags and radios. However, the probability of moving between core and peripheral areas is assumed to be the same for animals regardless of m a r k type, because model parameters do not appear to be identifiable otherwise. Also note that there is no detection probability for radio relocations that do not involve capture, because the associated probability is assumed to be 1. Models for this situation can be divided into two components. The portion of the likelihood for the recapture data of standard tags is exactly the same as
487
that used in standard multistate modeling (Section 17.3.1). For example, consider the expected value
E(m11] R 1) _ r)1c1(1).1.11~.,1(1) 1\2~
u?2 K 3
"
In order to appear in m23,11animals must be released from location 1 at time 2, survive from time 2 to 3, remain in location 1 (not move between times 2 and 3), and be captured in time 3. The survival and capture parameters are superscripted with "1," indicating animals with standard tags. The multistate capture-recapture portion of the likelihood includes the possibility of m o v e m e n t to location 2 (the peripheral area), but this state is unobservable, so all encounters occur in location 1, the core area. An example expectation for radioed animals that are caught in the core area is
E(a214]A 2) ,
=
A a c a ( 2 ) ~ l , a I r , I(2) ~3~ 't'3 /-'4 9
Animals in the above statistic must be released from the peripheral area at time 3, survive until time 4, move from the peripheral area to the core area between times 3 and 4, and be captured at time 4. The c o m p l e m e n t a r y expectation for radioed animals that have relocated to the core area but have not been caught there is
E(b2!4, [ A2) = ~A2c2(2),1,2111 - P1 ( 2 ) ] ~3~ '4'3
9
The above statistic is observable only because of the radio. Animals m a r k e d with standard tags can only be observed w h e n they are captured. Radioed animals also can be observed (but not caught) in the peripheral area, as in the following expectation:
E(b~,2IAI)
= a l ~ l ( 2 ) , l , 12 ~
'4'1
9
Note that there is no need for the complement of a capture probability in the above expectation, because capture probability is k n o w n to be 0 in the peripheral area. Finally, radioed animals can be located w h e n dead, leading to expectations of the following type:
E(d21 A2) = A2~[1 - $2(2)]. In the above example, neither capture nor m o v e m e n t parameters are needed, because survival is associated with location at the beginning of the interval and capture of dead animals is not possible. We k n o w from Section 17.3.1 that the capture-recapture portion of the likelihood can be written as the product of conditional (on releases) multinomials, with one multinomial for each group of releases. Similarly, the radio portion of the likelihood can also be written as the product of multinomials that are conditional on
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Chapter 17 Estimating Survival, Movement, and Other State Transitions
the releases A~ and A 2 in each location at each time period. For each release group, we have four possible fates: caught in core area, relocated (i.e., detected by radiotelemetry) but not caught in core area, relocated in peripheral area, and died. All four fates are observable for radioed animals, but only "caught in core area" is observable for tagged animals. 17.5.3.3.
Assumptions
The model of Powell et al. (2000a) requires the standard capture-recapture assumptions (Section 17.3.1) about homogeneity of rate parameters (survival and capture probabilities are similar for all animals with a particular mark type; movement probabilities are similar for all animals regardless of mark type) among individuals, and independence of fates. Standard assumptions also are required for survival and movement rate estimation from telemetry data (Sections 15.4 and 15.5). In addition, the model of Powell et al. (2000a) assumes that (1) movement between times i and i + 1 depends only on location at time i (the Markovian assumption) and (2) emigration from the two-patch system is not possible. In the most general sampling situation, assumption 2 means that the peripheral area must represent "the rest of the world," and satellite tracking would be necessary to sample such a peripheral area. In reality, there may be certain sampling designs (certain kinds of areas at particular seasons of the year) for which a well-defined and easily sampled peripheral area will be reasonable. 17.5.3.4. E s t i m a t i o n
Maximum likelihood estimation was carried out by Powell et al. (2000a) using a modified version of MSSURVIV (Hines, 1994). Estimation was based on the product multinomials for the summary statistics representing numbers of animals observed with the different possible fates. The parameterizations for survival and capture probabilities permit direct estimation of radio effect parameters. For example, define a parameter reflecting a radio effect on survival as
OL-~ sr(2)/S r(1). Then the survival probability for radioed animals can be rewritten as sr(2) __ Otsr(1).
and this parameterization can be used to estimate a radio effect directly, where oL = 1 denotes no radio effect on survival and 0 < oL < 1 indicates a negative
effect of radios (relative to standard tags) on survival probability. 17.5.3.5. Alternative M o d e l i n g
Various reduced-parameter versions of the model of Powell et al. (2000a) are of interest. For example, one can test for radio effects with a model for which survival and capture probabilities of radioed animals are constrained to be equal to those for animals with standard tags. Stationarity in survival and capture probabilities is also of interest. The models also can be made more general with the incorporation of group (e.g., sex) effects, age effects, and possible capturehistory dependence for the component marked with standard tags. Modeling various rate parameters as functions of time-specific or location-specific covariates should be possible as well. We envision many different types of combined-data models that are tailored to specific sampling designs. In some situations, it will not be possible to radiolocate animals in peripheral areas, but only to state with certainty that the radioed animals are no longer in the core area. In this case, emigration from the core area is a possible fate for radioed animals, but there is no group of animals in the noncore area on which to base the estimation of parameters associated with the noncore location. Several approaches to modeling this situation can be taken, depending on a variety of factors, such as whether emigration is viewed as temporary (as in Powell et al., 2000a) or permanent. In considering the joint use of telemetry and capture-recapture data, the investigator must assess the value of the additional information provided by telemetry data. In particular, telemetry data can be useful in permitting separate estimation of survival and emigration probabilities, which frequently are confounded in capture-recapture studies. The ability to locate animals off the study area, to detect all animals on a study area with certainty, and to locate dead animals, at least on the study area, all should contribute in various ways (depending on sampling design) to the separation of mortality and emigration probabilities. In addition, telemetry data should prove to be useful in increasing the precision of parameter estimates. For example, we have seen little use of telemetry data to aid in the estimation of capture probability. In standard capture-recapture modeling, a "0" at the end of a capture history is ambiguous, in that it can correspond to an animal's absence (death or permanent emigration) or to its presence and noncapture. With radios, an animal is always known to be present (or not) in an area exposed to capture efforts, so that the capture or non-
17.6. Study Design capture of an animal known to be in the capture area can be viewed as the outcome of a Bernoulli trial with associated capture probability. 17.5.3.6. Model Selection, Estimator Robustness, and Model Assumptions The approach to model selection described in Section 17.1.8 is applicable to models of both telemetry and standard tag data, and was used by Powell et al. (2000a). We are aware of no specific goodness-of-fit tests developed for such models and tentatively recommend the Pearson chi-square statistic at this time. Discussions of estimator robustness and model assumptions presented in Sections 15.4, 15.5, and 17.1.9 should be relevant to models with combined telemetry and capture-recapture data. The use of radioed animals that can be detected with certainty should reduce problems associated with heterogeneous capture probabilities with standard tags.
17.6. S T U D Y D E S I G N Designs of open population capture-recapture investigations can benefit from the general advice provided for model development in Chapter 3; that is, study design should be tailored to the questions being addressed and the parameters to be estimated. Issues such as replication and spatial and temporal variance components lead to important design recommendations, which are dealt with (at least generally) in Chapters 4-6 of this book. The focus here is on aspects of study design that are especially relevant to the conditional capture-recapture models for open populations. Given a narrow focus on estimation, it is important to tailor study designs to estimation-related study objectives. In the past, such tailoring necessarily was tied to one of a small number of estimation models and sampling methods. The available estimation methods involved closed-form estimators and variances developed by biostatisticians (e.g., Cormack, 1964; Jolly, 1965; Seber, 1965, 1970b). However, the development of flexible software for computing estimates based on user-defined models (e.g., White, 1983; Lebreton et al., 1992; White and Burnham, 1999) has dramatically changed this situation. In the preceding sections, we frequently have focused on specific models in order to illustrate model development, while also stressing alternative models and approaches in an effort to emphasize flexibility. Biologists now can develop a study design and associated model set for a wide range of estimation problems.
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The general recommendation to tailor a design to the specifics of the biological or management question makes the job of providing general design suggestions more difficult. However, some general design suggestions focus on three questions that are relevant to study designs: (1) what parameters are to be estimated, (2) how can assumption violations be minimized, and (3) how can precise estimates be obtained?
17.6.1. Sampling Designs and Model Parameters It should be clear from material in the previous sections of this chapter that the estimable parameters are determined largely by study design. For example, single-site capture-recapture studies and their associated models (Sections 17.1 and 17.2) can be used to estimate local survival rate, the complement of which includes both permanent emigration and mortality. If primary interest is in separation of these two components of loss, then additional data are needed. In particular, the models making use of various kinds of auxiliary observations (Section 17.5) should be selected for this purpose. The kind of auxiliary data obtained will depend on the specifics of the study, including the location of the primary study area (Does the study area cover most of the population's range? Is it isolated from other potential habitat?), the status of the study organism as a harvested species (Can recoveries of harvested animals be used?), and the ability of the organism to carry a radio transmitter (Can the animal carry a radio without adverse effects?). The robust design (Chapter 19) provides another source of auxiliary data that can be used to estimate parameters not estimable otherwise. Estimation of the probabilities of moving between locations requires sampling at the locations of interest. Simultaneous sampling at all sites is preferred, but if simultaneous sampling is not possible, then the investigator should try to approximate this situation to the degree possible. Consider two possible designs for sampling four sets of mist nets (at four different sites) for birds. In one, the investigator samples one site for 1 day, then moves to a different site and samples the next day, etc., finishing the rotation in 4 days. The next month, this rotation is repeated. In the other design, the investigator samples the first site for 1 day, waits a week, then samples the second site for 1 day, waits another week, etc. Although both designs sample all four locations each month, the first design more closely approximates simultaneous sampling. If there is little movement among sites over the 4 days of sampling,
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Chapter 17 Estimating Survival, Movement, and Other State Transitions
then multistate models can be used with data from this design. The use of multistate models with physiological or behavioral state variables requires the assignment of every captured animal to a state, though sometimes it is possible to do this for one state but not another. Reproductive activity is a characteristic about which interesting biological questions frequently are addressed. In many sampling designs, it will be simple to assign some animals unambiguously to a "breeder" category based on observations at a nest, e.g., with new young, during copulation, etc. However, observations of animals not engaged in these activities may not necessarily mean that the animal is not also a breeder. A useful research topic involves the use of probabilistic or imperfect state assignments in multistate modeling.
17.6.2. Model Assumptions 17.6.2.1. Homogeneity of Rate Parameters Previous investigations of capture-recapture assumptions have indicated that heterogeneous rate parameters can lead to biased estimates (Carothers, 1973, 1979; Gilbert, 1973; Pollock and Raveling, 1982; Nichols et al., 1982b; Nichols and Pollock, 1983b; Johnson et al., 1986; Rexstad and Anderson, 1992; Burnham and Rexstad, 1993), so it is important to design studies in ways that minimize heterogeneity. One aspect of design that is relevant to virtually all studies involves the information recorded for each animal on capture or resighting. As noted in Section 17.1.2, potentially relevant information includes group variables such as sex, attributes such as age that change nonstochastically, and attributes such as physiological condition and reproductive activity that vary stochastically. If the study area includes different habitat types, then geographic stratification also may be useful. By selecting the appropriate model from among the various possibilities presented in this chapter, it is possible to incorporate such information into an investigation. Model selection a n d / o r formal testing procedures then can be used to decide whether or not the selected variables are relevant to variation in demographic rate parameters (e.g., survival or movement probabilities) or sampling probabilities. If the variables are indeed associated with variation in model parameters, then group-specific or state-specific parameters should be retained in the model structure as a means of reducing problems associated with heterogeneity. If they are not relevant, the group-specific or state-specific parameters can be constrained to be equal across groups or states, with no loss of precision in estimation. Heterogeneous capture probabilities can be induced
by heterogeneity in sampling intensities, which should be at least partially under the control of the investigator. With regard to spatial sampling, it is important to sample all areas of a study area with similar intensity when practicable. When traps, mist nets, or other stationary sampling devices are used, it is important that all animals in the study area be exposed to these devices. This usually is ensured by a uniform spatial placement of devices, as in a trapping grid. In such spatial arrangements, the distance between adjacent devices should be smaller than the average daily movements or average home range radius of the species being sampled. Regardless of the exact nature of the spatial arrangement of sampling devices, the intention should be to have at least one trap or device (preferably more) within an animal's home range, so that animals should encounter at least one device each sampling occasion. If the number of sampling devices is not adequate to cover a study area in this manner, then the devices should be moved to randomly selected locations within the sampled area at each sampling occasion (Pollock et al., 1990). The use of multiple sampling methods is a way to reduce heterogeneity. The basic idea is that if certain animals behave in a manner that renders them especially difficult to encounter with one sampling method, then they may be more vulnerable to being sampled by an alternative sampling method. In this respect the combined use of physical capture and resighting as methods for obtaining samples should be useful. Similarly, the models for which capture-recapture data are augmented by auxiliary observations (Section 17.5) should provide ways of reducing problems associated with heterogeneity. A final component of study design that reduces problems associated with heterogeneous capture or resighting probabilities is sampling intensity. In general, heterogeneous capture probabilities lead to larger biases in parameter estimates when the probabilities are relatively small. For example, a capture-recapture study of a population in which half the animals have capture probabilities of 0.10 and the other half have capture probabilities of 0.35 should produce estimates with larger bias, compared to a study on a population in which half the animals have capture probabilities of 0.70 and the other half capture probabilities of 0.95. Thus, extra effort to increase capture probabilities should reduce problems associated with heterogeneous capture probabilities. Trap response in capture or survival probabilities is usually undesirable. Although models have been developed to deal with capture-history dependence, these models require extra parameters and therefore result in reduced precision in their estimates. Certain
17.6. Study Design forms of trap response cannot be modeled simply and present problems in inducing estimator bias. Traphappy responses are often associated with the use of baited traps. Such responses can be reduced or eliminated by the practice of prebaiting, placing baits either beside closed traps or in traps that are locked open. With prebaiting, animals are expected to become accustomed to traveling to the trap site in order to feed, so that when the traps are initially set, animals should then be caught with high probability. Trap shyness can also occur in some sampling situations. Mist-net studies of birds frequently encounter problems with net avoidance. If net avoidance results from birds learning net locations, then frequent relocation of nets within study areas may be useful. Use of trapping and handling methods that produce minimal stress on animals not only can minimize trap shyness but also possible trap response in survival probabilities. Use of reobservation methods that do not require physical capture of animals (as in studies based on resighting) should be useful in reducing trap responses in both capture and survival probabilities.
17.6.2.2. Tag Retention Selection of a marking method is an important aspect of design of survival studies. Clearly, it is desirable to select a mark that does not influence the animal's survival probability or even behavior, yet is likely to persist with negligible rates of loss. Larger marks, such as those used in resighting studies (neckbands, patagial tags) and radiotelemetry studies, are the most likely to result in changes in animal survival. Pilot studies of captive or semicaptive animals often provide inferences about the potential for marks affecting survival. Actual field studies with two kinds of marks [e.g., radios and legbands, as in the example of Powell et al. (2000a) (see Section 17.5.3)] can provide direct estimates of tag-related reductions in survival probability. The problem of tag loss does not appear to be adequately appreciated in many studies. It is not uncommon to see rather large studies that appear to have nonnegligible tag loss, yet with no means to estimate the magnitude of loss. In studies of animal survival, this problem is critical, because the parameter estimates correspond to tag survival rather than animal survival. Any capture-recapture survival study that shows even minimal potential for tag loss should include at least a sample of double-marked animals by which to estimate tag loss and animal survival (Arnason and Mills, 1981; Nichols et al., 1992a; Nichols and Hines, 1993). We have been involved with studies in which tag types have been changed over time (e.g., Spendelow et al., 1994; Fabrizio et al., 1999). Even with
491
tag loss estimates from double-tagging, the use of multiple tag types in survival studies makes analyses very complicated, at best. Our recommendation is to change tag types as infrequently as possible.
17.6.2.3. Instantaneous Sampling Another assumption that appears to be underappreciated is that of instantaneous sampling. Though this assumption is never met completely, it often is possible to select sampling periods during which animals experience negligible mortality. Indeed, the rule of thumb is to select sampling periods such that the time between sampling periods and, more importantly, the mortality likely to occur between successive sampiing periods, are large relative to the duration of the sampling period and the mortality occurring during this period (see discussion in Section 17.1.2). Selection of an appropriate sampling schedule thus involves both the season of the year and the duration of the sampling period. Whenever possible, it is best to avoid sampling during seasons of the year of suspected high mortality (e.g., harvest seasons; periods of severe weather, such as winters in some areas). Similarly, it is best to try to achieve high capture probabilities with intense sampling over a short period, rather than less intense sampling over a long period. As noted in Section 17.1.9, however, it is sometimes possible to model mortality during the sampling period (e.g., Tavecchia et al., 2002). 17.6.2.4. P e r m a n e n t Emigration
The assumption that all emigration is permanent is commonly listed for capture-recapture estimators for open populations. However, as noted by Burnham (1993), Barker (1995, 1997), and Kendall et al. (1997) (also see Sections 17.5.1 and 19.5.1), random temporary emigration produces no bias in survival estimates, but changes the interpretation of capture probability. However, temporary emigration may sometimes follow a first-order Markov process, such that animals have different capture probabilities depending on whether they were in the area exposed to capture efforts in the previous sampling period. Temporary emigration of this sort can produce biased estimates of parameters of interest (Kendall et al., 1997). Markovian temporary emigration is best handled using the robust design (Chapter 19), although open models with trap dependence in capture probabilities (Sandland and Kirkwood, 1981; Pradel, 1993) (also see Section 17.1.6) sometimes can be used to approximate Markovian temporary emigration. Auxiliary observations permit estimation under a model of Markovian temporary
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Chapter 17 Estimating Survival, Movement, and Other State Transitions
emigration when the emigration probabilities are constant over time (Barker, 1995) (also see Section 17.5.2).
17.6.3. Estimator Precision Various aspects of study design are relevant to the precision of resulting estimates. Good precision (i.e., small variances and coefficients of variation) is an important determinant of test power and inferential strength, and should be a major consideration in study design. Pollock et al. (1990) presented information about the influence of study design on precision of survival estimates. A general conclusion is that for any set of conditions (e.g., fixed survival probability and population size, and fixed number of sampling periods), cv(q~) decreases as capture probability increases. Certainly, any design modifications that can increase capture probabilities will be useful. These modifications could include increases in the density of capture devices, or the number of observers trying to resight marked animals, or the number of consecutive days on which traps are set or observations are taken. Of course, longer sampling periods deviate more strongly from the instantaneous sampling assumption, so that a tradeoff exists between meeting this assumption and increasing capture probability. Another general inference is that for any set of conditions, estimator precision tends to increase as the number of sampling occasions increases. This is true for time-specific estimates q~i, but the relationship is even stronger when time-invariant parameters are estimated (e.g., q~). For any set of conditions, cv(q~i) decreases as population size increases. Because the study population frequently is defined by the investigator, population size is under investigator control, at least to some extent. A study design involves a tradeoff between size of the sampled population and sampling intensity, in that effort (expressed in terms of numbers of traps, number of person-hours of observations, etc.) can be either spread out over a larger area to sample a larger population, or it can be concentrated in a smaller area to produce a higher capture-resighting probability. In many situations, increases in precision resulting from increased capture probability may be larger than those resulting from increased population size, but the important point is to keep this tradeoff in mind when considering the specific design of a new study. A final determinant of estimator precision is the magnitude of the survival rate. Initially, it might appear that this quantity is not under control of the investigaton However, the quantity of interest is not survival rate scaled to some arbitrary time (e.g., 1 year), but the probability of surviving the interval between succes-
sive sampling periods. Thus, sampling frequency should be tailored to the organism under study. Sampling meadow voles (Microtus pennsylvanicus) at annual intervals, for example, would represent a poor design. Because few marked animals in year i would survive to have a chance of being recaptured in year i + 1, there would be little information for estimating either survival or capture probability. It generally is a good idea to select sampling intervals that provide a relatively high survival probability (e.g., S i > 0.5). However, probability estimators associated with rare events have their own difficulties. For example, the numerical algorithms used in capture-recapture software (e.g., MARK) (White and Burnham, 1999) often have difficulties with parameter estimates near boundaries (e.g., probabilities near 1 or 0). Thus, it is useful to avoid intervals that are so short that deaths are rare and survival probabilities approach 1.
17.7. D I S C U S S I O N Because they allow for mortality, migration, and recruitment, open populations require models that include biological attributes not found in closed population models. The need to include parameters for these attributes means that open population models are considerably more complicated and in consequence the precision of parameter estimates is comparatively lower than for closed populations. This is yet another manifestation of the tradeoff between complexity and precision (see Section 7.1). The modeling approach in this chapter builds on the CJS model, which extends the closed population models in Chapter 14 by incorporating nonstationary survival probabilities. Data for the CJS model consist of marked and unmarked captures at each of a number of sampling occasions, which are summarized in terms of particular capture histories or as summary statistics in an mij -array. Numbers of recaptures in the model are conditioned on the number of releases at each sampling occasion of either unmarked animals (for capturehistory data), or the combination of marked and unmarked animals ( mij-array data). The model therefore represents only a part of the information available in a sample, and a fully efficient use of data requires the modeling of the unmarked captures. The statistical form of the CJS model consists of a conditional product-multinomial distribution of recaptures, from which maximum likelihood estimates and their variances can be derived. Closed forms for the maximum likelihood estimators can be seen as multitemporal analogs of the Lincoln-Petersen estimator (see Section 14.2).
17.7. Discussion A large part of this chapter has dealt with extensions of the CJS model to allow for a cohort structure, which can be incorporated in the model via cohort-specific survival and capture probabilities. Thus, Pollock's (1981b) model (Section 17.2) recognizes age cohorts, and the multistate models of Section 17.3 accommodate both phenotypic and geographic cohorts. The inclusion of both temporal and cohort variation in the probability structure generates a wealth of special cases in model parameterizations, whereby any number of constraints involving stationarity conditions, equality of parameters across cohorts, and other parameter restrictions can be imposed on model parameters. Because the constrained models include fewer parameters than unconstrained models, a collateral benefit is increased estimator precision in the resulting parameter estimates. Though the conditional models in this chapter focus on the estimation of survival and capture probabilities, we used the artifice of reverse-time modeling in Section 17.4 to address recruitment to the population. The approach reverses the direction of time and replaces the focus on survival in the CJS model with a focus on recruitment, building on the recognition of Pollock et al. (1974) that a backward process with recruitment and no mortality is statistically equivalent to a forward process with mortality and no recruitment. Thus, the survival probabilities of a forward process are reinterpreted in a backward process as "seniority parameters"
493
(Pradel, 1996) that inform the recruitment process via Eqs. (17.35) and (17.36). Finally, we explored methods to combine capture-recapture data with information collected from other sources such as band recoveries, radiotelemetry, and resightings of marked animals between capture occasions. In each case we saw that the additional information provides opportunities to improve estimator performance and to estimate new parameters, but at some cost in the mathematical intricacies of data management and analysis. There are numerous opportunities for additional modeling and analysis with these and other combined approaches, as their statistical analysis and application are quite new and yet to be fully explored. The models in this chapter all are based on multinomial distributions of recaptures, conditioned on initial captures. A fully efficient use of capturerecapture data requires the statistical modeling of initial captures as a component, along with the multinomial recapture distributions, of a comprehensive statistical model for the sampling process. In the next chapter we add additional stochastic features to the probability models described here, so as to account for random variation in initial captures. This added feature allows us to focus on a broader suite of biologically informative parameters, including a simultaneous accounting of recruitment and mortality as well as the estimation of population size.
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C H A P T E R
18 Estimating Abundance and Recruitment with Mark-Recapture Methods
18.1. DATA STRUCTURE 18.2. JOLLY-SEBER APPROACH 18.2.1. Model Structure 18.2.2. Model Assumptions 18.2.3. Estimation 18.2.4. Alternative Modeling 18.2.5. Model Selection, Estimator Robustness, and Model Assumptions 18.2.6. Example 18.3. SUPERPOPULATION APPROACH 18.3.1. Model Structure 18.3.2. Model Assumptions 18.3.3. Estimation 18.3.4. Alternative Modeling 18.3.5. Model Selection, Estimator Robustness, and Model Assumptions 18.3.6. Example 18.4. PRADEUS TEMPORAL SYMMETRY APPROACH 18.4.1. Model Structure 18.4.2. Model Assumptions 18.4.3. Estimation 18.4.4. Alternative Modeling 18.4.5. Model Selection, Estimator Robustness, and Model Assumptions 18.4.6. Example 18.5. RELATIONSHIPS AMONG APPROACHES 18.6. STUDY DESIGN 18.6.1. Parameters to Be Estimated 18.6.2. Model Assumptions 18.6.3. Estimator Precision 18.7. DISCUSSION
In this chapter we consider the estimation of population size and recruitment using capture-recapture data for open (to gains and losses between sampling occasions) populations. The relevance of population size and recruitment in this book should be clear, because population size is a state variable of interest in most of the population models that have been discussed, and recruitment of new animals is one of the processes responsible for population change (see Chapters 7 and 8). Methodologically, this chapter can be viewed as an extension of Chapter 17. We consider exactly the same kinds of data on animals that are marked, released, and recaptured at discrete sampling periods throughout the course of a study. In Chapter 17 the modeling of capturehistory data was discussed in terms of survival and recapture or resighting probabilities. The standardtime (as opposed to reverse-time) models of Chapter 17 were developed by first conditioning on animals that are marked and released, and then writing capture probabilities for each capture history in terms of survival and capture parameters. These conditional probability models are incorporated as components of the more comprehensive models presented in this chapter. In Sections 18.2, 18.3, and 18.4, three classes of models representing different parameterizations for the same data are presented and discussed. Each parameterization permits estimation of abundance N i and quantities related to recruitment. In particular, the models in Section 18.4 exploit the temporal symmetry in capture-recapture data that was noted in Section 17.4 on reverse-time modeling. One reverse-time pa-
495
496
Chapter 18 Estimating Abundance and Recruitment
rameterization permits direct estimation of the finite rate of population increase, ~'i -- Ni+l/Ni, as a model parameter.
18.1. DATA S T R U C T U R E For standard capture-recapture sampling, the data collected are identical to those used for the models of Chapter 17. However, for capture-resighting studies there is an important difference between models of this and the previous chapter. Estimation of survival and recapture probabilities using the models of Chapter 17 was shown to depend only on reobservations of marked individuals. On the other hand, the estimation of abundance and recruitment using models in this chapter requires information on the number of unmarked animals that are caught or sighted in sampling efforts. In standard capture-recapture studies, unmarked animals that are captured are given tags permitting individual identification, and the number of these unmarked captures is important in estimating abundance. In studies in which reobservations are obtained primarily by resighting, an effort must be made to count the number of unmarked animals encountered during the resighting sampling efforts. These counts of unmarked animals, which are not needed for survival rate estimation with the models of Chapter 17, play a key role in estimation of population size and recruitment. Model development in this chapter focuses on data for a single age class of animals (e.g., adults) and thus is closely related to the models of Section 17.1. The reason for this focus is that multiple-age models cannot estimate abundance of young animals based on count statistics, such as the number of young caught. This inability is directly related to the fact that capture probabilities cannot be estimated for an initial age class in age-stratified models (see Section 17.2). Most of the formal development for abundance estimation has focused on "adult" animals, or at least on single-age models (e.g., Jolly, 1965; Seber, 1965, 1982; Brownie et al. 1986; Pollock et al., 1990; Schwarz and Arnason, 1996). We note that the estimation of abundance for age classes other than the first is possible based on the models in Section 17.2, as described in Section 18.2.4. In addition, the estimation of abundance for even the initial age class is possible using the robust design described in Chapter 19. The classical conditional model for single-age class data was labeled in Chapter 17 as the CormackJolly-Seber model. Cormack (1964) focused directly on conditional modeling of survival and capture probabilities, whereas the modeling of Jolly (1965) and Seber (1965) also included additional model components for estimation of population size and recruitment. Thus,
it is appropriate to refer to the classical model that includes abundance and recruitment, as well as survival and capture probability, as the Jolly-Seber model. The data structure for models of this chapter is identical in most cases to that presented for single-age conditional models in Chapter 17. We can again think of two kinds of summary statistics, capture-history data and mq-array data. The capture-history data are the numbers of animals exhibiting each observable capture history (Table 17.1). For example, Xl01 denotes the number of animals in a three-period study that exhibited capture-history 101 (caught in periods 1 and 3, but not in period 2). In studies based on resightings, the numbers of unmarked animals observed during resighting efforts are simply treated as animals seen but not released back into the population. For example, the entry "0010 - 3 5 " in a table such as Table 17.1 would indicate that 35 unmarked animals were seen on sampling occasion 3 of a four-period study. The minus sign simply indicates that these 35 animals were not released into the population with marks (see Section 17.1.1). We introduce x~, to accommodate the removal of individuals from the sampled population at some point in the study time frame. Thus in a threeperiod investigation xi-10 is the number of animals exhibiting capture history 110 and not released following the final capture (sampling period 2 in this case). The mq-array data are summarized as in Table 17.2. In addition to information in the mq-array, abundance estimation requires the number of unmarked animals captured or sighted at each occasion (denote this number as ui). Recall from Chapter 17 that mij is the number of animals released at time i (members of R i) that are next caught or resighted at period j. The number of marked animals caught at period j can thus be computed as j-1 mj = ~,
mq.
i=1
Let n i be the total number of animals (marked and unmarked) caught at time i (n i -- m i 4- ui). If there are no losses on capture or, more generally, if all animals encountered at i are released back into the population with tags, then the number of animals caught equals the number released (n i -- Ri). In this case, the number of unmarked animals does not need to be recorded separately in the mq-array, because it can be simply computed as u i = n i - m i, where n i = R i. Two additional statistics (also defined in Section 17.1.2) are required: r i = ~]=i+1 mij is the number of animals released at i ( R i) that subsequently are recaptured; z i is the number of animals caught before sample period i, not caught in i, and recaptured at some period after i. In addition, deaths on capture can be modeled. Thus,
18.2. Jolly-Seber Approach let d i and d~ be the numbers of m i and ui, respectively, that are not released back into the population at i. The numbers of animals released into the population following each sampling period i thus can be written as R1
U 1
--
d{
R i = m i + ui
--
d i - d~
=
and
for i = 2, ..., K. It is worth emphasizing that in this chapter, u i represents the number of previously unmarked individuals that are captured or observed at time i, including those that are released after capture and those that are not. This differs from the meaning of u i in Chapter 17 (see Section 17.1.2), where ui was restricted to individuals that are both captured and released. By allowing for the possibility of different fates (release or removal) following capture, additional stochastic elements are introduced in the models discussed here that do not appear in the Cormack-Jolly-Seber model of Chapter 17.
18.2. JOLLY-SEBER APPROACH 18.2.1. Model Structure Parameters required for the Jolly-Seber model include the capture probability (Pi) and its complement (qi = 1 - Pi), the survival probability (q~i), and the probability of not seeing an animal again following period i [• see Eq. (17.1)], all of which were defined formally in Section 17.1.2. In addition, the parameters "l]i and ~ represent the probabilities of release for marked ( m i) and unmarked (u i) animals caught at i. Thus, E(di]mi) = mi(1
-
,l-]i )
and E(d~]ui)
=
ui(1 - n l ) .
The following parameters are u n k n o w n random variables, the values of which are to be estimated: Ni
Mi
U i =
Bi
N i - M i
The total number of animals in the population exposed to sampling efforts in sampling period i. The number of marked animals in the population just before sampling period i. The number of unmarked animals in the population just before sampling period i. The number of new animals joining the population between samples i and i + 1 and present at i + 1.
497
As with the conditional modeling of the previous chapter (e.g., Section 17.1.2), it is possible to consider models based on capture-history data or on m q-array summary statistics. Consider the modeling of animals caught in the first sampling period, Ul, and their subsequent capture histories for a three-sample study. We use the notation {x~, xoT} to denote the set of possible capture histories (in this example, observable histories for animals caught in period I of a three-period study), where the " - " superscript again indicates animals not released following final capture: P[{x+, x~}] = ul!(ul - ul)! p~',(1 - pl) U.... ul! 1
{
Ul!
(18.1)
X ['II(x-)!(xo; )! (~1{X1)..... (1 - xl{)...... (.q~1P2.q2x2)Xllo[.q{~plp2(1 _ ~i2)]/110 o. x [~1~q~1(1 - p2)q)2P3-q3] ..... [n'l~P1(1 - p2)qo2P3( 1 - n3)] T M X ('q{q~lP2"rl2qo2P3~3) .... [~plP2.q2qo2P3( 1 _ ~3)jxm},
where the index ~o ranges over the capture histories 100, 110, 101, and 111. Equation (18.1) differs from its counterpart under the conditional Cormack-Jolly-Seber approach [Eq. (17.2)] in two important respects. First, Eq. (18.1) does not condition on the new releases in period 1 (the R1), but instead includes an initial binomial term that involves the capture of u I animals from the available population of U1 animals. Second, even the modeling of the subsequent histories of the u I animals caught at time 1 differs from Eq. (17.2) in including the "11iand TI} and thus in modeling the process by which animals are not released (trap deaths, investigator removals, etc.). The complete model for data from the entire threesample capture-recapture study is written as the product of three expressions such as Eq. (18.1), with an expression for each group of unmarked animals that are caught, u 1, u 2, and u 3. We also present the approach of Seber (1982) and Brownie et al. (1986) for modeling such data using the m q-array summary statistics. This approach decomposes the distribution function for the observed variables, {/,/i}, {di, dl}, {m/j}, into three components as P({Ui}, {di, d~}, {mij })
-- {Pl({Ui} l {Ui}, {pi})] {P2({di, d~} ] {m i, ui}, {'1-]i, '1-]~})]
(18.2)
X ~P3({mij}[{ai}, {q0i, pi})].
The first component deals with the capture of unmarked animals and can be written as Pl({Ui}]{Ui}, {Pi}) =
(18.3) 1-I i=1
i! _ pui(1 _ p i ) U i - u i U (U i ui)!
498
Chapter 18 Estimating Abundance and Recruitment
(Seber, 1982). The second component of Eq. (18.2) concerns marked and unmarked animals that are caught but not released back into the population: Pa({di, dl} l {mi, bli}, {~qi, 11~}) __ i.=~ 1 d~l(ui. bli!- d~)W. tlli! z ,~ui-d~ p (1 -- Tlir)d;" --
X =
(18.4)
mi ! d i ! ( m i _ di)! (Tli)mi-di(1 -- ~qi)di .
The third component of the distribution of Eq. (18.2) is simply the conditional probability distribution written for the mq in Eq. (17.6).
18.2.2. Model Assumptions The assumptions for the Cormack-Jolly-Seber model listed in Section 17.1.2 also are required for the Jolly-Seber model. However, assumption (1), that every marked animal in the population at sampling period i has the same probability of being recaptured or resighted, must be modified for application to Jolly-Seber modeling. Under the models presented in Eqs. (18.1), (18.2), and (18.3), the capture probability parameters Pi also apply to unmarked animals. Thus, for application to the Jolly-Seber model, assumption (1) must be modified to state that every animal (marked and unmarked) in the population at sampling period i has the same probability Pi of being captured or sighted. The discussion of model assumptions presented in Section 17.1.2 is relevant to the Jolly-Seber and related models. Much of this discussion was directed at assumptions (1) and (2) involving homogeneity of the rate parameters ~i and Pi. The revision of assumption (1) to include unmarked animals leads to additional possibilities for assumption violations. For example, permanent trap response in capture probability refers to the situation in which different capture probabilities apply, depending on whether the animal is marked or unmarked (Nichols et al., 1984b) (see Section 17.1.2). Because estimation of survival and capture probabilities in conditional models depends only on recaptures of marked animals, it does not matter for these models (Chapter 17) that animals may exhibit increases or decreases in capture probability following initial capture. However, under the Jolly-Seber model [see Eqs. (18.1 )-(18.3)], the capture probabilities estimated using recaptures are assumed to apply also to unmarked animals, and permanent trap response renders this assumption false. The suggestions presented in Section 17.1.2 for dealing with model assumptions all should be relevant to abundance estimation. For example, when variation in capture probability is associated with state variables that are both static and discrete (e.g., sex), stratification
of capture-history data into groups frequently is useful. For deterministically dynamic state variables such as age, special models can be developed as in Section 17.2 (also see Section 18.2.4). Multistate models can again be used to deal with state variables that are discrete, yet stochastically dynamic. Given special attention to homogeneity of rate parameters for marked and unmarked animals under the models of Chapter 18, the discussion of assumptions presented in Section 17.1.2 should be adequate for the models of Chapter 18 as well. However, there are important differences between the models of Chapters 17 and 18 as to the robustness of estimators for the rate parameters identified in Chapter 17 and the additional parameters and unknown random variables of Chapter 18. These robustness issues are discussed in Section 18.2.5.
18.2.3. Estimation Equation (18.1) represents the probability distribution for capture histories of new (unmarked) animals released in period 1. The probability distribution for all capture histories resulting from an entire study is written as the product of K such expressions, one for the unmarked animals caught at each sampling occasion of the study. Equation (18.2) represents the probability distribution for the rely-array summary statistics over an entire study. Viewing the right-hand sides of Eqs. (18.1) and (18.2) as likelihood functions, it is possible to obtain the maximum likelihood estimates of model parameters q~i, Pi, ~qi, and ~ , using the methods of Chapter 4. Specifically, maximum likelihood estimation of these parameters is based on the portions of Eqs. (18.1) and (18.2) that do not include the unknown random variables Ui. For example, the P2 component of Eq. (18.2) specified in Eq. (18.4) can be used to estimate ~i and ~q~ based on the numbers of marked and unmarked animals that are caught and the numbers in these groups that are released. Similarly, the P3 component of Eq. (18.2) is specified in Eq. (17.6) and is used to estimate q~iand Pi based on the capture histories of animals that are caught and released, as described in Section 17.1.2. Closed-form maximum likelihood estimators based on the Cormack-Jolly-Seber model for parameters q~/and Pi were presented in Section 17.1.2 [Eqs. (17.7), (17.8), and (17.10)]. The focus in this chapter is on the unknown random variables N i and B i, estimators for which can be obtained in multiple ways. For example, Seber (1982) notes that the conditional expectation E ( n i ] N i) = N i p i
(18.5)
can be used to obtain a moment estimator for abundance by 1~ i -- Yli/Pi.
(18.6)
18.2. Jolly-Seber Approach Estimator (18.6) corresponds to the canonical estimation approach of Section 12.2, in that it is simply a count statistic divided by the corresponding estimate of detection probability. Using t h e Cormack-JollySeber model estimator Pi = m i / M i for capture probability [Eq. (17.7)] in conjunction with Eq. (18.6) yields (18.7)
1Qi -- l ~ i n i / m i .
This estimator of population size in Eq. (18.7) still requires an estimate M i of an u n k n o w n random variable Mi, and the estimator of Eq. (17.10) is typically used for this purpose. A reduced-bias version of the estimator in Eq. (18.7) is frequently used under the Jolly-Seber model with time-specific survival and capture probabilities (Seber, 1982; Pollock et al., 1990). Closed-form estimators for var (/~i) and cov(/~i,/~j) are presented by Jolly (1965), Seber (1965, 1982), and Pollock et al. (1990). The abundance estimators in the preceding development are presented for the general Jolly-Seber model (e.g., see Seber, 1982; Pollock et al., 1990). However, slightly different approaches sometimes can be used for reduced-parameter models. For example, Jolly (1982) and Brownie et al. (1986) estimated abundance as the sum of the estimated numbers of marked and unmarked animals alive in a given period:
/Qi = Mi q- ~/i.
(18.8)
Jolly (1982) and Brownie et al. (1986) used the relationships E ( m i + zi l M i ) = M i ( 1 - qixi)
and (18.9)
E ( u i l U i) -- Uip i
to develop the estimators 1~ i __
(18.10)
mi + Z i (1 - dtiy(i)
and
499
This expression is similar to Eq. (17.34) in that population size at time i + 1 is viewed as the sum of two components: (1) new recruits not present in the population at i (B i) and (2) survivors from the previous period [ q ~ i ( N i - yl i q- Ri) , where the t e r m - - t l i q- R i simply subtracts the number of animals removed from the population during sampling efforts]. The conditional expectation of Eq. (18.12) leads to the estimator
/~i-- ]Qi+I --
~i(1Qi-
Hi q- Ri)
(18.13)
for the number B i of new recruits. On reflection this estimator is intuitively reasonable. The number of recruits between i and i + 1 is expressed as the difference between estimated abundance at i + 1 and the estimated number of survivors from the previous period. This recruitment estimator is available for sampling periods i = 2, ..., K - 2. Estimators for variances and covariances associated with /3i are presented by Jolly (1965), Seber (1965, 1982), and Pollock et al. (1990). An estimator with reduced bias also is available (Seber, 1982; Pollock et al., 1990).
18.2.4. Alternative Modeling The material presented in Sections 18.2.1-18.2.3 concerns single-age models (e.g., for adult animals) that are parameterized with time-specific capture and survival probabilities. Indeed, most of the work on abundance estimation has involved this specific model (also see Section 18.3). Historically, two classes of alternative models for abundance estimation have received attention: (1) partially open models in which only gains or only losses to the population can occur and (2) timeconstant models in which capture a n d / o r survival parameters are assumed constant over time. 18.2.4.1. P a r t i a l l y O p e n M o d e l s
(18.11)
CIi -- u i / P i ,
where qi -- 1 - Pi. This approach to abundance estimation is used in programs JOLLY and JOLLYAGE (Brownie et al., 1986; Pollock et al., 1990). Note that all of the above abundance estimators require estimates of capture probability ]9i and are thus available for the sampling periods i - 2, ..., K - 1 for which capture probability can be estimated (Section 17.1.2). Under reduced-parameter models (e.g., capture probabilities assumed to be constant over time; Pi -- P), capture probability sometimes can be estimated for additional periods (e.g., periods 1 and K) as well (Section 18.2.4). Estimation of recruitment under the Jolly-Seber model is based on the relationship E ( N i + I [ N i , Bi) = B i + ~ P i ( N i - d i - d l )
= Bi+ ~i(Ni-
n i + Ri).
(18.12)
These models were considered by Darroch (1959), who provided estimators for the case of no losses on capture, and later by Jolly (1965), who viewed these models as special cases of his general open model (also see Seber, 1982). The death-only model can apply to isolated populations not subject to immigration, if sampling is restricted to a sufficiently short time period that new recruits resulting from reproduction are not added to the population. Other sampling situations include those in which timing of the study rules out the possibility of new recruits (see Haramis and Thompson, 1984) or those in which recruits can be identified and excluded from the analysis. The death-only model assumes no recruitment into the population over the course of the K sampling periods (B i -- 0 for i = 1, ..., K - 1). Thus, an animal first captured at any sampling period during the study (a
500
Chapter 18 Estimating Abundance and Recruitment
member of u i) is known (by assumption) to have been alive at all previous periods (
~,
j=i+l
Uj
be the number of animals not caught at i that are known to be alive at i, because they are seen later. Using an intuitive argument similar to that used for estimation of M i under the Cormack-Jolly-Seber model (Section 17.1.2), we have
( zl E Ni _
)~E(F~)
(18.14)
Yli
The denominator of the ratio on the left-hand side of Eq. (18.14) reflects the number of animals in the population on sampling occasion i that were not caught on that occasion, whereas the denominator on the right-hand side reflects the animals caught and released at i. The numerators of the two ratios reflect the numbers of animals in each group (caught and not caught at i) that are caught on subsequent occasions. Under the Jolly-Seber assumptions, these expectations should be approximately equal. Because N i is the only u n k n o w n random variable in Eq. (18.14), the expression can be rearranged to obtain the estimator 1~ i -- Yl i -}-
Riz~/r i.
(18.15)
Note that /~/i in Eq. (18.15) is defined for periods i = 1, ..., K - 1. A bias-adjusted estimator for N i under this model is presented by Seber (1982; also see Pollock et al., 1990). Survival and capture probabilities can be estimated under this model, and variance and covariance estimators for all of the above estimators also are available (Jolly, 1965; Seber, 1982; Pollock et al., 1990). For reasons of completeness, we note that estimators also have been developed for the unlikely situation in which there is recruitment to the population, but no death or emigration (so q~i = 1 for all i). Births-only models have been considered by Darroch (1959), Jolly (1965), and Seber (1982). In this situation, the number of marked animals in the population just before any sampling period i is given by the sum of the new releases into the population over all prior sampling periods: i-1
Mi = ~
(Rj-mj).
j=l
If capture probabilities are equal for marked and unmarked animals, then the following approximate equality should hold: \Ni!
E
.
(18.16)
Equation (18.16) is then solved for abundance to yield the estimator Xi
=
Mini/mi.
(18.17)
This estimator closely resembles the Lincoln-Petersen estimator for abundance of closed populations and is defined for periods i = 2, ..., K. Equation (18.17) is also similar to the standard Jolly-Seber estimator [Eq. (18.7)], with the difference that the number of marked animals in the population at any time i is known for the no-death model, whereas it must be estimated in the completely open Jolly-Seber model. The number B i of new recruits then can be estimated by /~i-~ /~i+1 -- /~i"
Variance and covariance estimators are also available under this model (Darroch, 1959; Jolly, 1965; Seber, 1982).
18.2.4.2. Reduced-Parameter Models Jolly (1982) and Brownie et al. (1986) considered abundance estimation under reduced-parameter models in which capture probability a n d / o r survival probability are assumed to be constant over time [also see Crosbie and Manly (1985), Schwarz and Arnason (1996), and Section 18.3]. Estimation of abundance and related quantities under these models is accomplished using the estimators of Eqs. (18.8)-(18.11) (Jolly, 1982; Brownie et al., 1986). Reduced-parameter models lead to gains in the precision of abundance estimates and increases in the number of sampling periods for which abundance can be estimated. For example, under model (q~,Pt) with survival constant over time, capture probability for the final sampling period (PK) can be estimated, permitting estimation of NK. Under model (q~, p), abundance can be estimated for all sampling periods. As with the partially open models, variance and covariance estimators under the reduced-parameter models were provided by Jolly (1982) and Brownie et al. (1986). Program JOLLY (Brownie et al., 1986; Pollock et al., 1990) implements the models considered by Brownie et al. (1986), and POPAN-5 (Arnason and Schwarz, 1999) is a flexible software package that can be used to estimate abundance for a large class of userdefined models.
18.2.4.3. Alternative Modeling Using Canonical Estimators Other than the partially open and reduced-parameter models described above, there has been little work on estimation of abundance using the alternative models described in Chapter 17. An exception to this statement is the work of Schwarz and Arnason (1996), to
18.2. Jolly-Seber Approach be discussed in Section 18.3. In principle, the estimation of animal abundance can be accomplished using any of the alternative models of Chapter 17. Recall that all the models in Chapter 17 included parameters for capture probability, which were viewed as nuisance parameters that are needed to estimate survival and movement probabilities. The specific role of capture probability in estimating survival can be seen most easily by recalling the closed-form survival estimator under the Cormack-Jolly-Seber model [Eq. (17.8)]. This estimator is based on two statistics obtained directly from the data (the numbers of marked captures m i and releases R i) and two estimated quantities [the numbers of marked animals alive at times i (M i) and i + 1 (~/Ii+1)]. The latter estimates can be obtained by applying the canonical abundance estimator of Section 12.2 to the number of marked animals caught:
Mi = mi/]9i.
(18.18)
Estimated capture probability thus is used to estimate the abundance of a specific class of animals, namely, those animals that are marked and therefore known to have been caught in previous sampling periods (also see Section 15.2). The canonical abundance estimator of Section 12.2 can be used to estimate the abundance of any group of animals for which a count is available and for which an associated detection probability can be estimated. The capture probability parameters of Chapter 17 are estimated using data on captures and recaptures of marked animals. However, in cases in which these capture probabilities logically apply to unmarked animals as well, it should be possible to estimate the abundance of marked and unmarked animals, as in Eq. (18.6). We consider below the estimation of abundance using the alternative models considered in Chapter 17.
18.2.4.4. Time-Specific Covariates In some cases, capture probability can be modeled as a function of time-specific covariates using the link functions presented in Chapter 17. For example, covariates such as sampling effort and weather conditions sometimes can be useful in modeling capture probabilities (Clobert and Lebreton, 1985; Clobert et al., 1985, 1987). In addition, survival probability sometimes can be modeled as a function of covariates (Section 17.1.4). Although some software used to fit these models does not provide estimates of abundance, the ]9i can be used in conjunction with Eq. (18.6) to estimate abundance. Again, this estimation approach is based on the assumption that the ]9i estimated from marked animals applies equally to unmarked animals. Covariate models can also be fit in POPAN-5 (Arnason and Schwarz,
501
1999), which provides the associated estimates of population abundance.
18.2.4.5. Group-Specific Covariates The details of abundance estimation using models with group-specific covariates depend on the manner in which capture probability is modeled. If capture probability for period i is described with the same parameter ]~i for the different groups, then abundance estimation for animals in each group separately, and for animals in all groups combined, can proceed with Eq. (18.6). If capture probability is modeled separately for the different groups (e.g., sex classes), then abundance for each group can be estimated separately as =
i/Pi,
n s
(18.19)
where s denotes group. Abundance for all groups combined (e.g., for the entire population) then is estimated by
/~i = ~_j /~i" s
(18.20)
Abundance estimation with group-specific covariates can be accomplished using POPAN-5 (Arnason and Schwarz, 1999).
18.2.4.6. Capture History
Dependence
Estimator (18.6) also can be useful for models incorporating capture-history dependence, but the nature of the dependence is critically important. For example, trap dependence in capture probabilities can be modeled using two different capture probability parameters in sampling period i, one for animals caught in the previous period, i - 1, and another for animals not caught the previous period (Section 17.1.6) (see Cormack, 1981; Sandland and Kirkwood, 1981; Lebreton et al., 1992; Pradel, 1993). Abundance of marked animals that were caught the previous period, and not caught the previous period, could be estimated separately as in Eq. (18.18). The key determinant of abundance estimation for the entire population of marked and unmarked animals is whether one of the capture probabilities for marked animals also is relevant to unmarked animals. In some cases it may be reasonable to assume that the capture probability for marked animals not caught the previous period would also be applicable to new, unmarked animals. Unfortunately, we know of no way using standard open-model data to test whether estimated capture probabilities based on marked animals also apply to unmarked animals (e.g., goodness-of-fit tests are not useful for this purpose). However, the robust design discussed in Chapter 19 may be useful, as capture-recapture models for closed populations can be used to test hypotheses
502
Chapter 18 Estimating Abundance and Recruitment
about capture probabilities of previously unmarked animals and to estimate abundance of groups with different previous capture histories. When trap response occurs in survival probabilities (Section 17.1.6), the applicability of Eq. (18.6) to abundance estimation depends on the reason underlying the trap response. For example, if unmarked animals exhibit lower survival probabilities compared to previously marked animals (e.g., because of trappingrelated or handling-related mortality), then it is possible that capture probability is similar for marked and unmarked animals. The number of marked animals present ( M i) will be smaller than if there was no trap mortality, but this reduction in abundance will be reflected in reduced numbers of animals caught (m i) and will not lead to problems in abundance estimation. However, when the different survival probabilities of marked and unmarked result from transients passing through the sampled area [Eq. (17.13)], then the investigator must decide if abundance estimates should include transients or just residents. If only residents are of interest, then abundance estimation can be based on the number of marked animals (which are resident by definition) caught, and the estimated fraction of unmarked animals caught that are resident (1 - ~-i): 1Q r = m i + ui(1
/~i
-
-
Ti)
'
(18.21)
where N~ denotes the number of resident animals. Equation (18.21) requires the assumption that marked and unmarked residents exhibit the same capture probabilities. If interest is in the combined abundance of transients and residents, then the estimator of Eq. (18.6) can be applied to both marked and unmarked animals, providing there is no difference in capture probability associated with mark status. The robust design (Chapter 19) may again be useful in testing assumptions about similarity of capture probabilities of previously marked and unmarked animals. 18.2.4.7. M u l t i p l e - A g e
Models
The Jolly-Seber model (Jolly, 1965; Seber, 1965) corresponds to a single age (usually "adults") and a single state (animals in different locations or physiological states are not considered separately). The alternative models discussed above all fit this single-age, singlestate model form. With respect to multiple-age models, we focus initially on the models of Section 17.2.2 (Pollock, 1981b; Stokes, 1984; Brownie et al., 1986). Recall that the sampling situation for these models is one in which l + 1 age classes can be distinguished for newly caught (unmarked) animals. Under the general model of Pollock (1981b), age-specific capture probability pa-
rameters, p!V), can be estimated for periods i = 2.... , K - 1, and for ages v = 1, ..., l, where the first age class is denoted as class 0. Recall that capture probability for age class 0 cannot be estimated in an open-model framework because there is never a subset of age-0 animals known to have been alive at any sampling period (see Section 17.2.2). Under this sampling design, it is possible to assign every captured animal to an age class, so it is possible to estimate age-specific abundance as l~i v) = n i(v)/j6!v) . 9
(18.22)
f o r i = 2 .... , K - l a n d v = 1,...,l. Totalabundance for all age classes other than 0 can be obtained by summing the /~,.v) over all ages. Abundance can be estimated for age class 0 using the robust design (Chapter 19) (see Stokes, 1984; Nichols and Pollock, 1990). Brownie et al. (1986) considered estimation of adult abundance for specific, reduced-parameter two-age models. However, the general approach using Eq. (18.22) should be applicable to virtually any multipleage models that fall within the general class of models in Section 17.2.2. Again, the age-specific capture probabilities p!~') estimated based on marked animals [ M (v) i ] must be applicable to unmarked animals [~v)], in order to estimate abundance using Eq. (18.22). In general, abundance estimation is not possible with the age-specific cohort models of Section 17.2.3. The critical difference between these models and those of Section 17.2.2 involves the manner in which age of an animal is determined. As noted above, the models of Section 17.2.2 (Pollock, 1981b; Stokes, 1984; Brownie et al., 1986) assume a sampling design in which new, unmarked animals that are captured can be unambiguously assigned to an age class. In the cohort models of Section 17.2.3 (Buckland, 1980, 1982; Loery et al., 1987), animal age is known only for marked animals. This approach is most commonly used for animals initially marked as young (age 0). If an animal initially caught in year i at age 0 is subsequently caught in year i + 2, then it is known to be in age class 2 at that time. It is possible to estimate the number of marked animals alive in each age class [ M i(v)] using the numbers of marked animals actually caught Lm 9 -(v)l i j and their estimated capture probabilities []~!v)]. The estimation of MI v) in turn permits estimation of age-specific survival under these models [see Eqs. (17.27) and (17.28)]. However, under this sampling design one cannot assign unmarked animals unambiguously to an age class. Thus, the unmarked animals caught in each sampling period, u i, are a mixture of animals from different age classes and cannot be assigned specific ages [u! v~] without resorting to restrictive assumptions about the age distribution for unmarked animals. Hence, abundance
18.2. Jolly-Seber Approach estimation generally is not possible with the cohort models of Section 17.2.3. On the other hand, the age-specific breeding models of Section 17.2.4 (Clobert et al., 1994) generally permit abundance estimation for the breeding component of the population. Recall that these models are based on animals marked at breeding colonies either as new young (age 0) or as breeders (denoted as age k+). Unmarked animals of age 0 can be distinguished from older animals, leading to the two statistics, u!~ and u} k+). All adult (age v >- k) animals caught at the breeding colony are assumed to be breeders, regardless of mark status. Thus, if capture probabilities are the same for marked and unmarked breeders [MIk+) and U}/k+), respectively], then abundance of breeders can be estimated as /~ik+) = n !k + ) / ~Ik+ ) ' where nl k+) = m! k+) + ul k+) (i.e., the number of breeding age animals caught at i is the sum of the marked and unmarked breeders caught, respectively). Abundance for animals of age 0 cannot be estimated using the models of Section 17.2.4, although estimation is again possible using the robust design (Chapter 19). We know of no way to estimate abundance of nonbreeding birds of age v > 0 using this class of models, even with the robust design.
18.2.4.8. Multistate Models Use of Eq. (18.6) in the context of multistate modeling (Section 17.3) is straightforward and uses estimators of the same form as used for grouped data [Eqs. (18.19) and (18.20)]. In both the Markovian and memory models considered in Section 17.3, capture probability for sampling period i was assumed to be specific to the state of the animal at that period. Thus, estimation of state-specific abundance is accomplished by _
i/Pi,
where ~ is the estimated number of animals in state s at sampling period i, n s is the number of these animals that are caught at i, and/3 s is the estimated capture probability for this group of animals. Total abundance for animals in all states (denote as N i) is simply estimated as the sum of these state-specific estimates:
s Once again, the critical assumption underlying this approach to estimation is that the marked and unmarked animals present in state s at sampling period i (M~ and ~/, respectively) exhibit the same capture probability.
503
18.2.4.9. Models Utilizing Auxiliary Data The estimation of abundance with Eq. (18.6) also applies to the models described in Section 17.5 that utilize auxiliary data. The key to successful application of this type of estimation is to be sure of a proper "match" between the estimated capture probability and the number of animals captured in the category of interest. Returning to the canonical estimator of Section 12.2, the point here is that the estimated detection probability must correspond to the count statistic (i.e., must estimate the probability that a member of the group of interest appears in the count statistic).
18.2.4.10. Variances and Confidence Intervals Darroch (1959), Jolly (1965), and Seber (1982) derived variance estimators under the partially open models considered above. Similarly, Jolly (1982) and Brownie et al. (1986) provided expressions for computing the information matrices, and thus the variance and covariance estimates, for the reduced-parameter models they considered. For the other models discussed above, there has been little previous work on abundance estimators and their associated variances (but see Arnason and Schwarz, 1999). In cases where Eq. (18.6) is used to estimate abundance, a bootstrap approach to variance and confidence interval estimation is recommended. If a closed-form estimator is needed, then we suggest the estimator va"'r(/Qi)
=
n2~r(pi) /~4
+
ni(1
-- Pi)
132
(18.23)
based on the delta method (Appendix F). In Eq. (18.23) the numbers of animals caught, ni, are statistics that come directly from the sampling, and the estimates and v'a'r(/3i) are computed by the software used to fit the particular model (e.g., MARK) (White and Burnham, 1999). Confidence intervals /Qi can be approximated using the approach of Chao (1989) [see also Rexstad and Burnham (1991) and Section 14.2.4]. The approach is based on the estimated number of animals not captured at sampling period i, foi = 1Qi -- rli" Here ln(f0 i) is treated as an approximately normal random variable, yielding the 95% confidence interval (n i - foil C, ti i q- foiC), where
]9i
for
C=exp
{1.96 [In ( 1 + var(/Qi)~] 1/2}. ~2i ]
(18.24)
The statistical properties of estimators such as those of Eqs. (18.6), (18.23), and (18.24) are not well understood, and additional work on abundance estimation for these models may well provide better estimators. Until then,
504
Chapter 18 Estimating Abundance and Recruitment
we recommend the above as reasonable approaches to inference about abundance under alternative models that cannot be fit using POPAN-5 (Arnason and Schwarz, 1999). 18.2.4.11. Individual Covariates
The one class of alternative models that does not fit nicely into the framework of canonical estimation is that in which capture probability is modeled using individual covariates. If survival is modeled using individual covariates, yet capture probability is modeled as a group or populationqevel parameter, then the above estimation approach [e.g., Eq. (18.6)] can be used to estimate abundance. However, if capture probability is estimated at the individual level based on covariates, then the animals captured on occasion i represent a heterogeneous mixture of capture probabilities. In theory one could use an average of these individual capture probabilities for all animals caught at i, but it is not clear how such an average would be computed in order to yield an unbiased estimate of abundance. A reasonable approach to estimation in the situation in which capture probability is modeled as a function of individual covariates involves an estimator of the type proposed by Horvitz and Thompson (1952). This approach was used by Huggins (1989) and Alho (1990) for estimation of abundance with closed-population capture-recapture models, with capture probability a function of individual covariates (see Section 14.2.2). The approach has been proposed by McDonald and Amstrup (2001) for use with open models, and we follow their recommendations here. Retaining the general notation of Section 17.1.7, let Jjm be an indicator variable that assumes a value of 1 if animalj is captured in sampling period m, and 0 if the animal is not caught during m. Let ]~jm be the estimated capture probability for animal j in period m, based on covariates associated with animal j and on an assumed relationship between capture probability and the relevant covariates. Abundance at period m then can be estimated as
n~ ~m
l~rn ~- ~ Pjm"
(18.25)
j=l
where n m is the number of animals caught at period m and N m is abundance at period m. The estimator in Eq. (18.25) is similar in appearance to the estimator used in the absence of heterogeneity [Eq. (18.6)]. However, because of the heterogeneity in capture probability and the ability to estimate an individual's capture probability as a function of measured covariates, abundance at period m can be estimated by summing the reciprocals of the estimated capture probabilities for animals that are caught at m. Note that if all animals
have the same value of the relevant covariate (i.e., if there is no heterogeneity), then Eq. (18.25) reduces to Eq. (18.6). McDonald and Amstrup (2001) investigated the properties of estimator (18.25) using simulation and concluded that it exhibited little bias. They also proposed an approximate variance estimator for N m, which performed well in simulations for small to moderate levels of heterogeneity, but not for large levels. McDonald and Amstrup (2001) suggested that bootstrap variance estimates might be useful. Although more work on this estimator should prove useful, the important point is that the Horvitz-Thompson estimator provides a reasonable approach to abundance estimation when capture probability is modeled as a function of individual covariates.
18.2.5. Model Selection, Estimator Robustness, and Model Assumptions In practice, model selection for the Jolly-Seber and related models is virtually identical to the process described in Chapter 17. As noted in Section 18.2.1, the conditional models of Chapter 17 (in particular, Section 17.1.2) can be viewed as the third component (P3) of the Jolly-Seber model [see Eq. (18.2)]. This component frequently is written as conditional on either the number of unmarked animals (ui) caught in each period [Eq. (17.5)] or the number of releases (R i) in each time period [Eq. (17.6)]. The subsequent capture-history data on marked animals provide the information needed for testing between competing models and for assessing model appropriateness and fit. Thus, between-model tests and goodness-of-fit tests for the Jolly-Seber and related models are usually based on the P3 component of the likelihood (e.g., Pollock et al., 1985; Brownie et al., 1986). As to the components P1 and P2 of the Jolly-Seber likelihood [Eq. (18.2)], P2 is essentially a binomial model of the number of removals of captured animals (e.g., trap deaths). Historically, Jolly-Seber modeling has not focused on the removal parameters (~1i, ~1~), although it sometimes is assumed that removal probabilities are the same for marked and unmarked animals ('qi = ~i). Models for these parameters could include tests for equality for marked and unmarked animals, tests for absence of temporal variation, and several other possibilities. However, under most capturerecapture sampling designs, removals are not viewed as a part of the natural population dynamics of interest to biologists. Removals are thus modeled in a general way with separate parameters ~i and ~q~ in Eq. (18.4) for marked and unmarked individuals that should not influence inferences about the more interesting processes. By assuming different removal parameters for
18.2. Jolly-Seber Approach each time, P2 effectively removes this model component and associated information from the assessment of Jolly-Seber model fit. The component P1 of the decomposed Jolly-Seber likelihood [Eq. (18.3)] models the number of unmarked animals ui caught at each sampling period conditional on the number Ui of unmarked animals in the population, with the latter treated as unknown random variables. Under this model, likelihood component P1 is used in the estimation of population size [Eqs. (18.8) and (18.11)] but is not useful in assessing model fit. In other parameterizations (Section 18.3) (see Crosbie and Manly, 1985; Cormack, 1989; Schwarz and Arnason, 1996) entry probabilities are used to model the entry of new, unmarked animals into the population. Under these alternative parameterizations, the ui can be useful in selecting appropriate models and assessing model fit. Model selection and goodness-of-fit testing under the Jolly-Seber modeling approach described in this section thus follows the procedures discussed in Section 17.1.8 for the Cormack-Jolly-Seber model, which is component P3 of the Jolly-Seber model. Therefore the goodness-of-fit tests of Pollock et al. (1985), Brownie et al. (1986), and Burnham et al. (1987) are appropriate. Model selection can proceed via likelihood ratio testing with nested models, although we recommend the alternative information-theoretic approach using AIC and its small-sample and quasilikelihood derivatives (Burnham and Anderson, 1998). The variance inflation factors ~ for lack of model fit should be appropriate for computing variance estimates of abundance [in this case the "corrected" variance is computed as ~ v~r (/~/i)]. The discussion of estimator robustness and model assumptions presented in Section 17.1.9 is applicable to the estimators for survival and capture probability under the Jolly-Seber model. As noted in Section 18.2.2 and repeatedly emphasized above, the critical additional assumption underlying abundance estimation with the Jolly-Seber and related models is that marked and unmarked animals exhibit the same capture probabilities. Here, we discuss robustness of open-model abundance estimators to deviations from underlying model assumptions. Equality of capture probabilities for all animals present in the population at any sampling period i is an assumption unlikely to be met exactly in any sampling situation. Heterogeneity of capture probability, in which different animals present at i exhibit different probabilities of being caught at i, can produce substantial bias in abundance estimates (Gilbert, 1973; Carothers, 1973; Nichols and Pollock, 1983b), depending on the form of heterogeneity. In the unlikely situation in which there is heterogeneity in capture probability, yet no covariation between capture probabilities of an indi-
505
vidual at different sampling occasions (i.e., an individual that has a relatively low capture probability at period i could just as easily have a relatively high capture probability at some other period i + j), there should be little bias in the abundance estimator. However, the more likely scenario is that individuals will tend to exhibit relatively low or high capture probabilities throughout the study (e.g., see Gilbert, 1973; Carothers, 1973; Nichols and Pollock, 1983b). In this case, animals in the population with high capture probabilities tend to be caught and become members of the marked component of the population, Mi, whereas animals with low capture probabilities tend to remain in the unmarked component of the population, Ui. As noted throughout this chapter, the estimation of capture probability is based on recaptures of marked animals. In the presence of heterogeneity of capture probabilities, the estimates Pi apply to the marked component (Mi, the animals with higher capture probabilities, on average), but are too high for the unmarked component of the population, Ui. Thus, the ]9i will be positively biased with respect to the average capture probability of the entire population, N i = M i + U i. Because ]~i appears in the denominator of the abundance estimator [Eq. (18.6)], the abundance estimator therefore is negatively biased. The magnitude of the bias in the population size estimator is dependent on characteristics of the heterogeneity (i.e., of the distribution of capture probabilities over individuals). Moderate to large degrees of heterogeneity [often expressed as the coefficient of variation of the distribution of capture probabilities over individuals, cv(p) (Carothers (1973)] tend to produce substantial negative bias in the Jolly-Seber estimator /~i. The exception to this tendency occurs when average capture probability is relatively high (e.g., >0.5), in which case heterogeneity in capture probabilities is relatively unimportant (Gilbert, 1973). The possibility of heterogeneous capture probabilities causing severe negative bias in Jolly-Seber estimates of population size has led to development of methods for bias reduction. Hwang and Chao (1995) used a sample coverage approach to address this problem. They derived estimators for the sample coverage (also see Section 14.2.3) and cv(p) (a statistic reflecting the degree of heterogeneity) and used these estimators to approximate bias of the Jolly-Seber/~i (as well as of the abundance estimators for the partially open models) and to provide new abundance estimators with reduced bias. These estimators have seen little use, though they have performed well in simulation studies. Pledger and Efford (1998) used simulation and inverse prediction (Carothers, 1979) to deal with heterogeneous capture probabilities for survival rate
506
Chapter 18 Estimating Abundance and Recruitment
estimation. Their approach used simulation to establish the functional relationship between the degree of heterogeneity [e.g., the coefficient of variation cv(p), or a metric linearly related to cv(p)] and the bias of the estimator, N i. Then, the cv(p) is estimated from the actual data, as is abundance. The abundance estimate is known to be biased, as it is based on the assumption of homogeneity in the capture probabilities. However, the magnitude of the bias can be estimated using the estimated cv(p) and the simulation-based relationship between cv(p) and bias of/~/i. Inverse prediction then can be used to obtain a new, bias-corrected estimator for abundance. Pledger and Efford (1998) used different estimators [including that of Hwang and Chao (1995)] for cv(p) and metrics related to cv(p) and concluded that the test statistic for heterogeneous capture probabilities developed by Carothers (1971, 1979; also see Leslie, 1958) led to the best estimates of abundance. The approach appeared to perform well in simulation studies that also included the Hwang-Chao estimator. This approach has seen little use because of its recent development. Trap response (a form of capture-history dependence) in capture probabilities also can influence abundance estimation. Temporary trap response can be dealt with via modeling (Section 17.1.6), and abundance can be estimated based on assumptions about the capture probabilities of unmarked animals (Section 18.2.4). Permanent trap response refers to the situation in which unmarked animals exhibit one capture probability and marked animals exhibit another. Such a response cannot be dealt with via modeling, because the information about capture probability comes from marked animals only. A trap-happy response, in which marked animals show higher capture probabilities than do unmarked animals, yields a positive bias in capture probability and produces negative bias in the abundance estimator (Nichols et al., 1984b). A trap-shy response occurs when marked animals exhibit lower capture probabilities than do unmarked animals. This response produces estimates of capture probability that are negatively biased, yielding abundance estimates that are positively biased (Nichols et al., 1984b). The biases in abundance estimates produced by permanent trap response can be substantial and are most severe when the population exhibits substantial turnover and the proportion of marked animals in the population is small (Nichols et al., 1984b). We know of no way to deal adequately with permanent trap response in Jolly-Seber type models other than to use the robust design (Chapter 19) to estimate size of the unmarked component of the population via closed capturerecapture models. Although homogeneity of capture probabilities is of obvious importance to abundance estimation, homoge-
neity of survival probabilities is also relevant. Although some research has been conducted on the robustness of Jolly-Seber survival estimates to heterogeneous survival probabilities (Section 17.1.9), there has been little to no work on effects of such heterogeneity on abundance estimators. Pollock et al. (1990) reported results for the situation in which survival probabilities are positively related within individuals (animals having high survival probability during one interval likely to show high survival probability for other intervals) but are independent of capture probability. They showed that this situation generally produces positively biased abundance estimates (Pollock et al., 1990). Robustness of abundance estimators to violations of the assumption of homogeneous survival probabilities merits further investigation. Tag loss is not as large a problem with abundance estimators as with survival estimators. When probabilities of tag loss do not vary as a function of tag age, Jolly-Seber abundance estimates remain unbiased (Arnason and Mills, 1981). Precision of abundance estimates will be reduced by tag loss, however, because there are fewer marked animals on which to base estimation of capture probability (Arnason and Mills, 1981). Because survival estimates are negatively biased in the presence of tag loss (Section 17.1.9), Jolly-Seber estimates of number of recruits, Bi [Eq. (18.13)], are positively biased by tag loss (Arnason and Mills, 1981). We are aware of no investigations of the effects of agedependent tag loss on abundance estimates, but this problem should cause heterogeneous "survival rates" (in the event of tag loss, open-model survival parameters correspond to tag survival) and thus has the potential to produce bias in abundance estimates. As discussed in Section 17.1.9, violation of the assumption of instantaneous sampling can result in heterogeneous survival probabilities. The problem arises because long sampling periods can result in populations that are open to losses (and gains) during the sampling period. Indeed, if the population size is changing during the sampling period, then it is not even clear what the true quantity of interest is (population size at what point during the sampling interval?). However, we are unaware of any work on the consequences of long sampling intervals for abundance estimates. If the population is thought to be open during the sampling period, then we recommend the robust design approach of Schwarz and Stobo (1997) and Kendall and Bjorkland (2001) in which mortality is modeled during the sampling period as well as between periods (also see Chapter 19). The assumption of no temporary emigration is relevant to abundance estimation and the interpretation of Jolly-Seber abundance estimates. In the presence of random temporary emigration (Section 17.1.9), the
18.2. Jolly-Seber Approach Jolly-Seber abundance estimate is positively biased for N i, the number of animals in the area exposed to sampling efforts during sampling period i (e.g., Kendall et al., 1997). However, Kendall et al. (1997) also considered a "superpopulation" of N~/animals that are associated with the area exposed to sampling efforts during sampling period i, in the sense that they have some nonnegligible a priori probability of being located in the sampled area at period i (see Chapter 19). In the case of random temporary emigration, the Jolly-Seber estimator for capture probability estimates the product of the probabilities of (1) being in the sampled area at i and (2) being caught, given presence in the sampled area. Thus, the Jolly-Seber abundance estimator provides an unbiased estimate of ~ (Kendall et al., 1997). Similarly, the Jolly-Seber estimator for number of recruits is unbiased for recruitment to the superpopulation. Markovian temporary emigration refers to the situation in which an animal's probability of being in the area exposed to sampling at time i depends on whether it was in the sampling area at time i - 1. Under Markovian temporary emigration, the JollySeber abundance estimator is biased with respect to both N i and N ~ and the magnitude of the bias is dependent on the form and nature of Markovian temporary emigration (Kendall et al., 1997). Finally, violation of the assumption of independent fates leads to unbiased abundance estimates, but biased estimates of variance (Section 17.1.9). In the case of dependent fates, with animals in pairs or family groups behaving similarly, variance estimates are negatively biased, but quasilikelihood procedures can be used for variance inflation and for adjusting likelihood ratio tests and AIC model selection statistics (Section 17.1.9). 18.2.6. E x a m p l e
We illustrate the Jolly-Seber model and estimators for abundance and recruitment with the meadow vole data from Section 17.1.10. We focus on abundance and recruitment of adult males over the 6 months of the study. As noted in Section 17.1.10, the CJS model, and hence the Jolly-Seber model, fit the data reasonably well. We thus based estimates on this model, using the bias-adjusted versions (Seber, 1982; Pollock et al., 1990) of the estimators in Eqs. (18.7) and (18.13), as implemented in program JOLLY (Pollock et al., 1990). Abundance estimates are available under the full Jolly-Seber model for all sampling periods except the first and last. For these data the estimates ranged from about 55 to 75 (Table 18.1). Jolly (1965) presented two variance estimators for abundance, and the associated standard errors for both are recorded in Table 18.1. The conditional standard error SE(I~ilN i) reflects variation associated with the
507
TABLE 18.1 Estimates of Meadow Vole Population Size (N i) and Number of Recruits (B i) under the General Jolly-Seber Model a Abundance Sample 9 period
1 2 3 4 5 6
Recruitment
/Vi [S-E(~,IN,),
Sampling dates
~E(/Qi)] b
6 / 27-7 / 1 8/1-8/5 8/29-9/2 10/3-10/7 10/31-11/4 12/4--12/8
74 59 62 55
mc (2.14, 3.89) (3.53, 5.84) (3.01, 5.66) (2.96, 5.61) c
I~i[~'E(Bi)] c 17 (2.56) 21 (2.71) 19 (2.82) me c
a For adult male meadow voles studied at Patuxent Wildlife Research Center, Laurel, Maryland, 1981 (data in Tables 17.5 and 17.6). b ~'E(l~ilNi ) is the conditional standard error including only error of estimation, whereas g'E(/Qi) is the unconditional standard error that also incorporates demographic stochasticity in the death process. c Quantity not estimable under Jolly-Seber model.
fact that all animals are not detected at each sampling period (i.e., Pi < 1). This component of variation is sometimes referred to as "error of estimation" (Jolly, 1965) or "sampling variation." The "unconditional" standard error SE(Ni) reflects both sampling variation and demographic stochasticity associated with the death process. The latter standard error actually is conditional on the number B i of new recruits at each sample period but incorporates stochasticity in the survival of these animals during subsequent time periods. The abundance estimates in Table 18.1 are relatively precise because of the high capture probabilities (see the ]9i in Table 17.7). Numbers of new recruits could be estimated for only three periods, but the estimates were similar, ranging from 17 to 21. Numbers of recruits were estimated with less precision than abundance (e.g., Pollock et al., 1990). In addition to the traditional Jolly-Seber modeling, abundance can be estimated using the group-specific canonical estimator in Eq. (18.19). In the example analyses of Section 17.1.10, the most appropriate model for the meadow vole data of both sexes was model (q~s+t, p). Using this model we estimated sex-specific abundance N~ for each period by =
where i = 1,..., K and s is an indicator of sex. Model comparisons suggested that a single capture parameter was appropriate for both sexes and all sample periods (Table 17.8), so that the single capture probability could be used for all n s. The AICc values of Table 17.8 also suggested that model (%+t, Ps,t) was a reasonable model for the data of Section 17.1.10. Under this model, abundance was estimated as 1Q~ = ?l si / p s
Chapter 18 Estimating Abundance and Recruitment
508
for i = 2, ..., K. Sex-specific recruitment in turn could be estimated based on a sex-specific generalization of Eq. (18.13), with the abundance estimates given by the above canonical estimators and survival estimates (see Fig. 17.2) coming directly from the respective models (q)s+t, P) and (q~s+t, Ps,t). Abundance estimates based on the above canonical estimators with models (q~s+t, P) and (q~s+t, Ps,t) are presented in Table 18.2. The associated standard errors were computed using a bootstrap approach (see Appendix F). The approach involves conditioning on estimated population size at the first sample period for which it can be estimated, and on the estimated numbers of new recruits for subsequent sampling periods. Capture histories for animals in these groups were simulated, with both capture and survival at each sample period treated as Bernoulli trials. The resulting capture histories were used to estimate capture probabilities under models (~s+t, P) and (q)s+t, Ps,t)" The capture probability estimate then was used with the n s for that bootstrap iteration to estimate abundance as above. The standard error of ~ was computed based on 1000 iterations as 1000
SE(/~/)
IEm=
'
m -
-
X s)
where Nim As is the estimated abundance from the ruth bootstrap iteration and N s is the mean of these estimates over the 1000 iterations: Ns=
E IO00 m = l /~zs'm
1000
18.3. S U P E R P O P U L A T I O N APPROACH
Several alternatives to the Jolly-Seber model have been developed in the years since its publication (Jolly, 1965; Seber, 1965). Most decompose the likelihood in a manner similar to Eq. (18.2), retaining the same general form for the P2 and P3 components of this likelihood, but focus on alternative models for component P1. The Jolly-Seber approach presented in Section 18.2 does not model recruitment B i to the population, but instead simply conditions on the number Ui of unmarked animals in the population at each sampling period. Alternative parameterizations for P1 have focused explicitly on the recruitment process (Crosbie and Manly, 1985; Cormack, 1989; Burnham, 1991; Schwarz and Arnason, 1996). Here we describe the parameterization of Schwarz and Arnason (1996), which is based on the earlier development of Crosbie and Manly (1985). 18.3.1. M o d e l
TABLE 18.2 Estimated Meadow Vole Abundances Based on the Canonical Estimator under Two Models a
Model (~Os+t, Ps.t)b
The canonical abundance estimates obtained under the two models (q)s+t, P) and (q)s+t, Ps,t) are similar to each other (Table 18.2) and to the Jolly-Seber estimates [model (q~s,t, Ps,t)] of Table 18.1. The bootstrapped standard errors based on model (q~s+t, P) tended to be slightly smaller than those based on model (q~s+t, Ps,t). The unconditional Jolly-Seber variance estimates SE(/~i) of Table 18.1 are based on approximations but are nonetheless similar to the bootstrapped estimates of Table 18.2.
Model
Sample period
~Qi
[~'~(~i)]c
~Qi
1 2 3 4 5 6
___e 75 62 62 55 88
j 3.31 5.41 5.34 5.70 6.60
62 80 54 63 51 85
(~Os+t,p)b
[~r 2.95 4.17 5.08 5.34 5.16 5.87
"See Section 17.10 and Table 17.7. Data are for capture-recapture data on adult male m e a d o w voles at Patuxent Wildlife Research Center, Laurel, Maryland (data in Tables 17.5 and 17.6). b Model fit using MARK with resulting estimates/~s used to estimate N [Eq. (18.19)]. c Standard errors estimated using a bootstrap approach with 1000 iterations. d Quantities not estimable under model (%+t, Ps,t).
Structure
Crosbie and Manly (1985) and Schwarz and Arnason (1996) reparameterized the Jolly-Seber model by directing attention to a new parameter, N, denoting the size of a "superpopulation" that serves as a source of individuals for the population of interest. Following the Schwarz-Arnason approach, K-1
N = ~
Bi
(18.26)
i=0
is the total number of animals available for capture at any time during the study, with B i the number of new animals in the population at sampling period i + 1 that were not present in the population at i. In the initial sampling period, all animals in the population the first sampling period are "new" with respect to sampling, i.e., B0 = N1. The random variables Bi are modeled with a multinomial distribution, whereby the members of the superpopulation are assumed to enter the sampled population at different times according to
18.3. Superpopulation Approach entry parameters ~i. Thus, recruitment {B0.... , BK_ 1} over time is distributed as a multinomial with parameters (N; 130, ..., [3K_l). Schwarz and Arnason (1996) then defined a new set of parameters ~Itl =
Schwarz and Arnason (1996) recommend the factorization
Pl({Ui} l N,{f3i},{Pi},{~i}) = {P{a(u.IN)} {P~b({Ui}l
~0
and
=
pi)q~i + f3i,
Xlfi+ 1 -- ~Iri(1 --
for use in modeling the numbers of unmarked animals caught at each sampling period. To illustrate, consider the parameter ~ 2 = [30(1 -- P l ) ~ 1 q- ~1" A member of the sampled population that is unmarked at sampling period 2 (1) could have been in the initial population (a member of B0 = N1), avoided capture in the first sampling period, and survived until the second period [with probability [30(1 - pl)q~l]; or (2) could have been a new recruit (a member of B1) in period 2 (with probability [~1)" The sum of these probabilities is the probability that an animal in the superpopulation is in the sampled population and unmarked at sampling period 2. Thus, the probability ~iPi corresponds to the capture of unmarked animals at sampling period i, and {u1, ..., u K} can be modeled as a multinomial with parameters (N; I ~ l P l , ..., t~KPK). The above development leads to the following expression for component P1 of the Jolly-Seber distribution function [Eq. (18.2)]:
P~({ui} l Xr{f3i},{pi},{~i}) =
N! Ul!U2! "'"
•
I1
-
UK!(N- U.)!
~
qd'ipi1 N-u (18.27)
i=1 K
• ri(qd, ipi)ui, i=1
where the P{ denotes the Schwarz-Arnason form of P1, and u. denotes the total number of unmarked animals caught in the study: K U = s /'/i" i=1
The parameters of P{ are N, to the constraint that
{~i}, {Pi}, and {q~i}, subject
K-1
s i=0
13i= 1.
509
{
u '(N9
[
u )! ~" i=1
qlripi
U.,{f3i},{pi},{~i}) }
1
(18.28)
-- s ~Iripi i=1
{
U K~ ~_~_ipi__d.~ui} .= X Ul!U2!...." ldK! i~11~K1 XIriPiJ 9 Component P{a concerns the members of the superpopulation that are and are not caught at any time during the study. Component P{b then concerns the temporal distribution of initial captures for those animals that are caught. We thus can retain model components P2 and P3 of the original Jolly-Seber model [Eq. (18.2)] and write the Schwarz-Arnason (1996) version of the entire distribution function as
P({ldi},{di, d~},{mij}) = [P~a(u.IX)] X [P{b({Ui}lU.,{f3i},{pi},{~i})] (18.29) X [P2({di,d~}l{mi,ui},{~qi,'rl~})] X [Ps({mij}l{Ri},{q~i,Pi})]. The components Pla and P~b model the captures of unmarked animals over the sampling periods as a function of the new entry probabilities as well as the capture and survival probabilities. Note that models (18.2) and (18.29) are differently parameterized likelihoods for the same capture-recapture data, with identical components P2 and P3 in each parameterization. Thus P1 from Eq. (18.2) and the product Pla X P~b from Eq. (18.29) are statistically equivalent expressions (see Section 18.5).
18.3.2. Model Assumptions Because the superpopulation approach embodied in Eqs. (18.27)-(18.29) is simply an alternative parameterization of the initial Jolly-Seber model, the assumptions required by the two approaches should be identical, or very nearly so. The discussion of assumptions in Section 18.2.2 is thus relevant to the superpopulation approach as well. The additional modeling in the superpopulation approach [Eqs. (18.27) and (18.28)] involves the unmarked animals that are caught during the study. The new parameters required for this modeling are the
510
Chapter 18 Estimating Abundance and Recruitment
entry probabilities, ~i, i.e., the probabilities that members of the superpopulation are unavailable for capture (e.g., not previously present on the study area) until sampling period i, and then enter the study population and are exposed to sampling efforts at period i + 1. The multinomial modeling assumes homogeneity of these entry probabilities. Thus all members of the superpopulation N that have not yet become available for potential capture as of sampling period i - 1 are assumed to exhibit the same probability of being in the group of animals exposed to sampling efforts at period i. The multinomial assumption about independence of fates is still required in the superpopulation approach, although the additional modeling with entry probabilities requires that "fate" refer not only to capture and survival, but also to entry of an animal into the population exposed to sampling efforts.
18.3.3. Estimation The approach to estimation recommended by Schwarz and Arnason (1996) for their superpopulation model involves numerically maximizing the product P~bP2P 3 in order to obtain estimates of the capture {]9i}, survival {q~i},and entry {~i} probabilities. As is true for the Cormack-Jolly-Seber model (Section 17.1.2) and the Jolly-Seber model (Section 18.2), the capture probability for the first sampling period cannot be estimated under the superpopulation approach. Similarly, q~K- lPK can be estimated only as a product. Finally, the constraint that the entry parameters ~i s u m to 1 leaves K - 1 of these parameters to be estimated, for a total of 3K - 3 parameters to be estimated under the full, time-specific superpopulation model (Schwarz et al., 1993b; Schwarz and Arnason, 1996). The superpopulation size can be estimated using Pla a s u 1Q = ~ K _ 1
'@i]~i
(18.30)
However, the lack of identifiability for Pl and PK cause difficulties in estimating the denominator of Eq. (18.30) under the full time-specific model (~t, Pt, f3t). Schwarz and Arnason (1996) recommend estimation based on the constraints Pl = PK = 1. The time-specific abundance N i and recruitment B i are viewed by Schwarz and Arnason (1996) as derived quantities. Recruitment can be estimated as /~i = ~i/~,
(18.31)
with abundance estimated in terms of
/~/1 = /~0, /~i+1 -= B i -ff ~Pi (1Q i -
n i if- a i ) ,
(18.32)
for i = 0, ..., K-1. Variance and covariance estimation for these quantities is discussed by Schwarz and Arnason (1996). Computations under the superpopulation approach are carried out by program POPAN-5, written by Arnason and Schwarz (1999). This software uses a logit link to estimate model parameters and permits the user to construct alternative models in much the same manner as MARK (White and Burnham, 1999). Finally, the Schwarz-Arnason superpopulation approach also permits estimation of population growth rate and related parameters, as discussed in Section 18.5.
18.3.4. Alternative Modeling It is possible to use the superpopulation approach in conjunction with various alternative models for the capture, survival, and entry parameters. The software POPAN-5 implements many of these alternative models. For example, the partially open models of Section 18.2.4 are easily obtained by constraining parameters of the general model (q~t, Pt, ~t). The birth-only model is obtained by constraining the survival parameters by % = q~2 . . . . . q~K-1 = 1, whereas the deathonly model is obtained by constraining the first entry parameter to 1 and the others to 0 ([3o = 1; ~1 -- ~2 = ....
~ K - 1 = 0).
Temporal constancy of capture, survival, and entry probabilities also can be modeled by constraining parameters of the general model. Because the initial entry probability ~0 includes all animals in the population at the beginning of the study, this parameter can be expected to differ from the other entry probabilities. Thus, a model might constrain ~1 = ~2 . . . . . ~K-1 and estimate f~0 separately. Schwarz and Arnason (1996) emphasize that identifiability problems can arise in models with temporal variation in capture probabilities, so models with constant capture probabilities should be considered whenever possible. Superpopulation modeling also can be extended to deal with multiple groups of animals (e.g., males and females). For example, constrained models can include time-specific entry probabilities that are equal for the different groups. As with the models in Chapter 17, it is also possible to model parameters as functions of time-specific covariates, such as relevant environmental variables (Arnason and Schwarz, 1999). Arnason and Schwarz (1999) have used POPAN-5 to introduce age specificity into model parameters, and it should be possible to build multistate models within this general superpopulation framework. Indeed, POPAN-5 permits estimation of abundance and recruitment for the full array of model structures that
18.4. Pradel's Temporal Symmetry Approach are available for conditional mark-recapture modeling (Chapter 17). 18.3.5. M o d e l S e l e c t i o n , E s t i m a t o r Robustness, and Model Assumptions
Model selection should follow the same basic procedures discussed for conditional models in Chapter 17 and outlined for the Jolly-Seber and related models in Section 18.2.5. The superpopulation models contain the extra set of entry parameters that permit additional modeling flexibility. In Section 18.2.5, we noted that virtually all of the information needed to assess model fit and select the appropriate model was found in component P3 of the Jolly-Seber model [Eq. (18.2)]. However, with the addition of the entry parameters it would seem that component P1 of the superpopulation models also should contain information that is useful for assessing model adequacy. The basic approach involving the use of AIC and likelihood ratio testing can be followed with superpopulation models just as with the conditional models of Chapter 17. POPAN-5 computes statistics for assessing model fit, testing between competing models, and selecting models from a candidate set (Arnason and Schwarz, 1999). The superpopulation models of Crosbie and Manly (1985) and Schwarz and Arnason (1996) have begun to see substantial use only recently. Thus, there has been virtually no work on assessing estimator robustness in the face of assumption violations for these models. Certainly, the underlying basis for estimation is very similar to that of the Jolly-Seber parameterization, so the previous discussion of Jolly-Seber estimator robustness (Section 18.2.5) should be relevant to estimation under the superpopulation models as well. However, it may be that the addition of the entry parameters leads to differences in effects of assumption violations on superpopulation estimators, and this topic requires further study.
the Jolly-Seber model (Table 18.1) led us to consider a model with recruitment constant over time. Thus, we fit superpopulation models with the entry parameters either time specific [model (q~t, Pt, ~t)] or constant over time [model (q~t, Pt, ~)]. Point estimates obtained under model (q~t, Pt, ~t) (Table 18.3) were similar to those of Table 18.1 based on the closed-form bias-corrected estimates of Section 18.2.6. However, the standard errors under the superpopulation approach were larger than even the unconditional standard errors of the Jolly-Seber approach, reflecting the assumption that the B i are random variables governed by underlying entry probabilities (Schwarz and Arnason, 1996). Point estimates Ni of abundance under the time-constrained model (q~t, Pt, ~) were similar to those under the more general model, and the stationary estimate of number of recruits (/~ = 24) was slightly larger than estimates under the timespecific model (Table 18.3). Variances for population size and recruitment estimates under the reducedparameter model were smaller than those for the more general time-specific model (Table 18.3).
18.4. P R A D E U S T E M P O R A L SYMMETRY APPROACH
The temporal symmetry of capture--recapture data was discussed in Section 17.4 on conditional reversetime models. As seen below, an important benefit of the reverse-time approach is that it permits estimation of parameters that are relevant to recruitment of new animals into the population. Pradel's modeling ap-
TABLE 18.3 Estimated Meadow Vole Population Size (N i) and Number of New Recruits (B i) under Two Superpopulation Models a Sample
period/ 18.3.6. E x a m p l e
We return to the example of adult male m e a d o w voles captured at Patuxent Wildlife Research Center during June through December, 1981. The data are found in Tables 17.5 and 17.6, and the sampling is described in Sections 17.1.10 and 18.2.6. We used the superpopulation approach of Schwarz and Arnason (1996) to fit two models using POPAN-5. Previous investigation of survival with the CJS model of Section 17.1.10 suggested the need to include temporal variation in both survival and capture probabilities, and we saw no reason to retest this inference. However, the similarity of point estimates for recruitment (/~i) under
511
1 2 3 4 5 6
Model (~ot, Pt, ~t )b
Model (~ot, Pt, ~)b
/Qi [S"E(/Q,)] /~, [~'E(]~i)] /~i [S'~'E(/~i)] ]~d [~"E(B)] __c 75 59 63 55 c
6.97 7.22 7.05 6.84
c 18 21 19 c c
4.82 5.14 5.12
c 69 63 68 63 c
24
(1.31)
4.94 4.96 5.23 5.37
From Schwarz and Arnason (1996). Data are for capture-recapture data on adult male meadow voles at Patuxent WildlifeResearch Center, Laurel, Maryland (data in Tables 17.5 and 17.6). bModels based on the superpopulation approach (f~-parameterization) of Schwarz and Arnason (1996) and fit using POPAN-5 (Arnason and Schwarz, 1999). CQuantities not estimable under model. dSingle estimate/~ applies to periods 1-5. a
Chapter 18 Estimating Abundance and Recruitment
512
proach also incorporates survival (q~i) and recruitment (1 - ~/i) parameters, thus permitting the direct modeling of population dynamics (Pradel, 1996). 18.4.1. M o d e l
P{0110101 last capture at period 5} = ~/s(1
Structure
The modeling described in this section differs from that described in the previous two sections in that it does not involve a decomposition of the overall likelihood used to model summary statistics. Instead, the approach is to develop a likelihood directly by modeling individual capture-history data, which then can be expressed in terms of sufficient statistics. The following definitions were provided in Chapter 17, but bear repeating here because they are important for parameterizing the temporal symmetry models. The conditional forward-time models use the following parameters: Pi is the probability that a marked animal alive and in the study population just before sampling period i is captured or observed during period i; q~iis the probability that a marked animal alive and in the study population just following sampling period i survives until period i + 1 and remains in the population (does not permanently emigrate); Xi is the probability that an animal alive and in the study population just following sampling period i is not caught or observed again at any sampling period after period i. For a K-period study, XK = 1, and values for other sampling periods (i < K) can be computed recursively [Eq. (17.1)] as Xi = (1 -
q~i) if- q~i(1 -- Pi+l)Xi+l.
As a reminder of the conditional, forward-time model structure of Section 17.1, consider the probability associated with the capture history 011010, indicating capture in periods 2, 3, and 5 of a six-period study: P{011010 i first capture at period 2} = q~2pgq~3(1
for i -- 2, ..., K and ~1 = 1. Again consider history 011010. For reverse-time modeling we condition on the final capture and model prior events in the capture history:
-
-
p4)q~4P5X5.
The statistical model for this history requires conditioning on the initial capture in sampling period 2 and then proceeds by modeling the events in the remainder of the capture history. The conditional reverse-time models of Section 17.4 require the following parameters: "~i is the probability that an animal present just before sampling occasion i was present in the sampled population just after sampling at occasion i - 1; p~ is the probability that an animal present just after sampling at time i was captured at i; ~i is the probability of not being seen at sampling periods before i for an animal present immediately before i. As with the forward-time parameters Xi, the reverse-time parameters ~i can be computed recursively as "Yi) nt- ~/i( 1
-
-
P;-1)~i-l"
P)
!
!
P4 ~/4PB~3P2~2
.
As described below, the temporal symmetry models of Pradel (1996) essentially use both forward- and reverse-time modeling simultaneously. The parameter definitions above make reference to times just before and just after sampling, because this separation becomes important for the modeling when there are losses on capture. Define NT and N +, respectively, as abundance just before and after sampling period i. Drawing on previous notation, where d i and d~ are the numbers of marked and unmarked animals that are caught at i and not released back into the population following sampling, we can write N + = N T - di - d~. For purposes of this section, we assume that marked and unmarked animals captured in period i have the same probability "i]i of surviving trapping and handling and of being released back into the population (e.g., in previous notation, g]i = "l] ~). This assumption is made for reasons of notational convenience and can easily be relaxed. The need for different capture probability parameters for forward-time (Pi) and reverse-time (p~) modeling also stems from losses on capture. The simultaneous use of reverse-time and forward-time modeling requires a relationship between the two capture probability parameters. Focusing on the capture probability for forward-time modeling, the relationship between N T and N + can be written as N + = N,:- [1
pi(1
-
-
1]i)].
(18.33)
Equation (18.33) simply indicates that a member of N T must survive the possibility of being caught and removed from the population in order to become a member of N +. Thus, the expected number of captures of individuals that are alive just after sampling at time i can be expressed as N + Pl = NT [1 - pi(1 - ~qi)]P~, with p~ the probability that a member of N + is captured in sampling period i. But this expected number also can be written in terms of parameters from forwardtime modeling as NTpiTIi . Equating these expressions then allows one to write Pl as -----
Pl
Pi
1 - pi(1 - Tli)
Similarly, the probability that a member of N + was not captured in sampling period i can be written as 1
~i = (1 -
--
-
Pi
1 - p ~ = 1 - p i ( 1 -~qi)"
18.4. Pradel's Temporal Symmetry Approach Note that when all animals are released following capture (TIi = 1), the forward- and reverse-time capture probabilities are equal: PI = Pi. Simultaneous forward-time and reverse-time modeling proceeds by conditioning on the number of animals in the population at the initiation of the study, N~ = B0. The expected number of animals in the population at later times is determined by considering the rate of population growth between successive sampling occasions. Expressions for population growth rate can be obtained by considering two alternative ways of writing the expected number of animals alive in two successive sampling occasions. Based on forward-time and reverse-time modeling we can write this expectation a s N+q~i ~ N~+1~/i+1. Solving this approximate equality yields an expression for population growth rate: ~'i = N ~ 1 / N~-
M = ~,Xh. h
The expected number of animals caught during a study can be written as the sum of the expected number of animals seen for the first time at each sampling occasion: K
E(M) = ~_~ ~iX-d-pi i=1 i-1
= S 1 i~l ~iPi( 1-I ~
Equation (18.34) is relevant to biological changes in the population, but does not account for animals that are captured and not released back into the population. To account for animals not released, we can write a modified rate of population change (~}) that also incorporates losses of animals during sampling: K} = N~+ 1~N-d-
"=
P(h) =
"Yi+I
Thus, the expected number of animals exhibiting capture history 011010 under Pradel's (1996) temporal symmetry model can be written as (18.36)
• (1 - P4) q~4Ps'rlsX5 9 The term N l k { gives the expected number of animals in the population just before sampling period 2, and ~2 is the probability that an animal in this group was not caught prior to sampling period 2 (i.e., was not caught at 1). The animals exhibiting this history were caught at period 2, and the associated probability is P2. They survived the sampling of period 2 to be released again (we know this because they were seen in subsequent periods), and the probability associated with surviving sampling is ~12. The subsequent (for sample periods >2) modeling is similar to that presented in Chapter 17, except that survival probabilities for the sampling process now are incorporated into the model. Thus, every capture event requires both a
\j=l
;)
(18.37) 9
Finally, the conditional probability (conditioned on the total M of animals caught) associated with a particular capture history [denoted as P(h)] can be obtained by dividing the expected number of animals with that history [e.g., as in Eq. (18.36)] by the expected number of total individual animals caught during the study [as in Eq. (18.37)1:
(18.35)
q~i[1 - pi(1 - T]i)]
E(XOllOlO]N1) = N1k~2P2~12q~2pgngq~ 3
capture probability Pi and a probability "l]iof surviving the sampling process. Equation (18.36) does not lead directly to a probability distribution, because the expectation contains the initial population size, N 1, an unknown random variable. Let x h be the number of animals exhibiting capture history h, and M denote the total number of animals caught in the entire study:
(18.34)
q)i/'Yi+l "
= hi(N +/N[-)
513
E(Xh) E(M) "
(18.38)
From Eqs. (18.36) and (18.37), the initial population sizes in the numerator and denominator of Eq. (18.38) cancel, leaving the conditional probabilities of interest expressed in terms of estimable model parameters. Then the likelihood L for the set of animals observed in a study can be written generally as the product of the conditional probabilities associated with all the individual capture histories: L = l-I P(h)Xh.
(18.39)
h
Pradel (1996) described likelihood (18.39) in more detail in terms of the model parameters and sufficient statistics. He suggested three different parameterizations for the above likelihood, each of which might be useful in addressing specific questions, all of which retain capture (Pi) and survival (q)i) probabilities. Of these, perhaps the most natural parameterization incorporates the reverse-time parameters %. Thus, Eq. (18.35) is substituted into the capture history expectations [Eqs. (18.36) and (18.37)], so that all probabilities (Ph) are written in terms of Pi, (Pi, and %.
514
Chapter 18 Estimating Abundance and Recruitment
A second parameterization uses population growth rate h i as a model parameter. Based on the definition in Eq. (18.34), the expression "~i = q~i-1/hi-1
is substituted for the "Yi of the original parameterization. A third parameterization is based on a measure fi of recruitment rate, which denotes the number of recruits to the population at time i + I per animal present in the population at i. This measure of recruitment is used in discrete-time matrix population m o d e l s m f o r example, in the single-age model, Ni+ 1 = Niq~ i + N i f i.
(18.40)
Equation (18.40) can be rearranged to yield Ni+I/Ni
= ~i if- fi
-- ~ i / ~ i + l .
Thus, the parameterization of a model with fi can be obtained by substituting r ~i-1 nt- fi-1
(18.42)
Ki "-- ~Pi/~i+l
= hi
'Yi --
parameters to the interval [0, 1]. For the h i parameterization, Pradel (1996) used a log transform for population growth rate (hi), in order to constrain it to be positive. All three parameterizations ([q~t, Pt, '~t], [q~t, Pt, ht], and [OPt, Pt, ft]) described in Section 18.4.1 have been implemented in program MARK (White and Burnham, 1999). Pradel's temporal symmetry models are relatively new and have seen only limited use. It appears that the numerical optimization algorithms may sometimes perform better (e.g., fewer convergence problems) with the y-parameterization than with the other two parameterizations. If primary interest is in population growth rate, it may be better to fit model (~t, Pt, "Yt) to data and then estimate population growth rate using Eq. (18.34) by
for i = 2, ..., K - 2. This estimator also is computed in program MARK (White and Burnham, 1999). The parameter fi can be estimated in a similar manner, based on estimates from model (q~t, Pt, ~t) and a rearrangement of Eq. (18.41):
(18.41)
for "Yiof the original parameterization. Equation (18.41) is an intuitive expression for the seniority parameter % Recall that this parameter is defined as the probability that an animal alive at period i is a survivor from the previous period, i - 1. All animals alive at i are either survivors from period i - 1 (expectation Xi_lq~i_ 1) or new recruits (expectation X i _ l f i _ l ) , so Eq. (18.41) is natural expression for the proportion of survivors.
fi = ~i(1 -- "Yi+I) "Yi+I
for i = 2, ..., K - 2. Future work on the models of Pradel (1996) should include detailed investigations of the identifiability of parameters under the different model parameterizations. Under the time-specific model with y-parameterization (oPt, Pt, ~/t), the parameters q~l, r
"", q~K-2;
"~3, ~4, "", "~K;
18.4.2. Model Assumptions Because Pradel's (1996) temporal symmetry models simply represent different ways to parameterize the original Jolly-Seber model, the basic assumptions are the same as for the Jolly-Seber and superpopulation approaches (see Sections 18.2.2 and 18.3.2). The general assumption of homogeneity of rate parameters now applies to Pradel's "~i as well as to the usual Pi and q~;.
Maximum likelihood estimates can be obtained for the likelihood of Eq. (18.39) or its analog based on sufficient statistics (Pradel, 1996). In Pradel's (1996) implementation of this model, he used a logit transform for q~i and '~i as a means of constraining these
P2, P3, "", PK-1; "Y2Pl;
q~K-lPK
can be estimated. Note that the list includes K - 2 survival parameters, K - 2 capture probabilities, K - 2 seniority parameters, and two product parameters with components not separately identifiable, yielding a total of 3(K - 2) + 2 = 3K - 4 parameters. Under the time-specific model with h-parameterization (OPt, Pt, ht), the parameters q~l, r
18.4.3. Estimation
(18.43)
"",
h2, h3, "", hK-2;
q~K-2; hl/Pl;
P2, P3, "", PK-1; ~K-lPK;
hK-lPK
can be estimated. This parameter list includes K - 2 survival parameters, K - 2 capture probabilities, K - 3 population growth rates, and three product parameters, again yielding a total of 2(K - 2) + (K - 3) + 3 = 3K 4 parameters.
18.4. Pradel's Temporal Symmetry Approach
18.4.4. Alternative Modeling Various types of alternative modeling should be possible using the basic models of Pradel (1996). For example, models with parameters constrained to be constant over time can be used to incorporate various hypotheses of potential biological interest. As noted in Section 17.4.1, models incorporating constancy of the ~/i (~/i = ~ for all i) reflect temporal similarity in the relative contributions of new recruits and old survivors to population growth. Models with stationarity of h i also are potentially useful for investigating population regulation and for testing the assumptions that underlie matrix population modeling (Chapter 8) based on stationary growth rates. One topic meriting consideration in reduced-parameter models that utilize these parameterizations involves the manner in which the h i and fi parameters are defined as functions of the parameters ~i that also appear in the model [e.g., see the estimators of Eqs. (18.42) and (18.43)]. Thus, modeling one set of parameters as temporally constant (e.g., q~i -- q~) may impose unintended constraints on the parameters h i and fi. Because of the lack of work on this topic, we simply recommend caution at this time. In cases where interest is focused on a parameter such as h i, a conservative approach might be to allow full time specificity in capture and especially survival probabilities when evaluating alternative models for the hi. However, whether this approach performs better than others is yet to be determined. The potential to describe parameters as functions of time-specific covariates offers interesting possibilities with these models. For example, it may be useful to model recruitment-related parameters (~/i and fi) as functions of environmental variables thought to influence either reproduction or immigration or both. The ability to model population growth rate as a function of environmental covariates also should prove useful. It often is of interest to investigate time trends in the hi, which is accomplished by modeling h i with time as a covariate. Though ecologists have long been interested in time trends, a focus on trends in the trend parameters (h is usually the quantity selected to express "trend" in population size) is relatively new (see Franklin et al., 1999). There may be large potential in using the h-parameterization in conjunction with data from other sorts of surveys (other than capture-recapture) in which count data are collected for the purpose of estimating trends in population size. For example, assume that we conduct capture-recapture studies on the same area where line transect counts also are collected (see Section 13.2) to estimate abundance. It should be possible to develop
515
joint likelihoods for the separate data types that share h i, thus combining information to better estimate population growth rate [e.g., see Alpizar-Jara and Pollock (1996, 1999) for a similar approach to a different problem]. In other situations, it may be possible to use simple count statistics for which no effort is made to estimate detectability. If count statistics have detection probabilities that are constant over time (see Chapter 12), then the ratio of counts Ci and Ci+ 1 in two successive years provides an estimate of population growth rate (i.e., Ci+I//Ci should estimate hi). One way to assess the reasonableness of this index assumption would be to use Pradel's (1996) models to model population growth rate using the counts as covariates (e.g.,)k i = ~Ci+ 1//Ci). If this model performs well (if it describes the variation in the data nearly as well as a model with no covariate model for h i, and if is near 1.0), then this can be taken as some evidence that the counts provide reasonable indices, at least over the period of study. In that case the model with count statistics as covariates should provide more precise estimates of population growth rate. It may be useful to consider using the h-parameterization in conjunction with individual covariate modeling. In studies of closed populations, or at least populations for which emigration and immigration are not important, the individual h values can perhaps be viewed as fitness estimates associated with individuals characterized by those covariates. It is possible to use a variance components approach (see Burnham et al., 1987; Skalski and Robson, 1992; Link and Nichols, 1994; Gould and Nichols, 1998) based on the conceptual framework of random effects modeling, to estimate the true temporal variance of h i . This variance is relevant to extinction probability (e.g., Lewontin and Cohen, 1969; Leigh, 1981; Goodman, 1987a) and emphasizes the potential utility of the direct estimation and modeling of h i for population viability analyses (see White et al., 2001) (also see Section 11.2.1). Finally, we note that most of the alternative modeling described above can be implemented using MARK (White and Burnham, 1999). In particular, MARK includes model parameterizations that incorporate ~/i, h i, and fi.
18.4.5. Model Selection, Estimator Robustness, and Model Assumptions Model selection should follow the same basic approach discussed for the conditional models of Chapter 17 and the other classes of models described above (Sections 18.2.5 and 18.3.5). As with the superpopulation models of Schwarz and Arnason (1996), the models of Pradel (1996) contain an extra set of parameters
516
Chapter 18 Estimating Abundance and Recruitment
(either seniority, population growth rate, or recruitment rate), providing additional flexibility in modeling. The recommendations of Chapter 17 apply here for the use of AIC, likelihood ratio testing, and quasilikelihood procedures. The goodness-of-fit tests recommended for conditional models (e.g., Pollock et al., 1985a; Brownie et al., 1986; Burnham et al., 1987) apply to these models, because the entries of new unmarked animals into the sample provide little additional information for assessing model fit, especially in the case of the models with time-specific parameters [e.g., model (q)t, Pt, 'Yt)]" Because they were only recently developed, these models have seen only limited use, and topics such as estimator robustness are yet to be investigated extensively. Hines and Nichols (2002) focused on the k-parameterization and investigated three possible sources of bias. The investigation was tailored to a particular set of analyses for the spotted owl, Strix occidentalis caurina (Franklin et al., 1999). However, some of the specific findings are likely to be relevant to other studies. The first potential problem investigated by Hines and Nichols (2002) involved expansion of the study area. Because this issue is not associated specifically with a model assumption, it has not been mentioned previously. Basically, capture-recapture estimates apply to a particular area under investigation, and if this area changes in size between sampling periods i and i + 1, the relevant rate parameters (e.g., q~i, ki) can be expected to reflect these changes. For example, we can envision certain study situations where there is a tendency to detect animals just beyond the periphery of the original study area and to target their capture and addition to the study. This tendency would be expected to result in increases in the size of the study area over time, with estimates of )ki that are larger than if there had been no expansion of study area. The estimates would not be biased, in the sense that they would reflect changes in numbers of animals on the expanding study area. However, if the interest is in growth or decline of a biological population, then an effort should be made to restrict attention to areas of similar size, so that inferences apply to biological processes and not changes in sampling area. If N;+ 1 is the number of animals exposed to sampling efforts in period i + 1 that were not exposed to efforts in period i, then the approximate bias (with respect to the original sampling area of period i) i n ~i is given by Bias(Ki) = E(K i) - )k i N;+ 1/Ni. Thus, expansion of the study area in sampling period i + 1 will result in positive bias in Ki with respect to
the population growth rate )ki o n the area sampled in period i. Another potential assumption violation considered by Hines and Nichols (2002) is permanent trap response, a violation known to produce biased estimates of abundance (Section 18.2.5) and seniority parameters (Section 17.4.1), but not survival rate (Section 17.1.9) (see Nichols et al., 1984b). Trap-happy response (higher capture probabilities for marked animals than for unmarked animals) produced a positive bias in K and Ki, whereas a trap-shy response led to a negative bias. The intuition underlying this result is based on the way of expressing [Eq. (18.34)] and estimating [Eq. (18.42)] population growth rate as a function of survival and seniority parameters, i.e., )ki-- q)i/'~i+l"
Survival rate estimates are not biased by permanent trap response because they are based on recaptures of marked animals only (Section 17.1.9). Estimation of seniority parameters is based on captures of marked and unmarked animals in previous periods, whereas estimation of capture probabilities is based on marked animals. In the case of trap-happy response, the estimated capture probability based on marked animals will be too high for unmarked animals, leading to seniority parameter estimates that are negatively biased (the estimated number of animals in i + 1 that were unmarked prior to i will be too small; see Section 17.4.1). If the ~i+1 a r e too small, then the ~i of Eq. (18.42) will be too large. Similarly, trap-shy response produces positive bias in seniority parameter estimates and negative bias in estimates of population growth rate. As expected, the magnitude of the bias in population growth rate was largest for the largest differences between capture probabilities of marked and unmarked animals. The bias also varied as a function of sampling period (Hines and Nichols, 2002). Under a trap-happy response with true )ki constant over time, for example, the )ki exhibited a negative time trend, with the largest positive biases occurring i n ~2 and the smallest biases occurring in )KK_ 2. O n reflection, this trend in estimator bias makes sense, because it involves the larger numbers of unmarked animals in the early sampling periods. The key point here is that time trends in )ki should be considered with caution in sampling situations when there is a possibility for trap response in capture probabilities. Heterogeneity in capture probabilities also was investigated as a potential source of bias in )~i. Although heterogeneous capture probabilities are known to cause serious bias in abundance estimates (Section 18.2.5), they appear not to cause problems for estimates of population growth rate (Hines and Nichols, 2002). If we consider estimating population growth rate as
18.4. Pradel's Temporal Symmetry Approach the ratio of abundance estimates in two successive sampling periods, then both the numerator and denominator will be negatively biased by heterogeneity. If the relative bias is similar for the two estimates, the estimate of their ratio (ki) can be expected not to show substantial bias, an expectation confirmed in the study of Hines and Nichols (2002). In the case of model (q~t, Pt, ~kt) with true population growth constant over time, the time-specific estimates k i showed a slight negative time trend, with positively biased estimates in the early time periods and negatively biased estimates in the later time periods. The combined effects of a trap-happy response and heterogeneity of capture probabilities also were examined by Hines and Nichols (2002). Results were similar to those obtained under trap response, the more important of the two assumption violations with respect to bias. Effects of tag loss and of sampling that is not instantaneous have not been investigated for the temporal symmetry models. Random temporary emigration (Burnham, 1993) should result in parameter estimates that are unbiased with respect to the superpopulation (Kendall et al., 1997), but not with respect to the population available for capture in the specific sampling periods. Markovian temporary emigration is expected to produce biased estimates of most parameters. Clearly, the topic of bias under the various parameterizations of Pradel's (1996) temporal symmetry models merits careful investigation. 18.4.6. E x a m p l e
We again use the m e a d o w vole data of Tables 17.5 and 17.6, in this case to illustrate the temporal symmetry approach of Pradel (1996). We applied program MARK (White and Burnham, 1999) to estimate parameters under all three parameterizations of the temporal symmetry approach. For each parameterization, we fit the full, time-specific model, as well as a reducedparameter model in which the parameters other than survival and capture probabilities (either seniority, population growth rate, or recruitment rate) are assumed constant over time. The natural parameterization with seniority parameters under model (oPt, Pt, "Yt) yielded estimates of seniority "Yi ranging from 0.60 to 0.71, indicating that about 60-70% of the adults in the population over the course of the investigation consisted of survivors from the same population in the previous month (Table 18.4). Monthly population growth rates were estimated as derived parameters from the r and ~/i, and ranged from 0.83 to 1.07 (Table 18.4). AICc favored the reducedparameter model (q~t, Pt, "Y), which produced an estimate for the proportion of survivors of 65% (Table
517
TABLE 18.4 Estimated Seniority Parameters (~i) and Population Growth Rates (h i) for Meadow Voles under Two Temporal Symmetry Models a Model (~t, Pt, ~t )b
Sample period
~/i
S"E(~i) ~i a
1
me
me
2 3 4 5 6
me 0.71 0.67 0.65 0.60
0.83 1.07 0.90 me e
0.068 0.070 0.075 0.063
Model (~ot, Pt, ~l)c
~(~i )
~f
0.105 0.138 0.123
0.65
S"E(~t) ~i a ~ ( ~ i )
me 0.031
1.31
0.094
0.87 1.10 0.92 me me
0.090 0.106 0.094
aBased on Pradel (1996). Data are for capture-recapture data on adult male meadow voles at Patuxent Wildlife Research Center, Laurel, Maryland (data in Tables 17.5 and 17.6). b Model (q~t, Pt, ~t ) AICc = 989.9. CModel (q~t,Pt, "Y) AICc = 984.8. e Estimated as the derived parameter Ki = ~i/~i+1" e Quantity not estimable under the model. fSingle estimate q corresponds to sample periods 2-6.
18.4). The derived estimates of population growth rate based on the estimates from the reduced-parameter model were similar to those under the time-specific model for the periods for which they could be estimated. The constant-parameter model (q~t,Pt, "Y) permitted estimation of an additional population growth parameter, ~1, and also yielded estimates with smaller standard errors than the time-specific model (q~t,Pt, "Yt). Estimates under the two )~-parameterization models are presented in Table 18.5. Estimates under the timespecific model (oPt, Pt, kt) are identical to those derived from model (q~t, Pt, "Yt) and displayed in Table 18.4. The estimates also can be compared with the derived estimates
Ki -- 1Qi+l / (1Qi -- di)
TABLE 18.5 Estimated Population Growth Rates Meadow Voles under Two Models a Sample
period
(~ki) for
Model (~t, Pt, ]kt)b
Model (%, Pt, k)c
~i
~e
~(~k)
1.04
0.042 e
1
d
2 3 4 5
0.83 1.07 0.90 d
S"E(~ki) d
0.105 0.138 0.123 d
a Based on Pradel (1996). Data are for capture-recapture data on adult male meadow voles at Patuxent Wildlife Research Center, Laurel, Maryland (data in Tables 17.5 and 17.6). bModel (q~t,Pt, ~'t) AICc = 989.9. CModel (oPt,Pt, k) AICc = 993.4. dparameters not estimable under the model. eSingle estimate )~applies to periods 1-5.
Chapter 18 Estimating Abundance and Recruitment
518
from the Jolly-Seber model (see Section 18.5), where the abundance e s t i m a t e s / ~ i are obtained using the biascorrected Jolly-Seber estimator as in Table 18.1 and the d i refer to trap deaths or losses on capture. The subtraction of trap deaths is intended to restrict inference about population growth to ecological (as opposed to investigator-related) processes. Estimates of population growth computed in this way are effectively identical to the estimates produced directly by the Pradel (1996) model (~t, Pt, )kt)" The reduced-parameter model (q~t,,,Pt, X) yields an estimated population growth rate of ~ = 1.04. However, a comparison of the AICc values for the two models suggests that time specificity in ]ki is needed to model the data adequately (Table 18.5). This result is logically consistent with the result that model (~t, Pt, ~/) was appropriate for modeling the same data. If survival is time specific and the proportion of new animals is time invariant, it then follows that population growth rate should vary over time. Finally, we note that use of model selection (AICc) or test (LR) statistics comparing models with timespecific vs. time-invariant population growth rate should be relevant to decisions about whether to use asymptotic rates of change from matrix population models as descriptions of population growth. The time-specific recruitment model (q~t, Pt, ft) yielded estimates of new recruits at i + 1 per animal at i ranging from 0.24 to 0.35 (Table 18.6). These estimates can be compared with the derived estimates
fi-- B i / ( l ~ i - di) obtained under the JollyzSeber model (see Section 18.5), where the /~i and N i are Jolly-Seber estimates and d i represents trap deaths. As was the case with population growth rate, the derived recruitment esti-
TABLE 18.6 Estimated Per Capita Recruitment Rates for Meadow Voles under Two Models a
(fi)
Sample period/
Model (%, Pt, ft )b
Model (%, Pt, f)c
fi
S'E(fi )
~e
S"E(f)
1 2 3 4 5
d 0.24 0.35 0.31 a
md 0.076 0.109 0.098 a
0.35
0.039
aBased on Pradel (1996). Data are for capture-recapture data on adult male meadow voles at Patuxent Wildlife Research Center, Laurel, Maryland (data in Tables 17.5 and 17.6). bModel (%, Pt, ft) AICc = 989.9. CModel (%, Pt, f) AICc = 987.6. aQuantities not estimable under the model. eSingle estimate f applies to periods 1-5.
mates are effectively identical to the estimates under model (q~t, Pt, ft). The AICc was slightly lower for the constant-f model, which yielded an estimate of f = 0.35 (Table 18.6). Note that the AICc values under the general models with all parameters time specific are identical for the three parameterizations of Tables 18.4-18.6. This is expected, because they are statistically equivalent ways of representing the same data. However, the AICc values for the reduced-parameter models with parameters constant over time are not identical for the different parameterizations. This again is expected, because the parameterizations in these models yield statistically different representations, with different consequences as to model fit.
18.5. R E L A T I O N S H I P S AMONG APPROACHES
The Jolly-Seber (Jolly, 1965; Seber, 1965), superpopulation (Crosbie and Manly, 1985; Schwarz and Arnason, 1996), and temporal symmetry (Pradel, 1996) approaches described above are simply three different ways of modeling the same data. In this section we attempt to clarify some of the relationships among the three approaches, as an aid to understanding the information contained in open-model capture-recapture data. In order to facilitate understanding, we consider the case of no losses on capture, though the arguments in this section remain intact absent this assumption (e.g., see Section 18.4.1). All three approaches include survival and capture probabilities, though capture probability parameters are viewed slightly differently under reverse time with losses on capture. However, the same Pi are used for all three approaches in the case of no losses on capture. In the discussion below, we first consider the Jolly-Seber quantities N i and B i and write them in terms of the superpopulation and temporal symmetry approaches. We then focus on the three parameterizations of Pradel (1996) and consider estimation of the associated parameters under the Jolly-Seber and superpopulation approaches. The Jolly-Seber approach treats population size ( N i) and number of recruits (B i) as unknown random variables to be estimated. Under the Schwarz-Arnason superpopulation approach, the expected number of recruits in any sampling period is simply given by the product of superpopulation size and the appropriate entry probability: E(BilN) = N ~ i ,
(18.44)
leading to the estimator in Eq. (18.31). Under the Schwarz-Arnason superpopulation ap-
18.5. Relationships among Approaches proach, the expected value of abundance in period i can be written as
E(Nil N) = N(~oq~lq~2 "'" q~i-1 if- ~1q~2q~3 "'"
(18.45)
X q~i-1 q- "'" q- ~i-1)
(see Schwarz, 2001). In Eq. (18.45), expected abundance in period i is written as the sum of the expected number of animals that first entered at each previous sampling occasion and survived until i. The recursive abundance estimator in Eq. (18.32) is based on Eq. (18.45). Under the temporal symmetry models of Pradel (1996), the expectation for B i can be written as
E(BilNi+I) = Ni+I(1 - '~i+1)-
(18.46)
Thus, the expected number of recruits at i + 1 is simply the product of population size at i + 1 and the proportion of these animals that are recruits. Equation (18.46) suggests the estimator Bi = /~/i+1( 1 - ~i+1)
(18.47)
for recruits between period i and i + 1. One way of writing the expected value for abundance in period i under the models of Pradel (1996) is to condition on abundance in the first sampling period and to multiply this abundance by the appropriate population growth rate parameters:
of the seniority parameter (the probability that an animal present at i+1 is "new" in the sense that it was not present at i) as the ratio of estimated new animals at i+1 to abundance at i+1. Now consider estimation of the three parameters (~i, Ki, fi) used by Pradel (1996) under the superpopulation approach of Schwarz and Arnason. Begin by writing the expected population size as a function of Pradel's (1996) per capita survival and recruitment parameters:
E(Ni+I[Ni) = Ni(q~i + fi)"
j=l
Ki-- ~i q- fi.
fi =
Pradel's (1996) per capita recruitment rate fi is defined as the expected number of animals in the population at time i+1 per animal in the population at time i. The natural estimator for this quantity under the Jolly-Seber approach is simply
fi = Bi/l~i.
(18.48)
Finally, the seniority parameter of Pradel (1996) can be estimated under the Jolly-Seber approach either as ~i+1 -" l~i~Pi/l~i+l
(18.49)
or as 1 - "Yi+I = Bi/l~i+l.
1[
~j=0
~i = /~i+l/l~i"
(18.50)
Estimator (18.49) is the ratio of estimated survivors from period i still present at i+1 to the estimated abundance at i+1. Estimator (18.50) shows the complement
(18.52)
Equation (18.52) expresses population growth rate intuitively, as the sum of survival and recruitment rates (see Section 8.1). Under the superpopulation approach of Schwarz and Arnason (1996), recruitment rate can be estimated as
~.j.
Estimation of population growth rate under the Jolly-Seber approach relies on the definition of ~ki as the ratio of two abundances and substitution of the appropriate estimates:
(18.51)
Equation (18.51) simply defines the expected population size in sampling period i + 1 as the sum of expected survivors and new recruits, written as the product of abundance and the sum of survival probability and recruitment rate. By rearranging Eq. (18.51), population growth rate written as
i-1
E(Ni[NI) = Nil-- [
519
~i
]
(18.53)
~j I-II=j+l i-1 ~Pl
(Schwarz, 2001). Estimator (18.53) is obtained by writing fi as a function of Jolly-Seber approach estimators (18.48) and then substituting the corresponding superpopulation estimators from Eqs. (18.44) and (18.45). The superpopulation estimator for population growth rate is then given by
~i = ~Pi if- fi' where the estimator for recruitment rate is based on Eq. (18.53). Finally, it is possible to estimate Pradel's (1996) seniority parameter by substituting the appropriate superpopulation estimators into Eq. (18.50), to obtain 1
-
qi+l
=
~i q- ~j=0 i-1 q~l] /-11 ~j II1=j+l
Last, we note that the superpopulation of Schwarz and Arnason (1996) can be written as the sum of the numbers of new recruits to the population over all sampling periods [e.g., see Eq. (18.26)]. Because the numbers of recruits can be estimated using both the Jolly-Seber [Eq. (18.13)] and temporal symmetry [Eq.
520
Chapter 18 Estimating Abundance and Recruitment
(18.47)] models, estimation of N under these approaches can be based on Eq. (18.26) (e.g., see Shealer and Kress, 1994). Thus, any quantity estimated using one of the approaches considered in this chapter can be estimated (although perhaps indirectly) via the other two approaches. At present, there is little basis for recommending one estimation approach over another based solely on estimator properties. The superpopulation and temporal symmetry approaches are sufficiently new that there are yet to be comprehensive investigations of estimator properties. However, if investigator interest is on a particular abundance or recruitmentrelated parameter, it seems reasonable to use the approach that permits direct modeling of that parameter.
18.6. S T U D Y D E S I G N As we have emphasized throughout this book, study design should always be tailored to the questions under investigation. In this section we follow the approach of Section 17.6 and focus on aspects of study design that are especially relevant to the estimation of abundance, recruitment, and related parameters. The models of Chapter 18 are flexible enough to allow one to focus on specific questions involving these parameters. Following Section 17.6, we discuss three considerations that are relevant to study design: what parameters are to be estimated, how assumption violations can be minimized, and how precise estimates can be obtained. Because the models of this chapter are obtained by adding components to the likelihoods of Chapter 17, virtually all of the design considerations discussed in Section 17.6 also are relevant to Chapter 18. The following discussion focuses on aspects of study design not covered in Section 17.6.
18.6.1. Parameters to Be Estimated One aspect of study design that is important to the estimators of this chapter but not to those of Chapter 17 involves the reobservation process. The model parameters of Chapter 17 can be estimated based solely on reobservations of marked animals. However, if abundance or recruitment is of interest then every sampiing occasion for which an abundance or recruitment estimate is desired must include sampling of unmarked animals. As discussed in Section 18.1, unmarked animals need not be captured and marked, but it is necessary to record the number of unmarked animals observed during the process of recording the identities of marked animals. Thus, one must be able to distinguish different unmarked individuals during the sampling process, so that an unambiguous count
of them can be made (see Hestbeck and Malecki, 1989b; Kautz and Malecki, 1990; Dreitz et al., 2002). Many studies of open populations using capturerecapture are designed to investigate population dynamics for relatively long periods of time, over which births and deaths contribute a substantial fraction of the population gains and losses. For example, studies of small mammals might span several years and multiple generations of animals. On the other hand, some studies use open models to investigate dynamics over very short periods in which movement would account for most gains and losses. For example, ornithologists sometimes are interested in estimating the number of migratory birds using migration stopover sites (e.g., Nichols, 1996; Nichols and Kaiser, 1999; Kaiser 1999), and fishery biologists are interested in the numbers of fish migrating seaward from spawning streams (e.g., Schwarz et al., 1993b; Schwarz and Dempson, 1994). In these short-term studies, the study areas are viewed as "flow-through" systems, and one objective is to estimate the number of animals going through the system over the course of the study. In such studies, the superpopulation size is the parameter of primary interest, and the estimation approach of Schwarz and Arnason (1996; also see Crosbie and Manly, 1985) is especially appropriate. An important design aspect of such studies is to be sure that the sampling occasions cover the entire period of interest. For example, if capture and recapture do not begin until after the arrival of birds at a migration stopover location, then the estimated superpopulation size N will not include birds that arrived and departed before the sampling began. Although this chapter has focused on single-age, single-stratum models, we noted in Section 18.2.4 that abundance and recruitment also can be estimated for multiple ages or locations or physiological states. For studies with geographic stratification, the design must include approximately simultaneous sampling at multiple locations, as discussed in Section 17.6.1. Similarly, animals must be assigned to age class or physiological state at each reobservation if age- or state-specific estimates are desired. In the case of multiple-age classes, it is possible to decompose recruitment into components associated with immigration versus in situ reproduction (Nichols and Pollock, 1990) (see Section 19.4). Because sampling design considerations necessary for application of this approach include the robust design, we defer its discussion until Chapter 19.
18.6.2. Model Assumptions 18.6.2.1. Homogeneity of Rate Parameters The discussion in Section 17.6.2 included various suggestions for dealing with heterogeneity of rate pa-
18.6. Study Design rameters (also see Pollock et al., 1990), and these suggestions should be just as applicable to the new parameters introduced in Chapter 18 (e.g., the entry parameters of the superpopulation approach; the seniority parameters of the temporal symmetry approach) as they are to capture and survival probabilities. Stratification by factors such as location, age, sex, size, reproductive state, and physiological state can be useful whenever rate parameters are thought to differ among strata. The key design issue is then to record data on the stratification factors (i.e., factors likely to be associated with variation in rate parameters). In addition to stratification, the use of multiple sampling methods is also a good approach for reducing the likelihood of heterogeneous observation probabilities. A design issue relevant to estimation of abundance and recruitment involves the use of resighting as a means of "recapturing" animals. As noted in Section 18.1, the use of resighting requires that the investigator record the number of unmarked animals encountered while resighting marked animals. The important element of such sampling is that marked and unmarked animals must have the same probabilities of being observed. Assume, for example, that Canada geese are being sampled, and that some birds are marked with neckbands. Assume further that a large group of birds is under observation. In this case, an effort should be made to scan the group for neck collars, recording the number of individuals whose necks are observed without collars as well as the identities of birds with collars. If a bird is seen to have a collar, but flies off before the collar can be read, one approach is to not record this bird (i.e., the bird does not add to either the marked or the unmarked group). Under this approach, it is important not to record birds seen to be unmarked that fly off before their band numbers could have been read had they been marked. Such a bird could be assigned unambiguously to the unmarked group, but such an assignment will lead to higher sighting probabilities for unmarked birds (because a marked bird seen for a similar length of time could not be identified). Thus, it is best to not record these birds in the sample. Similarly, any birds in the group whose necks are not examined would not be recorded in the sample at all. This example is just one of many possible field situations, but it illustrates the point that every effort should be made to ensure that marked ( M i) and unmarked (U i) animals in the sampled population have similar probabilities of entering the sample and being included in the associated count statistic (i.e., of appearing as a member of m i or u i, respectively). Spatial sampling with devices such as traps should be conducted in a way that ensures each animal in the area of interest is likely to encounter at least one sampling device (Section 17.6.2). As noted previously,
521
high sampling intensities [e.g., capture probabilities :>0.5; see Gilbert (1973)] can reduce the effects of heterogeneous capture probabilities on abundance estimates. Finally, the various methods for reducing traphappy or trap-shy responses (Section 17.6.2) should also be considered. The above recommendations concern the design of a study so as to minimize violations of the assumption of homogeneous rate parameters. An alternative approach for studies focused on abundance estimation is to implement the robust design (Chapter 19). The original motivation for this design involved the ability to use closed population models (Chapter 14) (Otis et al., 1978) for estimation of abundance (Pollock, 1982). Closed population models and estimators have been developed for situations in which capture probabilities vary among individuals (heterogeneity) and between marked and unmarked individuals (behavioral response models). Indeed, the "robust design" was so named because it provides the ability to obtain estimates of abundance (as well as survival and recruitment) in the presence of nonhomogeneous capture probabilities (Pollock, 1982).
18.6.2.2. Tag Retention As noted in Section 18.2.5, tag loss does not produce bias in Jolly-Seber abundance estimates but does lead to reduced precision of these estimates. Jolly-Seber recruitment estimates are biased by tag loss. The best design advice regarding tag loss is to use doubletagging (at least for a subset of animals) in cases where it is suspected (Section 17.6.2), because this permits estimation of loss rates and thus provides some ability to deal with any resulting problems of estimator bias (e.g., with respect to survival and recruitment estimators).
18.6.2.3. Instantaneous Sampling Violation of this assumption causes problems with estimation of abundance and recruitment. Indeed, if the population is open to gains and losses during the sampling period, then it is not even clear what is meant by "abundance during period i." The appropriate design recommendation is to select the seasonal timing and duration of the sampling periods in an attempt to reduce the possibility of nonnegligible mortality, immigration, and emigration (Section 17.6.2).
18.6.2.4. Temporary Emigration Markovian temporary emigration can result in biased estimates of abundance and recruitment (Kendall et al., 1997). Perhaps the best way to deal with this
522
Chapter 18 Estimating Abundance and Recruitment
possibility is to include in the study design a way to either (1) estimate the time-specific conditional capture probabilities for animals in the study area and exposed to sampling efforts or (2) estimate rates of migration to and from areas surrounding the primary sample area. The robust design (Chapter 19) has been proposed as a means of estimating conditional capture probabilities for animals in the study area (Kendall and Nichols, 1995; Kendall et al., 1997; Schwarz and Stobo, 1997). The direct estimation of movement rates can be accomplished by establishing another stratum (e.g., the area surrounding the principal study area) to be sampled via capture-recapture methods (e.g., using multistraturn models; Section 17.3). An alternative approach is to mark a subset of animals with radios and use telemetry to estimate directly rates of temporary emigration (e.g., Pollock et al., 1995; Powell et al., 2000a).
18.6.3. Estimator Precision Under the Jolly-Seber approach, abundance and especially recruitment tend to be estimated relatively less precisely (e.g., larger coefficients of variation) than survival probability (see Pollock et al., 1990). Precision is thus an especially important consideration for studies directed at estimation of abundance and recruitment. All of the design recommendations provided in Section 17.6.3 are relevant for abundance and recruitment estimation as well. As with survival rate estimation, increases in capture probability lead to increases in precision. The sample size figures of Pollock et al. (1990) plot cv(Ni) and cv(Bi) as functions of capture probability, and knowledge of this relationship often is useful in study design. Basically, all of the tradeoffs and considerations previously discussed (Section 17.6.3) are even more important for abundance and recruitment estimation because of the inherent tendency of these estimates (especially recruitment) to be relatively imprecise.
18.7. D I S C U S S I O N The basic approach for the models of Chapter 17 was to condition on release of a marked animal in a specific sampling period and then to model the remainder of its capture history as a function of capture and survival probability parameters. In the models of Chapter 18, we relaxed the conditioning in Chapter 17 by adding components that account for the entry of unmarked animals into the population. The primary motivation for adding these new components is to esti-
mate additional quantities such as population size, recruitment, and related parameters. The Jolly-Seber, superpopulation, and temporal symmetry approaches described in this chapter are simply three different ways of parameterizing the extra model components. The equivalence of the three approaches is emphasized in Section 18.5, where we show that any quantity estimated using one approach can be estimated (although perhaps indirectly) via the other two approaches. For example, the Jolly-Seber approach focuses on the direct estimation of numbers of animals (N i) and numbers of recruits (Bi). These quantities are treated as unknown random variables to be estimated after the modeling of survival and capture parameters, absent model parameters for abundance or recruitment. On the other hand, the superpopulation approach considers the total number of animals found in the study area during at least one sampling period of the entire study and the probabilities that a member of the superpopulation entered the sampled population at each of the sampling periods during the study. The temporal symmetry approach incorporates simultaneous backward and forward models for capture history data and utilizes seniority parameters (probability that a member of the population at sampling period i + 1 is "old" in the sense of having been in the population the previous period). Alternative parameterizations for the temporal symmetry models use either population growth rate or recruitment rate. It is remarkable that a simple capture history matrix (a vector of ls and 0s for every animal caught during a study) provides the information needed to estimate all these quantities. Time-specific estimates of abundance, survival, recruitment, and the various derivative parameters provide a very detailed description of the dynamics of the studied population. Of course, certain questions require auxiliary data (e.g., decomposition of losses into deaths and movement; decomposition of gains into recruits resulting from in situ reproduction and immigration), but the basic demographic bookkeeping associated with changes in numbers of animals on a predefined study area can be accomplished using the data from a simple capture history matrix. In Chapter 19 we deal with capture-recapture data obtained at two different temporal scales, thereby permitting simultaneous use of both closed (Chapter 14) and open (Chapters 17 and 18) models. We will see that the use of this "robust design" permits not only the estimation of quantities that could be estimated using either closed or open models separately, but also the estimation of quantities that could not be estimated without both in combination.
C H A P T E R
19 Combining Closed and Open Mark-Recapture Models: The Robust Design 19.1. DATA STRUCTURE 19.2. A D HOC APPROACH 19.2.1. Combining Open and Closed Models 19.2.2. Estimation Based Solely on Closed Models 19.3. LIKELIHOOD-BASED APPROACH 19.3.1. Models 19.3.2. Model Assumptions 19.3.3. Estimation 19.3.4. Alternative Modeling 19.3.5. Model Selection, Estimator Robustness, and Model Assumptions 19.4. SPECIAL ESTIMATION PROBLEMS 19.4.1. Temporary Emigration 19.4.2. Multiple Ages and Recruitment Components 19.4.3. Catch-Effort Studies 19.4.4. Potential for Future Work 19.5. STUDY DESIGN 19.6. DISCUSSION
dynamics (population size), the rate of change in that state variable, and the vital rates responsible for that change. This chapter represents a synthesis of capturerecapture approaches to the estimation of population size and vital rates, by combining in a single model the advantages of both open- and closed-population methods. Here we view the long-term study of an open population as a sequence of short-term studies of closed populations. Several advantages accrue to population sampling at two distinct temporal scales, including more robust estimation of the parameters considered previously and estimation of certain parameters not otherwise estimable with either open or closed models when considered separately. Both advantages are a direct consequence of the additional information provided by the short-term capture-history data. In one sense, the robust design can be considered to be a special case of using auxiliary data (Section 17.5) produced from short-term sampling. The original motivation for the robust design was a concern about estimator robustness, especially as relates to heterogeneity in capture probabilities. Previous to his formulation of the robust model, Pollock (1975) extended the work of D. S. Robson (1969) by incorporating certain kinds of capture-history dependence (see Section 17.1.6) in the context of Jolly-Seber models. However, other kinds of variation in capture probability, notably heterogeneity among individuals and permanent trap response, could not be dealt with adequately in an open-model setting. Although the survival estimators for the Jolly-Seber model are relatively robust to these sources of variation (e.g., see Carothers 1973, 1979), its abundance estimators are not
In Chapters 14 and 16-18 we focused on the estimation of population parameters based on studies of marked animals. In Chapter 14 we saw that capturerecapture models can be used to estimate population abundance over short periods of time during which the population is assumed to remain unchanged in size and composition. In contrast, open-population models (Chapters 16-18) allow one to include population gains and losses between sampling periods and thus to estimate population size, population rate of change between successive sampling periods, and rates of survival, recruitment, and movement between sampling periods. In terms of system dynamics, these quantities include the principal state variable for population
523
Chapter 19 The Robust Design
524
(e.g., Gilbert, 1973; Carothers, 1973). On the other hand, capture-recapture models for closed populations were developed to deal with trap response and heterogeneity in capture probabilities, leading to robust estimates of abundance under these conditions (Pollock, 1974; Burnham and Overton, 1978; Otis et al., 1978). Building on both approaches, Pollock (1981a, 1982) suggested sampling at two temporal scales, with periods of short-term sampling over which the population is assumed to be closed and longer term sampling over which gains and losses are expected to occur (also see Lefebvre et al., 1982). In particular he recommended that closed models be used to estimate abundance, with data arising from each short-term sampling episode (see Chapter 14). These data then can be pooled (with each animal recorded as caught if it was observed at least once during the closed population sampling) to estimate survival based on the Cormack-Jolly-Seber estimators (see Chapter 17). With the abundance estimates from the closed models and survival estimates from the open models, recruitment in turn can be estimated as in Eq. (18.13). Pollock (1982) suggested that such a sampling design should provide estimators that are robust to various sources of variation in capture probabilities.
stricted to include a fixed number of secondary occasions over all the primary occasions (l i = l for all i = 1, ..., K). As an example, a small mammal population might be trapped for five consecutive days every 2 months. Capture-recapture data from the robust design can be summarized in several ways. Perhaps the most basic summary is analogous to the X matrix of Section 14.2.1, with Xgij an indicator variable reflecting either capture (Xgij = 1) or no capture (Xgij = O) for individual g in secondary sampling period j of primary sampling period i. For example, a study with K = 4 primary sampling periods and l = 5 secondary sampling periods within each primary period would correspond to an X matrix with 20 columns. A row vector corresponding to a particular animal might be 01101 00000
A schematic representation of the robust sampling design is presented in Fig. 19.1. The design consists of K primary sampling occasions, between which the population is likely to be open to gains and losses. At each primary sampling occasion, a short-term study is conducted, with the population sampled over l i secondary sampling periods, during which it is assumed to be closed [although this assumption can be relaxed; see Schwarz and Stobo (1997)]. Though one can have a different number of secondary sampling occasions for each primary occasion, the design also can be re-
Secondary Periods
1
1
/ I N...11 2
10111,
consisting of four groups of five capture values. The first group of five numbers gives the capture history over the five secondary periods of primary period 1, showing that the animal was captured on occasions 2, 3, and 5 of primary period 1. The second group of numbers indicates that the animal was not captured at all during primary period 2. In primary period 3, it was captured on the third secondary occasion, and in primary period 4, it was captured on secondary occasions 1, 3, 4, and 5. The X matrix consists of all such capture history vectors for all animals caught at least once during the study. For example, the X matrix for the male meadow voles used in the example analyses of Sections 17.1.10, 18.2.6, 18.3.6, and 18.4.6 is presented in tabular form in Table 19.1. Note that a " - " designation in the final column of Table 19.1 indicates that the animal was not released back into the population following the last capture (the last "1") in the record. The individual capture history data also can be collapsed into various kinds of summary statistics. Here we follow the general notation of Kendall et al. (1995),
19.1. D A T A S T R U C T U R E
Primary Periods
00100
2
1
.
/ I N 12 2...
.
.
K
1
2...1K
FIGURE 19.1 Schematicrepresentation of Pollock's (1982) robust design for capture-recapture sampling. Primary sampling periods i = 1..... K are separated by relatively long time intervals over which the population is likely to be open to gains and losses. At each primary period i, sampling is conducted at l i secondary sampling periods. Secondary periods are separated by relatively short time intervals over which the population may be closed to gains and losses. Models for open populations are used for the capture history data summarized at the level of primary periods, whereas either closed or open models (as appropriate) are used for data summarized at the level of secondary periods within each primary period.
19.1. TABLE 19.1
525
Data Structure
C a p t u r e - R e c a p t u r e Data for A d u l t M a l e M e a d o w Voles a Primary sampling period b
Identification number
1
2
3
4
4321
00100
00000
00000
5311
00000
00000
00010
7701
11011
11100
7720
11110
7725
11111
7736
5
6
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00110
11111
11111
11111
11111
11111
11101
11111
11111
11011
11111
7745
11101
01100
00000
00000
00000
00000
7752
10101
11011
00000
00000
00000
00000
7762
00000
00100
00000
00000
00000
00000
7764
11111
11100
11000
11111
11111
11111
7772
10110
00000
00000
00000
00000
00000
7773
10101
01110
00000
00000
00000
00000
7775
11111
11100
00000
00000
00000
00000
7782
00000
00000
00010
00000
00000
00000
7785
11100
01110
00000
00000
00000
00000
7786
01100
11000
11111
00000
00000
00000
7792
01101
01100
01011
11111
11101
11101
7796
11111
11100
00000
00000
00000
00000
7811
11111
11110
11101
11111
11111
01001
7824
11111
11110
00000
00000
00000
00000
7828
11011
00000
00000
00000
00000
00000
7832
00001
11000
00000
00000
00000
00000
7840
01100
11100
00000
00000
00000
00000
7846
11101
10000
00000
00000
00000
00000
7847
11000
01100
00000
00000
00000
00000
7853
11011
10100
00000
00000
00000
00000
7855
10100
00000
11001
11110
00000
00000
7856
10000
01100
00000
00000
00000
00000
7857
11111
01100
11001
10111
11110
00000
7858
10101
11100
00010
10100
00000
00000
7860
01010
00000
00000
00000
00000
00000
7863
01011
01000
10010
11111
00000
00000
7865
01000
01000
10000
00000
00000
00000
7866
01000
00000
00000
00000
00000
00000
7867
01111
00000
00000
00000
00000
00000
7868
01110
01100
00000
00000
00000
00000
7869
01100
11100
11110
11111
11110
11010
7871
01111
00000
00000
00000
00000
00000
7872
00000
01110
00000
00000
00000
00000
7874
01011
00000
00000
00000
00000
00000
7875
01111
11100
00000
00000
00000
00000
7879
00101
00010
01000
11110
00010
01010
7882
00100
11110
11101
11111
00000
00000
7887
00100
11110
01000
00000
00000
00000
Not released ( - )
(continues)
526
C h a p t e r 19
The Robust Design
T A B L E 19.1
(Continued)
Primary sampling period b
Identification number
1
2
3
4
5
6
7890
00010
00000
00000
00000
00000
00000
7891
00010
11100
00000
10000
10100
10000
7892
00000
10100
00000
00000
00000
00000
7894
00010
11000
00000
00000
00000
00000
7895
00010
11100
00000
00000
00000
00000
7896
00001
11100
10000
11110
11111
10100
7901
00000
11100
00000
00000
00000
00000
7904
00000
11100
00000
00000
00000
00000
7905
00000
11101
01010
11000
00000
00000
7906
00000
11000
01000
00000
00000
00000
7907
00000
11100
11110
00000
00000
00000
7910
00000
11100
00010
10001
00000
00000
7912
00000
01100
00000
00000
00000
00000
7913
00000
01000
00100
00000
00011
01010
7918
00000
00000
11100
11001
00000
00000
7919
00000
01010
01100
00000
00000
00000
7920
00000
01000
10110
00001
11100
11011
7921
00000
01010
00000
00000
00000
00100
7922
00000
01000
00000
00000
00000
00000
7925
00000
00100
00000
00000
00000
00000
7930
00000
00100
00000
11000
00000
00000
7932
00000
00100
00001
01000
00000
00010
7935
00000
00010
00000
00000
00000
00000
7936
00000
00010
00000
00000
00000
00000
7937
00000
00010
00000
00000
00000
00000 00000
7938
00000
00010
00010
10010
00000
7940
00000
00010
00000
01011
01001
01111
7941
00000
00010
01000
01001
00000
10000
7944
00000
00000
10001
00000
00000
00000
7945
00000
10100
11111
11111
11111
11110
7946
00000
00000
11111
11111
00000
00000
7948
00000
00000
01011
00000
00000
00000
7949
00000
00000
01010
00111
11111
11111
7953
00000
00000
01000
00000
00000
00000
7954
00000
00000
00000
01011
00000
00000
7957
00000
00000
01000
00000
00000
00000
7958
00000
00000
00100
00000
00000
00000
7964
00000
00000
00010
10110
00000
00000
7967
00000
00000
00001
00001
11011
11101
7969
00000
00000
00000
10000
01111
11111
7970
00000
00000
00000
11111
00000
00000
7974
00000
00000
01000
10011
11111
11110
7975
00000
00000
00000
10001
00000
00000
7976
00000
00000
00000
00000
11101
11111
Not released ( - )
(continues)
19.1.
527
Data Structure
T A B L E 19.1
(Continued)
Primary sampling period b
Identification number
1
2
3
4
7978
00000
00000
00000
00000
01101
00110
7980
00000
00000
00000
10000
00000
00000
7983
00000
00000
00000
01110
00000
00000
7986
00000
00000
00000
01000
00000
00000
7990
00000
00000
00000
00000
00000
01000
7992
00000
00000
00000
00000
00000
00111
7995
00000
00000
00000
01000
11111
11011
7999
00000
00000
00000
01100
11110
11111
8002
00000
00000
00000
00110
00000
00000
8003
00000
00000
00000
00100
10100
10100
8007
00000
00000
00000
00100
00000
00000
8008
00000
00000
00000
00100
00000
00000
8009
00000
00000
00000
00100
00000
00000
8010
00000
00000
00000
00011
10111
00000
5
6
8014
00000
00000
00000
00000
01010
11011
8016
00000
00000
00000
00010
01101
00000
8017
00000
00000
00000
00000
01111
00010
8019
00000
00000
00000
00000
00010
00010
8022
00000
00000
00000
00001
00000
00000
8027
00000
00000
00000
00001
00000
00000
8028
00000
00000
00000
00001
00000
00000
8029
00000
00000
00000
00001
00111
00000
8032
00000
00000
00000
00000
00000
11111
8033
00000
00000
00000
00000
00000
10011
8034
00000
00000
00000
00000
00000
10000
8036
00000
00000
00000
00000
10111
11111
8038
00000
00000
00000
00000
11000
10110
8040
00000
00000
00000
00000
00000
01110
8044
00000
00000
00000
00000
10100
00000
8045
00000
00000
00000
00000
10001
00110
8046
00000
00000
00000
00000
00000
01011
8048
00000
00000
00000
00000
00000
01111 01010
8050
00000
00000
00000
00000
00000
8051
00000
00000
00000
00000
00000
11100
8052
00000
00000
00000
00000
01010
00000
8055
00000
00000
00000
00000
01010
11111
8056
00000
00000
00000
00000
00000
11110
8058
00000
00000
00000
00000
00000
11111
8061
00000
00000
00000
00000
01110
00100
8062
00000
00000
00000
00000
00000
00110
8064
00000
00000
00000
00000
01000
11011
8069
00000
00000
00000
00000
00100
00010
8070
00000
00000
00000
00000
00000
00001
8074
00000
00000
00000
00000
00100
00000
Not released ( - )
(continues)
528
C h a p t e r 19
The Robust Design
T A B L E 19.1
(Continued)
Primary sampling period b
Identification number
1
2
3
4
5
6
8080
00000
00000
00000
00000
00010
00000
8087
00000
00000
00000
00000
00001
01000
8090
00000
00000
00000
00000
00000
11100
8092
00000
00000
00000
00000
00000
10000
8093
00000
00000
00000
00000
00000
10100
8095
00000
00000
00000
00000
00000
10110
8097
00000
00000
00000
00000
00000
11000
8099
00000
00000
00000
00000
00000
10110
8100
00000
00000
00000
00000
00000
10110
8225
00000
01000
00000
01001
11111
11111
8421
00000
00000
00000
00000
00000
11101
8601
00000
00000
00000
00000
00000
01000
8602
00000
00000
00000
00000
00000
01000
8604
00000
00000
00000
00000
00000
01000
8606
00000
00000
00000
00000
00000
01001
8608
00000
00000
00000
00000
00000
01000
8610
00000
00000
00000
00000
00000
00100
8613
00000
00000
00000
00000
00000
00110
8616
00000
00000
00000
00000
00000
00010
8619
00000
00000
00000
01000
00000
00011
8620
00000
00000
00000
00000
00000
00001
8621
00000
00000
00000
00000
00000
00001
8624
00000
00000
00000
00000
00000
00001
8633
00000
11000
10011
00101
01001
11111
8645
00000
00000
00000
00000
00000
00010
8652
00000
00000
11111
11101
11101
11110
9321
10000
00000
00000
00000
00000
00000
9322
11111
11100
11101
11111
11011
10010
9334
11111
01100
00000
00000
00000
00000
9343
11111
11101
00000
00000
00000
00000
9345
11110
11100
10011
00000
00000
00000
9350
11111
10100
00000
00000
00000
00000
9359
11101
11000
00000
00000
00000
00000
9362
11110
11100
11011
11110
00000
00000
9381
11111
11110
11111
11101
11111
01110
FIB5
00000
00100
10100
00001
00000
00111
TCB1
00000
00000
00100
00000
00000
00000
TCB2
11111
01110
11110
00000
00000
00000
TCF7
11111
10100
00000
00000
00000
00000
Not released ( - )
a At Patuxent Wildlife Research Center, Laurel, Maryland, June-December, 1981. Data follow Pollock's robust design, with the columns under each primary period representing the five consecutive days of trapping each month. bInitial date in 1981 of each primary period: 1, 6/27; 2, 8/1; 3, 8/29; 4, 10/3; 5, 10/31; 6, 12/4.
529
19.2. Ad Hoc Approach who used the notation of Chapters 17 and 18 for the open-model portion of the capture-history data: ui
mi
Yli ~ U i -}- m i
Ri
mhi
The number of unmarked animals caught on at least one secondary occasion within primary period i. The number of animals marked previous to primary period i that are caught on at least one secondary occasion within primary period i. The total number of animals caught on at least one secondary occasion within primary period i. The number of n i that are released back into the population following primary period i. The number of R i that are recaptured at some primary period following i. The number of animals caught in primary period i that were last caught in primary period h.
In addition, the following statistics are associated only with the robust design: X~i
X)~i
x.~
The number of animals from u i that exhibit capture history 00 e f~ over the l i secondary periods of primary period i (where f~ is the set of all possible sequences of 0s and ls over the l i secondary periods). The number of animals from mhi that exhibit capture history co e f~ over the l i secondary periods of primary period i. The total number of animals caught in primary period i that exhibit capture history 00 over the l i secondary periods: i-1
X~ :
E
X~ i"
h=0
This notation allows us to partition the individuals captured at time i into (1) those previously captured at primary period h = 0, 1..... i - 1, and (2) those with secondary capture history oJ. We again designate animals not released following the final capture of the secondary capture history with a " - " preceding the number of animals exhibiting the history.
19.2. AD H O C APPROACH An ad hoc approach to the robust design typically involves a combination of open and closed models. However, it also is possible to develop ad hoc estimators
for abundance, survival probability, and recruitment using only capture-recapture models for closed populations. In what follows we describe both approaches, but emphasize that the former approach is by far the more commonly used.
19.2.1. Combining Open and Closed Models 19.2.1.1. M o d e l s
In his pioneering work, Pollock's (1981a, 1982) robust capture-recapture design involved three different approaches to estimation: (1) estimation of abundance with closed models using secondary capture-history data, (2) estimation of survival rates using standard open models with capture-history data reflecting captures in each primary period, and (3) estimation of the number of new recruits using the closed-model abundance estimates and open-model survival estimates, in conjunction with Eq. (18.13). Thus, the modeling proceeds via independent selection of an open model that incorporates survival and capture probabilities for the primary periods, and a closed model that incorporates abundances and capture probabilities for the secondary periods. The independent modeling of data from the primary and secondary periods distinguishes the ad hoc approach from a likelihood-based approach, in which both types of data are modeled simultaneously within a single likelihood (see Section 19.3). Under the most general ad hoc approach, model selection can be carried out independently for each of the K closed-model data sets (one for each primary period). This can lead to different closed models for different primary periods within a single analysis. Unless there are a priori reasons for expecting different closed models, we recommend the use of a single closed-population model for all K data sets. One reason for this recommendation concerns the magnitudes and directions of biases associated with the abundance estimators of the different models. Any biases are likely to be of similar direction and magnitude if the same model is used for estimation with all K data sets (e.g., see Skalski and Robson, 1992), and this similarity will yield more reasonable estimates of recruitment when the closed- and open-model estimates are combined [see Eq. (19.3)]. A second reason for recommending the use of a common closed-population model involves the imperfect nature of all model selection algorithms, including that of Otis et al. (1978) and Rexstad and Burnham (1991). It often is reasonable to expect similar processes (environmental variation, genetic variation)
530
Chapter 19 The Robust Design
to affect capture probability throughout a study, and it is appropriate under these circumstances to use model selection results from all K data sets to select a single model for the study. For example, if we have K = 8 primary periods and the model selection algorithm of CAPTURE (Rexstad and Burnham, 1991) indicates selection of model Mh for six periods, model M 0 for one period, and model Mth for one period, it may be reasonable to select model Mh for use with all eight data sets. Of course, this approach is ad hoc and without a sound theoretical basis, but it nonetheless seems reasonable. Note that the models for the robust design must account for two kinds of capture probabilities, corresponding to the two different temporal scales. The capture probabilities Pij associated with secondary sampling periods refer to the probability that an animal is captured on secondary occasion j of primary occasion i, given that it is in the population on that occasion. On the other hand, the capture probabilities p* for primary sampling periods refer to the probability that an animal is caught at least once in primary occasion i (i.e., on at least one of the l i secondary occasions), given that the animal is in the population during that sampling period. The latter probabilities are the same parameters used in Chapters 17 and 18, where the "," superscript is used here to avoid confusion with capture parameters for the secondary periods. Here we follow the approach of Kendall et al. (1995), who considered different models for these two different kinds of capture probability. In particular, they allowed for multiple sources of variability over the secondary periods, including temporal variability, behavioral response (i.e., dependence of capture probability on previous capture within the primary period), and heterogeneity (i.e., different capture probabilities among the animals in the population in a primary sample). Models and estimators for these sources of variation in capture probability were discussed in Section 14.2.2 (also see Otis et al., 1978). Recall from Chapter 17 that open models for capture probability also can include time and, in a limited sense, behavioral response, but not individual heterogeneity. Permanent trap response can be included in open models (see Section 17.1.9) because survival estimates are based on marked animals only and exhibit no bias in the face of permanent trap response (Nichols et al., 1984b). Note, however, that permanent trap response at the primary level imposes requirements on models for the secondary sampling period data. Permanent trap response indicates that animals that are unmarked just before primary sampling period i have one capture probability and marked animals have a different capture probability. In order to deal with this response in the closed-population modeling of the secondary periods, animals that were and were not marked prior to primary period i are
placed in two groups and analyzed separately with closed models. Thus, models incorporating behavioral response at the level of primary periods require the fitting of a single open model and (2K - 1) closed models, one closed model each for the marked and unmarked animals in each primary period 2, ..., K, and a single closed model for primary period I (when all animals are unmarked). Kendall et al. (1995) designated models for the robust design as M~, where 13 specifies the model for the primary period capture probabilities and oL specifies the model for the secondary period capture probabilities. A "0" in the subscript or superscript indicates no variation in the specified capture probabilities. Kendall et al. (1995) considered models in which the data within all primary periods exhibit the same sources of variation in capture probability. Thus, they did not consider independent model selection for all K closed models, but instead required a single closed model for use with all K data sets. Note that this formulation leaves unspecified the modeling of survival probabilities between the primary periods. Possible robust design models representing combinations of open (from Chapter 17) and closed (from Chapter 14) models are shown in Table 19.2. Because the model for the primary periods is fitted to the data independently of the fitting of closed models to the secondary period data, the models underlying the ad hoc approach are nothing more than simple combinations of open and closed models. However, given that there is no temporary emigration and all assumptions underlying both models are met, there is a mathematical relationship between capture probabilities at the primary and secondary sampling periods: li
1 - p~ = I-[(1 - P ij) j=l or
xPi =
li
1 - l-I(1 -Pij)
(19.1)
j=l
for i = 1, ..., K. In words, these expressions essentially say that the probability of noncapture on the primary time scale is given by the product of noncapture probabilities on the secondary time scale. Under the simplest model, M ~ the closed model M 0 (Section 14.2.2) (see Otis et al., 1978) is fitted to each of the secondary period data sets, and a model with constant capture probability [e.g., model (q~t,P)] is fitted to the primary period data. Note that model M ~ imposes the constraint that Pij = Pi., i.e., secondary capture probabilities are equal within each primary period.
19.2. A d Hoc Approach TABLE 19.2
Possible Models for Capture Probability under the Robust Design a Source(s) of variation in capture probability
Model
Secondary periods
Primary periods
M~
None
None
Mt
None
Time
M~'
Time
None
M{
Time
Time
M~
Behavior
None
M~
Behavior
Time
Mb
Behavior
Behavior
M~b
Behavior
Time, behavior
M~
Heterogeneity
None
M{~
Heterogeneity
Time
Mt~
Time, behavior
None
M~b
Time, behavior
Time
Mbb
Time, behavior
Behavior
Mttb
Time, behavior
Time, behavior
Mt~
Time, heterogeneity
None
Time, heterogeneity
Time
M~h
Behavior, heterogeneity
None
M~h Mbh
Behavior, heterogeneity
Time
Behavior, heterogeneity
Behavior Time, behavior
M{~b
Behavior, heterogeneity
o Mtb h M~bh Mbbh
All
None
All
Time
All
Behavior
tb
All
Time, behavior
a Following Kendall et al. (1995).
If all primary periods contain the same number of secondary capture periods (l i - l for all i), expression (19.1) then becomes p* = 1 - (1 - p)~
havioral response and heterogeneity for secondary periods. Behavioral response at the secondary sampiing level but not the primary level indicates that behavioral response is temporary, in that marking in previous primary periods is not relevant to capture probability, but marking in a previous secondary period within the same primary period does confer a different capture probability. 19.2.1.2. Model Assumptions
M~h
Mtbh
531
(19.2)
for all periods i = 1, ..., K. Thus, model M ~ in theory links capture probability on the secondary time scale with capture probability on the primary time scale (assuming the absence of temporary emigration; see Section 19.4.1). However, the independent model fitting of the ad hoc approach fails to impose the constraints in Eqs. (19.1) or (19.2). This inadequacy is addressed with the likelihoodbased models of Section 19.3. Other models include model M~, which denotes the robust design model with variable capture probabilities over both secondary and primary periods, and model Mtbh, which indicates time-specific capture probabilities at the level of primary periods, and both be-
The assumptions underlying the above models include those for the respective closed and open models. For example, the assumptions for modeling secondary samples within each primary period are that (1) the population is closed to gains and losses during the period [though this assumption can be relaxed; see Schwarz and Stobo (1997)], (2) marks are neither lost nor incorrectly recorded, (3) capture probability over the secondary periods varies according to specifications dictated by the structure of the selected model, and (4) the fate of each animal is independent with respect to capture probability. These assumptions are discussed in more detail in Section 14.2.3 and need not be revisited here. The assumptions of the ad hoc robust design also include those underlying the open modeling of primary period data: (1) the conditional probability of surviving from primary period i to i + 1 is the same for all animals, (2) the conditional probability of being caught at each primary period is the same for each marked animal in the population at that time, and (3) the fates of animals with respect to both survival and capture are independent. In addition, the closed model assumptions of marks retained and correctly recorded, and closure during the primary period [the equivalent of the instantaneous sampling assumption (4) of Section 17.1.2], are also required by the open models. These assumptions, and ways of dealing with their violation, are discussed in Section 17.1.2. 19.2.1.3. E s t i m a t i o n Estimation under Pollock's (1981a, 1982) ad hoc robust design uses open models for survival rates (~i), closed models for abundance (/~/i), and a combination of both for the recruitment estimator /~i = /~i+1 -- ~Pi(l~i-
Yli -}- Ri),
(19.3)
with (n i - R i) the number of animals caught during the primary period but not released back into the population. An approximate variance estimator for B i is presented by Pollock (1982) and Pollock et al. (1990), on assumption that the survival and abundance estimators are independent. Note that Eq. (19.3) has the
532
Chapter 19 The Robust Design
same appearance as expression (18.13), the only difference being the derivation of the abundance estimates. Estimation of ourvival rates was described in detail in Chapter 17, and abundance estimation with closed models was discussed in Chapter 14. The combined use of open and closed models under the robust design permits estimation of some parameters that are not identifiable using the standard, openmodel approach to estimation. For example, under the Jolly-Seber model, abundance can be estimated only for periods 2 through K - 1, because of an inability to estimate p~ and p~. However, under the robust design, the information needed to estimate these capture probabilities comes from the secondary samples, so capture probabilities and abundance can be estimated for periods I and K. The ability to estimate N 1 in turn permits estimation of B1 with Eq. (19.3). The combined approach also allows one to estimate separately the final capture and survival probabilities, which otherwise can be estimated only as the product q~K-1 P~ in the Cormack-Jolly-Seber model (see Section 17.1.2). There are several ways to estimate q~K-1 using the robust design. One such approach depends on the ability to estimate p~: using data from the secondary periods of primary period K based on Eq. (19.1 ). Thus, under the assumption of no temporary emigration (see Section 19.4.1), p~ can be estimated as
IK /~ = 1 - 1 - [ ( 1 - PKj).
(19.4)
j=l
Given this estimate of p~, a natural estimator for ~K-1 is formed by dividing the estimate of q~K-1 P~ by the estimate of p~:: ~K-lP~ q~K-1 = ~ ,
(19.5)
where the estimate of the product parameter is obtained via maximum likelihood in the same manner as the other survival estimates (e.g., see Section 17.1.2). Two additional approaches to estimation of ~K-1 are presented in Section 19.2.2. The availability of estimates of q~K-1 and N K under the robust design also permits estimation with Eq. (19.3) of recruitment BK_ 1 for the final primary period. Thus, the robust design permits estimation of the quantities of interest for all primary sampling periods: /~1, "",
I~K;
]91, "",
FK;
~1, "",
~K-1;
B1 ..... BK-1.
Kendall and Pollock (1992) present a good discussion of alternative estimators using the ad hoc approach under the robust design.
19.2.1.4. Alternative Modeling The ad hoc approach to estimation under the robust design can accommodate virtually any of the models described for open (Chapter 17) and closed (Chapter 14) populations. Because the models for primary and secondary periods are independent, the discussions of alternative modeling in these previous chapters are directly relevant to the robust design as well. For example, reduced-parameter models, covariate models, and models with capture-history dependence all can be used in conjunction with the robust design. In the case of temporary emigration, where an animal in the population of interest is not present in the area exposed to sampling efforts during a particular primary sampling period i, the capture probabilities for conditional modeling of open populations (e.g., Chapter 17) reflect the product of the probabilities of being present in the area exposed to sampling efforts and of being caught given presence in this area (Kendall et al., 1997). However, capture probabilities based on the secondary samples within a primary period reflect only the conditional probability of capture, but not temporary emigration. Under some forms of temporary emigration, this difference in the interpretations of the two kinds of capture probabilities can be used to estimate the probability of an animal being a temporary emigrant. This topic is sufficiently important that it merits separate discussion in Section 19.4.1. The discussion thus far has been in terms of singleage models, but multiple-age modeling is possible as well (see Pollock and Mann, 1983; Nichols and Pollock, 1990; Nichols and Coffman, 1999). The robust design also can be used for reverse-time modeling, and in fact, age-specific modeling in reverse time actually requires the robust design (Nichols et al., 2000a). Because the robust design with age specificity (using both standard-time and reverse-time approaches) permits estimation of quantities of special biological interest, these models are discussed separately in Section 19.4.2. Multistate modeling of data from the primary periods also can be used in the robust design. In multistate modeling, the closed modeling of the secondary period data should be stratified by animals in the different observable states (Nichols et al., 1992b; Nichols and Coffman, 1999). Finally, the robust design is useful in studies that include auxiliary data (e.g., band recoveries), a topic that is discussed briefly in Section 19.4.4.
19.2.1.5. Model Selection, Estimator Robustness, and Model Assumptions Model selection follows the principles discussed in Chapters 14 and 17 for closed and open models, respec-
19.2. Ad Hoc Approach tively. If permanent trap response is believed to operate at the level of primary sampling periods, then the secondary data within each primary period should be stratified into animals caught and not caught in previous primary periods. However, within each stratum, model selection can proceed as usual. Estimator robustness in the face of variable capture probabilities for individuals was the primary motivation for development of the robust design (Pollock, 1981a, 1982). As discussed in Section 17.1.9, survival estimates based on open models tend to be quite robust to variation among individuals in capture probability (also see Carothers, 1973, 1979). Abundance can be estimated using the suite of closed-population models developed specifically to deal with individual heterogeneity in capture probabilities (Chapter 14) (Otis et al., 1978; Pledger, 2001). Under certain assumptions (no temporary emigration, equivalent behavioral responses in capture probabilities at the primary and secondary levels, etc.), robust design models may impose additional logical constraints on the capture probability parameters of the closed and open models [e.g., Eqs. (19.1) and (19.2)]. These constraints typically are expressed in terms of a relationship between the two kinds of capture probability parameters [e.g., Eq. (19.1)]. However, the independent fitting of models under the ad hoc approach was not designed to impose these constraints, and they are best handled via likelihood-based estimation (see Section 19.3).
19.2.2. Estimation Based Solely on Closed M o d e l s For an open-model treatment of primary period data, Kendall et al. (1995) excluded from their list of models those with heterogeneous capture probabilities (see Table 19.2). However, it often is reasonable to think of animals as having innate tendencies to exhibit relatively high or low capture probabilities, with these tendencies extending over the duration of a long-term study and perhaps for the life of the individual. There are no open models that permit heterogeneity in capture probabilities, so survival estimation has proceeded by assuming similar capture probabilities for all animals within the group being investigated. In this section we note that it is possible to estimate survival probabilities using ad hoc estimators based entirely on closed-population models. If models including heterogeneity are used, it then becomes possible to estimate survival rate in the presence of heterogeneous capture probabilities. Here we consider approaches to the estimation of survival probability that are based entirely on the use
533
of closed models. These estimators can be used as alternatives to Eqs. (19.4) and (19.5) to estimate ~PK-1 (or if desired, all the survival parameters). One approach is based on the closed-form estimator
/~i+1 ~i = 1 ~ i _ mi + Ri
(19.6)
from the Cormack-Jolly-Seber (CJS) open model [Eq. (17.8)]. A slightly simpler estimator was presented by Nichols et al. (1992b): ~Pi "- ~'IR-~I / a i ,
(19.7)
where/VI,R_~I denotes the number of individuals in R i that are estimated to be alive at i + 1. Estimator (19.7) simply conditions on the animals released at i and estimates the number still present at i + 1. Note that Eq. (19.7) is the standard CJS estimator [Eq. (19.6)] for sample period 1, because there are no previously marked animals at that time. The estimator in Eq. (19.7) should be less efficient than that of Eq. (19.6), and we present it only because it is easily computed, with bootstrap confidence intervals, using program COMDYN, developed by Hines et al. (1999) for community-level analyses (Chapter 20). Under the CJS approach, the estimates/~i in Eq. (19.6) are based on the open-model estimator of Eq. (17.10). However, under certain conditions Mi also can be estimated using a closed-model estimator for the probability p* of capture in primary period i. Thus, an estimator for the number of marked animals in the population just before sampling in primary period i is 1Vii = m i l ~ * .
(19.8)
Equation (19.8) is simply the canonical estimator in expression (18.6) applied to a special subset of animals (in this case, marked animals that are caught in period i). If/~I~ ;-1 is desired [e.g., for Eq. (19.7)], then the m i in Eq. (19.8) is simply replaced by m Ri-1, the number of animals caught at i that also were caught at i - 1. An estimator for p* in Eq. (19.8) is obtained in a manner similar to that used for p~ (Eq. 19.4). Let Pij be the time-specific capture probability for secondary periods under model M t, or the time-specific capture probability for an animal not previously caught in primary period i under model Mtb. Then p* can be estimated as li
fi* = 1 -
l-I(1 -fiij).
(19.9)
j=l
On the other hand, if a heterogeneity model (e.g., M h o r Mth) is used for the secondary-period data, the
Chapter 19 The Robust Design
534
p* can be estimated as the average probability of being caught at least once during primary period i:
scribed above may not be particularly useful with standard capture-recapture data. However, in special cases in which heterogeneity is believed to be extreme (e.g., in community studies; see Chapter 20), we do recommend this approach. Finally, we note that the mixture models (see Section 14.2.3) developed by Norris and Pollock (1995,1996) and Pledger (2000) to deal with heterogeneity may prove to be especially useful for both ad hoc and likelihood approaches to the robust design.
(19.10)
~* = ni/lxl i,
where/~i is based on the selected heterogeneity model (e.g., Mh or Mth) and n i is the number of animals caught at least once during primary period i. We note that the use of Eq. (19.8) to estimate the number ( M i or M Ri-1) of marked individuals in the set of interest involves estimation of p* using all animals caught during primary period i [Eqs. (19.9) and (19.10)]. The approach thus assumes equal capture probabilities for animals that were marked before primary period i and those that were not. An alternative approach that does not require the assumption of equal capture probabilities for previously marked and unmarked animals focuses only on animals that were caught before primary period i (the m i o r mRi-1). The capture histories in primary period i for this subset of animals can be used directly with a closed model estimator (e.g., program CAPTURE) to estimate M i (or M~;-1). This approach to estimation is somewhat more conservative than that of Eqs. (19.8)-(19.10) and is probably most reasonable when heterogeneity models are being used (especially when high or low capture probabilities are thought to extend over the duration of the study). In any case, estimation of M i o r M Ri-1 using any of the above methods provides the estimates needed to estimate survival [Eqs. (19.6) and (19.7)]. Thus, it is possible to estimate all of the quantities of interest (q~i, Pi, Ni, Bi) using only closed models in conjunction with capture histories over the secondary periods. Because open-model survival estimators are robust to heterogeneity of capture probabilities, the ad hoc approach de-
TABLE 19.3
Example
The robust design capture-history data presented in Table 19.1 for adult male meadow voles at Patuxent Wildlife Research Center are used to illustrate the robust design. Use of these data permits comparison with the open-model approaches of Sections 17.1.10, 18.2.6, 18.3.6, and 18.4.6. Here we present results using a robust design that combines results of modeling with both closed- and open-population models. Under the original robust design approach (Pollock, 1981a; Pollock, 1982), we used the CJS survival estimates from Table 17.7. Abundance was estimated using program CAPTURE (Otis et al., 1978; Rexstad and Burnham, 1991) with the capture-recapture data from the five secondary periods within each primary period. The discriminant function model selection algorithm of CAPTURE indicated that model Mh was appropriate for primary periods 1 and 3-6, but not for period 2. The data from primary period 2 provided strong evidence of temporal variation and behavioral response. During primary period 2, a raccoon (later caught and removed) disrupted traps on the last 2 days of sampling, leading to very small numbers of captures. We thus reanalyzed truncated capture histories from the first 3 days of trapping during primary period 2, and
R e s u l t s of S e l e c t e d Tests f r o m Program C A P T U R E for A d u l t M a l e M e a d o w Voles a Test for heterogeneity (M o vs. M h)
Closure test Primary sample period (i)
1 2b 3 4 5 6
M h goodness of fit
z
P
X2
df
P
X2
df
P
-1.30 - 1.18 0.97 0.15 -0.84 -1.78
0.10 0.12 0.84 0.56 0.20 0.04
43.31 pc 22.12 67.05 19.39 50.04
2 _ 1 2 2 2
<0.01 D <0.01 <0.01 <0.01 <0.01
4.31 7.00 7.42 2.36 1.68 8.64
4 2 4 4 4 4
0.37 0.03 0.12 0.67 0.79 0.07
aAt Patuxent Wildlife Research Center, Laurel, Maryland, June-December, 1981. Capture-history data are from Table 19.1. bOnly the first 3 days of trapping were used, because a raccoon disrupted most traps on days 4 and 5. cInsufficient data for test.
535
19.3. Likelihood-Based Approach
in that a full likelihood is described for data from both secondary and primary periods. The full likelihoods are written as products of components corresponding to the two types of data, with mathematical relationships among the capture parameters of the components.
M h was a reasonable selection for these data. Results
of the closure test and tests regarding the appropriateness of M h are presented in Table 19.3. The number of new recruits each period, B i, was estimated using the closed-model/Qi and the open-model ~i [see Eq. (19.3)]. The closed-model abundance estimates obtained under the robust design (Table 19.4) can be compared with the Jolly-Seber open-model estimates of Table 18.1. In the face of heterogeneous capture probabilities, as indicated by the closed-model analyses, we expected negative bias in the Jolly-Seber estimates of abundance. Thus, we expected the abundance estimates of Table 18.1 to be smaller than those of Table 19.4. This expectation held for the estimates from primary periods 3 and 4, with the closed-model estimates being >10 animals larger than the estimates based on open models. For periods 2 and 5, however, the estimates based on the two approaches were quite similar, with the closed-model estimates actually being slightly smaller than those based on open models (Tables 18.1 and 19.4). The robust design permits estimation of abundance for periods 1 and 6, whereas N i cannot be estimated for these periods under the JS model. The differences in the estimated population sizes for the robust design and standard JS approach lead to differences in the estimated numbers B i of recruits as well. Finally, B1 can be estimated under the robust design, but not under the JS open model.
19.3.1. Models The likelihood-based approach can include virtually any of the open models of Chapters 17 and 18, though the closed-model components are restricted to models for which maximum likelihood estimators can be identified. For example, the likelihoods under the heterogeneity models (M h, Mbh, Mth, Mtb h) are typically overparameterized and thus are not useful for purposes of maximum likelihood estimation (Chapter 14) (Otis et al., 1978). Estimators for the latter models are based on ad hoc approaches like the jackknife (Burnham and Overton, 1978) or the concept of sample coverage (Chao et al., 1992). Models incorporating heterogeneous capture probabilities must be implemented using an ad hoc approach rather than a likelihood approach [though the mixture models of Norris and Pollock (1996) and Pledger (2000) are likelihood based and should change this situation]. As an example of a likelihood for the robust design, consider model M[ denoting temporal variation among primary and secondary sampling periods. To simplify notation we assume there are only two secondary periods for each primary period (ll = 12 . . . . . IK = 2) and no losses on capture. We then can write the probability distribution for the numbers of animals exhibiting all capture histories, {x~i} and {x~i}, as the product of three components:
19.3. LIKELIHOOD-BASED APPROACH The likelihood-based approach to the robust design (Kendall et al., 1995) differs from the ad hoc approach T A B L E 19.4
Estimates of A b u n d a n c e (Ni), Survival (~i), and Recruitment (/~i) for A d u l t M a l e M e a d o w V o l e s a Abundance b
Sample period (i)
Ni
Not released
Survival c
Recruitment d
S"E(Ni)
di
(r
S"E(~oi)
Bi
S"E(/~i)
1
64
5.22
3
0.86
0.052
21
6.87
2
73
4.15
3
0.58
0.066
37
11.63
3
78
10.42
1
0.71
0.072
18
11.70
4
73
7.19
1
0.59
0.069
8
7.27
5
49
3.27
1
e
..__e
6
97
8.32
1
e
___e
a At Patuxent Wildlife Research Center, Laurel, Maryland, June-December, 1981. Data from Table 19.1 were used with Pollock's robust design combining estimates from open (q~i) and closed (N i) models. b Estimated using the jackknife estimator for closed model M h (Burnham and Overton, 1978, 1979). c Estimated using the CJS open model (q~t, Pt); see Table 17.7. d Estimated from survival and abundance estimates [Eq. (19.3)]. e Quantity not estimable.
536
Chapter 19 The Robust Design
P({x~i},{x'~i}) = IPl({Ui} [ {Ui}, {p'~})j x [P2({mhi} I {Ri}, {q0i},{p*})]
(19.11)
X [Pg({X'~i}, {x'~i} I {//i}, {mhi}, {pq})]. The terms P1 and P2 in Eq. (19.11) are the components for the unconditional open-population model [expression (18.2)]. The first component deals with the capture of unmarked animals and is written as in expression (18.3):
Pl({Ui}[{Ui}, {p*}) = K [
I-[ i=l
]
ui ui)! (p,)Ui(1 ui!(Ui_
(19.12)
_ p~)Ui-ui
.
The second component is simply the conditional probability distribution of the recapture summary statistics mij, as in Eq. (17.6): K-1
P2({mij} [ {Ri} , {r
Ri! (mi, i + 1)!(mi,i+2)! . (miK)!(R i _ ri)W . . .
{P~}) = H i=1
~- mi, i+l [q~i(1 -- p'~+l)q~i+lP'~+2] mi'i+2 ... X (q~iPi+l)
(19.13) X [q~i(1 -- P * + I ) " ' "
q~K--lP~(]mi'Kx Ri-ri,
where Xi is the probability that an animal alive in the study population at primary sampling period i is not caught or observed at any primary sampling period subsequent to i. This probability is defined recursively as a function of q0i and p* [see Eq. (17.1)]. The third component involves the modeling of data across the secondary periods in all of the different primary periods: PB({x'oi}, {x~i} [ {ui}, {mhi}, {Pij}) _
(
xol-Xo1.~ol-
xo,( 10
//1 ! Pllq12 101 011-111 \ p~-
/
lion( 01
qllPl2
P11P12
\ P~ /
ixo,
\ p~ /
11
(19.14)
mals are captured, and animals in each of these groups exhibit one of the three possible capture histories (01, 10, 11) over the two secondary periods within primary period 2. At primary period i = 3, unmarked animals (u 3) and two groups of marked animals (m13, m23) are captured, and each animal again exhibits one of the three possible histories over the secondary periods. At the final primary period (i = K), K - 1 groups of marked animals can be caught (mlK, m2K, ... , mK_I,K), in addition to the unmarked animals, u K. The modeling of the capture histories over the two secondary periods within each primary period follows the conditional Lincoln-Petersen modeling of Section 14.1. We emphasize that although the above expressions pertain specifically to M~ with two secondary samples per primary sampling period, similar expressions can be written for any model that includes capture probabilities that are constant, or stationary, or exhibit behavioral response (Table 19.2). It is useful to consider the advantages of likelihoodbased modeling [Eqs. (19.11)-(19.14)] over the ad hoc approach presented in Section 19.2. In our view, the primary advantage involves the ability to incorporate relationships among the model components P1, P2, and P3. Under the ad hoc approach, the modeling of capture history data for the secondary and primary periods is conducted separately, and the results then are combined. The independent modeling of the data from secondary and primary periods precludes full use of the information contained in these data, and the resulting estimators are less than optimal in their statistical properties. In contrast, the likelihood approach takes full advantage of information in the data, producing estimators with a number of optimal properties (see Section 4.2.2). Under the full likelihood approach, the modeling of the two separate data types is linked through the shared capture probability parameters. Though the capture probability parameters differ for the two types of modeling (p* for primary, Pij for secondary), they are related via [i
X i= I - [2! \
l0 T 011.11 XOi "Xoi "J'Oi !
X1101a-011-11! i "'"1i ""~li
X (Pilqi2~ x!i~ (qilPi2~ x~ (PilPi2 1
--" /
10 T 01 W 11 \ X i-l,i" X i-l,i" X i-l,i !
x,:
where qij = 1- Pij. The first trinomial of Eq. (19.14) models the data for the two secondary periods within primary period 1. Because this is the first primary period, all animals in this group are unmarked and are members of u 1. At primary period i = 2, both marked (members of m12) and unmarked (members of u2) ani-
p* = 1 - 1-[(1 -Pij),
(19.15)
j=l
which forms the basis of the estimator in Eq. (19.9). In words, an animal must be missed (not caught) in each of the secondary periods of primary period i in order to be missed in primary period i. The p* appear explicitly in components P1 and P2 of Eq. (19.11) and as a function of the Pij in component P3 [e.g., see Eq. (19.14)]. Equation (19.15) thus forms the basis for the joint modeling of robust design data.
537
19.3. Likelihood-Based Approach
19.3.2. Model Assumptions
19.3.4. Alternative Modeling
The assumptions underlying the ad hoc approach of Section 19.2.1 also are required for the likelihood-based approach of this section. In addition, expression (19.15) linking capture probabilities across time scales must be satisfied in order for the joint likelihood approach of Kendall et al. (1995) to yield reasonable estimates. One way for the relationship in Eq. (19.15) to be false involves temporary emigration during the primary sampling periods. Under temporary emigration, the capture probabilities associated with the open portion of the model [e.g., p* of P2 in Eq. (19.11)] are products of the probability of being in the sampled area (i.e., of not being a temporary emigrant) and the probability of capture conditional on exposure to sampling efforts. However, the capture probability parameters associated with the closed portion of the model [e.g., the Pijof P3 in Eq. (19.11)] reflect only the probability of capture conditional on exposure to sampling efforts. Hence, the relationship underlying the joint likelihood [Eq. (19.15)] no longer holds, and estimates under joint models such as Eq. (19.11) are biased (Kendall et al., 1997). Likelihood-based and ad hoc approaches for dealing with temporary emigration are presented in Section 19.4.1.
In Table 19.2 and in Section 19.3.1, we noted that a variety of models for capture probability can be implemented using a likelihood-based approach. In addition, models can be developed with capture probabilities that are functions of time-specific or individual covariates. All the options specified in Section 17.1 for modeling survival probability (e.g., reduced-parameter models, time-specific and individual covariates, capture-history dependence, multiple groups) can be included in models of the robust design. It also is possible to treat temporary emigration (Kendall and Nichols, 1995; Kendall et al., 1997; Schwarz and Stobo, 1997), as discussed in Section 19.4.1. As with the ad hoc approach, likelihood-based robust design models can accommodate multiple-age modeling (see Pollock and Mann, 1983; Nichols and Pollock, 1990; Nichols and Coffman, 1999), though we are unaware of examples in the published literature. The robust design in the case of age specificity is discussed separately in Section 19.4.2. Multistate modeling of data from the primary periods under the robust design has been implemented by Nichols and Coffman (1999) and Coffman et al. (2001). The portion of the likelihood dealing with recaptures over the primary periods is identical to the conditional multistate likelihoods described in Section 17.3. The component dealing with the secondary recapture data includes stratification of animals by the different observable states (Nichols and Coffman, 1999). For the component P3 of Eq. (19.11), instead of conditioning on the ui and mhi as in the single-state case [Eq. (19.14)], rs w h e r e r a n d s one must condition on u S i a n d t h e mhi, denote states (see notation in Section 17.3). For multistate models Eq. (19.15) must be modified for state-specific capture probability parameters:
19.3.3. Estimation Estimation of survival under the likelihood-based approach is based on the product P2 x P3 in expression (19.11), which is maximized numerically. Because of the ability to estimate all of the capture probabilities for all of the sampling periods (i = 1..... K; j = 1, ..., l i) using data from the secondary sampling periods, all survival rates (i = 1, ..., K - 1) are identifiable. The numerical maximization and estimation can be carried out by program MARK (White and Burnham, 1999) and program RDSURVIV, which was developed for likelihood estimation under Eq. (19.11) by Kendall and Hines (1999). In addition to the parameter estimates, these programs compute variance-covariance estimates as well as AIC, goodness-of-fit, and likelihood ratio statistics. Estimation of abundance can be achieved using the canonical approach of Eq. (18.6): 1 ~ i "-
ni/p*
(pS), = 1 - 1--[ (1 -pSj), j=l
where the superscript s again denotes state. Finally, the likelihood-based approach to the robust design has been used for reverse-time modeling (Nichols et al., 2000a) and also can be used in studies involving auxiliary data. These applications are discussed in the context of multiple-age models in Section 19.4.
(19.16)
for i = 1, ..., K. The number of new recruits B i is then estimated as a function of the estimates for abundance and survival, just as for the ad hoc approach [Eq. (19.3)]. Recruitment parameters B i can be estimated for periods i = 1 , . . . , K - 1.
19.3.5. Model Selection, Estimator Robustness, and Model Assumptions Model selection and testing follow the principles discussed in Section 17.1.8 for conditional open mod-
538
Chapter 19 The Robust Design
els. Because the models discussed above are based on likelihoods, use of AIC and likelihood ratio tests for model selection is straightforward. Pearson chi-square tests and bootstrap approaches also seem reasonable, though completely satisfactory goodness-of-fit tests are yet to be developed for these models. The models for which a full likelihood can be developed permit variation in capture probability with respect to time and animal behavior, but not heterogeneity. Because of the ad hoc nature of most abundance estimators in the presence of heterogeneous capture probabilities, likelihood-based models must assume the absence of heterogeneity. Although we are aware of no studies of robustness to heterogeneity, we anticipate that the estimators behave in a manner similar to those for separate closed and open models. Thus, survival probability estimators are likely to be robust to heterogeneity, but estimated capture probabilities are likely to be positively biased and abundance estimates [e.g., based on Eq. (19.16)] in turn are likely to be negatively biased. The likelihood-based mixture models of Norris and Pollock (1996) and Pledger (2000) for analyzing closed-population capture-recapture data hold great promise for developing robust design models that deal with heterogeneity. Still another approach for dealing with heterogeneous capture probabilities involves the use of capture frequency data from previous primary periods as individual covariates to model capture probability. This idea was proposed independently by Fletcher (1994) and R. Julliard and N. Yoccoz (personal communication) and is briefly discussed in Section 19.4.4. The relationship in Eq. (19.15) between capture probabilities for secondary and primary sampling periods is required by the likelihoods of this section, but not by the ad hoc approach. A common violation of this assumption involves temporary emigration, which can result in biased estimates. Kendall et al. (1997) investigated the bias of estimators based on model M~ in the face of two types of temporary emigration: "random," in which each animal has the same probability of being a temporary emigrant at a sampling period, and "Markovian," in which animals that were temporary emigrants at period i - 1 have one probability of being a temporary emigrant again at i, and animals that were not temporary emigrants at i - 1 have a different probability of being absent at i. Under random emigration, they found that survival and capture probability estimates were negatively biased, whereas abundance estimates showed positive bias. For Markovian temporary emigration, the magnitude and direction of bias depend on the nature of the Markov process (e.g., whether temporary emigrants at i - 1 are more
or less likely to be temporary emigrants at i). In the numerical examples of Kendall et al. (1997), survival and capture probability estimates were again negatively biased under different forms of Markovian temporary emigration. The models described above can be viewed as special cases of temporary emigration models, in which the probability of temporary emigration can be constrained to be equal to 0. Temporary emigration is addressed in Section 19.4.1, which includes an example analysis using the likelihood-based approach to estimation under the robust design.
19.4. SPECIAL ESTIMATION PROBLEMS
19.4.1. Temporary Emigration The likelihood-based approach described in Section 19.3 is dependent on the assumed relationship between the capture probability parameters for the secondary and primary sampling periods [Eq. (19.15) ]. In the presence of temporary emigration, the relationship does not hold, and the models discussed in Section 19.3 yield biased estimates (see Section 19.3.5). In this section we discuss estimators and models developed by Kendall and Nichols (1995) and Kendall et al. (1997) for use in the presence of temporary emigration. Temporary emigration can be introduced by interactions between biology and sampling design. In the case of sampling via stationary traps or nets, it may be that the areas traversed by animals do not correspond exactly to the sampled areas. In situations in which animal ranges overlap sampling areas only partially, animals may or may not be present during the times of sampling. Many migratory animals (e.g., migratory birds) are sampled once every year either on the wintering or breeding grounds. In these situations, animals may not breed in some years or may visit alternative wintering or breeding grounds, again leading to temporary emigration (e.g., Hestbeck et al. 1991; Spendelow et al., 1995). Some animals enter torpor or otherwise become inactive during extremely dry or cold weather, causing them not to be exposed to sampling efforts (Kendall et al., 1997). Temporary emigration can be viewed as producing extreme heterogeneity of capture probabilities, because temporary emigrants exhibit capture probabilities of 0. The age-specific breeding models of Section 17.2.4 were developed to deal with a specific form of temporary emigration, namely, nonbreeding until attainment of a certain minimum age. Interior zeros in capture
19.4. Special Estimation Problems histories (zeros occurring between the time of marking as a newborn animal and first observation as a breeder) reflect both failure to detect the animal during sampling (the usual interpretation of 1 - Pi) and absence from the breeding grounds (temporary emigration). Animals that return to breed are assumed to breed as adults thereafter, and "true" capture probabilities (conditional on presence in the sampled area) thus are estimated using known breeders. These estimates provide the basis for estimating age-specific temporary emigration for prebreeders. Under the robust design, extra information about capture probability comes from the capture-recapture data over the secondary sampling periods. The robust design thus can be used to estimate breeding probabilities even in the case in which adults skip breeding in some years, and in this sense it is more flexible and robust than the approach of Section 17.2.4. Kendall et al. (1997) introduced random and Markovian models for temporary emigration, both of which are based on the concept of a "superpopulation" of N O animals. The idea is that animals are "associated" with the area sampled at period i, in the sense that they have some nonnegligible probability of being in the area exposed to sampling efforts during period i. Some number N i of these animals are actually in the area and therefore are available for possible capture with probability p*. The models of Kendall et al. (1997) assume that the population is closed to gains and losses (including temporary emigration) over the secondary periods of primary period i, but this assumption can be relaxed if necessary (Schwarz and Stobo, 1997; Kendall and Bjorkland, 2001). In addition to the notation of this and previous chapters (14, 17, and 18), the following unknown random variables are needed: M ~ is the number of animals marked before primary period i and in the superpopulation during period i(i = 2, ..., K; M ~ = 0); B~ is the number of animals entering the superpopulation between primary periods i and i + 1 and still in the superpopulation at i + 1 (i = 1, ..., K - 1). The values M i and B i represent the numbers of M ~ and B~ that are in the area exposed to sampling efforts during primary period i. Also define a new capture probability parameter: p0 is the probability that a member of the superpopulation at primary period i (one of the N Oanimals in the superpopulation) is captured during primary period i. The capture probability parameter p* under the robust design now corresponds to the probability that an animal exposed to sampling efforts at i (one of the N i animals) is captured during i. This capture probability can thus be viewed as conditional on presence in
539
the sampled area. Survival rate reflects the probability that a member of the superpopulation at time i is still alive and a member of the superpopulation at time i + 1.
19.4.1.1. Random Migration Model The model for random temporary emigration (also see Burnham, 1993) requires parameters "Yi representing the probability that a member of the superpopulation at period i is not in the area exposed to sampling efforts during i (i.e., is a temporary emigrant). Thus, E ( N i [ N ~ = (1 - ~/i)N~
Note that ~/, which elsewhere (e.g., Section 19.4.2.2) denotes seniority, is used here to denote the probability of temporary emigration. We can specify the relationship between the capture probabilities for animals that are exposed to sampling efforts at i (p*) and for those in the entire superpopulation, regardless of whether or not they are exposed to sampling efforts at i (p0): p0 = (1 - ~/i)P*.
(19.17)
Equation (19.17) simply specifies that in order for a member of the superpopulation to be caught at any period i, it must be in the area exposed to sampling efforts and then be captured. Equation (19.17) can be used to obtain an ad hoc estimator for ~/i. Recall that under the robust design, p* can be estimated using the capture probabilities (Pij) corresponding to the secondary sampling periods [see Eq. (19.9)], typically based on closed-population models. The other capture probability parameter, p0, is estimated by standard open-population models (e.g., Chapter 17) in the case of random emigration (Burnham, 1993; Kendall et al., 1997) (see Section 17.1.9). Using the estimator/3* from Eq. (19.9) based on secondary period data and ]~0 based on primary period data, we can write an estimator for the probability of temporary emigration as "Yi
=
1 - ]~0/]~,.
(19.18)
An estimator for the approximate variance of the estimator in Eq. (19.18) is provided by Kendall et al. (1997). Equation (19.18) is based on capture probability estimates obtained from both closed and open models. Both closed and open models for capture-history data can deal with temporal variation and with various forms of capture-history dependence in capture probabilities (see Chapters 14 and 17). However, heterogeneity in capture probabilities cannot be accommodated in a satisfactory manner by open models (e.g., Chapters 17 and 18) (Seber, 1982; Pollock et al., 1990). Thus, in the
540
Chapter 19 The Robust Design
presence of unexplained heterogeneity (heterogeneity that cannot be modeled as a function of covariates), the numerator of Eq. (19.18) is positively biased (Section 18.2.5) (see Carothers, 1973), and "Yiin turn is negatively biased. Kendall et al. (1997) recommended an alternative ad hoc estimator based entirely on estimates from closed model M h for use in this situation. A full likelihood approach to estimation under the random temporary emigration model is possible when capture probabilities include only temporal or behavioral variation, but not capture heterogeneity. Consider the analog of model M~ [expression (19.11)] in the presence of random temporary emigration. Component P3 of expression (19.11) deals only with animals that are not temporary emigrants, and so remains unchanged from the case with no temporary emigration. In addition, component P1 of expression (19.11) typically is not used to estimate capture probability. Thus, the only modification to expression (19.11) in the presence of temporary emigration involves component P2 dealing with data over the primary periods. Denoting the second model component under the model for random temporary emigration as P~, we have
P~_({mij}l{Ri},
{q~i}, {p*}, {'yi})
K-1 _ ~ "_~_
a/! ...
(mi i+l)!(mi i+2 )I p
9
9
(miK)!(R i - ri)t
~li+l)P'[+l
•
q~i(1 --
X
q~i[1 -- (1
(19.19)
~i+l)Pi+llq~i+l(1 -- ~/i+2)p}~+2
...
X {q~i[1-- (1-- ~/i+l)P~+l] "" ~K_l(1-- ~/K)p~}mi'K(xe) Ri-ri, where Xe is the probability that an animal alive in the superpopulation at primary period i is never seen again during the study. Under the random temporary emigration model, this quantity can be defined recursively in terms of other model parameters as Xe=l
- q~i{1- [1-(1-~/i+l)P~+l]Xe+l}
for i = 1, ..., K - 1, and X~ = 1. Note that the model in Eq. (19.19) assumes that survival probability is the same for temporary emigrants and animals that are exposed to sampling efforts. Kendall et al. (1997) refer to the model defined by the product P1 X P~ • P3 as model (q~t, Ptt, ~/t), thus emphasizing the Pij as the fundamental capture probability parame-
ters [the p* are expressed as functions of the Pij via Eq. (19.15)]. In Eq. (19.19) the probability of capture and probability of not being a temporary emigrant always occur together as the product (1 - ~/i)P~. For this reason p* and 1 - "Yicannot be estimated separately under standard open models (Chapters 17 and 18). However, under the robust design the information required to estimate capture probabilities is contained in the secondary period capture histories. These data are modeled in component P3, which is also parameterized with p*. This permits the separate estimation of p* and 1 - "Yi in the product ( 1 - ~/i)P* in model component P~. Under the model of Eq. (19.19), capture probabilities can be estimated for all primary periods, i = 1, ..., K. Temporary emigration can be estimated for all primary periods except the first and last, i.e., for i = 2, ..., K - 1, and survival can be estimated for all periods except K - 1. The final temporary emigration and survival parameters are confounded in the product parameter q~K_l(1 -- ~/K)- Estimation under this model can be accomplished using program MARK (White and Burnham, 1999) and program RDSURVIV (Kendall et al., 1997; Kendall and Hines, 1999). The model (q~t, Ptt, "Yt) c a n be generalized to add features such as trap response (see Kendall and Nichols, 1995) and can be made more specific by, e.g., introducing temporal constancy in parameters. In particular, we can obtain model M~ (also denoted by [q~i, Pij]) as a special case of model (~t, Ptt, "~t) simply by imposing the constraint ~/i = 0 for all i. Finally, we note that if the closure assumption is violated over the secondary periods within a primary period, then a full likelihood can be written that uses an open-population approach for component P3. Schwarz and Stobo (1997) and Kendall and Bjorkland (2001) describe this approach using the superpopulation modeling of Schwarz and Arnason (1996) (see Section 18.3).
19.4.1.2. Markovian Emigration Model Kendall et al. (1997) also developed a more general model for temporary emigration, in which the probability of being a temporary emigrant at primary period i depends on whether the animal was a temporary emigrant at time i - 1. Specifically, let ~/; denote the probability that a temporary emigrant in primary period i - 1 (i.e., an animal included in N ~ Xi-1) is also a temporary emigrant at time i. Let ~/~' denote the probability that a nonemigrant at i - 1 is a temporary emigrant at i. Temporary emigration is thus modeled as a first-order Markov process. We denote this general model a s (q~t, Ptt, ~/;) and note -
-
19.4. Special Estimation Problems that the random temporary emigration model (q~t, Ptt, ~lt) can be obtained from it as a special case by imposing the constraint ~/; = ~/~. The Markovian model is obtained from the robust design model (q~t,Ptt) of Eq. (19.11) by again modifying the second model component (denoted as pM under the Markovian model). The modifications are rather tedious to describe in general, but are illustrated by recapture expectations for three time periods: (19.20)
E(m12) = R1q~1(1 - ~/2JP2,""-* E(m13 ) = Rlq~lq~2p~[~/~(1 - ~/~)
+ (1 - ~/~)(1 - p~)(1 - ~/~)],
541
A 2 • 1 vector of probabilities of an animal being in the study area in primary period j, given that it is either outside (row 1) or inside (row 2) the study area in primary period j 1 and survives to period j:
[1 1
dj=
~;'_]"
If data are summarized in standard mq-array format (Table 17.2), the corresponding cell probabilities under component P2M of the Markovian emigration model can be written as in Table 19.5. Estimation under this model requires additional constraints, such as ~/~<= ~/K-1 to ensure that ~/i is identifiable for i = 2, ..., K, and ~/;< = ~/K-1 to ensure that ~/; is identifiable for i - 3, ..., K. The parameter ~/~ cannot be estimated because there are no marked animals in the superpopulation in primary period 2 that were not in the sampled area in period 1. We note that the ~/; parameters typically are quite difficult to estimate and have large standard errors, because of the need to condition on animals not observed the previous period. Finally, we emphasize that Markovian models for temporary emigration are especially relevant to populations for which breeding does not occur regularly on an annual basis. We noted above that some populations are sampled on their breeding grounds and that temporary emigration corresponds to nonbreeding status. In some long-lived vertebrates (e.g., sea turtles, some marine mammals, and some large seabirds), reproduction is sufficiently costly of time a n d / o r energy that a female reproducing in one year cannot breed the following year. This life history pattern has presented problems for standard capture-recapture analyses (e.g., Weimerskirch et al., 1987), because biennial breeding introduces extreme heterogeneity in capture probabilities in traditional open models. Indeed, true biennial breeding with annual sampling requires the constraint ~/~ = 1. The use of Markovian models for temporary emigration is especially appropriate in this situation (Kendall and Bjorkland, 2001). PP
E(ma3) = Raq~2(1 - ~/~)p~.
The expression for E(m13) in Eq. (19.20) includes two possibilities, the probabilities for which are added together inside the brackets. The first possibility is that the animal released at period 1 was a temporary emigrant at period 2. The second possibility is that the animal was not a temporary emigrant at period 2, but was simply not caught then. These two possibilities require two different temporary emigration parameters for period 3, reflecting the different emigration status at period 2. The second component of the Markovian emigration model, P2M, is most conveniently written in matrix notation (Kendall et al., 1997). Because this model can be viewed as a general starting point for robust design models, we briefly review it here. Letting q* = 1 - p*, the following notation is used for animals released following primary period h and recaptured in primary period s: fh+l
A 1 x 2 vector of probabilities of not being captured during the first primary period after release (period h + 1), given that an animal survives from primary period h to h + 1 and is in the superpopulation at h + 1:
fh+l_ Gi
--
(1
-
+1 ~/~+1)q~+1
"
A 2 • 2 transition matrix of probabilities that an animal is outside the study area (column 1) or inside the study area but not captured (column 2) in primary period i = h + 2, h + 3, ..., K - 1, given that it is outside (row 1) or inside (row 2) the study area in primary period i - 1, survives to period i, and is in the superpopulation in both periods: G, = r ~/; -
I!
!
(1
-
(1 -
~i)q*J"
Example We illustrate these models using capture-recapture data for white-footed mice, Peromyscus leucopus, trapped by Nichols in a lowland beech-maple hardwood forest at Patuxent Wildlife Research Center during the fall and winter of 1980-1981 (also see Kendall et al., 1997). We focus here on one of two trapped grids, a 14 • 14 checkerboard grid with adjacent trapping stations in each row or column separated by 15.2 m. A single Sherman live trap containing cotton bedding and baited with corn was placed at each station. Traps
Chapter 19 The Robust Design
542 TABLE 19.5
Cell Probabilities Underlying the Primary Period Capture-Recapture Data for Markovian Temporary Emigration a Primary period of next recapture
Primary period of release
2
3
1
q~l(1 - ~/~)p~
qol_f2q~2d3p~
2
4
5
q~lf2q~2G3q~3d4p~ ~2f3q~3d4p~
q02(1 - ~/~)p~
q~lf2qo2G3q~3G4q~4d5p~ q~2f3~3G4qo4d5p~ q~3_f4q~4d5p~ q~4(1 - ~/5)P5
q03(1 -- ~/~)p~
3
rr
4
aSummarized in m/j-array format (Table 17.2) for component L2M of the Markovian temporary emigration model (q~i, pq, ~/~) under the robust design. in response to the cold (e.g., see Hill, 1983). We used likelihood-based model testing and selection procedures to distinguish between these alternatives and to provide estimates of relevant parameters. Trapping data from 28 September 1980 through 1 March 1981 were selected for analysis, because this period included three cold trapping periods (beginning on 6 December 1980, 3 January 1981, and 31 January 1981) as well as several periods that were not especially cold (see capture data in Table 19.6). The capture-recapture data from two consecutive days of trapping within each monthly p r i m a r y period were modeled using the Lincoln-Petersen (model M t) approach (see Chapter 14). The full likelihoods representing models with and without t e m p o r a r y emigration and including both closed and open components were fit using program RDSURVIV (Kendall and Hines, 1999). In some
were baited and set one evening, checked the next day and reset, and then checked the following day and closed. Captured animals were m a r k e d with individually coded monel fingerling tags in their ears. Juvenile animals, identified by their entirely gray pelage, were omitted from these analyses, and males and females were pooled for analysis. These data were selected for analysis because of the a p r i o r i prediction of increased t e m p o r a r y emigration during cold periods (e.g., overnight temperatures <0~ This prediction was based on casual observations of smaller catches following cold nights during trapping on this grid since 1978. Smaller catches on cold nights could result either from (1) reduced activity, and hence reduced capture probabilities of all animals, or (2) t e m p o r a r y emigration with some animals staying u n d e r g r o u n d in burrows, and perhaps entering torpor,
TABLE 19.6
Capture-Recapture Statistics for Peromyscus leucopus under the Robust Design a
Sample period
Number released
Number unmarked
(h)
(R h)
(uh)
28 Sep 1980
1
20
4 Nov 1980
2
15
6 Dec 1980
3
6
3 Jan 1981
4
6
31 Jan 1981
5
7
1 Mar 1981
6
20
20 (14,2,4) 5 (1,1,3) 4 (3,0,1) 4 (0,0,4) 3 (2,1,0) 6 (2,4,0)
Sampling date
Number of R h next captured at period i (mhi) i=2
3
4
5
6
7
10 (2,1,7)
0
0
0
0
2 (0,0,2)
2 (0,0,2) 0
0
1 (1,0,0) 4 (2,2,0) 1 (0,1,0) 2 (1,1,0) 6 (4,2,0)
2 (1,0,1) 2 (0,1,1)
1 (1,0,0) 1 (0,1,0) 0 0 13 (10,1,2)
a Caught on woodlot grid 2 at Patuxent Wildlife Research Center, September 1980-March 1981. Numbers in parentheses indicate the frequencies of the three observable two-period capture histories (11, 10, 01) within each primary sampling period i, for new captures u i and recaptures mhi.
19.4. Special Estimation Problems cases the estimated variance-covariance matrix was not positive definite, which required us to replace the estimated standard errors from RDSURVIV with a parametric bootstrap approach (Buckland, 1980; Buckland and Garthwaite, 1991 ). For this approach we simulated 200 data sets based on point estimates from the original data, estimated parameters using each simulated data set, and obtained standard errors empirically using the replicate point estimates obtained from the simulated data. We fit several models that included no temporary emigration, random temporary emigration, and Markovian temporary emigration. The low-AIC model was (~t, Ptt, ~/t), including time-specific capture probabilities, time-invariant survival, and time-specific probabilities of temporary emigration (Table 19.7). In this model temporary emigration was of the random type (not Markovian), and the model fit the data adequately (X25 = 16.3, P = 0.36). We concluded that temporary emigration was indeed high during the cold months, and that it could be adequately modeled as a random process. Despite our a priori prediction about the existence of temporary emigration, this analysis was largely exploratory rather than experimental. As such, we believe it most reasonable to draw inferences as above based on model selection procedures (AIC) (Burnham and Anderson, 1998). However, for those who prefer, we could instead view the analysis as a sequence of hypothesis tests. In order to illustrate this alternative possibility, we present several such tests in Table 19.8. Specific likelihood ratio tests provided strong evidence of temporary emigration, yet no evidence that this emigration was Markovian (Table 19.8). Estimated probabilities of ternTABLE 19.7.
~AIC Statistics for Selected Models Fit to
Peromyscus leucopus Capture-Recapture Data a Emigration descriptor
Model
Number of parameters
&AIC
No temporary
(q~t, Ptt')
20
15.4
emigration
(q~, Ptt')
15
10.9
(q~t, Ptt', ~t)
25
5.0
(q)t, Ptt', ~)
21
11.8
(q~, Ptt', "~t) (q~, Ptt', ~/) (q~t, Ptt', ~/;) (q~t, Ptt', "Y') (q~, Ptt', ~;) (q), Ptt', ~/')
21
0.0
16 29
5.7 7.9
22
11.4
24
2.4
17
6.1
Random temporary emigration
Markovian temporary emigration
a From woodlot grid 2, Patuxent Wildlife Research Center, September 1980-March 1981.
543
porary emigration were small for periods 2 and 6-7, yet large for periods 3-5, the three periods for which higher temporary emigration was predicted (Table 19.9). We also computed estimates of temporary emigration using the ad hoc approach of Eq. (19.18), which matched estimates under the general model fairly well (Table 19.9). Estimated monthly survival probability was 0.81. 19.4.2. M u l t i p l e A g e s a n d Recruitment Components
The multiple-age model of Pollock (1981b) for open populations (see Section 17.2.2) permits estimation of age-specific survival probabilities for all age classes and age-specific capture probabilities for all age classes except the first one. Capture probability for the initial age class cannot be estimated using open models because there is no group of animals in this class that is known to have been alive and available for capture based on previous capture. Certainly young animals may be captured in the future, but it is not possible for them to have been caught before their appearance as a young capture. Thus capture probability, and hence abundance, cannot be estimated for young animals (Section 18.2.4). However, the robust design provides data in the secondary samples with which to estimate capture probability and abundance for this age class. Under some sampling designs, the ability to estimate capture probability and abundance for young animals permits separate estimation of recruitment components (Nichols and Pollock, 1990; Pollock et al., 1990, 1993). Specifically, it may be possible to estimate components associated with (1) in situ reproduction on the study area and (2) immigration from outside the study area. The motivation for this work is ecological, in that the relative contributions of these two components are important to understanding population dynamics (e.g., Connor et al., 1983; Pulliam, 1988). The timing of sampling relative to growth and maturation of the study organism is critically important (Nichols and Pollock, 1990), in that the interval between primary sampling periods must correspond to the period required for a young animal to mature into an adult. Thus, all animals classified as young in primary period i must have made the transition to adult status by period i + 1. In addition, new adults on the study area in period i + 1 that were not young on the area at time i are assumed to be immigrants from outside the study area. This assumption is likely to be met if reproduction is sufficiently discrete in time that there are no births between primary periods i and i + 1. If reproduction is not discrete to that degree, then the time between primary periods must be sufficiently
544
Chapter 19 The Robust Design TABLE 19.8
Likelihood Ratio Test Statistics for Selected Temporary Emigration Models of P e r o m y s c u s l e u c o p u s a Test statistic
Restricted model
General model
(H o)
(H a)
X2
df
P
Ecological hypothesis tested
(~, Ptt') (q~, Ptt')
(q~, Ptt', ~t) (q~, Ptt', ~/;)
(q~, Ptt', "~t) (q~, Ptt', ~/) (q~, Ptt', ~t)
(~, Ptt', ~/;) (q~, Ptt', "~t) (~t, Ptr, ~t)
22.4 26.1 3.6 15.7 3.0
6 9 3 5 4
<0.01 <0.01 0.31 <0.01 0.55
No temporary emigration vs. random temporary emigration No temporary emigration vs. Markovian temporary emigration Random temporary emigration vs. Markovian temporary emigration Constant temporary emigration vs. time-specific temporary emigration Constant monthly survival vs. time-specific survival
a
Data from woodlot grid 2, Patuxent Wildlife Research Center, September 1980-March 1981.
p r i m a r y period i; and B~I)'~ the n u m b e r of adults in the p o p u l a t i o n at p r i m a r y s a m p l i n g occasion i + 1 that entered the p o p u l a t i o n as i m m i g r a n t s b e t w e e n prim a r y periods i and i + 1. Originally the estimation of recruitment used an ad hoc a p p r o a c h (Nichols a n d Pollock, 1990), t h o u g h multiple-age m o d e l s have been d e v e l o p e d using the likelihood-based a p p r o a c h as well (J. Nichols, R. Hinz, and J. Hines, unpublished). U n d e r the ad hoc approach, survival rates for y o u n g (q~!0)) and adults (~I 1)) are estim a t e d using the open m o d e l s of Section 17.2 with data over the p r i m a r y s a m p l i n g periods. A b u n d a n c e is estim a t e d for y o u n g (N~i~ a n d adults (N~/1)) using closed m o d e l s with data over the s e c o n d a r y s a m p l i n g periods within each p r i m a r y period. The two recruitment comp o n e n t s then are estimated as
short that animals born in the s t u d y area b e t w e e n prim a r y periods i and i + 1 m u s t still be identifiable as y o u n g at period i + 1. These aspects of s a m p l i n g design are discussed by Nichols a n d Pollock (1990) and Yoccoz et al. (1993). 19.4.2.1. S t a n d a r d - T i m e
Approach
Here w e consider the case of two age classes, but the extension to more than two classes is straightforward. As noted above, the a d v a n t a g e of the robust design is p r i m a r i l y as a m e a n s of dealing with the initial age class. The a p p r o a c h uses the general notation described above, with the addition of superscripts to denote age class. In w h a t follows, the superscript v takes a value of 0 or 1, as in Section 17.2. Define the two recruitment c o m p o n e n t s as B!1)', the n u m b e r of adults in the p o p u l a t i o n at p r i m a r y s a m p l i n g occasion i + 1 that were y o u n g animals in the p o p u l a t i o n at
TABLE 19.9
/~11)' = ~l~176
Model (~t, Ptt', ~[t )a
Model (~o,Ptt', ~t )a
period
~
1 2 3 4 5 6 7
0.81
(19.21)
Estimated Rates of Survival and Temporary Emigration for P e r o m y s c u s l e u c o p u s on Woodlot Grid 2 (Patuxent Wildlife Research Center) under Models of Completely Random Emigration
Survival
Sample
- nl~ + RI ~
S'E(~) 0.045
Survival
Temp. emigration
~i
S"E(~i)
. . . . . . <0.01 0.149 0.87 0.086 0.85 0.162 0.72 0.185 0.16 0.110 0.16 0.126
~i
Temp. emigration
S"E(4~i)
0.60 0.144 0.82 0.173 0.94 0.146 0.75 0.172 0.91 0.109 . . . . . . . . . . . .
"~i
.
. . . . <0.01 0.86 0.71 0.70 0.18 . . . .
S'E(~/i)
Ad hoc estimator b
~i
. . . . . . . . 0.158 <0.01 0.094 0.85 0.261 0.88 0.161 0.70 0.112 0.18 . . . . . . .
S"E(~i)
0.313 0.108 0.090 0.143 0.112
a Variance-covariance matrix, as computed by RDSURVIV, was not positive-definite, so standard errors were estimated using a parametric bootstrap approach (200 simulations). bBased on capture probability estimates from closed (Lincoln-Petersen) and open (Jolly-Seber) models used in conjunction with the estimator of Eq. (19.18) and the associated variance estimator of Kendall et al. (1997).
19.4. Special Estimation Problems and
]~!1)" = /~/1+)1 -- ~11)(/~l.1)- r/f1)q -
RI1)) (19.22)
-q~!o)(/~/o) - nlO)+ R!~ where ~-i R(v) and ni(v) are simply the age-specific analogs of the statistics defined in Section 19.1. Thus, the estimated number of recruits resulting from in situ reproduction [Eq. (19.21)] is simply the expected number of surviving young from the previous sampling period. The estimated number of immigrants at period i + 1 is the difference between the estimated number of adults at i + 1 and the expected numbers of surviving young and adults from the previous sampling period [Eq. (19.22)]. Approximate variances for these estimators are presented by Nichols and Pollock (1990) and Pollock et al. (1990). The estimators can be seen to decompose the usual estimator for recruits in Eq. (19.3) into contributions from two separate components. Because the usual estimator typically is imprecise (Pollock et al., 1990) (see Section 18.6.3), the separate estimates of recruitment tend to be imprecise as well. The estimation of recruitment components can be extended to a system of multiple patches or study locations. In this case, recruitment components for a particular location include young from the same location, young from the other sampled locations, adults from the other sampled locations, and animals from outside the multiple-location study system. In this case the recruitment estimators represent a straightforward extension of the estimators for the single-location case. Nichols and Coffman (1999) presented recruitment estimators for a simple two-patch system.
19.4.2.2. Reverse-Time Approach Biological considerations about population dynamics often focus on the relative contributions of different demographic components to population growth rate (e.g., Pulliam, 1988; Nichols et al., 2000a), rather than absolute numbers of animals in the different components. For example, the definitions of source and sink populations depend on the relative contributions of immigration and in situ reproduction to population growth rate (see Section 11.2.3). Of course, it is possible to compute estimates of relative contributions of these two types of recruitment using estimates of absolute numbers of recruits from Eqs. (19.21) and (19.22). We have seen, however, that reverse-time capturerecapture modeling permits direct estimation of the relative contributions of demographic components to
545
population growth (see Section 17.4), absent a need to estimate the actual size of population cohorts. To apply a reverse-time approach in the robust design, define the following probabilities: Yi+l'(lv)is the probability that an adult animal (age = 1) at time i + I was in a particular age class (young, v = 0; adult, v = 1) on the study area at time i. Thus, ,y~11), ~/~10), and (1 ~/!11) _ ~!10)) represent the respective probabilities that an adult at time i was an adult on the study area at time i - 1, or a young animal on the study area at time i - 1, or an immigrant between i - 1 and i (i.e., not on the area at i - 1). In this section, we use the reversetime modeling approach to estimate the parameters ~/(lv) ; . We caution the reader not to mistake the use of the symbol y in this section (to denote seniority) with its use in Section 19.4.1 (to denote the probability of temporary emigration). We chose to retain y for both attributes because of its use in the literature to denote both. Because of the inability to estimate capture probability and abundance for animals of age O, open models cannot be used by themselves in reverse-time, multipie-age modeling. Instead, reverse-time modeling with multiple ages is essentially a multistaee problem (see Section 17.3), and estimation of y110) under the reversetime approach requires information about capture probabilities in each state, including p!~ 1. Thus, captures and recaptures of young from secondary samples under a robust design are needed in order to estimate capture probability for young (also see Nichols and Pollock, 1990). This requirement explains the need for the robust design when using reverse-time modeling with data from multiple age classes. Before turning to estimation, consider the interpretation of the yl iv) parameters. Define age-specific abundance for primary period i + 1 as N~/~)1, where v = 1 for adults and 0 for young. The finite rate of increase for the entire population, including both age classes (denoted with a "v." superscript) can be written as /
t
\
~klv') -- x~iO)l ~/0) -ff _+_ ~1))1. .
(19.23)
The finite rate of increase for the adult component of the population is given by
~k!1) = x~il) 1/X~i 1) .
(19.24)
For a population at stable age distribution (Chapter 8), these two growth rates are equal, --i x(v') = h!1), but they can differ substantially for a population exhibiting transient dynamics or temporal variation in vital rates. The number of adults at time i + 1 can be written as the sum of three demographic components with respect to time i:
x~il) 1 -- n~l)+ Elm+ BI 1),
(19.25)
546
Chapter 19 The Robust Design
where LI1) is the number of adults present in the population at time i that survive and are still present at time i + 1, L!~ is the number of young animals present at time i that survive to become adults at time i + 1, and B!1) denotes the number of immigrants between i and i + 1 that are present as adults at time i + 1. As in the single-age modeling (Section 17.4), we can view the components of Eq. (19.25) as random variables following a trinomial distribution, conditional on X~/?l and the parameters }'i+l"(lv)The probability distribution of these components at i + 1 can be written as
P(C~~ C~1), B!1) ] N~/I+)I) (N~/1+)1)! =
f (lO)~ Ll~
(L!O))!(L!I))!(B!I))!~li+I
(19.26)
'(11)~L11)( 1 - Ti+I X ,\/ / [I,i+l/ ,(10) -- Ti+I) ,(11) R(1)-i
In this two-age situation, we can use the Yi+I 0(lv) to decompose the adult population at time i+1 into relative contributions of adults at i, young at i, and immigrants. The corresponding decomposition for the growth rate of the adult component [defined in Eq. (19.24)1 is E(K!1))
E(L~~
+
E(L!1)) -tE(N~/1))
E(BI 1))
(lO)]kT(1) -t- ,(11)~/(1) q- (1- ,(lO) ,(11),~M(1) i+1~i+1 }'i+1~i+1 fi+l -- f i + l / ~ i + l
E(N~/1)) which expresses the adult growth rate ~.I1) in terms of adults, young, and immigrants at time i + 1, with the seniority parameters Ti+1'(1~for young and }'i+1'(11)for adults playing the role of proportionality factors that scale contributions to the adult population N (1) i+1" To illustrate, consider a population in which fi+1'(11)_-0.7, fi+l'(10) = 0.2, and 1 - Yi+l'(ll) _ Yi+l~176= 0.1. Under this scenario, one concludes that recruitment resulting from in situ reproduction made twice the contribution to adult population growth over the interval i to i + 1 as recruitment from immigration. These quantities can be used to draw inferences about the change in population growth of the adult component, }k!1), t h a t would have resulted from a change in a demographic component at i. For example, a proportional reduction oLin young survival [i.e., survival is given by (1 - o0q~l~ produces a proportional reduction in adult population growth of Ot~i+l (lO). ( 1 - '~ ,(10)/k/(1) q_ ,(11)R/(1) if_ ( 1 ,(10) ,(11)~RT(1) v'/Ti+l*~i+l }'i+1~i+1 }'i+1 - - Y i + l / * ~ i + l 1)
N~/
}k}1 ) ~,(.1)(1
--
"'i ' , " - "
(lO)x ffl) Ot~i+IB/~+I
N~/1)
__
,-,, #(10)"~ '-~fi+l/.
If interest is focused on the growth rate of the entire population (young and adults), then we can use expression (19.23) for Xlv). Relative effects of changes in vital rates now involve the age composition of the population at i + 1, as well as the relative contributions to the adult component at i + 1, and the computations initially seem more complicated than for X!1). However, in many sampling situations, young animals in a population at i + 1 can be viewed as the product of number of adults at i + I and a per capita reproductive rate for adults. Thus, a proportional reduction of ot~!l?? in the adult population at i + 1 produces an identical proportional reduction in the number of new young at i + 1. The consequence of this reasoning is simply that (10) gives the proportional reduction in b o t h }k!1) and Ot'Yi+l Xlv) that would have resulted from proportional reduction c~ in young survival between i and i + 1. To examine the influence of a reduction in reproductive rate, or one of its multiplicative components (e.g., probability of an adult breeding, clutch size, nest success), on population size or growth rate, one can focus on X~/~1 in Eq. (19.23). Proportional reduction of magnitude oL in reproductive rate, or one of its components, just prior to i + 1 should lead to a proportional reduction in Xlv) of od~i~ 1/(/~i~ if- ~/1)1). In the case of changes in reproductive rate, we would simply compute effects of such a change on X!v) using the agespecific estimates of population size, without resorting to reverse-time estimation. Given this interpretation of the seniority parameters y(lv) / , we now focus on their estimation. To begin, recall that the standard models for estimating age-specific parameters (Pollock, 1981b; Lebreton et al., 1992) condition on animals released in the different age categories and model their capture histories separately. For example, under a two-age model with young and adult, consider an animal caught in periods 1 and 3 but not in period 2 (capture-history 101). First assume that the animal was an adult (age, v = 1) when released at period 1. Conditional on release in period 1, the probability associated with the remainder of the capture his(1)_P(31), where the superscript again tory is q}{1)(1 - P2(1),)q:}2 denotes age. N o w assume that a young animal (age, v = 0) was released at time 1. Conditional on this release, the corresponding probability for capturehistory 101 is q~{~ p(21)){p(21)p(31).The only difference between the probabilities associated with this capture history for young and adults involves the initial survival probability, corresponding to the age of the released animal. After the interval between periods 1 and 2, the young animal becomes an adult and experiences the same survival and capture probabilities as the animal marked as an adult. Furthermore, the transition between young and adult is deterministic (all young at i are adults at i + 1). -
19.4. Special Estimation Problems When the time order of the capture-history data is reversed, it is clear that we cannot simply use the standard multiple-age capture-recapture models. Instead, we condition on animals that are all adults at time i + 1 and then ask (in the two-age case) what proportion of these animals were adults at time i, young at time i, or immigrants entering between i and i + 1 [see Eqs. (19.25) and (19.26)]. So rather than conditioning on animals in two distinct age classes and estimating their respective probabilities of appearing in a single age class in the next time period, we condition on animals in one age class (adults) and estimate the probabilities of their having been in one of three different classes (young, adult, potential immigrant) in the previous time period. The reverse-time approach to multiple-age modeling thus requires multistate modeling. An example is instructive. Consider the capture of an adult in the third period of a study, with capturehistory 101. If the animal was an adult in period 1, the reverse-time model of its capture history is @311)(1 p (21~)~/(211~p~1~. If the animal was a young animal in period 1, the reverse-time model of its capture history is .(10)p~0)" Note that, unlike the previous ~(311) (1 - P2(1)~'~'2 modeling with multistate models, the probability structure associated with a "0" in the capture history is modeled as a single path rather than as the sum of alternative possible pathways. This is because an animal that is young in period 1 cannot also have been young at period 2 (an animal grows deterministically from young to adult in a single interval). Just as with single-state models, reverse-time estimation using multistate models can utilize either an ad hoc approach (Pollock, 1982) (Section 19.2) or a joint likelihood (e.g., Kendall et al., 1995, 1997; Schwarz and Stobo, 1997) (Section 19.3). The likelihood-based approach is appropriate when models for the secondary sample data lead to maximum likelihood estimators of model parameters (e.g., closed models involving behavioral response and time). On the other hand, the ad hoc approach of Section 19.2 can be used when maxim u m likelihood estimates for the closed model are not available (e.g., the model includes heterogeneity in individual capture probabilities). Extinction under an ad hoc approach requires the following additional statistics: ,~(10~ " ' i , i - 1 , the members of n}1~ that were caught as young animals in primary period i - 1", and _(11~ Irti, i - 1 , the members of n}1) that were caught as adult animals in primary period i - 1. Estimates for the following two unknown random variables also are required" ~^j(10} v ~ i , i _ l , the members of n }1~that were in the sampled area as young animals in primary period i - 1 (but not necessarily captured in i - 1); and M"~l,l (.11~ - 1, the members of n}1) that were in the sampled area as adult animals in primary period i - 1 (but not necessarily captured in i - 1).
547
Estimation of the seniority parameters Yi-(lv) requires estimates of the M (iv) i , i - 1 (see below), which can be accomplished using either of two approaches. One approach is simply to condition on the m i(iv) , i - 1 animals that were caught in time periods i and i - 1. The capture histories of these animals over the secondary trapping periods of primary period i - 1 can be used to estimate M i(lv) , i - 1 using any estimator appropriate for the secondary period data (e.g., Otis et al., 1978; Rexstad and Burnham, 1991; Lee and Chao, 1994) (see also Chapter 14). In this manner we obtain an estimate of "population size" for a subset of the animals captured in state v at i - 1, namely, those that also were captured at i in state 1 9 For example, estimation of ~^j(10) w i , i - 1 would condition on all the animals caught as adults (state 1) at time i and as young (state 0) at time i - 1 (the m(10) i , i - l J "~ The capture histories of these animals over the secondary periods of primary period i - 1 would be used with a capture-recapture estimator (typically from a closed model), and the resulting abundance estimate would correspond to ~j(10) ~ v . i , i _ 1. The other approach to estimation o f ,lVli, j(lv~ i--1 is to focus on all animals of age v caught at i - 1, regardless of whether they were also caught at time i (we have denoted these animals as n !V~l). Use of the capture histories of these animals over the secondary periods of primary period i - 1 with a closed-model estimator yields an abundance estimate for the number of animals of age v in period i - 1 (/~/~1). The average probability that an animal of age v at primary period i - 1 is caught at least once during i - 1 (we denote this probability a s p}V)1) is then estimated as P}V)I =
(v) //~/v) 1.
hi-1
(19.27)
j(lv) We then can estimate lvli, i_ 1 as ]~(lv)
i,i-1
(lv) - "(v)
-- mi,i-1/Pi-1,
(19.28)
where the age-specific capture probability estimate is computed as in Eq. (19.27). The estimator in Eq. (19.28) should be more efficient than direct estimation of M(lV) i,i-1, as d e s c r i b e d
above.
Irrespective of which approach is taken, the estimates of the l~(lv~ v l i , i - 1 then can be used to estimate ~/}lv~ In the two-age problem, the seniority parameters ~/I1~ a n d "y111)thus are estimated as ,~!'v) __ ]~(lv) / n ( 1 ) 9
i,i-1
i
f o r v = 0, 1. Finally, we note that estimation of the relative contributions of different ages and demographic components to population growth can be extended to systems of multiple study locations (Nichols et al., 2000a). The logic underlying this extension is similar to that de-
Chapter 19 The Robust Design
548
scribed by Nichols and Coffman (1999) to estimate recruitment components using standard-time analyses. Reverse-time analyses that incorporate both agespecific and spatial variation essentially represent a straightforward combination of the methods described in this section and in Section 17.4.2. Example The data for this example involve meadow voles trapped at Patuxent Wildlife Research Center during the summer and fall of 1981 (Nichols et al., 1984a). The data were collected according to the robust design with primary sampling conducted at six monthly periods. A monthly interval between sampling periods was thought likely to meet the assumptions for separating recruitment resulting from in situ reproduction and from immigration (see Nichols and Pollock, 1990; Yoccoz et al., 1993). A 10 x 10 grid of trapping stations was used, with 7.6-m trap spacing. The study used Fitch traps (Rose, 1973) baited with whole corn and containing hay. Traps were set in the late afternoon and early evening of one day and then run the morning of the next day. Traps were locked during the day and opened again each afternoon and evening. This procedure was repeated for five consecutive trap nights. Animals were marked with monel fingerling tags in the ears, and all captures of marked animals were recorded. Captured animals were weighed and reproductive characteristics ascertained. Adults were
TABLE 19.10
Number of adults caught
Number of young caught
(h )
(n(h1))
(n(h~
12/4-12/8
6
149
10/31-11/4
5
10/3-10/7
4
8/29-9/2
3
8/1-8/5
2
6/27-7/1
1
1 -
p* =
(1 -
pl)(1
-
P2),
one of the period-specific capture probability parameters was then rewritten as a function of the overall probability p* of being caught in at least one of the two periods (see Kendall et al., 1995, 1997), because it is this latter parameter that corresponds to the openmodel portion of the likelihood. Closed-model data (obtained over secondary periods) were stratified by age (young, adult) in order to estimate capture probability for each age class. The young data were adequate to estimate capture probability well for primary periods 4 and 5, but not for periods 1-3. Thus, the secondary capture-recapture data for young voles in periods
Capture-Recapture Statistics for Young and Adult M e a d o w Voles a
Sample period
Sampling dates
defined as voles ___22 g as in the previous examples, and animals of smaller mass were designated as young. Analysis of the adult data in Section 17.1.10 failed to provide strong evidence of sex-specificity in either survival or capture probabilities, so we combined the sexes in a follow-up analysis (data and analysis are summarized in Table 19.10, and both can be found at the internet address http://www.mbrpwrc.usgs.gov/ pubanalysis). We developed a single likelihood for both closed and open components (Kendall et al., 1995, 1997). For the modeling of closed data, we used a LincolnPetersen approach considering the first 2 days of trapping as "period 1" and the second 3 days as "period 2" (see Menkins and Anderson, 1988). From
Number (n~1)) of adults most recently captured as young (mhi)10 or adult (mhi) 11 , in primary period i i=5 27, 83
101
32
(73, 9, 19)
(12, 6, 14)
102
28
(66, 13, 23)
(9, 4, 15)
102
3
(58, 16, 28)
(0, 2, 1)
127
4
(75, 26, 26)
(0, 2, 2)
106
8
(72, 5, 23)
(3, 1, 4)
4
3
2
1
4, 4
0, 0
0, 1
0, 0
18,59
0,2
0,0
0,0
1,66
0,7
0,0
2,68
0,5 8, 84
a Trapped under the robust design at Patuxent Wildlife Research Center, Laurel, Maryland, June-December, 1981. Data are presented in reverse-time order mhi-array format. Numbers in parentheses indicate the frequencies of the three observable two-period capture histories (11, 10, 01) within each primary sampling period, where days 1 and 2 are grouped as period 1 and days 3-5 are grouped as period 2 [general approach of Menkins and Anderson (1988)].
549
19.4. Special Estimation Problems 1-3 were modeled with the same parameters (assumed to be constant over the three primary periods) used for young in periods 4 and 5. The open-model portion of the likelihood was based on a multistate model. Because we used a reverse-time approach, certain transitions were not possible, and their transition probabilities were set equal to 0 (a young animal at i could not have existed at time i - 1, so 'y101) = ,y!00) = 0). Population sizes were estimated using Lincoln-Petersen estimates of abundance by age (e.g., Seber, 1982; Menkins and Anderson, 1988) (see Section 14.1.2). We used these population size estimates to estimate two different rates of increase, one corresponding to only the adult component of the population (KI 1)) a n d the other corresponding to the sum of adults and young
ranged from 0.15 to 0.25. Recruitment of adults was dominated by immigrants in some periods (3 and 4) and by surviving young in others (5 and 6), suggesting temporal variation in the mechanisms responsible for population growth. A knowledge of temporal change in the source of new recruits is important both for understanding animal population dynamics and regulation, and for managing animal populations. Additional information often can be obtained by a comparison of results of a reverse-time analysis, which focuses on proportionate contributions to reproduction, with results based on a standard-time analysis of the same data directed at the absolute number of immigrants (Nichols and Pollock, 1990).
(v) ()ki 9).
The multistate model for the robust design approach was implemented using program SURVIV (White, 1983) coded in a manner similar to that used in MSSURVIV (Hines, 1994). The fit of the vole data to the model was judged to be acceptable. Some reduced-parameter models appear to be appropriate for these data as well, but we present estimates under the general model for illustrative purposes. Estimated rates of increase ranged from 0.81 to 1.51 for adults and from 0.80 to 1.24 for adults and young combined (Table 19.11). The reverse-order modeling indicated that surviving adults were the largest contributor to population growth rate, constituting 0.60 to 0.74 of the adult population throughout the study (values of ,~!11) in Table 19.11). The proportion of adults that were young on the study area in the previous period was estimated to range from 0.02 to 0.25 (values of ql 1~ in Table 19.11). The estimated proportion of adults that were new immigrants [values of ( 1 - ~111)- "y110)) in Table 19.11]
19.4.3. Catch-Effort Studies Here we describe a robust design for catch-effort studies (Gould and Pollock, 1997a), with removals occurring over the secondary sampling periods within each primary period. The idea is to utilize the relationship between capture probability and sampling effort to improve the estimation of both survival and abundance. We first consider the partially open model in which animals are lost to natural mortality and permanent emigration between primary sampling periods. All population losses over secondary periods are assumed to occur as a result of known removals, and the population is assumed to experience no gains over the study. The model can be viewed as an extension of the catch-effort models presented in Section 14.4. We follow the general notation of Gould and Pollock (1997a), which differs somewhat from that used in the previous portions of this chapter:
TABLE 19.11 Estimates of Population Change of Meadow V o l e s a Trapping period
Dateb
-i ~(1)c
~-E[~,~,)]c
-, ~(.v.)c
~[~.~v.)]
-i 4(")d
S'E [~1~TM]
.~(,0)d Yi
~[~10)]
[1- ~/i~(11)_ ~/lO)]d.
s'~E[1 _ ,~11) _ ,~10)]
1
6-29-81
1.24
0.039
1.20
0.052
2
8-03-81
0.81
0.036
0.80
0.045
0.68
0.042
0.11
0.051
0.22
0.058
3
8-31-81
0.97
0.041
1.24
0.066
0.74
0.045
0.03
0.025
0.23
0.047
4
10-05-81
0.97
0.030
1.00
0.048
0.74
0.046
0.02
0.018
0.25
0.047
5
11-02-81
1.51
0.041
1.13
0.044
0.60
0.049
0.25
0.056
0.15
0.052
6
12-06-81
0.62
0.040
0.22
0.044
0.16
0.041
.
.
.
.
a For both a d u l t s (KI1)) and adults + y o u n g ( h^(~) i ' ) ; also s h o w n are relative contributions to adult population growth of adult survival ('~111)), fecundity and y o u n g survival (~/11~ and immigration from outside the study area (1 - -y111) - ~/110)) for m e a d o w voles t r a p p e d at Patuxent Wildlife Research Center, Laurel, Maryland, June-December, 1981 (data s u m m a r y in Table 19.10). b Midpoint of 5-day trapping period. c Estimated as ratios of Lincoln-Petersen a b u n d a n c e estimates. a Estimated using reverse capture-recapture with multistate models u n d e r the robust design.
550
Chapter 19 The Robust Design The population size just before the first sample in primary period i(i = 1, 2, ..., K). The number of animals removed at the time of the jth secondary sample in the ith primary period (j = 1, 2,
Ni
rift
Eh=l
nhi
The cumulative catch in primary period i prior to secondary sample j ( j = 2 . . . . , li+l; X l i -- 0).
fji
f ji
=
j-1
Eh=l
NI!
n11[,~
~
~n21
.. ~
P{nji} = I-IK_I H! i P l l ~,~/llF21' i = 1 nji!(N1 -- Xli+l,i )! X (qllq21 "'" qll-l,lPlfl)nll'(qllq21
"'" qlllq~lP12 )n12
9.., li).
j-1 xji---
mary sampling periods follows the multinomial distribution
fhi
The units of capture effort expended in secondary sample j of primary sampling period i. The cumulative effort in primary period i prior to secondary sample j ( j = 2, ..., li+l; Fli = 0).
X (qllq21 "'" qlllq~lq12P22 )n22 "'" X (qllq21 "'" qlllq~lq12q22 "'" q12-1,2P122 )n122 "'" X (qllq21 "'" qlfl~Plq12 "'" q~K-lPlK )nlK "'" • (qllq21 -" qlllq~l "'" ~ K - l q l K
(19.29)
"'" qlK--1,KPIKK )nl~K
X IIqllq21 ... q111{1 - q~l[ 1 - q12q22-" q122
X (1 - q~2II - q13q23"'" {1 -
q~3 ... K
X [1 - q~K-l(1--qlKqaK"" qIKK)"'" l}])l}]]N'-,~,%+'';)The population data are thus the rift statistics, with the fji statistics available for use as model covariates. Note that the parameter f differs in its meaning here from previous uses in this book, in particular in Chapter 14 where it is used to denote capture frequency, in Chapter 16 where it denotes recovery probability, and in Chapter 17 where it denotes a resighting probability. We have retained the use of f for each of these attributes in order to facilitate cross-referencing between material in this book and the biological literature. Define the following model parameters:
Pji = 1 -
e -kifji
The catchability coefficient for primary period i (the capture rate corresponding to one unit of e f f o r t ; i = 1.... ,K). The probability of capture for f j i units of effort, with qji = 1 - Pji -~ e-kifji.
P~i
q~i = e
-- Pq(ti+l -- ti)
The instantaneous rate of natural mortality between primary p e r i o d s / a n d / + 1 (i = 1, ..., K-l). The probability that an individual alive at time t i (just subsequent to primary sampling period i) will survive to time ti+ 1 (i = 1,..., K - 1).
We note that survival can be written as either a finite or instantaneous rate parameter, and the choice carries no real consequences for estimation. However, the modeling of capture probability as a function of catchability and effort is essential for the estimation of model parameters. The joint distribution of the catches {nji} under a partially open model permitting deaths between pri-
Though complicated, this parameterization is intuitively reasonable. For example, consider the cell probability corresponding to the number n22 of animals removed at secondary period 2 of primary period 2. The initial part of the product ( q l l q 2 1 "'" ql11) simply indicates that in order for an animal to have been removed in secondary period 2 of primary period 2, it must not have been removed during the l1 secondary sampling periods of primary period 1. The q~l term indicates that the animal had to survive the interval between primary periods 1 and 2. The q12 term indicates that the animal was not removed during the initial secondary period of primary period 2, and P22 corresponds to its removal in the secondary period 2 of primary period 2. The full parameterization in Eq. (19.29) involves writing the Pji and qji a s functions of the catchability coefficient and effort (e.g., Pji - 1 - e-kifji). Similarly, Gould and Pollock (1997a) prefer to write survival as a function of the instantaneous mortality rate and time (q~i -- e-~i(ti+l-ti)), although this is not necessary except for certain reduced-parameter models. The model has 2K parameters: N 1, k1, ..., k K, and ~1, " " , ~K-I" Gould and Pollock (1997a) present closed-form estimators for a special case of Eq. (19.29), but in general the maximum likelihood estimates must be found iteratively. Program SURVIV (White, 1983) was adapted for the purpose of obtaining estimates under this model (Gould and Pollock, 1997a). Asymptotic variance estimates also can be computed by SURVIV, although Gould and Pollock (1997a) recommend use of parametric bootstrap techniques. Because the above model is partially open, in that population losses (but not gains) can occur between primary periods, the only abundance parameter is the initial population size N 1. The model in Eq. (19.29)
19.4. Special Estimation Problems essentially follows the reduction in this initial number of animals over time. In order to deal with both mortality and recruitment, Gould and Pollock (1997a) recommended release of marked animals at the beginning of the study, prior to the first removal sampling. Estimation of survival rate and capture probabilities for marked animals essentially involves substitution of the number of marked animals for N 1 in Eq. (19.29). Survival is assumed to be the same for marked and unmarked animals, and capture probability for unmarked animals can be estimated using the secondary-period removals within each primary period. This approach thus provides the information needed to estimate numbers of unmarked and total animals at each period, as well as recruitment (Gould and Pollock, 1997a). Simulation studies have indicated that the maximum likelihood estimates for this model are superior to estimates based on least-squares regression methods (e.g., Chapman, 1961; Seber, 1982).
19.4.4. Potential for Future Work 19.4.4.1. "Unconditional" Closed-Population Modeling The likelihood-based models in expressions (19.11)(19.14) focus on the modeling of data over primary periods as described in Chapters 17 and 18, and the modeling of data over secondary periods following the "conditional" approach for closed populations of Chapter 14. Models for the secondary periods are conditional on the number of animals captured in each primary period, with the multinomial cell probabilities scaled via division by p* [see Eq. (19.14)]. An exciting area of future modeling involves the incorporation of "unconditional" closed-population models for the secondary-period data of the robust design. These models include the Ni as parameters directly in the likelihood. For example, component P3 of Eq. (19.11) would be written as
P3({x~i},{x~i} I {ui},{mhi},{Pij},{Ni}),
(19.30)
in which population sizes N i and capture probabilities Pij are the parameters of interest. The conditioning on captures of marked and unmarked animals in Eq. (19.30) is not necessary under some models, but is needed for others that incorporate certain forms of capture-history dependence. With this more general expression, the open-model data from the primary periods could then be modeled using the K-parameterization of the temporal symmetry approach of Pradel (1996) (see also Section 18.4). Such a parameterization of data from both secondary and primary periods would provide an opportunity
551
for the sharing of model parameters other than p*. One possible approach would be to retain N 1 as a model parameter for the secondary period data and use i-1
Ni = N1 I-I kj 1"=1
for all subsequent Ni, i = 2, ..., K (Nichols and Hines, 2002). Under this parameterization, the modeling of capture history data from secondary and primary periods would share not only the nuisance parameters p*, but also parameters for underlying population dynamics, ki. As abundance and changes in abundance are difficult to estimate precisely, the use of additional data (from secondary periods) for this purpose is likely to be useful. An interesting benefit emerging from such models extends beyond estimator precision. One possible advantage of the robust design is that different data sources provide estimates of time-specific abundance and survival probability that exhibit little (hopefully negligible) sampling covariation, thereby permitting inferences about density dependence in relationships between, e.g., abundance and survival (Nichols et al., 1984a; Kendall and Pollock, 1992). For example, Leirs et al. (1997) estimated abundance of the multimammate rat (Mastomys natalensis) on a Tanzania study site using closed capture-recapture models with data over secondary sampling periods. These abundance estimates then were used to categorize primary sampling periods with respect to abundance, and survival probabilities were modeled using this categorization of sampling periods (Leirs et al., 1997). The incorporation of the N i as model parameters allows for the possibility of direct modeling, e.g., survival probability as a function of N i, because both sets of parameters could appear in the likelihood. It would seem that this type of modeling might provide a good means of addressing questions about the functional relationship between abundance and parameters such as survival, seniority, and per capita reproductive rate [the latter from Pradel's (1996) temporal symmetry approach; see Section 18.4]. It also would provide a means of dealing with sampling variances and covariances in a manner that should be^preferable to the use of separately estimated N i as covariates in a survival (or other) model.
19.4.4.2. Robust Design with Other Data Sources In studies of harvested populations, it is possible to combine robust design capture-recapture and band recovery models (Nichols, 1996). This type of modeling essentially combines the approaches of Section 17.5.1 (also see Burnham, 1993) with those of Section 19.4.1 (also see Kendall et al., 1997), to estimate true survival
552
Chapter 19 The Robust Design
as well as permanent and temporary emigration probabilities. Lindberg et al. (2001) have developed models and computer software for this situation. Similarly, it should be possible to use the robust design in conjunction with studies that include sightings or observations between primary sampling periods (Section 17.5.2) or that include telemetered animals (Section 17.5.3). In all cases, designs that include secondary samples provide the ability to estimate capture probabilities. This ability typically increases precision of estimates and frequently permits estimation of additional parameters (e.g., temporary emigration). 19.4.4.3. Capture Frequency Data as Covariates
A possible likelihood-based approach for dealing with heterogeneous capture probabilities in the robust design involves the use of capture frequency data from previous primary periods as individual covariates to model capture probability. This idea was proposed independently by Fletcher (1994) and R. Julliard and N. Yoccoz (personal communication). Because of the difficulty in dealing with heterogeneous capture probabilities in an open-model framework, Fletcher (1994), Julliard, and Yoccoz considered information coming from secondary occasions in previous primary sampling periods as individual-level covariates. For example, consider the modeling of capture probability in primary period i. For each animal caught in period i, it should be possible to use the number of secondary periods in which the animal was captured in primary period i - I as a covariate to model capture probability for that animal. This approach appears to work well as a means of modeling sighting probability for Hector's dolphins (Fletcher, 1994). However, full implementation still requires several important decisions. For example, it is not obvious how to treat the observation of 0 captures or resightings in the previous primary period. If the 0 indicates presence of the animal but no capture or sighting, then the appropriate covariate value is 0. However, if the 0 indicates absence of the animal (temporary emigration), then 0 is likely not the appropriate value for the covariate, and it might be more reasonable to use the mean covariate value for all animals that were detected, for example. Despite such uncertainties, we believe that the approach holds promise as a means of dealing with heterogeneity in implementation of a likelihood-based approach.
19.5. S T U D Y D E S I G N Because the robust design combines capture-recapture modeling for open and closed populations, most
of the relevant design issues have been discussed in previous chapters. In particular, design features of closed-model capture-recapture studies for estimation of abundance were discussed in Sections 14.1.4 and 14.2.6. Designs for the estimation of survival and seniority parameters with open-population models were discussed in Section 17.6, and the extension of these models to estimate abundance and recruitment was discussed in Section 18.6. We note that the number of secondary periods chosen for each primary period represents a tradeoff between (1) the need to increase estimator precision and improve the discrimination among models and (2) the need to reduce the number of periods so as to increase the likelihood of population closure and thus to permit the use of closed-population models. If the closure assumption is shown to be false, then open models can be used (Schwarz and Stobo, 1997; Kendall and Bjorkland, 2001). However, open models do not deal adequately with certain sources of variation in capture probability (e.g., see Section 18.2.5), so efforts should be made to satisfy the closure assumption. Perhaps the main topic of interest associated with sampling design involves the identification of study objectives for which a robust design is especially useful. For example, a study involving temporary emigration might focus on the estimation of survival and abundance, which in turn relies on the handling of temporary emigration in modeling capture histories. On the other hand, temporary emigration may correspond to a quantity of substantial biological interest in its own right (e.g., breeding probability in some sampling situations). In either case, the modeling of Markovian temporary emigration can be accomplished using either radioed animals or the robust design [estimation is even possible under some completely openmodel designs, but this work is quite new and not yet fully developed (H. Caswell, M. Fujiwara, and W. Kendall, personal communication]. As discussed in Section 19.4.2, estimation of the number of young or the contribution of young to recruitment requires information on the capture probability of young animals, and this information is not available under standard, open-model designs. The robust design provides a convenient means of estimating these probabilities and thus should be the preferred design for any capture-recapture study directed at inferences about reproductive output and recruitment. In Section 19.4.4, we briefly discussed attempts to investigate possible density dependence of survival rates and other quantities such as seniority and reproductive rate (Section 18.4). In nearly all cases in which investigations involve retrospective analyses, and even when direct experimentation is used, inferences are
19.6. Discussion strongest when sampling covariances among the quantities under investigation are negligible. Kendall and Pollock (1992) found that sampling covariances between survival and abundance estimates frequently were small or negligible under the robust design. We believe that the robust design should be strongly considered for use in studies directed at questions about density dependence in vital rates. We noted in Section 19.2 that some quantities not otherwise estimable under full, time-specific models (e.g., the final survival rate q~K-1, the first and last capture probabilities Pl and PK, and the first and last abundances N 1 and N K) can be estimated using the robust design. For studies that require estimates of these quantities, the robust design may be the most convenient approach to estimation. Finally, we note that a by-product of the robust design is increased capture probabilities for primary periods. The use of multiple days of sampling (the secondary periods) simply increases capture probability [see Eqs. (19.1) and (19.2) for the form of the relationship] and hence results in more precise estimates, more powerful tests, and better discrimination among statistical models.
19.6. D I S C U S S I O N The methods in this chapter combine the advantages of models for open and closed populations and thus are appropriate for sampling designs that include both short-term and long-term sampling. Chapter 14 documented approaches to analysis of data for closed models, and Chapters 17 and 18 described conditional (Cormack-Jolly-Seber) and unconditional (JollySeber) models for open populations. The robust design in this chapter involves a combination of both open and closed models, with sampling over both primary (long-term) and secondary (short-term) time frames. The approach allows one to obtain robust and more precise estimates of quantities in the simpler designs and also allows one to estimate parameters otherwise not estimable when restricted to only one of the simpler designs. Because the robust design involves sampling at both primary and secondary time scales, it is clear that the design can be much more sampling intensive than the CJS and JS designs. However, there are obvious and substantial payoffs for this extra sampling effort. With the robust design one can estimate population parameters more precisely (via the inclusion of additional secondary samples), relax assumptions that otherwise restrict the use of capture-recapture methods (e.g., estimation of population size in the face of nonhomogene-
553
ity in capture probabilities), and estimate parameters that otherwise are not obtainable with standard CJS and JS estimation methods (e.g., the parameters Pl, PK, q~K-1, N1, NK, B1, and BK_ 1 in the JS model; see Section 19.2.1 ). By combining the estimates of capture probability for both open and closed models, the robust design also allows one to recognize and measure temporary emigration and to distinguish births from immigration in annual recruitment. Finally, the design contributes to an analysis of population growth rate via the simultaneous estimation of seniority parameters for immigration, reproduction, and aging. By informing the estimation and analysis of these and other attributes, the robust design advances materially the analysis and understanding of population dynamics. We note that the chapters in Part Ill represent a progression, with the later chapters representing increased information and fewer assumptions than the earlier chapters. Thus, Chapter 14 focused on the estimation of animal abundance with data from shortterm studies, based on an assumption of population closure to gains and losses between sampling periods. Chapters 15 through 18 dealt with open populations, in which gains to and losses from the population can occur between sampling periods. Chapter 15 introduced the estimation of vital rates and focused on survival estimation based on studies of marked animals (or other entities, e.g., nests) in which detection probability is I (animals can be located at will). Chapter 16 concerned the special class of studies in which animals are initially marked and then recovered as dead, usually in animal harvests. Even this minimal amount of information permits estimation of survival probabilities and harvest rates associated with the sampling (exploitation) process. Chapter 17 then considered studies of marked animals in which multiple recaptures or resightings are possible. The models of Chapter 17 conditioned on the releases of marked animals and dealt with the modeling of subsequent capture-history data as functions of survival and detection probabilities. Chapter 18 extended the models of Chapter 17 by applying the detection probabilities of marked animals to unmarked animals, permitting estimation of abundance and rates of change in abundance, in addition to the quantities of Chapter 17. Finally, Chapter 19 combines methods for open and closed populations in the robust design. We believe that the sampling designs in this chapter, describing the most flexible and least restrictive of the models in Part III, will prove to be extremely useful. As a general rule, the robust design should become the standard for mid-term and long-term capturerecapture studies. It provides the ability to obtain more precise and robust estimates of quantities in the open
554
Chapter 19 The Robust Design
models of Chapters 17 and 18. But it also has the added advantage of permitting estimation of quantities that cannot be estimated using standard open designs. Finally, it provides estimates that should be more useful in addressing questions about density dependence, by
virtue of small sampling covariances between estimates of abundance and vital rates. We anticipate that the robust design will provide a great many opportunities for improved analysis and understanding of animal populations in the future.
C H A P T E R
20 Estimation of Community Parameters
20.1. AN ANALOGY BETWEEN POPULATIONS AND COMMUNITIES 20.1.1. State Variables and Vital Rates 20.1.2. Count Statistics, Detection Probabilities, and Inferences 20.1.3. Abundance and Species Detection 20.2. ESTIMATION OF SPECIES RICHNESS 20.2.1. Quadrat Sampling 20.2.2. Multiple Sampling Occasions 20.2.3. Empirical Distributions of Species Abundance 20.2.4. Field Sampling Recommendations 20.3. ESTIMATING PARAMETERS OF COMMUNITY DYNAMICS 20.3.1. General Approach 20.3.2. Temporal Variation at a Single Location 20.3.3. Geographic Variation at a Single Time 20.3.4. Variation over Time and Space 20.3.5. Assumptions 20.4. DISCUSSION
ber of species in the community, and (2) species evenness, reflecting the relative abundances of different species. Species richness and evenness frequently are used to characterize communities in the conservation and management of biodiversity and in the investigation of human disturbance on biodiversity (e.g., Karr, 1991; Scott et al., 1993; Conroy and Noon, 1996; Keddy and Drummond, 1996; Wiens et al., 1996). The information required for both aspects of biodiversity is found in a species abundance distribution, depicting the number of individuals for each species in the community. From a methodological perspective, communitylevel studies of relative abundance can be viewed as collections of simultaneous population studies, requiring the use of population-level estimation methods. Thus, questions about the distribution of species abundance, in particular species evenness, can be addressed with the capture-recapture methods presented earlier for estimating abundance. On the other hand, an assessment of species richness, the other component of species diversity, relies on estimates of species presence/absence rather than abundance, and therefore requires adaptations of these methods. Recognizing that few sampling programs for animal communities provide censuses, ecologists early on considered quantitative methods to extrapolate from the number of species observed in a sample to the total number of species in the sampled community (e.g., Fisher et al., 1943; Preston, 1948). However, this early concern for sampling issues was later replaced with a tendency (1) to equate species richness with the number of species enumerated in a sample and (2) to identify the relative abundance of two species with the ratio of sample counts. The assumption underlying the first
Thus far we have dealt primarily with the population level of biological organization, focusing on aggregations of sympatric, interacting individuals of the same species. In this chapter we shift our attention to biological communities and focus instead on sympatric, interacting populations of different species. In what follows we show that with some modifications, the statistical modeling and estimation methods described earlier for single species also can be used to address questions for multispecies communities. Special emphasis will be given to species diversity and associated metrics. Species diversity typically is thought of in terms of (1) species richness, or the num-
555
556
Chapter 20 Estimation of Community Parameters
tendency is that all species are detected, and the assumption underlying the second is that all species in the community are sampled with equal probability. These assumptions are unlikely ever to be true, and in any case they should be tested before being accepted as true. Among others, capture-recapture studies of small mammals and mist net studies of bird communities offer interesting opportunities to develop multispecies capture-recapture models and to test assumptions about species-specific capture probabilities. Recognition that not all species are detected by sampiing efforts has led some community ecologists to apply capture-recapture estimators to the problem of estimating species richness (Burnham and Overton, 1979; Derleth et al., 1989; Karr et al., 1990; Palmer, 1990; Coddington et al., 1991; Baltanas, 1992; Bunge and Fitzpatrick, 1993; Hodkinson and Hodkinson, 1993; Colwell and Coddington, 1994; Dawson et al., 1995; Thiollay, 1995; Walther et al., 1995; Nichols and Conroy, 1996; Boulinier et al., 1998a,b). In this chapter we discuss some of this work, starting with an analogy between populations and communities that informs the estimation of community-level parameters. We use this analogy to develop estimators for species richness, focusing on different sampling situations and the models and estimators likely to be useful in these situations. We also define some community-level vital rates (e.g., local extinction probability, local turnover, local immigration rate) and provide estimators for them. We then discuss spatial variation in community attributes such as species richness and composition, and provide some relevant estimators. The chapter concludes with a discussion of the assumptions underlying communitydynamic estimators.
primarily because of the time scale of ecological studies. We typically deal with a time horizon in which events such as species origination and global extinction events are unlikely to occur. However, an evolutionary time scale allows us to examine fossil data over geologic time, with possible inferences about taxonomic origination and extinction using the same models as presented here for local communities (see Rosenzweig and Duek, 1979; Rosenzweig and Taylor, 1980; Nichols and Pollock, 1983a; Conroy and Nichols, 1984; Nichols et al., 1986a).
20.1.2. Count Statistics, Detection Probabilities, and Inferences
20.1. A N A N A L O G Y
In Chapter 12 we discussed the estimation of abundance using count statistics, e.g., the number of small mammals captured on a trapping grid in the northcentral United States, the number of ovenbirds seen and heard at a point count in Maryland, or the number of red kangaroos counted from an airplane in southern Australia. In each case the problem is to estimate the total number of individuals in a population based on only a sample of them. By analogy, community-level studies treat species as individuals in a population study, so that a sample consists of the number of different species counted in some area of interest. Just as population-level sampling methods seldom permit complete counts of all animals present in an area, community-level sampling methods seldom permit complete enumeration of all species. Let Ci be the count statistic (the number of species counted) and N i be the true number of species, both at time-location i. Define Pi as the associated detection probability, i.e., the probability that a member of N i appears in the count statistic Ci. Here Ci is a random variable, with expectation
BETWEEN POPULATIONS
E(Ci) = NiPi.
AND COMMUNITIES
20.1.1. State Variables and Vital Rates The material presented in this chapter is based on an analogy between animal populations and communities, recognizing that the attribute of concern in studies of community ecology is species richness rather than population abundance. Recall that changes in population size occur as a function of population vital rates, i.e., rates of survival, reproductive recruitment, and movement (emigration and immigration). Similarly, changes in species richness occur as a function of community-level vital rates, i.e., rates of local extinction, local turnover, and local immigration and colonization. We focus here on local rather than global changes,
(20.1)
If we can devise some method to estimate Pi, then an estimate of species richness is obtained by dividing the count statistic by the estimated detection probability: 1~ i -- C i / P i .
(20.2)
The need to consider detection probability in estimating community vital rates is analogous to the case for population-level studies. For example, assume that we identify R r different species in location r at time i, with the intent of estimating how many of these species are still present in location r at time i + 1. We cannot simply count the number of species detected at time i + 1, because this statistic likely fails to include all the members of R~ that are present at i + 1. Thus, detection
20.2. Estimation of Species Richness probability for time i + 1 must be estimated and incorporated into our estimator of species survival (e.g., see subsequent material on estimating local species extinction probabilities). The analogy between populations and communities provides a framework to consider what information is needed to estimate detection probability. In capturerecapture studies of animal populations, individual animals must be identified, either from natural markings (e.g., Karanth and Nichols, 1998) or, more typically, from marks applied by investigators. For communitylevel studies, the focus is on distinguishing among different species, and this nearly always is accomplished using natural markings. In fact, it is the difficulty in distinguishing individuals of the same species that motivates the marking of animals. Because marking typically is not needed in studies of species richness and related parameters (it is much easier to distinguish individuals of two different species than to distinguish two individuals of the same species), estimation of community parameters is likely to be logistically easier than estimation of population attributes.
20.1.3. Abundance and Species Detection It is useful to consider the influence of abundance on one's ability to recognize species presence. In the standard use of capture-recapture models, the capture probability Pik denotes the probability that animal k is caught during sample period i. Variation in this probability among individuals is a result of factors such as movement patterns relative to traps or nets, and wariness with respect to capture devices. On the other hand, the use of capture-recapture models to investigate communities requires a definition of detection probability Pij as the probability that at least one individual of species j is detected in period i. If Pijk denotes the probability of detecting individual k of species j in sampling period i, we can write the species detection probability as
Pij
=
1 -
nj II (1 k=l
-
Pijk),
-
p~)nj.
the sampling techniques likely to be used in most community surveys, we expect variation in species detection probabilities to be quite large (see Sauer et al., 1994; Cam et al., 2002). Indeed, the variation in detectability among species may exceed by a substantial margin the variation in detectability among individuals within a species.
20.2. ESTIMATION OF SPECIES RICHNESS The particular approach used to estimate species richness depends on the type of community sampling that is conducted. Here we discuss three sampling approaches (see Nichols and Conroy, 1996): (1) quadrat sampling using spatial replication, (2) sampling on multiple occasions at the same location (temporal replication), and (3) sampling for the distribution of species abundance, using information on the number of individuals detected per species. For technical details about likelihoods, estimators, and model assumptions we refer the reader to Chapter 14, where capture-recapture models for closed populations were introduced.
20.2.1. Quadrat Sampling Quadrat sampling involves the subdivision of an area of interest into a number of quadrats, or small sampling units, and the selection of a random sample of quadrats (Fig. 20.1). The investigator identifies species found on each selected quadrat, using some sampling method [e.g., direct observation of animals and their sign (tracks, scats, nests, etc.), auditory identification, trapping and netting with different trap and net types]. It is best to use the same sampling methods and expend a similar level of effort on each of the sampled quadrats, although models permitting variation in species detection probabilities among quadrats are available. The result of the sampling effort is a list
(20.3)
where nj is the number of individuals of species j exposed to sampling efforts. If all individuals of the species have the same detection probability Pij, * then the species detection probability is
Pij-- 1 - (1
557
L5 [--]
L3 VI
L1 N
(20.4)
Thus, variation in species-specific detection probabilities arises from variable numbers of individuals in the species, as well as other factors (e.g., movement patterns, size, secretiveness, vocalizations, wariness). For
FIGURE 20.1 Illustrationof a samplingdesign permittingestimation of species richness using spatial subsampling. Sampling produces species lists for each quadrat (quadrats denoted as L1, L2..... L5).
558
Chapter 20 Estimation of Community Parameters
of the species detected on each quadrat and, in some sampling situations, the number of individuals detected for each species. To illustrate, assume that an investigator randomly selects two quadrats from an area of interest and produces a species count in each quadrat. Let the total number of species identified in quadrat 1 be nl, the number identified in quadrat 2 be n2, and the number identified on both quadrats be m. Consider the detection probability Pl associated with quadrat 1. If all species have equal probabilities of detection on both quadrats, then we should be able to estimate Pl as the fraction of species detected on quadrat 2 that also were detected on quadrat 1" ]~1 = m / n 2.
(20.5)
Equation (20.3) then can be used to obtain the following estimator for species richness: /(/ -- nl/]~1
(20.6)
= nlrl2/m.
Alternatively, n 2 can be viewed as the count statistic and m / n I as the estimate of P2, resulting again in Eq. (20.6). This is simply the Lincoln-Petersen estimator for estimating population size (see Section 14.1). We recommend that investigators using this estimation approach substitute the bias-adjusted estimators of Chapman (1951) (see Section 14.1.2) for Eq. (20.6). The problem with using an estimator such as Eq. (20.6) (or its multiquadrat analog) for species richness is that it relies on the assumption of equal probabilities of detecting all species. There invariably are large differences in species detection probabilities, with some species being readily detected and others being very difficult to detect. Detection probabilities vary with the difficulty of catching or observing individuals of different species, and with the relative abundances of different species within the area of interest. Unequal species detection probabilities produce a negative bias (i.e., the estimate N tends to underestimate the true number of species). However, even in the presence of heterogeneous detection probabilities among species, the estimator in Eq. (20.6) will be less biased than a naive estimator such as n I 4- n 2 - m, which simply records the total number of different species detected on both quadrats. Several different estimators have been proposed for use with quadrat species lists when detection probabilities are heterogeneous among species (Burnham and Overton, 1979; Heltsche and Forrester, 1983; Smith and van Belle, 1984; Chao, 1987; Mingoti and Meeden, 1992; Bunge and Fitzpatrick, 1993). These estimation approaches are based on models similar to Mh, the closed-
population capture-recapture model permitting heterogeneous capture probabilities among individuals (see Section 14.2). The minimum value for these estimators is the total number of different species detected on all quadrats, and the distribution of individual species detections across the quadrats provides the information needed to estimate the total number of species (both detected and undetected) in the area. Computer simulations (Alpizar-Jara et al., in review) and empirical studies with communities of known richness (Palmer, 1990, 1991) support the use of the Burnham and Overton (1978, 1979) jackknife estimation procedure for model M h (see Section 14.2). Although other estimators have smaller root-meansquared errors for certain combinations of parameter values, the Burnham-Overton jackknife performs reasonably well over a wide range of parameter values (Alpizar-Jara et al., in review). The first-order jackknife estimator of Burnham and Overton (1978, 1979) [also discussed by Heltshe and Forrester (1983) specifically for the purpose of estimating species richness] is given by /r
S + ( t - 1)fl t '
(20.7)
where S is the number of species found as a result of sampling t quadrats, and fl is the number of species found on only one quadrat. In addition to Eq. (20.7), Burnham and Overton (1978, 1979) presented higher order jackknife estimators, as well as an algorithm for selecting the proper order and computing interpolated estimates. As noted in Section 14.2, the jackknife estimators are based on the seemingly weak assumption that the detection probabilities corresponding to the N different species in the community represent a random sample from some unspecified distribution. However, the jackknife estimators do require that the detection probability for a given species be the same on all sampled quadrats (hence the desirability of equal-size quadrats and similar sampling efforts in the different quadrats). An estimator proposed by Chao et al. (1992) in the capture-recapture context permits relaxation of the M h assumption of equal detection probabilities for all quadrats. In their model Mth (see Section 14.2), detection probability varies not only among different species, but also among the different quadrats (note that t denotes different sampled quadrats, rather than sampling periods). Such a situation could easily apply if randomly located quadrats are in different microhabitats. Computer program CAPTURE (Rexstad and Burnham, 1991) can be used to compute both the jackknife
20.2. Estimation of Species Richness estimator of Burnham and Overton (1978, 1979) for model Mh and the Chao et al. (1992) estimator of species richness under model Mth. The use of program CAPTURE for the purpose of estimating population size from animal capture-recapture data was described in Section 14.2. Here we briefly describe its use in estimating species richness with presence-absence data from quadrats. The relevant data for estimating species richness can be summarized in the form of an "X-matrix" (Table 20.1), containing one row for each species identified in any of the quadrats. The species name (analogous to an animal identification or tag number in the capture-recapture context) is followed by a string of ls and 0s indicating whether the species was found (1) or was not found (0) on a particular quadrat. Again, different quadrats are analogous to different trapping occasions in the capture-recapture context. Recall that program CAPTURE computes estimates under several different models, though we have discussed only the models Mh and Mth. These particular models are expected a priori to be most useful for estimating species richness from quadrat data, because they account for variable detection probabilities among different species. Of course, other models in program CAPTURE may be useful if detection probabilities are similar among species. The model selection procedure of CAPTURE can aid the investigator in selecting the most reasonable model for a particular data set. We recommend quadrat sampling only within areas for which the concept of an animal community makes sense. For example, an area encompassing an elevational or other habitat gradient might include different animal communities along the gradient. Under these conditions one should restrict the use of quadrat sampling to a part of the gradient with sufficiently homogeneous area to sustain a single community. This does
T A B L E 20.1
559
not mean, however, that species cannot exhibit spatial variation across the area, such that some species tend to be found in certain locations and not in others. The idea of animals being exposed to sampling efforts in some quadrats and not others may appear to violate the underlying assumption of community "closure." However, the key concept with quadrat sampling is that the quadrats are selected randomly from the area of interest. We view detection probability as the product of two components: (1) the probability that at least one individual of a species is present in at least one sampled quadrat and (2) the probability that at least one individual of a species is detected, given presence of the species on at least one sampled quadrat. Spatial variation in species distribution within the area of interest then becomes another source of heterogeneity in detection probability, but should not cause more serious problems.
20.2.2. Multiple Sampling Occasions 20.2.2.1. Single Investigator Instead of dividing the area of interest into quadrats and sampling a subset of these, an investigator may attempt to survey the entire area, but do so on multiple occasions (Fig. 20.2). As with quadrat sampling, the surveys may involve a variety of sampling methods, each leading to the identification of different species. The same methods should be used throughout an investigation, and the time between the first and last sampling occasions should be sufficiently short that the community composition (number and identity of species) is not expected to change. The data can be summarized in an X-matrix (Table 20.1), with columns now representing sample periods rather than sample quadrats.
S a m p l e X-Matrix for S p e c i e s List Data for U s e w i t h Program C A P T U R E to Estimate S p e c i e s R i c h n e s s a Quadrat b or sampling occasion c Species
1
2
3
4
Carolina chickadee, Poecile carolinensis
1
1
1
0
1
Tufted titmouse, Baeolophus bicolor
0
1
1
0
0
W h i t e - b r e a s t e d n u t h a t c h , Sitta carolinensis
0
0
0
1
0
Carolina w r e n , Thryothorus ludovicianus
1
1
1
0
1
Eastern bluebird, Sialia sialis
1
1
0
1
0
i
i
i
i
i
i
i
i
!
i
a See Rexstad a n d B u r n h a m (1991) for p r o g r a m CAPTURE. b Five different q u a d r a t s are s a m p l e d w i t h i n s o m e large area of interest. c The s a m e location is s a m p l e d on five different occasions b y the s a m e or different observers.
5
560
Chapter 20 Estimation of Community Parameters T1 T2 T3 T4 T5
FIGURE 20.2 Illustrationof a sampling design permitting estimation of species richness using temporal subsampling. Sampling produces species lists for each of five consecutive occasions (e.g., days, denoted as T1, T2..... T5). Under this kind of sampling, we believe it likely that detection probability for a particular species will increase following initial detection, because the observer will know what sign to look for, what specific areas to search, and so forth. If this change in detection probability indeed occurs, then the appropriate model is likely to be the generalized removal model Mbh (see Section 14.2) (see also Otis et al., 1978; Pollock and Otto, 1983). The critical data for this model are the number of different species first detected in each successive sample period (e.g., 1, 2 .... , K). The idea underlying the estimator for model Mbh (Section 14.2.3) is that the number of undetected species decreases over time, and the resulting change in number of detections of new species over time provides information about the number yet to be detected. Model Mbh permits variable detection probabilities among species, but assumes similar sampling effort on different sampling occasions. Two estimators for this model, developed by Otis et al. (1978) and Pollock and Otto (1983), are computed by program CAPTURE. Of course it is always possible that some other model is more appropriate for the data, and the model selection algorithm of CAPTURE (Rexstad and Burnham, 1991) should be useful for exploring that possibility. Data collection on multiple occasions produces data that sometimes have been referred to as "species accumulation curves," because species are "accumulated" over time. This name also has been used for sampling that involves increases in the number of species with increasing area sampled. Many different models have been used to describe species accumulation curves (Flather, 1996). Under the scheme of multiple temporal samples, we believe that model Mbh provides a natural way to model such data, and we recommend the approach as being substantially more flexible than other parametric approaches (Cam et al., 2001).
20.2.2.2. Multiple Investigators Another means of estimating species richness in the absence of quadrat sampling involves the use of multi-
ple investigators. For example, assume that a team of biologists is sent to estimate species richness for an area of interest. One biologist could sample the area of interest for 1 or 2 days using whatever methods he or she chooses and develop a species list. Then a second biologist could enter the area and develop a species list using his or her own sampling methods (not necessarily the same methods as those of the first biologist) for a day or so. These independently collected species lists (Fig. 20.2) then can be used to estimate species richness. Sampling data for such a study are of the same form as in Table 20.1, but with the columns of the X-matrix now corresponding to biologists. Model Mth (Chao et al., 1992) is most likely to be useful in this situation, because it permits variation in detection probability among different species and among different observers (i.e., there is no assumption of equal sampling efforts by the different observers). Estimates can be computed with program CAPTURE. If the biologists attempt to standardize their sampling by using the same methods and expending the same effort, it may even be possible to use the Mh estimators for such data.
20.2.3. Empirical Distributions of Species Abundance Instead of sampling at different points in space or time, an investigator can develop a species list based on catches or observations over the entire area of interest without regard to time or sampling occasion. If the number of different individuals encountered for each species can be recorded (e.g., animals are removed, or caught and marked for future recognition, or the area is traversed in such a manner as to ensure that no animals are counted twice), then the resulting data (number of individuals encountered for each species) can be viewed as an empirical species abundance distribution (Fig. 20.3). A variety of theoretical distributions have been considered as possible models of species abundance distributions (e.g., Engen, 1978). If the distribution underlying a particular data set is known, it frequently is possible to estimate species richness. Of course, the "true" underlying distribution is never known, and it often is difficult to distinguish which member of a set of competing models corresponds most closely to a particular data set. In fact, it is common for several different models to fit such data well but yield very different estimates of species richness [see Cormack (1979) in a capture-recapture context]. In addition, an empirical species abundance distribution usually differs substantially from the true species abundance distribution, in large part because of interspecific variation
20.3. Estimating Parameters of Community Dynamics
1
2
3
4
.
# Individuals
F I G U R E 20.3 Illustration of a sampling design permitting estimation of species richness based on an empirical species abundance distribution (number of species for which exactly one individual, exactly two individuals ..... are detected).
in detection probabilities. Thus, we do not recommend the use of theoretical species abundance distributions for estimating species richness or comparing richness in different communities. Burnham and Overton (1979) suggested a nonparametric approach to estimating species richness from empirical species abundance distribution data using a limiting form of their jackknife estimator under model Mh. They noted that in some studies there is no concept of trapping occasions (or sampling units). Thus, they considered the limiting value of their jackknife estimator as t (number of quadrats or sampling occasions) becomes infinite. The data required for an estimate of species richness with this approach are the total number of species encountered and the numbers of species for which 1, 2, 3, 4, and 5 individuals are encountered. Program SPECRICH selects the appropriate order jackknife (after Burnham and Overton, 1979) and computes the resulting estimate of species richness (Hines et al., 1999). 20.2.4. F i e l d S a m p l i n g
Recommendations
Few field methods are universally applicable for investigation of species richness, and the selection of a particular method should be dictated by factors such as investigator experience, prior knowledge of the fauna, and the nature of the sampled habitats. In recognition of the variety of field sampling methods, we emphasize a few general estimation approaches that require minimal assumptions. Under the quadrat approach, different quadrats should be sampled using a standard protocol if possible, so that Mh can be used.
561
If different biologists prepare species lists on the same general area, then the use of similar methods increases the potential applicability of M h. However, if sampling methods vary among different quadrats (or different biologists), then Mth can be used. If the entire area is surveyed by a single biologist (or team of biologists) on different sampling occasions, then similar sampling protocols should be used and the same effort expended on the different occasions, so that Mbh can be used. If the entire area is to be surveyed by a single biologist (or simultaneously by a team of biologists) to compile an empirical species abundance distribution for use with the limiting form of the M h estimator, then sampling methods must yield counts of individuals per species. It is important to avoid field methods that could lead to counting the same individual more than once. None of the estimation approaches described above requires the assumption that different species are encountered or detected with equal probability. This assumption almost certainly is not met when different methods are used to observe or catch different species. Even if attention is restricted to a subset of animals that are susceptible to the same sampling method, it is highly unlikely that different species will be sampled with equal probabilities. Though detection probabilities typically are ignored in community studies, when they are estimated it is common to find evidence of differences [e.g., capture probabilities for different species of small mammals (Nichols, 1986); detection probabilities associated with point counts of different species of birds (Nichols et al., 2000b)]. It thus is important to account for such a possibility in one's sampling and estimation approach.
20.3. E S T I M A T I N G
PARAMETERS
OF C O M M U N I T Y D Y N A M I C S With the exception of our own work (e.g., Boulinier et al., 1998b, 2001; Nichols et al., 1998a,b), incorporation of species detection probabilities has not extended beyond species richness to community dynamics. For example, some proposed methods for estimating species extinction rates and related metrics require that species are detected in different samples with certainty (e.g., Pimm et al., 1993; Clark and Rosenzweig, 1994; Rosenzweig and Clark, 1994; Burkey, 1995; Cook and Hanski, 1995). Work on rates of colonization and turnover is based on species counts over time (e.g., Hinsley et al., 1995), though these counts are subject to variable detection probabilities. As variability in detection probabilities constitutes a potentially important source of variation in species count data (Boulinier et al.,
562
Chapter 20 Estimation of Community Parameters
1998a), analyses that ignore it can lead to invalid inferences about species richness over time and space. Here we introduce some estimators that should be useful for studying changes of animal communities over time and space. Specifically, we consider estimation of rate of change in species richness, species extinction probability, species turnover rate, species colonization rate, and indicators of community similarity for two different locations. The estimators are based on capture-recapture methodology as described above for species richness. Community sampling procedures and capture-recapture models for richness estimation should be useful for addressing questions about community dynamics over space and time.
20.3.1. General Approach 20.3.1.1. Pollock's Robust Design
It is reasonable to think of a sampled community as being "closed" to local extinction and colonization for the relatively short periods over which species presence-absence data are collected. However, in order to estimate quantities concerning community change over time or space, we require a more general approach that permits change in the community between sampling periods. Standard capture-recapture models for open populations (e.g., Pollock et al., 1990; Lebreton et al., 1992) allow for change, but do not deal adequately with heterogeneous detection probabilities. Thus, we focus below on the robust design of Pollock (1982) (see Chapter 19) as a means of developing robust estimators for quantities associated with community change. Pollock's (1982) robust design involves sampling at two different temporal scales and is applied here to a single location. Thus, the primary sampling periods
Time 1
are separated by times that are sufficiently large to allow changes in the community from one primary period to the next. For example, we might compute quantities relevant to bird community change using primary sampling periods spaced 20 years apart, with some number of secondary spatial samples (Fig. 20.4) or secondary sampling periods chosen within each primary period (see Section 20.3.2). Numerous approaches are available for estimating quantities of interest from data collected under the robust design (e.g., Kendall and Pollock, 1992; Nichols et al., 1992b, 1994; Kendall et al., 1995, 1997). In their application to community dynamics, the parameter estimates are based on closed-model estimators of species richness computed over the secondary samples within each primary sampling period. Then estimates of parameters relevant to community change can be computed as functions of the closed-model richness estimates. In addition to community dynamics at a single location, one often is interested in spatial variation. In what follows, Section 20.3.3 deals with geographic variation at a single point in time, and Section 20.3.4 addresses both temporal and geographic variation. For quantities associated with spatial variation, we extended Pollock's robust design to allow the primary sampling to be conducted at different locations in space rather than at different times (Nichols et al., 1998b). At each location, secondary sampling is conducted, which might involve (1) spatial or quadrat sampling within each primary area (Fig. 20.5), (2) short-term temporal sampling within each primary area (Fig. 20.6), or (3) the collection of an empirical species abundance distribution at each primary location (Fig. 20.7). Estimation is based on closed-model estimators for species richness, computed over secondary samples within each primary sample location.
Time 2
FIGURE 20.4 Illustration of a sampling design permitting inferences about community change between two sampling times (primary sampling periods). The design uses spatial subsampling, with quadrats (TILi, T2Li) representing the secondary sampling units of the robust design. Sampling produces species lists for each quadrat, and these lists can be used to compute estimates of interest.
20.3. Estimating Parameters of Community Dynamics
Area A
563
Area B
FIGURE 20.5 Illustration of a sampling design permitting inferences about differences between two communities inhabiting areas A and B (primary sampling areas). The design uses spatial subsampling, with quadrats (ALl, BLi) representing the secondary sampling units of the robust design. Sampling produces species lists for each quadrat, and these lists can be used to compute estimates of interest.
20.3.2. Temporal Variation at a Single Location
20.3.1.2. Variance E s t i m a t i o n
In this development we present only point estimators for quantities of interest in community dynamics. Because estimates of parameters relevant to community change are computed as ratios or other functions of closed-model richness estimates, it is possible to use the delta method (see Appendix F) and other approximations to obtain estimators of associated variances. However, these approximations are necessarily based on variance estimates for the closed-model richness estimators, e.g., the variance estimators for the Mh jackknife estimates of richness. Because the latter estimators do not always perform well (e.g., Burnham and Overton, 1979; Otis et al., 1978), approximations that are functions of them are not expected to perform well. For this reason, we have chosen to compute bootstrap variance estimates for all of the estimators presented below. An outline of computational methods for the bootstrap estimators of variance is presented by Nichols et al. (1998a) (see also Appendix F).
The problem addressed here is to characterize changes in a community over time based on Pollock's (1982) robust design. The necessary data are obtained from sampling a single location at two different times (i.e., two different primary sampling periods). The attributes of interest include change in species richness, extinction rate, species turnover, and colonization rates. The primary reference for this material is Nichols et al. (1998a).
20.3.2.1. Rate of Change Rate of change in species richness between two sampling periods i and j can be estimated as ~kij = l~j/l~i,
(20.8)
where N k denotes species richness at time k (typically the index j corresponds to the more recent of the time periods). The estimates of species richness are obtained
AT1 AT2 AT3 AT4 AT5
BTI BT2 BT3 BT4 BT5
Area A
Area B
FIGURE 20.6 Illustration of a sampling design permitting inferences about differences between two communities. Sampling of the communities is conducted in locations A and B. Design uses temporal subsampling; species lists AT1 ..... Av3 are prepared on five consecutive occasions at sampling location A, and BT1..... BT5 at sampling location B.
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Chapter 20 Estimation of Community Parameters
1
2
3
4
.
.
.
# Individuals
1
2
3
4
.
# Individuals
Area A
Area B
F I G U R E 20.7 Illustration of a sampling design permitting inferences about differences between two communities. Sampling of the communities is conducted in locations A and B but without any spatial or temporal subsampling. Instead, an empirical species abundance distribution (number of species for which exactly one individual, exactly two individuals .... are detected) is obtained at each site and forms the basis for estimation using the limiting form of the Burnham and Overton (1979) jackknife estimator.
using species detection data in conjunction with closed-model capture-recapture estimators such as those for M h. If Pi = Pj, that is, the average species detection probabilities are the same for the two periods, then an alternative estimator for rate of change is
Kij = Rj/Ri,
(20.9)
where Rk denotes the number of species actually observed during sampling efforts in period k. Estimators such as Eq. (20.9) are based on the species counts, and therefore should have smaller variances than estimators such as Eq. (20.8) that are based on estimated quantities (e.g., Skalski and Robson, 1992). However, the estimator in Eq. (20.9) is biased if Pi ~ Pj" Under model Mh the hypothesis of equal average detection probabilities for two samples can be tested using frequency data fh (i.e., the number of species detected in exactly h secondary samples). Thus, a 2 • l contingency test can be used to test the null hypothesis that proportions of species found in h = 1, ..., l secondary samples are similar for the two primary sampling periods i and j (Skalski and Robson, 1992). Note that the rate of change in species richness concerns only species numbers and does not provide information about possible changes in species composition (the identities of the species). However, estimators of local extinction, turnover, and colonization do permit inferences about changes in community composition.
In addition to estimating rate of change in species richness, there may be interest in estimating temporal variance in species richness as a metric reflecting community "stability" (Boulinier et al., 1998b). However, measures of temporal variance based on numbers of species detected are likely to be biased because (1) the richness metrics are negatively biased and (2) the variance of these metrics reflects both true variation in richness and sampling variation associated with detection probabilities less than 1. If richness is estimated using capture-recapture models, then the average sampling variance can be subtracted from the variance computed from point estimates of richness, to estimate true spatial or temporal variance in richness (Boulinier et al., 1998b; also see Appendix F). 20.3.2.2. Local Extinction Probability We define local extinction probability as the probability that a species present in the community during primary sampling period i is not present at some later period j. We can use the estimation approach of Kendall and Pollock (1992) and Nichols et al. (1992) to estimate this quantity, using the reasoning underlying open capture-recapture survival estimators (Jolly, 1965; Seber, 1965). If R i is the number of species observed in period i and (Mj[R i) is the number that are still present in period j, local extinction probability is estimated as 1
-
~ij
=
1 - (]~'ijlai)/ai,
(20.10)
20.3. Estimating Parameters of Community Dynamics where q~/j is the probability that a species present in period i is still present in period j. The estimation of (MjlR i) is based on closed models with species occurrence data from period j and is obtained with either of two approaches. The most conservative approach is to use only the occurrence data for period j for species also observed at period i [i.e., use the members of R i that are also seen at time j, (mjlRi) , with a closed population estimator to estimate (Mjt Ri)]. The second approach requires the additional assumption that the ( M j l R i) species present in both i and j, and the N i - ( M j ] R i) species present in period j but not in period i, share the same average species detection probabilities at time j. If this assumption holds, then we can estimate the number of R i that are still present at j by
(l~[j l a i) - (mj l ai)/Pj,
(20.11)
where pj is the average detection probability of all species present at j. The value pj is estimated with occurrence data from all species observed at period j(Rj), in conjunction with the use of program CAPTURE to estimate total species richness at j(Nj):
pj = Rj/
.
(20.12)
The first approach for estimating ( M j l R i ) should have smaller bias but larger variance, and the second approach should have larger bias but smaller variance. A decision about the appropriateness of the second approach can be based on a test for equality of the average detection probabilities for species present in primary period j that were detected and undetected in period i. This test can be conducted with a 2 • 5 contingency table of the values of fh for the two groups of species detected in primary period j, i.e., those detected in primary period i, (mjl Ri) , and those not detected in primary period i [Rj - (mjlRi)]. The resulting chi-square statistic provides a test of the null hypothesis of equal proportions of species detected from the two groups. However, the group Rj - (mj IRi) is likely to be relatively small, and the contingency test for equal detection probabilities is not likely to be very powerful. We therefore recommend the use of the first approach for estimating ( M j l R i) in Eq. (20.10). 20.3.2.3. Local Species Turnover
The literature of community ecology contains a number of definitions of turnover. Most are inadequate, in that they are based on the results of sampling rather than a specified underlying parameter of interest. Here we follow an approach adopted in previous work with fossil data (Nichols et al., 1986a), by defining turnover between two times i and j (i < j) as the proba-
565
bility that a species selected at random from the community at time j is a "new" species (not present in the community at time i). This turnover parameter, which arises naturally in the modeling of capturerecapture data (Pollock et al., 1974; Pradel, 1996; see also Section 17.4), is a function of rates of extinction and colonization and reflects dissimilarity between communities at two different points in time. The parameter achieves a maximum value of 1 when all the species present in period j are new (not present at i) and a minimum value of 0 when all the species present at j are survivors from period i. Estimation of turnover is based on an observation by Pollock et al. (1974) about the temporal symmetry of capture-recapture data for open populations (also see Section 17.4). Specifically, if capture history data are viewed in reverse time order (treating the final sampling period K as the initial period and sampling period 1 as the final period), then the standard Jolly-Seber estimator for survival between two periods estimates the fraction of species in the more recent period that are "old" or survivors from the previous period. The complement of this estimator estimates the proportion of species that are new and were not present in the previous period (Pollock et al., 1974; Nichols et al., 1986a, 2000a; Pradel, 1996, see Section 17.4). The estimation of community turnover thus is accomplished by using the extinction probability estimator of Eq. (20.10) with data placed in reverse time order. The notation is the same as that for the survival estimator [Eq. (20.10)], except for a change in the subscripting to denote the change in temporal ordering. We estimate turnover as 1
-
~ji-
1 - (iVIi I Rj)/Rj
(20.13)
for i < j. The key to the estimation involves (~/[jlRi), the estimated number of species present in period i that later are observed in period j. Estimation is accomplished as with extinction probability, by conditioning on the subset of species actually observed at j and then estimating the number of these that previously were present in i. As was the case with extinction, the estimation of (Mil Rj) can be accomplished in either of two ways. The most conservative approach, requiring fewest assumptions, uses species occurrence data for the members of Rj that were also observed in period i. The other approach involves estimating average detection probability Pi for all species observed at i and then applying this estimate to the number of species observed at i that were also later observed at j [see Eqs. (20.11) and (20.12)]. As was noted in the discussion of extinction probability estimators, the first approach should have smaller bias and larger variance than the ^
566
Chapter 20 Estimation of Community Parameters
second approach. At this point we recommend estimating turnover based on the first approach. Although we advocate the use of expression (20.13) for the study of community turnover (Nichols et al., 1986a; 1998a), we note that many other turnover statistics have been suggested. It is possible to use the general ideas on richness estimation presented here to compute improved (in the sense of dealing with p 1) versions of these other metrics. For example, the widely used turnover index of Diamond (1969) can be computed readily using the estimates of species richness, number of local colonists, and local extinction probability presented above.
20.3.2.4. Number of Local Colonizing Species Denote as Bij the number of species not present in a local area at time i, that colonize the area between times i and j and are still present at time j. This quantity can be estimated using either of two approaches. The first approach is taken directly from the robust design estimator for number of new recruits entering the studied population between two sampling periods (Pollock, 1982; Pollock et al., 1990). Thus, the estimated number of surviving species from a previous time i is subtracted from the estimated species richness at time j:
Bij = N j - ~ijlCqi,
(20.14)
where ~ij denotes the probability that a species present at time i is still present at j. Estimation of species richness and local species survival probability in Eq. (20.14) have been discussed above. Although Eq. (20.14) is the community analog of the standard approach to estimating recruitment in a single-population capture-recapture study, another approach may be more efficient. Thus, the number of colonizing species can be estimated as the product of species turnover (proportion of all species present at time j that were not present at previous time i) and species richness (total number of species at time j):
eij - (1 - ?Pji)l~j.
(20.15)
The estimator of Eq. (20.15) may perform better than the estimator of Eq. (20.14) because Eq. (20.15) requires an estimate of species richness from only a single time period, whereas Eq. (20.14) requires richness estimates from two periods. 20.3.2.5. Annual Extinction and Recolonization Probabilities
Though the estimator I - q~qin Eq. (20.10) estimates the probability that a species present at time i is absent at some later time j, it says nothing about the detailed
process leading to the extinction event. It sometimes is of interest to distinguish between different processes underlying community change. Thus, in order to survive between two primary periods i and j, a species present at i may survive every year between i and j, or it may go locally extinct and then recolonize during this period. For example, consider the changes within a bird community between years i and i + 2. Then ~i,i+2 --
~i,i+lq~i+l,i+2 q- (1
-
~i,i+1)'~i+1,i+2,
(20.16)
"~i+1,i+2 is the probability that a species present in the community at time i but not at time i + 1 recolonizes the area during the interval (i + 1, i + 2) and is present in the community at i + 2. The first term on the right-hand side of Eq. (20.16) corresponds to population survival over both years, and the second term corresponds to local extinction and recolonization. Estimates of each of the three survival probabilities in Eq. (20.16) can be obtained via the general estimator presented in Eq. (20.10) and substituted into Eq. (20.16) to obtain where
~i,i+2- (~i,i+l~Pi+l,i+2) q/i+1,i+2 "-
(20.17)
1 -- ~i,i+1
(]~4i+2 ] R i) Ri
t 1-
(A~i+l I
ai)
Ri
Parameterizations such as that in Eq. (20.16) may prove useful for the estimation of community-level quantities with open models, which utilize capturehistory data with row vectors of ls and 0s, representing detection and nondetection, respectively. In the usual capture-recapture framework, a "0" appearing between ls (e.g., the 0 in capture history 101) indicates an animal that was present but not caught in the sampling period. However, in community studies, an interior 0 in a detection history can indicate either of two events: (1) a species was present but not detected or (2) a species was not present (locally extinct) yet recolonized at a later time. In this respect, the modeling of species detection history data in the presence of local extinction and recolonization is similar to the modeling of capture-recapture data in the presence of temporary emigration. The robust design (Pollock, 1982) provides the information needed to estimate quantities of interest in the presence of temporary emigration (Kendall et al., 1997) and may provide a basis for estimating community parameters from detection history data using parameterizations such as that of Eq. (20.16).
20.3.2.6. Example Analyses We illustrate some of our estimators with data collected as part of the North American Breeding Bird
20.3. Estimating Parameters of Community Dynamics TABLE 20.2
567
Species Detection Statistics for Maryland BBS Route 25, 1970 and 1990 Species detected on exactly h of the five groups of stops (fh)
Species group
Species detected
fl
f2
f3
f4
f5
Total species detected (1970), R70 Total species detected (1990), R90 Members of R70 detected in 1990, (m90 I R70) Members of R90 detected in 1970, (m70 I R90)
65 55 48 48
15 15 10 7
8 8 6 6
10 13 13 6
12 12 12 9
20 7 7 20
Survey (BBS). This survey is carried out every spring on permanent survey routes that are randomly located along secondary roads throughout the United States and southern Canada. Each route is 39.4 km long and consists of 50 stops spaced at 0.8-km intervals. Because the stops are not randomly selected from some larger area of interest, we view the area of interest as the area covered by the series of 50 circles, each with a radius of 0.4 km. The observer drives along the route, exiting the vehicle at each stop to record all birds seen and heard within 0.4 km of the stop during a 3-minute observation period (Robbins et al., 1986; Peterjohn and Sauer, 1993). In the BBS files, data are summarized by groups of 10 stops. Hence, there are five s u m m a r y records for each survey route, and for each s u m m a r y record (each group of 10 stops) there is a species list and the number of individuals counted for each detected species. For the examples reported here, the data used to compute estimates on each survey route are simply the species lists for each of the five groups of stops. We thus treat each group of 10 stops along a survey route as a "quadrat" that samples the area covered by the entire survey route. These five quadrats are the secondary samples of our robust design approach. Though it is possible to apply our methods to any subset of total species (e.g., defined by taxonomy, foraging habit), here we include all avian species. All computations were conducted using program COMDYN (Hines et al., 1999). Given the general applicability of model Mh to BBS data (Boulinier et al., 1998a), all the estimators in program COMDYN utilize the jackknife estimators of Burnham and Overton (1978,
TABLE 20.3
1979), with variance estimation accomplished via the bootstrap approach outlined by Nichols et al. (1998a). COMDYN also includes goodness-of-fit tests of the detection frequency data, and tests of the null hypothesis that two sets of detection frequency data are produced by the same average detection probability. We selected Maryland route 25 and Wisconsin route 1 from the BBS (Nichols et al., 1998a) and computed estimates for community dynamics between 1970 and 1990. The raw data (Tables 20.2 and 20.3) include the number of species detected on each of the five route segments, for all species observed in 1970 (R70) and 1990 (R90), all species observed in 1970 that were detected in 1990 (m90 I R70), and all species observed in 1990 that were detected in 1970 (m70 ] R90). Model Mh adequately fit all four sets of frequency data in Table 20.2 for Maryland BBS route 25 (P > 0.10). The richness estimate for 1990 was smaller than that for 1970, although the 95% confidence intervals for the two estimates overlapped substantially (Table 20.4). The estimated rate of change based on these richness estimates was 0.88, but the confidence interval for rate of change included values >1.0. The average detection probability of 0.83 for 1970 and 0.79 for 1990 provided little evidence of a difference (X2 = 5.86, P = 0.11), justifying the use of Eq. (20.9) for estimating rate of change. The resulting rate estimate of 0.85 was similar to that based on Eq. (20.8) but was substantially more precise. The confidence interval for the estimate was (0.73, 1.0) (Table 20.4), providing evidence that avian richness declined on the route between 1970 and 1990.
Species Detection Statistics for Wisconsin BBS Route 1, 1970 and 1990
Species group
Species detected
Total species detected (1970), R70 Total species detected (1990), R90 Members of R70 detected in 1990, (m90 I R70) Members of R90 detected in 1970, (m70 I R90)
66 80 57 57
Species detected on exactly h of the five groups of stops (fh) fl
f2
f3
f4
f5
17 23 9 10
19 15 10 17
9 9 6 9
10 12 11 10
21 21 11
11
568
Chapter 20 Estimation of Community Parameters TABLE 20.4
Estimates of Quantities Associated with Community Dynamics Based on Avian Species Seen on Maryland BBS Route 25, 1970 and 1990 Naive
Quantity (0)
"estimates "a
Estimator
Species richness (1970)
65.0
Species richness (1990)
55.0
95% c o n f i d e n c e interval
0
S-E(0)
N70
78.5
10.4
66.9-104.1
/~/90
69.4
11.1
55.6-94.3
^
Members of R70 present in 1990 b
48.0
(/~90 I R70)
54.7
13.2
35.0-86.0
Members of R90 present in 1970 b
48.0
(/~70 I R90)
51.1
6.9
38.5-67.1
Complement of extinction probability
0.74
q~70,90
0.84
0.15
0.54-1.00
Complement of turnover
0.87
q~90,70
0.93
0.09
0.70-1.00
Rate of change in richness [Eq. (20.8)]
0.85
K70, 90
0.88
0.18
0.61-1.28
Rate of change in richness [Eq. (20.9)]
0.85
K70, 90
0.85
0.07
0.73-1.00
Number of colonizing species [Eq. (20.14)]
870, 90 -~ P70
3.3
Average detection probability (1970)
7.0 1.0 c
0.83
0.10
0.62-0.97
Average detection probability (1990)
1.0 c
-~ P90
0.79
0.11
0.58-0.99
12.2
a "Estimates" based on the assumption that all species are detected. b Confidence intervals for (l~/ljla i) can include values <(mjIRi), because of variation associated with the extinction process:
0.0-40.9
(MjlR i) --,
B(R~, ~ij). c Perfect observability by assumption.
The complement of extinction probability for Maryland route 25 was estimated to be 0.84, and the 95% confidence interval included 1.00 (Table 20.4). The complement of turnover indicated that an estimated 93% of the species present in 1990 were also present in 1970, reflecting an estimated species turnover of 7% (Table 20.4). Consistent with this fairly low turnover, the estimated number of new species colonizing between 1970 and 1990 and present in 1990 was small (fewer than five species) for both estimators of B70' 90 (Table 20.4). In general there was weak evidence of a decline in species richness on this route, with the number of colonizing species not quite balancing the number of local extinctions. Model M h adequately fit (P > 0.10) all of the data sets in Table 20.3 for Wisconsin route 1, 1970 and 1990. Estimated species richness was greater in 1990 than 1970, and there was little overlap between the respective confidence intervals (Table 20.5). Both estimates for rate of change in species richness were >1.2, and neither confidence interval included 1.0 (Table 20.5), providing evidence for an increase in richness between 1970 and 1990. The test for similar distribution of detection frequencies (X2 = 3.37, P = 0.50) provided no evidence for different detection probabilities between 1970 (estimate, 0.85) and 1990 (estimate, 0.80). The estimate for q~70,90 reflected only a 7% local extinction probability between 1970 and 1990, but the estimate for q~90,70 indicated that 22% of the species present in 1990 were new (not present in 1970). Similarly, both estimators [Eqs. (20.14) and (20.15)] indicated about 28
species as local colonists between 1970 and 1990 (Table 20.5). Thus, the data for this route indicated an increase in the number of species, with the number of local colonists exceeding the number of local extinctions. The naive "estimates" presented in Tables 20.4 and 20.5 represent the common approach to treating such species list data, based on the assumption that detection probabilities are 1 (i.e., all species present are assumed to be detected). But these naive "estimators" are negatively biased for species richness and positively biased for extinction probability and turnover. Bias for estimators of rate of change in richness and number of colonizing species can be either positive or negative, depending on the specifics of the situation. 20.3.3. G e o g r a p h i c V a r i a t i o n at a Single Time
Many of the parameters discussed above have spatial analogs that may be useful in addressing questions about geographic variation in species richness and community structure. In this section we define these parameters and provide their respective estimators (also see Nichols et al., 1998b). As in the previous section, we use subscripts to denote time and include superscripts here to denote geographic location. 20.3.3.1. Relative Species Richness
We define relative species richness for two locations x and y at time i, as )~xy = N ~ / N x, (20.18)
20.3. Estimating Parameters of Community Dynamics TABLE 20.5
569
Estimates of Quantities Associated with Community Dynamics Based on Avian Species Seen on Wisconsin BBS Route 1, 1970 and 1990 Naive
Quantity (0)
"estimates "a
Estimator
Species richness (1970)
66.0
N70
Species richness (1990)
80.0
/~/90
Members of R70 present in 1990 b
57.0
(/~90 [ R70)
Members of R90 present in 1970 b
57.0
(](/I70 ] R90)
S"E(0)
95% c o n f i d e n c e interval
77.2
5.5
68.6-89.8
99.9
11.4
85.5-129.9
61.6
7.7
47.2-74.4
62.5
13.8
39.4-84.7
0
^
Complement of extinction probability
0.86
q~70,9o
0.93
0.09
0.72-1.00
Complement of turnover
0.71
~90, 70
0.78
0.16
0.49-1.00
Rate of change in richness [Eq. (20.8)]
1.21
~70, 90
1.29
0.17
1.06-1.70
Rate of change in richness [Eq. (20.9)]
1.21
~,70, 90
1.21
0.08
1.05-1.37
Average detection probability (1970)
13.0 1.0 c
B70, 90 -~ P70
Average detection probability (1990)
1.0 c
-~ P90
^
Number of colonizing species [Eq. (20.14)]
27.9
14.1
8.4-62.7
0.85
0.06
0.73-0.96
0.80
0.08
0.61-0.94
"Estimates" based on the assumption that all species are detected. b Confidence intervals for (I~j]R i) can include values ((mj]Ri), because we consider variation associated with the extinction process:
a
(Mj ]Ri) "" B(Ri, ~ij). c Perfect detectability assumption.
where N/y denotes species richness at location y in time i. Relative richness can be estimated as the ratio of richness estimates from the two locations: ~xy = / ~ / / ~ x .
(20.19)
As with the rate of increase in species richness, relative species richness can be estimated more efficiently as the ratio of observed species, when average detection probabilities are equal for the two areas, i.e.,
~xy = RYi / R x
(20.20)
if p~ = ~x. As noted in the discussion of Eqs. (20.8) and (20.9), a 2 x l contingency table based on the frequencies fh of the two samples can be used to test the null hypothesis of equal species detection probabilities for the two locations. Note that l is the number of secondary samples, assumed equal for the two locations. Spatial variance in species richness has been proposed as a metric useful in testing the so-called "niche limitation" hypothesis (e.g., Palmer and van der Maarel, 1995; Wilson, 1995) and in estimating the degree of "saturation" of communities on groups of islands (MacArthur and Wilson, 1967). However, measures of spatial variance based on numbers of species detected are likely to be biased because (1) the richness metrics are negatively biased and (2) the variances of these metrics reflect both true variation in richness and sampling variation associated with detection probabilities. If richness is estimated using capture-recapture models, then true spatial variance in richness can be
estimated by subtracting the average sampling variance from the variance computed from point estimates of richness (Appendix F). Spatial variance in population size has been investigated in this manner by Skalski and Robson (1992).
20.3.3.2. Species Co-occurrence In some cases it is of interest to ask what proportion of species found at one location are found at a second location. For example, one location may have experienced an environmental change (human-induced or otherwise), and it may be of interest to know the proportion of the species found at another unaffected site, or in a species pool, that also is found in the affected location (Cam et al., 2000a). We can use an analog of the survival/turnover estimators of Eqs. (20.10) and (20.13) for this purpose. Define q~xyas the probability that a species present at location x in time i is also present at location y at that time. We can estimate this probability as
~xy = (l~4Yi l RX)/R x,
(20.21)
where (M~]R x) denotes the number of species observed in location x at time i (i.e., members of R x) that also are present in location y at that time. The number (M~ I RX) of shared species is estimated by focusing on the species observed at location x at time i (the members of R x) and then using occurrence data for these species at location y with one of the
Chapter 20 Estimation of Community Parameters
570
richness estimators. We currently recommend this approach for general use. The alternative approach of estimating average detection probability based on all of the species [Eq.(20.12)] and then dividing this estimate into the number of the species observed at both y and x [Eq. (20.11)] may also be useful in some cases [see discussion of Eqs. (20.11) and (20.12)]. Expression (20.21) and associated estimators have been used to estimate the species richness of different areas relative to respective species pools (mj]Ri). For example, an investigation of avian communities in Maryland provided evidence of lower relative richness in areas with more urban habitat (mj]R i) (Cam et al., 2000a). Estimator (20.21) has also been recommended for use in studies of community saturation and ecological integrity. Historical investigations of community-level spatial dynamics have used similarity measures such as Whittaker's Coefficient of Community (Whittaker, 1975; Farley et al., 1994). This quantity (and related statistics) can be readily computed from estimates of species richness and the proportion of species present at location x that are also present at location y [Eq. (20.21)]. However, the use of capture-recapture estimates to compute such indices avoids the assumption of unit species detection probabilities and also provides an estimate of sampling variance. This approach thus should constitute an improvement over previous methods of determining the indices.
20.3.3.3. Number of Species at One Location That Do Not Occur at Another Location Some definitions of colonization rate are conditional on species that are present in location y at time i, but not present at location x at that time (denote this number as Bxy). These species represent potential colonists of location x (see below). Estimation of this quantity proceeds as for the Bij in the previous section. One such estimator is /~xy =/~/y_ q~xY/~i,
T A B L E 20.6
(20.22)
and an alternative estimator is /~xy = (1 - q~/yx)/~.
(20.23)
Equations (20.22) and (20.23) can be thought of as spatial analogs of Eqs. (20.14) and (20.15), respectively. The estimators provided in Eqs. (20.21)-(20.23) should prove useful in tests involving community nestedness and nested subset analysis (e.g., Patterson and Atmar, 1986; Lomolino, 1996; Worthen, 1996). Tests about these features seem especially vulnerable to misleading inferences when based on raw count statistics for which detection probabilities are less than unity. An example of the application of these estimators to nested subset analysis was presented by Cam et al. (2000b).
Example To illustrate the estimators for geographic variation, we selected two North American Breeding Bird Survey (Robbins et al., 1986; Peterjohn and Sauer, 1993) routes in Maryland that were fairly close geographically, but that represented substantial differences in the relative degree of urbanization (see Nichols et al., 1998b). We used digital land-use and land-cover data from the United States Geological Survey (U.S. Department of the Interior, Geological Survey, 1987) to estimate the proportion of area surrounding the BBS routes that was "urban" (C. Flather, personal communication; Flather and Sauer, 1996). During the mid-1970s, 23% of BBS route 29 was characterized as urban, whereas only 11% of BBS route 31 was considered urban. We computed estimates of quantities reflecting geographic variation between these two routes using species detection data for the year 1974 (Table 20.6). The species richness estimate for route 29 was about 65 avian species, whereas that for route 31 was 83 species, although the 95% confidence intervals overlapped (Table 20.7). The relative richness estimate ~29,31, computed directly from the richness estimates, was 1.28 (i.e., route 31 had an estimated 28% more species than route 29), but the 95% confidence interval
Species Detection Statistics for Maryland BBS Routes 29 and 31, 1974 Species detected on exactly h of the 5 groups of stops (fh) Species detected
fl
f2
Total species detected (rt. 29), R249
57
13
13
Total species detected (rt. 31), R341
69
19
15
Members of R249detected on Ft. 31, (m31]
50
7
9
Members of R 31 detected on rt. 29,
50
8
11
Species group
R74)29 (m249i R74)31
f3
f4
f5
8
7
16
17
13
5
16
13
5
8
7
16
20.3. TABLE 20.7
Estimating Parameters of Community Dynamics
571
Estimates of Quantities Relevant to Comparative Species Richness on Maryland BBS routes 29 and 31, 1974
Quantity (0)
Estimator
0
S"E(0)
95% confidence interval
Species richness (rt. 29) Species richness (rt. 31) Members of R249present on rt. 31 Members of R31 present on rt. 29 Proportion of rt. 29 species present on rt. 31 Proportion of rt. 31 species present on rt. 29 Relative species richness, rts. 29 and 31 [Eq. (20.19)] Relative species richness, rts. 29 and 31 [Eq. (20.20)] Speoes present on rt. 31 but not on rt. 29 [Eq. (20.22)] Speoes present on rt. 31 but not on rt. 29 [Eq. (20.23)] Speoes present on rt. 29 but not on rt. 31 [Eq. (20.22)] Species present on rt. 29 but not on rt. 31 [Eq. (20.23)] Average detection probability (rt. 29) Average detection probability (rt. 31)
" 29 N74 " 31 N74 " 31 29 (M74 ] R74)
65.1 83.0 52.9 53.7 0.93 0.78 1.28 1.21 22.7 18.4
5.7 10.2 7.0 12.2 0.10 0.17 0.19 0.08 12.8 15.0
57.6-81.0 71.4-112.2 39.5-63.3 33.3-73.6 0.69--1.00 0.47--1.00 0.99-1.74 1.07-1.38 5.0-55.2 0.0-48.6
"31 29 B74'
0.5
9.9
0.0-31.1
"31 29 B74'
4.7
7.0
0.0-21.7
"29 P74
0.88
0.07
0.69--0.99
"31 P74
0.83
0.09
0.61-0.97
" 29 31 (M74 ] R74)
for this q u a n t i t y included values slightly less than 1. The point estimates of detection probability for the two routes differed s o m e w h a t (Table 20.7), a n d the test for similar distribution of detection frequencies p r o v i d e d evidence of different detection probabilities (X24= 11.02, P = 0.03). Because of this difference in detection probabilities, w e d i s r e g a r d e d the relative richness estimator in Eq. (20.20) based on the n u m b e r s of species observed. The estimated fraction of route 29 species that were present on route 31 w a s 93%, w h e r e a s the fraction of route 31 species found on route 29 w a s only 78% (Table 20.7). So an estimated 22% of the species present on route 31 in 1974 were not present on route 29. Similarly, the two estimates of u74/~29'31indicated about 20 species present on route 31 that were not present on route 29, w h e r e a s fewer than five species were estimated to be present on route 29 and not also on route 31 (Table 20.7). Thus, the estimates of relative richness, proportion of shared species, and n u m b e r of species present on one route but not the other all indicated that the route with the smaller fraction of u r b a n area (route 31) had greater species richness, and a larger n u m b e r of species that were not found on the more u r b a n route, than vice versa. 20.3.4. V a r i a t i o n o v e r T i m e a n d S p a c e
20.3.4.1. Colonization R a t e Define ~/~xas the probability that a species in location y, but not location x, at time i, colonizes location x
",29 31 q~74' ,,31 29 q074' " 29 31 )k74' "29, 31 ~k74 "29 31 B74' "29 31 B74'
b e t w e e n times i and j and is still present in x at time j. This p a r a m e t e r can be estimated using estimates of the t e m p o r a l [Eqs. (20.14) and (20.15)] a n d spatial [Eqs. (20.22) and (20.23)] B quantities: ~/~x = / ~ / ~ x y .
(20.24)
The estimator in Eq. (20.24) a s s u m e s that all n e w species colonizing area x b e t w e e n p r i m a r y s a m p l i n g periods i and j were m e m b e r s of the species pool (defined as y) in period i (i.e., the estimator a s s u m e s that the species pool is the source of all colonizing species). A l t h o u g h this kind of colonization rate is relevant to a n u m b e r of interesting h y p o t h e s e s in c o m m u n i t y ecology, w e suspect that the estimator in Eq. (20.24) will usually have a large variance.
20.3.4.2. Relative Change T h o u g h the relative richness estimators presented in Eqs. (20.19) and (20.20) concern two locations at a single point in time, it is also possible to consider relative change b e t w e e n two points in time for two different locations. Let 0~. denote some p a r a m e t e r (e.g., rate of change )~. in species richness, or the c o m p l e m e n t q~. of species extinction probability) associated with c o m m u n i t y change b e t w e e n two p r i m a r y s a m p l i n g periods i and j, at location x. Then w e can estimate the relative rate of c h a n s e b e t w e e n two time periods for two locations as 6/~/0,~. Such an estimator m a y be of use, for example, w h e n e n v i r o n m e n t a l change has occurred b e t w e e n periods i a n d j on one area but not the
572
Chapter 20 Estimation of Community Parameters
other, and attention is directed at relative changes in the respective animal communities over this same period.
20.3.5. Assumptions At the time the above community-dynamic estimators were being developed (Nichols et al., 1998a,b), it was recognized that the effects of heterogeneity on estimators such as local extinction probability, local turnover, and proportion of species could be a potentially serious problem. The methods proposed above were developed specifically to deal with heterogeneity of detection probabilities; however, one must recognize that the probability of detection of a species depends on the number of individuals representing that species (see Section 20.1.3). Unfortunately, local extinction probability and the probability of having been present at some previous time period also are likely to depend on the number of individuals in the species. Investigations with a variety of different population-dynamic models suggest that extinction probability decreases as the number of individuals increases (Bailey, 1964; MacArthur and Wilson, 1967; Goel and Richter-Dyn, 1974). In fact, this relationship is the basis for the idea of a minimum viable population size (Section 11.2.1) (see Gilpin and Soule, 1986; Boyce, 1992; Burgman et al., 1993). Our estimators for local species survival q~are conditioned on the number of species actually observed at a specific period or location [Eqs. (20.10), (20.13), and (20.21)], and they essentially estimate how many of these species are present at a different time or location. If detection probability for a species is closely tied to number of individuals in that species, then the species on which one conditions the estimates (e.g., the members of R i) will tend to have more individuals, on average, than species that are present but not observed (e.g., N i - Ri). Thus, the species on which the estimates are conditioned tend to have greater probabilities of being present in some other sampling period or, in some cases, some other location. This positive covariance between p and q~within species tends to produce a positive bias in the estimates q~. Heterogeneous survival probabilities also cause problems in estimating some parameters for animal population models (Pollock and Raveling, 1982; Nichols et al., 1982b; Johnson et al., 1986; Rexstad and Anderson, 1992; Burnham and Rexstad, 1993), and the covariance between survival and recapture-recovery probability is an important determinant of estimator performance (e.g., Nichols et al., 1982b). The problem of a covariance between detection and, e.g., local extinction probabilities has been investigated by Alpizar-Jara et al. (in review). They conducted anal-
yses of BBS data by partitioning species into two groups characterized by relatively high and relatively low detection probabilities. As anticipated, the group with low detection probabilities indeed showed higher local extinction probabilities. The authors developed an ad hoc estimation approach to reduce bias in local extinction probability estimates, and evaluated with simulations the performance of this new estimator, as well as the estimators introduced above [Eqs. (20.10), (20.13), and (20.21)]. Bias and mean-squared error were found to be fairly small for the estimators (20.10), (20.13), and (20.21), but with slightly smaller bias for the new two-group estimator. Based on mean-squared error, Alpizar-Jara et al. (in review) concluded that the original estimators performed well and should be retained. Underlying the estimation of species richness and related parameters is the initial selection of an assumed model for detection probabilities. Model Mh was found to be the most frequently selected model by a wide margin in analyses of numerous BBS data sets (Boulinier et al., 1998a), so we developed program COMDYN to utilize this model and its estimators. In cases in which M h does not fit the data well, program CAPTURE (Otis et al., 1978) can be used for model selection. The resulting estimates of species richness then can be used in the estimators presented above, because these community-dynamic estimators are general and can be used with virtually any closed-population estimation model. If no model fits the data well, then we recommend a cautious reliance on the general robustness of the jackknife estimator for model Mh. It may be reasonable to use a quasilikelihood (see Burnham et al., 1987; Lebreton et al., 1992) approach, computing variance inflation factors based on the goodness-of-fit test results (Burnham and Anderson, 1998). We emphasize that lack of model fit is not an adequate reason for abandoning an estimation approach and resorting to use of ad hoc estimators, because model-based estimates are likely to perform better than ad hoc approaches even when model assumptions are not met (e.g., see Nichols and Pollock, 1983b).
20.4. DISCUSSION As noted earlier in this chapter, the concept of species diversity is frequently decomposed into the two components of species richness and species evenness. Because species evenness deals with abundances of individuals in the different species in a community, we believe that studies of evenness must necessarily consist of collections of studies of sympatric populations. As such, these studies must rely on the models
20.4. Discussion and estimation methods presented throughout Part III of this book for estimates of population size and related parameters. On the other hand, the richness component of diversity can be studied without detailed population-level studies. In this chapter, we have drawn an analogy between population and community levels of organization. The analogy includes state variables (the number of individuals in the population vs. the number of species in the community), vital rates (e.g., rates of mortality, recruitment, emigration, and immigration at the population level vs. rates of local extinction and colonization at the community level), count statistics (the number of individuals detected vs. the number of species detected), and inference methods (the use of capture-recapture approaches to estimate detection probabilities at both the population and community level). Based on this analogy, we have described methods for estimating species richness and quantities associated with the dynamics of richness over time and space. Our intent here has been to provide some suggestions for extending the capture-recapture estimation and modeling framework to the study of animal community dynamics. As seen in Tables 20.4 and 20.5,
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in which naive count-based "estimates" were presented together with our model-based estimates, the bias in count-based estimates can be substantial. Also, we note that these examples were based on data collected via the 50-stop sampling protocol of the BBS, whereas we expect naive estimates based on less intensive sampling schemes to exhibit greater bias. The estimation of species richness with capturerecapture methods is not new [e.g., see Burnham and Overton (1979) and the review of Bunge and Fitzpatrick (1993)], yet it still is not widely used by ecologists studying animal communities. On the other hand, development of methods for community dynamics is fairly new (Nichols et al., 1998a,b; Boulinier et al., 1998b, 2001; Cam et al., 2000a,b), and we expect rapid development in this area over the next decade. The methods presented here represent a substantial improvement over ad hoc approaches that currently dominate the literature of animal community ecology. Additional theoretical work along with practical experience with these methods should bring about many refinements and may produce a group of methods that will be broadly applicable for the investigation of animal communities.
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P A R T
IV D E C I S I O N ANALYSIS FOR ANIMAL POPULATIONS
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C H A P T E R
21 Optimal Decision Making in Population Biology
21.1. OPTIMIZATION AND POPULATION DYNAMICS 21.2. OBJECTIVE FUNCTIONS 21.3. STATIONARY OPTIMIZATION UNDER EQUILIBRIUM CONDITIONS 21.4. STATIONARY OPTIMIZATION UNDER NONEQUILIBRIUM CONDITIONS 21.5. DISCUSSION
In optimization, it is especially the case that biological complexity begets analytic complexity. Thus, single-species systems can be analyzed much more easily than multispecies assemblages; populations with only one cohort are easier to handle compared to populations with age structure, stage structure, a n d / o r spatial structure; linear systems typically are far easier to optimize compared to nonlinear systems. Of course, a profoundly complicating factor in optimization is temporal variation in decision making, whereby decisions at each point in time influence system dynamics throughout the remainder of the time frame. The generic optimization problem is to choose values for a set of decision instruments, so as to maximize some explicit objective that is expressed in terms of the instruments. The decision instruments can include system parameters, system state variables, state variable trajectories, control trajectories, or combinations of these. Optimization objectives incorporate values that are based on system states at specific times, or aggregates of these values across time, or functions of system controls over time, or elements of the time frame, or factors that are linked to system behaviors, etc. Examples of optimization with biological systems include the following kinds of problems.
In Chapter 7 we discussed a conceptual framework for biological decision making that included management actions at multiple points in time, along with forecasts of system responses to management. Several examples were given in Chapters 7 and 8 of systems involving populations and their environments, and we saw there that the inclusion of biological detail could lead quickly to formidable complexities in system representation and behavior. Recognizing that mathematical models inevitably simplify these highly complex biological systems, it is wise to account for only the biological structures necessary for systems analysis and management. In Part IV we focus on decision making in managed systems, specifically on optimal decision making. The notion is that biological populations and their environments are subject to manipulation through time, with decisions at any point in the time frame influencing system dynamics at subsequent times. It often is useful to identify a trajectory of decisions over time that maximizes benefits a n d / o r minimizes costs associated with management actions and system responses. Framed in this context, the management of populations is amenable to optimal decision theory for dynamic systems (Stengel, 1994).
9 Manage the habitat (and thus the competition coefficients) of three competing populations, so as to minimize a quadratic function of the population equilibrium states. Problems of this sort might utilize classical optimization procedures to identify a solution. 9 Choose a fixed harvest rate that maximizes total harvest of a population over some discrete time frame. Problems of this sort involve temporal equality con-
577
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Chapter 21 Optimal Decision Making
straints to characterize population dynamics, and classical programming methods may be applicable. 9 Allocate limited resources among recruitment, survivorship, and other management needs, so as to maximize long-term biological productivity of a population. This type of problem often involves inequality constraints, which can be handled by nonlinear programming. 9 Manage a population to attain a given stock size while minimizing costs associated with both population size and its rate of change over time. Problems of this sort often are amenable to the calculus of variations. 9 Choose a bounded trajectory of harvest rates to maximize accumulated harvests as a function of population size and harvest rate. An approach to this problem might utilize Pontryagin's maximum principle. 9 Choose a trajectory of stocking rates to attain a given population size, while minimizing expected accumulated costs for a population subject to stochastic influences over time. An approach to problems of this sort is stochastic dynamic programming.
variables Z(t). The population (or cohort) state variables in N(t) define a state variable trajectory {N(t)} -{N(t)It = t o, ..., tf} over the course of some time frame T (see Chapter 7). Similarly, the control variables in U(t) and environmental driving variables in _Z(t) define trajectories {U(t)} = {U(t)It = t 0, ..., tf} and {Z(t)} = {Z(t)It = t o.... , tf} respectively. Controls typically involve either direct resource removal or stocking, or the alteration of rates of flow among system components (see Chapter 7). As indicated in the examples above, an objective function can take many forms, depending on the nature of its optimality index, the amount of aggregation across time, the inclusion of a terminal value function, and other features. A frequently used formulation for dynamic optimization consists of time-specific costs and benefits (either economic or biological) that accrue as a result of management actions. Costs and benefits are measured by an optimality index I(N, U, t), which, when aggregated over the time frame T, defines an objective function
In the chapters that follow we develop a number of approaches to optimization problems such as these. We will see that the best approach for a particular problem depends on the mathematical nature of the decision instruments, the optimization objectives, and the biological system under investigation. As in earlier chapters, we begin with simple optimization problems that can be addressed with familiar optimization approaches, and add structure (and therefore complexity) as we progress.
l = ~ I(N, U, t)+ FI[N(tI)],
t~ t=to
where Fl[N(tf)] is a value corresponding to population size N(tf) at the end of the time frame {in many biological applications Fl[N(tf)] = 0}. Then the optimization problem focuses on an optimal selection of system controls from a set of allowable strategies: maximize
J
{U(t)} { U m
21.1. O P T I M I Z A T I O N A N D POPULATION DYNAMICS Because this book emphasizes dynamic biological systems, we focus the discussion below on systems that potentially change through time. A framework for optimization of dynamic systems requires (1) models describing system dynamics, (2) an objective function by which to evaluate prospective management strategies, and (3) a set of allowable management strategies, from which a particular strategy is to be chosen (Williams, 1989). To illustrate, consider the management of populations (or population cohorts) that change through time according to N(t + 1) = N(t) + [(N, Z, U, t), where population dynamics are influenced by management controls U(t) and environmental driving
subject to N(t + 1) = N(t) + [(N, Z, U, t), N(t0) = No, {N(t)} ~ N, where U and N define the set of feasible control strategies and population trajectories for the problem. Stochastic effects in either {Z(t)} or {U(t)} are handled naturally in this formulation, by substituting the expectation of the objective function and allowing for stochastic transitions. An analogous formulation for continuous-time systems is obtained by using differential transition equations and the integral form l =
f
tr to
Z(N, U, t) dt
+ Fl[X(tf)]
21.3. Stationary Optimization under Equilibrium Conditions of the objective function, wherein the summation across time is replaced by integration. In this case the problem is expressed as maximize
]
{U(t)} ~ U m
subject to d N / d t = N(t) + [(N, Z, U, t), N_(t o) = N O
{N(t)} e N, with dN/dt the time derivative of N(t). The optimization process can be viewed as a mathematical interaction between a dynamic population model and an objective function. An optimal value for the objective function is dependent on system responses to the corresponding control strategy and thus is dependent on the mathematical structure of the model used to describe system dynamics. On the other hand, optimal dynamics of the population model are dependent on the objective function, in the sense that the use of different functions lead to different optimal strategies and thus to different model behaviors. In essence, the optimal control problem involves mathematical feedback between the dynamic model and the optimization criterion. This feedback is a distinguishing feature in applications of dynamic optimization.
21.2. OBJECTIVE FUNCTIONS Applications of optimization often focus on harvests a n d / o r stocking and other forms of population enrichment, with optimality indices that include costs and benefits accruing to management. A simple optimality index for a single-age population accounts for biological yields I(N, U, t) = U(t),
or, in terms of yield rates, I(N, U, t) = u(t)N(t).
For populations with age structure, the optimality index might consist of yields for each cohort:
I(N, U, t) = ~, U~(t)
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Lotka-Volterra system, except that the summation would be over species instead of age classes. A simple extension would use net economic value of yield and might include the costs of control and other opportunity costs associated with population levels: I(N, U, t) = pU(t) - ClU(t) - c2N(t),
where p is the marginal economic return for the biological yield and cI and c2 are the unit control and opportunity costs, respectively. Yield can be characterized in terms of biomass by including a weighting factor w(t), which can vary through time to accommodate physiological growth, and the economic value of yield can be represented in terms of price p(N) and cost c(N, U) per unit of yield. Then an optimality index for a singlespecies, single-age model would have the form I(N, U, t) = [p(N)w(t) - c(N, U)]U(t),
with the corresponding objective function ti l = ~
[3t[p(N)w(t) - c(N, U)]U(t)
t=to
for a discrete-time model. The discount factor f~ < 1 effectively reduces the value of benefits and costs that occur later in the time frame. Note that if c(N, U) = c(N) the optimality index is linear in U(t). The significance of linearity in controls is discussed in Chapters 22 and 23.
21.3. STATIONARY OPTIMIZATION UNDER EQUILIBRIUM CONDITIONS In a generic sense, dynamic optimization applies to systems for which decisions vary over time, and the problem is to choose a strategy {U(t)} of time-specific actions that optimizes a recognized objective J. We begin the discussion of how such a strategy can be identified by considering the simpler problem of finding a stationary (time-invariant) action _U that maintains system equilibrium at some optimal level. Under these conditions the optimal control problem is described as
i
maximize
or, in terms of yield rates,
J
U
I(N, U, t) -- ~
ui(t)Ni(t).
subject to
i
The same general form would be appropriate for systems with more than one population, such as a
N(t + 1) = N(t) + [(N, Z, U, t), N ( t + 1) = N(t).
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Chapter 21 Optimal Decision Making
From the equilibrium constraint we have [(N, Z, U, t) = 0, which establishes a mathematical relationship between different values of U and the corresponding equilibrium population sizes N. If there are, say, m decision points in T, then
Assuming an objective of maximum harvest yield over some arbitrary number of years, m, i.e., J = mU, the optimal control problem under equilibrium conditions simplifies to maximize
with the maximum obtained by differentiation with respect to N and setting the result to 0 (see Section 22.2):
t
= mI(N, U) + FI(N) ,
and the optimal control problem simplifies to maximize u
mI(N, U) +
U = rN(1 - N / K ) ,
N
J = ~ , I(N, U) + F I ( -~
dU/dN-
FI(N)
r(1-2N/K)
=0, or
subject to
N * = K/2.
[(N, Z, U, t) = O.
On assumption that the terminal value F I(N) vanishes, the problem simplifies further, and the optimal value for U is obtained as a solution of the control problem maximize u
From Eq. (21.1) it follows that the optimal harvest of a logistic population under equilibrium conditions is U * = (rK/2) (1 - K/2K) = rK/4
I(N, U)
= rN*/2.
subject to [(N, Z, U, t) = O. m
Finally, if N is expressed as a function g(U) based on the constraint function [(N, Z, U, t) = 0, the optimal control problem can be described in terms of unconstrained optimization, as maximize
These results indicate that the optimal population size is one-half the carrying capacity, and the annual harvest needed to sustain that size is one-half the product of the optimal population size and the instantaneous rate of growth.
21.4. S T A T I O N A R Y
I[g(U), U].
OPTIMIZATION UNDER NONEQUILIBRIUM
U
Alternatively, if U is expressed as a function h(N), the problem becomes maximize
I[N,h(N)],
N
with optimal U given from the optimal value for N. Example
Consider a population for which annual change is described by a logistic equation of the form
CONDITIONS The problem of the previous section can be generalized somewhat by relaxing the equilibrium constraint N(t + 1) = N(t), thereby allowing for population dynamics over the time frame. Under these conditions the challenge is to identify a stationary optimal action U to be taken over the time frame, in the absence of equilibrium. Thus the optimal control problem is described as maximize
N(t + 1) = N(t) + rN(t)[1 - N(t)/K] - U(t),
u
J
m
where r and K are instantaneous growth rate and carrying capacity, respectively, and U(t) specifies the postreproduction harvest of individuals in the population during year t (see Section 8.2). Equilibrium conditions obtain when harvest exactly balances population growth, that is, when U = rN(1 - N / K ) .
(21.1)
subject to N(t + 1) = N(t) + [(N, Z, U, t) __N(to) = N o. Here the system state N(t) changes through time, but the management action U does not. The objective func-
21.5. Discussion tion J accumulates benefits and costs associated with U and N(t), and the challenge is to choose a stationary value U that maximizes J over the time frame. Sometimes N(t) can be expressed as a function of U and N Odirectly, without reference to the transition equations N(t + 1) = N(t) + [(N, Z, U, t). It then becomes unnecessary to carry the transition equations as constraints, and unconstrained optimization procedures can be used to identify an optimizing value for U.
Example Consider an exponential population for which the population's instantaneous rate of growth r can be increased through efforts to restore available habitat. Assume that the growth rate is controlled according to r = log(U + 1), so that r increases from 0 as restoration effort U increases. Population dynamics are described by
N(t + 1) = N(t) + rN(t) = [1 + log(U + 1)IN(t), so that N(1) = [1 + log(U + 1)IN0, N(2) = [1 + log(U + 1)]2N0,
N(t) = [1 + log(U + 1)]tN0 . A strategy to maximize population size over a time frame [0, tf] clearly involves the application of maxim u m allowable effort at each point in time. Conversely, a strategy to minimize population size would call for zero effort, so as to eliminate population growth over the time frame.
21.5. D I S C U S S I O N The general situation in optimal decision making involves the use of nonstationary strategies that induce population change, with actions at each time influencing system behaviors throughout the remainder of the
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time frame. Conceptually, one might consider evaluating the objective function J for each strategy in the opportunity set __Uand then choosing the strategy for which J is maximum. However, the number of potential strategies in ___Uis almost always too large to allow for such a case-by-case assessment, and nonenumerative approaches are required to solve the control problem. Six approaches to optimization are discussed in the following chapters: 9 Classical programming 9 Nonlinear programming 9 Linear programming 9 Variational mathematics 9 Dynamic programming 9 Heuristic approaches The first three approaches, originally developed for nondynamic problems, are discussed in Chapter 22. The last three were designed specifically for dynamic systems with time-varying controls, and they constitute the subject matter of Chapter 23. Nonlinear programming, dynamic programming, and variational mathematics produce bona fide optima when they are applicable. Linearization also produces an optimal solution, but in this case the system must be defined in linear terms, usually at considerable cost in realism. Heuristic approaches such as simulation gaming consist of several suboptimal approaches in which models are used to evaluate certain well-chosen options. Finally, classical programming is a special case of nonlinear programming that is applicable to systems with equality constraints. We describe each of these approaches in some detail in Chapters 22 and 23 and illustrate them with examples. We also provide a comparison of methods and indicate conditions in which one approach is likely to be more useful than another. In Chapter 24 we focus on optimal management under multiple sources of uncertainty, while accounting for the potential for learning through management. Uncertainties about biological process are described under the rubric of adaptive optimization, and variants of the algorithms of stochastic dynamic programming are proposed as a means to identify optimal adaptive strategies.
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C H A P T E R
22 Traditional Approaches to Optimal Decision Analysis
22.1. THE GEOMETRY OF OPTIMIZATION 22.1.1. Convexity Requirements 22.2. UNCONSTRAINED OPTIMIZATION 22.2.1. Univariate Decision Problem 22.2.2. Bivariate Decision Problem 22.2.3. General Multivariate Decision Problem 22.2.4. Solution Algorithms for Unconstrained Optimization 22.2.5. Summary 22.3. CLASSICAL PROGRAMMING 22.3.1. Bivariate Classical Programming 22.3.2. Multivariate Classical Programming 22.3.3. Sensitivity Analysis 22.3.4. Summary 22.4. NONLINEAR PROGRAMMING 22.4.1. Nonlinear Programming with Nonnegative Constraints Only 22.4.2. Nonlinear Programming with General Inequality Constraints 22.4.3. Solution Algorithms for Constrained Nonlinear Optimization 22.4.4. Summary 22.5. LINEAR PROGRAMMING 22.5.1. Kuhn-Tucker Conditions for Linear Programming 22.5.2. The Dual Linear Programming Problem 22.5.3. Using Linear Programming to Solve Nonlinear Problems 22.5.4. Simplex Solution Algorithm 22.5.5. Summary 22.6. DISCUSSION
ment strategies, and (3) an objective function that records benefits and costs for use in evaluating strategies. The general optimization problem is to identify a trajectory of decisions over time, i.e., a control strategy, that maximizes benefits a n d / o r minimizes costs, recognizing that decisions made at one time influence system dynamics at later times. Not surprisingly, solution approaches for the optimal control problem often involve mathematically complicated, computerintensive methods. Under certain simplifying assumptions about the control strategy and system state, optimization of dynamic systems can be handled with techniques that were developed for nondynamic optimization problems. In Chapter 21 we gave some examples of assumptions about management controls and population status that can simplify an optimization problem. Often one can take advantage of relationships between control and state variables that derive from such assumptions, to focus on optimization of either set of variables (or both). Indeed, classical optimization typically treats both control and state variables as decision instruments. Many, perhaps most, examples of optimization in biology involve the use of traditional techniques for nondynamic systems. In the following sections we simplify notation to accord with the usual conventions of constrained and unconstrained optimization methods. Thus, the optimization problem is described in terms of (1) a vector of x' = (x 1, ..., xn) of n decision variables, (2) an opportunity set X that is defined by constraints on the decision variables, and (3) an objective function F(x) by which to evaluate different combinations of decision variables in the opportunity set. The issue here is to select from
In Chapter 21 we described optimal decision making for animal populations in terms of (1) models of population dynamics, (2) a set of potential manage-
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Chapter 22 Traditional Approaches to Optimal Decision Analysis
the opportunity set X that combination of decision variables x that maximizes the objective function F(x)" maximize
a set of equality a n d / o r inequality constraints of the general form gl(x) = a 1
F(x)
subject to L gr(X) -- a r
x~X.
Note that the optimization problem is not framed in terms of temporal variability. The idea is simply to choose among alternative combinations of the decision variables, without explicit reference to time in that choice. Nor do we need to distinguish between control and state variables. To give added emphasis to the operational equivalence of control and state variables and to the lack of temporal variability, we denote the objective function in what follows by F(x), rather than ]. Note also that this optimization problem is invariant to linear transformation of the objective function, in that the same value of x is identified by maximizing F(x), maximizing a + bF(x) for b > 0, or minimizing a + bF(x) for b ~ 0. Thus, it is possible to transform a problem of the form minimize
F(x)
subject to xeX
into the equivalent problem maximize
-F(x)
subject to xeX. This transformation does not affect the optimizing solution, because both problems produce the same optimal value x*. In what follows we occasionally use this fact to transform problems involving, e.g., cost minimization, into maximization problems.
22.1. THE GEOMETRY OF OPTIMIZATION The identification of optimizing values involves a search among alternative values of x that lie in the opportunity set X. The opportunity set typically is defined by nonnegativity constraints x -> O, along with
and hi(x) ~ b1 9
o
hs(x) <_ b s
Several approaches are available for solution of the optimization problem, depending on the mathematical structure of the objective function and the constraints. Thus, classical programming is used if the opportunity set is defined only by equality constraints. On the other hand, linear and nonlinear programming are used in the presence of inequality constraints. Linear programming can be used when both the objective function and the constraints are linear forms in x. Nonlinear programming is required if the objective function or the constraints (or both) are nonlinear. The optimization problem is illustrated in Fig. 22.1 for an objective function that involves only a single decision variable x and the constraint x -> 0. In Fig. 22.1a the opportunity set includes all nonnegative numbers, and the objective function displays both a local maximum at x** and a global maximum at x*. Imposing the inequality constraint x -> b restricts the range of allowable values for x, as shown in Fig. 22.1b. In this case only one of the maximizing values is retained in the opportunity set. Unconstrained optimization involving two decision variables and a nonlinear objective function is displayed in Fig. 22.2. The nonlinear nature of the objective function is indicated by contours, i.e., curves defined by {x I F(x)_ = constant}, and by the gradient vector cgF/Ox = (3F/OXl,0F/3x 2) indicating the direction of steepest ascent in the objective function. Starting at some point x in the opportunity set, an efficient search procedure for locating the optimal value x* would move in the direction of 3F/Ox. Figure 22.3 displays a nonlinear objective function and nonlinear equality constraint g(x) = 0. In this case the feasible values for x* consist of points on the curve corresponding to the constraint. An optimal solution is found at the point of tangency between the constraint and a contour of the objective function (the point at which the gradients 0F/0x and 0g/0x are coincident). m
22.2. Unconstrained Optimization
585
In w h a t follows we make liberal use of differential methods, involving differentiation of both the objective function and the system constraints. As a matter of notational convenience we adopt the convention that differentiation of F(x) with respect to the column vector x produces a row vector OF/Ox = (OF/Ox 1, 3F/Ox2, ..., OF~ 0xn). Conversely, differentiation of G(_K)with respect to the row vector )~ = ()~1, ~2, ..., )~n) produces a column vector OG/O)~ = (0G/0)~ 1, 3G/OK2, ..., 3G/O~m)'.
22.1.1. Convexity Requirements
FIGURE 22.1 An optimization problem involving a single decision variable x. (a) Over the range of all positive values for x, local maxima can be identified at x* and x**.(b) Over the more restrictive range of values x - b, a single maximum can be identified at x*.
At that point the value of the objective function is greater than at any other point on the constraint curve. In Fig. 22.4 a nonlinear objective function is constrained by a set of linear inequality constraints. Figure 22.4a shows an optimizing value at a b o u n d a r y of the opportunity set. Under these conditions a change in the active constraints w o u l d lead to a different optimizing value x*. Figure 22.4b shows an optimizing value x* that is interior to the opportunity set X. In this case a change in the boundaries of the opportunity set w o u l d not affect the optimal choice of x*. Finally, Fig. 22.5 exhibits a linear objective function along with linear inequality constraints. Either a unique value x* maximizing F(x) is found at a vertex of the opportunity set X (Fig. 22.5a), or else optimal values for _x are found at any point along a b o u n d i n g surface of X (Fig. 22.5b). In either case a change in the active constraints w o u l d affect the choice of an optimizing value for x.
It is useful to describe the property of convexity as it applies to sets and functions, preparatory to developing the optimization procedures. If certain geometric properties involving convexity obtain for the objective function and opportunity set, then an optimization problem is assured of having a solution. Furthermore, u n d e r certain convexity conditions a local solution is guaranteed to be global over the opportunity set. Set convexity captures the idea that if x I and x 2 are two points in X, then any point on the line joining them also is in X. The property is expressed mathematically as follows: the set _X is convex if axl + (1 - a)x 2 _X for all x I E Xr X2 (~ X~ and a ~ [0,1]. Thus, there can be no "depressions" along the surface of a convex set, for then it w o u l d be possible to exit the set in moving along a line from one side of the depression to the other (Fig. 22.6). Function convexity also concerns the geometry of lines joining two points in X (Fig. 22.7). Thus, the function F(x) is convex if its value at points along a line segment in X is less than the corresponding average of values at the segment endpoints. In mathematical parlance, F(x) is convex on X if F[ax I + (1 - a)x 2] aF(x 1) + (1 - a)F(x 2) for all x__1 E Xr X2 E Xr and a e [0,11 (Fig 22.7a). By analogy, a concave function is defined by simply reversing the inequality sign: G(x) is concave on X if G[ax I + (1 - a)x 2] -~ aG(x 1) + (1 - a)G(x 2) for all X1 E----XIX2 (~ Xl and a e [0,11 (Fig. 22.7b). In mathematical p r o g r a m m i n g problems, concavity (convexity) in the objective function over convex X is sufficient to guarantee that a local m a x i m u m (minimum) is also global.
22.2. UNCONSTRAINED OPTIMIZATION In this section we describe optimization procedures for problems in which a vector of decision variables is chosen to maximize an objective function F(x), given that allowable values for the decision variables are not constrained. We assume in w h a t follows that the
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Chapter 22
Traditional Approaches to Optimal Decision Analysis
FIGURE 22.2 An optimization problem involving two decision variables x 1 and x2 and a nonlinear objective function F(x). The nonlinear nature of F(x) is indicated by contours of the form F(x) = c. m
o b j e c t i v e f u n c t i o n is t w i c e d i f f e r e n t i a b l e w i t h c o n t i n u ous derivatives. \ \ \
22.2.1. Univariate D e c i s i o n Problem
\ \
\ \
W e b e g i n w i t h t h e o p t i m i z a t i o n p r o b l e m of c h o o s i n g a v a l u e of a s i n g l e d e c i s i o n v a r i a b l e x to m a x i m i z e F ( x ) . A s s u m i n g t h e a b s e n c e of c o n s t r a i n t s o n x, t h e firstorder stationary condition
X X\
\ \ \
m
\
\
~ L..
~ --- -._ ~ x_.)=o. . . .
~- -"- - F(x)=cI F(x)--c3
x_*
FIGURE 22.3 An optimization problem involving two decision variables xI and x2, a nonlinear objective function F(x), and a nonlinear equality constraint g(x) = c. The optimizing value of x on the constraint is located at a point of tangency between the constraint and a contour of F(x).
d~ (x*) = 0 dx m u s t b e s a t i s f i e d for x* to b e m a x i m i z i n g ( A p p e n d i x H). H o w e v e r , f i r s t - o r d e r s t a t i o n a r i t y is n o t b y itself s u f f i c i e n t to e n s u r e a m a x i m u m , b e c a u s e t h e d e r i v a t i v e a l s o v a n i s h e s for l o c a l m i n i m a a n d i n f l e c t i o n p o i n t s (Fig. 22.8). A d d i n g t h e s e c o n d - o r d e r c o n d i t i o n d2F d x 2 (x*) < 0
(22.1)
22.2.
Unconstrained Optimization
587
X2
x2
a
\
\
\
x
\
\ \
x
\
a
x \
\
\
x \
\
\ x
\
\
N \
\
"\
\
N
: X\\
"'.i
\ \
\
\
XX x
\
\
\
,
---..
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\\
"'F(x)=c'
X''"
\
\
.
X
f(x)=c2
\
\
! \\ i: \ \\ \
\
\
-i..,, \ " \, , i \
""""-.-... F(x)=c3
i
\\
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\\
,, \
\
\
\\
\
_x
_X
\,
x2
b
\
x2
\
b \
\
\
---------_______
\
\ \
\
\ \
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\
\
\
\
\
"\\~NN N \\\
_LL-_LL-. _," /
..------~
/ /
..---...
\\\ \
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\
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,,
X
\\
"',iN "q \\ ", \\ ? \\X"
\
.
._L
...``/ / /
k x*
F I G U R E 22.4 An optimization problem involving two decision variables x I and x 2, a nonlinear objective function F(x), and a set of linear inequality constraints {g(x) < c}. (a) An optimizing solution at a boundary of the opportunity set. (b) An optimizing solution in the interior of the opportunity set.
F I G U R E 22.5 An optimization problem with linear objective function and linear inequality constraints. (a) The objective function is maximized at a point of intersection of the boundary constraints. (b) F(x) is maximized by any point along the edge of a boundary constraint.
Example
eliminates the possibility of a minimizing value or inflection point, so that together the first- and secondorder conditions guarantee that x* is a local maximum. Note that a local m a x i m u m identified by first- and second-order optimality conditions is not guaranteed to be a global m a x i m u m (Fig. 22.1). However, if (d2F/dx2)(x)
Consider an exponential population that is subject to postreproduction harvest U at each time t in a time frame {t = 1, ..., T}. Population status over time can be expressed as N(1) = (1 + r)N o - U,
N(2)
= (1 + r ) 2 N 0 -
N(t) =
(1 +
r)tNo-
(1 + r ) U -
U,
U [ ( 1 + r) t-1 + (1 + r) t-2 + ...
+ 1], or
N(t) = (1 + r ) t N o -
U[(1 +
r) t -
1]/r,
which establishes a mathematical relationship between
588
Chapter
22
Traditional Approaches
to O p t i m a l
Decision Analysis
x,
a
I I I
I I I
x-=axI+( l-a)x 2
x2
F(x) b
x 2
b
/
I
I x1 F I G U R E 22.6 T w o - d i m e n s i o n a l convex a n d n o n c o n v e x sets. (a) The line s e g m e n t connecting x 0 a n d xf is c o m p l e t e l y contained in the convex set _X. (b) Part of the line s e g m e n t connecting x 0 a n d Xf lies outside the n o n c o n v e x set X. m
F I G U R E 22.7 T w o - d i m e n s i o n a l concave a n d convex functions. (a) For the convex function F(x), the v a l u e Flax I + (1 - a)x 2] is less t h a n the a v e r a g e v a l u e aF(x 1) + (1 - a)F(x2). (b) For the concave function G(x), the v a l u e G[ax I + (1 - a)x 2] is greater t h a n the a v e r a g e value aG(x 1) + (1 - a)G(x2).
harvest level and population size over the time frame. If T is large, this relationship at the end of the time frame is well approximated by N(T) = (1 + r ) T ( N o - U / r ) .
(22.2) Hx)
Now assume that any stock remaining at the end of T periods is to be removed at some cost, and the population is subsequently to be restocked. An objective of management might be to maximize a quadratic expression in both total harvest and final population size, i.e., F = (TU) 2 - N 2. Differentiation of the objective with respect to U yields dF - 2TUdU ~. 2 T U +
/
dN 2N7i- ~
2(1 + r) 2T r
= 2U(Tq (1 + r)2T~ 72 ~
--/
[N O - U/r] --
(1 + r) 2T 2~N~
and a candidate optimal harvest level U* is found by setting d F / d U = 0, so that
X2
X3
F I G U R E 22.8 A differentiable function F(x) w i t h a local maxim u m x 1, inflection point x 2, a n d local m i n i m u m x 3. The derivative dF/dx v a n i s h e s at all three values of x.
22.2. Unconstrained Optimization (1 U n'-- 1"
+
r) 2T
(1 + r) 2T -
r 2 T NO.
We also have d2F dU 2
(1 + r) 2T = 2 T - 2 ~ r2
_2 -
r~[Tr 2 -
(1
589
As in the univariate problem, a search for maxima can utilize first-order stationarity to identify potential maximizing values, which then can be screened for negative definiteness of the Hessian matrix. Values that satisfy both conditions are locally maximal. Example
+ r) 2T]
<0 for large values of T, which guarantees that U* is in fact a maximizing value for the objective function. For large T, the optimal harvest U* is approximated by U * = r N o, suggesting that if the time frame is large, the optimal fixed harvest level reduces the population each time by an amount that is only slightly larger than the population growth increment. 22.2.2. Bivariate D e c i s i o n P r o b l e m
Suppose that the decision problem now involves decision variables x I and x 2, and a twice differentiable objective function in both variables. A requirement for x* to be maximal is the vanishing of the gradient ( O F / { } x ) ( x ) of F at x*, i.e.,
Consider a population with exponential population dynamics and constant harvest, as in the previous example. We assume here that both the annual harvest U and the initial population size N Oare under management control. Initially about 100 individuals can be handled efficiently, and costs are associated with stocks that deviate in size from that number. However, stock capacity is projected to grow to about 1000 individuals in T years. At that time there will be costs associated with stock sizes both above and below the target. If N T is the stock size after T years, an objective function for the problem might be F ( N o, U ) = - ( N
(
OF = _2 (N r -
1000)
dU /
= [(3F/OXl)(X*) , (3F/Ox2)(x*) ]
= 2(N T 1,
=0' (Appendix H). However, this condition is not sufficient by itself to guarantee a maximum at x*, because firstorder stationarity is satisfied for other points such as minima and saddle points. A stationary value x* is guaranteed to be locally maximal if the Hessian matrix
a2F 32F (x) = r 2 -
1000) 2 - (N O - 100) 2,
which is to be maximized by appropriate choice of N O and U. From above, necessary conditions for an optimal solution to this problem are
ou (OF/Ox)(x*)
T -
OXT (X)
32F (x)] {}XlX2 -- I
O2F | a2F (x)3-~2(x) J
-
{)X2Xl --
is negative-definite at x*, i.e., if
ax, L0x2 (x*) Ax < 0 for arbitrary nonzero Ax (see Appendix B for a discussion of the negative-definite property for matrices). Note that a vanishing gradient and negative definite Hessian are equivalent to the first- and second-order optimality conditions for the univariate decision problem. Negative definiteness of the Hessian over all values of x ensures that a local optimum is also global. m
1000)(1 +
r) T
=0, from Eq. (22.2), and OF
_ -
dN
-2(N
T -
1000)~-~0-2(N 0 - 100)
= -2(N
T -
1000)(1 +
ON o
r) T
- 2 ( N 0 - 100)
--0. The first condition requires that N* = 1000, which in turn implies N 8 = 100, from the second condition. Substituting this value back into Eq. (22.2) then yields
U* = r
1000 ] 1 0 0 - (1 +r)TJ"
It is straightforward to show that the Hessian matrix for F(N 0, U) is negative definite at (N~, U*), so that the optimal solution is in fact maximal. Note that the second term in the expression for U* must be less than 100 for the solution to be meaningful, and thus T must be large enough to ensure this condition. The form of the solution indicates that the optimal initial population size is 100 individuals irrespective of T, and with
590
Chapter 22 Traditional Approaches to Optimal Decision Analysis
very large T the optimal harvest level U allows for only slight growth in the population. On the other hand, a more abbreviated time frame requires a harvest level that is reduced below the annual growth increment, to allow more rapid population growth toward the population target of 1000 individuals. Example
In Chapter 9 we described the use of unconstrained optimization for systems identification and provided in Section 9.1 an illustration of the fitting of an exponential model to field data by maximizing the mean squared error
and k
k
k
a~, A i + b~
i=1
A2 = ~
i=1
A i l ~ i.
i=1
These are the well-known normal equations of statistical regression theory (Draper and Smith, 1981), with solution k Ei= l (Ai - --a)(1Qi - ~]) ~-'~= l (Ai - ~)2
and d=N-bA,
F ( N o, r) = ~ [ N ( t
i) - 1(q(ti)]2/k
ti~S
via the selection of model parameters N Oand r. Optimization in the context of model fitting also can be used to identify relationships among biological variables. To illustrate, consider a relationship
where A and N are the means of the data in {[Ai,/~/(ti)]" i = 1, ..., k}. Based on these data, the relationship is estimated as
a + t;A. 22.2.3. G e n e r a l M u l t i v a r i a t e Decision Problem
N i = a + bA i
between equilibrium population size N i of a local small m a m m a l population and a measure A i of unfragmented habitat for the population. Replicate studies over a region of interest have produced the data set {(A t,/Qi): i = 1.... , k} which can be used to estimate the relationship via least-squares optimization. The objective function for this problem is k F(a, b) = ~[/~(ti) - (a + b A i ) ] 2 / k , i=1
In the general multivariate case, a vector x* = (x~, x~ .... , x*)' is sought for which F(x*) is maximized in a neighborhood of points about x*. The gradient for this problem is a vector
cOX -
+ bAi)]/k
Ob
a2F
CO2F
-
" 32 F ---~(x) 0x~ --
(x) COXlcoX2--
. . . . (x) COXlcoXn --
co2F (x) _ coXncoX1
c92F (x) COXncoX2
..
CO2F ~ 2 (x) =
and OF
1
"'" OXn -
of dimension n, and the Hessian matrix at x is
and differentiating F with respect to a and b produces k OF = 2 E[/~/(ti) -(a COa i=1
-
9
CO2F COx2 (x) _
k = -2
~ , A i [ N ( t i) - (a + b A i ) ] / k .
i=1
As before, a necessary condition for a m a x i m u m at x* is the vanishing of the gradient, i.e.,
Necessary conditions for optimization are given by COF / COa]
OF (x*) COx-
[~
ka -+- b Z
i+1
k A i "-- E
i+1
(22.3)
In addition to first-order stationarity, a negative definite Hessian matrix is needed to guarantee that x* is maximal, i.e.,
or
k
0. -
l~i
,a2F Ax ~ (x*)kx < 0
(22.4)
22.2. Unconstrained Optimization for arbitrary nonzero Ax. Values x* for which conditions (22.3) and (22.4) are satisfied are locally maximal. Negative-definiteness of the Hessian over X establishes that F is concave over X_ and therefore guarantees global maximization (see Section 22.1.1). The gradient vector and Hessian matrix play prominently in the theory of optimization and are mentioned frequently throughout the remainder of this chapter. For notational convenience, in what follows we use aF _,) VF(_x*) = ~xx(x to denote the gradient of F at x*, and
591
proximations of x* [or successive approximations of F(x*)] no longer exceeds a minimum stopping criterion. To verify that a local maximum has been found, it is necessary to determine whether aF/ax = _0', and whether the Hessian matrix is negative-definite. If either condition fails to be met, the search is continued, utilizing larger neighborhoods, more intensive sampling within neighborhoods, or other rules for recognizing change in F(x). Nondifferential approaches include sequential univariate searches, simplex methods, conjugate direction methods, and other techniques (see Appendix H).
22.2.4.2. Differential Approaches
a2F
H~(x*) = 7 x2 (x*)
to denote the Hessian at x*.
22.2.4. Solution Algorithms for Unconstrained Optimization Procedures for identifying x* in an unconstrained optimization problem can take advantage of the fact that an optimal value must satisfy Eq. (22.3), so that the search for optimal values is reduced to a search for zeros of the system of equations aF/ax = 0'. In a limited number of cases analytic approaches can be used to solve this system of equations. In other cases involving only a few dimensions, nonderivative simultaneous search methods sometimes can be used to identify a maximizing value x* via an iterative search through a subset of points in X._ More typically, one must use iterative search procedures, with the direction and size of the search step in each iteration determined from results of the previous step. Key issues involve (1) choosing the "best" direction at each step and (2) determining the length of step to be taken in that direction (Miller, 2000b). Both differential and nondifferential search methods are available. m
22.2.4.1. Nondifferential Approaches Derivative-free methods involve a systematic search for a value _x* that satisfies aF/ax_ = _0', while avoiding the often difficult task of differentiating F(x). The search begins with repeated evaluations of F(x)_in a neighborhood around some starting value x 0, followed by movement in a direction of change in F(x). A step of appropriate size is taken in that direction, and a value x I is identified. Then x 0 is replaced by x1, and another search centered on x I produces a new value x 2. This process can be repeated until the optimal value x* is found, or until the difference between successive ap-
A number of methods utilize first and second derivatives in an iterative search for maximizing values x*. As with derivative-free approaches, the idea is to move in sequential fashion toward an optimum along the "best" direction possible. The iterations often involve (1) modification of the gradient in defining the "best" direction at each step and (2) determining the length of step to be taken in that direction. Iterative procedures such as the method of steepest ascent, conjugate gradient, and Newton's method (see Appendix H.1), all take advantage of the derivatives of F with respect to x. For example, an algorithm for the method of steepest ascent chooses Xk+l according to Xk+ 1
=
X k --
BkVF(Xk)
--
Xk
{[VF(Xk)][HF(Xk)][VF(Xk)]'}
--
-1
• {[VF(Xk)][VF(Xk)]'}VF(Xk). This algorithm is effective when the Hessian HF(x k) is positive definite over the operating range of values x k. A second approach is Newton's method, an iterative search algorithm based on Xk+ 1
=
X k --
[HF(Xk)]-lVF(Xk)
'.
Assuming the Hessian is positive definite over the operating range of values Xk, repeated application of Newton's method generates a sequence of values {xk} that converges to x*. We discuss steepest ascent and Newton's method in more detail in Appendix H. Though derivative-based methods can be quite efficient in converging to an optimal solution, the required differentiation and matrix inversion sometimes can be laborious. In particular, a comparison of Newton's method and the method of steepest ascent shows that the former is a more computer-intensive algorithm, in that it not only relies on the computation of first and second partial derivatives at each iteration but it also requires the inverse of Hr(Xk). On the other hand, the
592
Chapter 22 Traditional Approaches to Optimal Decision Analysis
c o m p u t a t i o n s involved in inversion are c o m p e n s a t e d by m u c h more rapid convergence in a n e i g h b o r h o o d of x* (Luenberger, 1989).
The direction of m a x i m u m change at x I is again given by the gradient V F ( x 1) =
(-
=
(0.82, 2.41)'
0.24, 0.47),
Example M a n a g e m e n t desires to limit the g r o w t h of two populations x I and x 2 with p o p u l a t i o n d y n a m i c s given the Lokta-Volterra competition equations (see Section 8.8.1):
dxl/dt = x1(1 - 0.3x I - 0.1x2) and
dx2/dt = 0.6x2(1 - 0.16x I - 0.16x2) An objective function for this problem is the s u m of the g r o w t h functions, or
F(x) = x1(1 - 0.3x I - 0.1x 2) 4- 0.6x2(1 - 0.16x I - 0.16x 2) -- Xl
4-
0.6X2 -- 0.3Xl2 -- 0.1X22 -- 0.2XlX2.
and a step of size
~1 = {[VF(x_.I)][HF(Xl)][VF(Xl)]'}-I{[VF(Xl)J[VF(x_I = 0.833
in that direction produces a n e w value x 2 = (1.02, 2.02)'. The process of m o v i n g in the gradient direction can be continued indefinitely, or until it converges to (x~, x~) = (1, 2)'. N e w t o n ' s m e t h o d also utilizes the gradient at each point in an iterative search, but the direction of search is adjusted by the Hessian matrix. As above, the a p p r o a c h starts at some arbitrary value x 0 and generates a sequence of values Xk that converge to x*. For the example above, the inverse of the Hessian is
Because minimization of F(x) and maximization of cF(x_) for c < 0 p r o d u c e identical optimal values for the decision variables Xl and x 2, the objective function for an equivalent optimization problem is
2
= [
F(x_) = 3x 2 + x 2 + 2XlX2 - 10x I - 6 x 2. The gradient for this function is OF/ax = (6x I + 2x 2 - 10, 2x I 4- 2x 2 - 6), and it is easy to solve OF/Ox = 0' for the stationary point (x~, x~) = (1, 2). However, one also could a p p l y the search procedures above to determine (x~, x~). The m e t h o d of steepest ascent searches iteratively for optima in the direction of the gradient OF/Ox. Starting a search at an initial value x 0 = (2, 3)' produces X__1 =
X0 --
2.41 based on the gradient direction VF(x 0) = (8, 4), the Hessian
and a resultant step size of 1
8 o = {[VF(x0)][HF(x0)][VF(x0)]' }- {[VF(x0)][VF(x0)] }
X__1 =
0.25-0.25] -0.25 0.75 '
X0 -- [HF(X0)]-lVF(x0)
= [23]-1/4I_
~ -131[: 1
['2] Because this particular objective function is quadratic, N e w t o n ' s m e t h o d converges to the optimal solution in one iteration. Typically, several iterations are required to identify a stationary value. Last, derivative-free approaches also can be taken to identify an optimal solution. One such a p p r o a c h is to evaluate the objective function at several points a r o u n d x 0 and choose a direction for x I based on the results. For example, one might evaluate F(x) at equispaced points on a circle of radius 0.5 a r o u n d x 0, and choose for x I that point for which F(x)_ is greatest. A set of eight such points a r o u n d x~ = (2, 3) is given by the four combinations of the coordinates x I = 2 + 1 / 2 and x 2 = 3 + 1/2, along with the four combinations of x 1 = 2 + 1/(2X/2) and x 2 = 3 + ( 1 / 2 V 2 ) . Evaluation of F(x) at each of these eight points indicates that the objective function decreases most rapidly in the m
= 0.147.
-1
and a first iteration of N e w t o n ' s m e t h o d starting at x 0 = (2, 3)' produces
80VF(x_0)'
= [2310147,[84] [082]
)-1'}
22.3. Classical Programming direction of (1.5, 2.5)', so a step of appropriate size in this direction defines a new starting point x I for followup investigation of points around it. Continuing this procedure for one more step results in convergence to (x~, x~_) = (1, 2). Applying the procedure at (1, 2) results in decreases in F(x) in all directions, confirming that (1, 2) is indeed a maximum point.
22.2.5. Summary In this section we have discussed optimization procedures for problems in which a vector of decision variables is to be chosen to maximize an objective function, given that the set of allowable values of the decision variables is not constrained. Necessary and sufficient conditions for local optima were identified on assumption that the objective function is twice differentiable with continuous derivatives in the decision variables. A solution algorithm involves the following steps: 9 Values of _x are identified such that the gradient of the objective function F vanishes [Eq. (22.3)]. 9 The Hessian of F is investigated for negative definiteness at each stationary point thus identified [condition (22.4)]. 9 As needed, computer-based search procedures, such as the method of steepest ascent, Newton's method, and derivative-free methods, can be used to identify stationary points of F. 9 On condition that the objective function is strictly concave for all x, a local optimum is also global.
593
where m < n. Population transfer functions N(t + 1) = N ( t ) + [ ( N , Z , U , t ) and initial condition N(t o) = N O serve as examples of such equality constraints. Other constraints might include requirements for growth in population size, specifications of per capita harvest rate, and other mathematical linkages among decision variables. A possible approach to the classical programming problem would be to use the constraints to express some decision variables in terms of others. The net effect is to reduce the dimensionality of x and simplify the optimization problem, eliminating a need for an explicit statement of the constraints. Assume, for example, that the decision variables can be reordered and grouped into n - m variables x I and m variables x 2, such that x 2 = h ( x 1) is derived from the equality constraints. Then the optimization problem is equivalent to maximize
F[x 1,
h ( x 1)],
and the problem is solved by identifying optimizing values for x I as in the previous section. Recall that in a previous example (see Section 22.2.1) we used the transition equations for a harvested population to derive a relationship between a stationary harvest amount and time-specific population size. For that particular example the relationship x 2 - - h(x 1) was easy to derive. Often it will be quite difficult, especially if the constraints are complicated nonlinear expressions in x.
22.3.1. Bivariate Classical Programming 22.3. C L A S S I C A L
PROGRAMMING Classical programming extends the unconstrained optimization problem of the previous section, by allowing for equality constraints on allowable values of the decision variables. Thus, we assume that the optimization problem involves a vector x of n decision variables, a twice differentiable objective function F(x), and one or more differentiable equality constraints gi(x_) = a i on x. The optimization problem is expressed as
A popular approach takes advantage of the differentiability of F(x) and g ( x ) to derive conditions for an optimal solution. Consider for example an optimization problem involving two decision variables x I and x 2, along with a single equality constraint g(x) = a. It is argued in Appendix H that optimizing values for x can be obtained by the method of Lagrangian multipliers, whereby the constraint is incorporated directly into an augmented "Lagrangian function"
m
maximize
F(x),
subject to
L ( x , h) = F(x_) + h[a - g(_x)],
where h is the Lagrangian multiplier for the constraint, and Xl, x2, and h are treated as independent (unconstrained) variables. Differentiating L(x, h) with respect to x and h yields the partial derivatives
g l ( X ) -- a l 0X 1
0X 1
OX1'
1
Lgm(X)'= a m
0L = OF OX2
0X 2
)8g,__ 0X 2
594
Chapter 22 Traditional Approaches to Optimal Decision Analysis
and OL
O--k = a - g(x), which vanish at o p t i m u m values x* and k*. The resulting equations are expressed in matrix notation as OF (x*) = k* 0g (x*) 3x -
(22.7)
It is straightforward to solve these three equations for the three variables Xl, x2, and k, with the result that (x~, x~) = (69, 73). This result suggests that resources should be allocated so as to maintain populations of approximately equal size. For _x*the available resources are fully utilized and the value of the objective function is L(x'~, x'~, k*) = F ( x ' t , x~) = 11,774.
3x -
22.3.2. Multivariate Classical Programming
and g(x*) = a.
(22.8)
Equations (22.7) and (22.8) establish necessary conditions for a maximizing value x*. A local m a x i m u m is guaranteed by a negative definite Hessian of F in x 1 and x2 for points that also satisfy the gradient condition [Vg(x*)l dx = 0. Because maximizing values x* along with k* must satisfy Eqs. (22.7) and (22.8), an operational approach to the constrained bivariate problem involves finding solutions to Eqs. (22.7) and (22.8) in the three unk n o w n s x 1, x 2, and k. The negative definiteness of the Hessian matrix of F ( x ) then can be ascertained at these solutions, and any value x* satisfying both stationarity and negative definiteness is a local maximum. If in addition the Hessian matrix is negative-definite for all values x satisfying the constraint, then a local maxim u m also is global over that set of values. m
The general multivariate problem with x' = (x 1, ..., x n) and m equality constraints gl(X)
al
Lxmix)J
a
(m < n) can be handled in m u c h the same w a y as the bivariate problem with a single constraint. Thus, feasible candidates for an o p t i m u m can be identified by the method of Lagrangian multipliers, with the constraints incorporated directly into a Lagrangian function by means of m Lagrangian multipliers _h = ()k 1, ..., )krn):
L ( x , k) = F ( x ) + k[a - g(x)],
(22.9)
where k i is a Lagrangian multiplier for the constraint
Example
Consider a situation involving two competing populations of sizes x I and x 2 that are constrained by the available resources needed to sustain them. Both populations are seen by managers as desirable, so both are to be maintained. The m a n a g e m e n t of both populations is to reflect the potential for population sustainability, the perceived value of individuals in each population, and the limitations on resource availability. These factors are accounted for in the objective function F ( X l , x 2) = 224x I + 84x 2 + XlX 2 - - 2 x 2 - x 2, which includes maintenance costs and resource benefits for both species. Limitations on resource availability are given by the constraint 3 x I + x 2 = 2 8 0 , s o that the Lagrangian is L ( X l , X2, ~k) -- 224x I + 84x 2 +
+ k(280 - 3x 1
XIX 2 --
--
X2)
2x 2 -
x2~
,
and differentiation with respect to x 1, x2, and k yields OL
= - 4 x 1 + x2 + 2 2 4 -
3k = 0,
Ox 1
OL
= - 2 x 2 + xI
-}-
84-
k = 0,
Ox 2
OL
O--X= 280 - 3x I - x 2
--
0.
g i ( x ) = a i and the elements of _x and _k are treated as independent (unconstrained) variables in L ( x , k). Dif-
ferentiation of L ( x , k) with respect to x and K_yields the required conditions
(x*) = x* g- (x*)
Ox -
-
Ox -
(22.10)
m
and g(x*) = a
(22.11)
m
for a local m a x i m u m x*. Equations (22.10) and (22.11) represent n and m equations, respectively, so that in combination they constitute a system of n + m equations in the n + m variables in x and _h. Equation (22.11) asserts that the equality constraints are satisfied at an optimal solution x*, and these equalities in turn establish that maximization of L ( x , k) at (x*, _h*)is equivalent to maximization of F(x) at x*. Note that Eqs. (22.10) and (22.11) are analogous to Eqs. (22.7) and (22.8) for the bivariate case with a single constraint. Note also that in the absence of equality constraints, Eqs. (22.10) and (22.11) reduce to condition (22.3) for unconstrained optimization. This confirms that unconstrained optimization is s u b s u m e d as a special case of classical programming.
22.3. Classical Programming Equations (22.10) and (22.11) are necessary conditions obtained from the Lagrangian for a local maximum. As with the bivariate case, an approach to maximization involves finding solutions to Eqs. (22.10) and (22.11) in the n + m unknowns in x and )~. A sufficiency condition can be stated either in terms of the Hessian of the original objective function or the Hessian HL(K, x) of the Lagrangian (see Appendix H). Thus, negative definiteness of the Hessian matrix of F(x) along with the gradient condition (Og/Ox)(x*) dx = 0 guarantees a local maximum. If these conditions obtain for all values x satisfying Eq. (22.11), a local maximum also is global.
595
The Lagrangian for this problem is L(xl, x2, x3, ~1, k2) = Xl + )~1Ix1 - 225 - 225x 2
1_
]
(225) 2
m
-ff
~33
'X2J q-
k21324 -
50X2 - x3]'
and differentiation yields OL 3x 1
0L
=l+k
1,
= )~1[-225 + (225)2]-50k2,x3 J
OX2
Example
(225) 2
3L
Consider a logistic population of 225 individuals located in an area that is subjected to human disturbance. Disturbances can be controlled at little cost, resulting in an increase in the population carrying capacity. The population also is subjected to considerable predation pressure, which is thought to be density independent over the range of reasonable population sizes. Predation has the immediate effect of depressing the instantaneous rate of growth below what it otherwise would be in the absence of predation. Predator control methods such as trapping and fencing are relatively expensive, but they can increase population growth rate substantially. An economic analysis suggests that it is 50 times as expensive to double the rate of growth through predator management as it is to double the habitat carrying capacity. Management wishes to allocate resources so that the population next year is as large as possible, given that available departmental resources are limited to $324. Let x 1, x 2, and x 3 specify the population size after 1 year, the rate of growth for the population, and its carrying capacity, respectively. These three decision variables are related by the logistic equation for population dynamics, X l = 2 2 5 + (x2)225[1 - 225], x3 J which gives the population size after 1 year for an initial population of 225. Predation control and disturbance management also are related through 50x2 + x 3 = 324, which expresses management's resource limitations. The optimization problem can be stated thus: maximize
0X 3
OL
=
--k I
X2
X 2 -- K 2 ,
(225) 2 = x 1-225x 2 + ~ - x 2 ,
0~. 1
OL
X3
= 324
-
50x 2 -
X3.
o3~ 2
From the first equation ~'1 = - 1 , from the third equation ~2 = x2(225/X3 )2, and substitution of these expressions into the second equation yields x32-225x3 = (50)(225)x 2. Using x3 = 324 - 50x 2 from the resource constraint, this simplifies to (324 - 5 0 x 2 )2 = (225)(324), or x2 = 1.08. Substituting back into the resource constraint then results in x 3 = 270. Thus, for optimal management of the population, resources should be allocated so as to produce a carrying capacity of 270 individuals and a rate of growth of 1.08. This allocation will allow the population to increase from 225 to 265 individuals over the year. A direct extension of the Lagrangian approach for problems over multiple time periods adds more decision variables (to characterize population size at each time) and additional Lagrangian multipliers (to include the transition equations as constraints for each additional period). For example, the same optimization problem as above but accounting for growth over 2 years is maximize
F(x) = x2
subject to
F(x) = x 1
xl = (x3)22511- 225],x4
subject to X1-- 2 2 5 + (x2)22511 - 225],
x3 J 324
= 5 0 x 2 + x 3.
X2 =
324
=
(X3)X 1
[ x4] 1 -
5 0 x 3 q- x 4,
,
596
Chapter 22 Traditional Approaches to Optimal Decision Analysis
where x I and x 2 now represent population size after 1 and 2 years, respectively, x 3 specifies the rate of growth, and x4 is the ca.rying capacity. The corresponding Lagrangian is
with values that are dependent on the constraint constants. By differentiating L(a) with respect to a, it is not difficult to show (see Appendix H.2.2) that _k* satisfies aL (x* k*)
L(x1, x2, x3, x4, }~1,)k2, ~.3) = x2 q- }~1
oa
-
'-
OF _,) = 7a (x
(22.12)
2252 ]
X Xl-225x3+-~4
x3j
x 2
q- }~2[X2-- XIX3 ff--if- ~3
[ 3 2 4 - 50x 3 - x4],
and differentiation yields aL c3X1
3L 0x 2
-~.1 + ~.2 I-x3 -}- 2Xlx31,x4 j
Example
- 1+k2,
[225+ 2252]+,2 [Xl+ x21 50,3 c~X3
X4
2252
X4J
x 2_ k3,
c~X4
X 2 X3
3L
2252 = x I - 225x 3 + ~ x 3 ,
3K 1
X4
3L
x2
m
3K 2
aL
X2- XIX3 q- X4 X3
= 324
-
Thus, a Lagrangian approach provides sensitivities of the optimal value F(x*) to changes in the constraint constants. In economics, the objective function often measures economic value as a product of price times quantity, with the constraint constants describing limitations in the availability of resources. Then the optimal Lagrangian multipliers measure marginal change in value as available resources increase and so are referred to as "shadow prices" (Hillier and Lieberman, 2001).
50x 3 -
X4 .
3}k3
Consider the preceding example involving allocation of resources to maximize the growth of a logistic population over 1 year. The Lagrangian multiplier k 2 corresponds to the constraint 324 = 50x 2 + x 3, which limits the resources available to enhance the growth rate and carrying capacity for the population. Because *) = (1.08, 270), we have ~'2 = X 2 (225/x3 )2 and (x~ , x 3 k~ = 1.08(225/270) 2 = 0.75. From Eq. (22.12) this means that a unit increase in the constraint constant a = 324 would result in an increase of 0.75 in the optimal value of the objective function.
22.3.4. Summary
Note that the Lagrangian now includes decision variables for population size in years 2 and 3, as well as Lagrangian multipliers for population transitions across two years. This gives a Lagrangian function in seven variables (an increase from five in the previous example) and results in a system of seven equations that must be solved to determine the optimal allocation of resources. The mathematical complexity of even this simple two-period problem requires the aid of computer-based search procedures (see Section 22.4.3).
In this section we have described procedures for classical programming, extending the unconstrained optimization problem of Section 22.2 by including equality constraints on the decision variables. The optimization problem in classical programming involves a vector x of decision variables, a twice differentiable objective function F(x), and one or more differentiable equality constraints on x. The goal is to identify a vector of decision variables that maximizes the objective function in the presence of equality constraints. A solution approach involves the following considerations:
22.3.3. Sensitivity Analysis
9 A set _k of Lagrangian multipliers corresponding to the set of equality constraints is used to define a "Lagrangian function" consisting of the original objective function and the sum of products of the Lagrangian multipliers and the constraints [Eq. (22.9)]. 9 Differentiation of the Lagrangian function with respect to _x and _k leads to a system of equations in the original variables and the Lagrangian multipliers [Eqs. (22.10) and (22.11)].
The optimal Lagrangian multipliers M, ..., X* provide useful information about the marginal influence of the corresponding constraints. If we consider the decision variables and Lagrangian multipliers as functions of the constraint constants in a, then the Lagrangian can be expressed as
L(a) = F[x(a)] + k(a){a -g[x(a)]}, B
22.4. Nonlinear Programming 9 The system of equations in the decision variables and Lagrangian multipliers is solved for stationary values x* and h*. 9 As needed, computer-based search procedures such as Newton's method can be used to identify stationary points (x*, h_*) that satisfy conditions on the constraint gradients and the Hessian of F (see Section 22.4.3 and Appendix H). 9 The optimal values _h* express sensitivities of the optimal value F(x*) of the objective function to changes in the constraint constants [Eq. (22.12)].
22.4. NONLINEAR PROGRAMMING Nonlinear programming extends the classical programming problem of the previous section, by allowing for inequality as well as equality constraints on allowable values of the decision variables. A generic statement of the problem is maximize
F(x)
597
and concavity of the objective function, an optimal solution x* to the nonlinear programming problem can be found in X, either in its interior or on its boundary.
22.4.1. Nonlinear Programming with Nonnegative Constraints Only A special case of the nonlinear programming problem restricts the constraints to nonnegativity conditions only: maximize
F(x)
subject to x~0. If x* is interior to X, i.e., x > 0, the same conditions for optimality obtain as for the unconstrained optimization problem, and in particular all first-order partial derivatives must vanish at x*. If, however, the maximizing value for one or more variables in _x is zero, then the vanishing of partial derivatives for these variables is no longer required. The conditions I
aF m (x*) -< 0',
subject to
3x-
-
(22.13)
gl(x) ~ bI aF
~ ( x * ) x ~ = 0,
(22.14)
J
I
kgm(X)" ~ b m
x* -> 0
x~0.
account for both possibilities (see Appendix H). In words, conditions (22.13)-(22.15) state that x* must be nonnegative, that first-order partial derivatives must be nonpositive, and that one of the terms x~ and aF/axj(x*) must be zero for all j = 1, ..., n. These conditions are required whether or not x~ is nonzero; however, they reduce to aF/axj(x*) = 0 for xj > 0, and to (aF/ax)(x*) = 0' for x* > 0.
The constraints are of two types, namely, nonnegativity conditions _x -> _0and the inequality constraints g(x) _ b. Note that this formulation can accommodate both equality and inequality constraints, because the equality constraint gi(x) = b i can be expressed as two inequality constraints, gi(x) ~ b i and - g i ( x ) ~ - b i. Note also that it is possible to express a free (unconstrained) variable, say Xl, as the difference Xll -- X12 o f t w o nonnegative variables. For these reasons both unconstrained optimization and classical programming can be seen as special cases of nonlinear programming. The nonnegativity conditions x -> 0 restrict the feasible solutions in nonlinear programming to the nonnegative orthant of the n-dimensional Euclidean space E". In general, feasible solutions are restricted to a subset of E n by each constraint gi(x) ~ b i. The opportunity set X consists of values x in the intersection of these subsets. There is no limit on the number of allowable constraints in nonlinear programming, but in general the inclusion of additional constraints reduces the size of the opportunity set. Assuming convexity of the opportunity set
(22.15)
Example
As part of its program to maintain fishing opportunities, management wishes to conduct an annual reservoir stocking program. Fish hatchery facilities are available for stock production, and two species of fish can be used. Species-specific costs associated with fry production, growth, and stocking depend on stock size. On the other hand, benefits such as alternative uses for stocks and facilities accrue to the maintenance of a stocking program. From a previous study, the cost of stocking (net of benefits) at levels of x I and x 2 is estimated to be g(x) = 2x 2 + 3x 2 - x l x 2 - 6x 1 + 13x 2 + 16. Thus, costs (net of benefits) can be reduced by maintaining a stocking program. A total of $20,000
598
Chapter 22 Traditional Approaches to Optimal Decision Analysis
is available annually for the stocking program, and u n u s e d funds can be reallocated each year to meet other m a n a g e m e n t needs. It is therefore useful to identify a stocking regime that minimizes costs or, equivalently, maximizes residual funds. Because the stocking decisions must meet nonnegativity conditions, this problem is amenable to nonlinear programming, with an objective function of F(x) = 20 - g ( x ) = 4 - 2 x 2 - 3 x 2 + x l x 2 + 6x I - 13x 2 and the inequality restrictions x I _> 0 and x 2 >_ 0. Partial differentiation of the objective function yields 3 F / O x I = - 4 x I + x2 + 6
and
constraints, with modifications to account for the nonnegativity constraints as in conditions (22.13)-(22.15). After some algebra (see Appendix H) the conditions for optimality can be written as OF (x*)
0x -
k* Og - 3x(X*)---0' b - g ( x * ) ~ O,
3F ag ] 7x (x*)_ - _x* ox-(X*) _x* = 0,
(22.16)
m
x*Eb - g(x*)] = 0, 3F/Ox 2 = xI
6X2 -- 13.
--
X* --> 0, m
The conditions (22.13)-(22.15) above present four possibilities, d e p e n d i n g on whether the optimal stocking levels x~ and x~ are nonzero: 1. If x~ = x~ = 0, then F(x*) = 4. 2. If x~ = 0 but x~ > 0, then condition (22.14) requires c]F/Ox 2 = O, so that Xl-6X 2 = 13 or x 2 = - 1 3 / 6 , violating the nonnegativity condition (22.15) for x 2. 3. If x~ > 0 but x~ = 0, then condition (22.14) requires c]F/Ox I = O, so that 4x 1 - x2 = 6 or x I = 1.5 with F(0, 1.5) = 8.5. 4. Finally, if x~ and x~ both are nonzero, then 3 F / O x 1 = 3 F / 3 x 2 = 0 and x2 = - 4 6 / 2 3 . Again, this violates the nonnegativity condition (22.15). E
It follows that the optimal stocking regime will involve the stocking of only one species, at a level of 1.5. The total cost for stocking at this level will be g(x) = $11,500, leaving a funding residual of $8500 for other uses.
22.4.2. Nonlinear Programming with General Inequality Constraints In this case the optimization problem is maximize
F(x)
subject to g ( x ) <- b, m
x~O. It is convenient to introduce "slack variables" s' = ($1, ..., Sm), one for each inequality constraint, so that the problem can be described in terms of equality constraints gi(x) - b i + s i -- 0 and nonnegativity conditions s -> 0 and x -> 0. A solution approach then combines the use of Lagrangian multipliers to handle the equality
k* ~ 0'. These equalities and inequalities are k n o w n collectively as the Kuhn-Tucker conditions (Kuhn and Tucker, 1951). Several points are w o r t h y of note: 9 The Kuhn-Tucker conditions involve 2n + 2m + 2 individual constraints (n nonnegativity conditions involving x*, m nonnegativity conditions involving _k*, n + m inequality constraints involving x* and _k*, and two equations). In particular, the inequality constraints and nonnegativity conditions in the original problem statement are included. 9 At x*, one or both of Xj a n d 3 F / 3 x j - ~ , i kiOgi/OX j must be zero for each j = 1, ..., n. The Kuhn-Tucker conditions require the latter expression to be nonpositive and to be zero if x~ > 0. Conversely, x~ must be nonnegative and is necessarily zero if O F / 3 x j - ~i )kiOgi/ c~xj < 0 atx*._ 9 Similarly, either k~ = 0 or gj(x*) = bj (or both) for each j = 1.... , m. The Kuhn-Tucker conditions require that gj(x*) <- bj; in addition, gj(x*) = by if k~ > 0. Conversely, k~ must be nonnegative and is necessarily zero if gj(x*) < bj. 9 On assumption that there are no inequality constraints, the Kuhn-Tucker conditions reduce to conditions (22.13)-(22.15) for optimization with nonnegative conditions only. This confirms that classical programming can be s u b s u m e d as a special case of general nonlinear programming. 9 If the constraints define a convex opportunity set and the objective function is concave, the Kuhn-Tucker conditions are sufficient to guarantee a global maximum. 9 As before, the optimal Lagrangian multipliers can be interpreted in terms of a marginal change in the
22.4. Nonlinear Programming objective function with respect to the constraint coefficients: OF (x*) = )~* 0b-"
(22.17)
599
at (x*, _k*) for each constraint j. It follows that the Kuhn-Tucker conditions for this problem are 300
-
2x I -
3
500 -- 4x 2 -- 5
m
Example
-
)kI = 0,
-- k I = 0,
- 2 3 + kI - k2 - 0 ,
Consider a situation in which two logistic populations are to be m a n a g e d in an animal caretaker facility. The annual population growth for each population is AX 1 -- 3x1(1 -- X l / 3 0 0 ) a n d AX 2 = 5X2(1 - X2/250), where x I and x 2 represent population sizes. The populations are to be maintained at constant size, and the growth increment is to be sold at $100 per individual each year. Each individual in population 1 requires about $3 a year for maintenance, and each in population 2 requires about $5 per year. Facility x 3 capacity can be e x p a n d e d to accommodate up to 400 individuals and rent and other annual facility costs are expected to total about $23 per unit capacity. M a n a g e m e n t needs to k n o w h o w large a facility to develop and w h a t the population sizes ought to be, in order to minimize net costs. The problem can be expressed as maximize
F(x)
= X 1 ( 3 0 0 -- X1) + X 2 ( 5 0 0 -
3x I
-- 5X 2
2X2)
- 23x 3
subject to X3 ~> X 1 + X2r X3 ~<
400,
X 1 ~> 0, X2 ~
0,
X3 ~ > 0 ,
-
X1) + X 2 ( 5 0 0 -
2X2) -- 3X1
2 3 x 3 + ~.l(X 3 -- x I -- x2) + k 2 ( 4 0 0 -
-- 5X 2 x3).
A s s u m i n g for now that the nonnegativity conditions on x are met by an optimum, the equalities in expression (22.16) require that
ag}
aF
COXi
E J
k2(400
-- X3) = 0, KI-->0, k2>--0.
An examination of possibilities for K 1 and k 2 s h o w s that they both cannot be zero, because that would violate the third condition above. Nor can we have k 1 -0 and k 2 > 0, because the third condition would then require that ~-2 -- --10. Consider the case in which K 2 = 0 and ~1 > 0. From the third condition we have K 1 = 23, so that x I = 142, x 2 = 103, and x 3 = 245 from the first, second, and fourth conditions, respectively. Thus, an optimal decision is to limit capacity below allowable limits and maintain populations at levels of 142 and 103 individuals. This example is unusual, in that it is possible to identify an optimal solution directly from the K u h n Tucker conditions. In general, the Kuhn-Tucker conditions tell us about the mathematical nature of an optimal solution, but by themselves are not particularly useful in helping to find one. It usually is necessary to take advantage of the differential properties of the objective function and constraints in a procedure that accounts explicitly for the constraints and leads in sequential steps to a local optimum.
22.4.3. Solution Algorithms for Constrained Nonlinear Optimization
and the Lagrangian is
L(x, k) = x 1 ( 3 0 0 -
K I ( - - X 1 -- X2 + X3) = 0,
)kJ~xi -" 0
at (x*, _k*) for each state variable xi, and Xj[bj - gj(x)]
= o
As with unconstrained optimization, the methods for constrained problems almost always involve an iterative search for an o p t i m u m x*, each step of which consists of choosing the "best" direction, and determining the length of the step to be taken in that direction. Though a mathematical description can be difficult, in concept the specification of such an iterative algorithm is straightforward. To illustrate, in w h a t follows we focus on gradient or "gradient-like" searches for constrained optimization, recognizing that these are but a few of m a n y approaches that are available (See Appendix H). At each iteration of a gradient-based procedure, the direction of search is initially chosen as the gradient
600
Chapter 22 Traditional Approaches to Optimal Decision Analysis
VF(xk) = OF(Xk)/Ox. A generic algorithm includes the following steps: 1. Identify a feasible starting value x 0 (this may or may not be a simple task, depending on the constraints). For opportunity sets that include the origin, a possible starting value (though not necessarily a good one) is 0. 2. Move in the direction of the gradient VF(x 0) for a distance determined by a selected step size 80 and thereby locate a new feasible point X.1 with F(Xl)>F(x0). 3. At iteration k, move in the direction of the gradient VF(xk) for a distance determined by step size 8k and identify a new feasible point Xk+1 with F(Xk+l)>F(Xk). 4. Repeat until a stopping criterion is satisfied. Key issues for such a search algorithm are (1) the choice of an appropriate step size at each iteration, and (2) the choice of a search direction that remains in the opportunity set. Difficulties arise when a step of size 8k in the direction of the gradient leaves one outside the opportunity set, or when movement along the gradient takes one immediately outside the set (e.g., x k is on the boundary of X and VF(xk) points away from X). The added complexity attendant to searching under constrained optimization is directly related to these situations. Three common approaches to them are the gradient projection method, the method of feasible directions, and the Lagrangian differential gradient method. Gradient projection is based on a suitable modification of gradient search (see Appendix H.1.4) to account for the constraints. It starts with an initial value x 0 in the opportunity set X_ and moves at each step in the direction of the gradient of F, provided that direction remains in _X. If at some step in the iteration, the gradient direction is infeasible [i.e., if x k is on a boundary of X and VF(xk) points away from X], the direction of movement is altered to follow the projection of the gradient vector on the tangent to the boundary of X. The corresponding step size is chosen to increase the value of the objective function while remaining in the opportunity set. Iterative application of the algorithm can be shown to converge to x*, provided the objective function is concave and the opportunity set is convex. An alternative approach is the method of feasible directions, which involves choosing a direction D k that deviates as little from VF(xk) as possible, while ensuring that at least some movement in that direction is possible. If the operative constraints are linear, under some rather mild conditions on the normalization of candidate directions D k, a feasible direction can be found at each step via linear programming (Luenberger, 1989). The corresponding step size in the direction of D k typically is determined by the nearer of (1) the point where
the direction vector leaves the opportunity set X, or (2) the point at which F(x) reaches a maximum in the direction of D k. Yet another approach to constrained optimization is the Lagrangian differential gradient method, which uses gradient search with the Lagrangian L(x, ~) rather than the objective function F(x)._ In this case the algorithm begins at an arbitrary initial value x 0 and moves from that point according to the gradient components OL/Ox and OL/3K of the Lagrangian; hence the name "Lagrangian differential gradient method." If the objective function is concave and the inequality constraints are convex, the procedure converges to optimizing values of x and _Kstarting at arbitrary values of these variables. Algorithms such as gradient projection, the method of feasible directions, and the Lagrangian differential gradient method require evaluation of partial derivatives of both the objective function and the inequality constraints each time a new search direction is determined. The corresponding computational requirements increase quickly with increasing numbers of decision variables and constraints and with increasing mathematical complexity. A further challenge is to ensure that a value x* thus identified is in fact a global maximum rather than a local maximum. Recognizing global maxima becomes much more difficult as the number of decision variables and constraints increases, and especially as the mathematical complexity of the problem increases. There is a wide variety of different approaches to constrained optimization, in addition to the methods mentioned above. Frequently cited procedures include (1) primal methods, in which the problem constraints are used to reduce the dimensionality of a search for an optimal value x*; (2) penalty and barrier methods, involving the approximation of a constrained optimization problem by an unconstrained problem, which then can be solved with procedures for unconstrained optimization; (3) dual methods that focus on the Lagrangian multipliers as the fundamental variables to be optimized, with the idea that determining optimizing values for the Lagrangian multipliers is tantamount (at least in some cases) to finding the optimal solution x*; and (4) Lagrangian methods that focus on simultaneously solving for the optimizing values of the decision variables and Lagrangian multipliers in the Lagrangian function. Many of these procedures are adapted from procedures for unconstrained problems (see Appendix H.1.4). In general, their rates of convergence are controlled by the structure of the Hessian matrix of the Lagrangian, much as convergence rates for unconstrained problems are controlled by the Hessian of the objective function (Luenberger, 1989). m
m
22.5. Linear Programming
601
22.4.4. Summary In this section we have described procedures for nonlinear programming, extending the classical programming problem of the previous section by including inequality constraints on allowable values of the decision variables. Thus, nonlinear programming procedures are used to identify a vector of decision variables that maximizes an objective function of them in the presence of inequality constraints and nonnegativity conditions. The objective function and constraints must be continuously differentiable in the decision variables. Solution approaches involve the following considerations: 9 Lagrangian multipliers _~ are introduced to account for the inequality constraints and are included along with the original objective function in a Lagrangian function. 9 Differentiation of the Lagrangian function with respect to x and _h leads to derivation of the Kuhn-Tucker conditions (22.16). 9 The optimal Lagrangian multipliers ~* describe sensitivities of the optimal value F(x*) of the objective function to changes in the constraint constants of the inequality constraints [Eq. (22.17)]. 9 On condition that the objective function is strictly concave and the constraints describe a convex opportunity set, a local optimum is also global. 9 Many approaches are available for finding an optimizing value x*, depending on the mathematical structure of the problem, the dimensionality of the decision space, and the nature of constraints defining the opportunity set. In most cases, computer-based search procedures must be used to identify values (x*, h_*) satisfying the Kuhn-Tucker conditions. m
22.5. LINEAR PROGRAMMING Linear programming is a special case of nonlinear programming, in which both the objective function and the constraints are linear combinations of the decision variables. A statement of the problem involving n decision variables and m constraints is maximize
a
x_>0, where c = (c 1, c2, Cn)' is a vector of constants in a linear objective function, b = (bl, b2, ..., bin)' is a vector of constraint constants, and
...
aln
a21
a22
...
a2n
am1
am2
...
amn
__
As before, the nonnegativity constraints _x -> _0 restrict feasible solutions to the nonnegative orthant of En. Additional restrictions are imposed by the linear constraints a i l x I + ai2x2 + ... + ainXn ~ b i
in A x <- b, each of which defines a set of points in E n on one side of a corresponding hyperplane m
a i l x I + ai2x 2 +
+ ainX n = b i.
The opportunity set X_consists of values x in the intersection of these subsets (Fig. 22.6). A comparison of the problem statements for linear and nonlinear programming makes it clear that linear programming is a special case of nonlinear programming. Thus, the objective function and constraint functions of nonlinear programming are simply F(x) = c' x
and 1/
g~(x) = ~, aijxj, j=l
respectively, for a linear programming problem. Of course, both problem statements allow for the nonnegativity constraints x -> 0. Example
Under certain conditions, linear programming can be applied to multiperiod problems, in which decisions made in one period influence decisions made in later periods. Consider a general situation in which population growth can be approximated by a linear growth function that is modified by stocking and harvest. The corresponding transition equation is of the form Xl(t +
A x <- b,
a12
m
c'x
subject to
all
1) = xl(t) + axl(t)
+ blxa(t) -
b2x3(t),
where x2(t) and x3(t) represent stocking and harvest, respectively. The multipopulation analog of this transition equation is xl(t + 1 ) = xl(t) + AXl(t) + Bx2(t),
...,
with xl(t) representing the sizes of k populations at time t, and x2(t) - [Xk+l(t), Xk+2(t)]' is a vector of time-
602
Chapter 22 Traditional Approaches to Optimal Decision Analysis
specific stocking and harvest rates. The elements of the k-dimension matrix A are population growth parameters and species interaction coefficients, whereas the elements of the k • 2 matrix B are weighting factors for stocking and harvest effort. Note that these transition equations are linear in the decision variables x I and x__2 at each point in time. A linear objective function in both sets of variables has the form w
Example
It is desired to manage the habitats in an area so as to sustain a balance of four coexisting species. Habitats in the area can be maintained either as forested lands, pastures, or wetlands. Within certain size limits, equilibrium population sizes are thought to be directly related to the amount of available habitats, N i -- oLiH1 4- ~iH2 4- ~/iH3,
F = ~[wl(t)'xl(t) 4- w2(t)'x2(t)], t
where wl(t) and w2(t) are weighting factors that measure time-specific utilities of population status, stocking levels, and harvest. That this problem is amenable to linear programming can be seen by the linear nature of the objective function and transition equations. Of course, other equality a n d / o r inequality constraints can be added as appropriate, so long as they also are linear in the decision variables. Example
Consider a situation in which the biological production of a bird population can be enhanced by efforts directed at either reproduction or survivorship. The balance equation for this population is x(t + 1) = x(t) + R(t) - D(t),
where reproduction R(t) and mortality D(t) are directly controllable at each point in time. To meet the anticipated growth in public demand for viewing this population, managers have set population goals of no less than a 5% growth in the population over time. Let cl(t) measure the cost of maintaining individuals in the population, and c2(t) and c3(t) represent unit costs associated with control of reproduction and mortality, respectively. An objective of management is to minimize the overall costs associated with management of the population. By relabeling xl(t) = x(t), x2(t) = R(t), and x3(t) = D(t), the problem can be expressed as
where N i represents the equilibrium population size for each species and Hj represents the available habitat of type j. The total area that is subject to management is fixed, so that H 1 4- H 2 4- H 3 = H. In order to sustain a balance in the biological community, management wants to constrain population sizes so that no population constitutes more than half the total number of individuals in the community. The benefits accruing from visitor fees, funding support, and donations are estimated to be directly related to the size and composition of the biological community, according to r l N 1 4- r2N 2 4- r3N 3 4- r4N 4. Annual costs associated with maintenance of the habitats a r e ClH 1 4- c2H2 4- c3H3. The objective of management then is to maximize returns net of costs, while meeting the constraints mentioned above. After relabeling population sizes by N i --- Xli and habitat amounts by Hj = x2j, the problem can be expressed as 4
maximize
3
__F'X1 - - C ' X 2 -- ~ l ' i X l i - ~ qX2j i=1 j=l
subject to X1 i -- OLiX21 4- ~iX22 4- "~iX23,
H = x21 4- x22 4- x23 , Xli ~ 0.5 Ex11 4- x12 4- x13 4- x14~, Xli ~ 0, x2i ~ O.
maximize
F = --~Cl(t)xl(t)
4- c2(t)Xa(t) 4- c3(t)Xg(t) ~
t
subject to Xl(t 4- 1) = Xl(t) 4- x2(t) -- x3(t) ,
Clearly, this problem meets the requirements of linear programming. A solution exhibits the appropriate species composition and mixture of habitat types to maximize net returns, while also satisfying biodiversity requirements and the other problem constraints.
Xl(t) ~ X1(0)(1.05) t,
x2(t) ~ 0, x3(t) ~ 0. A solution to this problem describes trajectories for population growth, reproduction, and survivorship that minimize costs while meeting the population growth requirements and other constraints.
22.5.1. Kuhn-Tucker Conditions for Linear Programming As a special case of nonlinear programming, the linear programming problem is amenable to a Lagrangian approach. The Lagrangian function is L(x, ~) = c'x + )~(b - A x ) ,
22.5. Linear Programming and combining the partial derivatives of L(x, k) as indicated in Eq. (H.16) of Appendix H yields the KuhnTucker conditions, c' - kA -< 0', w
b-Ax>_O,
[c' - k A ] x = O, X[b
Ax]
-
=
603
22.5.2. The Dual Linear Programming Problem As a consequence of linearity in the objective function and inequality constraints, one can easily define a second, or dual, linear programming problem simply by switching the role of the Lagrangian multipliers and the decision variables. The dual problem involves minimization of an objective function in the Lagrangian multipliers:
0,
minimize
x~0,
subject to
k --> 0',
hA -> c',
that must be satisfied at (x*, k_*). Several points about these conditions are noteworthy: 9 The Kuhn-Tucker conditions _b - A x >- 0 simply reaffirm the constraint statement A x <- b. 9 The constraints x > 0 and k _> 0' assert that all decision variables in x and all Lagrangian multipliers in _k must be nonnegative. 9 The two Kuhn-Tucker equations [c' - k A ] x = 0 and k[b - A x ] = 0 can be rewritten as c'x - K A x
and kb
=
kb
kAx,
from which we get c'x = kb.
But c'x is just the value of the objective function at x; thus, the vector product h b of the Lagrangian multipliers and constraint constants reproduce the value of the objective function at x* and _k*: F(x*) = k*b.
k ~> 0', B
m
where h A->c ' specifies n linear constraints on the m decision variables in X. Note that the constraint constants of the original (or primal) problem are used here as objective constants, the objective constants in the primal problem are now the constraint constants, and the direction of the inequalities has changed in the dual problem. It can be shown (see Appendix H.4) that the Kuhn-Tucker conditions for the dual problem are identical to the primal problem. This remarkable result means that the primal and dual optimization problems yield the same values _x* and _h*. Thus, if a solution _x* to the primal problem and a solution _X*to its dual problem can be found, taken together the pair (x*, _h*) is guaranteed to meet the Lagrangian conditions and thus to solve the extended Lagrangian optimization problem. Solution procedures (e.g., the simplex algorithmm see below) are available to determine optimizing values of the decision variables for a linear optimization problem. By applying such a procedure to both the primal and the dual problems, we can determine the full solution (x*, _h*) for a Lagrangian formulation of the optimization problem. From Eq. (22.18), the values in _~* represent sensitivities to marginal changes in the constraint constants b in the primal problem: u
m
aL (x* X*) ab
The sensitivities
-
'-
aF =
(x*)=
B
aF (x* h*) = K* ab-'-
(22.18)
to changes in the constraint constants follow directly therefrom. 9 Because the objective function and constraints are linear and therefore concave, (x*, _k*) is guaranteed to be a global solution for the linear programming problem.
Thus, by solving both the primal and dual optimization problems we identify not only the optimizing values x*, but also the sensitivities h* to changes in the constraint constants b in the original optimization problem.
22.5.3. Using Linear Programming to Solve Nonlinear Problems Under certain conditions, linear programming can be used to good effect to address problems that are
604
Chapter 22 Traditional Approaches to Optimal Decision Analysis 2
nonlinear in their objective functions or constraints (or both). One such class of problems is defined by objective functions and constraints of the form
~_j[(OLj 4- ~jXj) - (Cj 4- djx]/2)~ j=l
subject to
F/
r(x)
maximize
f,(xj)
=
X1 4- X2 ~
j=l
and
250,
C1 4- C2 4- dlX~/2 4- d 2 x 1 / 2 ~
2500,
x<_O.
F/
m
m
gij(xj ) <-- bi, j=l
This is clearly a separable programming problem.
i = 1,..., m. This structure defines the class of separable programming problems, the defining characteristics of which are that both the objective function and constraints can be written as linear combinations of functions fj(xj) and gij(xj) of individual decision variables. A formal definition is t/
maximize j=l
22.5.3.1. Solution Approach A general approach to separable programming is to linearize the separable components of the objective function and constraints, thereby transforming the nonlinear problem into one that is linear. Three generic steps are involved.
22.5.3.1.1. Partition of the Constraint Space
subject to
x>_O.
As a first step in linearization, the opportunity set must be partitioned. Partitioning initially involves the identification of endpoint values aj and by such that aj <- xj <- bj for each of the decision variables. Endpoint values for each decision variable often can be identified by straightforward investigation of the constraint inequalities. The values aj and bj provide the linear extent of decision variable xj in the opportunity set, and the partitioning is defined by a set {Pjl, ..., Pjs} of partition points for each decision variable, with Pjl -aj and Pjs = by.
The separable programming problem can be seen as a special case of the general nonlinear programming problem (see Section 22.4). However, the linear structure of separable programming provides an opportunity to simplify the search for optimal solutions.
22.5.3.1.2. Linearization of the Objective Function and the Constraints The functions fj(xj) and gij(xj) are approximated over the partition {Pjl, ..., Pjs} by
t/
glj(Xj ) ~ b 1 j=l
H
gmj(Xj ) ~ b m ..j= l
.. m
fj.(Xj) -- ~fj(Pjr) 4- (1
Example A 250-hectare wildlife preserve is to be managed by water control structures to provide wildlife habitat for a broad range of wildlife species. The preserve managers must decide how much of the preserve to maintain as wetlands (x 1) and how much to maintain as upland (x2). Both habitat types involve habitat-specific costs, which are expected to increase linearly with area according to CI(X1) OtI 4- ~IX1 and C2(X2) Ot2 4- ~2X2, respectively, for two habitat types. On the other hand, benefits to wildlife are anticipated to increase with the available area, according to a l ( x 1) -- c 1 4- dlX~/2 and R2(x 2) = c 2 4- d2X1/2. A maximum of $2500 is budgeted for habitat management on the preserve, which the preserve seeks to allocate optimally for net ecological benefit. Thus, the problem is =
" - -
-
~)fj(Pj, r+l)
and gij(xj) = ~)gij(Pjr) 4- (1 - ~))gij(Pj,r+l)
for r = 1, ..., s - 1, where 0 -< 8 <- 1. Thus, the function
fj(xj) in the objective function is replaced over the partition segment [PjF, Pj,r+l] by a line segment between the points [PjF, fj(Pjr)] and [Pj,r+l, fj(Pj,r+l)], whereas gij(xj) in constraint i is replaced by a line segment between the points [PjF,gij(Pjr)l a n d [(Pj, r+l, gij(Pj,r+l)]" More formally, the linearization offj(xj) and gij(xj) over the full partition is described by S =
r=l
22.5. Linear Programming and S
fi)ij(Xj) = Z ~jrgij(Pjr) r=l
with the additional constraints S
Z ~jr--
1,
r=l S
Z ~jrPjr -- Xj ' r=l
~jr ~
0,
for j = 1, ..., n. The additional constraints ensure that the variables ~jr a r e all nonnegative and less than unity, and that any value within the feasible range of decision variable xj is represented in terms of the set {Pjl .... , Pjs} of partition points.
22.5.3.1.3. Adjacency Condition To ensure efficiency in the search for optimal solutions, an adjacency condition must be imposed on the ~jr values. Thus, we require that at most two of these values can be positive for each j, and if two are positive then they must be adjacent (i.e., if ~jr is positive then positive values are allowed only for ~j,r-1 and ~j,r+l)" This ensures that the search for optima only considers points along the linearized form of the objective function and constraints. The linearized forms of the objective function and constraints can be used along with the adjacency and other conditions shown above to transform the separable programming problem into the approximating problem n
maximize
~
~(xj)
j=l
subject to tl
Z Xlj(Xj ) ~ bl j--1
n
Z gmj(Xj ) ~ bm .j= 1
~G=I, r=l S
Z ~jrPjr -- Xj ' r=l
~jr ~ O,
605
along with the adjacency condition. Because the requirement x -> 0 follows from the nonnegativity of the values 8jr, it has been omitted in the problem statement. Indeed, the variables xj become intermediate variables that are defined in terms of the ~jr i n the equality constraints. Because the values ~jr n O W play the role of decision variables in the approximating problem, if desired, the xj can be removed completely from the problem statement. The approximating problem can be solved with a variant of the simplex algorithm, with restrictions on the values ~jr t o meet the adjacency condition. For a maximization (minimization) problem, if each function ~(xj) in the objective function is concave (convex) and each constraint function gij(xj) is convex (concave), then a solution of the approximating problem with the ordinary simplex algorithm will automatically satisfy the adjacency condition. In either case, the refinement of the partition of the opportunity set toward continuous coverage leads to asymptotic convergence of the approximating solution to the original separable programming problem (Bazaraa and Shetty, 1979).
22.5.4. Simplex Solution Algorithm A search for optimizing values of x can take advantage of linearity in the objective function and inequality constraints. Geometrically it is easy to see that an optimizing value of x must occur at a boundary vertex of the opportunity set (in unusual cases, optimizing values can occur along the intersection of boundary hyperplanes or in the boundary region of a hyperplane; see Fig. 22.6). Because there are only finitely many vertices on the boundary of the opportunity set, an efficient procedure involves searching among the limited number of vertices for an optimal value. The simplex algorithm and its variants essentially describe procedures for systematically searching the boundary vertices of the opportunity set X, starting at an arbitrary vertex in the set (Dantzig and Thapa, 1997). The algorithm moves from the starting point in a direction in which the objective function increases and continues thereafter to move from vertex to vertex until the objective function no longer increases in value. When a vertex is located such that movement to any other vertex decreases the value of the objective function, a unique optimal value _x* has been located. If a further increase in value is not possible but movement to another vertex does not decrease the value of the objective function, then both vertices (and all boundary points between them) are optimal. Because there are only finitely many points to interrogate, the simplex algorithm must identify an optimal solution in finitely many steps (assuming the objective function is bounded on X). Mathematical details of the basic algoB
606
Chapter 22 Traditional Approaches to Optimal Decision Analysis
rithm and variants of it are documented extensively in the literature of systems analysis and operations research (see, e.g., Dantzig and Thapa, 1997; Hillier and Lieberman, 2001).
22.5.5. Summary Linear programming is a special case of nonlinear programming, in which both the objective function and inequality constraints are linear in the decision variables. The linear programming problem involves the identification of a vector of decision variables that maximizes a linear objective function of them in the presence of linear inequality constraints and nonnegativity conditions. A solution algorithm consists of the following: 9 The objective function and constraints for the problem are described as linear combinations of the decision variables in x, along with nonnegativity constraints on x. 9 The dual of the linear programming problem is obtained by switching the roles of constraint constants and optimality constants in the objective function, and searching for a minimizing value of the decision variables in the resulting problem. The linear programming solution of the dual problem produces the optimal values of the Lagrangian variables that would have been identified by solving the Kuhn-Tucker conditions of the original optimization problem. These values are the sensitivities of the original objective function to changes in the constraint constants. 9 The simplex algorithm, or some modification of the simplex algorithm, can be used to search the boundaries of the constraint set for a maximizing value of x. 9 Nonlinear problems can be addressed with linear programming via linearization of the objective function and inequality constraints. Under fairly mild conditions, an approximating linear programming problem can be solved by straightforward application of the simplex algorithm, and the resulting solution converges to that of the corresponding nonlinear problem as the partition of the opportunity set is refined. m
m
22.6. DISCUSSION In this chapter we have dealt with optimization approaches for which the time dimension is inessential in technique applications. We have described the optimization problem generically, in terms of a set of decision variables, an objective function of costs and benefits, and procedures by which to identify values
of the decision variables that maximize the objective function. Traditional approaches are distinguished from the more modern dynamic approaches to optimization covered in the next chapter by the fact that explicit recognition of the time dimension is unnecessary, and no distinction is made between control and state variables. However, classical optimization techniques such as linear, nonlinear, and classical programming can be used with a large number of important biological problems, including many for which temporality is implicit in the problem formulation. There is a natural sequence in the presentation of the optimization approaches in this chapter, with the mathematical complexity of solution algorithms tracking an accretion of mathematical structure in the problem under investigation. Thus, the simplest problem to investigate, and the method of investigation that is the least demanding analytically, involves unconstrained optimization with a mathematically wellbehaved objective function. In this case, the optimal solution can be identified without great difficulty, typically by differentiating the objective function and finding the zeros of the resulting stationarity equations. Additional complexity attends the imposition of equality constraints for the problem. However, an optimal solution still can be obtained fairly easily, by incorporating the constraints directly into the objective function via Lagrangian multipliers and then treating the problem as if it is one of unconstrained optimization. A more complicated problem involves inequality constraints, requiring analysis of the Kuhn-Tucker conditions. Finally, the imposition of linearity conditions on both the objective function and problem constraints defines a linear programming problem, which can be addressed with methods based on the simplex algorithm. Because the Kuhn-Tucker conditions for the primary and dual problems of linear programming are identical, one can derive shadow prices by simply applying the simplex algorithm to the dual problem. Though the problem set that is amenable to classical optimization techniques is quite large, there are many important problems not included in the set that require an explicit accounting of time a n d / o r a distinction between system states and controls. By incorporating these features, the optimal control methods described in the next chapter allow for a more elegant treatment of change in system status and a straightforward accounting of system responses to controls. Optimal control approaches also allow one to handle temporal constraints on system states and controls, as well as stochastic influences on system dynamics. These and other features will be dealt with in the next two chapters.
C H A P T E R
23 Modern Approaches to Optimal Decision Analysis
23.1. CALCULUS OF VARIATIONS 23.1.1. Euler's Equation 23.1.2. Transversality Conditions 23.1.3. Particular Forms of the Optimality Index 23.1.4. General Multivariate Problem 23.1.5. Constraints 23.1.6. Summary 23.2. PONTRYAGIN'S MAXIMUM PRINCIPLE 23.2.1. Unconstrained Optimal Control 23.2.2. Constraints on the Control Trajectory 23.2.3. Special Cases of the Maximum Principle 23.2.4. Sensitivity Analysis 23.2.5. Discrete-Time Maximum Principle 23.2.6. Summary 23.3. DYNAMIC PROGRAMMING 23.3.1. Deterministic Dynamic Programming 23.3.2. Stochastic Dynamic Programming 23.3.3. Summary 23.4. HEURISTIC APPROACHES 23.5. DISCUSSION
absence of a formal treatment of system dynamics, we saw that a number of interesting and important dynamic resource problems can be addressed with traditional optimization. In this chapter we also deal with approaches to systems optimization, but here we make explicit the time dimension and recognize a notational distinction between control variables and system state variables. In what follows, the system state at time t is designated by x(t) to include elements such as time-varying population a n d / o r cohort sizes, habitat conditions, or other indices of a natural resource system. System controls at time t are specified by U(t), and the general optimal control problem for discrete-time systems is tf
maximize {U(t)}eU
~, I(x, U, Z, t) + Fl[X(tf)] t=to
subject to x(t+l) = x(t) + [(x, Z, U, t), x(t0) = x0,
In Chapter 22 we considered approaches to optimal decision analysis in which the dimension of time is absent, or is otherwise not an integral element of the optimization procedure. The optimization problem was described as the selection of values for a set of decision variables that maximize an objective function. We utilized a notation for objective functions, system constraints, and decision variables that is conventionally used for nondynamic optimization problems. In particular, we suppressed the distinction between state and control variables, considering either or both to be instruments in the optimization process. Even in the
x(t) eX, where I(x, U, Z, t) is an optimality index that measures time-specific utility, Fl[X(tf)] is a terminal value function that assigns value to the system state x(tf), and U and X represent admissible control strategies and feasible state trajectories, respectively. In continuous time the problem is maximize {U(t)}~U D
607
tr I(x, U, Z, t) + Fl[X(tf)] to
608
Chapter 23 Modern Approaches to Decision Analysis
subject to d x / d t = [(x, Z, U, t),
x(t0) = x0, x(t) ~ X.
In words, we seek a control trajectory {U(t)} that maximizes an objective functional, conditional on system dynamics and any relevant system and control constraints. Note that the system transition equations essentially act as equality constraints, along with the system initial conditions and other operating constraints. It is the explicit accounting of system dynamics as they are influenced by the choice of controls that distinguishes the optimal control problem here from the traditional optimization problems described earlier. In the sections that follow, we discuss the optimal control problem in terms of variational mathematics, dynamic programming, and heuristic approaches such as simulation gaming. We begin with the calculus of variations, a well-established approach that addresses a limited but important class of control problems. This is followed by a description of the maximum principle, which extends the calculus of variations to handle a more general class of control problems. We then introduce dynamic programming as an alternative approach to the optimal control problem and follow dynamic programming with a discussion of some heuristic approaches to systems control. To simplify notation, in what follows we suppress the exogenous variables in Z, unless they are otherwise needed to characterize stochastic environmental influences.
where 2 = dx/dt. Note that the problem is stated in terms of deterministic systems in continuous time, with the objective functional accumulating values of an optimality index I(x, 2, t) over an interval of time with fixed endpoints x0 and xf. We will see below that this problem statement can be generalized by relaxing the endpoint constraints and by including equality, inequality, a n d / o r integral constraints. The calculus of variations is especially useful for optimization problems in which costs a n d / o r benefits are functions of the system time rate of change. Note that decision making involves the selection of a state variable trajectory rather than a control trajectory, and in fact no mention is made in the above problem statement of system controls. However, one often can recognize the calculus of variations as a special case of the generic optimal control problem. For example, it sometimes is appropriate to treat the time rate of change as if it is directly controlled, i.e., 2 = U(t), so that the control problem can be expressed as
y
t~
maximize {x(t)}
I(x, U, t) dt
to
subject to
~-u, x(t o) = x0, x(tf) = xf.
More generally, the control problem often can be expressed as maximize
tfo[P(x, t) + Q(x, t)g(x,
U)] dt
{U(t)}
23.1. C A L C U L U S OF V A R I A T I O N S The problem addressed by the calculus of variations is analogous to the classical programming problem of choosing decision variables to maximize an objective function, except that here one chooses a function x(t) rather than a fixed vector x. In its classical univariate formulation, the calculus of variations seeks a piecewise differentiable function x(t) that maximizes an integral objective of the function, its time rate of change, and time: t~ maximize I(x, 2, t) dt
subject to 2 = T(x) + W(x)g(x, U), x(t o) = x0, x(tf) = xf,
m
{x(t)}
f
to
subject to
where the optimality index and transition equation both include the expression g(x, U). In this formulation the transition equation can be rewritten as g(x, U) = [2 - T(x)]/W(x)
and substituted into the objective functional, thereby eliminating any reference to U. The problem statement then becomes
x(to) = x0,
maximize x(t~) = x i,
{x(t)}
f t~lP(x, t) + Q(x, t)[2 to
T(x)]/W(x)} dt.
23.1. Calculus of Variations subject to
609
ditions (see below), and in addition there are convexity and continuity requirements to guarantee that a solution is maximal (see Appendix H.5).
x(t o) = x0, x(tf) = xf,
Example
which is consistent with the calculus of variations problem as described above.
Example Consider a nuisance population with population dynamics given by
2 = R(x) - D(x) - u(t)x(t),
Direct control is to be applied to a pest population in an effort to reduce population size, while also minimizing costs associated with pest control. The objective function incorporates population size and control costs, which are expressed in terms of a quadratic function of effort U. The control problem over a period of two units of time is 2
where reproduction R(x) and mortality D(x) are both density dependent, and u(t) is the harvest rate at time t. An objective for management is to maximize the accumulated value of harvests over time, while also minimizing density-dependent costs associated with animal damage:
maximize {U(t)}
2=u, x(0) = 3, x(2) = 0.
where Q(x) represents the value associated with each harvested individual and C(x) is the cost associated with animal damage. The transition equation can be expressed as x(t)u(t) = R - D - 2, and substitution into the objective function results in the problem statement {x(t)}
a 0
subject to
l = f lio[Q(x) u (t)x(t) - C(x)] dt,
maximize
- ~ ( x + U2) dt
By substituting the transition equation into the objective functional, the problem can be expressed as 2
maximize
- ~ (x + 2 2 ) d t
{x(t)}
J 0
subject to x(0) = 3,
f lio[Q(x)( R - D - 2) - C(x)] dt
x(2) = 0.
subject to
Euler's equation for this problem is
x(t o) = x0, aI Ox
x(tf) = xf. In this form the problem now is recognizable as a calculus of variations problem.
23.1.1. Euler's Equation Though the mathematics of the calculus of variations can be rather complicated, requirements for an optimal solution are not difficult to describe. Absent additional constraints, an optimizing function x(t) must satisfy Euler's equation
OI Ox
d ( OI ~ dt\-~xJ = 0
d(0/) =_1
+2~=0,
dt
or ~/ = 1/2. This equation is solved by x(t) = t2/4 + clt + c2, with Cl and c2 determined by the initial and terminal conditions: x(0) = 3 = c2 and x(2) = 0 = 22/4 + 2c I + c2, from which we have Cl = - 2 . Thus, the optimal population trajectory is x(t) = t2/4 - 2t + 3, and the corresponding optimal control strategy is U(t) = 2 = t / 2 - 2. In words, the optimal population trajectory exhibits quadratic declines to extinction as a result of initially large applications of pest control that decline linearly over [0, 2].
Example (23.1)
(see Appendix H.5 for a derivation). Thus, the search for an optimal trajectory for the classical calculus of variations problems reduces to a search for solutions of Eq. (23.1), with boundary conditions given by the initial and terminal conditions x(t o) = x o and x(tf) = xf. The endpoint constraints require transversality con-
Assume for the previous example that values of the optimality index are time discounted, to allow costs earlier in the time frame to be weighted more heavily than costs incurred later. The control problem then becomes, e.g., maximize {x(t)}
-
f 2 e -t/4(X + 22) dt 0
610
Chapter 23 Modern Approaches to Decision Analysis costs are to be taken into account, as are increasing opportunity costs over time. An objective function that includes these factors is
subject to x(0) = 3, x(2) = 0, where the factor e - t / 4 essentially discounts values of the optimality index as time progresses. Euler's equation for this problem is OI Ox
d ( OI ~ = dt \-~x ] -1
2 + 2Y - -~ = 0
or
= 1/2 + 2/4.
is a general solution, with kI and k2 determined by initial and terminal conditions (see Appendix C). The final solution for x(0) = 0 and x(2) = 3 is e t/41/2
e
U2/2-
tx
= --22/2 -- 2 x -
2-
tx.
The optimization problem thus can be described in terms of the minimization of the objective functional 2
l = f [22/2 4- x x 4- 2 4- tx] dt
1
-1
with the quadratic terms representing population benefits and control costs, respectively, and the linear term representing increasing o p p o r t u n i t y costs over time. From the transition equation we have U(t) = 2 + x + 1, and substituting this expression into the optimality index yields
= (X + 1 ) 2 / 2 - - [2 + X + 112/2-- tx
x(t) = 4kl et/4 - 2t + k 2
=
[(x + 1)2/2 - U 2/2 - tx] dt, 0
I = (X + 1 ) 2 / 2 -
It is straightforward to show that
x(t)
l-f
2
-2t
0
+ 3.
It is instructive to compare this solution to x(t) = t 2 / 4 - 2 t + 3 for the corresponding undiscounted problem. Population trajectories for both problems are positive and monotonically decreasing over [0, 2], and both satisfy x(0) = 3 and x(2) = 0. However, the discounted solution is uniformly larger than the undiscounted solution for 0 < t < 2. For example, x(1) = 1.44 for the discounted problem, whereas x(1) = 1.25 for the undiscounted problem. In general, the effect of discounting is to avoid a large expenditure of effort early on, in response to the increased weighting of costs at the beginning of the time frame. This in turn allows for a larger pest population (and therefore greater pest damage) over the time frame. Example
Consider an exponential population that experiences a negative instantaneous rate of growth because of relatively poor habitat conditions, with additional losses from emigration as a result of improving habitat conditions elsewhere. Population dynamics are characterized by
over [0, 2], given x(0) = x 0 and x(2) = xf. Partial derivatives of the optimality index are 3I/Ox = - 2 - t and 01/02 = - 2 - x - 1, so that Euler's equation for this problem is 8I Ox
d(oI) dt -~x
= - (2+t) + (5( + 2)
=5~-t =0. A solution to 5~ = t is x(t) = t 3 / 6 + clt 4- c2, with the constants with c I and c2 determined by initial and terminal conditions: x(0) = c2 = x 0, and x(2) = xf = 4 / 3 4- 2c I 4- x 0 or c I = (xf - x 0)/2 - 2/3. For example, if (x0, xf) = (1, 3) then the function x(t) that minimizes J is x(t) = t 3 / 6 + t / 3 + 1. The corresponding optimal control strategy U(t) = 2 + x + 1 = (t 3 4- 3t 2 4- 2t + 14)/6 follows immediately from x(t). Thus, an optimal strategy has effort U(t) increasing rapidly over [0, 2], which results in accelerating population growth from x 0 to Xy over the time frame.
2 = [-x(t) - 1] + U(t),
23.1.2. Transversality Conditions
where the first term in brackets accounts for a negative growth rate and the second term accounts for losses from emigration. M a n a g e m e n t of this population involves habitat restoration so as to sustain the population over the next 2 years while also minimizing the costs of restoration. Population benefits and restoration
The calculus of variations problem can be generalized to allow for unspecified initial and terminal states, by means of the transversality conditions 0 to
23.1.
Calculus of Variations
Equation (23.2) states that either (1) the variational function ~](t) m u s t vanish at the e n d p o i n t s (a requirem e n t w h e n initial a n d / o r terminal conditions are specified) or else (2) 01/02 m u s t vanish w h e n terminal conditions are not specified [see A p p e n d i x H.5 for a formal definition of ~](t) a n d derivation of Eq. (23.2)]. For example, the terminal time constraint x(tf) = xf forces the variation ~](t) to vanish at tf, thereby allowing (OI/32)(tf) to be free. O n the other hand, the absence of a terminal time constraint allows the variation ~(t) to be free at tf, so that (OI/02)(tf) m u s t vanish. It often is the case in wildlife biology that an initial state for the resource s y s t e m can be identified, but it is unnecessary or undesirable to specify a terminal state. U n d e r these circumstances an optimal trajectory satisfies
Ox
dt
611
state trajectories, and this larger set provides an opportunity to increase the o p t i m a l value of the objective functional (Fig. 23.1). A further generalization allows for relaxation of the terminal time, by specifying a relationship b e t w e e n the s y s t e m state a n d the terminal time. Rather than identifying a fixed value of tf in the p r o b l e m statement, one can i m p o s e the restriction that feasible trajectories m u s t terminate on the curve [tf, C(tf)]. Then the problem becomes one of choosing a function x(t) that satisfies x(tf) = C(tf) while m a x i m i z i n g the objective functional over the interval [t 0, tf] w i t h tf thus constrained. This situation allows for variations ~]t(tf) in the terminal time, w h i c h leads in turn to the optimality condition
= O,
(t4) = o,
x(t) a
and x(0) = x0. The net effect of leaving the terminal state u n d e c l a r e d is to replace one b o u n d a r y condition, x(tf) = xf, with another, n a m e l y (OI/32)(tf) = 0. In either case Euler's equation can be solved for an optimal state trajectory.
Example Consider the previous e x a m p l e of a declining exponential p o p u l a t i o n for w h i c h I = (x + 1)2/2 - U2/2 - tx over [t 0, tf] = [0, 2]. A s s u m e that the initial condition is given as x 0 = 1, but the terminal condition is unspecified. From above w e have x(t) = t3/6 + clt + c2 as a m a x i m i z i n g function, with x(0) = c2 = 1. The transversality condition for the terminal time is
x(t) I
b f
/
OI 02
"i I I
/
m=-(2+x+
J
1)
I
,,,,"1 = -(t2/2
+
c 1) -
(t3/6 + clt + 1) - 1
j
=0, which, w h e n evaluated at t = 2, gives 3c I -- - 1 6 / 3 or c I = - 1 6 / 9 . Thus the optimal solution is x(t) = t3/6 16t/9 + 1. A c o m p a r i s o n with the solution x(t) = t3/6 + t/3 + 1 for a specified value x(2) = 3 at the terminal time indicates the potential influence of this b o u n d a r y condition. As expected, the value J* for the o p t i m i z a t i o n p r o b l e m w i t h o u t a terminal constraint exceeds the c o r r e s p o n d i n g value in the presence of a constraint. In essence, r e m o v a l of the terminal state constraint allows one to consider a larger set of feasible -
t0
t,
FIGURE 23.1 Multiple state trajectories over [0,tf], starting at x(0) = x0. (a) Terminal state x(tf) = xf identified in the problem statement. (b) Terminal state unspecified in the problem statement. By limiting the range of feasible state trajectories, a terminal state condition restricts the choice of an optimizing trajectory.
612
Chapter 23 Modern Approaches to Decision Analysis
for a candidate point [tf, x(tf)] satisfying x(tf) = C(tf). This condition essentially says that either the terminal time is fixed [so that Tit(tf) vanishes and the condition in brackets is free] or else the terminal time is unspecified [so that ~t(tf) is free and the expression in brackets must vanish]. Example It is desired to minimize the costs of maintaining a population through time, given that there are both fixed and variable operations costs. An objective function that emphasizes variable costs is based on the optimality index I = [1 + U2] 1/2. M a n a g e m e n t is flexible about the time frame for population maintenance; however, for logistical reasons, a smaller population is required at the end of a more extended time frame, according to xf = 2 - tf. The optimization problem is thus minimize {U(t)}
cally, the objective functional measures the distance along x(t) from (0, 1) to (t, 2 - t) (Fig. 23.2). But the minimizing path from (0, 1) to the line x = 2 - t is a straight line that is perpendicular to x = 2 - t and intersects it at t = 1/2.
23.1.3. P a r t i c u l a r F o r m s o f t h e Optimality Index Depending on the mathematical form of the optimality index, several special cases of Euler's equation can be identified" 9 I = I(x, 2). In this case the objective functional depends on both the system state and its time rate of change, but does not d e p e n d explicitly on t: a I / a t = 0. Expressing Euler's equation as
d( +I)
ftf [1 + U2] 1/2 dt 0
subject to
-
0 -
o-7
=
it is easy to see that this condition implies
2 - - Mr I
x(0) = x0, with x(tf) lying on the curve C(tf) = 2 - t f . After substitution of 2 = U into the optimality index, Euler's equation is
Ox
dt\O2J = ~
(1 + 2) 1/2
(1 + 2) 3/2
0I. a -xx = c,
(23.3)
where the constant c is determined from initial and terminal conditions. Example. Consider a previous example of an exponential population with negative growth rate and losses from emigration. We assume here that opportunity costs are independent of time, so that I(x, 2) = 22/ 2 + x2 + 2 + x. A trajectory {x(t)} is desired that maximizes the accumulation of these index values over
=0, so that x(t) = a t + b and x(0) = a(O)+b = 1. Thus, the minimizing trajectory for this problem is linear in t over a time frame from t = 0 to the point of intersection with the terminal curve C(tf) = 2 - tf. The point of intersection is given by the transversality condition
2 ] q" (1 + 9~2)1/ 2
( C _ 2)O/ &/: + I : ( - 1 - 2) (1 + X--2-1/2 ) -2+1 (1 + 22) 1/2 =0,
so that 2 = 1. It follows from x ( t ) = at + 1 that a = 1, and therefore x(t) = t + 1. The terminal time tf then is obtained by combining x(t) = t + 1 with the terminal condition x(tf) = 2 - tf to get tf + 1 = 2 - tf, or tf = 1/2. On reflection, this result is intuitive. Geometri-
x(t) 2
,
//""\>,, I/d/---"
II
//
tl / / . / Xo~ ~ /
-%\ \
\.
\
\
\
\
FIGURE 23.2 Multiplestate trajectories from (0,1) to the terminal curve x(t) = 2 - t. The trajectory of minimum length is a straight line that is perpendicular to the terminal curve.
23.1.
[0, 2]. Because the optimality index is of the form I(x, ic), Eq. (23.3) is operative. We then have OI/Oic = Yc + x + 1, and c = I
613
C a l c u l u s of V a r i a t i o n s
ic = - t + U(t) that the optimal control is linear over time: U(t) = 2 t + 1. 9 I = I(x, t). In this case the objective functional d e p e n d s on the s y s t e m state, but not on the change in s y s t e m state. Then Oi/Oic = 0 so that Euler's equation simplifies to
0I. a-x
= ( i c 2 / 2 4- xic 4- ic 4- x) -- ic(ic 4- x 4- 1)
OI/Ox = O.
= - - i c 2 / 2 4- x.
Differentiation of this expression with respect to x yields 5/= 1, so that x(t) = (t2/2) 4- clt 4- c 2. The initial and terminal conditions can be used to d e t e r m i n e c 1 a n d ca, by x(0) = ca = x 0, and x(2) - xf = 2 + 2c 1 4x 0 or c I = (xf - x0)/2 - 1. For the particular case in which, e.g., (x 0, xf) = (1, 3), this gives x(t) = t 2 / 2 + 1 as an o p t i m i z i n g p o p u l a t i o n trajectory for the problem. From the transition equation ic = [ - x ( t ) - 1] + U(t) the c o r r e s p o n d i n g optimal control strategy is given by U(t) = ic + x + 1 = t 2 / 2 + t + 2.
9 I = l(ic, t). In this case the objective functional d e p e n d s on the time rate of change in s y s t e m state, but not on the s y s t e m state. Then 0 I / O x = 0, and Euler's equation becomes o
(23.5)
But this is s i m p l y an u n c o n s t r a i n e d o p t i m i z a t i o n problem, involving the choice of x to o p t i m i z e the value of I. Thus, an optimality index of the form I(x, t) allows one to solve the calculus of variations p r o b l e m by solving a series of traditional o p t i m i z a t i o n problems, one for each time in the time frame. Example. M a n a g e m e n t seeks to m i n i m i z e the deviations from a target p o p u l a t i o n trajectory a(t), while also m i n i m i z i n g time-specific costs c(t)x(t) associated w i t h p o p u l a t i o n size. An optimality index for this p r o b l e m is I = Ix(t) - a(t)] 2 + c(t)x(t), a n d because it is of the form I(x, t), w e use Eq. (23.5) to get OI/Ox = 2[x - a(t)] + c(t) = 0 or x(t) = a(t) - c ( t ) / 2 . This suggests that the optimal trajectory tracks the target a(t), with modifications at each point in time based on per capita costs
c(t).
or
23.1.4. General Multivariate Problem OI
Oic
= c,
(23.4)
In its classical multivariate form, the calculus of variations p r o b l e m is
w h e r e c is a constant that is d e t e r m i n e d from the initial and terminal conditions.
f'
maximize Ix(t)}
Example. Consider a p o p u l a t i o n that declines in the absence of control as a linear function of time: ic = - t + U(t). A control trajectory is desired that will minimize costs over [0, 2], according to objective functional
J =
f
I(x, 2, t) dt
to
subject to x(t 0) = x0,
2 U2
x(t ) - x ,
- ~ dt. o
w h e r e x' = ( X l , . . . , Xk). Note that the m u l t i v a r i a t e nature of the p r o b l e m allows considerable flexibility in the form of the objective functional. For example, the optimality index can be a function of some, all, or none of the state variables a n d / o r their time rates of change. O p t i m a l i t y conditions for the multivariate p r o b l e m are completely analogous to those for the univariate problem. In particular, the multivariate version of Euler's equation is w
Substituting U(t) = ic + t into the optimality index produces I = ic2/2 4- tic + t 2 / 2 , w h i c h is of the form I(ic, t). We therefore use Eq. (23.4) to get OI/Oic = ic + t = Cl, or x(t) = - t 2 / 2 + clt 4- c 2. As before, initial and terminal conditions can be used to d e t e r m i n e c I a n d c2: x(0) = c2 = x 0 and x(2) = xf = - 2 4- 2c 1 4- x 0 or c I = 1 + ( x f - x0)/2. For (x 0, xf) = (1, 3), this gives x(t) = - t 2 / 2 + 2t + 1 as an o p t i m i z i n g p o p u l a t i o n trajectory for the problem. It follows from
Ox
dt
_
-"
614
Chapter 23 Modern Approaches to Decision Analysis
involving k equations, one for each of the state variables. The corresponding transversality conditions for initial and terminal times are
which can be combined with the constraints to characterize a solution. Example
0
(23.7)
for t = t o, tf. A state variable trajectory {x(t)} that satisfies Euler's equation and the initial and terminal conditions x(t 0) = x 0 and x(tf) = xf is called an extremal, and the optimal solution for a calculus of variations problem with specified b o u n d a r y conditions is necessarily extremal. Note that an extremal trajectory in the calculus of variations plays a role analogous to that of a stationary point satisfying aF/ax = 0 in mathematical programming.
23.1.5. Constraints It is possible to incorporate certain kinds of constraints in the calculus of variations problem. In particular, equality, inequality, and integral constraints can be handled by straightforward extensions of Euler's equation.
A population with linear transitions ~ = 2x + U / 2 is to be m a n a g e d so as to minimize l =
-~- dt, o
while ensuring that the population grows from x 0 to xf over 1 year. One approach is to use the transition equation to transform this problem into the standard calculus of variations format, as described above. Another is to treat the control variable U as another state variable, with x = x I and U = x 2. Then the system transition equation is 21 = 2x I 4- x 2 / 2 , which can be handled as an equality constraint and incorporated into the objective functional with a time-varying Lagrangian multiplier: L = x 2 / 2 + h(2x I 4- x 2 / 2 -
21).
Euler's equation for the problem then becomes
23.1.5.1. Equality Constraints
ax
dt
~_x
A statement of the optimization problem that includes equality constraints is
=
k x 2 4-
h/2
=0, which gives
maximize {x(t)}
I(x, 2, t) dt ~( = -2X
to
subject to
X 2 -"
-X/2
21 = 2X 1 + X 2 / 2 .
g(x, 2, t) = a, x(t 0) = x0,
x(t;~) = x~. As in mathematical programming, a solution approach involves a set of Lagrangian multipliers K_ = (X1, ..., kin), one for each of the constraints in g(x, 2, t) = a. The a u g m e n t e d optimality index is
The first equation yields X - Cl e-2t, SO that x 2 = -- c l e - 2 t / 2 from the second equation and 21 -- 2X 1 -- c l e - 2 t / 4 from the third. The latter equation is solved by x(t) = [16c2e2t - cle-2t]/16, with the parameters c 1 and c2 determined by the initial and terminal conditions: x 0 = c2 - Cl/16 and xf = c2e 2 - c l e - 2 / 1 6 .
m
L(x, 2, X, t) = I(x, 2, t) - )~[a - g(x, 2, t)],
Example Consider the optimal control of the linear system_x = A x + B U to minimize a quadratic objective functional
and a solution is obtained by maximizing J' =
f'
J = 1/2 g(x, 2, )t, t) dt.
to
As before, this leads to Euler's equation,
Ox
d-t\~_~/
-'
[ U ' R U + x ' Q x ] dt, to
subject to the constraints x(0) = x 0 and x(1) = x f (assume without loss of generality that R and Q are symmetric matrices). The Lagrangian for this problem is L = (U' R U + x ' Q x ) / 2
+ h(Ax + BU-
2),
23.1. Calculus of Variations and Euler's equations are o, Ox
dt\O2_J
x,
615
and augmenting the objective functional by means of time-varying Lagrangian multipliers gives Q + h A + ~_ =
o,
1
f0 [x2/2
+ ~.1(x2- 21) 4- ~.2(x3- 22) ] dt.
and Euler's equation then is o, OU
a_(oL dt\OU]
= u'R
+
= o' -"
OL'
OX
The optimal solution is therefore characterized by the system of linear differential equations (see Appendix C)
dt\02]
~'1 4- ~2 X 3 4 - K2
--
=0,
2=Ax+BU,
)~A,
~. = - Q x -
or ~.1 =
0,
with the control trajectory for _U in this system given in terms of the Lagrangian multipliers:
J~2 = --~1,
U = - R -1 BK'.
X3 = -- h 2.
A minimizing solution depends on the existence of an inverse for _Rand also requires that _Rand Q be positive definite. On condition that an initial value x(t o) is specified but x(tf) is not, the transversality condition specifies that (OL/O2)(tf) = K(tf) = 0', and identification of an optimizing control requires the solution of a twopoint b o u n d a r y value problem.
From the first of these equations ~'1 -- Cl, from the second equation ~'2 -- --clt + c2, and therefore the third equation gives x 3 = clt - c 2. From the transition equations we then have x 2 = c l t 2 / 2 - c2t 4- c 3 and x 1 = c l t 3 / 6 - c2t2/2 + c3t + c 4. The initial and terminal conditions can be used to solve for the constants c 1, c2, c3, and c4 in the equation for x 1, producing xl(t) = 3t 3 - 5t 2 + t + 1. The first derivative of x I gives the instantaneous rate of growth as x2(t) = r(t) = 9t 2 10t + 1, and the second derivative of x I gives the optimal control as x 3 = U = 18t - 10. From r(t) = (9t - 1)(t - 1) it is easy to see that the instantaneous rate of growth decreases from r = 1 to r = 0 at t = 1/9, declines yet further to r = - 1 4 / 9 at t = 5/9, and then increases to zero at t = 1 (Fig. 23.3). In response, the population increases for t ~ [0, 1/9] and then decreases to zero at t = 1. At first glance it may seem counterintuitive that an optimal strategy to eliminate the population w o u l d allow it to increase over part of the time frame. Recall, however, that the population was ass u m e d to be increasing initially, with 2(0) = 1. T h o u g h the optimal population growth rate begins immediately to decline from unity, a small increment of time is necessary before the growth rate becomes negative and the population begins to decline.
Example
Consider an exponential population for which initial and terminal growth rates are 1 and 0, respectively. The population rate of growth is to be controlled directly, with an objective of eliminating the population in 1 year. Thus, effort U(t) is to be applied over the interval [0,1] to influence the rate of change r according to d 2 x / d t 2 = d r / d t = U. Note that this is a s o m e w h a t different formulation of the control problem, in that the instantaneous rate of growth parameter is controlled rather than the population. The objective is to minimize l =
flu2
-~- dt
0
subject to initial and terminal conditions on both x and 2, as specified by x(0) = 2(0) = 1 and x(1) = 2(1) = 0. The problem can be formulated in terms of the calculus of variations by changing notation to x I = x and x 3 = U and introducing another variable x 2 = r such that 21 --- X2 and 22 = x 3. It is easy to see that (d/dt)(21) = X2 = X3 or x1 -- X3, which is equivalent to d2x/dt 2 = U in the original problem statement. System dynamics are expressed in matrix form by
[~:] = [~
10][;:] 4- [~] x3,
23.1.5.2. Inequality Constraints A Lagrangian approach can accommodate inequality constraints of the form g(x, 2, t) <- b, through the identification of Lagrangian multipliers for each of the inequality constraints and their incorporation into an a u g m e n t e d optimality index L(x, 2, )~, t) = I(x, 2, t) + )~[b - g(x, 2, t)]
616
Chapter 23 Modern Approaches to Decision Analysis d2x/dt 2
6
x(t) a
1.0
dx/dt
0.8
4
1.0 0.6 0.4
2
0.5
0.2 0.0
0.0 0.2
0.4
0.6
0.8
J
i
|
0
I
,
1.0
.6
-0.5
0.8
1.0
-2
-1.0 -4
-1.5 -6
-2.0
F I G U R E 23.3 Optimal state and control trajectories for the system d2x/dt 2 = U with quadratic optimality index and boundary conditions x(0) = ~(0) = 1 and x(1) = ~(1) = 0. (a) Population size is given by x(t) = 3t 3 - 5t 2 + t + 1. (b) The instantaneous growth rate r(t) is given by the time derivative ~(t) = 9t 2 - 10t + 1 of population size. (c) The optimal control trajectory U(t) is given by the time derivative f(t) = 18t - 10 of the instantaneous of growth.
as before. The resulting optimality conditions are
Ox
dt
maximize population size Xl(t f) at the end of the time frame. System transitions again are given by
0 - '
I~12] -- I ~
10]IXX12]-}-I~IX3 '
g(x, Yc, t) <_ b,
)~-~0, m
K[b - g(x, 2, t)] = 0.
Note that the first set of conditions simply expresses Euler's equation for the augmented optimality index L, whereas the conditions that follow Euler's equation are the Kuhn-Tucker conditions described in Section 22.4. Example
A treatment of inequality constraints can be illustrated by imposingbounds Xmi n ~ X3 < Xma x o n the allowable range of values for x 3 in the previous example, assuming in this instance that the objective is to
subject t o x l ( t 0) = Xo, x2(t 0) = Vo, a n d x2(tf) = vf. B o u n d s on the control can be handled by defining a new variable a by which to change the bound inequalities into an equality constraint: (X3 -- Xmin)(Xma x -- X3) = a 2.
It is clear that this equation can be satisfied only if Xmin ~ X3 ~ Xmax, for otherwise the product would be negative. The extended objective functional
ftf
{~kl(X 2 _ 9c1) -ff ~k2(x3 _ 22) q- )k3
to
X [(X 3 -- Xmin)(Xmax -- X3) --
a2]}dt + xl(t f)
23.1. Calculus of Variations n o w includes three time-varying Lagrangian multipliers, and Euler's equation yields
617
The problem then is solved by maximizing this functional with respect to x and minimizing with respect to h.
~1 = 0 ,
Example
~2 = ~1' h2--
h3(2X3--Xmax--Xmin),
a h 3 = O.
The transversality condition corresponding to xl(t f) is 0L c921(tf) -- --1 -- hl(t f) - 0,
so that the optimal control trajectory is given as a solution of
T h e r a t e of growth for an exponential population with initial size x 0 at time 0 is to be controlled so that it is of size x T at time T. Allowance is to be m a d e for variation in the population size over [0,T]; however, an average population size of N is desired over the time frame. Control is to be imposed so as to minimize a measure of the effort U(t) over [0,T]. This optimization problem can be stated as T maximize - f U2 dt {u(t)}
0
subject to
21 -- X2 '
~=u,
22 = X3, T
f ox dt = N / T ,
~1 - - 0 , ~'2 = - - h l ,
x(to) = Xo,
a h 3 = O,
x(T) = x T.
h 2 = h3[2X3--Xmax--Xmin] , a2=
(X3--Xmin)(Xmax--X3) ,
The a u g m e n t e d optimality index is L = - 2 2 + Xx, and Euler's equation is
Xl(t0) = Xo,
OL 3x
x2(to) = V0, or
x2(t f) -- Vf,
~/= -h/2.
)kl(t f) = --1.
This is a nonlinear two-point b o u n d a r y value problem, requiring the aid of a computer to determine its solution. However, if we assume that X m i n - - - 1 and Xmax = 1, it follows that the constant a must be zero. The optimal control then takes the value x3(t) = 1 w h e n h 2 > 0 and x3(t) = - 1 w h e n )k2 < O.
23.1.5.3. Integral Constraints A third set of constraints in the calculus of variations involve integral constraints of the form
ft~ G(x, 2, t) dt = c, to where G(x, • t) is assumed to be differentiable over T. A Lagrangian approach again is appropriate, with the product of a Lagrangian multiplier and the integrand G(x, • t) a d d e d to the optimality index:
ftf to
[I(x, 2, t) + KG(x, 2,
d 'OL~ dt ,32] = h + 2 5 i = 0,
t)] dt.
A solution to the latter equation is x(t) = - h t 2 / 4 + clt + c2, with q , c2, and X determined by the constraints X(0) -- C2 "- X0, x(T) = - K T 2 / 4
+ C l T + c 2 = XTr a n d
- h T 3 / 1 2 + CLT2/2 + c2T = N / T . As a case in point, l e t T = 1, x 0 = 1, x T = 2, a n d N = 2. T h e n c 2 = 1, c 1 = 6 N - 8 = 4, h = 1 2 ( 2 N - 3) = 12, and the optimal population trajectory is x(t) = - 3 t 2 + 4t + 1. The corresponding optimal control strategy is given by U(t) = 2 = - 6 t + 4. Thus, linear control induces quadratic population growth to a m a x i m u m population size of 7 / 3 at t = 2/3, followed by population declines to N = 2 a t t = 1.
23.1.6. Summary The calculus of variations is analogous to classical programming, in that both optimization procedures involve the choice of a decision instrument [a vector of decision variables x in classical programming; a function x(t) in the calculus of variations] to maximize an objective. In the classical formulation of the calculus
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Chapter 23 Modern Approaches to Decision Analysis
of variations, a piecewise differentiable function is sought that maximizes an integral expression of the function, its time rate of change, and possibly time. A solution approach involves the following considerations: 9 Euler's Eq. (23.6) is used to identify a general form of the solution. 9 Transversality conditions (23.7) are used to determine the specific solution, depending on whether initial conditions, terminal conditions, and the terminal time are specified in the problem statement. 9 Equality, inequality, and integral constraints can be included in the calculus of variations by incorporating them into the optimality index via Lagrangian multipliers. 9 The system of differential equations resulting from Euler's equation and the transversality conditions can be difficult to solve, depending on the number and mathematical complexity of the equations. From the foregoing, it is clear that the calculus of variations can handle a number of interesting problems involving dynamic systems with several kinds of operating constraints on the system state variables. However, it is difficult for the approach to handle problems with complex constraints on the control set. To find solutions for optimization problems with general control constraints, more recent developments, such as dynamic programming and the maximum principle, are available.
of the calculus of variations. Note also that the objective functional can incorporate a terminal value function Fl[x(tf)], usually in the absence of a terminal time constraint. This formulation can be extended to include both initial and terminal time values. A solution to the control problem can be obtained with a Lagrangian approach as before, by introducing multipliers_X(t) = [~l(t), ..., )~k(t)] for each of the equality constraints x_"- / ( x , U, t) = 0. Recall that we previously introduced time-varying Lagrangian multipliers, which here are called costate variables, as a way of dealing with constraints in the calculus of variations. In the following development we define the Hamiltonian to be the sum of the optimality index I(x, U, t) and the product _X(t)[(x, U, t) of costate variables and transfer functions: H(x, U, )~, t) = I(x, U, t) + _X(t)~(x, U, t).
The augmented objective functional then has the form J' =
f'
[S(x, U, X, t ) -
maximize {U(t)}, U D
m
f'
I(x, U, t) dt + f l[x(tf)l
to
subject to Yc = fix, U, t),
x(t 0) = x0, x(t ) = x ,
where x = ( x I . . . . , Xk)'. Note that the allowable controls in {U(t)} are constrained to be in a control set U, and system change ~ is influenced but not directly controlled by U(t). These characteristics distinguish the control problem of the maximum principle from that
X• dt + Fl[x(tf)l.
to
23.2.1. Unconstrained Optimal Control A special case of the control problem allows the control trajectory {U(t)} to be any piecewise continuous function of t. After some rather detailed mathematics (see Appendix H.6), it is possible to show that in the absence of constraints on U(t), optimal trajectories {U(t)}, {x(t)}, and {_X(t)}must satisfy
23.2. P O N T R Y A G I N ' S MAXIMUM PRINCIPLE Pontryagin's maximum principle (more familiarly, the maximum principle) allows one to extend beyond the calculus of variations problem, by including complex constraints on the control variables in U(t). The general control problem is
(23.8)
OH =0, aU -
to <- t <- tf;
O H _ ~,
t o <- t <- tf;
3X
--
aF1]ax = 0, X _ - 3x -
t = tf.
m
The latter condition is essentially a transversality condition, in which either the variation 8x or the form [X_.- OF1/3x] must be zero at tf, depending on specification of a terminal time condition. A search for an optimal solution can be limited to the investigation of trajectories meeting these conditions. Pointwise stationarity of the Hamiltonian, expressed by the equation a H / a U = 0, depends on the fact that there are no constraints on U at each point in time, so that a value of U(t) can be found where the derivative O H / a U vanishes. With fixed initial conditions and free terminal state, the optimal control is given by the solution of the system Yc = [(x, U, t)
(23.9)
23.2. Pontryagin's Maximum Principle
619
or U = k/2. The Euler-Lagrange Eqs. (23.9) and (23.10) are
and it = - O H / Ox,
(23.10)
-aH/Ox
D
with x(t 0) = x 0, k_(tf) = OF1/3X(tf), and U given in terms of state and costate variables by O H / O U = 0. This is a two-point b o u n d a r y value problem, wherein integration of the state equation proceeds forward in time starting at x(t0), and integration of the costate equation proceeds backward in time starting at k_(tf). These integrations are complicated by the fact that x(t) appears in the costate equation, and _k(t) appears in the state equation [through U(t), a function of _k(t) and x(t)]. Equations (23.9) and (23.10) are k n o w n as the EulerLagrange equations. Though required for an optimal solution of the control problem as stated above, they are not sufficient by themselves to guarantee an optimizing control strategy. A control strategy is guaranteed to be optimal, at least in a neighborhood of the strategy, if the Euler-Lagrange conditions are satisfied at each point in the time frame, and the Hessian matrix for H with respect to U(t) is negative definite at each point in the time frame. Typically, pointwise stationarity of the Hamiltonian can be used to simplify the Euler-Lagrange equations. Thus, one can use O H / O U = 0 to derive the form of the optimal control as a function of the state and costate variables. Incorporating this control function into the state and costate transition equations then defines a system of equations in the state and costate variables, absent any direct reference to the controls. m
Example
Consider a previous example in which direct control is to be applied to a pest population in an effort to control population size. The objective of m a n a g e m e n t is to reduce the population size while also minimizing the costs associated with pest control. As before, control costs are expressed in terms of a quadratic function of effort U, so that the control problem is
and OH~Ok = U = it.
From the stationarity condition we have U = k / 2 , and thus the Euler-Lagrange equations are X=I and = k/2. The first Eu|er-Lagrange equation produces k = f + c, and substituting this expression into U = k / 2 produces the o p t i m a l control U = t / 2 + c 1. From the second Euler-Lagrange equation we have ~ = f / 2 + c1, or x(t) = t2/4 + clt + c 2. The constants c~ and c2
are given by the initial and terminal conditions; thus, x(0) = 3 = c I and x(2) = 0 = 22/4 + 2c I 4- c2, from which we have c2 = - 2 . Note that this is the same optimal trajectory identified by the calculus of variations for this problem (see Section 23.1.1). Example
Consider a simple exponential model for a pest population with an intrinsic growth rate of unity and a potential for continuous harvest over the time frame [0,1 ]: ~ = x - U. A s s u m e that the population at t = 0 is 1 and that the objective of m a n a g e m e n t is to minimize harvest effort as a quadratic function of U so as to extinguish the population at t = 1. The problem statement is maximize
-f
{u(t)}
flu2
-~- dt
0
subject to 9~ -- X m U~
x(0) = 1, x(1) = 0.
(x + U 2) dt 0
-
{U(t)}
2
maximize
= 1 = it
The Hamiltonian for this problem is H = - U 2 / 2 + Mx - U), so that
subject to
aH/OU=
- U-k=0,
0-
~=U, -OH/Ox
= -
k = it,
O <- t <- 1;
x(0) = 3, x(0) = 1,
x(2) = 0.
x(1) = 0. The Hamiltonian for this problem is H ( x , U, k) = - x U 2 + kU, and differentiation with respect to U gives -
OH/OU = -2U
+ k = 0
The stationary condition produces U = - k , so that the Euler-Lagrange equations are ~(=-X
620
Chapter 23 Modern Approaches to Decision Analysis and c2 = - - 1/(e 2r - - 1), and the optimal trajectory a n d control are given by
and 2=x+X.
er(2-t)
From the costate equation, w e get h = Cl e - t , and substituting this expression into the state equation gives = x 4- Cl e - t . It can be seen that x(t) = - c l e - t / 2 4c2et is a solution to the latter equation, with c I and c2 d e t e r m i n e d by the initial a n d terminal conditions: x(0) = - C l / 2 + c2 1 and x(1) = - c l e - 1 / 2 + c2e = O. Thus, c I -2eZ/(e 2 - 1) a n d c2 - 1 / ( e 2 - 1), a n d the optimal state and control are given by
x(t) =
e 2r-
U(t) =
2re2r 2r
~ ert
1
and
e
-1
- rt e
- -
=
- -
e2-t
x(t) =
These control and state trajectories reduce to the solutions given in the previous example w h e n r = 1. Example
_ et
Consider a logistic p o p u l a t i o n with transition equation 2 = x(1 - x ) - U and b o u n d a r y conditions x(0) = x 0 and x(tf) = xf. The H a m i l t o n i a n for a quadratic objective functional as above is
e2 - 1
and 2e 2
U(t) =
e
2
e-t
-1
H = -U2/2
"
This s h o w s that an optimal harvest strategy involves exponentially declining effort over the time frame.
+ h(x-
Example
-OH/Ox=
The previous example can be generalized by allowing for intrinsic g r o w t h rate in the transition equation to be p a r a m e t e r i z e d by r: 2 = rx - U. The p r o b l e m statement is
-
{u(t)}
f
U),
and differentiation w i t h respect to the state and costate variables p r o d u c e s OH/OU = -U-
maximize
x 2-
-h
h = O,
O<_t<_tf;
o<_t<_t~;
+ 2 X x = h,
x(0) = x0,
x(t~) = x~. Again w e have U = - h from the stationarity condition, so the state a n d costate equations are expressed by
l U2 - ~ dt
2=x-x2+h
0
and
subject to
J~ = 2 h x ~
--- t'X b
h
Mr
= h(2x - 1). x(0) = 1, x(1) = 0. The H a m i l t o n i a n for this p r o b l e m is H = - U 2 / 2 h ( r x - U), so that
+
Differentiation of the state equation p r o d u c e s X = 2 2 + 2x2, and substituting this expression along, w i t h the equation for X from the state equation into X/h = 2 x - 1 produces =
OH/OU=
-U-
h = O,
2x 3-
3x 2 +
x
0_
-OH/Ox
= -rh
= h,
0_t<_l;
x(0) = 1,
a solution for w h i c h is an optimal trajectory for the problem.
x(1) = 0. Again w e have U = - X from the stationarity condition, but n o w X = Cl e - r t from the costate equation and 2 = yX 4- Cl e - r t from the p o p u l a t i o n transition equation. A solution is x(t) = - c l e - r t / 2 r 4- c2 ert, with c I a n d c2 d e t e r m i n e d by x(0) = - C l / 2 r + c2 = 1 and x(1) = -cle-r/2r 4- c2 er = 0 . In this case cl = - 2 r e 2 r / ( e 2r - 1)
23.2.2. Constraints on the Control Trajectory In this case the feasible control trajectories are constrained to be in some b o u n d e d control set U, and the constraints translate into b o u n d s on U(t) at each point in time. It can be s h o w n (see A p p e n d i x H.6) that an optimal control trajectory {U(t)*} m u s t m a x i m i z e the
621
23.2. Pontryagin's Maximum Principle Hamiltonian at each time in the time frame: H(t)* H(t) for any feasible trajectory {U(t)}. This is the ~ellknown "maximum principle," so named because it establishes pointwise maximization of the Hamiltonian as the optimizing "principle" for solution of the optimal control problem. Note that an optimizing control is determined by direct inspection of the Hamiltonian H, rather than by identification of the zeros of aH/OU. This allows the maximum principle to handle a much larger class of bounded control problems. Of course, the identification of optimal values of U via the equation aH/aU = 0 for unconstrained problems is consistent with the maximum principle, since optimal values of H satisfy the stationary condition if the problem is unconstrained. For constrained optimization problems with ~ t representing the constraints at time t, the Hamiltonian is to be maximized by choosing the appropriate value U(t) in Fit. The optimization problem then is to m
H(x,U,t)
maximize U ~ l~t
for all t ~ T. An optimal value is obtained either at an interior point of f~t, in which case aH/aU vanishes, or at a boundary point of f~t. Irrespective of the nature of the constraints, it is easy to see from Eq. (23.8) that the partial derivatives all~ ak of the Hamiltonian reproduce the time rate of change of the state variables. Thus, a general solution of the optimal control problem consists of trajectories {x(t)}, {U(t)}, and {_Mt)}for which the Hamiltonian is maximized o v e r ~-~t at each point in the time frame, and the canonical equations n
~_ = all~OK_,
x(t 0) = x0; (23.11)
it = -OHlax,
k_(tf) = aF~/axf
are satisfied. Maximizing the Hamiltonian with respect to U(t) typically allows one to identify the optimal control variables as functions of the state and costate variables, so that the canonical equations can be expressed in terms of 2k state and costate variables without reference to controls. Thus, the optimal state and costate trajectories solve a system of 2k ordinary differential equations with split boundary values. The optimal control trajectory subsequently is identified by using the optimal state and costate trajectories in the functional relationship derived from the Hamiltonian.
Example Consider an exponential pest population with the potential for continuous control: :t = ax + bU over some time frame [t0, tf], with x(0) = x 0. Management desires to minimize accumulated pest damage and con-
trol costs over [t0, tf], and an appropriate objective functional is ti l = 1/2 [rU 2 + qx 2] dt + px2(tf),
f
to
in which r, q, and p are positive constants. Assume that the allowable values of U are constrained by Umi n U ~ Uma x. The Hamiltonian for this problem is
H = 1/2[rU 2 + qx 2] + X[ax + bU], which is to be maximized at each point in the time frame. Differentiation with respect to U yields
3H/OU = rU + bk, and the maximum principle indicates that U(t)* must satisfy U(t) = - b X / r if this value is in the interval (Umin, Umax). Otherwise, U(t)* is chosen to be either Umin or Umax, depending on the sign of aH/OU. Note that an optimal control strategy is maximal, because 32H/3U 2 = r > O.
23.2.3. Special Cases of the Maximum Principle We mention here some formulations that often arise in applications of the maximum principle.
23.2.3.1. Autonomous Problems In general, differentiation of the Hamiltonian with respect to time yields
dH dt
-
=
aH ax-
~2
+
aHld OU--
+
aH~ aH +~ -~at
OH "_k) 3H ~ + [(x, U, t) + aHUaLI--+ ~'at
That the first term in this expression vanishes follows from the canonical equations (23.11). The second term vanishes because either the derivative 3H/aU vanishes (at an interior point of fit) or ~ vanishes (at a boundary of fit). It follows that
dH/dt = 3H/at. If time t does not appear explicitly in either the optimality index or the state transfer equations, i.e., both functions are autonomous, then dH/dt = 0 and the value of the Hamiltonian must be constant over the time frame. In that case an optimal control strategy identified by the maximum principle is necessarily global over the control set U.
622
Chapter 23 Modern Approaches to Decision Analysis
Systems
23.2.3.2. Control of Linear
and the costate equations are
Consider a control problem for which the optimality index and transfer functions are linear in the controls. Assume, for example, that the univariate control U(t) can vary over the interval [U 0, U 1] and the optimality index and transfer functions are expressed as I(x, U, t) = II(X, t) + aU(t)
it = - O H ' -
3X
__ [KI( -- X2-- bu) q- ~k2x2] ~.2(Xl
-
1)-
~klX 1
with
and _ (tp =
[(x, U, t) = [l(X, t) + bU(t),
respectively, where b = [b1 -.. bk]'. Then the Hamiltonian is H(x, U, t)
=
t) + _Mt)/l(X, t)] + [a + h_(t)b]U(t),
[II(X,
which is maximized by U(t) = U o
if
a + Mt)b < O
U(t)
if
a + Mt)b > O.
and =
U 1
Because the controls fluctuate between the maximum and minimum allowable values for U(t), depending on the sign of the switching function a + K(t)b, a solution is termed bang-bang control. Example
Consider a predator-prey system in which prey numbers are to be controlled through the use of a pesticide. Control is expressed as a percentage of the prey population that is targeted for removal, with adjustments for effectiveness. Population transitions are described by a modified version of the Lotka-Volterra equations X1 =
X1(1
- x 2)
9C2 - - X 2 ( X 1 - -
-
bx lu,
1),
where the control term indicates that pesticide application is linear in its effect on prey. In the absence of pesticide application, the population fluctuates about the equilibrium point (x 1, x2) = (1, 1), with amplitudes determined by the initial system state (see Appendix C). An objective of management is to bring the populations to equilibrium with minimum application of the pesticide, i.e., to minimize J =
f,,
to
23.2.3.3. Singular Controls u dt
with 0 -< u -< U m a x over the time frame. The Hamiltonian for this problem is H = u + ~1[x1(1
- - X2) - -
Because the control u does not appear in OH/au = 1 - - K l b X l , setting the latter to zero does not provide a solution to the control problem. However, the maxim u m principle indicates that the optimal control must maximize H at each point in the time frame, which in turn means that u takes a value of either 0 or U m a x , depending on the sign of 1 - K l b X 1. Determination of the actual switching strategy depends on the trajectories of Mt) and x(t), which are difficult to derive based on the canonical equations. In this particular instance, however, it is possible to deduce the appropriate strategy from knowledge about the behavior of the Lotka-Volterra system. When forced by maximum pesticide application, the system oscillates indefinitely about a new equilibrium point that is defined by the parameter b (see Appendix C). The pattern of these oscillations of course depends on the state of the system at the time when pesticide application begins. It is easy to see that only one oscillation pattern for the forced system will include the equilibrium point (1,1) of the unforced system (Fig. 23.4). Provided Umax is sufficiently large, that oscillation pattern will coincide with oscillations of the unforced system in at least one point. Assuming the initial system state is not a point of intersection between the forced and unforced trajectories, the optimal strategy is to leave the system unforced until it evolves to a point of intersection and then apply the pesticide at Umax until equilibrium (1,1) is attained. Because of the oscillatory nature of the system, optimal application must continue at the level U r e a x u p to the time at which equilibrium is attained; otherwise, predator and prey numbers will immediately begin a new pattern of indefinite oscillations and never attain equilibrium.
bXlU ] +
Ka[X2(Xl
--
1)],
In the previous example of a predator-prey system, the switching function is nonzero almost everywhere in T, so that a bang-bang control strategy is optimal over the entire time frame. However, in many cases the switching function can vanish over a nonzero interval
23.2. Pontryagin's Maximum Principle
forced system
250
\f ~,
_[
623
N--~
o "~ 200 r i.. El.
~-~...___._____.~
150
0
,
200
4;0
600 prey
8;0
10'00
,
1200
,,
N
1
FIGURE 23.4 Oscillations for forced and unforced Lotka-Volterra predat o r - p r e y systems. Starting at N 0, the optimal strategy for a m i n i m u m effort objective is to leave the system unforced until it evolves to N 1, and then switch to m a x i m u m control until the equilibrium N* is attained.
of time. Under these circumstances the control problem is said to be singular, and extremal control values are insufficient to describe an optimal control strategy over the time frame. Solution approaches to singular control problems typically require the interpretation of the state a n d / o r costate equations (23.11) to identify the pattern of optimal controls.
Example Previous investigations suggest that the population carrying capacity of a habitat undergoes periodic fluctuations, which can be approximated by the trajectory a(t). Management wishes to control population changes so as to minimize deviations from this trajectory, while recognizing limits on the level of available control. The optimal control problem can be stated as maximize
- ~ T [ x ( t ) - a(t)] 2 dt d o
U m i n -< U -< U m a x
subject to ~=U. The Hamiltonian for this problem is H = - I x - a] 2 + KU, with the state and costate equations given by ~ = OH/3K = U and ~, = - 3 H / O x = 2(x - a), respectively. The Hamiltonian is maximized by
U(t)
= Umi n
if
~,(t) < 0
U(t)
=
if
k(t) > 0.
and Uma x
However, if Mt) = 0, the Hamiltonian becomes H = - [x - a] 2, and the appropriate level of control sustains the equality x(t) = a(t) and thereby causes the Hamiltonian to vanish. Under these conditions the transition equation ~? = U yields U(t) = k = li. An optimal control trajectory therefore consists of intervals of maximum and minimum controls, along with intervals of nonextremal control to track the population carrying capacity exactly. Interval lengths in the optimal control trajectory are determined by the trajectory of K(t). From the costate equation ~, = 2(x - a) we have t
k(t) = 2 / Ix(s) - a(s)] ds, 0
with M0)
-
X(T) =
0
[endpoint conditions on Mt) follow from the transversality condition above, given that endpoint values for x(t) are not specified]. The optimal control strategy therefore adjusts the population rate of change to ensure coincidence of the population and carrying capacity, over intervals in which the necessary change does not exceed allowable limits of control. From the above equation, these intervals are characterized by K(t) = 0. If control limits are exceeded by the changes needed to track the carrying capacity exactly, the optimal strategy calls for maximum allowable controls for K(t) > 0 and minimum allowable controls for K(t) < 0. The locations of interval endpoints for maximum, minimum, and
624
Chapter 23 Modern Approaches to Decision Analysis
target control rates can be seen to be uniquely determined, as a result of the condition
f 0TEx(t) -- a(t)] dt = O. 23.2.4. Sensitivity Analysis As with the Lagrangian multipliers in nonlinear programming, the costate variables can be interpreted in terms of the sensitivities of the objective functional to certain parameter changes. In particular, it can be shown that the initial value K_*(t0) of the optimal costate trajectory expresses the sensitivity of the objective functional to a change in the system initial state:
As with nondynamic optimization, these sensitivities can be given an economic interpretation. Thus, the objective functional sometimes measures total economic value, in terms of price times quantity accumulated over time. If the system state represents resource availability, then the costate variables at t o can be interpreted as the marginal change in economic value with respect to a change in the resource inputs in x(t0). Hence the use of the term shadow price in reference to the optimal costate values.
23.2.5. Discrete-Time M a x i m u m Principle It is possible to derive a version of the canonical equations for problems in which the time frame is discrete. The relevant optimization problem in discrete time is tf - 1
{u(t)} ~ u
~ I[x(t), U(t), tl t=to
+
Fl[X__(tf)]
subject to x(t + 1) = x(t) + ~(x, U, t), x(t0)
x 0.
=
As before, Lagrangian multipliers can be used to incorporate the transition equations and initial conditions into the objective functional, producing a discrete-time version H[x(t), U(t), X(t + 1), t] = m
~
First we consider a situation in which there are no constraints on the control trajectory, i.e., the vector U(t) of control variables can be any point in E k. After some rather complicated mathematics it can be shown (see Appendix H.6) that an optimal trajectory must satisfy aH/OU(t) = 0', k(t) = O H / a x ( t ) ,
and x(t) = OH/OMt)
at each point in the time frame, along with the transversality condition
oJ*lox(to) = _~*(t0).
maximize
23.2.5.1. U n c o n s t r a i n e d O p t i m i z a t i o n w i t h Discrete Time
3F1 -- k(tf)]T](tf) = 0 x(tf)
-
-
and initial condition x(to) = Xo.
Note that these optimality conditions are analogous to the Euler-Lagrange Eqs. (23.9) and (23.10) for continuous problems. In both cases the optimization problem reduces to a two-point boundary value problem, typically requiring the solution of a system of nonlinear transition equations in state and costate variables. In general, both discrete-time and continuous-time problems must be solved by iterative techniques. In most instances the solution of a discrete-time problem converges to its continuous-time analog as the partitioning of the time frame becomes increasingly fine. Example
Management desires to eradicate a pest population over a 10-day period, while minimizing the cost of removal. A unit U(t) of effort on day t results in the removal of o~U(t) individuals, at a cost of U(t)2/2. Based on survey data the population size is estimated to be 100 individuals, so the optimization problem is 9
maximize
- ~ , U(t)2/2
{u(t)}
t=0
subject to x(t + 1) = x(t) - o~U(t),
m
I[x(t), U(t), t] + _~(t + 1)[[x(t), U(t), t]
x(0) = 100, x(10) = 0.
of the Hamiltonian. Optimal controls then can be found by appropriate choice of U(t) to maximize the Hamiltonian at each point in the time frame.
The Hamiltonian for this problem is
/
H = - U ( t ) 2 / 2 + )~(t + 1)Ix(t)- oLU(t)],
23.2. Pontryagin's Maximum Principle so that
625
so that ~U (t) = - U(t) - )t(t + 1)
~U (t) = - U(t) - ~)~(t + 1)
=0,
=0,
)t(t) = -~xH(t)
)~(t) = ~ ( t )
= )~(t + 1)Jr + x(t)/K],
= )~(t + 1), and
and
x(t + 1) = aH/a)t(t + 1) = rx(t) - x ( t ) 2 / 2 K -
x(t + 1 ) = - ~all. ( t + 11
U(t).
From the optimality condition on H we have U(t) = -K(t + 1), so that
= x(t) - oLU(t).
From the costate equation we have )~(t) = c, so that U(t) = -oLc from the optimality condition on H. Substituting this expression for U(t) into the state transition equation then gives x(t + 1) = x(t) + oL2c. It follows from x(0) = 100 that x(t) = 100 + tc(x 2, and in particular x(10) = 0 = 100 + 10col2. Then c = -10/oL 2, and the optimal control and state trajectories are given by U(t) = 10/c~ and x(t) = 100 - 10t, respectively. Thus, the m i n i m u m - c o s t control strategy calls for a u n i f o r m effort over the time frame, which results in a linear decline in the p o p u l a t i o n to extinction at time t = 10.
x(t + 1) = rx(t) - x ( t ) a / 2 K + K(t + 1).
B o u n d a r y conditions for the problem are the initial population value x(0) = 10 and ~(3) = 1 from the transversality condition. Utilizing the costate equation, we can step b a c k w a r d from the terminal time to obtain )~(2) = )~(3)[r- x ( 2 ) / K ] = r -
x(2)/K,
)~(1) = M2)[r - x ( 1 ) / K ] = [ r -
x(2)/K][r-
x(1)/K],
M0) = )~(1)[r- x(O)/K] = [ r -
x(2)/K][r-
x(1)/K]
• [r - x(0)/K]. Example
Consider the optimal harvest of a logistic population x(t + 1) = x(t) + r'[x(t) - x ( t ) 2 / K '] - U(t) = rx(t)
x(t) 2
2K
x(2) = x ( 1 ) [ r - x(1)/2K] + h(2) = x ( 1 ) [ r - x(1)/2K]
2 {U(t)}
-~
x(1) = x ( 0 ) [ r - x(O)/2K] + )~(1) = 1 0 [ r - 5 / K ] + [ r - x(2)/K] [ r - x(1)/K],
U(t)
over T = {0, 1, 2, 3}, where harvest cost at each point in time is a quadratic function of effort, and a terminal value is ascribed to the population size at the end of the time frame. A s s u m i n g an initial population size of 10 individuals, the optimization p r o b l e m is maximize
On the other hand, we can utilize the state transition equation to step forward from the initial time, to obtain
U(t)2/2 + x(3)
t=0
+ [r - x(2)/K],
x(3) = x ( 2 ) [ r - x(2)/2K] + K(3) = x ( 2 ) [ r - x(2)/2K] + 1. The latter system of equations can be solved for x(t), and the resulting values then can be used in the costate equations to identify )~(t). The costate values in turn can be used to identify the optimal control sequence, by U(t) = h(t+l).
subject to x(t + 1) = rx(t) - x ( t ) 2 / 2 K -
23.2.5.2. Discrete-Time Optimization with Constraints on Controls
U(t),
x(O) = 10.
In this case there are constraints on the vector U(t), i.e., U(t) e f~t. As before, the Hamiltonian
The H a m i l t o n i a n for this problem is
H[x(t), U(t), )~(t + 1), t] = _
H = -U(t)2/2
+ )~(t + 1) rx(t) - x ( t ) 2 / 2 K -
U(t)],
_
l
I[x(t), U(t), t] + _K(t + 1)[[x(t), U(t), t]
626
Chapter 23 Modern Approaches to Decision Analysis
is to be maximized by choosing the appropriate value U(t) in f~t: maximize
H(x, U, t)
U(t) ~ ~t
for all t e T. An optimal value is obtained either at an interior point of f~t, in which case 3 H / O U vanishes, or at a b o u n d a r y point of f~t. A general solution of the optimal control problem consists of trajectories {x(t)}, {U(t)}, and {K(t)} for which the Hamiltonian is maximized o v e r ~t at each point in the time frame, and the discrete canonical equations are satisfied:
x(t) = OH/OK_(t),
x(t 0) = x0;
K_(t) = OH/Ox(t),
K_(tf) = cgF1/cgX f.
Substitution of the optimality condition on u(t) into the costate equation produces Mt) = UmaxE1 -- K(t + 1)(1 + r)] + X(t + 1)(1 + r) (1 + r)Mt + 1)
u(t) = Umax, u(t) = O.
Because neither a terminal state nor a terminal value is specified, the transversality condition is X(T)
OF1 / 3 X ( T ) = 0,
=
and u(T - 1) = Umax from (1 + r)MT) < 1. The costate equation then gives K ( T - 1)
=
Umax[1
-
K(T)(1
+
r)~ +
K(T)(1
+ /')
and, from the optimality condition,
Example Previous population surveys indicate that within certain limits, a h u n t e d population exhibits postharvest exponential growth over time. If u(t)x(t) is the harvest in year t and y(t) = [1-u(t)]x(t) is the postharvest population size, then population change over time is given by
x(t + 1) = (1 + r)y(t) = (1 + r)[1 - u(t)~x(t), where r > 0 is the population instantaneous rate of growth. M a n a g e m e n t seeks to maximize the total harvest over T time periods, assuming an initial population of size x(0) = x 0 and a range of harvest rates between 0 and Ureax. The optimization problem is
u(T-
2) =
Umax,
if (1 + r)K(T - 1) < 1. This backward stepping process using Umax and X(t + 1) to identify k(t) continues until (1 + r)X(t*) >1, at which time the optimal harvest rate goes to 0. The harvest rate then remains at 0 for t < t*, because X(t - 1) = (1 + r)X(t) > 1 over that time. Thus, the optimal harvest strategy calls for abstention from harvest early in the time frame to allow the population to increase in size, followed by m a x i m u m harvest throughout the remainder of the time frame. The appropriate time to switch from u = 0 to u = l/max is determined by the intrinsic growth rate r and the value Umax, according to the difference equation X(t)
=
Umax[1
-
X(t + 1)(1 + r)] + K(t + 1)(1 + r).
T
maximize
~
u(t)x(t)
u(t), [0, Umax] t = 0
subject to
x(t + 1) = (1 + r)E1 - u(t)]x(t), x(0) = x0. The Hamiltonian for this problem is
H = u(t)x(t) + k(t + 1)(1 + r)[1 - u(t)]x(t), which is maximized by
u(t) =
Ureax 0
(1 + r)X(t + 1) < 1, (1 + r)K(t + 1 ) > 1.
The Euler-Lagrange equations are
K(t) = ~x(t) = u(t) + K(t + 1)(1 + r)[1 - u(t)] and
x(t + 1) = - ~OH. ( t + 1) = (1 + r)E1 - u(t)lx(t).
23.2.6. Summary The m a x i m u m principle extends beyond the calculus of variations, to include complex constraints on the control variables in U(t). The problem formulation includes possible initial and terminal value functions, and also allows for constraints other than the system transition equations. A solution approach involves the following: 9 Time-varying costate variables are included as weighting factors for the state transfer functions in an augmentation of the optimality index k n o w n as the Hamiltonian. 9 An optimal control strategy is obtained by maximizing the Hamiltonian with respect to the controls at each time over the time frame of the control problem. Pointwise maximization defines the " m a x i m u m principle." 9 Differentiation of the Hamiltonian with respect to the state variables produces the time rate of change
23.3. Dynamic Programming of the costate variables. Differentiation with respect to the costate variables reproduces the state transfer functions. 9 Differentiation of the Hamiltonian with respect to the controls often can be used to identify the form of optimal controls as a function of the state and costate variables. Incorporating the control function into the state and costate transition equations then defines a system of equations in the state and costate variables, absent any direct reference to the controls. 9 Optimal state and costate trajectories are derived as solutions of this system of equations, subject to initial boundary conditions on the state variables and terminal conditions on the costate variables. 9 Solving the canonical equations usually involves backward integration of the costate equations and forward integration of the state equations. The optimal control strategy subsequently is identified as a function of the optimal state and costate trajectories. 9 In some cases, most notably when the Hamiltonian is linear in controls, other methods besides the use of stationarity conditions must be used to identify the optimal control strategy. 9 Initial values of the optimal costate variables express the sensitivities of the optimal value of the objective functional to marginal changes in the initial state variable values. 9 A maximum principle can be formulated for discrete-time problems, with optimal controls derived from Euler-Lagrange equations that are essentially identical to those for continuous systems.
23.3. D Y N A M I C P R O G R A M M I N G As with variational mathematics, the general control problem for dynamic programming is to choose a control strategy {U(t)} from some constrained set ___Uthat maximizes an objective functional of system states, controls, and possibly time. For deterministic systems with continuous time frames the control problem is identical to that of the maximum principle: maximize {U(t)} ~ U B
f t~ I(x, U, t) dt + Fl[X(tf)] to
subject to
2_ = [(x, U, t), x(t0) = x0, x(t~) = x~.
627
An analogous statement for discrete-time systems replaces the integral with a summation over the time frame, i.e., ti l = ~,, I(x, U, t) + Fl[X(tf)], (23.12) t=to
and utilizes difference equations of the form x(t + 1) = x(t) + f(f, x, U, t)
m
m
to express system transitions. Dynamic programming also is amenable to stochastic problems, although the objective functional is stated in terms of expected values"
J = g
[t0
}
I(x, U, t) + fl[x(tf)] ,
(23.13)
where the expectation is with respect to random elements Z(t) that influence system behaviors by x(t + 1) = x(t) + [(x, U, Z, t), with {Z(t)} a time series stochastic process (see Chapter 10). Though dynamic programming and the maximum principle share a common statement of the control problem for deterministic systems, dynamic programming constitutes a substantially different approach to its solution. Rather than incorporating the state transition equations into the objective functional by means of costate variables, the approach here is to use the "Principle of Optimality" (Bellman, 1957) to derive a partial differential equation or difference equation, the solution of which solves the dynamic programming problem. The Principle of Optimality is stated as follows: An optimal policy has the property that, whatever the initial state and decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.
The meaning of this principle is illustrated in Fig. 23.5, which describes the decomposition of an optimal state trajectory into two parts: an initial trajectory that starts at an initial condition x(t o) and evolves to x(tl), and a subsequent trajectory that starts at x(t 1) and evolves to the terminal point x(tf). Essentially the Principle of Optimality says that if the overall trajectory is optimal with respect to the system initial condition, then the second trajectory is optimal in its own right, relative to its initial condition. Thus, the optimal behavior of the second trajectory is independent of how the system came to be at the second starting point.
628
Chapter 23 Modern Approaches to Decision Analysis
x(t) x(t~)
x(t,)
J
X(to)
I I I I
to
t,
tr
F I G U R E 23.5 B e l l m a n ' s Principle of Optimality. For o p t i m a l trajectory {x(t)} o v e r [t 0, tf], the trajectory {x(t)} over [t 0, tf] m u s t be o p t i m a l relative to the initial c o n d i t i o n x(t 1) at t = t 1. After Intriligator (1971).
In both its discrete and continuous forms, dynamic programming can be seen to apply to systems for which the past behavior of the system is unimportant in assessing optimal controls. Utilizing a term from stochastic processes, we characterize systems for which future behavior is independent of past trajectories as Markovian (see Sections 10.3-10.6). The future history of a Markovian system at any point in time is influenced only by the system state (and controls) at that time. Indeed, the Principle of Optimality is an expression of the Markovian nature of control systems with no time lags. In order to apply dynamic programming, a Markovian system description is necessary. Of course, the statement of the optimal control problem used here includes system transitions that are Markovian.
23.3.1. Deterministic Dynamic Programming We consider here the application of dynamic programming to problems with nonstochastic behaviors. We may write the optimal value of the objective function (23.12) as J*[x(t), t] to emphasize its dependence on time and the starting point of the state trajectory. Application of the Principle of Optimality yields the fundamental recurrence relation J*[x(t), t] =
max
[I(x, u, t)At
{u(t)} ~ ut m
+ J*(x + Ax, t + At)],
(23.14)
a form that is appropriate for solution of control problems with discrete states and time frames. Additional smoothing assumptions ensuring the continuous differentiability of J[x(t)] yield the Hamilton-Jacobi-Bellman (HJB) equation -ol*/ot =
max [I(x, U, t) + (ol*/ox)~(x, u, t)], {U(t)}, m
ut
(23.15)
which, along with the boundary condition
l,[x(tj), t~] =
Fl[X(tf)],
provides the analytic framework for solving the optimal control problem for continuous systems. From the above discussions, it is clear that the application of the principle of optimality requires one to account for all possible states of the system at each point in the time frame. Thus, dynamic programming describes a field of values for the objective functional, one for every state at every time in the time frame, with each value produced by a state-specific and timespecific optimal control strategy. The net effect is to generate a feedback control rule, wherein the state and time can be fed back as arguments in an optimal control function U*[x(t), t]. This function identifies an optimal action for state x(t) at time t, on assumption that future state and control histories will follow optimal trajectories over the remainder of the time frame. 23.3.1.1. Applications in Continuous Time Few applications of dynamic programming in renewable natural resources use a continuous-time formulation, primarily because of the formidable difficulties in analyzing the HJB Eq. (23.15) in continu-
23.3. Dynamic Programming ous time. The equation involves partial derivatives for k + 1 variables (k state variables in x and time t) and is extremely difficult to solve, even with the aid of highspeed computers, for all but a handful of problems. D
Example
Consider the example discussed earlier, in which direct control of population change is to be applied in an effort to eliminate a pest population over [0, 2], while also minimizing control costs. The problem is to 2
maximize
f
IU(t)}
( x + U2) dt
Jo
subject to
~=u, x(0) = 3, x(2) = 0.
629
trajectory identified previously by the calculus of variations and the maximum principle. 23.3.1.2. L i n e a r - Q u a d r a t i c C o n t r o l in C o n t i n u o u s T i m e
A useful application for which the HJB equation can be solved involves a quadratic objective functional and linear system transitions. To illustrate, consider the expenditure of resources to control a community of pests. Pest population sizes x(t) and the effort U(t) directed to pest control both are assumed to influence population changes in a linear fashion, though pest control efforts have differential impacts, depending on the affected species. Pest damage is measured by x ' Q x , and the cost of pest control through time is U ' R U. The objective of management is to minimize accumulated costs as measured by a quadratic function of these factors, or, equivalently, to maximize the negative of this accumulation. A formal statement of the problem is ti
The HJB equation for this problem is
minimize {U(t)} m
1/2
f to
[ x ' Q x + U ' R U] dt
subject to Yc=Ax
with minimization of the term in brackets given by differentiation with respect to U:
0[
s
(y -}-
U2) q-
J
~r
x(tp = x~,
03"~"
with Q and _R negative-definite matrices. If the time frame is unlimited, a potential solution of the HJB equation has the form ]* = x'P x/2, with P a symmetric matrix that solves the system of equations
or
U
x(to) = Xo,
O]*U] = 2U + 0]*= O, OY
_ _ _
+ BU,
lal* 2 Ox
Q + PA + A'P - PBR-IB'P
Substitution of the expression for U* into the HJB equation produces o3j, : X q- l ( ~ at
-4\ Ox )
2 -- l ( ~
(see Appendix H.7). The optimal control strategy is expressed by
2 U*(t) = - R - 1 B ' P x ( t ) ,
2 \ Ox ) '
which describes a linear feedback strategy in x(t). Substituting U*(t) back into the transition equations then produces
or l(a]*~ 2
4\-O-~xl+
= 0
Ol___~*a=t 0. Yc = A x + B U *
This equation is solved by J* = - x t / 2 - x 2 / t + t3/48, as seen by substitution of the partial derivatives of J* back into the equation. From oJ*/ox = - t / 2 - 2 x / t we obtain the optimal control U* = - ( o J * / O x ) / 2 = t / 2 + x/t, and also the transition equation ~ = t / 2 + x/t. This in turn gives x*(t) = t 2 / 4 + clt + c 2. The constants Cl and c2 are of course determined by the initial and terminal conditions: x(0) = 3 = cI and x(2) = 0 = 22/4 + 2c I + c2, from which we have c2 = - 2 and thus x*(t) = t 2 / 4 + 3t - 2. Note that this is the same optimal
= [A - B R - 1 B ' P ] x
with a solution k x(t) = ~ , ci(vie.Xit), i=1
where k 1, ..., k k are the eigenvalues of A - B R -1 B ' P and __v1, ..., __vk are the corresponding eigenvectors (see Appendix B).
630
Chapter 23 Modern Approaches to Decision Analysis
Example Consider an earlier example in which the growth rate of an exponential population x(t) is to be controlled directly: f = U(t). Recall that the population dynamics for this situation were handled by introducing a second state variable to account for the control of growth, with x(t) r = x2(t), and ~1 = x 2 = U(t). Population dynamics then are given by
Substituting this expression back into the transition equations produces
['~.~2] 1 = {I 00 10]--[~]l-O 1] [42 ~]} [X12]
=i0 1][,]
= xl(t),xI =
E12]= Ea 101[x:] Assume for this example that the objective is to maximize
l = -fto
E2x2+ 2j2],t
to account for both population and control costs. For this problem R = r = 1, B = (0,- 1)', m
Q=
-2
-2
x2
"
It is straightforward to show that the eigenvalues for this system are complex, so that the optimal state trajectory is oscillatory (see Appendix C). Thus, the optimal control strategy is linear in both the population state and its growth rate, and optimal control induces oscillatory population levels. Example Assume for the control problem of the previous example that there are separate controls for the two state variables, with Ul(t) representing direct control of the population and U2(t) representing control of the population rate of growth:
0'
0]ix:] 1
and
_A=
Id:2 =
[0 ~] 0
with an objective functional
"
, = St,~0Ix2 + ~2+ ~2],t
For p=[Pl --
P2] P2 P3
As above, a solution for this problem is given in terms of the matrix P satisfying
we have
Q + PA + A'P - P B R - 1 B ' p
Q + PA + A ' P --
Pl - P2P3] PBB'P/r = [ 4-p2 LPl - P2P3 2p2 - p32J
= [
1_p21_ p2 Pl -- PlP2 -- P2P3
/
Pl -- PlP2- P2P3| 2p2 _ p 2 _ p2 J
=0.
-- Or m
from which it follows that
It is straightforward to show that
22]
P=
and
I 1/2 X/3/2
X/'3/2] 1/2 J
satisfies this equation, and substitution of P into U*(t) = - B ' P x ( t ) / r
= --[xl(t),x2(t)][ 4
= 2xl(t ) + 2x2(t).
U*(t) =
22] I_~]
-R-1B'Px(t)
defines an optimal linear feedback strategy with U'((t) = [V3xl(t) + x2(t)]/2
23.3. Dynamic Programming and
U~(t) = [xl(t) + x2(t)]/2. 23.3.1.3. Applications in Discrete Time Most applications of dynamic p r o g r a m m i n g in renewable natural resources use a discrete-time formulation. The operative form of the HJB Eq. (23.14) is J*[x(t), t] =
[I(x, U, t)&t + J*(x + Ax, t + &t)~,
max {U(t)} m
~
Ut
in which the time increment for system change is At. As a matter of convenience, the time interval in most applications is taken to be 1, so that the HJB equation is expressed by /*[x(t), t] = max {I(x, U, t) +/*[x(t + 1), t + 1]}. (23.16) {U}, Ut m
Several points are worthy of note. First, this formulation displays clearly the stagewise character of the optimization process, in which the problem of selecting an optimal control trajectory over an entire time frame is decomposed into a series of single-stage optimization problems. Thus, the maximization in Eq. (23.16) is with respect to potential actions taken at time t only. Of course, one must identify a maximizing action for every possible state x(t) of the system at time t. Second, maximization at time t requires one to project forward to time t + 1 the consequences of actions at time t. Thus, an action taken at time t with the system in state x(t) engenders a transition to state x(t + 1) at time t + 1. The optimization must account not only for the utility of the action at time t, but also the effect of that action on system dynamics (and therefore on future utilities) from that time forward. Third, maximization at time t requires optimal values at time t + 1 in order to select a maximizing action at time t. When the system is in state x(t), one must k n o w the optimal values associated with state x(t + 1) to which transfer is made from x(t). And this in turn requires the optimal control strategies for states at time t + 1. The optimization at time t simply augments the field of state-specific control strategies and values for time t + 1, by adding additional actions for time t to create a new field of state-specific strategies and values for time t. Fourth, time-specific and state-specific optimal values and controls can be identified by means of a sequence of stagewise optimizations that progress backward from the end of the time frame. The following algorithm can be used for stagewise dynamic prog r a m m i n g problems: 9 Beginning at tf, an action is chosen that maximizes the terminal v a l u e Fl(Xf) , given that these values are
631
influenced by the choice of an action. Otherwise, the value J*[x(tf), tf] for state x__fat time tf is J*[x(tf), tf] = Fl(Xf). 9 The second step utilizes the state-specific values J*[x(tf), tf] for tf to determine state-specific actions at time tf - 1 according to Eq. (23.16). The optimization produces state-specific values l*[x(tf - 1), tf - 1] for a two-stage problem. 9 The third step utilizes the values J*[x(tf- 1), t f 1], to determine state-specific actions at time tf - 2. The optimization produces state-specific values J*[x(tf - 2), tf - 2l for a three-stage problem. 9 The algorithm continues to step backward through the time frame until the initial time t o is reached. At the completion of this process a field of optimal control trajectories has been identified for all possible states of the system at all times in the time frame. To determine the appropriate control strategy, one need only identify an initial system state and initial time. The corresponding strategy then consists of a predetermined sequence of optimal actions requiring no additional information. Such a strategy is k n o w n as openloop control, so named because the strategy is selfdetermined once the initial state and time are specified (Intriligator, 1971). The dynamic p r o g r a m m i n g approach of decomposing a multistage optimization problem into a series of single-stage optimization problems yields tremendous gains in computational efficiency. To illustrate, consider a simple problem of finding the least-cost sequence of state transitions over [0, T], given that the transition from a state at time t to any other state at time t + 1 is possible. If initial and terminal states are fixed and k states are available at all other times, it is straightforward to show that there are k T-1 possible state trajectories for this problem. For a given trajectory, one must perform T - 1 additions to determine the accumulated transition costs. Thus, a search for the optimal trajectory via comparison of these costs for all possible trajectories would entail (T - 1)k T- 1 additions. On the other hand, the stage-specific optimizations of dynamic p r o g r a m m i n g require k2 additions at each stage between t = 1 and t = T - 1, plus k additions for the transition from t = 0 to t = 1. A total of only (T - 2)k2 + k additions therefore is required to determine the optimal trajectory. Even for very small problems, the computational savings are extraordinary. A problem with five possible states (k = 5) and six time periods (T = 6) requires only (4)(52) + 5 - 105 additions for a dynamic p r o g r a m m i n g solution, as opposed to (5)(5 s) = 15,625 additions for complete enumeration. It thus would be 149 times more costly to enumerate
Chapter 23 Modern Approaches to Decision Analysis
632
all possible trajectory costs than to use dynamic programming for the optimization. A comparison of the formulas (T - 1)kT-1 and (T - 2)k2 + k shows that the relative efficiency of dynamic programming increases exponentially with an increase in the number k of possible states and the length T of the time frame. Thus, expanding the number of states to k = 6 and the number of periods to T = 7 in this example increases the relative cost of enumeration from 149 to over 1500. Though computing demands sometimes are heavy for applications of dynamic programming, they pale in comparison to the demands for enumeration. Example
Managers of a seasonal fishery wish to minimize accumulated costs of stocking and maintenance, given a requirement to satisfy sport-fishing demand during a 4-month fishing season. Let x(t), u(t), and z(t) represent the size of the fish population, the level of stocking, and anticipated angler demand each month during the season. Assume for simplicity that population sizes, stocking events, and angler demand are measured in units of 10,000 fish. The size x(0) of the population, estimated via capture-recapture methods, is available prior to the season opening. There is no reproduction during the fishing season, and natural mortality is negligible over that time. On the other hand, population increases can occur through stocking, and angler demand for fish take is expected to fluctuate over the season. Population dynamics during the season are given by x(t + 1) = x(t) + u(t) - z(t),
where monthly stocking levels u(t) are under management control and monthly demand z(t) = {1, 3, 2, 4} is assumed known at the beginning of the season based on previous angler surveys. Stocking levels are limited by the hatchery capacity, assumed to be Umax - 5 , and population sizes are constrained by available fish habitat to be x(t) -< 5. Costs are incurred as a result of stocking at the beginning of each month, according to the cost function Cl[U(t)] = 0 for u[(t) = 0 and cl[u(t)] = 3 + u(t) otherwise. Costs associated with population and habitat maintenance accrue at the end of each month, according to c2[x(t + 1)] = [x(t) + u ( t ) - z(t)]/2. The objective functional accumulates stocking and maintenance costs over the fishing season. Thus, the optimization problem is 3
minimize
3
I[x(t), u(t)] = ~,{Cl[U(t)] + c2[x(t + 1)1} t=0
t=0
subject to x(t + 1) = x(t) + u(t) - z(t),
t = 0, 1,2,3;
O <- x(t) -< 5, 0 <- u(t) -< 5.
Because the objective functional, transition equations, and constraints all are linear functions of the state and control variables, this optimization problem could be solved with linear programming. However, optimal stocking also can be determined with the backward iteration algorithm of dynamic programming. Because the objective functional does not include a terminal value function, the HJB equation for t = 3 is simply J*[x(3), 3] = min I[x(3), u(3)], u(3)
with ~3u(3)/2 + x(3)/2 + 1 I[x(3), u(3)] = [x(3)/2 - 2
if u(3) > 0, otherwise,
where the allowable values of u(3) must satisfy the problem constraints for a given population size. For example, when the population size is x(3) --- 1, stocking levels of 3, 4, and 5 meet nonnegativity conditions and other constraints on demand and capacity. The corresponding costs are 6, 7.5, and 9, respectively (Table 23.1), and u(3) = 3 can be seen to minimize accumulated costs over the remainder of the time frame. Similarly, when x(3) = 2, stocking levels of 2, 3, 4, and 5 are feasible, and the corresponding costs are 5, 6.5, 8, and 9.5. In this case the stocking level u(3) = 2 minimizes accumulated costs. Optimal stocking rates and associated costs are determined for all population sizes at t = 3, and these values are retained for use in determining optimal values for t = 2 (Table 23.2). At time t = 2 the optimal stocking strategy is found by minimizing the total of current (t = 2) and projected (t = 3) costs, /*[x(2), 2] = min{I[x(2), u(2)] + J*[x(3),3]} u(2)
over all values u(2) that satisfy the problem constraints. For example, when the population size is x(2) = 0, stocking levels of 2, 3, 4, and 5 meet nonnegativity conditions and other constraints on demand and capacity. The corresponding accumulated costs are 12, 12.5, 13, and 13.5 (Table 23.1), so that u(2) = 2 minimizes accumulated costs over the remaining 2 months of the time frame. Similarly, when x(2) = 1, stocking levels of 1, 2, 3, 4, and 5 are feasible, and the corresponding costs are 11, 11.5, 12, 12.5, and 10. In this case the stocking level u(2) = 5 minimizes accumulated costs.
TABLE 23.1 Projected Fishery Population Size" u(t) = 0 x(t)
1=3
x(t
+ 1)
I
,
u(t) = 1 x(t
+ 1)
I
,
u(t) = 2 x(t
+ 1)
I
,
2 3 4
0.0
0.0
0.0
0.0
5.0
5.0
0.0
6.0
6.0
1.0
6.5
6.5
,
0.0
7.0
7.0
1.0
8.5
8.5
1.0
7.5
7.5
2.0
9.0
9.0
2.0
8.0
8.0
3.0
9.5
9.5
4.0
10.0
10.0
+ 1)
=4
+ 1)
I
4.0
4.0
1.0
5.5
5.5
2.0
7.0
7.0
3.0
8.5
8.5
4.5
2.0
6.0
6.0
3.0
7.5
7.5
4.0
9.0
9.0
0.0
5.0
12.0
1.0
6.5
12.5
2.0
8.0
13.0
3.0
9.5
13.5
0.0
4.0
11.0
1.0
5.5
11.5
2.0
7.0
12.0
3.0
8.5
12.5
4.0
10.0
10.0
1.0
4.5
10.5
2.0
6.0
11.0
3.0
7.5
11.5
4.0
9.0
9.0
4.0
8.0
8.0
7.0
3
1.0
0.5
6.5
2.0
5.0
10.0
3.0
6.5
10.5
4
2.0
1.0
6.0
3.0
5.5
9.5
4.0
7.0
7.0
0.0
5.0
17.0
0 1
0.0
6.0
18.0
1.0
7.5
17.5
2.0
9.0
16.0
1.0
6.5
16.5
2.0
8.0
15.0
3.0
9.5
16.0
4.0
10.0
16.0
4.0
10.0
20.5
0.0
4.0
16.0
1.0
5.5
13.5
2.0
7.0
14.0
3.0
8.5
15.0
3
0.0
0.0
12.0
1.0
4.5
14.5
2.0
6.0
13.0
3.0
7.5
14.0
4.0
9.0
15.0
4
1.0
0.5
10.5
2.0
5.0
12.0
3.0
6.5
13.0
4.0
8.0
14.0
0.0
4.0
20.0
1.0
5.5
20.5
2.0
7.0
21.0
3.0
8.5
20.5
1
0.0
0.0
16.0
1.0
4.5
19.5
2.0
6.0
20.0
3.0
7.5
19.5
4.0
9.0
19.5
2
1.0
0.5
15.5
2.0
5.0
19.0
3.0
6.5
18.5
4.0
8.0
18.5
3
2.0
1.0
15.0
3.0
5.5
17.5
4.0
7.0
17.5
4
3.0
1.5
13.5
4.0
6.0
16.5
0
uit) = 5 x(t
4.5
0.0
2
,
I
u(t)
x(t
1.0
0.0
2
1=0
,
=3
0.0
0 1
I=1
I
u(t)
0 1
1=2
xit + 1)
a The population size x(t + 1), optimality index [[x(t), u(I)], and accumulated cost ][x(t), u(t)] for combinations of population size x(t) and stocking level u(t), for a fishery subjected to sport harvest and periodic stocking during a fishing season of 4 months. Rows represent population size at time I, and columns represent stocking level. Table entries for the pair [x(t), u(t)] are given by x(t + 1) = x(t) + u(t) - z(t) for population change, with ][x(t), u(t)] = [[x(t), u(t)] + J[x(t + 1), I + 1] from the Hamilton-Jacobi-Bellman equation. Numerical values are displayed for all combinations [x(t), u(t)] satisfying nonnegativity conditions and other system constraints. The optimal stocking decision for a population of size x(t) at time I is given by the minimum value ][x(t), u(t)] in the row corresponding to x(t).
634
Chapter 23 Modern Approaches to Decision Analysis TABLE 23.2
O p t i m a l Fishery S t o c k i n g Strategy a
t=o
x(t)
t=l
u*
t=2
u*
J*
J*
u*
0
1
20
5
16
1
0
16
4
15
2 3 4
0 0 0
15.5 15 13.5
3 0 0
14 12 10.5
0 0 0
t=3
J*
u*
J*
2
12
4
7
5
10
3
6
2 1 0
5 4 0
7 6.5 6
a For a fishery subjected to sport harvest and periodic stocking during a fishing season of 4 months. Rows represent population size, columns represent stocking times, and table entries are optimal stocking levels and accumulated costs for each population size at each time during the fishing season. Optimal stocking rates and associated costs are determined for all population sizes at t = 2, and these values are retained for use in determining optimal values for t = 1 (Table 23.2). In like manner, the algorithm continues to step backw a r d in time through two more iterations, until the beginning of the time frame is reached. At each stage, size-specific stocking levels are identified according to the HJB equation, along with accumulated costs over the remainder of the time frame (Table 23.2). On completion of the backward iterations, a field of optimal stocking rates and associated costs is identified for all population sizes and all times in the time frame. To use this strategy, a m a n a g e r need only (1) determine the size x(0) of the population at the beginning of the fishing season, (2) identify from Table 23.2 the optimal level of stocking for x(0), (3) calculate the transition to a new state according to x(1) = x(0) + u(0) - z(0), (4) identify from Table 23.2 the optimal level of stocking for x(1), (5) calculate the transition to a new state according to x(2) = x(1) + u(1) - z(1), and so on. Several noteworthy patterns can be seen from examination of Table 23.2: 9 Optimal stocking at t = 3 calls for eliminating the population, irrespective of its size. In essence, there is a cost but no gain in stocking beyond w h a t is needed to meet d e m a n d at the end of the time frame. The optimal strategy avoids unnecessary costs by stocking only as needed to meet that demand. 9 Similarly, the optimal strategy calls for stocking only as necessary to meet d e m a n d in the first month, irrespective of the initial population size. Thus, the optimal strategy calls for stocking one unit if x(0) = 0, and 0 units otherwise. This allows d e m a n d for t = 1 to be met without incurring additional stocking costs. 9 Accumulated costs increase as the time left in the fishing season increases. This reflects the obvious fact
that m a n a g e m e n t costs accrue through time, and accumulated costs therefore increase as more time is available for m a n a g e m e n t actions. 9 Variation in the optimal stocking level reflects variable d e m a n d over the fishing season. Thus, higher stocking levels generally are seen in the second and fourth months w h e n d e m a n d is high, and reduced stocking levels are seen in the first and third months w h e n d e m a n d is reduced. 9 With some exceptions, the optimal stocking level varies inversely with population status. Thus, stocking generally is increased w h e n populations are low and is reduced w h e n populations are high. 9 Stocking levels at each point in time are influenced by anticipated as well as current demand. For example, stocking in period 2 at the level u(2) = 5 for a population of size x(2) = 1 not only satisfies d e m a n d z(2) = 2 in period 2, but also meets the d e m a n d z(3) = 4 in period 3 without a need for any additional stocking. These general patterns are intuitively appealing, and on reflection are s o m e w h a t obvious. Less obvious are the specific minimum-cost stocking levels at each time during the fishing season. Even for problems as simple as this example, optimal m a n a g e m e n t strategies and their associated costs cannot be identified by cursory inspection and instead must be obtained via analysis in the context of optimal decision making. Example
Assuming that the state transition equations and optimality index are differentiable over T, the stagewise dynamic p r o g r a m m i n g algorithm can be applied to problems with continuous state variables. To illustrate, consider a single-age harvested population over a discrete time frame, with the transition equation x(t + 1 ) = G[y(t)],
where y(t) - x(t)[1 - u(t)].
To simplify mathematical notation, the population state and the growth function have been combined into a general function G, which is expressed in terms of the postharvest population y(t). The function G is ass u m e d to be concave in y, in that the second derivative of G with respect to y is negative. The harvest rate u(t) is necessarily b o u n d e d between 0 and 1, so that y(t) must lie in the interval [0, x(t)]. We take the optimality index for this problem to be I(x, u, t) = ~ ( t ) v ( x ) u ( t ) x ( t ) = e~(t)v(x)[x(t) - y(t)],
635
23.3. Dynamic Programming where v(x) is the marginal value for a unit of harvest. The optimal recurrence relation for this problem is
J*[x(t), t] = max E{I(x, U, t) + J*[x(t + 1), t + 1]}, {U(t)} m
(23.17)
e Ut m
J*(x, t) = max J(x, y, t),
where the expectation refers to the stochastic structure of the exogenous environment or the control trajectory or both. Several points are worthy of note:
J(x, y, t) = v(x)[x - y] + f3tJ*[G(y), t + 1],
9 Stochastic dynamic programming is used frequently in biological applications, in part because of the influence of the many stochastic factors that influence biological system dynamics. In fact, the degree and complexity of stochastic variation in biological systems distinguish their management from that of many engineering systems and motivate the use of procedures that account for stochastic variation. 9 We assume here that the transition functions are known, so that the expectation in the HJB equation need not account for uncertainties about the system structure. Nor need it account for partial system observability, as the state of the system is assumed to be observed without error at each point in time. We deal more fully with structural uncertainties and partial observability in Chapter 24. 9 Biological applications of stochastic dynamic programming typically utilize discrete system states in some manner. Many applications involve infinite time horizons, with the intent of orienting optimal control strategies strongly to the future impacts of present actions. In that case, the objective functional in Eq. (23.13) can become infinite, depending on the optimality index and the nature of the transitions. Comparisons among feasible control strategies then become impossible, as does the identification of an optimal strategy. 9 If the optimality index in Eq. (23.13) is time discounted with discount factors that are less than unity, the objective functional is necessarily finite. General conditions that guarantee finite values for the objective functional include, but are not limited to, the discounting of utilities (Williams, 1988). The time average
O<-y<-x
where
with ~t the discount factor for the period [t, t + 1]. Under conditions that ensure differentiability of J(x, y, t), maximization results in an optimal strategy defined by the relationship 3
~t ~{l*[G(y), t + 1]} = ~t aJ*[G(y),aGt + llG,(y ) = v(x). This equation can be solved iteratively to produce timespecific and state-specific values yt(x), starting with Ytf (x). The corresponding optimal strategy is then given by
~yt(x), y*(t) = ~ x(t),
yt(x) < x(t); yt(x) >- x(t),
indicating zero harvest if x(t) is below the critical level yt(x), or harvest to a level prescribed for x(t) if x(t) is above the critical level. This is a switching strategy of the kind seen previously in applications of the maxim u m principle. It calls for harvest levels at each point in time to drive the system as quickly as possible to a predetermined state trajectory. Having achieved this trajectory, the optimal strategy tracks the trajectory as closely as possible. The trajectory is determined by the mathematical structure of the optimality index and the transition equation.
23.3.2. Stochastic Dynamic Programming Many applications of dynamic programming in natural resources incorporate stochastic effects, almost always in a context of discrete time frames. Randomness associated with environmental variation, limitations in system controllability, and other factors induce stochastic system behaviors (see Section 7.5). System transitions can include a stochastic element Z(t) to account for such random factors: x(t + 1) = x(t) +/(x, U, Z, t), where Z(t) usually represents an uncorrelated white noise process with a stationary (i.e., time invariant) distribution. The corresponding control systems are known as Markov decision processes (see Section 10.6), and the appropriate formulation of the HJB equation for a unit time step is
l =
,Tim{ E
(T + 1) -1 E ~, I(x, U, Z, t) t=0
]}
is appropriate for problems with undiscounted optimality indices, in that it avoids problems with potentially infinite objective functionals. An optimal control strategy for a time-averaged objective functional maximizes the average single-step system gain. It also identifies state-specific optimal values (Puterman, 1994) (see Section 10.6). 9 The iterative algorithm described above for deterministic systems can be modified for stochastic systems, by incorporating the expectation operator as appropriate:
636
Chapter 23 Modern Approaches to Decision Analysis
Beginning at tf, an action is chosen that maximizes the terminal value E[Fl(Xf)], given that the values are influenced by the choice of an action. Otherwise, the value l*[x(tf),tfl for state x(tf) at time tf is l*[x(tf), tf] = E[Fl(Xf)]. The expectation in this expression allows for stochastic effects that influence utility. The second step utilizes the optimal state-specific values for tf to determine state-specific actions at time tf - I according to Eq. (23.17). Randomness in the transition from x(tf - 1) to x(tf) is inherited from Z(tf - 1), and the expectation essentially averages over the values I(x, U, Z, tf - 1) + l*[x(tf), tf]. The optimization utilizes these averages to identify optimal state-specific actions and optimal values J*[x(tf- 1), tf - 1] for a two-stage problem. The third step utilizes the values J*[x(tf- 1), tf - 1] in Eq. (23.17) to determine state-specific actions at time tf - 2. Randomness in the transition from x(tf - 2) to x(tf - 1) is accounted for by the expectation, which averages over the values I(x, U, Z, tf - 2) + l*[x(tf), tf - 1]. The optimization utilizes these averages to identify optimal actions and optimal state-specific values J*[x(tf - 2), tf 2] for a three-stage problem. The algorithm continues to step backward through the time frame until the initial time t o is reached. 9 Repeated application of the algorithm eventually stabilizes on a set of optimal state-specific actions that are independent of time. For times sufficiently removed from tf, one need k n o w only the system state, but not the time, to recognize the optimal action. 9 Repeated application for discounted optimality indices produces optimal values J*[x(t), t] that converge to stationary state-specific values. On the other hand, repeated application with undiscounted optimality indices produces optimal values J*[x(t), t] that diverge over time. However, the time average of values converges to the system gain over time (Williams, 1982).
with randomness in Z(t) inducing stochastic behaviors in population dynamics. For purposes of illustration, we assume here that the population size each year is either 5000, 10,000, or 15,000 individuals, that annual harvest rates are restricted to 0.1, 0.2, or 0.3, and that reproduction-recruitment rates are poor, average, or good, depending on environmental conditions. We also assume that environmental conditions fluctuate randomly and without discernible trend from year to year. The transitions from x(t) to x(t + 1) are driven by different combinations of population size x(t), harvest rate u(t), and environmental effect Z(t). For example, if population size is x(t) = 10 and harvest rate is u(t) = 0.1, then the population either grows, declines, or remains at 10 depending on environmental conditions. These outcomes inherit probabilities from the environmental variation. Because of nonlinearities in the reproduction-recruitment and mortality functions, the outcome probabilities vary with both population size and harvest rate. It is possible to tabulate the transitions and their associated probabilities, based on the population model and stochastic distribution of Z(t) (Table 23.3). The information in Table 23.3 can be used to determine an optimal harvest strategy for this population, based on the backward stepping algorithm described above. For an arbitrary terminal time T, we begin by recognizing that u = 0.3 maximizes harvest yield for x(T) = 5, 10, and 15, with J*(x, T) - 1.5, 3, and 4.5, respectively. At time T - 1, the HJB equation is J*[x(T- 1),T-
max E{I(x, u, T - 1) +/*Ix(T), T]}. u ~ {0.1,0.2, 0.3} Assuming x ( T - 1 ) = 5, the transition probabilities from Table 23.3 can be used to evaluate the expectations on the right hand side of this expression:
Example
TABLE
Consider an objective to maximize accumulated harvest for a population that is subjected to hunting each year. Reproduction/recruitment R(t) is a function of population size x(t) and environmental conditions Z(t), and mortality D(t) is influenced by both the population size and harvest rate u(t). The general transition equation for this population is
II =
23.3
Single-Step
Transition
Probabilities
a
x(t + 1) u = 0.1 5
10
u = 0.2 15
5 0.2 0.5 0.3
x(t) 10 0.2 0.3 0.5 15 0.1 0.3 0.6
5
10
u = 0.3 15
0.5 0.3 0.2 0.2 0.5 0.3 0.2 0.4 0.4
5
10
15
0.7 0.3 0.0 0.6 0.3 0.1 0.3 0.5 0.2
x(t + 1) = x(t) + R(t) - D(t), R(t) = fix(t), Z(t)], D(t) = g[x(t), u(t)],
For a harvested population with three states and three harvest rates. Rows represent population size at time t, and columns represent population size at time t + 1. a
23.3. Dynamic Programming u = 0.1,
637
T A B L E 23.4
(0.1)(5)+ (0.2)(1.5) + (0.5)(3) + (0.3)(4.5) = 3.65;
Optimal
Harvest Rates
u*(t) a n d V a l u e s J*[x(t)] a
u = 0.2,
(0.2)(5)+ (0.5)(1.5) + (0.3)(3) + (0.2)(4.5) = 3.55;
u = 0.3,
(0.3)(5)+ (0.7)(1.5) + (0.3)(3) + (0.0)(4.5) = 3.45.
u*(t)
From these expected values the o p t i m a l action for x ( T - 1) = 5 is seen to be u = 0.1, w i t h a value J*[5, T 1) = 3.65. A similar exercise for x ( T - 1) = 10 exhibits the expectations u =0.1,
(0.1)(10) + (0.2)(1.5) + (0.3)(3) + (0.5)(4.5) = 4.45;
u = 0.2,
(0.2)(10) + (0.2)(1.5) + (0.5)(3) + (0.3)(4.5) = 5.15;
u = 0.3,
(0.3)(10) + (0.6)(1.5) + (0.3)(3) + (0.1)(4.5) = 5.25;
revealing that u = 0.3 is optimal for x ( T - 1) = 10 a n d J*[10, T - 1) = 5.25. Finally, the expectations for x ( T - 1 ) = 15 are u =0.1,
(0.1)(15) + (0.1)(1.5) + (0.3)(3) + (0.6)(4.5) = 5.25;
u = 0.2,
(0.2)(15) + (0.2)(1.5) + (0.4)(3) + (0.4)(4.5) = 6.3;
u = 0.3,
(0.3)(15) + (0.3)(1.5) + (0.5)(3) + (0.2)(4.5) = 7.35;
a n d it follows that u = 0.3 is o p t i m a l for x ( T - 1) = 15, w i t h J*[15, T - 1) = 7.35, respectively. To d e t e r m i n e the o p t i m a l actions a n d values for time T - 2, w e again use the transition probabilities in Table 23.3 along w i t h the optimal values for T - 1 in the HJB equation. For x ( T - 2) = 5 w e h a v e u =0.1,
(0.1)(5) + (0.2)(3.65) + (0.5)(5.25) + (0.3)(7.35) = 6.06;
u = 0.2,
(0.2)(5) + (0.5)(3.65) + (0.3)(5.25) + (0.2)(7.35) = 5.87;
u = 0.3,
(0.3)(5) + (0.7)(3.65) + (0.3)(5.25) + (0.0)(7.35) = 5.63;
d e m o n s t r a t i n g that u = 0.1 is o p t i m a l for x ( T - 2) = 5. A similar exercise for x ( T - 2) = 10 reveals that u = 0.2 is optimal, a n d J*[10, T - 2)=7.56. For x ( T - 2) = 15, the action u = 0.3 is optimal, a n d J*[15, T - 2) = 9.69. The process can continue in this m a n n e r indefinitely. Thus, the o p t i m a l actions a n d values' are obtained for time T - 3 by once again utilizing the transition probabilities in Table 23.3 along with the optimal values for T - 2. The optimal actions identified for x ( T - 3) = 5, 10, a n d 15 are u = 0.1, 0.2, a n d 0.3, respectively, w i t h /*(5, T - 3) = 8.4, J*(10, T - 3) = 9.9, a n d / * ( 1 5 , T 3) = 12.03. Several points of e m p h a s i s can be m a d e from these results. First, the iterative a p p r o a c h p r o d u c e s an optim a l action for every state of the s y s t e m at every decision stage in the time frame. Thus, one can identify from Table 23.4 the o p t i m a l control a n d c o r r e s p o n d i n g o p t i m a l value of the objective functional, no m a t t e r
T-3
T-2
l*[x(t)] T-1
T
T-3
T-2
T-1
T
5
0.1
0.1
0.1
0.3
8.4
6.06
3.65
1.5
10
0.2
0.2
0.3
0.3
9.9
7.56
5.25
3.0
15
0.3
0.3
0.3
0.3
12.03
9.69
7.35
4.5
a For three population sizes and four time periods. Population size x(t) takes values of 5, 10, and 15. The index T - k represents k periods prior to the terminal time T.
w h e r e or w h e n in the time f r a m e one begins. H o w e v e r , at a n y given point in time an optimal sequence of controls over the r e m a i n d e r of the time f r a m e cannot be d e t e r m i n e d a priori, because it is not possible to d e t e r m i n e s y s t e m transitions with certainty. Instead, at each decision stage the state of the s y s t e m m u s t be ascertained, so that the a p p r o p r i a t e action for that state can be used. A control strategy requiring sequential interrogation of the s y s t e m state as the strategy is being a p p l i e d is k n o w n as closed-loop control (Intriligator, 1971). This contrasts w i t h o p e n - l o o p control strategies identified in deterministic d y n a m i c p r o g r a m m i n g , for w h i c h specification of initial state x(t 0) is sufficient to d e t e r m i n e the o p t i m a l sequence of actions t h r o u g h o u t the r e m a i n d e r of the time f r a m e (see Section 23.3.1). Second, Table 23.4 illustrates the t e n d e n c y of an optimal h a r v e s t strategy to be exploitative w h e n the time f r a m e for decision m a k i n g is abbreviated. Conversely, less exploitative actions are o p t i m a l w h e n m o r e time r e m a i n s for decision m a k i n g . A p a t t e r n of increasing harvest as the e n d of the time f r a m e a p p r o a c h e s is clear for x = 5 a n d x = 10. This accords w i t h the intuitive notion that long-term resource m a n a g e m e n t focuses on conservation a n d future resource o p p o r t u nities, w h e r e a s m a n a g e m e n t over the short t e r m emphasizes i m m e d i a t e returns w i t h little accounting of the effect on resource d y n a m i c s . Third, h a r v e s t rates tend to be m o r e exploitative for larger p o p u l a t i o n sizes. T h o u g h this p a t t e r n n e e d not a l w a y s arise in nonlinear systems, it typically occurs if mortality rates increase m o n o t o n i c a l l y w i t h h a r v e s t a n d if r e p r o d u c t i o n / r e c r u i t m e n t rates decrease m o n o tonically w i t h p o p u l a t i o n size. Fourth, the h a r v e s t strategy stabilizes after t w o iterations, b e c o m i n g i n d e p e n d e n t of time from then on (Table 23.4). A n optimal strategy for this p r o b l e m includes u = 0.1, 0.2, a n d 0.3 for x = 5, 10, a n d 15, respectively, for all times further than t w o stages from the end of the time frame.
638
Chapter 23 Modern Approaches to Decision Analysis
Fifth, optimal values of the objective functional increase without limit as the number of iterations increases. However, the stationary system gain of 2.34 has been attained by time T - 3. This is seen in the increase of 2.34 in the optimal value of J* from T - 3 to T - 2, irrespective of system state (Table 23.4). The stationary gain is given by J*[x(t - 1), t - 1] - J*[x(t), t] for all states and all times further than two stages from the end of the time frame. This is indicative of a general pattern in backward iteration, and is consistent with the convergence to a stationary gain that is found with time-averaged objective functionals (Williams, 1982).
23.3.3. Summary Dynamic programming addresses the general problem of choosing a control strategy from some constrained set of feasible strategies, to maximize an objective functional of system state, control, and possibly time. The approach is broadly applicable to continuous-time and discrete-time systems with either stochastic or deterministic behaviors, over time frames that are either finite or infinite, with or without constraints on state and control trajectories. Solution approaches involve the following: 9 A particular control problem is imbedded in a larger class of problems, and the Hamilton-Jacobi-Bellman (HJB) equation is derived by application of Bellman's Principle of Optimality. The solution of the HJB equation provides a general solution of the control problem for any system state at any point in the time frame. 9 For continuous systems, the HJB equation is a partial differential equation in the state variables and time, and as such is usually very difficult or impossible to solve for nonlinear systems. Numerical procedures almost always are required. 9 For discrete systems, the HJB equation essentially describes a backward iteration procedure, in which the optimal action for a given state at each time is based on the sum of the optimality index at that time and a value corresponding to an optimal strategy for the rest of the time frame. 9 For stochastic systems, the objective functional is expressed in terms of expected values, and the transition equations essentially describe a Markov decision process.
23.4. HEURISTIC APPROACHES It should be clear that theoretically based procedures such as dynamic programming, variational mathemat-
ics, and nonlinear programming impose rather stringent mathematical requirements on the transition equations, objective functions, and operating constraints on an optimal control problem. As indicated above, these approaches can be. useful in biological investigation, but only under specific circumstances involving reasonably well-behaved systems, typically involving small dimensions, limited controls, and uncomplicated boundary conditions. The problem is that biological systems rarely meet these requirements; thus, only a small suite of biologically informative optimization problems can be addressed formally. Fortunately, other computer-aided approaches are available to investigate patterns of control in biological systems. These utilize modeling less to solve an optimal control problem than to explore it, and really consist of an amalgam of procedures involving simulation models in combination with optimization methods to search for optimal strategies. Whereas each of the techniques mentioned previously yields a strategy that is optimal over some specified constraint set, simulation gaming is more exploratory and less directed at genuine optima. The approach generally involves the construction of a response surface using a simulation model, followed by the analysis of the surface with some optimization procedure or other heuristic aid. Often the analysis consists of "what if" gaming, in which several management strategies and corresponding system responses are simulated and then assessed with analytic tools such as response surface analysis, mathematical programming, and other mathematical/statistical techniques. Because it is heuristic rather than rule-driven, and because it enlists the power of computer simulation, simulation gaming suffers few restrictions on the size or structure of the model that can be used. In particular, the allowable biological complexity in this approach is virtually unlimited. For example, an application might involve the simulated dynamics of a population possessing a complex age structure, with each cohort relating through reproduction and mortality to a suite of dynamic habitat conditions that are stochastic. More generally, a simulation model might include a large community of interacting species, each with complicated cohort structures for both size and age, each possessing its own reproduction, survivorship, and migration patterns over a heterogeneous landscape, each cohort relating to other cohorts and to a complex milieu of dynamic habitat conditions through cohort-specific functions, with the model incorporating a broad range of dynamic habitat features along with random environmental variation and other stochastic influences. Management in such a situation might involve habitat
23.5. Discussion manipulation, selective harvest, and stocking, and its objective might involve dynamic equilibria for the community to meet biodiversity goals over the landscape, while maintaining acceptable levels of harvest opportunity and controlling the costs of stocking and habitat manipulation. Without dramatic simplification of such problems, it is not possible to use formal optimization procedures to identify optimal controls for them. If the value of simulation gaming lies in the use of such large a n d / o r complex models to organize and explore information, its weakness is that it constitutes a suboptimal analysis, and in most instances one cannot determine how the results compare to a truly global optimum. The same complexity that motivates computer simulation also renders a simulation model impossible to analyze formally with the optimization tools described above. This conclusion is not unexpected, because it expresses in another context the tradeoffs among generality, realism, and accuracy that were identified in Chapter 7. Example
A simple illustration of the use of simulation models involves a logistic model for population growth for a single species without age structure, which is to be harvested over some finite time frame: x(t + 1) = x(t) + r(1 - x/K) - U(t),
where the growth rate r and the carrying capacity K are stochastic. Harvest functions of the form U(t) = ax(t) + b are considered, with the objective function given by average total discounted harvest:
J = E
~, t = to
]
639
the objective function, the "response," exhibited on the third axis. This surface then can be explored by a search procedure to identify that point on the surface for which the objective function is maximum. Other heuristic searching methods have received much attention, and some of these may hold promise for obtaining "quasioptimal" solutions to complex natural resource management problems. For example, genetic algorithms (GA) are searching procedures patterned on the mechanisms underlying natural selection (Goldberg, 1989). In GA, a universe of possible strategies (modeled by "genotypes") is sampled to obtain candidates ("parents") to produce new combinations of strategies ("offspring"), which are evaluated with respect to an objective function ("fitness"). The random search is directional because of "selection" for strategies providing the best "fitness," and local optima are avoided by the randomizing effects of "recombination," "mutation," and "crossover." GA can provide efficient, near-optimal solutions to very large problems (i.e, problems containing many decision, state, and random variables). However, as with simulation, there can be no general statements about the optimality (or lack thereof) of solutions thereby obtained. Simulated annealing (SA) (Kirkpatrick et al., 1983), based on the physical problem of the cooling of a molten solid, also uses randomization to allow the system to move away from local suboptima. Like GA, no general conclusions can be made about the optimality of SA solutions. Nevertheless, work on the applications of heuristic methods like GA and SA to natural resource decision making (e.g., Moore et al., 2000) suggests that heuristic methods may be valuable adjuncts to the optimization approaches emphasized in this book.
f3tu(t) ,
23.5. D I S C U S S I O N where 13discounts future returns to present value and the expectation accounts for stochastic variation in returns attendant to the harvest of a stochastic population. The idea is to identify the values of a and b that maximize this objective. In this case, the simulation model can be used to simulate population dynamics for various combinations of the parameters a and b, with randomly chosen values for r and K. For given values of a and b, the values of the objective function are averaged across combinations of r and K that arise from their joint distribution, producing the average called for in the objective function. This procedure is repeated for many values of a and b, effectively producing a three-dimensional response surface with parameters a and b on the horizontal axes and the value of
An important advantage in the application of linear programming, classical optimization, simulation gaming, and even nonlinear programming is that these techniques do not share the size limitations of dynamic programming and variational mathematics. However, they have other limitations not shared by the modern approaches. For example, linear programming requires both the system model and the objective function to consist of equations and inequalities that are algebraically linear in state and control variables. This is an unlikely mathematical structure for ecological systems, one that is often inadequate to represent essential ecological behaviors. Simulation gaming, an amalgam of procedures producing suboptimal results, typically of-
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Chapter 23 Modern Approaches to Decision Analysis
fers no indication of how well a control strategy thus produced approximates a true optimal solution. Classical optimization procedures typically limit the class of feasible control strategies to those characterized by a set of parameters, again with no indication of the degree to which such solutions approximate a bona fide optimal control strategy. And the search procedures of nonlinear programming often have convergence problems and thus fail to identify global optima. As with most analytic techniques, one faces tradeoffs in performance among approaches to the optimization of dynamic biological systems. Comparison of these dynamic optimization methods is problematic, because the methods generally apply to systems with different mathematical characteristics and distinct formulations of the optimal control problem. Nevertheless, it is worthwhile to consider at least a heuristic comparison with respect to some general model characteristics. Three scales are useful for this comparison: a "complexity" scale, by which methods are compared by the complexity of the models to which they are applied; a "size" scale, by which methods are compared by the size of the model; and a "precision" scale, by which the techniques are compared according to the precision of their optimal solutions (Fig. 23.6). The applicability of the methods with respect to system complexity is displayed in Fig. 23.6a for systems ranging from simple to complex. Simulation gaming procedures apply to the most complex systems, because the procedures involve no necessary model
restrictions. At the other extreme, linear programming applies only to very limited systems, namely, those with linear transitions, constraints, and objectives. Classical methods occupy a midrange position, because they are applicable to complicated models but usually involve simplifying restrictions on model behaviors or admissible control strategies or both. Nonlinear programming procedures require fewer assumptions and generally allow for more complicated mathematical structures compared to classical procedures, but are not as applicable to complex systems compared to simulation procedures. Dynamic programming, because of its ability to incorporate stochastic effects and system constraints, generally applies to more complex systems than do variational methods. However, this conclusion does not necessarily extend to continuous models, because the ordinary differential equations produced by variational methods typically are less difficult to analyze than the partial differential equations of dynamic programming. Figure 23.6b shows the comparison of methods relative to system size, ranging from smallest to largest system dimensions. Without question, linear programming applies to the largest models, with virtually limitless numbers of state and control variables allowable in the procedure. Simulation gaming procedures also allow for large numbers of states and controls. However, limits can be imposed by simulation costs, which can become high for a thorough optimality analysis. Classical methods, though applicable to large systems, usually involve restrictions to a limited set of system
a SIMPLE
LP
DP
~
vA
v A
J
DP v A
b SMALL
C IMPRECISE
~
VM
CO v
v A
A
NL v A
VM v A
HM
LP
v A
v A
...
CO v A
NL
v A
HM v A
HM V
A
LP 'A v
CO
NL
VM
DP
v A
v A
v A
v A
J
COMPLEX
~
LARGE
~
PRECISE
F I G U R E 23.6 Heuristic scalings of six dynamic optimization procedures: linear programming (LP), dynamic programming (DP), variational mathematics (VM), classical optimization (CO), nonlinear programming (NL), and heuristic methods (HM). (a) Comparison of procedures with respect to system complexity. (b) Comparison of procedures with respect to system size. (c) Comparison of procedures with respect to precision of the optimal solutions.
23.5. Discussion parameters. Dynamic programming, variational mathematics, and nonlinear programming usually are restricted to much smaller systems. Dynamic programming is useful with discrete systems having only a few dimensions, and sometimes with continuous systems having a single dimension. The system size appropriate to variational methods depends in large measure on the mathematical structure of the model, but usually is restricted to a small number of state and control variables. Size limitations for nonlinear programming are associated with the number of system constraints and the number of decision points in the time frame. In general, only a limited number of state and control variables are included in a nonlinear programming problem. Figure 23.6c shows the comparison of methods relative to the precision of the optimum solutions obtained. Variational methods and dynamic programming both produce global optima when the corresponding differential or difference equations can be solved. Typically, however, the numerical methods used for continuous problems introduce errors that are proportional to the problem size and complexity. Classical methods, by imposing restrictions on the system or the controls, produce an optimal solution only for the restricted problem. Linear programming offers an extreme example of such restrictions, with system behaviors represented in linear terms only. Nonlinear programming procedures also can include restrictions on the model, including limitations on its time frame. Beyond that, the iterative procedures of nonlinear programming converge to local optima, which can be quite dissimilar from global optima for nonconvex Lagrangian functions. Finally, simulation gaming cannot be placed easily in the comparison, because it is not possible to compare the strategies resulting from the procedures to the true optimal strategies. Some general properties emerge from these comparisons: 1. Variational mathematics is likely to be a fruitful approach for small systems that are continuous in states, controls, and time. This is especially true for systems without stochastic components and without severe nonlinearities. 2. Dynamic programming often is applicable for small systems that are discrete (or can be made discrete) in state, control, and time. This is especially true for systems that include stochastic effects, system constraints, or nonlinearities. In particular, system constraints are of benefit in limiting the computational requirements of dynamic programming. 3. Linear programming is appropriate for large systems with stable, linear transition structures and dis-
641
crete time frames. The approach is useful for analysis of "sustained-yield" problems with infinite time horizons and steady-state constraints, but is of limited value for many systems with a high degree of nonlinear behavior that cannot be represented adequately via linear approximation. 4. Nonlinear programming is useful for systems with numerous constraints and with well-behaved differential properties, whereby iterative gradient search procedures are applicable. Though this approach can produce bona fide optimal solutions, in many cases the rates of convergence to a solution can present problems, and often the approach will identify local rather than global optima, but will prove unable to identify true global optima. 5. Classical optimization is applicable to systems for which system behaviors a n d / o r controls are characterized by a small set of parameters a n d / o r equality constraints that strongly restrict the available options. In most instances, continuous differential properties are required for the objective functional and state transitions. 6. Simulation approaches often are fruitful for large, complex systems with nonlinearities, stochastic behaviors, and discontinuities in objective functions and transition equations. Indeed, for systems of this kind, no other approach appears to be possible. However, one usually cannot determine the degree to which the resulting control strategies approximate genuine optimal solutions. From these characterizations, one can recognize obvious advantages in describing a natural resource system as economically as possible. Systems with few stochasticities, nonlinearities, and other complicating structures are amenable to more elegant analyses and greater precision in determining the optimal control strategies. However, this economy typically involves a sacrifice of completeness (e.g., by limiting the number of state variables describing the system) or realism (e.g., by limiting the transition equations describing system dynamics to linear forms). Among the scientific disciplines, biological and ecological science may be the most vulnerable to these tradeoffs. The appropriate degree of resolution and mathematical detail cannot be determined a priori, and is influenced by, among other things, the objectives of the investigation and the intrinsic behaviors of the resource system. Thus the "best" approach for the optimal control of a biological resource system remains problematic, with the biologist required to decide when gains in mathematical tractability compromise the value of the optimization results. Such decisions involve the weighting of often conflicting objectives, and this conflict, as much as the
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Chapter 23 Modern Approaches to Decision Analysis
intrinsically mathematical nature of the problem, presents an enduring challenge to develop population models that are informed both mathematically and biologically. Indeed, a goal of this book is to narrow
the gap between purely biological and purely mathematical perspectives, and thereby promote the intelligent use of models for management of biological resources.
C H A P T E R
24 Uncertainty, Learning, and Decision Analysis
24.1. DECISION ANALYSIS IN NATURAL RESOURCE CONSERVATION 24.1.1. Accounting for Environmental Variation 24.1.2. Incorporating Process Uncertainty 24.2. GENERAL FRAMEWORK FOR DECISION ANALYSIS 24.2.1. Multiple Options, Outcomes, Hypotheses 24.2.2. Multiple Decision Times 24.3. UNCERTAINTYAND THE CONTROL OF DYNAMIC RESOURCES 24.4. OPTIMAL CONTROL WITH A SINGLE MODEL 24.5. OPTIMAL CONTROL WITH MULTIPLE MODELS 24.6. ADAPTIVE OPTIMIZATION AND LEARNING 24.7. EXPECTEDVALUE OF PERFECT INFORMATION 24.7.1. Identical Values for All Models 24.7.2. Identical Optimal Policies for All Models 24.8. PARTIALOBSERVABILITY 24.9. GENERALIZATIONSOF ADAPTIVE OPTIMIZATION 24.10. ACCOUNTING FOR ALL SOURCES OF UNCERTAINTY 24.11. "PASSIVE" ADAPTIVE OPTIMIZATION 24.12. DISCUSSION
able. At several points we have suggested approaches for dealing with random environmental variation and other stochastic factors, through the use of probability modeling, dynamic optimization, and statistical assessment. We have seen that the addition of random variation can complicate the analysis of a biological system and increase the difficulty of identifying and implementing optimal long-term strategies for it. There are a number of factors in addition to environmental variation that can contribute to uncertainty about optimal resource management, and each is distinct in its patterns and its influence on system behaviors. One of the most pervasive is the limited recognizability of system state at each point in time. For example, population size is almost never known with certainty and must be estimated based on an incomplete record of individuals in the population. It seems intuitive that the utility of decision making can be compromised by inaccuracies and imprecision in estimating system state. Another source of uncertainty is a lack of knowledge about important biological processes influencing resource dynamics, which often is expressed as confusion (or disagreement) about the appropriate mathematical descriptions of these processes. Yet another is a disparity between actual and intended management controls, which occurs frequently with the use of indirect control methods such as harvest regulations. In combination, these and other uncertainty factors complicate the analysis of resource systems and present serious challenges in developing optimal management strategies for them. In this chapter we describe a unified framework for the optimal management of wildlife populations under uncertainty. The framework incorporates multiple
Thus far in this book we have discussed the formulation and analysis of biological models, the use of data to inform our knowledge of biological structure and function, and the optimal management of biological populations and their habitats. We have seen that added complexity attends the incorporation of cohort structures, multispecies interactions, habitat effects, and other factors. Throughout the book we have described the estimation of population attributes through sampling when populations are only partially observ-
643
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Chapter 24 Uncertainty, Learning, and Decision Analysis
sources of uncertainty and recognizes the potential value of learning through management. We note that in recent years biologists increasingly have focused on uncertainty, and its reduction when possible, as an aid to improved wildlife management. New technologies involving remote sensing, pattern analysis, and survey design have enhanced substantially our capability to monitor wildlife populations and the habitats they use (Bookout, 1994). Developments in adaptive resource management (Holling, 1978; Walters, 1986), information theory (Burnham and Anderson, 1998), and related topics allow wildlife managers to make more informed and effective decisions in the face of uncertainty about biological processes. These and other advances have enabled biologists to better recognize and monitor environmental and biological variation, and to more effectively manage the biological processes contributing to that variation. However, there remains a need for a systematic treatment of the forms and influences of uncertainty, especially as they affect the optimal management of populations.
24.1. D E C I S I O N A N A L Y S I S IN NATURAL
RESOURCE CONSERVATION Here we describe a process of decision analysis that incorporates multiple decision options, utility functions of the decisions and their outcomes, and uncertainties or "risks" associated with these outcomes. The idea is to calculate for each alternative decision an average utility based on the associated risks, so as to identify a decision that maximizes average utility. We introduce these ideas by means of a hypothetical conservation problem. Thus, consider that a decision maker is faced with the potential loss of an endangered species, and two alternatives are available: the design and management of a 100-km 2 reserve (action a 0 = 1), and a "do nothing" alternative (action a 0 = 0). Because the concern is to sustain an extant population, the population state can be characterized at time t by a binary state variable x t, with x 0 = I indicating species presence just prior to the decision. Given either of the alternative decisions, there are two possible outcomes: the population persists and possibly increases in abundance (Xl = 1), or the population declines to local extinction (Xl = 0). The sequence of possible decisions and outcomes involves four branches, resulting from two alternative decisions and two outcomes for each decision (Fig. 24.1). Each combination of a decision and outcome has associated with it potential costs (e.g., reserve construction) and benefits (e.g., species persistence), which are
incorporated into a utility function R(a 0, xl) that depends on both the decision and its outcome. From the point of view of endangered species management, population persistence is always preferable to extinction, and the best situation might involve persistence (x 1 = 1) along with no action (a0 = 0). On the other hand, the worst situation might involve no action (a0 = 0) and extinction (x I = 0), because of the ramifications (e.g., lawsuits, negative publicity) of inaction followed by extinction. Otherwise, population persistence without conservation is preferable to persistence with conservation, because of construction and maintenance costs. These relationships can be captured with values for R(a 0, x 1) that emphasize the benefits of population persistence (Fig. 24.1). In the event that decision-specific outcomes are certain (i.e., the consequences of conservation and inaction are known), a determination of the optimal decision becomes a straightforward comparison of the corresponding utilities. Assume, for example, that reserve construction and management are certain to result in population persistence, whereas inaction is certain to lead to extinction. Because the respective utilities are R(1, 1) = 0.7 and R(0, 0) = 0.0, respectively (Fig. 24.1), the optimal decision is to build and maintain the reserve.
24.1.1. Accounting for Environmental Variation Of course, the outcome of decision making with biological systems is almost never certain. Assume here that environmental factors influence population dynamics through their effects on reproduction and mortality. The essentially random nature of environmental variation induces uncertainty in population dynamics and thus in the likelihood of persistence. At least conceptually, these uncertainties can be represented by assigning probabilities to the possible outcomes, based on an assumed model of population dynamics and available field data. In a later treatment we will describe in some detail a data-based approach to the sequential identification of model uncertainties. For our present example, we let p(xl l ao) represent the probability of outcome x I given that action a 0 is taken. There are four such probabilities, corresponding to the four combinations of a 0 and x 1. Then an optimal decision can be identified simply by comparing the average utility
-R(a o) = ~ , p(xl]ao)R(a o, x 1)
(24.1)
X1
for the "no action" and conservation decisions. For example, assume that persistence and extinction are
24.1. Decision Analysis in Natural Resource Conservation
645
Persistence R(1,1) = 0.7
~
Extinction '-'.O
X0= 1
_ ~,
No action
., ~39
4~
XI = 0
R(1,0) -- 0.2
Persistence R(0,1) = 1.0
L__J R(0,0) = 0.0
F I G U R E 24.1 Branching d i a g r a m of possible decisions a n d o u t c o m e s for a h y p o t h e t i c a l e n d a n g e r e d species. The v a l u e x 0 = 1 indicates species presence prior to the decision. The v a l u e x I represents species status after the decision: x 1 = 1 c o r r e s p o n d s to species persistence, a n d x 1 = 0 to extinction. The v a l u e a 0 represents the c o n s e r v a t i o n decision: a 0 = 1 c o r r e s p o n d s to reserve d e v e l o p m e n t , a n d a 0 = 0 to the " n o action" alternative. Here, p(xlla o) is the probability of being in state x, at time t = 1, given that decision a 0 is i m p l e m e n t e d prior to t = 1; a n d utility R(a0, x 0) is associated with decision a 0 a n d o u t c o m e x 1. Rectangles a n d circles represent species status a n d decisions, respectively, a n d a r r o w s are u s e d to connect decisions a n d species status.
equally likely, no matter w h a t decision is made: p(Olao) = p(lla 0) = 0.5, irrespective of the value of a 0 (Fig. 24.2). Then the average utilities are given by
R(a o)
= 0.5[R(a 0, 0) + R(a 0, 1)],
(24.2)
and a p p l y i n g Eq_ (24.2) with the utilities s h o w n in Fig. 24.2 results in R(a o) = 0.5 for the "no action" alternative a 0 = 0, and R(a o) = 0.45 for the conservation alternative a 0 = 1. Because the former utility exceeds the latter, the optimal decision is not to establish the reserve. Persistence and extinction probabilities influence the choice of an optimal action for this problem, according to Eq. (24.1). If we had a s s u m e d an extinction probability of 0.75 rather than 0.5, then the average utilities w o u l d be R(a o) = 0.25 and 0.325 for a 0 = 0 and 1, respectively, and the optimal decision w o u l d have been to establish the reserve. On reflection, these results are intuitively appealing, in that they suggest a proactive a p p r o a c h to conservation w h e n the risk of extinction is high and a "hands-off" strategy w h e n it is low. A s s u m i n g that extinction probabilities are identical for both options, i.e., p(011) = p(010) = P0, Eq.
(24.1) can be used to determine a "break-even" value for extinction risk. Based on the utilities in Fig. 24.1, conservation is w a r r a n t e d w h e n e v e r extinction is sufficiently probable to ensure that R(1) = 0.2p0 + 0.711 - P0] is greater than R(0) = 0.0p0 + 1.011 - P0], that is, w h e n P0 > 0.6. In words, an extinction probability greater than 0.6 justifies the construction of a reserve, based on the a s s u m e d utilities. For extinction probabilities less than 0.6, a "no action" strategy is appropriate. These results of course d e p e n d on the particular utility function that is used. To see the influence of the utilities, a s s u m e that construction costs are projected to be higher than originally anticipated, so that the utilities R(1, 0) and R(1,1) associated with conservation are reduced from 0.7 and 0.2 to 0.6 and 0.1, respectively. U n d e r these conditions, conservation is w a r r a n t e d w h e n extinction is large e n o u g h to guarantee that R(1) = 0.1p0 + 0.611 - P0]
646
Chapter 24 Uncertainty, Learning, and Decision Analysis
R(1,1) = 0.7
w
R(I) = 0.45 R(I,0) = 0.2
X0= 1 R(0, l) : 1.0 R(0) = 0.50
xl=0
R(0,0) = 0.0
FIGURE 24.2 Branching diagram of possible decisions and outcomes for a hypothetical endangered species. Here, p(x 1 a0)is the probability of state x1at time t = 1, given that decision a0 is implemented prior to t = 1; R(a o) is the average utility corresponding to decision a0.
is greater than R(0) = 0.0p0 + 1.011 - P0], i.e., w h e n P0 > 0.8. Thus, the extinction probability n o w must be 0.8 or higher to justify the development of a reserve. In essence, increased construction costs reduce the value of conservation, thereby increasing the required extinction risk before reserve development is justified. This is consistent with patterns often exhibited in decision making, whereby conservation is discouraged w h e n conservation costs are high relative to projected benefits and encouraged w h e n costs are low relative to benefits. Just as optimal decision making is influenced by the utility function, so also is it d e p e n d e n t on patterns in the outcome probabilities. With utilities as in Fig. 24.2 and with equiprobable outcomes for both options, we saw that R(0) = 0.5 and R(1) = 0.45 from Eq. (24.2), so that an optimal decision is not to develop a conservation reserve. If, however, the conservation option is assumed to increase the persistence probability from p(111) = 0.5 to p(1[1) = 0.65, then the utilities become R(0) = 0.5 and R(1) = 0.525 from Eq. (24.1). Under these conditions the optimal decision would then be to establish and maintain the reserve (Fig. 24.3). Again, these results are intuitively appealing, in that they jus-
tify conservation actions that otherwise w o u l d be too costly to undertake, if those actions lead to an improved chance of species persistence.
24.1.2. Incorporating Process Uncertainty Thus far we have assumed that there is a single, distinctive relationship between a decision and its outcome, recognizing uncertainties attendant to r a n d o m environmental variation. Our decision analyses have assumed that either (1) persistence and extinction are uninfluenced by decisions (Fig. 24.2), or else (2) conservation leads to an increased probability of persistence (Fig. 24.3). Each of these two hypotheses allows for the potential effect of environmental variation, but neither takes into account the uncertainties about which hypothesis more appropriately represents the relationship between a decision and its outcomes. This is a key limitation, because uncertainty about the linkages between decision making and its consequences lies at the heart of m a n y controversies in natural resource conservation. One w a y to handle process uncertainty is simply to incorporate alternative hypotheses directly into the decision analysis, with hypothesis weights or "likelihoods" representing a decision m a k e r ' s confidence in
24.1. Decision Analysis in Natural Resource Conservation
647
R(1,1) = 0.7
m
R(1) = 0.525 R(1,0) = 0.2
x I =0
X0=
1
xl=
]
R(0, ~) = ~.0
m
R(0) = 0.500
R(0,0) 0.0 =
FIGURE 24.3 Branchingdiagram of possible decisions and outcomes for a hypothetical endangered species. The probabilities corresponding to conservation action a0 = 1 differ from those for the "no action" alternative a0 = 0.
the hypotheses. Consider, for example, the two hypothesized responses mentioned above: (1) persistence and extinction are equiprobable and unaffected by the decision (hypothesis H1; see Fig. 24.2) and (2) persistence is more likely with conservation than without it (hypothesis H2; see Fig. 24.3). Recall that the optimal decision u n d e r hypothesis H 1 is to take no action, whereas the optimal decision u n d e r hypothesis H 2 is to establish and maintain a reserve. An optimal decision that accounts for both hypotheses can be identified by comparing average utility values for each decision, except that now the averaging includes likelihood weights p(H i) representing the hypothesis uncertainties:
E[R(ao) ] = ~ p(Hi)-Ri(ao) i
(24.3)
action" utility is larger, the optimal decision is not to establish and maintain a reserve. Not surprisingly, this decision analysis is sensitive to the model likelihoods in Eq. (24.3). If extra weight [say, p(H2)-0.8] is given to the hypothesis that persistence is more likely u n d e r conservation, the average utilities from Eq. (24.3) become E[R(ao)] = 0.5 and 0.51 for the "no action" and conservation alternatives, respectively. In that event, the decision to establish a reserve becomes optimal, based on the larger value for the conservation option. This supports the intuitively appealing notion that conservation actions are justified as one becomes more confident of their potential benefits. On the other hand, inaction is justified w h e n these benefits are considered less likely to accrue. We note that Eq. (24.3) can be rewritten so as to identify likelihood values p' (H i) for t = 1. Thus,
= E,. p(Hi)[~Xl pi(xlla~176 where the subscript i in -Ri(ao), Ri(ao, Xl), and pi(xl]ao) indicates that the probabilities and utility functions in Eq. (24.3) are hypothesis specific (Fig. 24.4). Assume that the two hypotheses are equally likely, that is, p (H 1) = p(H 2) = 0.5. Applying Eq. (24.3) for the two options in our example, we then have E[R(ao)] = 0.5 for the "no action" alternative a0=0, and E[R(ao)] = 0.488 for the conservation alternative a 0 = 1. Because the "no
i
Xl
= E -p(xllao)-R[ao , xllp '(H1), P '(H2)], Xl
where
R[ao,xl]p'(H1 ), P'(H2)]
= E i
P'(Hi)Ri(ao,xl)
648
Chapter 24
Uncertainty, Learning, and Decision Analysis
R(1,1) = 0.7
m
R,(i) = 0.450 R(I,0) = 0.2
R(1,1) = 0.7 R2(I ) = 0.525 E[R(I)] = R(1,O) = 0.2
/ . ~
~
xj=l
R(0,1)=I.0
(-,
E[R(O)] =
~?~
R,(0) = 0.5
R(o,o)
=
o.o
_
R2(0) = 0.500
x I =0
R(0,0)= 0.0
FIGURE 24.4 Branching diagram of possible decisions and outcomes for a hypothetical endangered species, showing two hypotheses linking decisions and outcomes. For hypothesis H1, the persistence probability p1(1 a0) is unaffected by decision a0; for hypothesis H2, the persistence probability p2(1, a0) is larger with the conservation option a0 = 1 than with the "no action" alternative a0 = 0.
and
p, (Hi) = P (Hi)Pi(Xl[%) ff(xllao) p(Hi)Pi(Xllao)
~i p(Hi)Pi(XlJao)" The latter expression is an application of Bayes' Theor e m (see Section 4.5 a n d A p p e n d i x A.3), by m e a n s of
w h i c h the likelihood values p(H i) at time t = 0 can be u p d a t e d to p' (H i) at time t = 1. Likelihood u p d a t i n g is b a s e d on s y s t e m transitions across time, leading to increases in the likelihoods of h y p o t h e s e s that c o n f o r m w i t h s y s t e m b e h a v i o r s a n d to decreases in the likelih o o d s of those that do not. This in t u r n leads to a reduction in process u n c e r t a i n t y over time a n d to the e v e n t u a l identification of the m o s t a p p r o p r i a t e h y p o t h esis. This " a d a p t i v e " use of s y s t e m responses for learning is explored in considerable detail below.
24.2.
General Framework for Decision Analysis
24.2. GENERAL FRAMEWORK FOR DECISION ANALYSIS Up to this point we have restricted attention to decision making in which there are only two outcomes (persistence and extinction), two options (conservation or no action), two hypotheses (outcome probabilities are either dependent or independent of the decision), and decision making at a single point in time. These restrictions can be relaxed to allow for a more comprehensive treatment of optimal decision making.
24.2.1. Multiple Options, Outcomes, Hypotheses Consider a system in which the outcome x I of a decision can be any of an indefinite number of possible states. For example, x 1 might represent population abundance, density, demographic structure, or geographic distribution after some management action is taken. The management action might be chosen from an indefinitely large number of optionsmfor example, the number of animals to be removed from a population by trapping, the size of a planned reserve, or an intended stocking density. A variety of hypotheses could be used in predicting system response to mana g e m e n t m f o r example, any of a number of possible relationships between harvest and mortality, physiological condition and survivorship, population density and viability, etc. It is straightforward to generalize the decision problem previously described to accommodate this more general situation. As before, optimal decision making under these circumstances involves computing an average utility
E[R(a0)] = ~,.
649
24.2.2. Multiple Decision Times It also is possible to generalize from decision making at a single time, to allow for situations in which decisions are made at more than one time. One might anticipate that multiple decision times can be handled as naturally as multiple options, outcomes, and hypotheses; however, the extension to multiple time periods requires a substantially different, and much more complicated, optimization analysis. Because many important resource problems involve sequential decision making as well as temporal variation in system responses, we describe here a framework for dynamic decision analysis, preparatory to a more comprehensive treatment of optimal management under uncertainty. Thus, consider a simple extension of the one-period problem to accommodate decision options at two points in time. As before, we designate with x t the resource status at time t, with x 0 the initial state of the resource at time 0. We u s e pi(xt+liat, x t) t o represent the probability of transfer from state x t to xt+ 1 assuming action a t at time t, with Ri(a t, Xt+IIX t) the corresponding utility. Note that pi(xt+liat, x t) and Ri(a t, Xt+l[Xt) are natural extensions of the notation pi(xliao) and Ri(aoixl), used previously for hypothesis-specific probabilities and utilities. Because the population was assumed earlier to be present at time 0 (though not necessarily at time 1), x 0 had no role as a conditioning factor at time 0 and therefore was suppressed in pi(xlia O) and Ri(ao]xl). In what follows, resource status is allowed to vary over time, and we also allow for the initial resource state to be any of a range of values. Hence the need for a more complex notation. The decision problem is described in terms of both time periods. In essence, we seek an optimal sequence of decisions, a 0 at time 0 and a I at time 1, that provides maximal utility as expressed by
P (Hi) I~Xl pi(xlia~176 E[-R(ao,al)] = ~, p (Hi)l~ pi(xl]ao Xo)[ai(ao, xlixo)
for each available option, and then choosing that option for which the average utility is maximum. The difference between a general problem and the endangered species problem considered earlier is that the summations now are over multiple values of i and x 1, and many more decision options can be considered in the analysis. Identifying and keeping track of large numbers of hypothesis likelihoods, decision options, outcome probabilities, and utilities leads to substantial increases in data management and computing. On the other hand, incorporating any or all these generalizations adds little if any complexity to the decision analysis.
z
L xl
-}- ~ pi(x2iaI, x1)ai(al, x2
(24.4)
x2ixi)]/. J
Several points are noteworthy: 9 By adding the utilities for a second step to those produced for the first step, Eq. (24.4) is an additive extension of Eq. (24.3) for the single-time step problem. Equation (24.4) thus reduces to Eq. (24.3) when an optimal decision is required for one time period only. 9 Equation (24.4) includes a 0 and a I as arguments for the average utility, and optimal decision making
650
Chapter 24 Uncertainty, Learning, and Decision Analysis
requires the identification of both. Provided the number of decision options is limited, an optimal sequence of decisions can be identified by examining all possible decision combinations. This approach quickly becomes unwieldy as the number of options, and the number of decision periods, increase. 9 Computations for the first term of the sum in Eq. (24.4) are the same as in Eq. (24.3) for the single-time step problem. However, computations for the second term require an averaging of Ri(al, X21X1) based on the conditional probabilities pi(x21al, Xl) , followed by a second averaging based on the conditional probabilities pi(xllao, Xo).
9 It is straightforward to extend Eq. (24.4) to include more than two decision periods. For a time frame of length T with decision points at integer steps, the appropriate utility function is
E[R(Ao) ] = ~ p ( H i ) E i
t=0 xt
,i X .la xt i at x,.lXt ]
where A 0 is a sequence {a0, al, ..., aT_l} of time-specific decisions and the expectation on the right-hand side connotes the sequential averaging of utilities Ri(at, Xt+llX t) based on the transition probabilities pi(xllao, Xo) , pi(x21al, Xl), ..., pi(Xt+llat, Xt). Note that when T = 2, this equation describes decision points at t = 0 and t = 1 only, and therefore reduces to Eq. (24.4) for the twoperiod case. 9 For all but the most elementary problems, identification of an optimal sequence of decisions requires a rather complicated treatment via dynamic optimization. As will be seen below, optimal decision making for dynamic systems under uncertainty relies on adaptations of the Hamilton-Jacobi-Bellman algorithms of dynamic programming (see Section 23.3.2 and Appendix H).
24.3. U N C E R T A I N T Y A N D THE C O N T R O L OF DYNAMIC RESOURCES In what follows we deal with the uncertainty factors mentioned above in a context of dynamic systems analysis and optimal control theory. We account for four common sources of uncertainty facing managers. The most ubiquitous is e n v i r o n m e n t a l variation, which is largely uncontrollable, possibly unrecognized, and often has a dominating influence on animal populations through factors such as random variability in
climate, unrecognized landscape heterogeneity, and unpredictable human impacts on the environment. Partial observability connotes uncertainty about resource status, as measured by sampling variation in the monitoring of resources. Partial controllability expresses the difference between targeted conservation in decision making and the actual implementation of conservation actions, leading to possible misrepresentation of conservation efforts and thus to inadequate accounting of their influence on population dynamics. Finally, structural u n c e r t a i n t y concerns a lack of understanding (or lack of agreement) about the structure of biological relationships that drive population dynamics. Environmental variation, partial observability, partial controllability, and structural uncertainty all limit a manager's ability to make informed management decisions (Nichols et al., 1995a; Williams, 1997). Structural uncertainty is characterized here with multiple models of population dynamics over a discrete time frame, along with model-specific measures of uncertainty about which one is most appropriate. Without loss of generality as to optimal management, we combine environmental variation and partial controllability into a single stochastic factor z t affecting population dynamics. For notational simplicity, population status is characterized by the size x t of the population, recognizing that x t could as easily be a vector of population attributes. Management action at time t is designated by a t, and policies describing actions over the remainder of the time frame are designated by A t . As a matter of notational economy, we use the subscript t to denote time, as with z t, x t, and at, recognizing the same meaning for these terms as z(t), x(t), and u(t) in earlier chapters. With these notational conventions, consider a biological population that annually is subjected to management, with management actions that are based on population size xt and the projected effects on future population size. Models depicting population responses play prominently in the assessment of impacts. Several models of the form Xt+l = Xt -}- fi(xt, at, Zt)
are available, where a t and z t represent management controls and random variation, respectively. It is assumed that one does not know which model most appropriately represents population change in response to management. This uncertainty is captured in a set {pi(t)} of likelihoods that express one's confidence in the models at time t. The notation pi(t), which plays the same role as p ( H i) in the previous section, allows for evolving likelihood values in response to accumulating information about management controls
24.4. Optimal Control with a Single Model and population responses. By affecting population dynamics, management can influence the evolution of the likelihoods and thereby promote learning. Benefits and costs attend the implementation of management controls over time, and these can be captured in a utility function that may be model specific. For simplicity we describe utilities as functions of current system states and actions, recognizing that the utility function might also represent an average of utilities across outcome states [see Eq. (24.3)]. Thus, Ri(atlx t) is the utility for model i if the population is of size xt and action a t is taken. An overall value for utility that accounts for model uncertainty is the average
a(atlxt, Pt)
where the expectation is with respect to stochastic environmental variation and partial controllability over the time frame. Decomposing the sum into current and future utilities, we have
wi(atlx t) = E
ai(atlx t) 4-
= ai(atlxt) 4- ~
I T • E
m
= ai(atlxt). Each of the population models characterizes transitions of the population over time, as influenced by factors such as survivorship, recruitment, and migration, along with the controls affecting them. These factors always are subject to environmental variation and other stochastic factors, including randomness in the effects of controls. Thus, the population size xt+ 1 that is projected by model i inherits a probability distribution pi(Xt+llXt, a t) from environmental and other sources of variation. The challenge is to choose controls that maximize aggregate utility in the face of stochastic effects, while also accounting for uncertainties about the biological processes that drive population dynamics.
24.4. O P T I M A L C O N T R O L W I T H A SINGLE MODEL Let policy A t specify an action for every population size x~ at every time in the time frame {t, t+l, ..., T}. In the absence of structural uncertainty, the associated policy value Vi(Atlx t) is given by the accumulated utilities
ai(a~lx ~) L=t
~
L
Ri(a~lx~)
T=t+I
= ai(atlxt ) 4- ~
-R(atlxt, Pt) -- pi(t)ai(atlxt)
xt}
pi(xt+llXt, a t)
Xt+l
i
based on model-specific utilities ai(atlx t) and model likelihoods pi(t). If there is only a single model under consideration, or if the likelihood is assumed to be pi(t) = 1 for model i, the utility corresponding to action a t simplifies to
ai(a.~lx,) x t T=t+l
- ~_j pi(t)ai(atlxt),
wi(atlx t) = E
651
Xtl}
pi(xt+llXt, at)Wi(at+l Xt+l).
Xt+l
Several points are noteworthy: 9 The notation wi(atlx t) indicates that the accumulation of utilities begins at time t, the start of the time frame for A t . Among other things this means that larger values can be obtained by extending the time frame, simply because more actions can occur (and therefore more utility can accumulate) when more time is available. The notation also indicates that the model-specific values wi(atlx t) a r e conditional on the population size xt at time t. Thus, Wi(Atlx t) c a n (and usually does) vary for different population sizes. 9 The transition probabilities pi(Xt+llXt, a t) are assumed to be stationary, in that they change through time only as a result of controls: conditional on action a t , the stochastic pattern of population change is constant over time. Stationarity is assumed here as a matter of convenience, but it is not a theoretical requirement. Indeed, the transition probabilities can vary through time as much or as little as needed to represent population dynamics. However, at each time the transition probability structure is assumed to depend only on the current system state and action. 9 A value Wi(Atlx t) for the aggregate utility can be obtained for every possible policy A t over the time frame. By proper choice of A t , these values can be optimized. A backward iteration algorithm to determine the optimal policy is given by the Hamilton-JacobiBellman equation V,(xt)
-
maxf
at
Ri(atlxt )
(24.5)
L
Jr- ~ Xt+l
Pi(Xt+l Xt, at)W~(Xt+l)} J
(see Section 23.3). This formula describes a straightforward stochastic dynamic programming problem (Bell-
652
Chapter 24 Uncertainty, Learning, and Decision Analysis
man and Dreyfus, 1962; Dreyfus and Law, 1977), which can be solved by iterative application of Eq. (24.5). 9 An optimal solution consists of a policy A~(x t) identifying actions for all population sizes at all times in the time frame, along with a field of optimal values V*(xt) for all population sizes and times. To implement the optimal policy at time t, one need only know the population size at a particular time and then apply the control specified by the policy for that size at that time.
so that
W(Atlxt,Pt) = -R(atlxt,Pt)+ ~.~ ~.~ pi(t 4- 1)p(xt+llxt, at)Wi(at+llXt+l) i Xt+l (24.7) = a(atlxt, Pt) 4- ~.~ P(Xt+llXt,at)~ Pi( t 4- 1)wi(at+llXt+l) i
Xt+l
= -a(atlxt, Pt) 4- ~.~ P(Xt+l[xt,at)w(at+llXt+l,Pt+i). Xt+l
24.5. OPTIMAL C O N T R O L WITH MULTIPLE MODELS
Several points are noteworthy:
Now consider the control of a population for which several models are available to describe population dynamics, but the most appropriate model is not known with certainty, i.e., pi(t) ~ 1. Policy value again is given in terms of accumulated utilities, except in this case the utilities are averaged over all models, based on the model likelihoods:
W(Atlxt, P_.t) = E
pi(t)
Ri(a~lx ~)
xt}
= ~.j pi(t)vi(at xt). i This expression can be further decomposed into current and future utilities by
V(At]xt, Pt)
9 In the computing form [Eq. (24.7)] linking successive policy values, the likelihoods pi(t) are used in the average -a(atlxt, Pt) of model-specific utilities, and the likelihoods pi(t + 1) from Eq. (24.6) are used in the average w(at+lIXt+l, Pt+l)9 The transition probabilities p (x t+l Ix't, at) represent an average of the model-specific transition probabilities, based on the likelihoods pi(t). They evolve through time, as a result of the sequential updating of likelihoods by means of Bayes' Theorem. This evolution is a key feature that distinguishes single-model and multiple-model optimization problems. 9 A value W(Atlxt, Pt) for the average accumulated utility can be obtained for every possible policy A t over the time frame. By proper choice of A t , these values can be optimized, and a backward iteration algorithm identifying the optimal policy is given by
"W*(xt, Pt) = m a x f -R(atlxt, Pt) --
at ~
(24.8)
--
4- ~a P (Xt +l lXt ' at)W*(xt +1, Pt + 1) t" J
Xt+l
This again is a stochastic dynamic programming problem, though complicated somewhat by the characterization of system state by (x t, Pt). Transitions for Pt are given by Bayes' Theorem as in Eq. (24.6), and the transitions for x t are given in terms of the nonstationary transition probabilities ~ (x t +llXt, at). Nevertheless, the optimization problem still can be solved by iterative application of Eq. (24.8). 9 An optimal solution consists of a policy A*(xt, Pt) that identifies a specific action for every combination (xt, Pt) of population size x t and likelihood state Pt, along with a field of optimal values V*(x t, Pt) for all population sizes and model likelihoods at all times in the time frame. To implement the optimal policy, at each time one must (1) determine the population size, (2) update the likelihoods with Bayes' Theorem, and (3) apply the control specified by the optimal policy for the population size and set of updated likelihoods. m
= ~_,pi(t){Ri(atlxt)4- ~ pi(xt+llxt, at)Vi(At+llXt+l)} i Xt+l = 'a(at[xt, Pt) 4- ~ pi(t)~ pi(xt+l[Xt, at)Vi(At+l[Xt+l) i Xt+l -- 'a(atlxt, Pt) 4- ~ ~ pi(t)pi(xt+llXt, at)Wi(at+llXt+l). i Xt+l The term pi(t)pi(xt+ 1 xt, a t) in the latter expression can be replaced by pi(t + 1)p(xt+l[Xt, a t) via Bayes' Theorem
pi(t 4- 1) =
pi(t)pi(xt+llXt' at) ~-Ji pi(t)pi(xt+l]Xt' at) pi(t)pi(xt+llXt, at) p(Xt+llXt, at) '
(24.6)
24.6. Adaptive Optimization and Learning 9 When pi(t) = 1, the optimal policy and values for (Xt, Pt) are the policy A~(x t) and values V~(X t) identified for a single model i. This intuitive result follows from the fact that if p i ( t ) = 1,-p(Xt+llXt, a t) = pi(Xt+lIXt, at) throughout the remainder of the time frame, so that the computing algorithm, Eq. (24.8), reduces to algorithm (24.5). m
The multimodel optimization problem is an example of adaptive resource management (Holling, 1978; Walters and Hilborn, 1978; Waiters, 1986), and algorithm (24.8) is a particular case of adaptive optimization (Williams, 1996a). Adaptive resource management often is described in terms of "dual control" (Sage and White, 1977; Stengel, 1994) in which learning (i.e., the reduction of process uncertainty) is facilitated through the decision-making process pursuant to other management objectives (Waiters, 1986). Uncertainty here is expressed through different models of population dynamics, along with model likelihoods pi(t) that measure confidence in them. In the approach described above, the model likelihoods are included in an optimization procedure as state variables, and they factor directly into the determination of optimal policies and values. Adaptive optimization thus exemplifies "dual control," with optimal actions identified on the basis of projected changes in the likelihoods as well as accumulated utilities.
24.6. ADAPTIVE OPTIMIZATION A N D LEARNING "Learning" in the context of adaptive optimization is represented by the iterative updating of model likelihoods pi(t) with Bayes' Theorem (Williams et al., 1996). It is instructive to show how the updating process can help to identify the most appropriate model. Consider a population that is subjected annually to sport harvest, with harvest regulations based on population status in the spring of each year. Assume that two models are under consideration, but only one model appropriately characterizes population dynamics. Initially the appropriate model is not known, and both models are considered equally likely to be adequate. An optimal policy having been identified by iterative application of algorithm (24.8), management proceeds as follows: 9 For the first year: (i) Specify initial likelihood values. For example, assume that the model likelihoods are equal, i.e., pl(0) -1 p2(0) = ~. (ii) Determine the population size x 0 by monitoring during the spring.
653
(iii)
Apply the control a 0 that is prescribed for a population of size x 0 and equal likelihoods. 9 For each succeeding year: (iv) Determine the population size xt by monitoring during the spring. (v) Identify the applicable transition probabilities pl(xt]xt- 1, at- 1) and p2(xtIXt_l, at_ 1) for the pair (Xt_l, Xt) , given the prescribed control at_ 1 for population size xt_ 1 at time t - 1. (vi) Use pl(XtlXt_l, at_ 1) and pa(XtlXt_l, at_ 1) to update the likelihoods to pl(t) and p2(t) with Bayes' Theorem, based on pl(t - 1) and p2(t - 1) from the previous year. (vii) Apply the control that is prescribed for a population of size x t and likelihoods pl(t) and pa(t). 9 Increment the time index t by 1, and repeat steps (iv)-(vii) each year until the end of the time frame.
Example A hunted wildlife population can be described by two population models that include quite different assumptions about the influence of hunting. Because both models are consistent with the limited population data that are available, initial model likelihoods are set at pl(0) = p2(0) -- 0.5. Preharvest monitoring yields an initial population size of x o, and action a 0 from the optimal harvest policy is taken. Assessment of population dynamics under the two models indicates that model-specific transitions have distributions pl(XllXo, a 0) and p2(XIIXo, ao). These distributions differ primarily in their modes, with modal values of 5 and 10 for which the respective probabilities are pl(51Xo, a O) = p2(101x0, a 0) = 0.02 (Fig. 24.5). In addition, nonmodal probabilities are given by pl(x1 -- 10Ix0, a 0) = pa(51Xo, a 0) = 0.005. Preharvest monitoring in year 1 yields a population size of, say, x I = 5. Then we have pl(51Xo, a 0) = 0.02 and p2(51Xo, a 0) = 0.005, and applying these values in Bayes' Theorem produces p1(1) = 0.8 and p2(1) = 0.2. Thus, a dramatic increase is seen in the likelihood for model 1, and a corresponding decrease in likelihood is seen for model 2. These results follow automatically from observing a population size that is more consistent with model 1 than model 2. If the observed population size had been x I = 10, then Pl (10Ix0, a 0) = 0.005, p2(10Ix0, a 0) = 0.02, and application of Bayes' Theorem would have produced p1(1) = 0.2 and p2(1) = 0.8. In this instance there would have been a rapid increase in the likelihood for model 2 and a decrease for model 1, driven by the fact that a population of size 10 was more consistent with model 2 than
654
Chapter 24 Uncertainty, Learning, and Decision Analysis
0.02
0.005
5
10
F I G U R E 24.5 Two distributions of population size at time t = 1, given the population has size x 0 at time t = 0 and action a 0 is taken. The distributions are based on alternative models of population response to management, and differences between them represent structural uncertainty as to which model is more appropriate to characterize population dynamics.
model 1. Proceeding with the management process as outlined above, subsequent steps include (1) selecting the appropriate harvest action for x I = 5 and [p1(1), p2(1)] = (0.8, 0.2); (2) monitoring in year 2 to determine x2; (3) updating of the likelihoods for year 2 based on x2; (4) selecting the appropriate harvest action for year 2 based on x 2 and P2; (5) monitoring in year 3 to determine x2; and so on. It should be clear that rapid learning occurs in this example because of large differences between the distributions pI(XIlXO, ao) and p2(XllXo, a0), which makes it easy to discriminate between models. In general, variation in model-specific stochastic structures will be reflected in the Bayesian updates of model likelihoods, with increasing variation leading to accelerated learning. Conversely, learning is not possible if pi(xtlXt_l, at_ 1) = pj(xtlxt_ 1, at_l), i.e., if the available models all have the same transition structures.
EVPI is essentially a comparison of the average of maximum values (the term in brackets) against the maximum of an average of values [V*(xt, Pt)]. That V*(xt, Pt) is the maximum of an average of values can be seen by inspection of Eq. (24.8)"
24.7. EXPECTED VALUE OF PERFECT I N F O R M A T I O N
~,~ pi(t)W~(xt ) = ~_j Pi(t) / m a x wi(atlxt)}. i i ! At
V*(xt, P t)
D
The value of knowing the most appropriate model can be measured by the expected value of perfect information (EVPI) (Waiters, 1986), which compares the average of the optimal values V*(xt), assuming each model i is appropriate, against the optimal value V*(xt, Pt), which accounts for model uncertainty: EVPI
pi(t)[W~(xt) -
= ~
W*(xt, Pt)]
i
= [~ pi(t)V~(xt)] - W~'(xt, Pt). i
=maXat
{ -~(atlXt" pt) -}- ~ P(Xt+llxt'at)W*(Xt+l" Pt+l)} Xt+l
= max ~ pi(t) E
At
i
= max ~
At
i
-
Ri(a,lxi)lxt, P_t
p~(t)gi(Atlxt).
That the term in braces is an average of maximum values is seen by
EVPI is the difference between these quantities. Because the average of maximum values is never less than a maximum of the average of values, EVPI = ~ pi(t)V~(xt) - W*(xt, Pt) ~ O.
i
The nonnegative character of EVPI allows it to be interpreted as a measure of the "importance" of knowing the appropriate model. For example, a value of 0 for EVPI suggests that there is no value in learning which model is appropriate, because the optimal value under process uncertainty is as high as would be ex-
24.8. Partial Observability pected with full knowledge of the model-specific optimal values. It is possible, albeit unlikely, for EVPI to vanish, though only under some rather stringent uniformity conditions on the optimal policies and policy values.
24.7.1. Identical Values for All Models EVPI vanishes if a policy produces identical values for all models: Vi(Atlx t) -- V(Atlx t) for all i. Then ~.~ pt(i)V*(xt) = ~ , pt(i) i i
vi(atlx t)
max
At
~ , pt(i) max V(Atlx t)
=
i
At
= ~_~ pt(i)V*(Atlxt) i
= V*(xt) and "W*(xt, Pt) =
max
At
~ i
pi(t)Vi(Ailxt)
= max ~ pi(t)V(Atlxt) At i =
max
At
V(Atlx t)
= W~(xt ),
so that EVPI = 0. Thus, if available policies produce the same aggregate utility across models, there is no need to account for uncertainty in determining optimal policies. The same optimal policy and values are identified by means of adaptive optimization irrespective of the likelihoods.
24.7.2. Identical Optimal Policies for All Models EVPI also vanishes if the optimal policies are identical for all models: A*(x t) = A*(x t) for all i. Then we have
V*(xt, P_t) = max ~ pi(t)Vi(Atlxt) At
i
~.j pi(t) i
max
At
vi(at]x t)
= ~.~ pi(t)Vi[A~(xt)Jxt] i
= ~,~ pi(t)Vi[A*(xt)lxt] = ~
pi(t)V*(xt) 9
655
But
w[a*(xt)lxt,
Pt] -- ~_j pi(t)wi[a*(xt)Ixt , Pt] i = ~
pi(t)V*(xt),
so that "w(atlxt, Pt) takes its maximum value for a*(xt), and EVPI = 0. Thus, if the same optimal policy is identified for all models, that policy also is optimal under adaptive optimization, no matter what are the likelihood values. Based on these arguments, the vanishing of EVPI means that accounting for uncertainty is irrelevant in policy determination. Thus, if policy values are the same for all models, or if optimal policies are the same for all models, then the likelihoods pi(t) in no way influence the choice of an optimal policy. Under these circumstances, any set of likelihoods will lead to the same optimal policy. On the other hand, accounting for uncertainty will be important in policy determination if (1) policyspecific values differ among models and (2) optimal policies differ among models. The likelihoods pi(t) will factor into the choice of an optimal policy only if both conditions are met. In a management context, the conditions suggest that efforts to learn are justified only to the extent that improved understanding leads to different decisions and increasing utility. Otherwise, there is no advantage in knowing which model is most appropriate, and one can use any one of the available models to develop and implement an optimal policy.
24.8. PARTIAL OBSERVABILITY Thus far we have considered optimal population management in the context of completely observable systems. Thus, population size x t is assumed to be known with certainty at each point in time, and in combination with the likelihoods Pt, this information is used to determine optimal policies with algorithm (24.8). In reality, population size almost never is known with certainty, and instead must be estimated at each time with field data. An estimate 9~t of population size inherits a distribution from data collected in the field, conditional on the field sampling design and the actual population size x t. Let Yt represent field data collected at time t, and Yt represent the accumulation of data up to t. Each year's monitoring effort adds to the accumulation of data, by Wt+l = {Wt, Yt+l}. Assume that an estimate xt of population size can be obtained as a function Xt = xt(YtlYt-1 ) of the data accumulated up to time t. Because Yt is conditional on xt, the estimate Xt inherits conditional distributions fl(X.tJX t) and f2(xt]xt)
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Chapter 24 Uncertainty, Learning, and Decision Analysis
from Yr. Then the transition from ~'t -- xt(YtIYt-1) to 9~t+1 = Xt+l(Yt+l]Yt) is given in terms of the model-specific probabilities
pi(Xt+l])Ct, at) = pi(xt+l]Xt, at)
pi(Xt+lI~'t, at) = (24.9)
~ f2(xt]x't)Pi(Xt+IIXt, at)fl(Xt+lIXt+l), Xt+l xt and a solution of the optimization problem for model i is obtained by iterative application of the Hamilton-Jacobi-Bellman equation
from Eq. (24.9), and the computing algorithm (24.10) reduces to algorithm (24.8). The extension to adaptive optimization with multiple models is straightforward. The Hamilton-JacobiBellman Eq. (24.8) now becomes
V*(9~t, Pt) = max { ~ f2(xt]x.t)-a(at]xt, Pt)
V~(x t) = max I ~ f2(xt]xt)ai(at]xt ) at !-77t
at
(24.10)
xt
q- ~ P(9~t+l]9~t, at)W*(Xt+l, Pt+l)), ~t+l
-Jr- ~ pi(x't +1]xt, at) V'~(Sct+1) t Xt+l
Xt, and the conditional associations between x t and Xt become trivial: f2(xt]dct) = f2(xt]x t) - 1 and fl(X.tIX t) -fl(Xt[Xt) = 1. It follows that
9
(Williams, 1996a). The following points are noteworthy: 9 The key feature in incorporating partial observability is the statistical association between xt and xt, from which are derived the conditional distributions fl(XtIXt) and f2(xt]Xt). These distributions derive from the stochastic structure of the statistic Xt = xt(Yt]Wt-1 ), which is parameterized by x t. 9 The distribution fl(X.t]Xt) arises naturally from 9~t = xt(Yt[Yt_l), based on sampling variation in Yt. On the other hand, the derivation of f2(xt[x.t) can b e q u i t e difficult and is a subject of considerable theoretical interest. The Kalman filter and its variants for nonlinear systems offer one approach to transitions among state estimates (Williams and Nichols, 2001). 9 Considerably more computing is required to find optimal solutions for partially observable systems. For example, the number of transition probabilities that must be considered for the transition from xt to Xt+l increases from 1 with complete observability, to a product of the numbers of nonzero probabilities in the distributions f2(xt]Xt) and fl(x-t[xt) under partial observability. Each transition probability requires products and sums as in Eq. (24.9). 9 Because there are additional elements of uncertainty in the optimization problem, there is a greater degree of stochastic variation in the transitions from 9~t to Xt+l. Added stochasticity in turn induces a tendency to smooth and therefore lessen the variability of state-specific values identified in algorithm (24.10). 9 In the absence of variation in 9~t (i.e., the population is censused), the optimal policy and values in algorithm (24.10) are the policy A~(x t) and values V*(x t) identified by algorithm (24.8) for a completely observable system. This follows from the fact that Yt records the population size x t when the population is completely observed, so that xt(Yt]Yt_l) -- xt(xtlYt_l) = m
where W*(9~t, Pt) and VX(9~t+l, Pt+l) are defined as before and
p(9~t+1[9~t, at) = ~
~ f2(xt[xt)p(xt+l xt, at)fl(x.t+l[Xt+l ). Xt+l xt
To identify the transition probabilities pi(x. t +1IX.t, at), one must determine the transition probability -Fi(Xt+l]Xt, a t) for every state x t for which there is a nonzero probability f2(xt]xt). One also must calculate the average utility -R(at]xt, Pt) for all states with nonzero probabilities. These requirements result in further increases in computations, well beyond what is required for a solution with a single model.
24.9. GENERALIZATIONS OF ADAPTIVE O P T I M I Z A T I O N Consider an optimization problem in which the objective function under consideration is
w(at]xt, Pt) -(24.11)
x ' xt,
P_t},
where the temporal index -r in pi('r) is used to indicate that the model weights vary over time. Equation (24.11) is distinct from the objective function
w(atIxt, Pt) = E
pi(t)~ Ri(a~Ix~) Xt, Pt} "r=t
for adaptive optimization, the key difference being the nature of the likelihoods. In adaptive optimization, the likelihoods pi(t) used in the averaging of accumulated utilities are specified a priori, and likelihoods for later times are determined iteratively with Bayes' Theorem
24.9. Generalizations of Adaptive Optimization based on the transition probabilities pi(xt+l]Xt, a t) between population sizes x t and xt+ 1. On the other hand, likelihood updating for the general optimization problem is based on sampling that can be targeted to attributes other than population size (e.g., population vital rates such as survival a n d / o r reproduction). Objective function (24.11) is the more appropriate criterion for optimization, at least in theory, when model likelihoods are determined annually by model comparisons based on data other than population size. For example, banding data can be used in band-recovery and capture-recapture analyses (Chapters 16 and 17) to discriminate among models with comparative statistics such as the AIC (Akaike, 1973, 1974; Burnham and Anderson, 1998). In this scenario, the data Yt include band recoveries in year t, and Yt includes the record of band recoveries up to year t. The model likelihoods then become functions of data by pi(t) = pi(Yt]Yt_l) and are subject to annual updating. The average transition probabilities are determined as before from the model transitions and model likelihoods pi(t), according to
the updated
likelihoods entering into
W(At+l[Xt+l, Pt+l) are now determined directly from field data, rather than by Bayes' Theorem. As before, the objective function for the general optimization problem can be written in terms of current and future utilities, and Eq. (24.11) becomes
9 The conditional expectation in Eq. (24.12) is with respect to the conditional distribution of Pt+l given PtThis distribution is based on the fact that Pt and Pt+l both are functions of the field data Yt. However, depending on the sampling design and computing formulas for Pt, it may not be possible to describe the conditional-distribution in a way that is amenable to the computation of optimal policies. 9 Equation (24.12) generalizes Eq. (24.7) through the distribution of Pt. Let
F(Xt+l, Pt+l]Xt,
at,Pt) -- fl(Pt+l] Xt+l, Pt)p(xt+l[Xt, at)
be the joint distribution of xt+ 1 and Pt+l, given x t, a t, and Pt, with Pt absorbed in p(xt+l]Xt, at) as in Eq. (24.6). For t-he general optimization problem, model likelihoods are identified directly from field data that induce stochastic variation in pi(t + 1). This variation can be represented by FI(Pt+IIPt) = Fl(Pt+l]Xt+l, Pt),
In contrast, adaptive optimization utilizes values pi(t+l) that are derived from xt+ 1 and pi(t), and no
other source of variation need be accounted for in updating the likelihoods. In this c a s e , Fl(Pt+llXt+l, Pt) is trivial, and
F(Xt+l, Pt+llXt, Pt) -" P(Xt+llXt, Pt).
v(atlxt, Pt)
Transitions from time t to time t + 1 therefore can be characterized in the objective function by the simpler expression p(Xt+llXt, Pt), and Eq. (24.12) reduces to Eq. (24.7). A comparison of
=E{~~.i~pi(.r)ii(Glx~)lxt, =E
The following points are noteworthy:
to emphasize its independence from the variation in Xt+l, and is accounted for in the objective function with the conditional expectation.
p(Xt+llXt, at) = ~_~ pi(t)pi(Xt+llXt, at). i However,
657
pi(t)ai(atlxt ) q- ~
~ Pi(T)ai(a,[Xr)]Xt, Pt
9=t+l i
{?
= a(atlxt, Pt) q- Ept+,lpt
(24.12)
~ pi(t)pi(xt+llxt,at ) Xt+l
W(Atlxt, Pt ) -- -R(atlxt, Pt ) q- ~ P(Xt+llXt, at)W(at+llXt+l, Pt+l) Xt+l
for adaptive optimization and
v(atlxt, Pt) -- -a(atlxt, Pt) q- Ept+llpt m
X E
"r=t+l i
Pi('r)Ri(a~x~)lxt+l, Pt+l
X {~
p(Xt+llXt, at)W(At+llXt+ 1, Pt+l)}
Xt+l
= -a(atlxt,Pt ) Jr- Ept+ljpt I~_a p(xt+llxt,at)
L
Xt+l
for the generalized optimization problem reveals that both functions include the average utility -R(atlx t, Pt) and both utilize an average transition probability ]9(Xt+llxt, at) based on the likelihoods in Pt" However, additional variation arising in pi(t + 1) from field sampling is accounted for by the conditional expectation D
X w(at+l]Xt+l, Pt+l)}.
658
Chapter 24 Uncertainty, Learning, and Decision Analysis
in the general problem. In the absence of field data, the conditional distribution of Pt+l given Pt becomes trivial, so that the general problem reduces to the adaptive optimization problem.
with
9 An optimal policy for the general optimization problem is obtained by iterative application of the Hamilton-Jacobi-Bellman equation
as before. Under these conditions Eq. (24.12) becomes
Wn(xt, Pt) --
maxI-R(at[xt, Pt) -}- Ept+llpt at L
~_, fl(xtIxt)p(xt+l[Xt, at)f2(xt+llXt+l)
Xt+l Xt
v(at[:rt, Pt) = R(atlxt, Pt) q- Ept+llpt u _ X {x~t+l
p(xt+llxt, at)W(At+llXt+l, Pt+l)},
(24.13)
X [~.j p(xt+llxt, at)V*(Xt+l, Pt+l)] }. Xt+l
This again describes a stochastic dynamic programming problem, with the likelihoods incorporated into the algorithm as state variables. As with adaptive optimization, the transition probabilities p(Xt+llXt, a t) vary over time as the probabilities in Pt change. 9 Optimization with objective function (24.13) generalizes the adaptive optimization problem, to allow for sample-based likelihoods. This follows directly from the generalization of objective function (24.7) to objective function (24.11), under a sampling regime that induces a conditional distribution for Pt. In the absence of random variation in Pt, the optimal policy for the general problem becomes the optimal adaptive policy. 9 Computing requirements for the general optimization problem are likely to be much greater than for adaptive optimization, primarily because of the very large number of possible transitions from Pt to Pt+l. On the other hand, the accumulation of data Yt can lead to more rapid learning than with the adaptive process. m
24.10. A C C O U N T I N G FOR ALL SOURCES OF UNCERTAINTY
The general optimization problem described above can be adapted easily to a situation in which the population size is not known with certainty and instead must be estimated. Assume that likelihoods Pt and the estimate Xt are based on field data Yt each year. Assume also that population size and the likelihoods are estimated independently, through statistically independent sampling efforts such as, e.g., population surveys and banding programs. Then
F(~:t+l, Pt+ll~'t,
p(Xt+llXt, at) = ~
at, Pt) = Fl(Pt+llX.t+l, Pt)p(xt+llX.t, at) = Fl(Pt+llPt)P(Xt+llXt, at)
and an optimal policy is identified by iterative application of the Hamilton-Jacobi-Bellman equation
W*(xt,Pt) = maXat { "-~(atl~t' Pt) q-" Ept+llpt
(24.14)
With this formulation of the optimization problem we have incorporated all identifiable sources of uncertainty. Thus, environmental variation and partial controllability are represented in the transition probabilities pi(Xt+llXt, at)', structural uncertainty is represented by multiple models and their likelihoods, via model-specific probabilities pi(Xt+llXt, a t) and the average transition probabilities p(Xt+l[Xt, at); partial observability is represented by the estimation of x t by xt; and data-based updating of the likelihoods is represented by the conditional expectation in algorithm (24.14). The aggregation of so many sources of variation in a single problem makes the finding of a solution extremely difficult for any but the simplest problems. For any useful optimization problem to be amenable to solution, simplification of some sort is necessary.
24.11. "PASSIVE" ADAPTIVE OPTIMIZATION Of the identified sources of uncertainty, the greatest computational demands attend structural uncertainty and the requisite tracking of changes in model likelihoods. "Passive" adaptive optimization offers one way to account for structural uncertainty, while also eliminating the need to carry model likelihoods as state variables in the optimization algorithm (Williams and Johnson, 1995; Williams, 1997; Johnson and Williams, 1999). Consider a situation as described above for adaptive optimization, with multiple models and timevarying likelihoods pi(t). To simplify notation, assume that the population is completely observable. As be-
24.11. "Passive" Adaptive Optimization fore, model-specific utilities Ri(atlxt, Pt) are aggregated into an average utility by
659
The following points are noteworthy:
D
-a(atlxt, Pt) :
~.~ pi(t)Ri(atlxt), i
(24.15)
and model-specific transition probabilities pi(xt +llXt, at) are aggregated into average probabilities by
F(Xt+IlX t, a t) : ~ pi(t)pi(Xt+llXt, at). i
(24.16)
These averages can be used in algorithm (24.5) as if they represent the utilities and transition probabilities for a single model:
V*(xt):max{-a(atxt)+~~_,pi(t)pi(xt+llxt,at)V*(Xt+l) at
Xt+l
}
i
(24.17)
:max{-R(atxt)+~,-p(xt+lxt,at)V*(Xt+l) }. at
V*(xt, Pt) = m a x
at
(atlxtPt) (24.18)
where V*(x t) is simply the accumulated utility for model i when the optimal policy a*(x t) is used. Of course, Eq. (24.18) can be rewritten by Bayes' Theorem as
W~(xt, Pt)
(24.19)
{-a(atlxt, Pt)+ ~ -fi(Xt+llxt, at)~ pi(t+l)V~(Xt+l)} Xt+l
Pt+l) -
Xt+l
A comparison of Eqs. (24.17) and (24.18) shows that they generally have the same form, but the values V*(x t) in Eq. (24.18) are model specific, whereas in Eq. (24.17) they are not. It is the model specificity of values that requires one to track not only the population x t through time, but also the likelihoods pt(t). On condition that Vi(x t) = V(xt) , i.e., the values for all models are identical, Eq. (24.18) reduces to Eq. (24.17). In this sense, passive adaptive optimization can be seen as a special case of active adaptive optimization.
~
i
pi(t+l)Wi(at+llXt+I )
are optimized as in Eq. (24.19). Variation in the values Vi(At +l lXt + 1) leads to the inequality of V(A t +l lXt + 1) and W(At+llXt+l, P t + l ) , with a result that V*(X t) in Eq. (24.18) differs from V*(xt, Pt) in Eq. (24.13). 9 Because passive adaptive optimization can be seen as a special case of active adaptive optimization constrained by Vi(x t) = V(xt), the optimal values associated with active adaptive optimization are never less than those produced by passive adaptive optimization. This intuitive result essentially follows from the fact that values produced by unconstrained maximization are at least as large as those produced by maximization under constraints (see Chapter 22). 9 The optimal policy identified in passive adaptive optimization does not coincide with the optimal policy for active adaptive optimization. This follows from the inequality of V(A t +l lXt+ 1) and V(A t +l lXt +1, Pt + 1), which in turn means that
w(atlxt) - -a(atlxt) -Jr- ~ -P(Xt+llXt, at)W(at+llXt+l)
i
-max{-a(atxt, Pt)+ ~_,-fi(Xt+lxt,at)W*(Xt+l,Pt+l)}. ai
are optimized in Eq. (24.17), whereas
w(at+llXt+l,
-t- ~_j ~_, pi(t)pi(xt+llXt, at)V~(Xt+l)}, i Xt+l
at
w(at+llXt+l) - ~.~ pi(t+l)W(at+llXt+l) i
Xt+l
The key difference between this approach, termed passive adaptive optimization in what follows, and that of active adaptive optimization as described above, is in the treatment of optimal values in the optimization algorithms. In active adaptive optimization we have
= max
9 Like active adaptive optimization, the passive adaptive approach accounts for multiple models and model-specific likelihoods, and thus is appropriately described as "adaptive." However, the passive approach does not account for the evolution of likelihoods in determining an optimal policy, whereas active adaptive optimization does. 9 The values identified in passive adaptive optimization do not coincide with the values identified in active adapti~,e optimization. This can be seen by recognizing that the values
Xt+l
differs from
w(atlxt, Pt) -~ -R(atlxt, Pt) -}- ~.~ -p(Xt+llXt, at)w(at+llXt+l,
Pt+l).
Xt+l
Thus, the policies maximizing V(Atlx t) and W(Atlx t, Pt) are potentially different. 9 Because differences in the policies under passive and active adaptive optimization are a function of differences in policy values Wi(Atlx t) among models, quite distinct policies can result from large difference in these values. On the other hand, policies for the two ap-
660
Chapter 24 Uncertainty, Learning, and Decision Analysis
proaches converge as model-specific differences in policy values vanish. Variation in policy values among models is tied to differences in model-specific transition probabilities. Thus, active and passive adaptive optimization algorithms recognize different optimal policies to the extent that there are substantial differences in population projections among models. 9 As with active adaptive optimization, learning also occurs with a passive adaptive approach, and Bayes' Theorem can be used to recognize it. Likelihood updates play an important role in implementation of passive optimization, through a sequence over the time frame of policy identification followed by the updating of likelihoods. Management in the context of passive adaptive optimization proceeds as follows: 9 For the first year: (i) Specify an initial set of likelihood values pi(O). (ii) Determine the population size x 0. (iii) Develop average utilities as in Eq. (24.15), and average transition probabilities as in Eq. (24.16), based on the likelihoods P0. (iv) Use the average utilities and transition probabilities to identify an optimal policy with algorithm (24.17). (v) Apply the control a 0 that is prescribed for a population of size x 0. 9 For each succeeding year: (vi) Determine the population size x t. (vii) Identify the applicable transition probabilities pi(xtlXt_l, a t - l ) for the pair (xt_ 1, xt), given the prescribed control at_ 1 for population size x t _ 1 at time t - 1. (viii) Use the transition probabilities pi(xtlXt_l, at_ 1) and the observed population size x t to update the likelihoods to pi(t) with Bayes' Theorem, based on p i ( t - 1 ) from the previous year. (ix) Develop average utilities as in Eq. (24.15), and average transition probabilities as in Eq. (24.16), based on the likelihoods pi(t). (x) Use the average utilities and transition probabilities to identify an optimal policy with algorithm (24.17). (xi) Apply the control a t that is prescribed for a population of size x t. 9 Increment the time index t by 1, and repeat steps (vi)-(xi) each year until the end of the time frame. Because the likelihoods Pt+l are determined only after the optimal policy is identified for time t, they
have no influence on the identification of that policy or on the action a t that is taken based on the policy. This compares with active adaptive optimization, in which optimal policies are based explicitly on projected changes in the likelihoods. Note that a passive approach requires the identification of a new policy at each point in time, based on the updated likelihoods. This is in contrast to active adaptive optimization, in which a single optimal policy is identified at the start of the time frame and is used thereafter to determine optimal actions at each time. The repeated updating of likelihoods in the passive adaptive approach allows one eventually to identify the most appropriate model under consideration. If the models differ substantially in their population projections, these differences will be reflected in the transition probabilities pi(Xt+llXt, a t) and thus in the model likelihoods as they are updated with Bayes' Theorem each year. Model likelihoods will increase for those models with predictions that are consistent with population size x t+l, and likelihoods will decrease for those models with predictions that are inconsistent with xt+ 1. In this way the most appropriate model can be identified as time progresses, i.e., learning is advanced over the time frame. Though learning can occur with both passive and active adaptive approaches, a key difference between them concerns the role of policy to promote learning. In passive adaptive optimization, learning is essentially an unplanned byproduct of implementation, with the likelihoods updated independently of the identification of optimal policies. In active adaptive optimization, evolution of the likelihoods is acknowledged in the objective function, and along with other management goals, it influences the identification of optimal policies. For this reason, active adaptive optimization identifies policies that are more responsive to learning and yield larger aggregate utilities over the time frame. However, the relative performance of an active vs. passive approach is yet to be determined, and there remains a need for comparative assessment of optimal values, policies, and computing costs.
24.12. D I S C U S S I O N Each of the uncertainty components discussed above adds complexity to wildlife conservation and makes the identification of optimal policies more difficult. Thus, the inclusion of environmental variation and partial controllability introduces random factors into population dynamics, complicating the stochastic transition structures both within and among models. The inclusion of partial observability requires conditional distributions linking actual and estimated popu-
24.12. Discussion
661
X!
~
r
E~pi(t) Vi(A,lx,)
Ti
Pi(X,+l l x~ a,*)
m
pi(t)
X l (I)
~
E'iPi(t+l)Vi(A,+lx,+l)
o~176
IPi(
~-J
t
+
(11
x; (2)
......
Y, iPi(t)Vi(A;"
x,
p,4x,+ 1 x, , a,
)
BT
~iPi(t+l) - i,..,+t l x,, I)
"'"
F I G U R E 24.6 A d a p t i v e decision making for (a) a single resource system and (b) two similar but spatially separated systems. (a) System state x t and model likelihoods pi(t) are used with modelspecific transition probablilites pi(xt + 1]Xt, a t) to determine an optimal decision a~ via Eq. (24.8). Decision a~ in turn produces return R~ = R(a'[ x t, Pt) and induces the transition probablities pi(xt+ 1]Xt, al). Based on the observed system state x t +1 at time t + 1, pi(xt + 1 Xt, a'~) and pi(t) are combined in Bayes' Theorem (BT) to determine n e w model likelihoods pi(t + 1), which in turn are used to produce a n e w optimal decision a~+l and return R~+I. (b) System state x~1) in stratum 1 and model likelihoods pi(t) are used with model-specific transition probabilities Pi (x(1)t+l ~'I1), a t) to determine an optimal decision al 1)*. Decision a~1)* in turn produces return RI 1)* = ~c~(1), ~-,~t ~.(1 ~t ),Pt) and induces the transition probablities ,.(1) at time t + 1, F?,'~t+l ~,,r XI1) , all),) and pi(t) pt(x t+l (1) ~~'I 1) , a~l)*). Based on the observed system state -~t+l are combined in Bayes' Theorem to produce new model likelihoods pi(t + 1), which in turn are used ,,(2)* and return with the observed system state -~t+1~'(2)in stratum 2 to provide an optimal decision ~t+l R (2), t+l"
lation sizes, which are needed to calculate average transition probabilities. The incorporation of structural uncertainty with multiple models requires an expansion of the state space to include model likelihoods as state variables. Finally, the use of sample-based likelihoods in determining optimal policies requires additional distributions to account for the transition of likelihoods over time. It is not difficult to see that a comprehensive treatment of uncertainty can render the optimal management of wildlife populations effectively impossible. A conventional approach to these problems has been to avoid altogether the treatment of uncertainty, or to treat only environmental variation or partial observability (though rarely both) in isolation from other uncertainty factors (Hilborn et al., 1995; Nichols et
al., 1995a). Fortunately, developments in adaptive control offer the prospect of an analytic treatment of structural uncertainties, although the simultaneous accounting of structural uncertainty and partial observability continues to present formidable challenges (Waiters, 1986; Williams, 1996a). Given the added dimensionality requirements of adaptive optimization, one quickly faces prohibitive computing demands when both factors are included in a problem. As mentioned above, passive adaptive management represents a compromise in adaptive optimization that retains an explicit accounting of uncertainty, yet avoids a requirement to include model likelihoods as state variables. In addition to the passive adaptive approach, a number of other suboptimal approaches to decision making can facilitate learning. For example, "probing"
662
Chapter 24 Uncertainty, Learning, and Decision Analysis
control strategies can be used for learning, whereby system response to management is probed with controls at the extremes of their allowable ranges. Alternatively, projected population responses for a particular model can be used to identify optimal controls, with likelihoods for multiple models updated subsequently through annual population monitoring. Any number of criteria can be used to select controls for political and other purposes in the absence of any model-based assessment, with follow-up monitoring of population responses to facilitate learning about biological processes. Of course, a requirement for learning with any approach is the availability of adequate monitoring, by means of which population responses can be compared to predicted responses. This chapter has emphasized natural resource problems involving sequential decision making that lend themselves naturally to adaptive management. For these problems, decision making is explicitly dynamic, with potentially rapid feedback of information gained from monitoring. However, many important natural resource decisions are made at a single, initial time, and new decisions are made only after long intervals of time, if at all. Examples of such problems include the installation (or removal) of dams on rivers, the establishment of reserves for biodiversity management, and the reintroduction of an endangered species to a natural area. It is not immediately obvious how such problems lend themselves to dynamic decision making, let alone adaptive management. However, we note that in each of these examples, decisions similar to the one currently under consideration often must be made in similar systems at future times. In most cases, monitoring can (and should) be conducted
prior to and following the decisions made in each system at each time. There often is substantial uncertainty about system processes, and plausible alternatives can be proposed for system responses to alternative decisions. Under these circumstances, information gained from actions taken at one system at a particular time t 1, including prediction and monitoring over (t 1, t2), presumably can be useful in reducing uncertainty about another system over (t2, t3), provided the systems share common dynamics. Figure 24.6 illustrates a conceptual extension of the single-system adaptive optimization problem (Fig. 24.6a) to a twosystem problem with common, underlying dynamics (Fig. 24.6b). This approach could potentially be generalized to very large, multiunit systems and is the basic idea behind ongoing work on spatio-dynamic optimization of forest reserves for wildlife conservation (Moore et al., 2000). As to a formal treatment of uncertainty, several computing approaches are available for identification of optimal policies (Williams, 1996b). A particularly useful computer program for stochastic dynamic programming (SDP) is reported in Lubow (1995). SDP has been used to good effect for wildlife problems incorporating environmental variation and partial controllability (Johnson et al., 1997), and the software has been extended to include partial observability of populations and to accommodate multiple models for adaptive optimization (Lubow, 1997). Enhanced programming capabilities of this sort, along with rapid advances in computer hardware and improvements in analytic capabilities, hold great promise for overcoming technical obstacles that have prevented more comprehensive treatments of uncertainty in the past.
C H A P T E R
25 Case Study: Management of the Sport Harvest of North American Waterfowl
25.1. BACKGROUND AND HISTORY 25.1.1. Early Development of Harvest Regulations 25.1.2. Toward a Modern Process 25.1.3. Recent Adaptations 25.2. COMPONENTS OF A REGULATORY PROCESS 25.2.1. Setting Harvest Regulations 25.2.2. Biological Monitoring 25.2.3. Predicting the Effects of Regulations 25.3. ADAPTIVE HARVEST MANAGEMENT 25.4. MODELING POPULATION DYNAMICS 25.4.1. Structural Uncertainty 25.4.2. Environmental Variation 25.4.3. Partial Management Control 25.5. HARVEST OBJECTIVES 25.6. REGULATORY ALTERNATIVES 25.6.1. Predicting Harvest Rates 25.7. IDENTIFYING OPTIMAL REGULATIONS 25.7.1. An Algorithm for Adaptive Harvest Management 25.7.2. Optimal Regulatory Prescriptions 25.8. SOME ONGOING ISSUES IN WATERFOWL HARVEST MANAGEMENT 25.8.1. Setting Management Goals 25.8.2. Monitoring and Assessment 25.8.3. The "Scaling" of Harvest Management 25.9. DISCUSSION
these conditions, a harvested population typically experiences increased reproductive output or decreased natural mortality, as more resources (e.g., food, nesting cover, refugia from predators) are available per individual. In the absence of random environmental fluctuations, population size eventually settles around a new equilibrium, and the harvest, if not too heavy, is thought to be sustainable without destroying the breeding stock. Resource managers often attempt to maximize the sustainable harvest by driving population density to a level that maximizes the instantaneous rate of population growth (Section 11.1) (see also Beddington and May, 1977). If the theoretical basis for harvesting renewable resources is fairly straightforward, the practice of harvest management is not. The record of harvest management is replete with cases in which randomly fluctuating environmental conditions, uncontrolled variation in harvests, naive assumptions about population response to harvest, and management policies with short time horizons have led to resource collapse (Ludwig et al., 1993). Sustainable harvest management has proved to be a complex and difficult challenge, requiring at a minimum a biological understanding of the harvested system, an accounting of the key sources of system variablility, and an articulation of management objectives that are consistent with the renewal capacity of the resource. Even with a firm commitment to longterm resource conservation, harvest managers must deal with the inherent complexity of dynamic biological systems, under conditions in which management controls are indirect and often severely constrained. In this chapter, we show how the principles of modeling, estimation, and optimal decision analysis currently are being used to regulate the sport harvest of
Much of the theory underlying the harvest of biotic resources is predicated on an assumption of densitydependent population growth (Hilborn et al., 1995), whereby population dynamics are held to be influenced by intraspecific competition for resources. In a relatively stable environment, unharvested populations exhibiting density dependence tend to fix on an equilibrium in which births balance deaths. Under 663
664
Chapter 25 Case Study
waterfowl in the United States. The regulatory process for waterfowl harvest represents a collaboration among federal and state governments, along with nongovernmental organizations and the public at large. It is one of the most visible examples of science-based natural resource management in the United States, with broad-scale biological and socioeconomic impacts. Each year, roughly 13 million waterfowl, principally mallards (Anas platyrhynchos), teal (Anas crecca; Anas discors), wood ducks (Aix sponsa), and Canada geese (Branta canadensis), are harvested by about 1.5 million sport hunters (U.S. Department of the Interior, 1988). Sport harvests can consist of as much as 25% of the postbreeding population size (Anderson, 1975a). The economic impacts of hunting on the economy are substantial; each year, waterfowl hunters in the United States spend over $500 million, and the total economic output is estimated at $1.6 billion annually (Teisl and Southwick, 1995). The authority of the United States government for establishing waterfowl hunting regulations is derived from treaties for the protection of migratory birds signed with Great Britain (for Canada in 1916), Mexico (1936), Japan (1972), and Russia (1978) (U.S. Department of the Interior, 1975). These treaties prohibit all take of migratory birds from March 10 to September 1 each year and allow for hunting seasons not to exceed 32 months in duration. Each year, the U.S. Fish and Wildlife Service solicits proposals for hunting seasons from interested parties, and after extensive public deliberations, establishes guidelines within which individual states select their hunting seasons (U.S. Department of the Interior, 1988; Blohm, 1989). Hunting regulations typically specify season dates, daily bag limits, shooting hours, and legal methods of take (e.g., see Martin and Carney, 1977; Rogers et al., 1979).
25.1. BACKGROUND AND HISTORY As with many natural resources, the development of a process for waterfowl harvest regulations has followed a trend in the amount and quality of information used in resource management. This development started early on before initiation of large-scale waterfowl monitoring programs and led naturally to the coupling of regulations to biological information and understanding.
25.1.1. Early Development of Harvest Regulations Enactment of the Migratory Bird Treaty Act marked the beginning of the modern era of waterfowl harvest
management in North America, and the regulatory framework in current use has evolved over the years out of initial efforts, subsequent to passage of the Act, to establish mechanisms for regulating harvest [see historical reviews in Anderson and Henny (1972), Nichols et al. (1995a), and Nichols (2000)]. During the early development of these mechanisms, few if any of the currently available monitoring and management tools were in place, and formal procedures for review, assessment, and public input were yet to be developed. In the absence of reliable data about populations and regulatory impacts on populations, regulations were set with a "seat of the pants" approach, with little or no monitoring of populations except at a local level. Thus, the setting of regulations was informed by only anecdotal information about harvest levels, population numbers, and the impacts of harvest (Fig. 25.1). The principal aims in harvest regulation during this time revolved around the maintenance of hunting opportunity. In the absence (or near-absence) of reliable information about regulatory consequences, regulations were based primarily on tradition and on the degree of hunter satisfaction. Under these conditions it was not possible to adjust regulations based on population status, and the prospects for accidental overharvest, with subsequent population declines, were high. Nor was it possible to use regulations "adaptively," to improve understanding of population dynamics and the impacts of harvest on populations. Through time, monitoring programs tracking population status and trends were initiated, including the Waterfowl Breeding Ground Survey, Waterfowl Production Survey, Waterfowl Harvest Survey, and Midwinter Waterfowl Survey (Anderson and Henny, 1972; Martin et al., 1979; Smith et al., 1989; Nichols, 1991a). As these programs were implemented, it became possible to monitor the status of waterfowl populations and to use the resulting information in the setting of regulations. Data on the status of waterfowl soon came to be incorporated into the annual regulatory process, which then allowed regulations to be updated based on population status. In essence, the anecdotal information of the early period was replaced by survey and monitoring data, which could be used to inform the regulations process (Fig. 25.2). In this scenario, regulations in a given year t influenced the level of harvest and, through harvest, the population status in year t + 1. Information about harvest and population status for year t + 1, collected by means of the monitoring program, subsequently could be utilized in the setting of regulations in the next regulations cycle. During this period, tradition continued to play an important but diminished role, at the same time that the role of information about population status was
25.1. Background and History
regs t
~
popt+l
regst+l N
--,
poPt+2
2'
regst+2 ",,
7
anecdotal inf~
665
anecdotal
1
inf~
2
FIGURE 25.1 The regulations cycle for sport hunting of migratory birds, during the early period following passage of the Migratory Bird Treaty Act, prior to implementation of large-scale monitoring programs. The process was informed primarily by anecdotal information.
ulation status was assumed to be directly proportional to harvest, which in turn was assumed to be directly related to harvest regulations). Over time, these models increased in complexity and realism, as characterizations of reproduction, survival, recruitment, and other components of the life cycle of waterfowl were refined. This in turn led to additional refinement in models of the relationships among population status, harvest levels, and harvest regulations. The goal was (and is) to represent the responses of a population to harvest regulations, based on long-term monitoring and re, search programs. By building on the information bases they were designed to represent, these models added yet another information component to the regulations process. Thus, monitoring data on the status and trends of populations now could be used to update population models, which in turn could be used to guide the regulatory process in the next cycle (Fig. 25.3). The models could be updated each year with new monitoring data, so that both the models and the information bases they represent are constantly evolving. In this expanded scenario, regulations have direct as well as indirect effects, and both are key to the effective regulation of waterfowl populations. First, regulations directly affect a regulated population by influencing the amount of harvest, and through harvest, the subsequent population status. This is indicated in Fig. 25.3 by arrows connecting regulations and population status. Second, regulations indirectly affect populations by influencing the information base that
growing in importance. Regulatory issues continued to focus on the maintenance of hunting opportunity, with an additional emphasis on the conservation of viable populations. The availability of population data allowed for more informed regulations pursuant to these goals. If, for example, regulations in year t were followed by dramatic declines in populations the next year, then regulatory decisions in year t + 1 could take these declines into account. This regulatory "feedback," by which the population status following regulatory actions could be used to adjust subsequent regulations, represented a great improvement in the regulatory process. However, the absence of a longterm data base meant that the process utilized only the most recent data about population status. There remained a need to acquire population and harvest data over an extended period of time and to incorporate the biological understanding implicit in these data into the regulatory process. 25.1.2. T o w a r d a M o d e r n P r o c e s s
The extensive base of information accumulated through monitoring and research led eventually to the development of population models characterizing population dynamics in terms of population size and distribution, as influenced by harvest regulations (reviewed by Williams and Nichols, 1990). Early renditions of these models represented the influence of regulations by means of simple relationships (e.g., pop-
regs t
-.
poPt+l
regst+l N
2'
d a t a t +1
-.
poPt+2
regst+2 N
--.
2'
d a t a t+2
FIGURE 25.2 The regulations process for sport hunting of migratory birds, after implementation of monitoring programs but prior to the developmentof an extensiveinformation base. Annual monitoring data replaced anecdotal information as the basis for informed regulations.
666
Chapter 25 Case Study
regs t -~
poPt+l
regst+l
N
&tat+ 1 -" m~
,;'
--,
regst,2
poPt+2 N
1
datat +2
7
-"
m~
FIGURE 25.3 The current regulations process for sport hunting of migratory birds. Longterm monitoringdata and research results are incorporated into models that aid in the annual promulgation of regulations.
is captured in the population model and subsequently used in the regulatory cycle. This is shown in Fig. 25.3 by arrows leading from the population to the data base, from the data base to the model, and from the model back to the regulatory process. It seems intuitive that informative regulations (in the sense of improving the models used in the regulatory process) are in some sense better than uninformative regulations. The regulatory scenario described in Fig. 25.3 recognizes the importance of information in management, via the annual updating of waterfowl data bases, the incorporation of these data into improved population models, and the use of this information for the setting of annual harvest regulations. A typical application would involve the use of population models to explore the impacts of a number of different regulations, with the idea of identifying regulations that maximize harvest (or harvest opportunity) while limiting the negative impacts on population status. The regulations thus identified can be incorporated into the decisionmaking process, thereby ensuring that harvest regulations account for current population status and potential impacts on future population status. Some regulatory strategies are likely to be more informative than others, in the sense that they lead to more informative data bases and improved models for describing the consequences of regulations. If one actively seeks through regulation to improve the information by which regulatory options are evaluated, the regulatory process can be described as actively adaptive (see Chapter 24). If, on the other hand, improved information is simply an unplanned byproduct of harvest regulations, the process is passively adaptive. With the single exception of the period from 1980 to 1984, when waterfowl harvest regulations were stabilized (Brace et al., 1987; Sparrowe and Patterson, 1987), waterfowl harvest management in recent years has been (and continues to be) of the passively adaptive variety.
25.1.3. Recent Adaptations By the late 1970s, harvest management was widely perceived to be approaching the limits of its capacity
for improvement, in large part because of structural uncertainties attendant to the management of populations for stability (i.e., the setting of regulations each year to control--and hopefully eliminate--population fluctuations). The logic for such a strategy is that managing to maintain steady-state conditions can avoid the twin failures of sacrificed hunting opportunity (overly restrictive regulations when birds are abundant) and overexploitation (liberal regulations when birds are scarce). Indeed, a steady-state regulatory strategy can be appropriate for a population, if its population dynamics, as influenced by harvest regulation, population status, and environmental conditions, are completely understood. In the more common circumstance in which our understanding of population dynamics is less than complete, regulating for steady-state conditions can sacrifice the information needed for sound management, in order to protect against over- and underharvest. One effect of such a strategy is to "chase populations with regulations," i.e., to set liberal regulations whenever populations appear to be abundant and restrictive regulations whenever populations are low. Another is to "manage on the margin," i.e., to effect small changes in a large number of regulatory controls (season lengths, bag limits, opening/closing dates, split seasons, zones, early/late seasons, etc.), with the idea of targeting specific cohorts of birds with specific harvest pressures. In both cases the consequence is to confound harvest regulations and environmental conditions and thereby lose one's ability to assess the impacts of regulations on population dynamics. Of course, that is simply another way of saying that information and understanding are sacrificed by the regulatory strategy. Recognition of this problem led federal and provincial agencies in Canada to initiate a program of stabilized hunting regulations in 1979, which the United States joined in 1980 (Brace et al., 1987; U.S. Department of the Interior, 1988). Season lengths and bag limits were stabilized at 1979 levels for hunting seasons through the 1984-1985 season. At the same time a number of large-scale waterfowl research studies were initi-
25.2. Components of a Regulatory Process ated on both the wintering and breeding grounds, in order to learn about the influence of environmental and other variation on duck population dynamics in the absence of regulatory change (McCabe, 1987). Some studies were directed at questions about the densitydependence of seasonal mortality in an effort to investigate possible mechanisms for compensatory mortality (e.g., Blohm et al., 1987; Reineke et al., 1987). Others examined continental survey data from the period of stabilized regulations in new retrospective analyses (e.g., Caswell et al., 1987; Reynolds, 1987; Trost et al., 1987) Although much was learned about waterfowl population dynamics during the period of stabilized regulations (see, e.g., McCabe, 1987), this body of information did not resolve uncertainties about biological processes, especially as concerns the influence of hunting on annual survival of ducks (Trost, 1987; Sparrowe and Patterson, 1987; U.S. Department of the Interior, 1988). Nevertheless, the stabilized regulations program represented a milestone in waterfowl harvest management, in that this was the first large-scale attempt to manipulate hunting regulations for the express purpose of learning about population dynamics. Immediately following completion of the stabilized regulations program, more restrictive harvest regulations were imposed in response to widespread drought conditions on the waterfowl breeding grounds in North America. Restrictive regulations were continued throughout the 1980s and early 1990s, as precipitous declines occurred in duck population numbers. This in turn led to decreasing public participation in sport hunting and to tensions among managers about the appropriate regulatory response. By the early 1990s, managers had become frustrated about the continuing uncertainty about biological process and the inability to come to consensus about an appropriate harvest strategy in the face of this uncertainty. Political intervention in 1994 threatened the integrity of the regulatory process and ushered in the current approach to waterfowl harvest management under the rubric of Adaptive Harvest Management.
25.2. COMPONENTS OF A REGULATORY PROCESS The federal government of the United States derives its responsibility for establishing sport-hunting regulations from the Migratory Bird Treat Act of 1918 (as amended), which implements provisions of the international treaties for migratory bird conservation. The Act directs the Secretary of Agriculture periodically to adopt hunting regulations for migratory birds, "having
667
due regard to the zones of temperature and to the distribution, abundance, economic value, breeding habits, and times and lines of migratory flight of such birds" (U.S. Department of the Interior, 1975). The responsibility for managing migratory bird harvests has since been passed to the Secretary of the Interior and the U.S. Fish and Wildlife Service. Other legislative acts, such as the National Environmental Policy Act, the Endangered Species Act, the Administrative Procedures Act, the Freedom of Information Act, and the Regulatory Flexibility Act, provide additional responsibilities in the development of hunting regulations and help define the nature of the regulatory process (Blohm, 1989). In general, the goals of harvest regulations are as follows: 1. To provide an opportunity to harvest certain migratory game bird populations by establishing legal hunting seasons. 2. To limit the harvest of migratory game birds to levels compatible with their ability to maintain populations. 3. To avoid the taking of endangered or threatened species so that their continued existence is not jeopardized and their conservation is enhanced. 4. To limit taking of other protected species when there is a reasonable possibility that hunting is likely to affect adversely their populations. 5. To provide equitable hunting opportunity in various parts of the country within limits imposed by abundance, migration, and distribution patterns of migratory birds. 6. To assist, at particular times and in specific locations, in preventing depredations on agricultural crops by migratory game birds (U.S. Department of the Interior, 1988).
25.2.1. Setting Harvest Regulations Most waterfowl hunting regulations are established annually, within a timetable that is constrained by the timing of biological data collection and the need to give states and the public an opportunity to influence regulations. Information on waterfowl population status, and on the outlook for annual production, is typically unavailable until early summer of each year. Some waterfowl hunting seasons open as early as midSeptember, so that the time available for interpreting biological data, developing regulatory proposals, soliciting public comment, and establishing and publishing hunting regulations is quite limited. Delays in the process can result in closed hunting seasons because proac-
668
Chapter 25 Case Study
tive regulatory action is required to allow any harvest of migratory birds. The annual regulatory process is documented in the Federal Register, which provides a detailed record of proposals, public comment, government responses, final regulatory guidelines, and hunting-season selections by individual states. The process includes two development schedules, dedicated to "early" and "late" hunting seasons. Early seasons are those opening prior to October 1, and they primarily focus on migratory birds other than waterfowl (Gruidae, Rallidae, Phalaropodidae, and Columbidae), but also include all migratory birds in Alaska, Puerto Rico, and the Virgin Islands. Late-season regulations pertain to most duck and goose hunting seasons, which typically begin on or after October 1. The early-season and lateseason processes occur concurrently, beginning in January and ending by late September of each year. The regulatory process begins early each year, when the U.S. Fish and Wildlife Service announces its intent to establish waterfowl hunting regulations and provides the schedule of public rule-making (Fig. 25.4). A Migratory Bird Regulations Committee presides over the process and is responsible for regulatory recommendations. The Committee convenes two public meetings during the summer to review biological information and to consider proposals from consultants representing the Waterfowl Flyway Councils (Fig. 25.5). The Flyway Councils and the state fish and wildlife agencies they represent are key partners in the management of migratory bird hunting. Following these consultations, hunting-season proposals are presented at public hearings and in the Federal Register for comment. The resulting framework regulations are Flywayspecific and identify the earliest and latest dates for hunting seasons, the maximum number of days in the season, and daily bag and possession limits. States select hunting seasons within the bounds of these frameworks, usually following their own processes for proposals and public comment. Final hunting regulations, including any state-imposed restrictions, are published in the Federal Register.
25.2.2. Biological Monitoring As indicated above, a key component of the regulatory process consists of data collected each year on population status, habitat conditions, production, harvest levels, and other system attributes of management interest (Anderson and Henny, 1972; Martin et al., 1979; Smith et al., 1989). This monitoring program is essential for discerning resource status and modifying hunting regulations in response to changes in environmental
conditions. The system of waterfowl monitoring in North America is unparalleled in its scope and is made possible only by the cooperative efforts of the U.S. Fish and Wildlife Service, the Canadian Wildlife Service, state and provincial wildlife agencies, and various research institutions. Surveys conducted from fixed-wing aircraft at low altitudes are a mainstay of waterfowl management. Among the most important are surveys conducted in the principal breeding range of North American ducks (Smith, 1995). Each spring, duck abundance and habitat conditions are monitored in over 5 million km 2 of breeding habitat, using 89,000 km of aerial transects (Fig. 25.6). The transects are distributed according to a stratified systematic design with double sampling (see Chapters 5 and 12), in which ground surveys are conducted on a subset of the aerial transects to estimate the proportion of birds that are undetected from the air (see Section 12.6). This attention to both spatial variation (via spatial sampling) and detection probability (via the double-sampling with ground counts) is rare for large-scale surveys and it provides duck biologists and managers with abundance estimates that are far better than those available for most animal species (Thompson et al., 1998; Yoccoz et al., 2001; Pollock et al., 2002). The central portion of the breeding range is surveyed again in midsummer to estimate the number of duck broods and remaining wetlands and to assess the progress of the breeding season. These surveys have been operational since the 1950s and provide key information for setting annual duck-hunting regulations. Waterfowl abundance also is determined during winter through a network of aerial and ground surveys in the United States and Mexico (Smith et al., 1989). These surveys originated in the 1930s and were the basis for establishing duck-hunting regulations prior to the development of breeding-ground surveys. Winter surveys are intended to provide a census of major waterfowl concentration areas, but they lack the rigorous statistical design of breeding-ground surveys. Estimates of winter waterfowl abundance thus lack measures of precision and are subject to error resulting from variation in the distribution of birds relative to surveyed areas. Nonetheless, winter surveys provide useful information about large-scale waterfowl distribution and habitat conditions, and they remain the primary source of information for setting harvest regulations for geese. Waterfowl also are monitored through a large-scale marking program in which individually numbered leg bands are placed on over 350,000 birds annually, usually just prior to the hunting season. The band inscrip-
25.2. Components of a Regulatory Process
669
March 11 - Proposed rulemaking with public comment periods ending July 27 for early-season regulations and September 7 for late-season regulations
May 21 - Supplemental proposed rulemaking
Early seasons
Late seasons
(opening before 1 October)
(opening after 1 October)
June 23 & 24 - USFWS Regulations Committee Meeting
August 4 & 5 - USFWS Regulations Committee Meeting
June 25 - Public Hearing on proposed early-season regulations
August 6 - Public Hearing on proposed late-season regulations
July 15 - Supplemental proposed rulemaking for early-season regulations with public comment period ending July 27
August 21 - Final early-season regulations
August 31 - Final rulemaking amending Title 50 CFR for early seasons
August 26 - Supplemental proposed rulemaking for late-season regulations with public comment period ending September 7
September 28 - Final late-season regulations
September 29 - Final rulemaking amending Title 50 CFR for late seasons
FIGURE 25.4 Approximate timetable used by the U.S. Fish and Wildlife Service for setting annual hunting regulations for migratory birds.
tion asks the hunter or finder of a dead bird to report the band number, date, and location to the U.S. Fish and Wildlife Service. Banding is the principal tool used to understand migratory pathways and was the basis for establishing the four administrative flyways (Lincoln, 1935). The banding program also is essential for
understanding temporal and spatial variation in rates of harvest and total mortality (see Chapter 16). The U.S. Fish and Wildlife Service also conducts hunter surveys to determine hunting activity, harvest by species, date, and location, as well as age and sex composition of the harvest (Martin and Carney, 1977).
670
Chapter 25 Case Study the species and demographic structure of the harvest can be estimated reliably. A complete record of waterfowl harvest in the United States extends back to 1962.
S ) ~Central~MiSsisSippi;~-Atla~tic
FIGURE 25.5 Waterfowlflyways,which are used for administering the regulations process for sport hunting of migratory birds.
This monitoring program is conducted via a mail questionnaire, which is completed by a sample of 30,000-35,000 waterfowl hunters across the United States. The sampling frame is derived from purchasers of federal Migratory Bird Hunting and Conservation ("duck") Stamps at randomly selected post offices or, more recently, directly from the sale of state hunting licenses. Questionnaire results provide the basis for estimating hunting effort and total waterfowl harvest. In addition to the questionnaire, about 8000 hunters send in wings or tail feathers of harvested birds so that
25.2.3. Predicting the Effects of Regulations Long-term data from monitoring programs are used to estimate key population parameters such as survival and reproductive rates, and to associate levels of harvest with various regulatory scenarios (Martin et al., 1979). These and other estimators are combined to produce dynamic population models, which describe how waterfowl abundance varies in response to harvest and uncontrolled environmental factors (Chapters 8 and 11) (see also Williams and Nichols, 1990). These models in turn are used to inform the regulations process, on assumption that population status is directly related to harvest and harvest can be predicted as a function of hunting regulations (Johnson et al., 1993). By building on accumulated monitoring data, the models reflect an evolving understanding of waterfowl population dynamics and the impacts of harvest. Unfortunately, the modeling of waterfowl populations and their harvest continues to be characterized by great uncertainty. In many cases, the sheer number and complexity of hunting regulations, combined with
FIGURE 25.6 Strata and transects of the Waterfowl Breeding Population and Habitat Survey, which is conducted annually by the U.S. Fish and Wildlife Service, the Canadian Wildlife Service, and state and provincial partners.
25.3. Adaptive Harvest Management
671
inadequate replication and experimental controls, has precluded reliable inference about the relationship between regulations and harvests (Nichols and Johnson, 1989). Managers know even less about the influence of harvest on subsequent waterfowl population size (reviewed in U.S. Department of the Interior, 1988; Nichols, 1991b; Nichols and Johnson, 1996). Particularly problematic in this regard are questions about the nature of density-dependent population regulation, which provides a theoretical basis for sustainable exploitation (Hilborn et al., 1995). Uncertainties about the relationships among hunting regulations, harvest, and population size constitute a principal source of controversy in the regulations-setting process.
present, four models are used, each developed from data bases that have accrued as a result of waterfowl monitoring and research programs. 4. Measures of reliability for the models, which are used in selecting harvest regulations. Reliability measures are used to weight the model outputs and are updated each year as additional data about population status and the impacts of regulation become available. The notion of reliability is included in the process as an acknowledgment that the "correct" model (i.e., the model best approximating system dynamics and responses to harvest) for evaluating regulatory options is not known with certainty, and this uncertainty should be incorporated somehow in the procedure for evaluating and selecting regulations.
25.3. A D A P T I V E HARVEST MANAGEMENT
Adaptive Harvest Management is framed in terms of sequential decision making under uncertainty, or more particularly in terms of adaptive control processes (Bertsekas, 1995). In this conceptual model, managers periodically observe the state of the resource system (e.g., population size and relevant environmental features) and take some management action (e.g., hunting regulations) (Fig. 25.7). An immediate return accrues as a result, which is expressed as a function of the benefits and costs that are relevant to the stated objectives of management. In response to the combined influence of management actions and uncontrolled environmental variation, the resource system subsequently evolves to a new state. The managers then observe the new system state, make a new decision, accumulate additional returns, and the system evolves to yet another state (Fig 25.7). And so on. The goal of management is to make a sequence of such decisions, each based on information about current system status, so as to maximize net benefits over an extended time frame. By taking advantage of the decision-making structure and predicted system behaviors, it is possible to characterize waterfowl harvest management as a Markov decision process (see Section 10.6). In this class of sequential decision processes, management actions, returns, and system transitions are described in terms of current system state and action. Given this simplifying constraint, computing algorithms and software are available for determining the optimal regulatory choice for the array of possible resource states (Chapter 24) (see also Puterman, 1994; Lubow, 1995, 1997; Williams, 1996b). An essential element of the optimization process is a set of state-specific and action-dependent transition probabilities, which are associated with possible management outcomes. It is these probabilities that reflect key stochastic effects and uncertainties in system dynamics.
Adaptive Harvest Management (AHM), the process currently in use for waterfowl harvest regulations, explicitly accounts for uncertainty and the value of information in the regulatory process. Along with an institutional regulatory framework and appropriate monitoring programs as described above, four elements are definitive of the process: 1. An array of regulatory options that are available for the regulation of waterfowl harvest. These options include various combinations of regulations representing, e.g., "restrictive," "liberal," and "moderate" regulations, with possible constraints on allowable fluctuations from year to year. The set of feasible regulatory options can be limited or expanded as the need and desirability to do so is recognized by management. 2. An objective function by which to evaluate and compare these options. The general form of the objective function is a weighted sum of harvests (or harvest utilities) over some recognized time frame. This is in keeping with traditional goals for waterfowl harvest management and ensures that the focus is on harvest and harvest opportunity. An extended time frame protects against overexploitation in the short term, by emphasizing the importance of sustainable harvests (and thus sustainable populations to support those harvests). 3. A set of waterfowl models representing an array of meaningful hypotheses about the influence of regulations on waterfowl populations. For example, the set currently in use includes models that incorporate the hypothesis of additive hunting mortality and others that incorporate the hypothesis of completely compensatory hunting mortality. These models are used to gauge the consequences of different regulations. At
672
Chapter 25 Case Study
decision(t) T
._lsystemstate vI
It)
retlrn It)
decision(t+2)
decision(t+1) t
J systemstate (t~l)
retuln (t+l)
T
J systemstate vI
It+2)
retuIn
(~-2)
F I G U R E 25.7 A sequential decision-making process, in which management decisions made over time (t) elicit an immediate return (benefits-costs) and, along with uncontrolled environmental factors, drive the resource system to a new state.
A major advantage of adaptive harvest management over traditional approaches is in the explicit acknowledgment of alternative hypotheses describing the effects of regulations and other environmental factors on population dynamics. The hypotheses are codified in a set of system models, each of which has an associated weight reflecting its ability to describe system dynamics. Each year the weights are updated by comparing the model-specific predictions of changes in population size with the actual change observed from the monitoring program. By iteratively updating model weights and optimizing regulatory choices, the process should eventually identify which model is most appropriate to describe the dynamics of the managed population. An adaptive approach to harvest management thus can be described as a four-step process: 1. Each year, an optimal regulatory decision is identified based on resource status and current model probabilities. 2. Once the decision is made, model-specific predictions for subsequent breeding population size are determined. 3. When monitoring data become available, model probabilities are increased to the extent that observations and predictions agree, and decreased to the extent that they do not agree. 4. The new set of model probabilities is used to start another iteration of the process. The overall system of monitoring, modeling, and regulations setting is designed to identify optimal regulatory choices for particular resource states and simultaneously to track measures of model reliability over time (Johnson et al., 1997).
25.4. M O D E L I N G POPULATION DYNAMICS A general formulation of population dynamics for waterfowl allows for a multidimensional resource system, with x t representing the system state (i.e., population size and indicators of habitat conditions). System transitions are given by Xt +1 = Xt q- Fi(xt" Ht, Zt),
(25.1)
where Ht represents time-dependent harvests, Z t represents time-dependent environmental effects, and the index i specifies one of a number of models used to represent population and habitat dynamics. Here we describe analyses that produced the suite of models currently in use in Adaptive Harvest Management. The modeling of populations, pond conditions, and harvest rates focuses on midcontinent mallards and follows closely the work described in Johnson et al. (1997).
25.4.1. Structural Uncertainty Structural uncertainty (see Section 24.3) can be expressed by means of a set of alternative models that are defined by model-specific survival and reproduction functions. Let x~(t) represent the number of adult mallards of sex s that are present in the midcontinent survey area and x2(t) represent the number of ponds in Prairie Canada in May of each calendar year t (U.S. Department of the Interior, 1994). In what follows, we consider two models for survivorship (i = 1, 2) and two models for recruitment (j = 1, 2). Denote by q~(t) the sex-specific survival rate of adults from May of year t through April of year t + 1, with q~s(t)' the model- and sex-specific survival rate of young from
25.4. Modeling Population Dynamics September of year t through April of year t + 1. Also, let y~(t) be the number of young of sex s in the fall population. Then the transition of population size over time (t, t + 1] is S S v x~(t+l) = XSl(t)q~(t) + yj(t)q~i(t),
where
y;(t) = gj[XSl(t), x2(t)] is a model-specific recruitment function [see Eqs. (25.2)-(25.4) below]. The combining of two forms for the survival function q~s(t) and two forms for the recruitment function y~(t) leads to four population models.
673
to justify age dependency or year dependency in rates of survival from nonhunting causes. To characterize the effect of exploitation on survival, additive and compensatory models of survival during the hunting season are considered (see Section 11.1.3). In the case of additive hunting mortality, sex-specific survival of adults and young during the hunting season shows linear declines with increases in harvest rate:
h~(t)
~(t) = 1
1 --C
and
hs(t)'
13~(t)' = 1 1
--
C~,
25.4.1.1. Survival
Annual survival of mallards currently is modeled as the product of survival from hunting and survival from natural mortality factors outside the hunting season. Sex-specific survival of adults and young for model i is described by
q~(t) = ~/%[3~(t) and S P q~s(t) P = ~/f3i(t),
respectively, where % is a sex-specific summer survival rate, ~s(t) and ~s(t)' are sex-specific hunting-season survival rates for adults and young, respectively, and ~/is winter survival rate. Nonhunting-season survival rates for the models were estimated using the methods of Smith and Reynolds (1992) and assuming a crippling loss of c = 0.2 (Anderson and Burnham, 1976) and a band-reporting rate of )t = 0.32 (Nichols et al., 1991). For the models currently used in AHM, the same data represented in Smith and Reynolds (1992) were fitted to a model that does not include geographic variation in nonhunting mortality rates (Johnson et al., 1997). Estimates of annual survival in the absence of hunting mortality were 0.81 (SE = 0.02) for males and 0.64 (SE = 0.01) for females. These rates were partitioned into winter and summer components based on the results from the Stabilized Regulations Study (U.S. Fish and Wildlife Service Office of Migratory Bird Management, Washington, D.C., unpublished data), with verification against other literature (Cowardin and Johnson, 1979; Reineke et al., 1987; Dugger et al., 1994) to ensure reasonable estimates. For males, a survival rate of % = 0.9 is used, with % = 0.71 for females. Winter survival for both sexes is ~/ = 0.90, absent empirical evidence
where h~(t) and h~(t)' are sex-specific harvest rates of adults and young, respectively. Inclusion of crippling loss (c) accounts for birds that are killed by hunters but not retrieved [Anderson and Burnham (1976) refer to hs(t)/(1 - c) and hs(t)'/(1 - c) as kill rates]. Compensatory mortality also is considered, with complete compensation for hunting mortality up to a threshold kill rate (defined as the annual rate of nonhunting mortality). Thus, annual survival is constant for kill rates below this threshold and declines linearly with increases in harvest rate beyond it: 1.0 13~(t) =
1
hs(t) <- 1 - %~/,
if
1 - c
if
1 - c
if
1 - c <- 1 - OLs~/,
hs(t) 1 --C
O~s~/
hs(t) > 1 - %?,
and
h~(t)
i .0 f3~(t)' =
h~(t) 1 - c C~s~/
if
hs(t) > 1 - C~s~/. 1 - c
For both the additive and compensatory models, nonhunting mortality during the hunting season is assumed to be negligible (e.g., Cowardin and Johnson, 1979; Reineke et al., 1987). 25.4.1.2. Recruitment
Recruitment models are based on estimates of the annual fall age ratios of female mallards originating from the region of North America surveyed in spring,
674
Chapter 25 Case Study
1961-1993 (U.S. Department of the Interior, 1994). Let A t be the ratio of young females to adult females in the preharvest population, as estimated from the age ratio of the harvest corrected for relative harvest vulnerability (young:adult ratio of direct recovery rates of banded females), in year t (Martin et al., 1979). Age ratios of the harvest were calculated from parts-collection surveys (Martin and Carney, 1977) in those portions of the Central Flyway and Mississippi Flyway that derive ->80% of their harvest from the mallard population of interest (Munro and Kimball, 1982). The harvest vulnerability of young relative to adults was estimated for each of eight banding reference areas (Anderson and Henny, 1972) within the breeding range, then averaged for a single estimate of relative vulnerability for each year, with estimates of population size within reference areas as weights. Age ratios are linked to population and habitat conditions via models describing A t as a linear function of mallard population size xl(t) and the number of ponds x2(t). An interaction between xl(t) and x2(t) also was considered, and the linear relationship was allowed to vary between two unspecified "epochs" within the period 1961-1993. Weighted least-squares regression was used to identify the models, with the values A t inversely weighted by the variance of the annual harvest age ratio, which was considered to be proportional to variability in A t . All possible regression models induced by interactive combinations of xl(t), x2(t), and epoch were fitted, with epoch boundaries (i.e., the first year of the second epoch) adjusted to each year between 1965 and 1990 (14 models/epoch partition • 26 epoch partitions + 5 "no epoch" models = 369 models). Model selection was based on the lowest value of the Akaike Information Criterion (AIC) (see Section 4.4) and checked against model residuals for conformity with least-squares regression assumptions (Draper and Smith, 1981). The model with the smallest of the 369 AIC values contained xl(t) and x2(t), but no interaction. It distinguishes an epoch boundary at 1970, where the regression coefficient for xl(t) changed from a pre-1970 value of -0.0874 • 10 -6 (SE -- 0.0622 • 10 -6) to a post1970 value of -0.0547 • 10 -6 (SE = 0.0225 • 10-6). That portion of the model corresponding to the most recent epoch was selected as a weakly density-dependent model of recruitment: Al(t ) = (0.8249 - 0.0547 x 10-6)x1(t) (25.2) + (0.1130 x 10-6)x2(t). To express uncertainty about the degree of density dependence in recruitment [i.e., the magnitude of the coefficient for xl(t)], a strongly density-dependent
model of recruitment also was considered, based on the minimum parameter estimate for the coefficient of xl(t) for the post-1970 period located on the 95% confidence ellipsoid for all the parameters (Draper and Smith, 1981). The minimum estimate was selected as an alternative to Eq. (25.2) based on the most likely mechanisms for density-dependent recruitment (e.g., spacing behavior of pairs) (Dzubin, 1969). Thus, the strongly density-dependent model of recruitment was A2(t) = (1.1081 - 0.1128 • 10-6)xl(t)
(25.3)
+ (0.1460 • 10-6)x2(t). The number of young females in the fall population was modeled as a product of the predicted age ratio and the number of adult females in the fall. The number of adult females in the fall was given in turn by the product of summer survival and the number of females in the spring, which was determined based on the May estimates of population size and assuming a constant sex ratio of 1.2 males per female (Anderson, 1975a). To determine the number of young males and females, a sex ratio of 1.0 was assumed for young birds in the fall (Bellrose et al., 1961; Hestbeck et al., 1989). Thus:
y~(t) = y[(t) = gj[XSl(t), x2(t)]
(25.4)
= Aj(t)oL~xl(t)/Z2. The combination of two survival hypotheses (i = 1, 2) and two recruitment hypotheses (j = 1, 2) resulted in four alternative models of mallard population dynamics: (1) additive hunting mortality and weakly density-dependent recruitment; (2) additive hunting mortality and strongly density-dependent recruitment; (3) compensatory hunting mortality and weakly density-dependent recruitment; and (4) compensatory hunting mortality and strongly density-dependent recruitment (Johnson et al., 1997).
25.4.2. Environmental Variation The number of wetland basins containing surface water (ponds) in the Prairie Pothole Region during the breeding season is an important determinant of mallard production (Pospahala et al., 1974). Since 1961, the number of ponds in Prairie Canada during May has varied from 1.443 million (SE = 0.075 million) in 1981 to 6.390 million (SE = 0.308 million) in 1974 (U.S. Department of the Interior, 1994). Managers involved in harvest management cannot predict with certainty the number of ponds (and thus the mallard production) in the future. However, it is possible to make probabilistic statements about temporal changes in pond abun-
675
25.4. Modeling Population Dynamics dance, thereby allowing managers to assess the future consequences of current regulatory decisions. Thus, the estimated number of ponds in Prairie Canada and records of monthly (1 June 1974-31 May 1992) precipitation (millimeters) from five weather stations in southern Alberta, Saskatchewan, and Manitoba were used to construct an autoregressive model (see Section 10.8.4) of pond abundance: x2(t+l) = -3835087.53 + 0.45xa(t)
(25.5)
+ 13695.47r(t), where r(t) is total precipitation during the 12-month period from time t to t + 1. Pond numbers predicted by this model are nearly identical to those of the model provided by Pospahala et al. (1974). Annual (1 Jan-31 Dec) precipitation records were examined for the period 1942-1991 from the same five weather stations. The hypothesis that annual precipitation r(t) was distributed normally (Shapiro-Wilk W = 0.97, P = 0.36, range 304-574 mm, 2 = 418 mm, SD = 56 mm) was supported by the data, and results were virtually identical to those reported by Pospahala et al. (1974). Preliminary analyses with several data sets (some of > 100 years) failed to provide strong evidence of precipitation cycles (J. R. Sauer, U.S. Geological Survey, personal communication), supporting the conclusion of Pospahala et al. (1974) that annual precipitation in Prairie Canada can be described adequately as a normally distributed, independent random variable. Random draws from this distribution provided stochasticity in pond abundance according to Eq. (25.5).
25.4.3. Partial Management Control Managers control hunting regulations rather than harvest rates directly, and accounting for uncertainty in the functional relationship between the two is important. Early on, partial controllability was incorporated based on band recovery data from preseason banding as applied to three regulatory options. Later, a more complex procedure was used that would allow for variable bag limits and season lengths (see Section 25.6.1). In this section we describe the derivation of distributions of harvest rates for midcontinent mallards under each of three regulatory options. These options, characterized as liberal, moderate, and restrictive, corresponded to the regulations in effect in 1979-1984, 1985-1987, and 1988-1993, respectively. Each regulatory option contained flyway-specific season lengths and bag limits. The analysis of harvest relied on direct recovery rates of mallards banded before the hunting season in a representative portion of the midcontinent region
(banding reference areas 3-5) (Anderson and Henny, 2 1972). First the mean [fAM,p] and variance [Stota 1 (f,4~,p)] of recovery rates were estimated from the point estimates of direct recovery rate for adult male mallards for each of the three time periods (p). The analysis focused on adult males because they generally had the largest banded-sample sizes. The variances of the direct recovery rates are composed of both temporal and sampling components. For the purpose of choosing hunting regulations, interest focused primarily on the temporal component, which is a measure of the variability in recovery rates that could be expected when the same regulations are used in different years (i.e., partial controllability). This temporal v a r i a t i o n [S2emp(fAM,p)] was estimated using the approach suggested by Burnham et al. (1987): 2 St2mp(fAM'p) = S t ~
--
~t~l
s2(fAM,p,t) I np
(25.6)
where np is the number of years in period p and s2(fAM,p,t ) is the estimated sampling variance. Periodspecific harvest r a t e s [hAM,p] and their temporal vari2 a n c e s [Stemp(hAM,p)] then were estimated using a constant band-reporting rate of X = 0.32 (Nichols et al., 1991): St2mpGM,p) St2mp(hAM,p) --
ha
.
(25.7)
Based on this analysis, mean harvest rates for adult males were 0.090 [Stemp(hAM,p) = 0.016] for the restrictive option, 0.120 [Stemp(hAM,p) = 0.022] for the moderate option, and 0.156 [Stemp(hAM,p) = 0.025] for the liberal option. A closed season also was considered, in which the mean harvest rate was assumed to be 0 with no variation. The vulnerability to harvest for each of the other age-sex cohorts (adult females, young males, young females) was specified relative to that of adult males. The mean relative vulnerability Wa,p of each cohort a during each period p was calculated by averaging the ratio of annual recovery rates within the specified period: E n p fa,p,t t = lfAM,p, t Wa,p -Tip
Mean relative vulnerabilities did not differ among periods (asymptotic normal test of general contrast) (Sauer and Williams, 1989) for adult females, immature males, or immature females. Thus, constant rates of differential vulnerability were used, with 0.480 for adult females, 1.310 for young males, and 0.868 for young females.
676
Chapter 25 Case Study
25-
20-
X
15-
..~ ""...,.
/
10-
/iX....."",. / '
5-
--...'..
\
g
~ "\'~;~. O'
-
,
0.00
"
~
0.03
...,.. ... ,
'3
0.06
0.09
0.12
0.15
0.18
0.21
The harvest m a n a g e m e n t objective for midcontinent mallards is to maximize cumulative harvest value over the long term, given an aversion to harvest decisions that result in an expected population size below the goal of the North American Waterfowl Management Plan (NAWMP) (Fig. 25.9). The value of harvest opportunity decreases proportionally as the difference between the goal and expected population size increases. This balance of harvest and population objectives results in a more conservative harvest strategy than one maximizing long-term harvest, but a more liberal strategy than one seeking to attain the N A W M P goal regardless of losses in hunting opportunity. The current objective uses a population goal of 8.7 million mallards, based on the N A W M P goal of 8.1 million for the federal survey area and 0.6 million for the combined states of
(hAM,p)
Minnesota, Wisconsin, and Michigan (U.S. Department of the Interior, 1994). The utility function expressing the model-specific value of harvest u[Hi(htixt)] has the form u[H~(htlxt)]
=
if Ei[xl(t+l)] >- 8,700,000, if Ei[xl(t+l)] <-4,000,000,
1.0 0.0 [, 8,700,000 - 4,000,000
OBJECTIVES
0.27
mallards
f Ei[Xl(t+l)]. - 4,000,000 25.5. HARVEST
"
0.24
FIGURE 25.8 Probability density function of harvest rate of adult under liberal (--), moderate (--.--), and restrictive (...) regulations.
Based on the period-specific means and variances from Eqs. (25.6) and (25.7), a two-parameter g a m m a distribution (see Appendix E.2.3) of adult male harvest rates was assigned to each regulatory option (Fig. 25.8). Distributions of harvest rates for the other age-sex cohorts were obtained by multiplying the constant rate of differential vulnerability for each cohort by the harvest rates for adult males.
, '~"--
if 4,000,000< Ei[xl(t+l) ] < 8,700,000,
where the model-specific expectation Ei[xl(t + 1)] depends on both x t and h t. This utility function represents 100
g 8o ~ 6o
>
m 40 population goal = 8.7
"r 20 0 0
1 --2
3
4
5
6
7
8
9
10
Expected population size next year (in millions) FIGURE 25.9 The relative value ~ of mallard harvest, expressed as a function of breeding population size expected in the subsequent year.
25.6. Regulatory Alternatives TABLE 25.1
677
Regulatory Alternatives Considered for the 1995 and 1996 Duck-Hunting Seasons Flyway
Regulation
Atlantic
Shooting hours Framework dates Oct 1-Jan 20 Season length (days) Restrictive 30 Moderate 40 Liberal 50 Bag limit (total/mallard / female mallard) Restrictive 3/ 3/ 1 Moderate 4/4/1 Liberal 5 / 5/ 1
Mississippi
Central a
Pacific b
One-half hour before sunrise to sunset for all flyways Saturday closest to October 1 and Sunday closest to January 20 30 40 50
39 51 60
59 79 93
3/2 / 1 4/3/1 5/ 4 / 1
3/3 / 1 4/4/1 5 / 5/ 1
4 / 3/ 1 5/4/1 6-7c/ 6-7c/ 1
The High Plains Mallard Management Unit was allowed 12, 16, and 23 extra days under the restrictive, moderate, and liberal alternatives, respectively. bThe Columbia Basin Mallard Management Unit was allowed seven extra days under all three alternatives. cThe limits were 6 in 1995 and 7 in 1996. a
a compromise over the range of population sizes below 8.7 million, in that neither the objective to maximize harvest nor the objective to maintain the mallard population at or above the plan goal w o u l d be realized fully (Fig. 25.9). Both the population size xl(t) and the capacity of available breeding habitat x2(t) to promote population growth during the interval t to t + 1 are considered in the determination of the optimal regulatory decision for x t. Thus, liberal hunting regulations could still be appropriate for a mallard population that is below the goal of 8.7 million, if current habitat conditions were expected to result in good production of young.
25.6. R E G U L A T O R Y
ALTERNATIVES W h e n A H M was first implemented in 1995, liberal, moderate, and restrictive regulations were defined based on regulations used during 1979-1984, 19851987, and 1988-1993, respectively (Table 25.1). These regulatory alternatives also were considered for the 1996 hunting season. However, in 1997 the regulatory alternatives were modified to include (1) the addition of a very restrictive alternative, (2) additional days and a higher duck bag limit in the moderate and liberal alternatives, and (3) an increase in the bag limit of hen mallards in the moderate and liberal alternatives. The basic structure of the regulatory alternatives has been u n c h a n g e d since 1997 (Table 25.2).
25.6.1. P r e d i c t i n g
Harvest Rates
Since 1997, harvest rates (and associated variances) for the A H M regulatory alternatives have been predicted using (1) linear models that predict total seasonal mallard harvest for varying season lengths and bag limits, accounting for numbers of successful duck hunters, and (2) adjustment of historical estimates (Section 25.4.3) to reflect differences in bag limit, season length, and trends in hunter numbers (Table 25.3). The adjustments are based on estimates of hunting effort and success from hunter surveys. The procedure utilizes linear models that predict total seasonal mallard harvest for varying regulations (daily bag limit and season length), while accounting for trends in numbers of successful duck hunters. Using historical data from both the U.S. Waterfowl Mail Questionnaire and Parts Collection Surveys, the resulting models allow one to predict total seasonal mallard harvests and associated harvest rates for varying combinations of season length and daily bag limits. The linkage between regulations and harvest rate involves two component models: a "harvest" model that predicts average daily mallard harvest per successful duck hunter for each day of the hunting season, and a " h u n t e r " model that predicts the n u m b e r of successful duck hunters. The "harvest" model uses as its d e p e n d e n t variable the square root of the average daily mallard harvest (per successful duck hunter), with independent variables that include the consecutive days of the hunting season (ignoring splits in the season), daily mallard bag limit, season length, and
Chapter 25 Case Study
678 TABLE 25.2
Regulatory Alternatives Considered for the 1999 Duck-Hunting Season Flyway
Regulation
Atlantic a
Mississippi b
Shooting hours
Central c
Pacific a
One-half hour before sunrise to sunset for all flyways
Framework dates
Oct 1-Jan 20
Season length (days) Very restrictive
Saturday closest to October 1 and Sunday closest to January 20
20
20
25
38
Restrictive
30
30
39
60
Moderate
45
45
60
86
Liberal
60
60
74
107
Bag limit (total/mallard/female mallard) Very restrictive 3/ 3/ 1 Restrictive
3/3/1
3/ 2/ 1
3/ 3 / 1
4/ 3/ 1
3/2/1
3/3/1
4/3/1
Moderate
6/4/2
6/4/1
6/5/1
7/5/2
Liberal
6/4/2
6/4/2
6/5/2
7/7/2
a The states of Maine, Massachusetts, Connecticut, Pennsylvania, New Jersey, Maryland, Delaware, West Virginia, Virginia, and North Carolina are permitted to exclude Sundays, which are closed to hunting, from their total allotment of season days. b In the states of Alabama, Mi3sissippi, and Tennessee, in the moderate and liberal alternatives, there is an option for a framework closing date of January 31 and a season length of 38 days and 51 days, respectively. c The High Plains Mallard Management Unit is allowed 8, 12, 23, and 23 extra days under the very restrictive, restrictive, moderate, and liberal alternatives, respectively. d The Columbia Basin Mallard Management Unit is allowed 7 extra days under the very restrictive, restrictive, and moderate alternatives.
the interaction of bag limit and season length. Terms for an opening-day effect, a week effect, and several other interaction terms also are included. Seasonal mallard harvest per successful duck hunter is obtained by back-transforming the predicted values from the model and summing the average daily harvest over the season length.
TABLE 25.3 Expected Harvest Rates (SE) of Adult Male Midcontinent and Eastern Mallards a Harvest rate (SE) Mallard population
Alternative
Midcontinent
Very restrictive
Eastern
1995, 1996
1997, 1998, 1999
N/A
0.053 (0.011)
Restrictive
0.067 (0.014)
0.067 (0.014)
Moderate
0.089 (0.020)
0.111 (0.027)
Liberal
0.118 (0.029)
0.131 (0.032)
N/A
0.121 (0.020)
Restrictive
0.133 (0.021)
0.135 (0.022)
Moderate
0.149 (0.023)
0.163 (0.025)
Liberal
0.179 (0.028)
0.177 (0.028)
Very restrictive
a Under different regulatory alternatives, based on mean hunter numbers during 1981-1995.
The "hunter" model utilizes information about the numbers of successful duck hunters (based on duck stamp sales information) from 1981 to 1995. Using daily bag limit and season length as independent variables, the number of successful duck hunters is predicted for each state. Both "harvest" and "hunter" models were developed for each of seven management areas: the Atlantic Flyway portion with compensatory days; the Atlantic Flyway portion without compensatory days; the Mississippi Flyway; the low plains portion of Central Flyway; the High Plains Mallard Management Unit in the Central Flyway; the Columbia Basin Mallard Management Unit in the Pacific Flyway; and the remainder of the Pacific Flyway, excluding Alaska. The numbers of successful hunters predicted at the state level were summed to obtain a total number (H) for each management area. Likewise, the "harvest" model results in a seasonal mallard harvest per successful duck hunter (A) for each management area. Total seasonal mallard harvest then is given by the product T=H•
To compare total seasonal mallard harvest under different regulatory alternatives, ratios of total harvest for different alternatives were formed for each management area and then combined into a weighted mean. Under the key assumption that the ratio of harvest rates realized under two different regulatory alternatives is
25.7. Identifying Optimal Regulations equal to the expected ratio of total harvest obtained under the same two alternatives, the harvest rate experienced under the historic "liberal" package (19791984) was adjusted by T to produce predicted harvest rates for the current regulatory alternatives. Harvest rates for each of the regulatory alternatives for 1999 were predicted assuming no change in the regulatory alternatives from 1997 and 1998 (Table 25.4). However, predicted harvest rates for 1997-1999 differ from those used previously as a result of revised analytical procedures, which rely on mean numbers of hunters during 1981-1995 rather than on short-term trends in annual hunter numbers. This change was made to prevent year specificity in harvest rates predicted for a given alternative and to better reflect uncertainty about hunter numbers in the future.
25.7. I D E N T I F Y I N G OPTIMAL REGULATIONS 25.7.1. A n A l g o r i t h m for A d a p t i v e Harvest Management
ai(dtlx t) -= ~ p(ht[dt){bl[Hi(ht[xt)3}, ht
(25.9)
i
= ~ Pi(t) ~ , p(htldt){u[Hi(htlxt)~},
i
ht
where pi(t) is the weight for model i [~,i Pi(t) = 1]. The n o t a t i o n a(dt]xt, Pt) in this expression indicates that the return accruing to decision d t depends on the model weights in Pt as well as system state x t. Similarly, system transitions can accommodate random effects and structural uncertainty. Thus, each of the transition models in Eq. (25.1) inherits stochastic behaviors from random environmental variation and partial controllability of harvests, on the basis of which a state transition probability structure can be derived. Let pi(x t +1 IX t, dt) represent the model-specific probability of transition from state x t to Xt+l, given regulatory decision d t. These transition probabilities can be aggregated across models into an average probability by
dt) = ~ pi(t)pi(xt+llXt, dt). i
(25.10)
The averages in Eqs. (25.9) and (25.10) are used in the Hamilton-Jacobi-Bellman algorithm (see Section 23.3) as if they represent the utilities and transition probabilities for a single model: V*(xt) = maxdt { -~(dtlXt'pt)+
~ ~ pi(t)pi(xt+ltXt'dt)g~'(Xt+l)} Xt+ 1
i "x
TABLE 25.4 Mean Harvest Vulnerability (SE) of Female and Young Mallards a Mean harvest vulnerability (SE) Young females
Young males
Midcontinent
0.748 (0.108)
1.188 (0.138)
1.361 (0.144)
Eastern
0.985 (0.145)
1.320 (0.264)
1.449 (0.211)
a
R(dtlxt, Pt) -~ ~_j pi(t)Ri(dtlxt)
(25.8)
where u[Hi(htlxt)] is the utility accruing to harvest Hi(htlx t) and p(htld t) is the probability of a specific harvest rate conditioned on the regulatory decision d t. The notation Hi(htIx t) indicates that harvest is a modelspecific function of system state x t and harvest rate ht, and the summation in Eq. (25.8) essentially averages harvest utilities over the possible harvest rates corresponding to regulatory decision d t. The model-specific
Adult females
utilities in Eq. (25.8) can be aggregated into an average utility by
F(Xt+I[Xt,
Implementation of a regulatory strategy for sport hunting yields annual benefits for waterfowl harvest, and the goal of management to provide as large a temporal sum of benefits as possible over an extended time frame. Assuming model i, the immediate harvest benefit at time t for state x t and regulatory decision d t is
Mallard population
679
Relative to adult males, based on band recovery data, 1979-1995.
(25.11)
~-a(dtlxt, Pt) + ~ -P(X_.t+llXt,dt)W*(Xt+l)~) dt I, xt+l
= max
(see Section 24.11). Algorithm (25.11) shows that an optimal regulatory strategy can be identified sequentially, in a manner that explicitly accounts for structural uncertainty, environmental variation, and partial controllability of harvests. After some finite number of iterations, continued application of Eq. (25.11) produces the same statespecific decisions, so that the decision structure becomes time independent. The result is a stationary open-loop feedback strategy (see Section 23.3) that identifies an optimal regulatory decision for each combination of population size and number of ponds on the breeding grounds. In 1995 the adaptive regulations process was implemented, and in that year the following steps were taken: 9 The population size and number of ponds were determined from the breeding grounds survey.
680
Chapter 25 Case Study
9 Initial model weights were chosen to be pi(O) = 0.25, thereby weighting each of the four models equally. 9 Average utilities and transition probabilities were identified as in Eqs. (25.9) and (25.10), based on these weights. 9 The average utilities and transition probabilities were used to identify an optimal policy with algorithm (25.11). 9 The appropriate regulatory decision was identified for the population and habitat conditions on the breeding grounds. The strategy identified in 1995 was specific to the set of equal model weights used in Eqs. (25.9)-(25.11). These weights have evolved over time, as information from the breeding grounds survey is used to compare model predictions against population status. Thus, in each succeeding year since 1995, the information from previous years has been incorporated in the process via the following actions: 9 Each spring the population size and number of ponds (xt+1) have been determined from the breeding grounds survey. 9 Population and habitat conditions have been used to identify the transition probabilities pi(Xt+llXt, dt). 9 The transition probabilities have been used to determine the model weights pi(t + 1) with Bayes' Theorem (see Section 24.5 and Appendix A), based on pi(t) from the previous year. 9 Average utilities and transition probabilities have been identified as in Eqs. (25.9) and (25.10), based on the weights in p(t + 1). 9 The average utilities and transition probabilities have been used to identify an optimal regulatory policy with algorithm (25.11). 9 The appropriate regulatory decision has been identified for the population and habitat conditions in xt+ 1. This sequence describes a "passive" adaptive approach to harvest management (see Section 24.11). The process involves incrementing the time index t by 1 each year, and then following the prescribed sequence of actions based on new information from the breeding grounds. This approach to harvest management has been in use since 1995. The sequence can be applied on a continuing basis, thereby allowing the model weights pi(t) to evolve over time. On assumption that the model set contains a model describing system dynamics appropriately, in theory the continued application of the sequence will lead to the convergence of model weights to unity for that model and to zero for the other models.
25.7.2. Optimal Regulatory Prescriptions The AHM process was implemented in 1995 based on equal weights for each model, and each year since then the strategy has been revised as the model weights have evolved. As shown in Tables 25.5-25.9, optimal harvest strategies for the 1995-1998 seasons have shown changes over time as model weights have changed (U.S. Fish and Wildlife Service Office of Migratory Bird Management, unpublished report). The 1999 AHM strategy is based on (1) regulatory alternatives that are unchanged from 1997 and 1998, (2) model weights for 1999, and (3) the dual objectives of maximizing long-term cumulative harvest and achieving a population goal of 8.7 million birds (Table 25.10). This strategy provides optimal regulatory choices for midcontinent mallards assuming that all four flyways use the prescribed regulations. Blank cells in Tables 25.6-25.9 represent combinations of population size and environmental conditions that are insufficient to support an open season, given current regulatory alternatives. In the case of midcontinent mallards, the prescriptions for closed seasons largely are a result of an emphasis on population growth at the expense of hunting opportunity when mallard numbers are below the NAWMP goal. Of course, a decision to close the hunting season always depends on both biological and sociological considerations, recognizing that limited harvests at low population levels might well impact long-term population viability only slightly, if at all. Population dynamics were simulated with the harvest strategy in Table 25.10 with the four population models and current weights, to determine expected performance characteristics. Assuming that regulatory choices adhere to this strategy, the results indicate the annual harvest and breeding population size would average 1.3 million (SE = 0.5) and 8.3 million (SE = 0.9), respectively. Based on a breeding population size of 11.8 million mallards and 3.9 million ponds in Prairie Canada from the breeding grounds survey in 1999, the table indicates that the optimal regulatory choice for midcontinent mallards is the liberal alternative.
25.8. SOME ONGOING ISSUES IN WATERFOWL HARVEST MANAGEMENT 25.8.1. Setting Management Goals Natural resource management is a process of using biological information to predict the consequences of management and sociological information to value those consequences (Lee, 1993). When managers agree
25.8. TABLE 25.5
Some Ongoing Issues in Waterfowl Harvest M a n a g e m e n t
681
Temporal Changes in M o d e l Weights ("Likelihoods") for Alternative M o d e l s of Midcontinent Mallard Population D y n a m i c s a Model weights
Mortality hypothesis
Reproductive hypothesis
1995
1996
1997
1998
1999
Additive mortality
Strong density dependence
0.2500
0.65479
0.53015
0.61311
0.60883
Additive mortality
Weak density dependence
0.2500
0.324514
0.46872
0.38687
0.38416
Compensatory mortality
Strong density dependence
0.2500
0.0006
0.00112
0.0001
0.0001
Compensatory mortality
Weak density dependence
0.2500
0.0001
0.0001
0.0001
0.007
a Four models are included in the model set, each including a different combination of hypotheses about (1) the effects of hunting on annual mortality density, and (2) the magnitude of density dependence in reproduction.
on b o t h goals a n d c o n s e q u e n c e s , m a n a g e m e n t decis i o n s c a n b e b a s e d o n a n e s t a b l i s h e d r o u t i n e of g a t h e r i n g a n d e v a l u a t i n g i n f o r m a t i o n . W h e n t h e r e is d i s a g r e e m e n t a b o u t m a n a g e m e n t g o a l s , a p r o c e s s of n e g o t i a t i o n a m o n g s t a k e h o l d e r s is n e c e s s a r y to d e v e l o p a c c e p t a b l e policy. O n t h e o t h e r h a n d , w h e n m a n a g e m e n t g o a l s a r e b r o a d l y a c c e p t e d b u t t h e r e is d i s a g r e e m e n t o r u n c e r t a i n t y a b o u t t h e i m p a c t s of m a n agement actions, adaptive management can be a useful t o o l for a d d r e s s i n g a n d r e s o l v i n g t h e conflicts. I n effect, a d a p t i v e m a n a g e m e n t a l l o w s m a n a g e r s to a g r e e o n
TABLE 25.6 Optimal Regulatory Choices for Midcontinent Mallards during the 1995 H u n t i n g Season a Ponds b Mallards c
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0
M L L L L L L L L L L L L L
M L L L L L L L L L L L L L
M L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
policy when they do not necessarily agree on the outc o m e s . O n t h e o t h e r h a n d , it is n o t b y itself a u s e f u l a p p r o a c h for a d d r e s s i n g d i s a g r e e m e n t o v e r m a n a g e m e n t goals a n d objectives. It s e e m s o b v i o u s t h a t a n y d e c i s i o n - m a k i n g p r o c e s s w i l l b e l i m i t e d in its e f f e c t i v e n e s s if t h e r e is a m b i g u i t y a b o u t t h e g o a l s o r o b j e c t i v e s of t h e p r o c e s s . Yet, m u c h of t h e h i s t o r y of w a t e r f o w l h a r v e s t m a n a g e m e n t in N o r t h A m e r i c a h a s b e e n m a r k e d b y a l a c k of explicit, u n a m b i g u o u s , a n d a g r e e d - u p o n o b j e c t i v e s ( N i c h o l s et al., 1995a). P e r h a p s b e c a u s e h a r v e s t e d w a t e r f o w l a r e not a commercial commodity, there always has been
TABLE 25.7 Optimal Regulatory Choices for Midcontinent Mallards during the 1996 H u n t i n g Season a Ponds b Mallards c
a This strategy is based on the regulatory alternatives for 1995, equal weights for four alternative models of population dynamics, and the dual objectives of maximizing long-term cumulative harvest and achieving a population goal of 8.7 million. R, Restrictive; M, moderate; and L, liberal. b Estimated number of ponds in Prairie Canada in May, in millions. c Estimated number of midcontinent mallards during May, in millions.
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0
1.5
R R M M L L L L L L L
2.0
R R M L L L L L L L L
2.5
R R M L L L L L L L L
3.0
3.5
4.0
4.5
5.0
5.5
6.0
R M L L L L L L L L L
R R M L L L L L L L L L
R R M L L L L L L L L L
R M L L L L L L L L L L
R R M L L L L L L L L L L
R M L L L L L L L L L L L
R M L L L L L L L L L L L
a This strategy is based on the regulatory alternatives and model weights for 1996 and the dual objectives of maximizing long-term cumulative harvest and achieving a population goal of 8.7 million. R, Restrictive; M, moderate; and L, liberal. b Estimated number of ponds in Prairie Canada in May, in millions. c Estimated number of midcontinent mallards during May, in millions.
682
C h a p t e r 25
TABLE 25.8 O p t i m a l R e g u l a t o r y C h o i c e s for M i d c o n t i n e n t M a l l a r d s d u r i n g the 1997 H u n t i n g S e a s o n a
Case Study TABLE 25.10 O p t i m a l R e g u l a t o r y C h o i c e s for M i d c o n t i n e n t M a l l a r d s d u r i n g the 1999 H u n t i n g S e a s o n a Ponds b
Ponds b Mallards c
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0
1.5
VR R R M M L L L L L
2.0
VR R R M M L L L L L
2.5
VR VR R M M L L L L L L
3.0
VR VR R M M L L L L L L
3.5
VR R R M L L L L L L L
4.0
VR R M M L L L L L L L
4.5
VR R M L L L L L L L L
5.0
VR R M M L L L L L L L L
5.5
VR R M L L L L L L L L L
6.0
VR R M L L L L L L L L L
a This strategy is based on regulatory alternatives and model weights for 1997 and on the dual objectives of maximizing longterm cumulative harvest and achieving a population goal of 8.7 million. VR, Very restrictive; R, restrictive; M, moderate; and L, liberal. bEstimated number of ponds in Prairie Canada in May, in millions. CEstimated number of midcontinent mallards during May, in millions.
TABLE 25.9 O p t i m a l R e g u l a t o r y C h o i c e s for M i d c o n t i n e n t M a l l a r d s d u r i n g the 1998 H u n t i n g S e a s o n a Ponds b Mallards c
4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0
1.5
VR R R M M L L L L L
2.0
VR VR R M M L L L L L L
2.5
VR VR R M M L L L L L L
3.0
VR R R M L L L L L L L
3.5
VR R M M L L L L L L L
4.0
4.5
5.0
5.5
6.0
VR R M L L L L L L L L
VR R M M L L L L L L L L
VR R M L L L L L L L L L
VR R M L L L L L L L L L
VR R M L L L L L L L L L L
a This strategy is based on regulatory alternatives and model weights for 1998 and on the dual objectives of maximizing longterm cumulative harvest and achieving a population goal of 8.7 million. VR, very restrictive; R, restrictive; M, moderate; and L, liberal. bEstimated number of ponds in Prairie Canada in May, in millions. c Estimated number of midcontinent mallards during May, in millions.
Mallards c
<5.0 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 ~9.0
1.5
VR R R M M L
2.0
VR VR R M M L L
2.5
VR VR R M M L L
3.0
VR R R M L L L
3.5
VR R M M L L L
4.0
4.5
5.0
5.5
6.0
VR R M L L L L
VR R M M L L L L
VR R M L L L L L
VR R M L L L L L
VR R M L L L L L L
a This strategy is based on regulatory alternatives unchanged from 1997 and 1998 (Table 25.2), current model weights, and on the dual objectives of maximizing long-term cumulative harvest and achieving a population goal of 8.7 million. VR, very restrictive; R, restrictive; M, moderate; and L, liberal. bEstimated number of ponds in Prairie Canada in May, in millions. c Estimated number of midcontinent mallards during May, in millions.
s o m e r e l u c t a n c e to c o n s i d e r t h e s i z e of t h e h a r v e s t as the m o s t r e l e v a n t p e r f o r m a n c e m e a s u r e . H u n t e r activity a n d s u c c e s s o f t e n a r e d e e m e d to b e m o r e a p p r o p r i a t e m e a s u r e s of p e r f o r m a n c e , n o t w i t h s t a n d i n g t h a t fact t h a t t h e p r o d u c t of t h e s e m e a s u r e s c o n s t i t u t e s h a r v e s t . I n t e r e s t i n g l y , h u m a n - d i m e n s i o n s s t u d i e s ( E n c k et al., 1993; R i n g e l m a n , 1997) i n d i c a t e t h a t h u n t e r p a r t i c i p a t i o n a n d satisfaction are not increased s u b s t a n t i a l l y b y r e g u l a t i o n s t h a t p r o v i d e for t h e m a x i m u m a l l o w a b l e h a r v e s t . M o r e d i s t u r b i n g is e v i d e n c e t h a t m a n a g e r s c o n t i n u e to o v e r e s t i m a t e t h e i m p o r t a n c e of a c h i e v e m e n t - o r i e n t e d f a c t o r s in s e t t i n g h u n t i n g r e g u l a t i o n s , w h i l e i g n o r i n g s o c i a l a n d a e s t h e t i c a s p e c t s of o u t d o o r r e c r e a t i o n ( R i n g e l m a n , 1997).
25.8.2. Monitoring
and
Assessment
A m a j o r a d v a n t a g e of a d a p t i v e h a r v e s t m a n a g e m e n t o v e r m o r e t r a d i t i o n a l a p p r o a c h e s is t h a t it m a k e s explicit t h e r o l e of r e s o u r c e m o n i t o r i n g in t h e f o r m u l a t i o n of h a r v e s t s t r a t e g i e s . A f o r m a l l i n k a g e b e t w e e n m o n i t o r i n g a n d m a n a g e m e n t a l l o w s m a n a g e r s to b e t t e r e v a l u a t e t h e u t i l i t y of v a r i o u s s u r v e y a c t i v i t i e s , t h u s e n s u r i n g t h a t b e n e f i t s a r e c o m m e n s u r a t e w i t h costs. G i v e n t h e c u r r e n t fiscal c l i m a t e in w h i c h t h e u t i l i t y of m a n y s u r v e y p r o g r a m s is b e i n g e x a m i n e d critically, it is i m p o r t a n t t h a t m o n i t o r i n g b e i n t e g r a t e d a l o n g w i t h a s s e s s m e n t a n d decision m a k i n g into an effective prog r a m for h a r v e s t m a n a g e m e n t .
25.8. Some Ongoing Issues in Waterfowl Harvest Management As well developed as they are, current waterfowl monitoring programs are deficient in providing useful information on landscape features and patterns that underlie population dynamics. A key challenge for a more informed approach to harvest regulation is to identify landscape features that are relevant to demographic processes and then to monitor these features regularly at appropriate spatial and temporal scales. Because waterfowl migrate long distances, large-scale, coordinated approaches are necessary to help identify cross-scale effects on population dynamics. Given the cost of such programs, managers increasingly will need to rely on remotely sensed data and geographic information systems (GIS) technologies (Johnson et al., 1996). The lack of information on the spatial and temporal dynamics of waterfowl habitats may account for the dearth of studies regarding the nature of density dependence in population regulation. Even if more habitat information were available, the mobility of waterfowl makes definitions of density seem tenuous at best. Nonetheless, it still is disturbing that a primary theoretical basis for sustainable exploitation (Hilborn et al., 1995) has received inadequate attention in waterfowl studies. On a more hopeful note, recent advances in banding and marking programs and associated dataanalytic methods (see Chapters 16-20) should stimulate construction of more spatially explicit population models that can be linked to the temporal and spatial patterns of landscapes (e.g., Dunning et al., 1992). A necessary component for both passive and active adaptive management is an agreed-upon procedure for updating the weights associated with alternative models of system dynamics. The updating procedure adjusts model weights each year, using weights from the previous year and the change in population size between years (see Section 24.5). This procedure contrasts with the use of long-term information bases such as banding data, for which a comparison of alternative models depends on analysis of historical information extending years into the past. A key difficulty in attempting to identify optimal strategies with the latter approach is that the optimization procedure becomes ever more complex as information accumulates over time, and this complexity can quickly overwhelm available computer resources (Waiters, 1986; Williams, 1996a). A major technical challenge is to develop procedures whereby historical information such as banding data can be folded effectively into the updating of model probabilities (see Section 24.9). Finally, we note that problems attendant to partial observability are yet to be handled in a satisfactory manner in adaptive harvest management. Recall that partial observability occurs when one must character-
683
ize a biological system with sample data representing only a part of that system. Because virtually all realistic examples in population biology utilize sampling to recognize system state, uncertainty as to the actual state of the system at any point in time is (or should be) an explicit element of biological management. In the context of adaptive harvest management, it thus is necessary to ascribe probabilities to the transitions between observed system s t a t e s 9~t and xt+l (rather than the actual states xt and Xt+l). But transitions between observed states are influenced not only by environmental variation and structural uncertainty as described above, but also by stochastic associations between the observed system s t a t e 9~t at each point in time and the corresponding actual state xt (see Section 24.8). This can present formidable technical challenges, especially as concerns the adaptive updating of the distribution p(xtlX t) [see Eq. (24.9)] (Williams and Nichols, 2001). A rigorous treatment is yet to be developed for adaptive waterfowl harvest management in the presence of partial observability.
25.8.3. The "Scaling" of Harvest Management The scale at which individuals are aggregated for management purposes is an arbitrary decision, but one that can strongly influence both the benefits and costs of management. Management systems that account for important sources of ecological variation yield the highest benefits, but also are characterized by relatively high monitoring and assessment costs. Determining the optimal scale for management depends critically on the criteria for evaluating performance and on the description of relevant ecological patterns. The description of ecological patterns in turn depends on the availability of data to explore variation across scales and to elucidate underlying causal mechanisms. Waterfowl managers currently are using the modelbased decision process described above to determine optimal levels of aggregation or "management scale" for regulating waterfowl harvests. The utility of this approach depends on one's ability to model temporal, spatial, and organizational sources of variation, so that the implications of aggregation across these sources of variation can be ascertained. Also needed is an explicit accounting of both management costs and benefits as a function of management scale. When costs are invariant to management scale, managers are driven toward extreme levels of disaggregation, which is accompanied by increasing complexity in harvest regulations (with complexity in turn inducing lower rates of learning, a cost of a different kind with real long-term impacts on management). When benefits are largely
Chapter 25 Case Study
684
invariant to management scale, cost considerations motivate managers to aggregate sources of ecological variation and manage across scales at a coarser level of ecological organization.
25.9. D I S C U S S I O N
The term "adaptive resource management" was coined in the 1970s by biologists who were concerned with the intrinsic uncertainties in environmental management (Holling, 1978). However, the concept of "learning by doing" (Walters and Holling, 1990) has been espoused for many years in many forms (e.g., see Beverton and Holt, 1956), often under the rubric of "management by experimentation" (MacNab, 1983) or "probing" (Waiters, 1986). In fact, waterfowl biologists often have advocated experimenting with regulations to help resolve uncertainty about the effect of harvest on annual survivorship (e.g., Hickey, 1955; Anderson and Burnham, 1976; Anderson et al., 1987; Conroy and Krementz, 1990). These recommendations have largely been ignored, not because the reduction of uncertainty is considered unimportant, but instead because of the short-term risks to harvest opportunity that experimentation might entail. In effect, implementation of a regulatory experiment means temporarily replacing traditional harvest objectives with an objective to learn (i.e., to discriminate among alternative hypotheses of system dynamics). As a consequence there is a potential loss of harvest opportunity (and increase in ecological risk) with experimentally based regulations. In contrast, the focus of adaptive harvest management as described above is on neither learning rates nor short-term harvest, but instead on regulations that provide an optimal balance of short-term and longterm harvest and conservation benefits. The theory, computing algorithms and software necessary to compute optimal, actively adaptive strategies have become available only recently (Williams, 1996a,b; Lubow, 1997). It is not yet clear how management actions for an actively adaptive strategy may differ from those in passive adaptation, but in either case, adaptive optimization presents opportunities to improve management performance (and political acceptance) above and beyond that of a classic experimental approach.
Although the ultimate fate of an adaptive approach to harvest management as described in this chapter remains uncertain, its ancillary benefits are becoming increasingly apparent. The process has provided an effective link between data and decisions by integrating monitoring, assessment, and decision making in a coherent framework. The explicitness demanded by an adaptive approach has helped focus attention on important biological and social issues and has guaranteed greater accountability in management decisions. Formal acknowledgment of management uncertainties, combined with more rigorous and focused assessments, has fostered a greater willingness among managers to challenge dogma and traditional beliefs. Nevertheless, the long-term use of a formal adaptive approach for regulating waterfowl harvests in the United States is by no means assured. As Walters (1997) suggests, the greatest challenges to the long-term viability of adaptive management programs likely are institutional rather than technical. For example, it is not yet clear that waterfowl managers will accept the limits to performance imposed by uncertainty. Nor is it clear that they are prepared to invest the same energy and resources in collecting information on resource users as they have invested on the resource, even if a failure to do so means that management objectives remain illdefined and unmet. Ultimately, the success of adaptive harvest management depends on a general agreement among stakeholders about how to value harvest benefits and how those benefits should be shared. Revisions to the Canadian Constitution, and subsequent efforts to amend treaties to legalize spring hunting for subsistence, portend important changes to the structure of the decision-making process. In effect, more stakeholders, including aboriginal peoples and provincial governments, likely will demand a stronger role in the decision-making process. Managers in both countries must address the concerns of these stakeholders, while ensuring acceptable allocations of the harvest. It is these unresolved value judgments, and the lack of an effective structure for organizing negotiations about them, that present the greatest threat to a continuation of a science-based adaptive approach to waterfowl harvest management.
A P P E N D I X
A Conditional Probability and Bayes' Theorem
An event-based approach to probability begins with the notion of an uncertain outcome of some action and an outcome space or sample space S = {O1, ..., On} of all possible outcomes. For example, for a flip of a coin, an outcome is the occurrence of either head or tail, and the sample space is S = {head, tail}. In the d r a w of a single card from a deck, an outcome is the drawing of any one of 52 cards from one of four suits, and the corresponding sample space is S = {13 clubs, 13 spades, 13 hearts, 13 diamonds}. An event E is defined by an attribute shared a m o n g some (but not necessarily all) outcomes. For example, an event might be the drawing of a d i a m o n d from a deck of cards, which can occur for 13 of the possible outcomes of a draw. The complement E of an event E is the portion of the sample space that does not contain the event. For example, for a coin flip E might be the event that the coin is heads, with E the event that it is tails. The complement of the d r a w of a d i a m o n d is a d r a w of any of the other 39 cards that are hearts, spades, and clubs. One can define probability in terms of the frequency of occurrence of events. Thus, the probability of event E, P(E), is the proportion of times the event occurs in a large n u m b e r of r a n d o m trials. If all outcomes in the sample space S are equally likely, then P(E) =
For the above example, the event E is the drawing of a diamond, E is the d r a w i n g of a n o n d i a m o n d , and therefore P(E) = 1 - P(E) = 1 - 13/52 = 3/4. We note that probability also can be defined in terms of the degree of belief in an outcome. Under this view of probability, P(E) = 1 if the event E is held to occur with certainty; P(E) = 0 if E is held to occur with certainty (that is, it is certain that E does not occur); and P(E) = 0.5 denotes equal belief (or uncertainty) in either E or E. As evidence accumulates, one's belief in E changes, in that the degree of certainty of its occurrence increases or decreases d e p e n d i n g on whether the evidence supports E or E. Conditional probability can be defined in terms of the joint occurrence of two events. Thus, the conditional probability of E 1 given E 2 is the probability that event E 1 occurs given the occurrence of event E2. The mathematical definition for conditional probability is m
P(EIIE 2) =
P(E1 n E2)
P(E 2)
where P(E1 n E2) is the probability that both E 1 and E 2 occur. This relationship can be rewritten as P(E1 n E2) - P(E2)P(E1]E2),
n u m b e r of outcomes in E e S total n u m b e r of outcomes in S
(A.1)
which expresses the probability of the joint occurrence of two events as the unconditional probability of one event, times the conditional probability of the second event. In fact, the probability of joint occurrence can be expressed in terms of the conditional probability of either event. Thus, the probability of E 1 and E 2 is given either by Eq. (A.1) or by
For example, in a large n u m b e r of draws from a deck of cards, the probability of d r a w i n g a d i a m o n d is P(E) = 13/52 = 1/4. The event E and its complement E are by definition mutually exclusive and exhaustive, so that m
P(E) = 1 - P(E).
P(E1 n E2) = P(E1)P(E2]E1),
685
(A.2)
686
Appendix A Conditional Probability
where
If we define the odds of event E as the ratio of the probabilities of the event and its complement, i.e.,
P(E2IE 1) =
P(E1 n E2)
O(E) =
P(E1)
The above relationship leads directly to a definition of independence: events E 1 and E2 are independent if and only if
P(E) P(-E)"
then Eq. (A.4) can be rewritten in odds form as
O(E2]E1) -
P(E2]E1) P(E2]E1)
P(E2)P(E1]E2) ./P(-E2)P(EllE2) P(E1) / P(E1)
P(E1 n E2) = P(E1)P(E2), that is, P(E2IE 1) = P(E 2) and P(E1]E 2) = P(E1). In words, E 1 and E2 are independent if the occurrence of either event provides no information about the probable occurrence of the other. Equations (A.1) and (A.2) provide the basis for Bayes' Theorem. Thus,
P(EllE2) = O(E2)~(EII-E2 ) 9 Thus, the conditional odds O(E21E1) of E2 is given by the unconditional odds O(E2), multiplied by the factor
P(EIIE2) /P(EIIE2). The odds ratio can be especially informative in a context in which E2 and E2 represent hypotheses H 0 and H a, and E1 represents a sample outcome (X). Then
P(E1 n E2) = P(E1)P(E2]E1) and
P(X[Ho) O(H0lX) - O(Ho)p(XlHa), P(E1 n E2) = P(E2)P(E1]E2),
so that
P(E1)P(E2IE1) = P(E2)P(EIIE2).
(A.3)
Rewriting Eq. (A.3), we have
P(E2[E 1) =
P(E2)P(EI[E2) P(E1)
(A.4)
and
P(EI[E 2) = P(E1)P(E2IE1).
(A.5)
P(E 2) Expressions (A.4) and (A.5) often are applied to situations in which the events E 1 and E2 occur sequentially in time. Thus, P(E 2) in Eq. (A.4) often is called the prior probability of E2 (i.e., it is not conditioned on E 1 having occurred) and P(E2]E 1) is called the posterior probability of E2 (i.e., it is conditioned on E 1 having occurred). Analogously, P(E 1) in Eq. (A.5) is called the prior probability of E 1 (i.e., it is not conditioned on E2 having occurred) and P(EI[E 2) is called the posterior probability of E 1 (i.e., it is conditioned on E2 having occurred). One particularly useful application of Bayes' Theorem involves an event and its complement. In this case Eq. (A.4) is expressed as
P(E2IE1) =
P(-E2)P(E1]-E2) P(E1)
(A.6)
in which case O(H o) is the prior odds (before data collection) that the hypothesis is true, O(H0]X) is the posterior odds (following data co|lection), and the quotient is the likelihood ratio or Bayesian updating factor. Equation (A.6) highlights the operational difference between Bayesian and maximum likelihood approaches to hypothesis investigation. As described in Chapter 4, a likelihood approach utilizes the available data to determine the ratio of likelihood functions L(Ho]X) = P(X]H o) and L(HaIX) = P(XIHa), with each evaluated at parameter values maximizing the respective likelihood. On assumption that H 0 is the operative hypothesis and H 0 is nested within H a, twice the natural logarithm of this ratio is approximately distributed as chi square, with degrees of freedom given by the difference between the numbers of parameters in H 0 and H a. This chi-square distribution can be used to test H 0 vs. H a. On the other hand, a Bayesian approach utilizes the sequential collection of data to specify transitions from prior probabilities P(H 0) and P(H a) = 1 - P(H 0) to the posterior probabilities P(Ho]X) and P(HaIX) = 1 P(Ho[X). This process is iterative, in that the posterior probabilities resulting from data collection in one sampling period become the prior probabilities for the next period. In this way, evidence for the hypotheses accumulates sequentially over the course of investigation. The weight of evidence for both hypotheses at any point in the investigation is reflected in the posterior odds, with large odds reflecting stronger support for H 0 than for H a.
A P P E N D I X
B Matrix Algebra
Matrix algebra is appropriate for the characterization of systems involving two or more variables and is useful for finding solutions of a broad array of multivariate problems. For example, matrix algebra sometimes can be used to describe and solve systems of linear equations, to describe dynamic systems of differential or difference equations, to characterize statistical dispersion matrices, optimization problems, and so on. In this appendix, we describe some features and operations of matrix algebra, as they pertain to the modeling and estimation of animal populations. We restrict our focus here to two-dimensional matrices.
single row. Matrices with the latter form are sometimes called row vectors. Another special case consists of square matrices, with the same n u m b e r of rows and columns. Because the numbers of rows and columns are identical, m • m matrices often are described as square matrices of order m. The transpose A' of a matrix _Ais obtained by switching positions of the rows and columns of A. For the 2 • 3 example above, [1 _A' =
= B.1. D E F I N I T I O N S
1
[121 2 -4.
-1
1
Note that if the order of a matrix is m • n, then the order of its transpose is n • m. In particular, the transpose of a square matrix of order m is again a square matrix of order m, and the transpose of a column vector (order m • 1) is a row vector (order 1 • m). A special class of square matrices consists of symmetric matrices, wherein a matrix is identical to its transpose: A' = A. For example, the matrix
In simplest terms, a matrix is a rectangular array of numbers, such that both the numerical values and their positions in the array are definitive of it. Thus, if m and n are the numbers of rows and columns in the array, then the matrix is specified by the element aij in the jth position of the ith row, where i = 1, ..., m and j = 1.... , n. For example,
A=
2-1],
2 -4
I1 2 -1] 2-4 1
[ 1 2 - 3 ] 2 4 -5 -3 -5 9
is a 2 • 3 matrix consisting of numbers in two rows of three columns. The n u m b e r of rows and columns of a matrix specify the order of the matrix. For example, a matrix with two rows and three columns is said to be of order 2 • 3. A special case is an m • 1 matrix, consisting of multiple rows and a single column. Such matrices are called vectors or column vectors, to distinguish them from matrices with multiple columns and a
is symmetric, because switching rows and columns leaves the matrix unchanged. It is easy to see that matrix s y m m e t r y requires that aij = aji, i.e., the pattern of upper off-diagonal elements must be reflected in the pattern of lower off-diagonal elements. In the example above, a12 = a21 = 2, a23 = a32 --= - - 5 , and a13 = a31 =
687
--3.
688
Appendix B Matrix Algebra B.2. M A T R I X A D D I T I O N AND MULTIPLICATION
Under certain conformability conditions on their orders, matrices can be a d d e d and multiplied.
the rows of A and d o w n the columns of B, for matrices A and B that are conformable for multiplication. Let A and B be matrices of orders m • s and s • n, respectively. The matrix result C = A B of multiplication of A and B is given by the elements
B.2.1. Matrix Addition The addition A + B of two matrices A and B consists of element-by-element addition aij + bij for all values i = 1, ..., m and j = 1, ..., n. Clearly, this definition requires that there are corresponding elements in both matrices, i.e., that the matrices are of the same row and column orders. If two matrices possess the same row and column orders, they are said to be conformable for addition. For example, the sum of the 2 • 3 matrices 1 2 -11 2-4 1
A= and B =
I
-1 -4 3 -1
21 -5
is given by
Cq- ~
aikbkj,
k=l
for i = 1, ..., m and j = 1, ..., n. Note that the computing form requires that the n u m b e r of columns in A _ and rows in B be identical. Note also that the resulting matrix is of order m • n, which need not conform to the order of either A or B. Matrix multiplication does not share with matrix addition the property of commutativity. Indeed, matrices that are conformable for the product A B m a y not be conformable for the product B A; for example, the product A B is defined for matrices of orders 2 • 3 and 3 • 4, but the product B A is not. Thus, matrix multiplication is not commutative. The m • m m a t r i x / , consisting of ls in its diagonal positions and 0s elsewhere, constitutes a multiplicative identity for matrices that are conformable for multiplication. Thus, if A and I are m • n and n • n, respectively, the product A I is simply m
A+B=
(1 - 1) (2 + 3)
(2-4) ( - 4 - 1)
( - 1 + 2) (1 - 5 )
m
m
5
2 5 - 4i"l
all
a12
...
aln
a21
a22
...
a2n
9
On the other hand, a 2 • 3 matrix A and a 2 • 2 matrix B are not conformable for addition, because the matrices do not possess corresponding elements in all positions. Because addition of real numbers is a commutative process (that is, a + b = b + a), it follows that matrix addition is also commutative: A + B = B + A. Also, the m • n matrix 0, consisting of zeros in all positions, constitutes an m • n additive identity, because A + 0 = 0 + A = A for all m • n matrices A.
B.2.2. Scalar Multiplication Multiplication of a matrix by a scalar consists of element-by-element scalar multiplication. Thus, the elements of kA are given by kaij for all i = 1..... m and j = 1.... , n. It follows that the matrix cA + dB has elements c(aij) + d(bij) for all i = 1, ..., m and j=l,...,n.
B.2.3. Matrix Multiplication The multiplication A B of two matrices consists of the multiplication and s u m m a t i o n of elements across
9
9
am1
o
o
am2
1 0 0
0
0
...
0
1
0
...
0
0
1
...
0
0
0
....
9
...
amn
all
a12
...
aln
a21
a22
...
a2n
9
9
am1
am2
1
o
...
amn
As before, this product is not commutative. However, the product I A also reproduces A for the identity matrix of order m. Note that the order m identity matrix I is both a right and left identity for square matrices of order m. Some matrix properties follow directly from the definitions of matrix symmetry, transpose, and matrix product. Thus, the transpose of a matrix product is the product of transpose matrices: (A B)' = A'B'. The transpose of the scaled matrix sum is the sum of the scaled transpose matrices: (aA . +. bB)'. = . aA' + bB'. Finally, the products A ' A and A A' are symmetric.
w
B.4. Inverse of a Matrix B.3. M A T R I X
DETERMINANTS
Associated with any square matrix A is a real n u m ber IA] called the determinant of A. The determinant characterizes the internal structure of a matrix by w a y of a series of multiplications and additions of its elements. Thus, the matrix A = [all --
La21
a12] a22]
has a determinant of ]A] = alia22 -- a12a21. Note that the determinant for the 2 • 2 case is simply the product of diagonal elements minus the product of off-diagonal elements. The determinant of a 3 • 3 matrix has the more complicated form 3
I
1=
-
1 ) i+jaijlA,jl,
i=1
for any value of j, or 3 [a[ =
~.~(-1)i+JaijJaijJ j=l
for any value of i. The value JAij [ is the determinant of the 2 • 2 matrix Aq obtained by eliminating row i and column j from _A.Thus, the element called the minor of element aij, is multiplied by either I or - 1 according t o ( - - 1 ) i+j t o produce the "cofactor" ( - - 1 ) i+j JAij[ o f aij. The cofactor in turn is scaled by the corresponding element aij, and the products are s u m m e d across all values of either the index i or j. C o m p u t i n g the determinant of a general square matrix of order m follows the same procedure as with a 3 • 3 matrix, except that the cofactors JAijl are based on ( m - 1) • ( m - 1)matrices Aq and the s u m m a t i o n involves more terms. Thus, the element JAij [ is scaled by aij, multiplied by either 1 or - 1 , d e p e n d i n g on the position of aij in the matrix, and the products are s u m m e d across all values of either index i or j:
I&l,
JAJ = ~ ( -
9 The determinant of a product of matrices is the product of determinants; i.e., ]A B] = ]AJ JB]. 9 The determinant of an n-dimensional matrix that is scaled by a is the determinant of the matrix scaled by a"; i.e., JaAI = anlAJ. 9 The determinant of a matrix and its transpose are identical: JA'J = IAI. 9 If any row (or any column) of a square matrix consists of 0s, the determinant of the matrix is 0. 9 If any two rows (or any two columns) of a square matrix are identical, the determinant of the matrix is 0.
B.4. I N V E R S E
1)i+Jaij[aij
OF A MATRIX
Let A be a square matrix of order m with nonzero determinant: IA] 4: 0. It can be shown that there exists a unique square matrix _B of order m such that A B = B A = I. This matrix, expressed as A -1, is determined by a three-step process: 1. The (-1) 2. The 3. The
elements aij of A are replaced by the cofactors
i+j JA,j I.
matrix of cofactors is transposed. transposed matrix is multiplied by JA]-1.
Some properties of matrix inverses are as follows: 9 The inverse of A exists if and only if IA_I 0. This follows immediately from the computing formula for A -1, which requires multiplication by IA1-1. 9 The inverse of a product of matrices is the product of inverses: ( A B ) - 1 = B - 1 A -1. 9 The inverse of a transpose matrix is the transpose of the inverse matrix: (A') -1 = (A-l) '. 9 The inverse of an inverse of A is A: (A-l) -1 - A. 9 The inverse of I is again I, because H = I. 9 The inverse of a diagonal matrix with diagonal elements aii , i = 1, ..., m, is again diagonal, with diagonal elements that are the reciprocals 1/aii , i = 1, ..., m. This is seen by -all
m
689
0
j
0
...
0
a22
...
0
0
...
-1/all 0
0
...
0
1/a22
...
0
0
...
i=1
for any value of j, or 9
m
JA] = ~, ( - 1)i+Jaij[aij j
,=
0
~
9
o
atom
o
0 m
m
,.
j=l
for any value of i. If JAqJ = 0, the matrix _A is said to be singular. Conversely, if JAq] 4: 0, the matrix is said to be nonsingular. Nonsingular matrices possess a matrix inverse, as discussed below. Other properties of determinants are as follows:
..,
1
0
...
0
0
1
...
0
9
o
9
0 m
. 9149
9
1 m
o
1/amm
-
690
Appendix B Matrix Algebra
9 The inverse of the 2 • 2 matrix A with elements (all , a12 , a21, a22) is given by switching positions for the two diagonal elements, multiplying off-diagonals by - 1 , and scaling the resultant values by the inverse of
Ia] = (alia22- a12a21): -1 Jail a12] = Ial-1 [ a22 -a12]. La21 a22J -a21 all That this matrix is in fact the inverse of A is seen by simple multiplication:
A-1A = ]AI-I[ a22-a12] [all a12]
. . . .
a21
0
0
9 The trace of a s u m of conformable square matrices is the sum of the respective traces: tr(A + B) = tr(A) + tr (B). 9 The trace of a scaled matrix is the trace of the orginal matrix, multiplied by the scaling factor: tr(kA) = ktr(A). 9 The trace of a direct product of conformable square matrices is the product of traces: tr(A B) = tr(A) tr(B). 9 The trace is invariant to the order of multiplication: tr(A B) -- tr(B A) for conformable matrices A and B. 9 If P is an orthonormal matrix, tr(A) - tr(P A P'). This follows directly from the fact that the trace is invariant to the order of multiplication: m
all La21 a22J
= ,al_l[alla22-a12a21
for the n • n matrix A. Several properties of the trace are noteworthy:
]
a11a22 --a12a21
=I.
m
Note that if the determinant of A is near zero, the elements of A-1, because they are scaled by [A[-1, tend to be large in magnitude. Under such circumstances A -1 is said to be unstable or ill-conditioned, in that small r o u n d i n g errors in computation of the determinant and cofactors can have large consequences in the c o m p u t e d values of elements in the inverse.
B.5. O R T H O G O N A L
tr(PAP')
= tr (A).
B.7. E I G E N V E C T O R S AND EIGENVALUES
O R T H O N O R M A L MATRICES D
tr(AP'P)
= tr(A/)
AND
Two n-dimensional vectors Pl and P2 are said to be orthogonal if their inner product p lP2 is 0. This is analogous to the geometric notion of orthogonality, whereby vectors oriented at right angles to each other are said to be orthogonal. By extension, a matrix _P with columns consisting of orthogonal vectors is said to be an orthogonal matrix. Because products of all pairs of column vectors Pi and pj (for i ~ j) are 0, P _ has the property that its transpose product is diagonal: P ' P = D, with D an n • n diagonal matrix. If the columns of P also are of unit length (i.e., p;p_i = 1, i = 1, ..., n), then the matrix P is said to be orthonormal. In this case P ' P = I, from which it follows that p-1 = p, and therefore P P' = I.
-
We introduce the concepts of eigenvectors and eigenvalues by focusing on square matrices. Thus, v is said to be an eigenvector (or sometimes a latent or characteristic vector) of the square matrix A if v satisfies the matrix equation Av=
Xv
for some scalar value )~. The value )~ is k n o w n as an eigenvalue (or sometimes a latent or characteristic value) corresponding to the eigenvector v. The eigenvector equation can be expressed as Av-
Xv=0
or B.6. TRACE OF A MATRIX
(A - )~/)v = 0, m
In addition to the determinant, another value that can be used to characterize square matrices is the trace, which is simply the sum of diagonal elements:
from which it can be seen that a nontrivial solution requires
n
tr(A) = ~
i=1
aii
tA-
x[I = 0
B.7. Eigenvectors and Eigenvalues This determinantal equation is known as the characteristic equation for matrix A. If A is of order n, the characteristic equation represents an nth-degree polynomial in X, with solutions hi, ..., hn. For each eigenvalue ~ki there corresponds an eigenvector v i that solves a v i = ~kiVi. These solutions can be arranged in matrix format by A ( v l , . . . , Vn) = ( ) t l V l ,
... ,
691
It follows immediately that tr(A -1)
-- ~ i
~ki-1"
9 The determinant of A is the product of the eigenvalues of A:
IAI--I__V
V'I
: Ivl I• Iv'l
KnVn)
= Ix__lI__v'l I_v[
or
=HNi. AV=
i
VK,
where V is an n • n matrix with eigenvectors for its columns and X is a diagonal matrix with the eigenvalues of A for diagonal elements. It can be shown that the eigenvectors of a matrix are orthogonal; i.e., V iV j = 0 for i 4= j. If they also are standardized to unit length (i.e., v ; v = 1), the eigenvector matrix V is orthonormal. Postmultiplication of the eigenvector equation by V' thus allows us to express A as
It follows that IA-11 = I I i ~k- 1 . In addition, the matrix A has an inverse if and only if all the eigenvalues of A are nonzero. This follows from the requirement that
w
A = A(VV') = (AV) V'
= VKV'.
This form is known as the spectral decomposition or singular value decomposition of A. Listed below are some useful properties of eigenvectors and eigenvalues: 9 If the matrix A is symmetric, all eigenvalues and eigenvectors are real. 9 A and A -1 have the same eigenvectors, and their eigenvalues are reciprocals: A - 1 = (V~K V ' ) - I = ( V ' ) - 1 ~k - 1 W - 1 -- V~K - 1 V ' , where
~-1 is a diagonal matrix with diagonal elements
X71. 9 The trace of A is simply the sum of the eigenvalues of A: tr(A) = tr(VX V') = tr(h V ' V )
IAI = Iii
)ki 4: 0.
9 If the eigenvalues of a square matrix A are Xl,..., h n, then the eigenvalues of powers of A_ are simply powers of the eigenvalues of A; i.e., the eigenvalues of A k are hik, ..., h~. This is shown by simply expressing A in terms of its spectral decomposition, multiplying A by itself k times, and using the fact that V is orthonormal. It follows that tr(A k) = ~ i ~kk, and ~_A__kl= I I i ~kk. 9 If the eigenvalues of _A are distinct, the corresponding eigenvectors are unique (up to a scaling factor). If k eigenvalues ~ki a r e equal, the identity of the corresponding eigenvectors is arbitrary (within certain limits). Thus, any set of k vectors satisfying the eigenvector equation with ~'i c a n be used to span a vector subspace, and any vector in that subspace is an eigenvector of A. 9 If A is square, nonsingular, and irreducible (see Section 10.3.2), with nonnegative elements, then the lead eigenvalue of A is real and positive, and the corresponding eigenvector has real positive elements (Gantmacher, 1959). Most age-classified population projection matrices (see Section 8.4.2) possess these attributes and thus are characterized by a nonnegative stable age distribution and positive population growth rate. 9 It sometimes is useful to describe A in terms of its left eigenvectors. Thus, the row vector w is a left eigenvector of A if w A = ~KLW.
The properties listed above for right eigenvectors also apply to left eigenvectors. Note that w A v can be written as w ( A v ) = (wv) X,
= tr(h) or as = E )ti.
( w A ) v = ~.L(WV).
692
Appendix B Matrix Algebra
Therefore ~kL = )k, demonstrating that associated left and right eigenvectors of A have the same eigenvalue.
B.8. LINEAR A N D Q U A D R A T I C FORMS ,
If A is an m • m matrix and x is an m • 1 vector, the vector elements of the product A x are said to be linear forms in x. Linearity is conferred by virtue of the fact that the vector product only contains linear terms in x:_ the elements of A x can be expressed as fi(x) = ~,j aijx j. On the other hand, the scalar product x ' A x is a quadratic form in _x,in that the product is composed of quadratic terms x 2 and XiXj: f(x) = x ' A x = ~,i,jaijxiX j. If linear and quadratic forms are combined into a form that consists of both linear and quadratic terms, the form is still referred to as quadratic. A standard expression combining both linear and quadratic terms is the quadratic form f(x) = (x - c)' A ( x - c). If A = ~ and c = ~, we have the familiar expression (x - ~)' ~(x - ~) that is found in the multivariate normal probability density function with mean I~ and dispersion ~.
9 A similar argument can be used to show that all eigenvalues of a positive-semidefinite matrix are nonnegative: )~i --- 0, i = 1, ..., m. 9 If A is positive-definite (positive-semidefinite), then A -1 is also positive-definite (positive-semidefinite). This follows directly from the fact that eigenvalues for A -1 are reciprocals of the eigenvalues for A. By analogy, an m x m matrix A is said to be negativedefinite if the quadratic form x ' A x is strictly negative for all nonzero real vectors x and negative-semidefinite if x ' A x is nonpositive for all nonzero real vectors x. The eigenvalues of a negative-definite matrix are all negative, and those of a negative-semidefinite matrix are nonpositive. The inverse of a negative-definite (negative-semidefintie) matrix is also negative-definite (negative-semidefinite). m
B.10. MATRIX DIFFERENTIATION Differentiation of vectors and matrices is a straightforward extension of univariate differentiation. Thus the derivative of a vector y(x)' = [yl(x), y2(x), ..., yk(x)] is defined by differentiation of its components:
d y / d x = (dy 1/dx, ..., dyk/dX).
B.9. POSITIVE-DEFINITE A N D SEMIDEFINITE MATRICES
Partial differentiation of a vector y(x)' = [yl(x), y2(x), 9.., yk(X)] is defined similarly:
Oy/Ox i = (Oyl/Oxi, ..., Oyk/OXi). An m • m matrix _A is said to be positive-definite if the quadratic form x ' A x is strictly positive for all nonzero real vectors x. A is said to be positive-semidefinite if x ' A x is nonnegative for all nonzero real vectors x. Some properties of positive-definite and positive-semidefinite matrices are as follows: 9 The matrix A is positive-definite if and only if its eigenvalues are positive. This follows from
Ordinary and partial differentiations of matrices also are defined in terms of the derivatives of the matrix elements. Thus, if Y is an m • n matrix with elements yq(x), the derivative of Y is given by
dY/dt
If the elements of Y are described in terms of a vector of variables x, then partial differentiation of Y defined by
3Y/Ox k = [3yq/OXk].
x ' A x = x' ( V k V ' ) x =
(V'x)X (V'x)
= y')~y = ~_j ~kiy 2.
i Thus the requirement that x ' A x va 0 for all nonzero x is tantamount to a requirement that ~'i ~kiY2 =/= 0 for all nonzero y. Because y2 > 0 for all values of Yi, the latter requirement is met only on condition that ~ki ~ 0, i = 1, ..., m.
= [dyij/dt].
Finally, differentiation of the elements in the n-dimensional vector y by elements in the m-dimensional vector x can be expressed in matrix form as =
ax
roy/] Lax/i 3yl / c~X1
...
3y 1/ (~Xm
9
LOYn)OXm
..i
~
Oyn)OXmJ
A P P E N D I X
C Differential Equations
In this appendix we describe procedures for analysis of some differential equations that arise in population ecology. These equations express differential change in population size over a continuous time frame, with population trajectories that are given by their solution. The procedures discussed below apply to multiple populations or to a single population with multiple cohorts (or both). Without loss of generality, we refer to N(t) as a vector of populations, recognizing that N(t) also may represent a vector of population cohorts. More detailed treatments of differential equations can be found in a large number of references, such as Tenenbaum and Pollard (1985), Coddington and Landin (1989), and Rainville et al. (1996). Consider a set of populations (or population cohorts) N(t) that experience change through time according to the equation dN/dt
is nonlinear. Under some quite general conditions, the growth function for a single population can be written as a Taylor series expansion oo
F(N) =
so that the models can be expressed as (possibly infinite) polynomials. For example, the exponential model in Section 8.1 requires only a linear term for an exact representation of its growth function, the logistic model in Section 8.2 requires linear and quadratic terms, and the Allee effect F ( N ) = alN + a2N2 + a3N3
expressing depensatory population change can be modeled with linear, quadratic, and cubic effects. The growth functions for most population models are complicated mathematical expressions, polynomial approximations for which may require many terms. For example, growth functions for the Gompertz model of human mortality and the Ricker and the Beverton-Holt models from fisheries biology involve exponential terms, and therefore require infinite series of polynomial terms for an exact representation. If F(N) contains only terms that are functions of N, the equation is said to be homogeneous; otherwise, it is nonhomogeneous. Thus, d N / d t = r N is a homogeneous differential equation, whereas
= F(N),
in which F(N) expresses differential change in population status at any particular point in time. This formulation represents population change with a first-order differential equation, i.e., a differential equation that includes only first derivatives. It is assumed here that the growth function F(N) is well behaved, in that the derivatives of F _ with respect to the population units in _N exist. If F(N) contains no terms of degree higher than 1, then the equation is said to be linear; otherwise, it is nonlinear. For example, dN/dt
= rN
dN/dt
is a linear differential equation, whereas the logistic equation dN/dt
= rN(1 -
akN k, k=0
= rN + c
is nonhomogeneous. Homogeneity obviates population change in the absence of individuals in the population, e.g., spontaneous generation (or independent
N/K)
693
694
Appendix C Differential Equations
immigration). A homogeneous growth function F(N) can be expressed by a Taylor series: F(N) = a l N + a2 N 2 + a3 N 3 + ... = N(a I + a2N + a3N2 + ...
One need not use the artifice of choosing an appropriate function to derive this solution, because Eq. (C.1) is simple enough that it can be solved by straightforward integration. Thus, Eq. (C.1) can be rewritten as d N / N = rdt, and integration yields
= NG(N).
f dN
The function G(N) is the instantaneous rate of growth or per capita rate of growth, which varies with N for all polynomial growth functions except the constant growth rate of the simple exponential. Finally, a constant coefficient differential equation contains coefficients that are invariant through time. For example, the logistic equation above is a constant coefficient differential equation, but replacement of r with an oscillatory term such as r(t) 4: r sin(t/4) produces a differential equation
= In(N)
-frdt =rt +c,
with the result that N(t) = kert,
where k = ec. Substituting (0, N 0) for [t, N(t)] in this equation produces k = N 0, so that N(t) = N o eFt.
d N / d t = r(t)N[1 - N / K ] , Example
with a time dependent growth coefficient r(t).
Consider a logistic population as in Section 8.2, with population dynamics given by
C.1. FIRST-ORDER LINEAR H O M O G E N E O U S EQUATIONS
d N / d t = rN(1 - N / K )
C.1.1. Population Dynamics for One Species
and initial population size N(0) = N 0. A closed form for the population trajectory can be solved by rewriting Eq. (C.2) as
Consider first a single population with no cohort structure, for which population dynamics are characterized by a single linear homogeneous differential equation. Several methods are available for solving this equation, including the use of Taylor series expansion, numerical integration, and mathematical analysis.
dN =rdt N(1 - N / K )
and recognizing that
1 N(1 - N / K )
Example
Consider the exponential model described in Section 8.2, d N / d t = rN,
=
1
t
N
1 K-
N
Then
(C.1)
T N(I
-
dN N/K)=
with initial population size N(0) = N 0. A simple approach is to assume a solution of the form N(t) = ke ~t, so that
dN dN f --N + f K - N
=In
(N) N-K
+C
= rt,
d N / d t = Mke xt)
so that
= XN.
Comparing this expression with Eq. (C.1), we have X = r and N(t) = ke rt, with the constant k determined by
N N-K
-- ce rt"
Substituting the initial condition N(0) = N O into this expression yields c = N o / ( K - No), so that
N(O) = ke ~
= N O. Thus, a complete solution for the model is N(t)
(C.2)
= N o ert.
N K-N
o
K-
rt
No e "
C.1. First-Order Linear Homogeneous Equations After some algebra, the population trajectory can be rewritten as N(t) = 1 + Ce -r(t - to),
with C = K/N o - 1. Differentiation of this equation demonstrates that it satisfies Eq. (C.2).
C.1.2. Population Dynamics for Two Species The mathematical situation is somewhat more complicated with two populations. Let the population dynamics again be specified by linear homogeneous differential equations dN1/dt = a11N1 + a12N2,
(C.3)
dN2/dt = a 2 1 N 1 + a 2 2 N 2.
Two approaches are available for the solution of this system. The first approach utilizes the fact that these equations can be combined into a single second-order differential equation, and the second approach utilizes matrix theory. C.1.2.1. Second-Order Equations
A pair of first-order differential equations can be combined into a single second-order differential equation, which then can be solved by straightforward algebraic procedures. For example, N 2 can be eliminated from the pair of equations in Eq. (C.3) by a second differentiation of the transition equation for NI: d2Nl= dt 2
a dN1 dN2 11---~- + a12 dt a
dN1
--- 11--~- q- a12(a21N1 q- a22N2)
11---~ q- a12a21N1 + a12a22 - - ~
- a11N1
)/
a12.
Thus, elimination of N 2 in Eq. (C.3) results in the second-order equation d2N1/dt 2 -
otdN 1/dt
+ ~ N 1 = 0,
(C.4)
with ot = all -t- a22 and [3 = alia22 -- a12a21. It is easily shown that the elimination of N 1 leads to a differential equation in N 2 with the same coefficients. To solve the system of differential equations [Eq. (C.3)], assume a solution of the form Nl(t) = ke ~'t for Eq. (C.4). Substitution of the first and second derivatives of Nl(t) into Eq. (C.4) results in ~2 _ OLd. + [3 -- O,
(C.5)
695
a quadratic equation that is satisfied for the values K -- 0.5[or -+- (or 2 -- 4 ~ ) 1 / 2 ] . Thus there are two solutions to Eq. (C.4) of the form Nl(t) = ke xt. Furthermore, any linear combination N l ( t ) = k11e~,1 t + k12 e~'2t
also is a solution. Population dynamics for Nl(t) thus are determined by the parameters )~1 = (o~ + V ~ ) / 2 and )~2 = (Or -- V ~ ) / 2 , where y = ot 2 - 4~ is the discriminant of Eq. (C.5). The parameters K 1 and K2 a r e either both real or both complex, depending on whether ~/>- 0. As shown in Fig. C.1, several possibilities arise: 1. The parameters are both positive but not identical. This situation occurs if oL and y are positive and c~ > Vyy, where the latter condition is equivalent to f~ > 0. Under these conditions, the population trajectory for Nl(t) is a linear combination of two exponential components, both of which increase (but at different rates) through time. The exponential term in )k I dominates the trajectory as t increases. 2. One parameter is positive and the other is negative (or 0), i.e., )~1 ~ 0 and )~2 -< 0. This situation occurs if V ~ - > levi, which holds if and only if ~ -< 0. On condition that )~1 and )~2 differ in sign, the population trajectory for Nl(t) is a linear combination of two exponential components, one of which decreases as the other increases through time. 3. Both parameters are negative but not identical. This situation occurs if the discriminant y is positive and V~y < -e~. With y > 0, sufficient conditions for ~1 and )~2 to be negative are ot < 0 and [3 > 0. If )~1 and )k 2 a r e negative, the population trajectory for Nl(t) is a linear combination of two exponential components, each decreasing through time. 4. The parameters are complex conjugates" ~'1 = O~ + iX/-8 and )~2 = Ot -- iVS, where i = ( - 1 ) - 1 / 2 and 8 = ]y]. This situation occurs if the discriminant is negative, in which case the solution of Eq. (C.4) can be expressed as Nl(t) = k11e~,lt + k12e~'2t = kll e~t {cos(St) + i sin(St)} + k12 e~'t {cos(St) - i sin(St)} = e~t(k11 + k12) cos(St) + ie~t(k11 - k12) sin(St) = e at {ClCOS(St) + c 2 sin(St)}.
This combination of sinusoidal terms satisfies Eq. (C.4) for any choice of cI and c2. It is possible to describe Nl(t) in real terms only, by Nl(t) = e at {C 1 cos(St)
+ C2
sin(St)}.
696
Appendix C Differential Equations
NI(O
N,(t)
F I G U R E C.1
Possible system trajectories arising from solutions ~kl,2 -- 0.5[Or -+- (Or2 -- 4~) 1/2] of the equation and k 2 are real; the system trajectory is decreasing only if k 1 and k 2 both are negative. a r e complex; the trajectory is oscillatory with either increasing, decreasing, or stationary
~2 _ Ot~k -f- ~ -- 0. (a) K1 (b) ~1 and amplitude.
K2
The corresponding population trajectory exhibits oscillations of period 2~r/8, with the magnitude of the oscillations increasing, decreasing, or stable through time depending on whether ot is positive, negative, or zero, respectively. 5. The parameters }kI and k 2 are identical. This situation occurs if the discriminant is 0, which is equivalent to all -a22 = -4a12a21.
In this special case, a solution to Eq. (C.4) is given by Nl(t ) = kll ext + k12te xt, with population dynamics exhibiting exponential growth (or decay) scaled by the factor kll + k12t. In the preceding development, the two equations in Eqs. (C.3) were combined so as to eliminate N2(t). It is easy to show that a companion differential equation in N2(t), obtained by elimination of Nl(t) in Eqs. (C.3), has a solution of the form N2(t) = k21eMt + k22ex2t, with the same values kl and ~k2 in the exponential terms and therefore the same patterns in the trajectory of
N2(t). The values k21 and k22 a r e related to kll and k12 by k21 = kll (~1 -
a11)/a12
k22 = k12(~ 2 -
a11)/a12 ,
and
with specific values of kll and k12 determined by the system initial conditions. In summary, the differential equations [Eqs. (C.3)] give rise to a range of possible system behaviors, depending on the values of e~, I3, and ~/: 1. Weighted exponential growth with distinct exponential rates of growth: ~ > 0, [3 > 0, ~ > 0. 2. Weighted average of exponential growth and decay:
~<0. 3. Weighted exponential decay with distinct exponential rates of decay: ot < 0, [3 > 0, ~/> 0. 4. Unstable (increasing) oscillation: a > 0, ~/< 0. 5. Damped (decreasing) oscillation: e~ < 0, ~/< 0. 6. Stable or neutral oscillation: a = 0, ~ < 0. 7. Exponential growth (or decay) with a single growth rate: ~/ = 0.
C.1. First-Order Linear Homogeneous Equations Because y is defined in terms of oLand [3, these system behaviors can be described in terms of the latter parameters alone. Thus, the plane defined by ((x, [3) can be partitioned into zones corresponding to the first six conditions described above, with condition 7 associated with the parabola oL2 = 4[3 (Fig. C.2).
697
C.1.2.2. M a t r i x A p p r o a c h
The system of equations shown in Eqs. (C.3) can be expressed in terms of matrices, by
aN1/at] dN2/dtJ
Example
Consider the population trajectories of N' = (N 1, N2) as defined by the transition equations
=
,21 [:12]
[all [a21
(C.6)
a22J
or d N / d t = A N. As above, assume a solution to this matrix equation of the form
[NI] iv1]
d N 1 / d t = 3N 1 - N2,
=
e at
N2
v2
d N 2/dt = 6N 1 - 4N 2.
Combining these transition equations results in the second-order differential equation
or N = ve ~'t. Then d N / d t = Mve xt) = A(ve~'t), resulting in the matrix equation A v = )~v or (A - )U)v = 0.
d2N1/ dt 2 4- d N 1/ dt - 6N 1 = 0.
Substituting first and second derivatives of ke ~'t into the latter equation yields )~2 + ) ~ _ 6 = 0 , with solutions is
~kI =
2 and
)k 2 - -
- 3 . The general solution
with specific values for k l l and k12 determined from the system initial conditions. Thus, the trajectory for the system is characterized by a combination of two components, one increasing exponentially and the other decreasing exponentially through time. Because = - 6 < 0, this behavior is consistent with condition 2 above.
"x asymptotic decrease
increasing oscillation /
X1, ~,2<0
neutral ----oscillation
J
asymptotic increase
~1<0,X2>0
This equation has a solution for values of X satisfying the characteristic equation I a - )k/I = )k2 _ ( a l l 4- a22)~k 4- ( a l i a 2 2 - a12a21 )
N l ( t ) = kll e2t 4- k12 e - 3 t ,
decreasing oscillation
(C.7)
/
/
/
/
(C.8) =0, with the corresponding vectors v produced from Eq. (C.7). The parameters X and v satisfying Eqs. (C.7) and (C.8) are the eigenvalues and right eigenvectors of A (see Appendix B). From above, the values of X solving Eq. (C.6) are ~kl, 2 -- (OL -4-
V~)/2,
where o~ = tr(A) = all 4- a22 and y = (x 2 - 4[3, with [3 = Ial = a l i a 2 2 - a12a21. W i t h s o m e algebra, it can be shown that for a12 4= 0,
vi
=
[1] ~ki -- a l l a12
. .. asymptotic asymptoti Increase
X1, ~2>0
asymptotic increase ;~l >0, ;L2<0
F I G U R E C.2 Partition of the (o~, y) plane defined by p a r a m e t e r s of the equation )~2 _ o& + [3 = 0, w h e r e ~/ = oL2 - 4[3. Each region c o r r e s p o n d s to a different kind of s y s t e m trajectory. The parabola is given b y OL2 5E 4 ~ . See text for further explanation.
is an eigenvector corresponding to h i. Either of the pairs (Xi, vi) corresponds to a solution N ( t ) = vi exit
to Eq. (C.6), and the general solution is given by
N t, = cl(vle lt) + c2(vae2t) From this expression it is clear that the trajectories for both populations are controlled by the values of )k I and )~2. For example, both populations decrease if ~kI ~ 0 and )k2 ~ O.
698
Appendix C Differential Equations and
Example
To illustrate, consider again the system
200 = cI
dN 1/dt = 3N 1 - N2,
N l(t) = 80e 2t + 20e-3t
expressed in matrix notation as and
-1 [36 ] IN12] 4 "
N2(t ) =
With an assumed solution of the form N - ve ~'t this system reduces to !
[3
6c2,
so that ( C l , C2) - - ( 8 0 , 20). The particular trajectories of Nl(t) and N2(t) therefore are
dN 2/dt = 6N 1 - 4N2,
dt [dN1/dt]= LdN2/
-ff
4 lIE Vlv2
c9,
8 0 e 2t +
120e -3t
Example
To illustrate oscillatory system behavior, consider the system of equations dN 1/dt = N 1/3 - N2/5,
which is satisfied for values of k such that
dN2/dt = N 1 / 5 - N2/3.
3-k I A - k/[ = 6
As above, an assumed solution of the form N = ve ~'t allows this system to be expressed in matrix form as
=(k-
-1 -4- k
m
[1/5-k 1/3
2 ) ( k + 3)
-1/3 ] [vii =0, -1/5k 122 -
(C.10)
-- 0.
which is satisfied for values of k such that Each of the two eigenvalues satisfying this equation corresponds to an eigenvector. For example, substituting k = 2 into Eq. (C.9) produces [1-1] 6 6
1/5- k -1/3 1/3 -1/5=
[ v i i i = 0, kV12 -
~k2 q- ( 4 / 1 5 )
2
=0. This equation is satisfied by ~'i = + 4i / 15, and substituting k = 4i/15 into Eq. (C.10) produces
or V l l - - V12 -- 0
[14
and
-- ~ l 1
6Vll
-- 6V12 =
(3 - 4/)Vll
Cl
1 1
e2 t + c2
n t-
C2
=
5V12.
-- 5v22.
Thus the eigenvectors corresponding to are
[113t 6 ' e
with cI and c2 determined from the system initial conditions. For example, if _N'0 = (100, 200), then 100 = c I
- - 0~, -
Likewise, substituting k = - 4 i / 1 5 into Eq. (C.10) produces (3 + 4/)v21
By setting v21 - - 1, one obtains v22 = 6. Thus, the general solution for the system of equations is
=
Vl
or
6V21 -- V22 -- 0 .
Nl(t)
I21
0.
Becuse these equations are redundant, one of the variables must be expressed in terms of the other. By setting V l l = 1, one obtains the value V12 -- 1. In analogous fashion, k = - 3 can be substituted into Eq. (C.9), to produce
[N2(t )] I]
1 ]i:1
--~ 1 4 --~ - - 1-~i
Vl
I53]-
v2
I3]+
and 5
hi =
+4i/15
699
C.1. First-Order Linear Homogeneous Equations If u(t) and w(t) are defined by
tion of the equations into a single kth-order equation, in the same manner as for two equations. This produces a kth-degree polynomial equation with k roots Xi, i = 1..... k, each of which corresponds to a solution c1 exp(Xit). The general solution is
u(t [35]cos( ,t [0] sinI4t) and
w(t
k
sin(l t; [:] cos( t;
Ni(t ) = ~
i=1
it is straightforward to show that a general real solution is
N(t) = Clu(t) + C2~__(t)
cos(4t)] sin(4t)] =C1
ill COS(4t) j + Ii sin(4t) j /
3sin(gt;] cosI4tt]
Cie~it,
with a specific solution determined by the system initial conditions. Alternatively, a matrix approach can be used to determine the population trajectories. For k populations, the equation d N / d t = A N is of dimension k, which results in a characteristic equation with k roots. Thus, there are k combinations (Xi, vi) of eigenvalues and eigenvectors that satisfy [A - XI]v = 0, and any ()ki,Vi) corresponds to a solution
+Cati5sin(4t)j_iicos( t)]t
N(t)
= vi exit
of d N / d t = A N. The general solution is given by
-I
3Clc~
1
5ClCOS(~t)+4C1sin(4t)+ 5C2sin(~t)- 4C2cos(4t)J ' with the coefficients C1 and C2 determined by system initial conditions. For example, if N' _0 = (12, 12), then
3C1 ] X(0) = 5C1 _ 4 C 2
so that
C1 -- 4, C2 =
2, and
N ( t ) = [11~] c o s ( ~ t ) +
[66] s i n ( 4 t ) .
The oscillatory nature of this trajectory is constrained by N(t)~0, and N2(t) attains a value of zero earlier than does N l ( t ) . Thus, the oscillations result in extinction for population 2.
N(t) = ~
i=1
(C.11)
c i v__ie~'it .
As above, the population trajectories are controlled by the values Xi, i = 1, ..., k. For example, the populations decline if all eigenvalues Xi are negative and increase if at least one eigenvalue is positive. Oscillations occur if there is at least one pair of complex conjugate eigenvalues. It is clear that the inclusion of additional populations into a system, increasing its dimensionality and thus increasing the number of eigenvalues, can lead to greater complexity in system behaviors. Because the exponential terms in Eq. (C.11) all have a value of unity when t = O, the population initial state is simply
k N(O) = E CiVi'
i=1
indicating that the constants c i are directly related to population initial conditions. This relationship can be expressed in matrix notation as m
C.1.3. Population Dynamics for Multiple Species The population dynamics of k species can be characterized by k first-order differential equations dN/dt
X(0)-- IvI v2 --- Vk]
= A N,
and two approaches can be taken to determine the population trajectories. The first involves transforma-
m
Cl C2
m
Ck g
=Vc.
so that c = V-1 N(0). On condition that A is symmetric,
700
Appendix C
Differential Equations
the eigenvectors v i are both real and orthogonal, in that -v-;j v = 0 for i #= j (see A p p e n d i x B). Then (V~Vl)-1 (W'V) -1 W'X0
=
9..
9
0
9
.
9..
( V ~cVk) - 1
Example
05][]
To illustrate, consider the system
-0.75 -1.75 - 0 . 5 -0.5 -2.0
X1
N2 X3
Assuming a solution of the form N(t) = ve xt, the system equations reduce to
[175 075 05][vl] -0.75 -0.5
-1.75 --0.5
)k
--0.5 - 2 - )k
V2 = 0, (C.12) V3
I
-1.75 - ~ -0.75 -0.5 1 -0.75 -1.75 - k 0.5 = 0. -0.5 -0.5 -2-
The latter is a polynomial equation with the three roots = - 1 , -0.5, and -0.25. Substituting these values back into Eq. (C.12) produces the eigenvectors v~ = !
!
v 3 -- (1, 1, - 2 ) ,
so that a
general solution is
[Nl1t [il E1] Ei] N2(t) N3(t)
= cI
e -t
Jr-
-
with c 1, c2, and c3 determined by the system initial conditions. For N(0)' = (250, 50, 100), the constants are given by
[Cl]=/v~/2/ rvJgl[250] C2
50
c3
k_v;/6_l 100
=
33]
00 , 17_]
An important extension concerns h o m o g e n e o u s differential equations that include nonlinear terms. In this case, the growth function F(N) in d N / d t = F(N) includes terms such as N 2, NiNj, and other mathematical expressions that are nonlinear in the population values N i. Familiar examples include the logistic model and the Lotka-Volterra models in Chapter 8. We restrict attention here to an analysis of population dynamics for populations that are "near" an equilibrium, with the idea of assessing equilibrium stability.
Consider the dynamics of a population with a nonlinear growth function F(N), for which derivatives exist over some operative range of population size. A s s u m e that the population is in equilibrium at a value of N*, so that F(N*) = 0. Then population dynamics can be expressed in terms of a Taylor series expansion of F about N*: dF , n 2 d2F F(N* + n) - F(N*) + n-d-~(N ) + - ~ ~ (N*) + ... (C.13)
with n = N - N* describing "small" deviations from N*. The higher degree terms in Eq. (C.13) are of negligible importance, leading to the simplified expression F(N* + n ) =
e -0.25t
C2 --1 e -0"5t + c 3
0
C.2. NONLINEAR HOMOGENEOUS EQUATIONS--STABILITY ANALYSIS
C.2.1. Stability Analysis for One Species
which is satisfied for values of ~ such that
(1, 1, 1), V 2 -- (1, - - 1 , 0), a n d
N3(t) = 133e -t - 34e -0.25t.
__W__N0
Thus, an eigenvector decomposition of A sometimes provides a convenient w a y to compute the constants in c corresponding to a set population of initial conditions.
-1.75 -0.75 -0.5
+ 17e -~
N2(t) = 133e -t - 100e -0.St + 17e -~
m
[
Nl(t) = 133e -t + 100e -~
0
=C.
FdN1/dt] |dN2/dt | = LdNB/dt]
and the corresponding population trajectories are
dF n-v-,. (N*).
Because dN/dt
= d(N* + n ) / d t = dn/dt,
the equation for population dynamics can be expressed in terms of deviations n = N - N* as dF , d n / d t = n - - ~ ( N ).
(C.14)
Thus, a nonlinear transition equation can be approximated by a linear differential equation in a neighborhood of N*. It follows that N* is a stable equilibrium if
701
C.2. Nonlinear Homogeneous Equations--Stability Analysis F' (N*) < 0 (because the trajectory of deviations exhibits exponential decay) and N* is an unstable equilibrium if F' (N*) > 0 (because the trajectory of deviations exhibits exponential growth).
the higher degree terms are of negligible importance, and Eq. (C.15) reduces to F k ( N * + n) = nlFkl(N *) + n2Fk2(N*).
As in the single-species case, we can express d ( ~ / d t
Example
as
Consider the logistic model d N / d t = d(N* + n ) / d t d N / d t = rN(1 - N/K)
from Section 8.2 with constant growth rate r > 0 and constant carrying capacity K > 0. The model has two equilibria, N* = 0 and N* = K, and deviations in a neighborhood of N* are given as in Eq. (C.14), by dF , dn/dt = n -~(N ) = rn(1 - 2N*/K).
Population dynamics around N* = 0 are specified by dn / dt = rn (1 - 2N* / K)
= dn/dt,
so that the equation for population dynamics can be expressed in terms of the deviations n = N - N* as d n k / d t = n]Fk(N *) + n2Fk(N*),
k=l, 2. Thus, the nonlinear transition equations can be approximated by linear differential equations in a neighborhood of N*. Equation (C.16) is written in matrix notation as [dnl/dt]
-- TTl,
d n / d t = rn(1 - 2 N * / K )
--
~
OF2
L~-~ (N*)
tl 1
n2
(C.17)
= J(N*)n, and the properties of J(N*), known as the Jacobian matrix, determine the equilibrium stability of the system. Assuming a solution of the form n(t) = ve ~t, Eq. (C.17) reduces to
= rn(1 - 2 K / K ) --
OF 1
F 3F1 |-~11 (N*)
Ldna/dtJ = ] OF2
which exhibits simple exponential growth away from 0. Thus, N* = 0 is an unstable equilibrium, in that positive deviations from 0 increase over time. On the other hand, population dynamics around N* = K are specified by
(C.16)
[J(N*) - M]v = 0,
TYl,
for which the characteristic equation which exhibits simple exponential decay toward 0. Thus, N* = K is a stable equilibrium, in that deviations from K lead to asymptotic declines in the deviations [and thus to asymptotic convergence of N(t) to K].
I!(_N*) -
is a polynomial of degree 2. Thus, either of the pairs ()~1, Vl) and (~2, v2) of eigenvalues and eigenvectors corresponds to a solution
C.2.2. Stability Analysis for Two Species As before, the addition of another state variable complicates the analysis. Consider two populations with nonlinear growth functions F(N)' = [FI(N), F2(N)]. A Taylor expansion about an equilibrium value N* is F k ( N * + n__) = F k ( N *) + nlFlk(N *) + n2Fk2(__*) 2
-Jr-
n 2F k22'__" rN,~ kl(X~') -}- T
(C.15)
x_ l = o
n(t) = vi e•
of d n / d t = J(N*)n, and a general solution is given by
n(t)
= Cl(Vl exit) + c2(v2eX2t).
From the analysis of linear differential equations, N* is a stable equilibrium if the roots ~'1 and K2 of Ij(N, ) _ )~ii = ~ 2 _ )kO~ -Jr-
=0 + (nln2)Fk12(N *) + ...
for k = 1, 2, where F/k(_N)= 3 F k ( ~ / O N i and F~(N) = 3 2 F k ( ~ / O N i 3Nj. For "small" deviations n = N - N*,
are both negative. For positive discriminant ~/= ~2 _ 4[3, this condition is equivalent to oL = tr[J(N*)] = all + a22 < 0 and [3 = IJ_(N*)I = a11a22 - a12a21 > 0 (see
702
Appendix C Differential Equations
Section C.1.2). Thus, tr[J(N] < 0 and IJ_(N*)I > 0 ensure that deviations n I and n 2 both decrease asymptotically to 0 [so that N(t) converges to N*].
Thus, the population dynamics for this system exhibit neutral stability in a neighborhood of (48, 80), with trajectories given by
Example
N(t) = N* + n(t)
Consider a pair of populations with dynamics governed by the system of equations
_[N~] I_N~J
dN1
N1
N2
dt
3
5'
I
dN2 7N2(1 - N2/35 ) N1 = 4 dt 75 5
OF 1
i
1
12cost4t) 2 sin(4t) + 48
OF2 -~2 (N*)
_1
and
7(1 - 2N~/35) " 5 75
12cos(4t) 26sint4t) + 80.
For deviations in a neighborhood of N* = 0, the Jacobian is
J(N*) =
.
Example Consider the Lotka-Volterra competition equations
1
[ Nlj t] [F1NII N1 a12 2 J l]
,
Ygj
dN2/dt
for which tr[/(N*)] > 0 and IJ(N*)I > 0. From the analysis of linear differential equations, it follows that N* = 0 is an unstable equilibrium, in that positive deviations from 0 exhibit growth away from 0. On the other hand, the Jacobian for N* = (48, 80) is 1
'
where the coefficients C 1 and C2 are determined by the initial population sizes. For initial population sizes of, say, N~ = (60, 92), it is easy to show that the system oscillations are described by
;~11(N_*) ;G2 (N_*) J(N*) = I OF2(N,) LON 1 --
1
+ 5ClcOs(~t)+4C 1sin(4t)+5C2sin(4t)-4C2cos(~t)
It is easy to show that this system has two equilibria, N* = 0 and N* = (48, 80). The Jacobian for the system is OF
3C1cos(4t)+ 3C2 sin(4t)
r2N2(K2 -
N 2 -
a21N1)/K2
from Section 8.8, for which there are two equilibria, N* = 0 and m
K 1 - a12K2 IN, I= LN~_J
1
1 - a12a21
K2 - a21K1
9
1 - a12a21 The Jacobian for this system is for which tr[J(N*)] = 0 and IJ(N*) i > 0. These conditions indicate that the deviations n(t) exhibit stable oscillations, with the deviation trajectories described by n(t) =
I
()
3~,cos/~tt+3~2sin(4tt
()()
(
5C1cos 4t +4C 1sin 4t +5C 2sin ~t -4C 2cos ~t
tl
J(N*) r OF1
9 OF1 / ~ <-~> ~<~ 9>/1 = /oF 2
[r1(1 -
I
,
aF 2
. /
2N'~/K 1 - a12N'~/K1) -r2a21N~/K 2
-rla12N'~/K1
r2(1 -
]
2N'~/K 2 - a21N'~/K2) ~"
C.2. Nonlinear Homogeneous EquationsfStability Analysis For deviations in a neighborhood of N* = 0, the Jacobian takes the values
of Section 8.7. This system has two equilibria, N* = 0 and
[
J(N*)
-[
703
X ~ ] __ [ d 2 / b 2 ]
r111 - O/K 1 - a12(O/K1) ] - r2a21(0/K 2)
-rlal2(O/K1) r211 -
]
OIK2- a2~(OIK2)]
=Er,0 r0]"
N~J
kblldlJ"
where N 1 and N 2 characterize prey and predator populations, respectively. The Jacobian of the system is
J(N*) =
Because
OF
9
OF 1
9 -I
I 3F 2
9
3F 2
,/
tr[J(N*)] = r I + r 2 > 0
pbl-alN2 L b2N2
and
I!(N*)I =
rlr2 > O,
it follows that _N* = _0 is an unstable equilibrium, in that positive deviations from _0 exhibit growth away from 0. On the other hand, the Jacobian for
-alN1 -I b2N1 - d2 ]"
For deviations in a neighborhood of N* = 0, the Jacobian takes the values
r bl - dl(~
l(m*) - --
L
b2(O)
-dl(~
-I
b2(O)-d2]
K 1 - a12K 2
IN~}= LN~J
1 - a12a21 K2-
a21K1
1 -
a12a21
This corresponds to the simple differential equations dn 1 / d t = bin 1
and
](N*) =
dn2/dt = -d2n 2
(1 -- a12a21) -1 r - r l ( K 1 - a12K2)
t r2a21(K 2 -
rla12(K 1 - a12K2)]
a21K1)
- r 2 ( K 2 - a21K1) J '
for which tr[J(N*)] = - r l ( K 1 - a12K2) + r2(K2 - a21K1) 1 - a12a21 and
I/(N*)I
in the deviations//1 and n 2, so that//1 (and therefore N 1) increases in a neighborhood of 0, while n 2 (and therefore N 2) decreases. This accords with the biological sense of predator-prey interactions, whereby small numbers of predators allow for growth of a prey population, and small numbers of prey lead to predator declines. On the other hand, the Jacobian for N*' = (d2/b2, b 1 / d 1) is j(N,)
= rlr2(K1 - a12K2)(K2 - a21K1).
b2(bl/dl)
If competition is not severe (that is, if K 1 - a12K 2 > 0, a21K 1 > 0, and 1 - a12a21 ~ 0), it follows that tr[J(N*)] < 0 and IJ(N*)I > 0. The latter conditions ensure that deviations in a neighborhood of the equilibrium converge to 0, so that the population returns to N*.
Example The analysis of stability provides a mathematical justification for the oscillatory patterns observed with the Lotka-Volterra predator-prey equations
Ibl-dlN1][ =
b2N2
_d 2
N1] N2
(C.18)
-dl(d2/b2) ] b2(d2/b2) - d 2
I0
K2 -
dN1/dt] dNaldt ~
= [bl - d l ( b l / d l )
bib 2
for which tr[J(N*)] = 0 and IJ(N*)I bid 2. At this equilibrium, the system eigenvalues are the complex conjugates K1,2 = 0.5(-bid2 )1/2. The corresponding deviation trajectories are sinusoidal, and the populations exhibit stable oscillations about the equilibrium. =
Example It is straightforward to show that the stable oscillations of a Lotka-Volterra predator-prey system are not
704
Appendix C
Differential Equations
maintained in the presence of density-dependent birth. Let the p r e d a t o r - p r e y system in Eq. (C.18) be modified by
aN1/at dN2/dt]
= [bl(1-N1/K) b2N 2
-diN1 -d2 ] [X12]"
for k = 1, ..., m. For small deviations n = N - N*, the higher degree terms are of negligible importance, and Eq. (C.19) reduces to
m OFk Fk(N * + t l ) = s n i - ~ i (N*). i=1
As above, we can write d ( N ) / d t as
Equilibria for this system are N* = 0 and
d N / d t = d(N* + n ) / d t = dn/dt, =
LN~._]
bl dll
bid2 ' b-~-lK_]
so that the equation for population dynamics can be expressed in terms of the deviations n = N - N* as
with d 2 < b2K a necessary condition for N~_ to be positive. The Jacobian matrix is [b I - (2blN~/K) -
J(N*)
L
baN ~
diN'2 -dlN'~ 1 b2N'~ - d2_]"
For N* = 0, the Jacobian is identical to the matrix for the unmodified Lotka-Volterra system, so the dynamics of n 1 and n 2 in a neighborhood of 0 are the same as in the previous example. However, the Jacobian at N*' = [d2/b 2, b l / d I - bld2/(b2dlK)] is
OFk ni ~ i (N*),
dnk/dt =
(C.20)
i=1
k = 1, ..., m. Thus, the nonlinear transition equations can be approximated by linear differential equations in a neighborhood of N*. Equation (C.20) can be expressed in matrix notation as
n
J(N*) =
bid2 b2K bib______22 bid2 dl dlK
d n l d t = !(N*)n, where n' = (n I .... , n m) and
J(N*) = I_-~i (N*) .
dl___d2 l
I'
A s s u m i n g a solution of the form n_(t) = ve ~t, this system reduces to [/(N*) - XI]v = 0,
for which tr[J(N*)] = -bld2/(b2K ) < 0 and IJ(N*)I = bid2[1 - d2/(b2K)] > 0. As argued previously, these conditions ensure that _N* is a stable equilibrium, so that deviations in a neighborhood of N* are eliminated as N(t) returns to N*.
C.2.3. Stability Analysis for Multiple Populations
for which the characteristic equation IJ(N*) - ~/I = 0 is a polynomial of degree m. Thus, there are m combinations (h i, vi) of eigenvalues and eigenvectors for which Eq. (C.20) is satisfied. A n y of these combinations corresponds to a solution n(t)
Consider m populations with nonlinear growth F ( N ) ' = [FI(N), F 2 ( ~ , ..., Fm(N)l. A Taylor expansion
=
vi e~it
of d n / d t = J(N*)n, and the general solution is given by
about an equilibrium value N* is n__(t) = ~ . ci vi e~#
m OFk Fk(N * + t l ) = Fk(N *) + ~.~ n i - ~ i (N*) i=1
+ ~
m
n2 c92Fk
i=1 2 a-~i2(N*)
m c92Fk + ~ , nin;_ (N*) + ... i,j=1 i c~mi cgXj --
9
i=1
(C.19)
As above, the deviation trajectories in a neighborhood of N* are controlled by the values hi, i = 1, ..., m. The trajectories decline if all eigenvalues hi are negative, and they increase if at least one eigenvalue is positive. Oscillations in the trajectories follow from the occurrence of complex conjugate eigenvalues. As with the single-species and two-species models,
705
C.3. Graphical Methods it is possible to describe stability conditions for a multispecies system in terms of the characteristic equation. Consider again the deviation model
dn/dt = J(N*)n, the eigenvalues for which are given by the characteristic equation IJ_(N*)
-
=
0.
This equation can be expressed as the polynomial Km + al Kin-1 4- a 2 ) t m - 2
+
""
+ a m - - O,
with a i given in terms of the coefficients of J(N*). Define m matrices Hj, j = 1, ..., m, such that Hj contains the elements
a21_k, 1, 0,
I
0 ~ 2l -
k ~ m;
21 = k; 2lk+
For example, H 1 =
H3
with
aI =
-(all
+ a22) a n d
a 2 = alia22 -
a12a21 . T h e
conditions a I ~ 0 and a 2 ~ 0 previously were shown to result in exponential declines. Finally, after some algebra, the system of equations,
IdN1/dt I [-1.75-0.75-0.5] dN2/dt = -0.75 -1.75 -0.5 dN3/dt -0.5 - - 0 . 5 - - 2 . 0
IN1] N2 , XsJ
which was shown in a previous example to exhibit exponential declines, can be seen to meet the Routh-Hurwitz criteria for equilibrium stability.
C.3. GRAPHICAL M E T H O D S
il
[ al =
a3
a2
aI
1
0]
a3
a2
al
a5
a4
a3
I
.
It can be shown that the equilibrium value N* is stable [that is, the real parts of all the eigenvalues for Eq. (C.20) are negative] if the determinant of each of the m matrices defined by Eq. (C.21) is positive:
]Hj] > O,
(C.22)
j = 1,..., m. The matrices Hj are called Hurwitz matrices, and conditions (C.22) constitute the Routh-Hurwitz criteria for stability. The Routh-Hurwitz criteria for systems of dimension m = 1, 2, 3, and 4 are 1" 2: 3: 4:
~. )k2 q- a l h + a 2,
(C.21) m.
and
= = = =
an - K a12 ~- )k2 -- (all + a22)h + (alia22 - a12a21 ) a21 a22- K
al,
S2
m m m m
tially if a I ~ 0. For two populations, the characteristic equation is
al>0; a I >0, a2>0; a I >0, a3>0, ala2>a3; a 1 > O, a3 > O, a4 > O, ala2a 3 > a2 + a2a4.
These conditions are in accord with the equilibrium conditions previously described. For example, the linear differential equation for a single population is d N / d t + alN = 0, with a corresponding characteristic equation of h + a I -- 0. The solution for this equation is
N(t) = No e-alt, so that the population trajectory decreases exponen-
One sometimes can obtain useful information about population dynamics without actually obtaining solutions for the corresponding differential equations. Often it is sufficient to recognize the direction of movement for a population of a given size at each point in time. The graphical representation of this information is called a direction field, consisting of direction vectors at each point in the (t, N) plane. A direction vector at (t, N) is simply the vector (1, dN/dt), with d N / d t = F(N) evaluated at (t, N). It represents the direction of change of the population in a neighborhood of (t, N). Curves of constant directional vectors in the (t, N) plane are given by F(N) = C, with different vectors specified by different values of C. This is illustrated in Fig. C.3 for the model d N / d t = N 2 - t. Note that the curves for which the directional vectors are unidirectional are given by N 2 - t = C (Fig. C.3a). Population trajectories coincide with points of tangency to the directional vectors (Fig. C.3b). If the differential equation is autonomous, i.e., if F(N) does not contain an explicit reference to t, then the direction vectors vary only with N over the direction field. The direction field for an autonomous growth function is illustrated in Fig. C.4a for the logistic equation d N / d t = N(1 - N). Because the direction field for an autonomous function varies with population size but not with time, one can essentially collapse the directional information in the direction field into a one-dimensional phase representation, with F(N) plotted against N. This is illustrated in Fig. C.4b for the logistic equation dF/dt = N(1 - N). Thus, the direction of change for N between 0 and 1 is positive (irrespective of the time at which the N achieves that
706
Appendix C Differential Equations
////
/////////..
\
\\\
\\\\\\\\\\\\\\\\\ \ \ \
\\\\\\\\\\
\ \
\ \
\\\\ ' // // // // / / / / , /
\\\
\ \\
N(t) a
\ K
\ \\\
\
\\\\ \\\\ \\ \\\ \\\
//Z
\\\~
//~/////,,
//X///~/////////// / /
/
/
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/
/
/
/
/
/
/
/
~
/
dN/dt
b
rKI4
F I G U R E C.3 Direction field for d N / d t = N 2 - t. (a) Directional vectors at each point [t, N(t)] are given by (t, N 2 - t). Directional vectors are constant along parabolic curves for which N 2 - t = C. (b) Population trajectories coincide with points of tangency to the directional vectors.
value), whereas the direction for N > 1 is negative. At the values 0 and 1, of course, the change is 0.
C.3.1. Stability Assessment with Null Clines The notion of a phase representation for autonomous differential equations can be extended naturally to two equations. In this case, a phase plane is described, with directional vectors at each point that are given by the growth functions F I(N) and F2(N) of the system of equations. Null clines are defined by the equation F I(N) = 0, which specifies curves in the phase plane for which the rate of change of N 1 is 0, and F2(N) = 0, which specifies curves for which the rate of change of N 2 is 0. System steady states are given by the intersections of the respective null clines. Vectors along the null cline F I(N) = 0 are of the form [0, F2(N)] and therefore are represented as
K/2
K
F I G U R E C.4 The direction field for logistic equation d N / d t = N(1 - N). (a) Because the logistic function is autonomous, its direction field varies with population size but not with time. (b) Onedimensional phase representation of the directional information in the direction field, with d N / d t plotted against N.
vertical arrows in a phase plane. Similarly, vectors along the null cline F2(N) = 0 are of the form [FI(N), 0] and are represented as horizontal arrows in a phase plane. Because the growth functions FI(N) and Fa(N) are assumed to be continuous in N, the direction vectors change smoothly along the null clines and therefore can change direction only at a steady state. Consider, for example, the system depicted in Fig. C.5 with null clines FI(N) = 0 and F2(N) = 0 and a unique steady state N*. Because the point P1 on the null cline FI(N) = 0 satisfies F2(N) > 0, all points on the null cline to the right of _N* must satisfy F2(N) > 0. Furthermore, the direction vector must reverse direction at N*, so that F2(N) < 0 for all points on the null cline to the left of N*. Similarly, FI(N) > 0 for the
C.3. Graphical Methods
707
F I G U R E C.5 Phase plane for a system with null clines FI(_N) = 0 and F2(_N) -- 0 and a unique steady state _N*. Both populations increase in region I, both decrease in region II, and the populations move in opposite directions in regions III and IV. These directional tendencies correspond to oscillatory system behavior.
III
G" eq
II
/ ,v
1
>4 1
/2
i
..q Nl(t)
point P2 on null cline F2(_N) = 0, and therefore FI(N) > 0 for all points on the null cline to the left of N*. Furthermore, the direction vector must reverse direction at N*, so that FI(_N) < 0 for all points on the null cline to the right of N*. An analogous logic can be applied to systems with multiple steady states, and in this w a y the pattern of direction can be deduced in a fairly straightforward way, with little calculation. The directions of the arrows along the null clines also are indicative of the direction of m o v e m e n t throughout a direction field. Thus, both populations increase in region I of Fig. C.5, both decrease in region II, and the populations move in opposite directions in regions III and IV. These directional tendencies correspond to oscillatory system behavior.
Example The use of direction fields can be illustrated with the Lotka-Volterra competition equations
d X l / d t ] __ [FINI(K1- X 1 --a12N2)/K1] dN2/dt Lr2N2(K2 - N 2 - a21N1)/K2] from Section 8.8. The null clines dN 1/dt = 0 for population N 1 are given by N 1 = 0 and N 1 = K 1 - a12N2, whereas the null clines dN2/dt = 0 for population N 2 are given by N2 = 0 and N 2 = K2 - a21N 1. The null clines N 1 = 0 and N 2 = 0 coincide with the axes of the (N 1, N 2) plane, limiting the operative values of N 1 and N2 to the set of nonnegative population values. The other two null clines are arranged in the (N 1, N 2) plane in one of four configurations, depending on the magnitudes of the carrying capacities and competition coefficients: Case 1. K 1 < a12K 2 and K2 > a21K 1. As s h o w n in Fig. C.6a, the null clines do not intersect, and the null cline for N 2 is to the right of the null cline for N 1. Direction vectors on dN1/dt = 0 point in the direction of growth for N 2, whereas direction vectors on
dN2/dt = 0 point in the direction of decline of N 1. The corresponding direction field suggests that population 1 will become extinct and population 2 will attain its carrying capacity. This accords with results highlighted in Section 8.8. Case 2. K 1 > a12K2 and K2 < a21K1. Again, the null clines do not intersect, but n o w the null cline for N 1 is to the right of the null cline for N 2. As s h o w n in Fig. C.6b, the direction vectors on dN2/dt = 0 point in the direction of growth for N 1, whereas the direction vectors on dN 1/dt = 0 point in the direction of decline of N 2. The direction field n o w has N 1 increasing and N 2 decreasing, so that population 2 becomes extinct as population 1 attains its carrying capacity. Again, this accords with results highlighted in Section 8.8. Case 3. K 1 > a12K2 and K2 > a21K1. In this case, the null clines intersect at an equilibrium point N* at which the direction vectors on the null clines switch direction. As s h o w n in Fig. C.6c, the direction vectors on dN 1/dt = 0 indicate growth in N 2 for points to the right of N* and indicate declines in N 2 for points to the left of N*. On the other hand, the direction vectors on dN 2/dt = 0 indicate declines in N 1 for points to the right of N* and indicate growth in N 1 for points to the left of N*. This partitions the (N 1, N 2) plane into four regions, one in which N 1 and N 2 both are increasing, one in which N 1 and N 2 both are decreasing, one in which N 1 is decreasing and N 2 is increasing, and one in which N 2 is decreasing and N 1 is increasing. The corresponding direction field suggests that the populations will converge on N* irrespective of initial population sizes (as long as both are positive). Case 4. K 1 < a12K2 and K2 < a21K1. The null clines again intersect at positive population sizes. As s h o w n in Fig. C.6d, the direction vectors on dN 1/dt = 0 indicate declines in N 2 for points to the right of _N* and growth in N 2 for points to the left of _N*. On the other hand, the direction vectors on dN2/dt = 0 indicate growth in N 1 for points to the right of N* and declines
708
Appendix C
Differential Equations
Kt/a12
K1/a12
~._
b
.
KI
K2/a21
K2/a21
K1
K2/a21
K1
K2,
C
K lla 1 2 ~ , , , ~
K1/a12~~~'~i.~ K2
.I-
<
~" ~
K1
K2/a21
F I G U R E C.6 Null clines and direction vectors for a system of two competing species. The null clines dN1/dt - 0 for population N 1 a r e given by N 1 = 0 and N 1 = K 1 - a 1 2 N 2 , and null clines dN2/dt = 0 for population N 2 are given by N 2 = 0 and N 2 = K2 - a21N1. The nontrivial null clines are arranged in the (N 1, N 2) plane in one of four configurations, depending on the magnitudes of the carrying capacities and competition coefficients. (a) The null clines do not intersect, and the null cline for N 2 is to the right of the null cline for N 1. Population 1 is driven to extinction and population 2 attains its carrying capacity. (b) The null clines do not intersect, but now the null cline for N 1 is to the right of the null cline for N 2. Population 2 becomes extinct as population I attains its carrying capacity. (c) The null clines intersect at an equilibrium point _N* at which the direction vectors on the null clines switch direction. This partitions the (N 1, N 2) plane into regions in which N 1 and N 2 both are increasing (region I), both are decreasing (region II), N 1 is decreasing and N 2 is increasing (region III), and N 2 is decreasing and N 1 is increasing (region IV). The populations will converge on N* irrespective of initial population sizes. (d) The null clines again intersect at positive population sizes, partitioning the (N1, N 2) plane into four regions, in which N 1 and N 2 both are increasing (region I), N 1 and N 2 both are decreasing (region II), N2 is decreasing and N 1 is increasing (region III), and N 1 is decreasing and N 2 is increasing (region IV). One of the populations will be driven to extinction, depending on population initial conditions. B
in N 1 for p o i n t s to t h e left of _N*. T h i s a g a i n p a r t i t i o n s t h e (N1, N 2) p l a n e i n t o f o u r r e g i o n s , o n e in w h i c h N 1 a n d N 2 b o t h a r e i n c r e a s i n g ( r e g i o n I), o n e in w h i c h N 1 a n d N 2 b o t h a r e d e c r e a s i n g ( r e g i o n II), o n e in w h i c h N 2 is d e c r e a s i n g a n d N 1 is i n c r e a s i n g ( r e g i o n III), a n d o n e in w h i c h N 1 is d e c r e a s i n g a n d N2 is i n c r e a s i n g ( r e g i o n IV). T h e c o r r e s p o n d i n g d i r e c t i o n f i e l d s u g g e s t s t h a t o n e of t h e p o p u l a t i o n s w i l l b e d r i v e n to e x t i n c t i o n , depending on population initial conditions. The line N2(0)
K 2 - a21K 1
K 1 - ~a12K2N1(O)
(C.23)
c o n n e c t i n g 0 a n d N* d i v i d e s t h e ( N 1, N 2) p l a n e i n t o
a r e a s of i n i t i a l p o p u l a t i o n extinction results. Thus,
s i z e s t h a t l e a d to d i f f e r e n t
K2 - a21KIM (0) N2(0) > K1 - a12K2~'1 l e a d s to t h e e x t i n c t i o n of p o p u l a t i o n N2(0) < K2 - a21K1
1, a n d
(0)
K 1 - a12K---22N1 l e a d s to t h e e x t i n c t i o n of p o p u l a t i o n 2. P o p u l a t i o n initial c o n d i t i o n s t h a t s a t i s f y Eq. (C.23) l e a d to c o n v e r g e n c e to N*.
A P P E N D I X
D Difference Equations
over one time period, whereas in Eq. (D.2) the change in population status is implicit. In what follows, we use either Eq. (D.1) or (D.2) to represent population dynamics, depending on the context. The distinction between linear and nonlinear equations applies to difference as well as differential equations. Thus, a growth function F(N) containing no terms of degree higher than I defines a linear difference equation; otherwise, the difference equation is nonlinear. Thus, linear difference equations are limited to those containing scalar multiples of the elements in N. For example,
In this appendix we describe procedures for analysis of difference equations that arise in population ecology. It is useful to think of these equations as expressing differential change in population size over a discrete time frame, with population trajectories defined by their solution. In what follows, we refer to N ( t ) as a vector of populations, recognizing that N(t) also may represent a vector of population cohorts or other resource entities. A more detailed treatment of difference equations can be found in references such as Goldberg (1986), Levy and Lessman (1992), and Elaydi (1999). Consider a set of populations (or population cohorts) N(t) that experience change through time according to the equation N ( t + 1) = N ( t ) + F ( N ) ,
AN = rN
is a linear difference equation, whereas
(D.1)
A N = rN(1 - N / K )
where F(N) expresses a change in population status from time t to t + 1. Population dynamics also can be expressed in somewhat simplified form by N ( t + 1) = F ( N ) ,
(D.3)
(D.4)
is a nonlinear equation. Under some quite general conditions, the growth function for a single population can be written as a Taylor series expansion
(D.2)
oo
where population size N(t) at time t is absorbed in the growth function F(N). Equations (D.1) and (D.2) characterize population change as a first-order difference equation, wherein population status N(t + 1) at time t + 1 is based solely on population status at time t, without reference to lags of greater than one time step. Equation (D.1) explicitly represents population change as
F(N) = ~_, akNk, k=O
so that the models can be expressed as (possibly infinite) polynomials. For example, the exponential model [Eq. (D.3)] requires only a linear term to represent growth, and the logistic model [Eq. (D.4)] requires linear and quadratic terms. The growth functions for most population models are complicated mathematical expressions, polynomial approximations for which may require many terms. For example, growth functions for the Gompertz model of human mortality and the
a N = N ( t + 1) - N ( t )
= F(N)
709
710
Appendix D Difference Equations
Ricker and the Beverton-Holt models from fisheries biology involve exponential terms and therefore require infinite series of polynomial terms for an exact representation. As with differential equations, homogeneous and nonhomogeneous difference equations are distinguished by the presence in the latter of terms in the growth function F(N) that do not include _N. Thus, Eq. (D.1) is a homogeneous difference equation, whereas
so that a complete solution for the model is N(t) = N0(1 4- r) t. The corresponding population trajectory increases through time for k > 1 and decreases through time f o r 0 < k < 1. Because of its simplicity, one can solve Eq. (D.3) by a straightforward repetition of differencing, as in N(1) = (1 + r)N 0, N(2) = (1 + r)N(1),
AN = rN + c
is nonhomogeneous. The growth functions of homogeneous equations often are expressed as F(N) = NG(N),
with G(N) the per capita rate of growth for the population.
D.1. FIRST-ORDER LINEAR HOMOGENEOUS EQUATIONS D.1.1. Population Dynamics for One Species Consider first a single population with no cohort structure, for which population dynamics are characterized by a single linear homogeneous difference equation. Several methods from numerical analysis are available for solving this equation, including graphical methods, computer simulation, and, in some simple cases, mathematical analysis. Example
Consider the linear homogeneous difference equation with constant coefficient from Section 8.1" AN = rN
with initial population size N(0) = N 0. An easy approach to its solution is to assume a solution of the form N(t) = kX t. Then ~N = kkt+l
__
kkt
= r[kk t] or
kk t+l = (1 + r)(kXt), so that k = 1 + r. The constant k is determined by N(0) = kk~
=No,
N(t) = (1 + r ) N ( t -
1).
Starting with k = 1, repeated substitution of the expression for N(k) into the expression for N ( k + 1) leads directly to N(t) = N0(1 + r) t.
D.1.2. Population Dynamics for Two Species The mathematical situation is somewhat more complicated with two populations. Let the population dynamics be specified by linear homogeneous difference equations Nl(t + 1) = a11N1(t) + a12N2(t), N2(t 4- 1) = a21N1(t) 4- a22N2(t),
(D.5)
where the terms aiiNi(t) absorb the population size Ni(t), as in Eq. (D.2). Two approaches are available for the solution of this system. The first approach utilizes the fact that the equations can be combined into a single second-order difference equation, and the second approach utilizes matrix theory.
D.1.2.1. Second-Order Equations A pair of first-order difference equations typically can be combined into a second-order difference equation, which then can be solved by straightforward algebraic procedures. For example, N 2 can be eliminated from the pair of equations in (D.5) by considering the transition equation for N 1 at time t + 2: Nl(t + 2) = a11N1(t + 1) + a12N2(t + 1) = a11Nl(t 4- 1) + a12[a21N1(t) 4- a22N2(t)] = a11N1(t 4- 1) + a12a21Nl(t) 4- a12a22[N1(t 4- 1) - a11N1(t)]/a12 .
D.1.
First-Order Linear Homogeneous Equations
Thus, elimination of N2 results in the single secondorder equation N l ( t + 2) - OtNl(t + 1) + f3Nl(t) = 0
(D.6)
with oL = all + a22 and [3 = alia22 - a12a21. It is easy to show that the elimination of N1 leads to a difference equation in N 2 with the same coefficients. To solve this system of difference equations, assume a solution of the form Nl(t) = kh t for Eq. (D.6). Substitution of the appropriate time-dependent expressions for population size into Eq. (D.6) results in ~.2 _ O~ q- ~ -- 0,
(D.7)
a quadratic equation that is satisfied for the values )~ = 0.5 [oL ~ (or2 - 4~)1/2]. If the discriminant ~/= ot2 4[3 is nonzero, there are two solutions of the form N l ( t ) = kK t, based on the solutions )~1 and )~2 from Eq. (D.7). Furthermore, any linear combination Nl(t) = k11~.~ + k12~.t)
(D.8)
also is a solution. Population dynamics for Nl(t) thus are determined by__the parameters )~a = (oL + V ~ ) / 2 and ~'2 = ( O ~ - V ' ~ ) / 2 . These parameters are either
711
both real or both complex, d e p e n d i n g on whether ~/->0. D.1.2.1.1. Discriminant y > 0
If the discriminant ~/is positive, then ~'1 and )~2 both are real, and therefore both components of Eq. (D.8) are as well. The behavior of each component depends on the m a g n i t u d e of the exponential term )kt (Fig. D.1). Thus: 9 For ~ > 1, ~.t grows exponentially. 9 For 0 < ~ < 1, )~t declines exponentially. 9 For - 1 < )~ < 0, ~t oscillates each time period between positive and negative values, with amplitudes that decline over time. 9 For K < - 1 , )~t oscillates each time period between positive and negative values, with amplitudes that increase over time. The trajectory N l ( t ) is influenced by both components in Eq. (D.9) and inherits transient characteristics from both. However, one component eventually dominates the trajectory over time. Values of the parameter pairs (ha,)~2) can be grouped according to the asymptotic
x(O b
x(t)
FIGURE D.1 Trajectoryof x(t) = )~t as influenced by the sign and magnitude of )~. (a) ~ is greater than 1. (b) is positive but less than 1. (c) ~ is negative with magnitude greater than 1. (d))~ is negative with magnitude less than 1.
712
Appendix D Difference Equations
behaviors of the corresponding trajectory (Fig. D.2). Because ~kI is always larger than )~2,feasible parameter combinations lie below the line )kI = ~k2,as shown in Fig. D.2. Four regions are defined. Region I. h I > 1, )kI > IK2I. In this region, K1 is greater in magnitude than ~'2, and K 1 exceeds unity (Fig. D.2). Thus the component h~ dominates )~t over time, and the trajectory exhibits asymptotically exponential increases. Region II. 0 < ~'1 <1, ~'1 > IK21. In this region, )kI is greater in magnitude than K2, but K 1 takes only positive values less than unity (Fig. D.2). The component )~ again dominates Kt2over time, and the trajectory exhibits asymptotic declines attendant to the small magnitude of ~'1" Region III. - 1 )~1.Here K1 is smaller in magnitude than ~2 (Fig. D.2), and ~2 takes negative values between - 1 and 0. The component h~ dominates )~ over time, and the trajectory exhibits declining oscillations of period 2 over time. Region IV. K2 < --1, IK2] > )kl. A g a i n )kI is smaller in magnitude than )k2 (Fig. D.2), but )k2 takes values less than - 1 in this region. The component )~t2again dominates h~ over time, and the trajectory exhibits increasing oscillations of period 2 over time. D.1.2.1.2.
Discriminant
iMS, where i = ( - 1 ) - 1 / 2 and B = lYl. The solution of Eq. (D.7) then can be expressed as N l ( t ) = k11~.~ 4- k12)kt2
= k11rt{cos(~pt)
4-
i sin(~pt)}
4- k 1 2 r t { c o s ( q ~ t ) -
i sin(q~t)}
= rt(k11 4- k12)cos(~pt) 4- irt(k11 = rt{kl cos(qvt) 4- ik 2
k12)sin(q~t)
sin(q~t)},
where q~ = tan-l(X/-B/o0. This combination of sinusoidal terms satisfies Eq. (D.7) for any choice of Cl and c2. It is possible to describe Na(t) in real terms only, by N l ( t ) = rt{cl
cos(q~t) + C2 sin(q~t)},
(D.9)
with the corresponding population trajectory exhibiting oscillations of period 2~r/q~. The magnitudes of these oscillations increase, decrease, or remain stable through time depending on whether r > 1, r < 1, or r=l. D.1.2.1.3.
Discriminant
9, = 0
If the parameters in Eq. (D.5) satisfy
y < 0
If the discriminant y is negative, K 1 and K2 a r e complex conjugates with ~ 1 - OL 4- iV~ and ~2 -- OL --
(all 4- a22 )2 = 4(alia22 - a12a21),
then the discriminant y = 0 and a solution to Eq. (D.6) is given by h = (all + a22)/2. Using this value of h in Eq. (D.6) produces
/ / /
\ \
/
\ \ \
/ \
,/ \
-1 /"
/
/ /
/
/
0 ,,
/
III
/
/
/
/
\
\
II \
\
I I I I I
I
2 ~ t+2 -- (all + a22)K t+l --- 0,
11 I I
2
-%
(t + 2)K t+2 - (all + a22)(t4-1))~ t+l 4- 0.25t(a~1 + a22)2Kt \ \
-2
F I G U R E D.2
-- (t4-2)h t+2 -- (all 4- a22)(t4-1)h t+l 4- (alia22 - a12a21)tK t \. \
Partition of the (~1, ~2) plane defined by solutions equation ~2 _ O~K q- ~ = 0, w h e n the discriminant y = oL2 - 413 is positive. In region I, )H > 1, )H > [K2I, and the system trajectory exhibits asymptotically exponential increases. In region II, 0 < K1 < 1, K1 > ]K2], and the trajectory exhibits asymptotic declines attendant to the small magnitude of )~1.In region III, - 1 < K2 < 0, ]K2 ] > K1, and the trajectory exhibits declining oscillations of period 2 over time. In region IV, ~'2 < - - 1 , ]~'2 ] > ~q, and the trajectory exhibits increasing oscillations of period 2 over time.
h l , 2 -- 0.5 [0~ -+ (OL2 -- 4 ~ ) 1/2] o f t h e
and combining these equations yields
'
\
IV
t(a11 + a22)h t+l 4- 0.25t(a11 4- a22)2h t = O.
From h = (all 4- a22)/2, we have
/ \
-2
th t+2 --
/
=0, demonstrating that tK t is a solution of Eq. (D.6), along with hr. The population trajectory thus is given by N l ( t ) = k11)k t + k12(t)~t),
with population dynamics that exhibit exponential change or oscillatory behavior as scaled by the factor kll 4- k12t.
D.1. First-Order Linear Homogeneous Equations
713
It is easy to show that the companion difference equation in N2(t), obtained by elimination of Nl(t), has a solution with a form analogous to that of Nl(t). For example, if y ~ 0, the trajectory of N2(t) is given by
nent that increases exponentially through time and a component that decreases exponentially through time.
N2(t) = k21)` ~ 4- k22)`t2,
The system of equations shown in Eq. (D.5) can be expressed in terms of matrices, by
with the same values )`1 and )`2 a s in the solution for Nl(t). Thus, the trajectory for N2(t) exhibits the same patterns as Nl(t), with the values k21 and k22 related to kll and k12 by k21 = k11()` 1 - a11)/a12
D.1.2.2. Matrix Approach
N2(t+l)j =
[all a121[ 11 a21
a22J
Nl(t)
N 2 ( t ) ] = [ vl V2
On condition that y = 0, the solution again includes t)` t along with )`t: N2(t ) = k21)` t 4- k22(t)`t),
with )` = (all 4- a22)/2 and with k21 , k22 given by the system initial conditions.
(D.10)
or N(t+l) = A N. As above, assume a solution to Eq. (D.10) of the form
and k22 = k12()` 2 - a11)/a12.
N2
])`t
or N(t) = v)` t. Then N(t+l) = )`(v)`t) = A(v)`t), resulting in the matrix equation A v = )`v or
(A - )`/)v = 0.
(D.11)
This equation has a nontrivial solution for values of )` satisfying the characteristic equation
IA_- _II =
-- (all 4- a22))` 4- (alia22 - a12a21 )
(D.12) =0,
Example Consider the population trajectories of _N' = (N 1, N 2) for two populations with interactions defined by the transition equations N l ( t + l ) = 2N1/3 + N2/3, N2(t+l ) = 2N1/3 + N 2. Combining the transition equations results in the second-order difference equation 9N1(t+2) - 15N1(t+1) + 4Nl(t) = 0,
with the corresponding vectors v produced from Eq. (D.11). The parameters )` and v satisfying Eqs. (D.11) and (D.12) are the eigenvalues and eigenvectors of A (see Appendix B). The values of )` solving Eq. (D.12) are )`1,2 = (0L "4- ~ / ~ ) / 2 ,
where oL = tr(A) = all 4- a22 and y = oL2 - 4 ~ , with = IA] = alia22 - a12a21. N o t e that t h e s e are the s a m e values produced from Eq. (D.7) above. With some algebra, it can be shown that, for a12 ~ 0,
and substitution of k)` t into the latter equation yields 9)`2 - 15), + 4 = 0, with solutions )`1 = 4 / 3 and )`2 -- 1/3. The trajectory Nl(t) for population 1 is therefore N l ( t ) = k11(4/3) t 4- k12(1/3) t,
with specific values for kll and k12 determined from the system initial conditions. An analogous derivation for population 2 yields a trajectory N2(t) with the same exponential components and with constants again determined by initial conditions. Thus, the population trajectories for this particular system include a compo-
vi
=[
a12 ]
(D.13)
) ` i - au
is an eigenvector corresponding to )`i" Either of the pairs ()`i, v i) corresponds to a solution _N(t) = vik~ of Eq. (D.10), so that a general solution is given by N(t)=
Cl(Vl)`~)4- c2(vaKt~).
From this expression, it is clear that the trajectories for both populations are controlled by the values of k 1 and k 2.
714
Appendix D Difference Equations
Example
selves complex conjugates of the form Vl, 2 a_ + _bi. Expressing )~ and kt2 as k~ = rt[cos(~pt) + i sin(~pt)] and kt2 = rt[cos(~pt) - i sin(~pt)] from DeMoivre's Theorem, a general solution to Eq. (D.10) may be written as ~-
To illustrate the matrix approach, consider two competing populations with population transitions defined by
N(t + 1)
Nl(t+l ) = N 1 - 0.25N2, N2(t+l ) = - N 1 + N 2,
=
ClVl~.~ if-
C2Va)ktp
= rt{cl(a+ bi)[cos(~pt) + i sin(~pt)]
which can be expressed in matrix notation as ~[_N2(t N l ( t ++ l )1)]
= [ _ 1 -01"25] [X12]"
With an assumed solution of the form N system reduces to [l-k-0.25] -1 1-h
=
_ _ [cos(q~t) -sin(q~t) i +c2(a-bi) ]}
vX t, this
=rtcl{acos(~pt)+a[isin(~pt)]
Iv1] =0 ' v2 -
+ b[i cos(~pt)]
- b sin(~pt)} +rtc2{acos(~pt)-a[isin(q)t)]
which is satisfied for values of k such that - b [ i cos(~p)t]- b sin(~pt)}, I A - kI[ = --
-
1 - k
-0.25 1-k
- 1
= k2-2k
with ~p = tan-l(o~/X/'8). Choosing cI = c2 = 0.5 yields the real solution
+ 3/4
= (2k-3)(2k=0.
whereas cI
From Eq. (D.13), the eigenvectors corresponding to ~k1 = 3/2 and )k2 = 1/2 are v~ = [1, -2] and v~ = [1, 2] respectively. Thus, the general solution for the system of equations is
N2(t)
= Cl
[ 1] -2
u(t) = a cos(~pt) - b sin(q)t),
1)/4
(1"5)t q-
c2
[12]
5 0 = C1 q- C2
produces
w(t) = a sin(q~t) + b cos(q~t). Because any linear combination of these expressions is a solution, N(t + 1) = rt[ClU(t) + C2w(t)]
(0"5)t'
with cI and c2 determined from the system initial conditions. For example, if _0 N' = (50, 80), then
-- --C 2 = 0 . 5 i
(D.14)
is a general real solution to Eq. (D.10).
Example Consider the system of equations
Nl(t + 1 ) = 2N 1 + 3N2/2,
and 80 = -2c I + 2c2,
N2(t + 1 ) = - 2 N 1 / 3 + N 2.
so that (c1, C2) = ( 5 , 45). The particular trajectories of Nl(t) and N2(t) therefore are
As above, an assumed solution of the form N = vh t allows this system to be expressed in matrix form as
Nl(t ) = 5(1.5) t + 45(0.5) t and N2(t) = -10(1.5) t + 90(0.5) t, with population 1 exhibiting exponential growth and population 2 quickly driven to extinction. Of particular interest are systems for which )k I and complex conjugates. With complex eigenvalues, the corresponding eigenvectors in Eq. (D.11) are them~'2 a r e
m
[2-k -2/3
3/2 ] [Vl] = 0 ' 1- k v2 -
which is satisfied for values of k such that IA-
_II =
2- k -2/3
3/2 1 - k
= k 2 - 3k + 3 =0.
(D.15)
D.1. First-Order Linear Homogeneous Equations The latter is satisfied by k = V ~ ( V ~ / 2 ___ i / 2 ) = V 3 [cos('rr/6) + i sin(-rr/6)], and substitution of these values into Eq. (D.15) produces Vl, V 2 -- a -4- bi
E3
E 3]i
Defining
E B1
=
ci]k~,
i=1
- 1 cos (6t) -
E3]
with a specific solution determined by the system initial conditions. Alternatively, a matrix approach can be used to determine the population trajectories. For n populations, the equation N(t + 1) = A N is of dimension n, which results in a characteristic equation with n roots. Thus, there are n combinations (ki, vi) of eigenvalues and eigenvectors that satisfy Eq. (D.11), any of which corresponds to a solution
sin(6t)
and w ( t ) = a sin(q~t) + b cos(q~t)
[:1 sin(6t, [ 31 cos6, from Eq. (D.14) a general population trajectory is given by N(t) = 3t/a[ClU(t) + C 2 w(t)]
with the coefficients C 1 and C 2 determined by system initial conditions. For example, if _N'0 = (40, 50), then
[ Cl ] _C 1 if- V~C2
[401 50'
C1
=40,
C2 =
30 V~, and
3j2(E40] 50
= Vik ~
of N(t + 1) = A N. The general solution is given by
[30] )
cos(6t) + X/3 _70
(D.16)
Ci Vi~. I .
]
(-C1 + C2X/3)cos (6t) - (C1V3 + C2) sin (6t) '
X(0)--
N(t)
N(t) -- i~
C1 cos (6t)+ C2 sin(6 t)
= 3t/2 [
--
and two approaches can be taken to determine the population trajectories. The first involves transformation of the equations into a single nth-order equation, in the same manner as for two equations. This produces an nth-degree polynomial equation with n roots k i, i = 1, ..., n, each of which corresponds to a solution cik ~. The general solution is Ni(t) = ~
u ( t ) = a cos(q~t) - b sin(q~t)
so that
715
As above, the population trajectories are controlled by the values ki, i = 1..... n. For example, the populations decline if 0 ~ ~ki < 1 for all eigenvalues, and increase if all eigenvalues are positive and at least one eigenvalue exceeds unity. Oscillations occur if there is at least one pair of complex conjugate eigenvalues a n d / o r at least one negative eigenvalue. It is clear that the inclusion of additional populations into a system, increasing its dimensionality and thus increasing the number of eigenvalues, can lead to greater complexity in system behaviors. Because the exponential terms in Eq. (D.16) all have a value of unity when t = 0, the population initial state is simply
sin(6t) .
tl
_N ( 0 ) = ~ The oscillatory nature of this trajectory is most easily understood in terms of oscillations about some equilibrium system state (see below). The trajectory then exhibits increasing oscillations about the equilibrium population levels, with an oscillation period of 12.
CiVi,
i=1 indicating that the constants c i are directly related to population initial conditions. This relationship can be expressed in matrix notation as c1
D.1.3. Population Dynamics for Multiple Species
c2
__N(0) = [Vl
The population dynamics of n species can be characterized by n first-order difference equations N ( t + 1) = A N ( t ) ,
v2
"'"
Vn] Cn
= Vc,
716
Appendix D Difference Equations
so that c = V - i N ( 0 ) . On condition that A is symmetric, it can be shown that the eigenvectors v i are both real and orthogonal, in that v' i vj = 0 for i ~ j (see Appendix B). For example, a system of three populations with symmetric transition equations has
Fv l .
.
i
=
C1
(-1) t
if- C 2
--
(--0.5)
t 4-
C3
(-0.25) t,
N3(t)J
V~_V2
1
~
[Cl] Ivy/a|i25o1
.
9
=
N2(t) /
with Cl, c2, and c3 determined by the system initial conditions. For N(0)' = (250, 50, 100), the constants are given by
v~vl
=
(1, 1, 1), v~ = (1, - 1 , 0), and v~ = (1, 1, - 2 ) , so that a general solution is
C2
:
C3
9 V3V 3
Lv;/6J
~1331
:[lOOl,
~
K2
and the corresponding population trajectories are Nl(t ) = 133(-1) t + 100(-0.5) t + 17(-0.25) t,
=)~,
N2(t ) = 133(-1) t -
from which it follows that K-iV'N0
L 00j
100(-0.5) t + 17(-0.25) t,
N3(t) = 133(-1) t - 34(-0.25) t. =c.
Thus, an eigenvector decomposition of A sometimes provides a convenient way to compute the constants in c corresponding to a set of population initial conditions. Example
It is instructive to note the similarities between this solution and that of an analogous example in Section C.1 for conditional time. Thus, the coefficients of the two solutions are identical, so that the only effect of a discretized time frame is that the exponential function e ~it in the continuous-time solution is replaced by the power function K~ in the discrete-time solution.
To illustrate, consider the system N2(t+l) N3(t + 1 )
=
[_1.75075 o5]rNlt] 0.75 0.5
1.75 0.5
0.5 2.0
|N2(t) .
LNB(t)
This system is analogous to an example in Section C.1, which there was described in terms of continuous time. For the present case, a solution is assumed to be of the form N(t) = v)~t, so that the system equations reduce to
[175 075 05][Vl] -0.75 -0.5
-1.75 - )~ - 0 . 5 -0.5 -2- K
v2 V3
= 0,
(D.17)
which is satisfied for values of )~ such that - ~ -0.75 -0.5 ] -0.75 -1.75- h -0.5 =0. -0.5 -0.5 -2-
D.2. NONLINEAR HOMOGENEOUS EQUATIONS~ STABILITY ANALYSIS An important extension concerns homogeneous difference equations that include nonlinear terms. In this case, the function F(N) in N ( t + 1) = F(N) includes terms such as N12,N i Nj, and other mathematical expressions that are nonlinear in the population values N i. Examples include the logistic and Lotka-Volterra models in Chapter 8. We restrict attention here to an analysis of population dynamics for populations that are "near" an equilibrium, for the purpose of assessing equilibrium stability.
-1.75
The latter is a polynomial equation with the three roots )~ = - 1 , -0.5, and -0.25. Substituting these values back into Eq. (D.17) produces the eigenvectors v~ =
D.2.1. Stability Analysis for One Species Consider the dynamics of a population with a nonlinear growth function F(N) for which derivatives exist over some operative range of population size. Assume that the population is in equilibrium at a value N*, so
717
D.2. Nonlinear Homogeneous Equations--Stability Analysis that F(N*) = N*. Then population dynamics can be expressed in terms of a Taylor series expansion of F about N*:
about the equilibrium are given by Eq. (D.19), with ( d F / d N ) ( N * ) determined by d ( l n F ) = 1 dF dN
N ( t + 1) = F(N* + n t) dF n2t daF = N* + nt-d-~(N*) + - - ~ - ~ ( N * )
1
=N
+ ...,
with n t = N ( t ) - N * describing "small" deviations about N*. The higher degree terms in Eq. (D.18) are of negligible importance, leading to the simplified expression
d--n
= 1-
dF nt--d-~(N*),
dF
nt--~(N
,
-
lnot.
Thus the equilibrium condition N* = In or/[3 is stable for all values of c~ such that I 1 - log ~1 < 1, in that small deviations from N* decrease through time to 0. Example
so that the equation for population dynamics can be written in terms of deviations n t = N ( t ) - N*, as --
-1
ot
Expressing N ( t + 1 ) = N* + nt+l, we have
lit+l
- ~ e-~N*
= a In oL
dF , = N* + n t -d--~(N ).
= N* +
f~"
Then
N ( t + 1 ) = F(N* + II t)
N* + lit+l
F dN
(D.18)
Consider the logistic model, Eq. (D.4), which can be reparameterized as N ( t + 1) = r N ( 1 - N / K ) ,
).
(D.19)
In this way a nonlinear transition equation can be approximated by a linear difference equation in a neighborhood of N*. It follows that N* is a stable equilibrium if F' (N*) < 1 (because the trajectory of deviations exhibits exponential decay a n d / o r damped oscillations) and N* is an unstable equilibrium if F' (N*) > 1 (because the trajectory of deviations exhibits exponential growth a n d / o r increasing oscillations).
with constant growth parameter I < r < 2 and constant carrying capacity K > 0. The model has two equilibria, N* = 0 and N* = K(1 - l / r ) , and deviations in a neighborhood of N* are given as in Eq. (D.19), by dF
nt+ 1 = n t - ~ ( N
,
)
= rnt(1 - 2N*/K).
Population dynamics around N* = 0 are given by F/t+ 1 - - t.litr
Example
A model of broad applicability for fish population dynamics is the Ricker model N(t+l) = otN(t)e -~N(t) with the parameter ot representing a m a x i m u m population growth rate and [3 inhibiting growth with increasing population size. Population steady state N* for the Ricker model is given by N(t+l) = N ( t ) = N*, so that N * = otN*e-f3N* or
1 =
~e-~N*.
After some algebra, N* = In ot/ [3 is seen to be a nontrivial equilibrium. The dynamics of small deviations
which exhibits simple exponential growth away from 0. Thus, N* = 0 is an unstable equilibrium, in that positive deviations from 0 increase in magnitude. On the other hand, population dynamics around N* = K(1 - l / r ) are given by nt+ 1 = rnt(1 -- 2 N * / K ) = rnt[1 - 2(1 - l / r ) ] = nt(2 - r),
which exhibits simple exponential decay toward 0. Thus, N* = K(1 - l / r ) is a stable equilibrium, in that deviations from K(1 - l / r ) lead to asymptotic declines in the deviations (and thus to asymptotic convergence of N ( t ) to K(1 - 1 / r ) . It is useful to consider the influence of the parameter
718
Appendix D
Difference Equations
r in the stability conditions N* = 0 and N* = K(1 1/r) for the logistic model. From nt+ 1 = rn t it follows that N* = 0 is an unstable equilibrium only for r > 1, because all other values of r produce declining (or negative) deviations and hence population extinction. On the other hand, N* = K(1 - 1/r) is a stable equilibrium only for values of r such that 1 < r < 3; for all other values, nt+ 1 = nt(2 - r) produces deviations that fail to converge to zero [and populations that do not to return to K(1 - l / r ) ] . In particular, as values of r increase from 3 to 4, the population exhibits stable limit cycles with increasing periodicity. It can be shown that values of r beyond 4 produce unstable behaviors that can lead to extinction (May, 1976).
D.2.2. Stability Analysis for Two Species The addition of another state variable complicates the analysis of system stability. Consider two populations with nonlinear growth functions F(N)' = [FI(N), F2(N)]. A Taylor expansion about an equilibrium va~ue N* is 1 k , k , F k ( m * nu Fit) = Fk(N *) + n tFl(N_ ) + n 2tF2(N )
(nl) 2 +
2
Fkl(N*)
-+-
(nt2) 2 2 Fk2(N*)
Ht+l
3F 1
L 3F
3F2
2
aVii("_*) ;G2("_*) = J(N*)nt,
and the properties of J(N*), k n o w n as the Jacobian matrix, determine the equilibrium stability of the system. For example, the analysis of linear difference equations above indicates that N* is a stable equilibrium if the r o o t s ~.1 and ~k2 o f I a - ~._/I -- ~.2 _ o/.K -+= 0 are both of m a g n i t u d e less than 1. If so, then deviations (n I, n 2) from N* will decay through time, and N* is a stable equilibrium. Determination of the stability properties of a nonlinear system does not require one to solve the determinantal equation above for K1 and )~2. Because ot = ~k1 nt- ~k2 = tr[](N*)] (see A p p e n d i x B), a necessary condition for stability is - 2 < oL < 2 or Ic~/21 < 1. Additional conditions are that 0.5(or + , ~ 1 / 2 ) ~ 1 if ot > 0, and 0.5(OL _ ~ / 1 / 2 ) > - 1 if ot < 0, where y = OL2 -- 4[3 and [3 = IJ(N*)I. The latter two inequalities can be combined into a single inequality ]o~/2[ + X/yy/2 < 1, which, after some algebra, simplifies to 1 + [3 > Ic~I. Because ~ = ~.lK2 = I ! ( X * ) l < 1, w e therefore have
for k = 1, 2, where Fk(N) = oFk(N)/ONi and Fk(N) = c92Fk(N)/ONi cONj. For "small" devi--ations __Ht = N(t) N*, the higher degree terms are of negligible importance, and Eq. (D.20) reduces to +
[ni+I]
3F
(D.20)
+ (n~n 2t)F12(N k , ) + ...
Nk(t + 1 ) = Fk(N *
neighborhood of N*. Equation (D.21) is written in matrix notation as
levi < 1 + ~ < 2 .
Example Consider a system of two populations with (scaled) dynamics given by
F/t)
Nl(t + 1) = Nl(t)exp{-0.2511 - N2(t)]}
= N~ + n lFk(N *) + r l 2t F 2k ( X , ). and As in the single-species case, we can express Nk(t + 1 ) = N ' ~ + n kt + 1 , SO that N~
+
F/k+l
--
g'[ + n~Fk(N *) + n t2f a (kN
,
),
so that the equation for population dynamics can be expressed in terms of the deviations F/t = X ( t ) - X * ,
The equilibrium condition _N (t + 1 ) = N* yields exp{0.2511 - N~]} = 1.0 and N~ = 0.5N~(3 - N'~/N1), with the resulting nontrivial equilibrium point N*' = (N~, N~) = (1.0, 1.0). The behavior of small deviations about N* is governed by
n,,1]:[ 1 0.25] rn'l
as 2 k , ntk+l = n t1F lk( N , ) + ntF2(N ),
N2(t + 1) = 0.5N2(t)[3 - N2(t)/Nl(t)].
(D.211
k = 1, 2. Thus, the nonlinear transition equations can be approximated by linear difference equations in a
I
n2+1
--0.5
0.5
Ln 2]
for n(t) = N(t) - N*. Because oL = ( a l l + a22) = 1.0 + 0.5 = 1.5 and ~ = (1.0)(0.5) - (-0.5)(0.25) = 0.625, we
D.2. Nonlinear Homogeneous Equations--Stability Analysis have Jo~J< 1 + ~ < 2 and the equilibrium (1.0, 1.0) is stable. This result is confirmed by an eigenanalysis of the transition matrix, which reveals that the system eigenvalues are complex conjugates of magnitude less than unity.
and so on. Each list of subscripted coefficients is shorter than the list that precedes it alphabetically, until there are only three quantities that relate to their predecessors by the rule qn
D.2.3. Stability Analysis for Multiple Species
qn-1 qn-2--
As above, a description of the population dynamics for n species requires n transition equations, one for each species. In theory the stability of small perturbations about an equilibrium point can be determined by linearization of the transition equations as above. Nevertheless, a stability analysis for n species still involves finding the zeros of a polynomial equation of degree n, a difficult task for large values of n. However, it is possible to specify necessary and sufficient conditions such that all zeros are of magnitude less than unity. Thus, consider the polynomial P(k)
=
~n
4- a l k n - 1
4- a 2 k n - 2
4- . . .
719
=
p2_
2 Pn-3,
= PnPn-1
-- Pn-3Pn-2,
PnPn-2-
Pn-3Pn-l"
Then necessary and sufficient conditions for all zeros of P(M to be of magnitude less than unity are as follows: 1. P(1) = 1 + a I 4- a 2 4- ... 4- a n _ 1 4- a n ~ O. 2. ( - 1 ) nP(--1) = ( - 1 ) n [ ( - 1 ) n 4- a 1 ( - 1 ) n - 1 4a2(--1) n-2 4- -'- 4- an - 1(-1) + an > O. 3.
lan] < 1,
[b l > [bl[, ]On]
>
]C2[,
]dn]
>
]ds],
Iqn]
~
Iqn-l[
an
4- a n _ l K 4-
of degree n. Let bi, ci, di, etc. be defined by bn = 1 - a 2,
c n = b2 - b2,
d,, = c2 - c2,
bn-1 - al - a,,an-1,
%-1 = bnb,-I - bib2,
dn-1
=
bn_ k = a k -
Cn_ k = b n b n _ k -
dn-k
= CnCn_ k -
anan_k,
bl = an-1 - an-1 - anal,
blbk+l,
c2 = bnb2 - blbn_l,
d3
=
CnCn-1
CnC 3
--
--
C2C3,
C2Ck+2,
C2Cn-1,
(Jury 1971). As an example, consider the zeros of the polynomial P(M = )k4 4- ~3 4- K2 4- ... 4- K 4- 1. Conditions (1) and (2) above are satisfied, because P(1) > 0 and (-1)P(1) = 1 (1 - 1 + 1 - 1 + 1) > 0. However, condition (3) fails, because a 4 = 1. Thus, there is at least one zero of P(M that is not smaller than unity, and the corresponding system equilibrium is not stable.
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A P P E N D I X
E Some Probability Distributions and Their Properties
parameters: the sample size n and probabilities Pl, ..., Pk-1 (conditional on Pl, ..., Pk-1, the parameter Pk is given by s pj = 1). Means and variances for the random variable x i of a multinomial distribution are ~i - - npi and 0-/2 = npi( 1 _ Pi), respectively, and the covariance between x i and xj is cov(xi, xj) = - n p i P j . If n is assumed known, the m a x i m u m likelihood estimate of Pi is ]9i -- x i / n . In some applications (for example, when the parameter n is identified as the size of the population rather than the size of the sample), n is u n k n o w n and must be estimated. If ~i is an estimator of n, the conditional m a x i m u m likelihood estimator (conditional on ~i) of Pi is given by Pi = Xi/19l" When there are only two categories of individuals in the population, the multinomial distribution reduces to the binomial distribution, with probability density function
In this appendix we describe some statistical distributions that often arise in modeling and estimation of animal populations. Because of the emphasis in this book on count data for estimating parameters, we describe several distributions that are appropriate for counts. We also focus on distributions that arise in the application of m a x i m u m likelihood estimation and likelihood testing procedures. Probability density functions for these distributions are described in terms of their moments, shaping parameters, and other relevant statistical properties. Where appropriate, parameter estimators also are given. A more detailed treatment of statistical distributions can be found in references such as Evans et al. (2000) and Johnson and Kotz (1969, 1970a,b). Distributions of linear and quadratic forms are covered in detail by Searle (1971) and Graybill (1976).
n~ f(xlp'n)
E.1. DISCRETE D I S T R I B U T I O N S
Consider a trial for which k distinct outcomes are possible, and denote the probability associated with each outcome as Pi, with s Pi = 1. Suppose there are n trials, and let x i denote the number of trials for which outcome i is observed. If the trials are independent, then the resulting probability density function is n Xlr
"'"
Xk
i=1
P)
yl m X
"
The binomial sometimes is denoted by B(n, p) or B(xln, p) to emphasize the roles of n and p. The parameter n determines the number of values that x can take, and p influences the probability mass associated with each of these values. The mean and variance for the binomial distribution are ~ = np and 0 -2 = np(1 - p). Figure E.la shows the binomial distribution for different values of n, and Fig. E.lb shows the distribution for different values of p.
E.1.1. Multinomial Distribution
f ( x l p , n) =
x!(n - x)! px(1
xi Pi ,
E.1.2. Poisson Distribution
with ~i Xi -- Yl. Note that if x 1, ..., Xk_ 1 are given, then the value of Xk is determined by ~i Xi = Y/. Note also that the distribution is parameterized by k independent
The Poisson is a discrete distribution that corresponds to the counting of occurrences of some event 721
722
Appendix E Some Probability Distributions and Their Properties
F I G U R E E.1 Binomial probability density function. (a) Effect of the parameter n (number of trials), for p = 0.7. (b) Effect of the parameter p (probability of success), for n = 10. The binomial mean varies with changes in both n and p, according to E ( x ) = tip.
(e.g., birth, death, or migration) over some continuous time frame T. Because a Poisson r a n d o m variable is restricted to nonnegative integer values, the Poisson is a candidate for the distribution of any counting process. For temporal processes it arises u n d e r the following conditions: 1. For an arbitrary time t in the time frame T, the probability of exactly one occurrence in a "small" interval [t, t + h] is approximately ~h:
3. The n u m b e r s of occurrences in n o n o v e r l a p p i n g time intervals are independent: if there are Xl occurrences in [tl, tl + hi], x2 occurrences in [t2, t 2 + h2], and t 2 ~ t I 4- h i , then Prob(Xl, x 2 ) = Prob(Xl) • Prob(x2). If these three conditions are satisfied, the n u m b e r x of occurrences in a period of length t has a Poisson distribution with probability density function f(x[h,) = e-"~.X/x!,
Prob(one occurrence in It, t + hi) = ~h + o(h), where o(h) is some value with limiting m a g n i t u d e of degree less than h:
o(h)
lim T h-~0
= 0.
2. The probability of more than one occurrence in [t, t +h] is negligible w h e n c o m p a r e d to the probability of a single event: Prob(two or more occurrences in It, t + hi) = o(h).
where K = ~t (h, is referred to as the m e a n rate of occurrence). A Poisson r a n d o m variable can take any nonnegative integer value and the distribution parameter K can a s s u m e any positive value. The p a r a m e t e r ~, influences the spread of the Poisson distribution, such that distributions with smaller values of ~, are more peaked (Fig. E.2). The Poisson sometimes is denoted by P(M or P(x[M to e m p h a s i z e the role of ~. The m e a n and variance of a Poisson r a n d o m variable x are identical, with E(x) = var(x) = ~,. For r a n d o m samples of size k, the m a x i m u m likelihood
E.1. Discrete Distributions
723
FIGURE E.2 Poissonprobability density function. Effects of changes in the parameter )~.
estimator of ~ is ~. count for sample j.
= ~,j xj/k,
where
xj
represents the
E.1.3. Geometric and Negative Binomial Distributions The geometric distribution represents the n u m b e r of failures before the first success in a sequence of independent Bernoulli trials (see Section 10.1). R a n d o m variables with geometric distribution take nonnegative integer values according to the probability density function
f(xlp)
= p(1 -
p)X,
where the parameter p can assume any value in the interval 0 < p < 1. As illustrated in Fig. E.3, the probability density function declines geometrically for all values of x _> 1, with the rate of decline specified by the parameter p. The mean and variance for the geometric distribution are given by E(x) = (1 - p)/p and var(x) = (1 - p)/p2. The m a x i m u m likelihood estimator for p is/~ = k/(k + ~,j xj), based on a sample of k r a n d o m variables. The negative binomial distribution is closely related to the geometric, in that the sum of independent geometric r a n d o m variables is distributed as a negative
binomial. The probability density function of the negative binomial distribution is
f(x[r'P) = (r + x - 1 )
pF(I -
where r is any positive integer, 0 ~ p ~ 1, and x can take any nonnegative integer value. The r a n d o m variable x in this distribution can be thought of as the n u m b e r of additional trials (beyond the m i n i m u m possible number, r) required to record r successes in a sequence of independent Bernoulli trials. As s h o w n in Fig. E.4, the parameter r influences the location of the m o d e of the distribution and the parameter p plays a similar role as in the geometric. The mean and variance of the negative binomial are given by E(x) = r(1 - p)/p and var(x) = r(1 - p)/p2. The m a x i m u m likelihood estimator for p is ]~ = kr/(kr + ~,j xj), based on a sample of k r a n d o m variables. As mentioned above, the sum of r identically distributed geometric r a n d o m variables has a negative binomial distribution: if the r a n d o m variables xj, j = 1, ..., r are distributed as geometric and y = ~,j xj, then y is distributed as a negative binomial. It follows that the geometric distribution is a special case of the negative binomial, in that the geometric distribution is simply a negative binomial distribution with r = 1.
FIGURE E.3 Geometricprobability density function. Effects of changes in parameter p.
724
Appendix E Some Probability Distributions and Their Properties
F I G U R E E.4 Negative binomial probability density function. (a) Effect of parameter r on the distribution (p = 0.3). (b) Effect of the parameter p (r = 2).
E.1.4. Hypergeometric Distribution Like the multinomial distribution, the hypergeometric distribution generally is applicable to sampling situations in which k distinct outcomes are possible. However, the hypergeometric differs from the multinomial in the size of the population a n d / o r the manner of sampling. A hypergeometric distribution is appropriate under the following conditions: 1. A finite population consists of k different categories with sizes M' = (M1, ..., Mk). It is assumed that every individual is in one and only one of the categories, i.e., ~ i M i -- N . Thus the population is assumed to be finite. 2. Sequential sampling of the population is without replacement; i.e., once an individual has been selected, the individual no longer is available for subsequent selection. The size of the sampled population thus is effectively reduced by one. It follows that the probability of selection of any individual is influenced by the selection of others, so that individual selections are not statistically independent events.
3. All combinations of n individuals are equally likely to arise in a sample of size n from the population. For example, any individual in the population is equally likely to be chosen in a sample of size 1; any combination of two individuals is equally likely to be chosen in a sample of size 2; and so on. Under these conditions, a hypergeometric distribution is defined as follows. For a random sample of size n, let x i denote the frequency of occurrence of individuals from category i, i = 1, ..., k. The vector x' = (x I .... , x k) of frequencies is described by the probability density function m
H:lt -
f(xlM) =
X i
where ~'i Xi -- n. Note that the distribution is parameterized by k + 1 independent parameters: the population size (N), sample size (n), and category sizes (M1, ...,
E.2. Continuous Distributions
Mk_ 1) for k - 1 of the categories (conditional on M 1, 9.., Mk-1, the parameter M k is given by ~,j Mj = N). If there are only two categories of individuals in the population, then the hypergeometric distribution with multiple categories reduces to the standard hypergeometric distribution, with probability density function
f(xln, N, M) =
t xl
725
and var(x) = n ( M ) t N
NM)(N
-11
9
Figure E.5a shows the standard hypergeometric distribution for different sample sizes, and Fig. E.5b shows the influence of category size M on the distribution.
E.2. C O N T I N U O U S In this expression, the parameter N is the population size, M is the size of one of the two population cohorts, and n is the sample size. The parameter n determines the n u m b e r of values that the hypergeometric r a n d o m variable x can take, and both N and M influence the relative probability mass associated with these values. The mean and variance for the standard hypergeometric distribution are M
E(x) = n m N
DISTRIBUTIONS E.2.1. Normal Distribution The normal distribution is appropriate for continuous m e a s u r e m e n t s with m e a s u r e m e n t frequencies that decline rapidly as the m e a s u r e m e n t s deviate from some central value. The normal also is a limiting distribution in the central limit theorem and arises in the theory of m a x i m u m likelihood estimation. It therefore is used extensively in statistical modeling and estima-
FIGURE E.5 Hypergeometric probability density function. (a) Influence of sample size (n) for fixed category size (M = 25) and population size (N = 100). (b) Influence of category size (M) for fixed sample size (n = 25) and population size (N =100).
726
Appendix E Some Probability Distributions and Their Properties
tion. The probability density function of the univariate normal distribution is
f(xll~, 0-) -
1 V'2"rr0-
exp
[1( )21 X--Ia,
-
,
size the role of the mean and variance in specifying the distribution. M a x i m u m likelihood estimates of I~ a n d 0 -2 a r e ~ = ~ i x i / F l a n d 4 2 -- ~ i ( X i -- t~)2/F/ for a r a n d o m sample of size n. Because 42 is biased, the adjusted estimator s 2 = ~ i ( x i - ~)2/(F/ -- 1) typically is used in its place. An intuitive extension of the univariate normal distribution involves m e a s u r e m e n t on individuals of two or more attributes instead of one. If the corresponding r a n d o m variables are normally distributed, then the vector of variables is said to have a multivariate normal distribution. For example, the bivariate normal distribution is defined by two r a n d o m variables, Xl and x 2, with probability density function
(E.1)
0-
which is parameterized by the population mean # and the standard deviation 0- (or equivalently, the variance 0-2). The distribution is bell-shaped, symmetric about IJ,, and more or less peaked d e p e n d i n g on 0-. The mean IJ, is a location parameter, in that it specifies the location (but not the shape) of the distribution. The variance is a shape parameter, in that it specifies the shape (but not the location) of the distribution. Small values of 0correspond to distributions that are highly peaked, with probability mass concentrated about the population mean. Figures E.6a and E.6b show the influence of D and 0- on the univariate normal. Typically the normal is denoted by N(I~, 0-2) o r N(x I ~ , O'2), to empha-
f ( x l ' Xa]lJt,1 ' ill,2, 0-2, 0-2, 0-12)
= (2,rr) -1 I ~ 1 - 1 / 2 exp{--(2i~l) -1 [0-2(X 1 -- IJbl)2 + 0-2(X2-
]Jb2)2 + 20-12(Xl -- ~ 1 ) ( X 2 -
f(x) a
p.=-I
~
0.4
0.3
/ 0.2
/
0.1
!
/i
i ,'
I "
/
l.l=O
/ //""\\
, A' \
/ \/ ,' ~ ,; /\./\ ,,' ,
/
/,,~,
iJ.=l
/
"-'..\ ,, ', ',,,
\
\
\
'\.
',
\,, \x
k
0.0
f(x)
0.4] b 0.3 o2=1
0.2
0.1
.:'Y/ .//// ././ /
/-"
\'%, \ x,/
a2=3
\ \~,"\ a2=2 X ",,~
/
0.0 -2
0
2
4
FIGURE E.6 Normal probability density function. (a) Influence of mean I~on the distribution. (b) Effect of the standard deviation or.
I-1'2)]}
E.2. Continuous Distributions
in Fig E.7 corresponds to a probability mass of I - oL, i.e.,
or
f(xlp_, Y-)
727
= ( 2 ~ ) -1
lY.I-~/2
exp [-(x
- ~_)' ~ _ - l ( x - ~_)/2],
P r o b [ ( x - D)'~-l(x
where Ix' = (I/,1, 1.1,2) is a bivariate vector of means for x I and x 2 and 0-1, 0-2, and 012 are the variances and covariance for x I and x 2, respectively. The expression ]~___1 = 2 2 0-10.2 0"22 represents the determinant of the dispersion matrix
_
2
O.12
0-2
l
consisting of the variances and covariance of x I and x 2. As shown in Fig. E.7, the bivariate normal corresponds to a bell-shaped surface that is centered at Ix, with the spread and orientation of the distribution determined by the variances and the covariance, respectively. The ellipse (X-
IJ,)'~-I(x-
__~) -
X2_.(2)
- la,) ~ X2_~(2)] = 1 - oL,
where X12_~(2) is the 1 - oL quantile of a chi-square distribution with 2 degrees of freedom (see below). M a x i m u m likelihood estimates (MLEs) of the means in ix and the variances in ~_ are given as above, and the MLE for covariance is 412 = ~ k ( X l k -- ~l)(X2k -- ~ 2 ) / n. Because 612 is biased, the bias-adjusted estimator $12 = ~k(Xlk~1)(X2kt3"2)/(n -- 1 ) t y p i c a l l y is used in place of ff12" Similar expressions hold for multidimensional systems for which the n u m b e r of variables is greater than 2. If samples are characterized by k variables, x' = (Xl, ..., Xk), the probability density function is given by f(xIth, E) = (2,rr)-k/2[~[ -1/2 e x p [ - ( x - I.I,)'E-I(x-
where IX' = (I Jr,l, ..., I.l,k), ~___is a k-dimensional dispersion matrix of variances and covariances, and [E] is the de-
F I G U R E E.7 Bivariate normal distribution. The 1 - oL probability ellipse for the bivariate normal random variables is parameterized by the 1 - oL quantile of a chi-square with 2 degrees of freedom, according to ( x - IX)' ~ _ - l ( x - IX) =
x~_o(2).
ix)/2],
728
Appendix E Some Probability Distributions and Their Properties
terminant of the dispersion matrix ~. By analogy with the two-dimensional case, a k-dimensional ellipsoid (X-
I,t,)'~-l(x-
___~) =
a
Axlk)
X2_o,(k) k=3
corresponds to a probability mass of 1 - oL: 0.2
P r o b [ ( x - }.l,)'~-l(x__- ____~)<-- X12_~(k)] = 1 - a. \k = 5
Maximum likelihood estimates for the distribution means, variances, and covariances are computed in analogous fashion to those of the bivariate normal.
0.1
E.2.2. Chi-Square Distribution
0.0
// \ \\ _. / \ / '~ N - ' ' k = 1 0 / /X~, \ \\ i / ~ \\ \., / / ,
....
The chi-square distribution arises naturally from the normal distribution, in that the square of a standard normal random variable has a chi-square distribution with one "degree of freedom," and the sum of squares of k independent, standardized normal random variables is distributed as chi-square with k degrees of freedom. The term "degrees of freedom" refers to the number of independent random variables in such a sum of squares. The chi-square distribution is a continuous, nonnegative distribution with probability density function
,-"2 "---Z~-'--
5
0
10
-7
....
15
T
..........
x
20
t~xlk,z) ~=1
k=5
0.10
_...~..~5\ /
//
~
N,,,. . . . K=10
0.05
-....
f(xlk) =
1
F(k/2)2k/2 x
(k-2)/2 - x / 2 e
/ . ~
.
(E.2)
It is nonnegative, unimodal, asymmetric about the mode, and skewed in the positive direction. With larger degrees of freedom, the distribution has greater probability mass in its tail and is less peaked at its mode (Fig. E.8a). The mean and vairance for the distribution are k and 2k, respectively. The chi-square distribution in Eq. (E.2) generalizes naturally to the noncentral chi-square distribution oo
f ( x l k , h) = ~
j=0
--
e xxJ
J!
x (k+2j-2)/2 e - x / 2 F[(k + 2 j ) / 2 1 2 j+(k/2)'
where h is a noncentrality parameter, k corresponds to the chi-square "degrees of freedom," and F(r) is the gamma function defined by F(F) = f ~ Z r-1 e - x dx. The parameters h and k have roughly similar effects on the location and shape of the noncentral chi-square distribution, with an increase in either parameter resulting in greater probability mass in the tail of the distribution, a decrease in its modal density, and a shift in the distribution mode in the positive direction (Fig. E.8b). The mean and variance for the distribution are k + 2h and 2(k + 4h), respectively. On assumption that a chi-square random variable
0.00 0
,
B
,
i
5
10
15
20
FIGURE E.8 Chi-square distribution. (a) Effect of degrees of freedom k for central (h = 0) chi-square distribution. (b) Effect of the noncentrality parameter h.
is composed of a sum of squares of k random variables with multivariate normal distribution, its noncentrality parameter reflects the influence of the multivariate mean and dispersion. It can be shown that if _x' = (Xl, ..., x k) has a multivariate normal distribution with mean tx and dispersion ~, the quadratic form x ' ~ - l x has a chi-square distribution with k degrees of freedom and noncentrality parameter h = Ix'~-I I,L//2. In particular, the square X2/O- 2 of a single normal random variable is distributed as chi-square with 1 degree of freedom and noncentrality ~2//(2o.2). The chi-square distribution arises frequently in statistics and is a limiting distribution in goodness-of-fit testing procedures, model comparison procedures, and analysis of categorical data. A typical application would investigate the fit of data to a hypothesized statistical distribution, via chi-square goodness-of-fit procedures (see Section 4.3.3). Likelihood ratio test statistics are asymptotically distributed as chi square (see Section 4.3.4), and the noncentral chi-square distribution provides a means of computing the power of such tests.
E.2. Continuous Distributions
E.2.3. E x p o n e n t i a l and Gamma Distributions The exponential and gamma distributions are continuous analogs of the geometric and negative binomial distributions, in that the former are appropriate for continuous (as opposed to discrete) waiting-time processes. The exponential and gamma distributions are candidates for statistical models of the time until occurrence of some event. The exponential is a continuous, nonnegative distribution that is parameterized by a single parameter k. Its probability density function is f(xlX) = he -xx, where x can take any nonnegative value, and the parameter h, which represents the mean number of occurrences per unit time, can be any positive value. The probability mass decreases monotonically as x increases from zero, with the rate of decrease dependent on the size of h. As illustrated in Fig. E.9, rapid declines in probability mass correspond to large values of h. The exponential distribution sometimes is denoted by Exp(h) or Exp(xlX) to emphasize the role of h. The mean and variance for the exponential distribution are given by E(x) = 1 / h and var(x) = 1 / h 2. The m a x i m u m likelihood estimator for h is ~, = k/Ejxj, based on a sample of k > 1 random variables. It can be shown that this estimator is biased; i.e., E(h) = [k/(k - 1)]k. A bias-adjusted estimator for h is given by h = (k - 1) / E j xj. The exponential distribution arises naturally in Poisson stochastic processes, which track the occurrences of some event subject to the Poisson conditions described in Section E.1.2. Thus, the number x of such occurrences in an interval of length t is distributed as P(K) with X = vt, whereas the time between occurrences
729
is distributed as Exp(v), with 1/v the mean time between occurrences (see Section 10.3 for a discussion of Poisson processes). The gamma distribution is closely related to the exponential, in that the sum of independent exponential random variables is gamma distributed. The probability density function of the gamma distribution is f(xlh, r)
where r > 0 and F(r) is the gamma function, defined as above. The gamma distribution is unimodal, with mode influenced by the parameter r. The parameter k plays a similar role as in the exponential. Figure E.10a shows probability density functions for gamma distributions with different values of r, and Fig. E.10b shows probability density functions for different values of k. The gamma distribution sometimes is denoted by F(k, r) or by F(xlk, r), to emphasize the role of k and r in influencing the shape of the distribution. The mean
flxlX, r) X=I 0.8
0.6 r=l O.4 \
0.2
\
r = 5
f--\
/V\
r=lO
r-15
--............. .",,f ~ - \ ............... ..-<.. -................
~',
0.0../ 0
......~<____~_ ......... ........................... 5
10
X=2 r=5
0.08
0.5 0.4 \ . . \ ~ j
/
~,=0.50
\.
006 004
~,=0.25
/ / /
0.1 o.oo,
.
FIGURE
2
E.9
4
6
8
.
. 10
i 0
. 12
14
Exponential distribution. Influence of p a r a m e t e r X.
s.~.2~\ \.....""~:'--..,
///
0.02
0.0
\ \\ \ \~.=3
//
0.2 ~ N N ~ . I
0
x
2O
15
~xl~,r)
>ffxlX)
0.3
k ~-C~(;kx)r-le-xX,
=
/
/
/"
\
\\
\
"~:,0 \ \ .............::.-.<...............................
"" """ " ' \ \ N
...... 10
----, 20
30
40
F I G U R E E.10 G a m m a distribution. (a) Effect of p a r a m e t e r r w i t h X = 1. (b) Effect of p a r a m e t e r h w i t h r = 5.
730
Appendix E Some Probability Distributions and Their Properties
and variance of the g a m m a are given by E(x) = r/h and var(x) = r / h 2. The m a x i m u m likelihood estimator for k is k = r(k/~,j xj), based on a sample of k > 1 r a n d o m variables. As with the exponential, the estimator k is biased. A bias-adjusted estimator of k is ~. =
r(k - 1)/~,j xj. As indicated above, the sum of k identically distributed exponential r a n d o m variables has a g a m m a distribution. Thus, if xj, j = 1, ..., k, is distributed as Exp(h) and y = Y_,jxj, then y is distributed as F(k, k). It follows that the exponential distribution is a special case of the gamma; i.e., F(k, 1) = Exp(k).
E.2.5. Student's t Distribution The Student's t distribution arises as the ratio of independent standard normal and chi-square r a n d o m variables. Thus, if Z has a standard normal distribution as in Eq. (E.1), U has a chi-square distribution as in Eq. (E.2), and X and U are independent, then the r a n d o m variable x = Zl~-Ulk
follows a t distribution function
with
f(xlk) = rF[(k { ~ - / - ~+) ~1)/2]] -~ (1
probability
+ x2/k)
density
-(k+1)/2 .
E.2.4. Beta Distribution The beta distribution is closely related to the g a m m a and exponential, and it has a n u m b e r of important applications, including Bayesian methods (e.g., as a prior distribution). The probability density function for the beta distribution is -1
f(x[a,b) =
The m e a n and variance for the distribution are p~ = 0 (for k > 1) and r = k/(k - 2) (for k > 2). The effect of an increase in the parameter k is to increase the model value and decrease the probability mass in the tails of the distribution (Fig. E.12a). By extension, the r a n d o m variable
[['(a)['(b)]xa-l(l_x)b-1 F(a + b)
Z+5
X-
V'U/k
with positive parameters a and b. The mean and variance for the distribution are
p~ = a l ( a
+ b)
y(xlk)
and
k = lOO - v 2 ~ \ 0 -2 =
//
ab (a + b + 1)(a + b) 2"
\~---k = s
0.3
respectively. The beta distribution can be used to model heterogeneity in a binomial process (e.g., animal capture or survival) by modeling the binomial parameter p as a r a n d o m variable following a beta distribution. Figure E.11 shows the probability density function for various values of a and b.
//
'\
/// i'/
Z G
0.0 -4
-2
0
2
4
Axlka)
.flxla,b)
5=1 2.5
k=5
0.3
2,0
/
/
(2,4) f ~
.
.
.x
//
1.5
N
(4,4) 9 .. /.
.t
(4,2) ~..
.."" ~
5=5 r\
"N
),.-
\
""...
0.2
'\ \
\\\
6 = J0
0.1
0.0
:C/
oo 0.0
,t 0.2
" 0.4
0.6
x
0.8
FIGURE E.11 Betadistribution. Influence of parametersa and b.
,
,
;
|
0
5
10
15
FIGURE E.12 Student's t distribution. (a) Effect of degrees of freedom k on central t distribution (8 = 0). (b) Effect of the noncentrality parameter 8.
E.2. Continuous Distributions follows a noncentral t distribution with k degrees of freedom and noncentrality parameter 8. The probability density function of the noncentral t distribution is kk/2exp[-82/2] ~ (k+i+l) f ( x l k , 8) = F ( k / 2 ) , r r l / 2 ( k + x2)(k+1)/2 s [" i=0 2 i i/2
Vq_X2
9
The primary effect of the noncentrality parameter 8 is to shift the distribution to the right and thereby increase the probability mass in its tail (Fig. 12b). The most frequent use of the student's t distribution involves the assessment of sample means. Thus, consider a set {x1, ..., x n} of statistically independent random variables with normal distribution N(IX, o.2). Then z i = (x i - Ix)/(r is normally distributed as N(0, 1), and therefore the sample mean ~ = ~i zi/rl is distributed as N(0, 1 / n ) (see Chapter 4). Furthermore, it can be shown that
.zi ,.2: [=.xi ,.21/,2
i
i
= ( n - 1)S2/O "2
and U 1 , U2 are independent, then the random variable X defined by
t= ~
) z)a/V'n- 1
is distributed as F, with probability density function
ka)/2](k1/k2)kl/2~
f F [ ( k 1 -t-
f(xlki,
k2)
F-(k;72)F(~/--2-)
= k • [1
+
] x(k1-2)/2
(kl/k2)x] -(k1+k2)/2.
Like the chi-square distribution, the F distribution is nonnegative, unimodal, asymmetric about the mode, and skewed in the positive direction. The mean and variance for the distribution are tx = k / ( k - 2) (for k > 2) and O"2 = 2~(k 1 + k2 - 2 ) / [ k l ( k 2 -2)2(k2 - 4)] (for k > 4). For fixed k2, larger values of kI correspond to smaller probability mass in the distribution tails and greater probability mass near the mode (Fig. E.13a). By extension, the noncentral F distribution arises as the ratio of a noncentral chi-square random variable to a central chi-square random variable, each again divided by its degrees of freedom. That is, if
y(x~,k..) /\
z
.k I = 15
/
0.8
\
i
k 2 = 30
\
/z~l\ \
\\
o
has a Student's t distribution. One therefore can use the Student's t distribution to describe confidence intervals and hypothesis tests for the parameter tx (see Chapter 4). The noncentral t distribution is useful in the estimation of power (Section 6.7.1), where the noncentrality parameter 8 is related to the difference between population means (e.g., due to an experimental effect) under null and alternative hypotheses. The effect of various values of the noncentrality parameter is similar to that for the noncentral chi-square distribution. Thus, large values of 8 correspond to a heavier tail and decreased modal density.
U1/kl U2/k2
X=
is distributed as • - 1), and the random variables 2 and s2 are independent (Graybill, 1976). It follows that the ratio
( x/s
731
.
0.2
0.0
x
0
1
2
3
4
.f(xlk,,kz,K) k~, k 2 = 5, 30 ;k=l
0.6
0.4 10
E.2.6. F Distribution The F distribution arises as the ratio of two independent chi-square random variables, each divided by its degrees of freedom. That is, if U 1 --- x 2 ( k l ) , U 2 ---. x 2 ( k 2 ),
II1\\\\ -~
0.2
ii/ 0.0
X=50
\ \ x ~'-... t I __.,~...
.
.
.
.
.
.
.
.
..
-.
~.
....
i
5
10
15
F I G U R E E.13 F distribution. (a) Effect of n u m e r a t o r degrees of freedom k I on central F distribution (k = 0) with d e n o m i n a t o r degrees of freedom k2 -- 30. (b) Effect of the noncentrality p a r a m e t e r k.
732
Appendix E Some Probability Distributions and Their Properties V 1 ~- x2(kl, ]k), W2 ~- x2(k2 ),
and V1 and V2 are independent, then the random variable X defined as X-
Wl/kl Va/k2
is distributed as noncentral F, with probability density function f(x kl, k2, K)
= ~ XJe ~F[(2j + k~ + k2)/21(kl/k2)(k'+2J )/2 x%.2j_2)/2 j=0 j!F[(2j + k1)/2]F(k2/2) x [1 + (kl/k2)x] -(k~+k2+2p/2. For particular values of k~ and k 2, the effect of the noncentrality parameter is similar to that for the
noncentral chi-square distribution, with large values of ), corresponding to a heavier tail and decreased modal density (Fig. 13b). The F distribution plays a prominent role in statistics, most familiarly in the analysis of variance. A typical application involves the examination of potential differences in central tendency among group means in an experiment. For example, on assumption that a onefactor experimental design is balanced (i.e., k experimental groups all have the same number n of samples) and the groups share a common variance, the statistic F=
Yl~i(X i -- X ) 2 / ( k ~.,ij(Xij - "xi)a/k(yl -
1) 1)
has an F distribution with noncentrality parameter given in terms of the differences in group means. Thus, the statistic can be used to test for differences among group means.
A P P E N D I X
F Methods for Estimating Statistical Variation
Proper estimation of variance (Section 4.1.3) is a key element in accounting for statistical uncertainty in parameter estimates that are included in biological models. Besides their obvious utility in estimating confidence intervals and other measures of parameter reliability, variances are needed in sampling and experimental design (Chapters 5 and 6) and in the incorporation of statistical uncertainty into models for decision making (Part IV). In this appendix we provide several alternative approaches for estimating variance. Most familiar are the distribution-based (Section El) and empirical (Section E2) estimates of variance. We also describe several alternative methods that may be suitable for particular applications and which increasingly are included as features of modern estimation software (Appendix G).
and
var(0) = ~
[6 - E(0)I 2 f(x)
x
for discrete distributions, and E(0) = f 0f(x) dx X
and var(O) = f [ 0 - E(O)I2 f(x)dx X
for continuous distributions. If 0 is a complicated function of x, evaluation of its moments in this way can be difficult. A second approach is to derive the distribution f0(0) of 0 from f(x) and then calculate the variance of 0 with f0(0). The distribution f0(0) can be found by means of variable transformation methods described in, e.g., Mood et al. (1974). Once f0(0) is obtained, it then can be used to compute the mean and variance 0 according to the usual formulas for distribution moments, i.e.,
F.1. D I S T R I B U T I O N - B A S E D VARIANCE ESTIMATION The estimator 0 = 0 (x) of a parameter 0 inherits its distribution from the random variables x on which it is based. Assuming the distribution f(x) is known, two distribution-based approaches are available for determining the variance of 0. First, one can evaluate E[0 - E(0)] 2 directly, utilizing f(x). Thus, the estimator is treated as a function of the random variables x, and its expected value and variance can be computed in the usual manner (see Section 4.1.3). This approach requires only the specification of f(x), which then can be used to evaluate the first and second moments of according to D
E(0) = ~
0 fo(0)
and
m
var(O) = ~ [ 0
- E(O)] 2 fo(O)
for discrete distributions, and
E(0) = ~ 0 f(x)
E(0) = f~ 0 fo(0) dO
x
733
734
Appendix F Estimating Statistical Variation The variance among the point estimates is estimated directly from the replicates as
and var(O) = r i o -
E(O)I2 fo(O)dO
k ( 0 i - 0)2/( k - 1).
v"d'r(0i) = ~ for continuous distributions. A problem with this approach is that the derivation of f0(0) from f(x) can be quite difficult for all but the simplest estimators.
F.2. E M P I R I C A L VARIANCE ESTIMATION Sometimes an empirical estimate of variance can be obtained with replicate information. Assume that the parameter 0 is estimated with 0 based on sampling data. If independent replicates xi, i = 1.... , k, of data are available, then k estimates 0i, i = 1, ..., k, of 0 can be obtained with these replicates. The variance of then can be estimated with
k(
V~(0)-- /~1"= 0 i -
/(k-
1),
where k 0 = ~ Oi/k.
i=1 These are essentially method-of-moments estimates of the mean and variance (see Section 4.2.2 for a description of parameter estimation with the method of moments). Note that the above estimates are "distribution free," in that the distribution for neither the data x nor the estimator 0 is required. Because they incorporate both replicate-to-replicate variation and within-replicate variation, empirical estimates can be useful in separating these two sources of variation. For example, assume that the k replicates represent different sampling locations and that the parameter of interest may differ among these locations (denote the true parameter as 0i for location i). In this case, it may be of interest to estimate spatial or replicate-to-replicate variation in the parameter of interest, which we denote as 0-2. Further assume that 0i is estimated using a model-based approach (e.g., maximum likelihood; see Chapter 4) so that a within-replicate estimate ~r(OilO i) of sampling variance (see Section E3) is available for each location. The expected value of this conditional sampling variance can be estimated using the mean over all replicates: k ~r(6i]Oi ) /~[var(~lO)] = ~ i = 1 k
i=1 This estimate represents the sum of two conceptually distinct components, the true spatial variance 0-2 and the sampling variance: var(0i)
=
E[var(610)].
0-2 nt_
The above expression permits estimation of the true spatial variance as 42-- v"a"r(0i) - E[var(0]0)]
- - - -
k ( 0 i - ~)2 ~i=1 k-1
~k n
i=1
V~(0i[0i) k
"
Although this estimator assumes no sampling covariance among the parameter estimates, sampling covariance terms can be similarly subtracted from the variance among point estimates (see Link and Nichols, 1994; Gould and Nichols, 1998). Also note that this general approach assumes that the conditional sampling variances, V~(6i[0i), are approximately equal. If this is not the case, then a modified iterative estimator is described by Burnham et al. (1987; also see White, 2000). The above variance components estimation approach can be used to estimate either spatial or temporal variation in parameters of interest (Burnham et al., 1987; Skalski and Robson, 1992; Link and Nichols, 1994; Gould and Nichols, 1998; White, 2000). Such sources of variation in underlying parameters are sometimes referred to as process variance (see Franklin et al., 2000; White, 2000), emphasizing that they reflect variation associated with underlying biological and environmental processes. It is important to distinguish this variation from sampling variance that does not usually involve interesting biology. For example, the construction of population viability models (e.g., Chapter 11) typically requires estimates of true temporal variation of the underlying vital rates, and a common mistake is to use variance estimates that also include a sampling component (White, 2000).
F.3. E S T I M A T I N G V A R I A N C E S AND COVARIANCES WITH THE INFORMATION MATRIX As with all data-based estimators, maximum likelihood estimators inherit a probability distribution from the data on which they are based. We discuss here a
E3. Estimating Variances and Covariances with the Information Matrix procedure whereby variances and covariances of maxim u m likelihood estimators can be estimated from the likelihood function itself. Assume for now that 0 is the MLE of a single parameter 0. On condition that 0 can be obtained as a solution of the likelihood equation, the asymptotic variance of is given by var(0) = -[E0(d 2 log f(x]O)/d02)] -1, where the symbol E0 is used to indicate that the expectation is conditional on the value 0. Note that the expectation in the formula is with respect to the distribution f(xl0) of the random variable x, rather than the distribution of 0. Thus, the varianceof 0 is determined by the following procedure: 1. Take the second derivative of the log likelihood to get d2[log f(xlO)]/dO 2. 2. Calculate the expected value of this expression with respect to the distribution f(xl0). 3. Invert the negative of the expectation to get the variance of 0.
of which are expressed in terms of the likelihood function: 032log L(0]x))
1. Determine the partial derivatives 032log L 030 i 030j
of the log likelihood for all pairs of indices (iz j). 2. Calculate the expected values of all partial derivatives with respect to the distribution f(x]0). 3. Arrange the negative of the expectations into an information matrix and invert to get the dispersion
of 6. Example Consider the maximum likelihood estimators ]91 and ]~2 for the parameters Pl and P2 in a trinomial distribution n ) p~1p~2(1-- p 1 - - p2) n-xl-x2, Xlz x2
from which a single sample of size n is taken. Following the procedure outlined above, we have 032
log L
_
xi p/2
m
d02
x -
p2
n - x (1 -- p ) 2 "
The expected value of this expression simplifies to - n / [ p (1 - p)], the negative inverse of which 8ives the variance of 0: var(0) = p(1 -p)/n. Because 0 = x/n for this example, we can derive the variance of ]~ from the definition of variance. It is easy to show for this example that the asymptotic result conforms to the theoretical variance of the MLE. If the number of parameters to be estimated is greater than one, large-sample variances and covariances still can be derived from the MLEs. Here we are concerned not only with estimator variances, but also with covariances among the estimators. When k parameters are estimated, estimator variances and covariances can be expressed in a k • k dispersion matrix E, with variances on the diagonal of ~ and covariances in the off-diagonal positions. Thus the ith diagonal element of ~ is the variance cr2 of 0i, and the off-diagonal element in position (i, j) is the covariance oij between 0i and 0j. The dispersion matrix can be derived as the inverse of the information matrix I(0), elements
"
The procedure for using the information matrix to determine dispersion is as follows:
f(Xl, X2) = (
d 2 log f(x]0)
030i 030j
[I(0)]q = - E
Example Consider the maximum likelihood estimator fi for the parameter p in a binomial distribution B(n, p), corresponding to a single sample of size n. Following the procedure above, we have
735
03p2i
n - xI - x2 (1 -
Pl -
P2 )2
and 032
log L
_
n -
(1
03Pl 03P2
xI -
x2
-- P l -- P2) 2"
The expectations for these terms are incorporated into the information matrix n(1 Pl( 1 -
-
P2)
Pl -
__n P2)
1
-- P l -- P2
n(1
n
1 -- Pl -- P2
1
.di
-- P l )
----"Pl ~
! /'
P2)3
which, on inversion, yields the dispersion matrix Pl( 1 - Pl) n
--PIP2 n
-PIP2 n
P2(1 - P2) " n
This indicates that the variances of ]91 and ]92 have the same form as the binomial variance, and the covariance between/~1 and ]92 is simply -plP2/n. These results
736
Appendix F Estimating Statistical Variation
conform to theoretically derived variances and covariance for the trinomial (see Appendix E).
function is g'(x) = 0.5X-1/2, SO that g'(l~) = 0.5~ -1/2. The delta method therefore yields var(y) = [g'(tx)l 2 var(x)
E4. A P P R O X I M A T I N G VARIANCE W I T H THE DELTA M E T H O D Often one wishes to determine the variance of a function of a r a n d o m variable with a k n o w n distribution. Consider the problem of determining variance of the function y = g(x) of the r a n d o m variable x, w h e n the distribution of x is k n o w n at the outset but the distribution of y is not. As above, one w a y to proceed is simply to evaluate ~g = E[g(x)] and E[g(x) - i~g]2, where the expectation is based on the distribution f(x). Another is to derive the distribution of y = g(x) from f(x), and then calculate the variance of y directly with the derived distribution. Though both methods produce exact values of variance, deriving the distribution of y a n d / o r evaluating the expectations can be difficult and time consuming for all but the simplest functions. Derivation and evaluation problems can be avoided with the well-known "delta m e t h o d " (e.g., see Seber, 1982). This approach uses a Taylor series approximation of g(x) to produce an approximation for its variance. Steps in the method are as follows: 1. Express y = g(x) as a "first-order" Taylor series expansion about the m e a n of x: y = g(x) = g(tx) + g'(l~)(x - t~), where g'(l~) is the derivative of g(x) evaluated at I~ = E(x). Note that the expected value of y based on this approximation is simply E(y) = g(l~). 2. Calculate the variance of y from
= 0-2/4~ for the approximate variance of y. The delta m e t h o d of approximating variance can be extended to functions of more than one r a n d o m variable. Consider the problem of determining the variance for y = g(x 1, x2), where x I and x 2 have a k n o w n joint probability distribution f(x 1, x2). The Taylor series approximation of g(x 1, x 2) n o w involves both variables, but the basic approach is the same: 1. Express y = f ( x 1, x 2) as a first-order Taylor series expansion about the means of x I and x2: y = g(xl, X2) = g(ta,1, I-1,2) if- COX/OXl(X1 -- ~1) if- 3g/3x2(x2
where Og/Ox i is the partial derivative of g(x 1, x 2) evaluated at (1~1, 1~2). Again, the expected value of y based on this approximation is simply g(txl, 1~2). 2. Calculate the variance of y from var(y)
= [g'(i.t,)]2 var(x). Thus, an approximate value for the variance of y = f(x) can be obtained from the variance of x, simply by multiplying the latter by the square of g'(l~). Under certain mild conditions involving b o u n d s on the size of var(x) relative to the second derivative g"(l~), this approximation can be shown to approximate the actual variance of y quite well.
=
(tgg/cOXl) 2 var(x 1) + (egg/cOx2) 2 var(x 2) + 2 0g / 0x I 0g/0x 2 cov(x I , x2).
Thus, an approximation of the variance of y can be obtained from the variance and covariance terms of the distribution f ( x 1, x2). Example Assume that x I and x 2 are counts from a trinomial distribution f(Xl, X2) = (
var(y) = E[y - g(i.t,)]2 = E [ g ' ( l ~ ) ( x - ~)l 2
-- 1-1'2),
n tp~1p~2(1-Xl~ x 2
Pl -- P2) n-xl-x2,
and the variance of the geometric m e a n y = g(x 1, X 2) = X/-X-XlX2 is desired. From probability theory it can be shown that the means and variances for Xl and x 2 are ~i -- npi and 0-2 = npi( 1 _ Pi), i = 1, 2, with the covariance between x I and x 2 of O-12 --- --nplp 2
Example Assume that a r a n d o m variable y is expressed by y -- g(x) --- xl/2~, where x is normally distributed with m e a n I~ and variance 0-2. The first derivative of this
(see Appendix E). The partial derivatives of y = g(x 1, x 2) evaluated at (1~1, i~2) are given by 3g/3x I
= 0.5~11/2~
1/2
E6. Bootstrap Estimation and 3g/Ox 2 = 0 . 5 ~ I / 2 ~ 2 1 / 2 ,
so that the approximate variance of the geometric mean is
2~
v a r ( y ) - ~20"2 4- ,'~10"2 44~ 1 4p~2
4
n2p2Pl( 1 - Pl) 4- n2plP2(1 - P2) 4npl
2nplP2
4np2
737
verge to the theoretical mean and variance of the estimator as sample size increases. It is possible to define higher order jackknife estimators based on deleting more than one observation at a time. For example, a second-order jackknife procedure involves deletion of two observations, thereby generating n(n - 1)/2 estimates 0_/j. These estimates can be used to produce pseudovalues, which in turn can be used to produce the estimator mean and variance as above.
1/
= ~[P2(1 -- Pl) + p1(1--p2) -- 2plP2].
F.5. JACKKNIFE ESTIMATORS OF MEAN A N D VARIANCE ,,,
The jackknife procedure was first developed as a method for reducing bias in estimators that were too complicated for theoretical treatment (see Quenouille, 1949, 1956; Tukey, 1958). However, the jackknife estimator also can be used to estimate variances. Consider a situation in which n observations are to be used for estimation of the parameter 0, based on an estimator = 0 (x)_ with unknown estimator bias and precision. The jackknife procedure for estimating bias and precision consists of the following steps: 1. The full set of n observations is used to estimate 0. Call the estimator utilizing all observations 0. 2. One observation is deleted from the observation set, leaving a total of n - 1 observations, which then are used to estimate 0. Designate as 0-i the estimate resulting from omission of observation i. This procedure is repeated for each observation, producing a set S = {0_i:i= 1.... , n } o f n e s t i m a t e s . 3. "Pseudovalues" are calculated for each element 0-i, according to 6 i = n0 - (n - 1)0_i. A total of n such values are calculated. 4. The "pseudovalues" are used to calculate an estimated mean and variance for 0, by
F.6. BOOTSTRAP ESTIMATION Bootstrapping (Efron, 1979; Efron and Gong, 1983) involves repeated sampling with replacement from a single sample of size n. Bootstrap sampling treats the original sample as a population of size n, from which multiple samples of size n are selected. Sample estimates are calculated for each sample, and an empirical estimate of variance is based on the variability among these sample estimates. The procedure is as follows: 1. The full set of n observations is treated as a population, and k samples of size n are selected with replacement from this population. Because sampling is with replacement, some observations appear more than once in a given bootstrap sample. 2. Each of the k bootstrap samples is used to estimate 0 by the same method as used for the original sample. This procedure is repeated for each bootstrap sample i, producing a set S = {0i: i = 1.... , k} of k bootstrap estimates. 3. The bootstrap estimates are used to calculate an estimated mean and variance for 0, by k
0 = ~_. Oi/k i=1
and m
k (0i -- _~)2 v
(6) = 2E
9
i=1
H
i=1
and 1"/
i=1
n-1
A confidence interval on 0 may be estimated using either of two basic approaches. In the first, which relies on the assumption that 0 is distributed normally, the bootstrap estimates of 0 and var (0) are used to calculate a confidence interval by Eq. (4.5). For example, a 95% confidence interval on 0 would be ( 0 - 1.96 N/v"gr(0), 0 + 1.96V'v"~r(0)),
For a very large class of distributions and estimators, the jackknife estimates of mean and variance can be shown to be asymptotically unbiased, in that they con-
where 0, v"a-r(0) are the bootstrap estimates obtained as described above. The second approach, which is
738
Appendix F Estimating Statistical Variation
distribution free, employs the empirical distribution of the bootstrap estimates to compute confidence intervals. The procedure is as follows:
2. Generate k replicate samples each of size n via Monte Carlo simulation from the assumed distribution, using 0 = 0. 3. Estimate _0i: i = 1, ..., k, from each of the replicate samples using moment or MLE procedures. 4. Compute the mean and variance of the parametric bootstrap estimates from ^
1. Generate the set S = {0i: i = 1, ..., k} of k bootstrap estimates, as above. 2. Order the k estimates and select as the endpoints of the confidence interval the two values that enclose the central 100(1 - o0% of the estimates. This "percentile approach" (and variations) are described in further detail by Manly (1998). A variation on bootstrapping known as parametric bootstrapping involves using the sample estimates to calculate moment or maximum likelihood estimates of a statistical distribution, which are then used to produce replicate samples via Monte Carlo simulation. The procedure is as follows: 1. Compute sample estimates of the parameters 0 of an assumed distribution, using moment or MLE procedures. m
m
k
0 = ~_j Oi/k i=1
and k
v'~(6) = ~
i=1
(6 i __ ~ ) 2
k-
1
A full discussion of the parametric bootstrap and the use of Monte Carlo methods is provided by Manly (1998).
A P P E N D I X
G Computer Software for Population and Community Estimation ware has been written. The approach based on multiple independent observers yields data that can be analyzed using models developed for closed-population capture--recapture (Section G.1.2). As noted in Section G.1.2, program CAPTURE (Rexstad and Burnham, 1991) is currently the preferred choice, with MARK (White and Burnham, 1999) and SURVIV (White, 1983) also being quite useful in cases in which general heterogeneity in detection probabilities is not an issue. For the case of two dependent observers, J. E. Hines has written program DOBSERV (Nichols et al., 2000b), which is simply a front-end for program SURVIV (White, 1983) that generates cell probabilities corresponding to the general double-observer model described by Cook and Jacobson (1979) and various reduced-parameter alternatives. The marked subsample approach described in Section 12.6.2 has led to the development of several estimators and at least two associated computer programs. NOREMARK (White, 1993) provides multiple estimators for the sampling situation in which the number of marked animals is known prior to each sighting survey. When the number of marked animals is not known, a program written by Arnason et al. (1991) can be used to compute abundance estimates. Several user-oriented packages have been developed for the distance sampling methods described in Chapter 13. Two early packages were TRANSECT (Laake et al., 1979; Burnham et al., 1980) and LINETRAN (Gates, 1979, 1980). Both packages computed estimators for grouped or ungrouped data and for either right-angle distance data or sighting distance and angle data. Both packages computed various parametric and nonparametric estimators. Burnham et al. (1980) emphasized the robustness of their Fourier series estimator (not computed by LINETRAN), whereas LINETRAN computed the spline estimator developed by
Although some of the methods for estimating population and community parameters described in Part III can be performed on a desktop calculator using the closed-form expressions provided in the text, this is not possible for many models, tests of fit, and model comparisons. However, several computer packages are available to perform these calculations, and most will run on desktop computers. In this section we list some software packages that are available for obtaining parameter estimates and conducting associated tests and provide some guidance as to which may be most useful for a particular application. Most of these packages have been cited in the sections of the book that deal with estimation and testing methods. The list is not exhaustive, and we have included only software with which we are familiar. A comprehensive software review was conducted by Lebreton et al. (1993) [also see reviews in North (1990) and Lebreton et al. (1992)], and we recommend these publications to the interested reader. In addition, detailed descriptions of specific programs for estimating animal abundance and related parameters can be found in the proceedings volumes of the European Union for Bird Ringing (EURING) meetings (e.g., see Conroy, 1995; Pradel et al., 1995; White and Burnham, 1999). Here we make no attempt to describe the programs or their operation in detail, but refer readers to the original sources for each program.
G.1. ESTIMATION OF A B U N D A N C E A N D DENSITY FOR CLOSED POPULATIONS G.1.1. Observation-Based Methods The observation-based methods of Section 12.6.2 include several approaches for which user-oriented soft-
739
740
Appendix G Estimation Software
Gates (1979, 1980) and not available in TRANSECT. Drummer and MacDonald (1987) developed program SIZETRAN (Drummer, 1986) for the situation in which animals are detected in clusters and cluster or group size can influence detection probability. The current software of choice for distance sampling is program DISTANCE (Laake et al., 1993; Buckland et al., 1993), which evolved from program TRANSECT. DISTANCE provides a wide selection of estimation models developed around the key function approach described in Chapter 13. DISTANCE provides several different approaches to model selection. It can deal with clustered populations and with right and left censoring. DISTANCE can compute stratified estimates of density and abundance, and it includes a bootstrap approach to variance estimation. In addition, DISTANCE can accommodate data from variable circular plot sampling and from cue count sampling as well as traditional line transect sampling.
G.1.2. Capture-Recapture Models The first comprehensive computer package developed for estimating abundance under closed population models (Chapter 14) was CAPTURE, developed by White et al. (1978) to accompany the monograph by the same authors (Otis et al., 1978). The authors considered eight different models corresponding to different ways of modeling capture probability. Estimates were computed for five of these models and all five estimators required iterative solutions (thus a computer program was necessary for standard use of the estimators). Statistics for goodness-of-fit tests and between-model tests were computed, and an innovative model selection procedure was implemented (see Chapter 14). CAPTURE remains an excellent program for analysis of data from closed population capture-recapture studies. The more recent version of CAPTURE (see Rexstad and Burnham, 1991) provides some new additional estimators for the five models for which estimation was possible in the original version, as well as estimators under two models for which estimation was not previously possible. Program MARK (White and Burnham, 1999) implements the models contained in program CAPTURE that do not involve individual heterogeneity of capture probabilities. The likelihood-based finite mixture models of Pledger (2000) recently have been incorporated into MARK to deal with heterogeneity. MARK also permits estimation under models in which capture probability is modeled as a function of individual covariates (Huggins, 1989, 1991; Alho, 1990) (also see Section 14.2), a class of models not included in CAPTURE. Of course, all likelihood-based estimators
for closed populations can also be implemented in SURVIV (White, 1983). Cormack (1979, 1985, 1993) has also had success modeling capture-recapture data from closed populations using GLIM. Arnason et al. (1996) developed program SPAS for estimation under a special class of stratified markrecovery experiments. Maximum likelihood estimation under catch-effort models (Section 14.4) can be accomplished using program LINLOGN (Hines et al., 1984), based on the approach of Pollock et al. (1984). Gould and Pollock (1997a,b) considered additional catcheffort models and implemented them in SURVIV (White, 1983). Udevitz and Pollock (1991) used iteratively reweighted nonlinear least squares to estimate abundance for the change-in-ratio method (Section 14.5) and provided SAS code for implementation. Finally, trapping web data can be analyzed using either a distance sampling approach (program DISTANCE) (Laake et al., 1993) or the geometric approach of Link and Barker (1994), for which some software has been written by Link.
G.2. ESTIMATION OF ABUNDANCE AND DEMOGRAPHIC PARAMETERS FOR OPEN POPULATIONS G.2.1. Band Recovery Models The first rigorous approach for estimating survival rates from band recovery data was developed by Seber (1970b) and Robson and Youngs (1971) for the singleage case (see Chapter 16). Although this original model yielded closed-form estimators for model parameters and their variances and covariances, computation was still tedious. Anderson et al. (1974) thus developed a FORTRAN program for computing these estimates and related quantities (geometric and arithmetic means, mean life span). This was the first user-oriented software developed by D. R. Anderson, who pioneered the development of user-friendly packages for implementing state-of-the-art estimation methods in animal population ecology (e.g., Anderson was instrumental in the development of programs CAPTURE and TRANSECT discussed above as well as additional software to be discussed subsequently). This early work led to the development of programs ESTIMATE and BROWNIE to handle single-age and multiple-age models, respectively (Brownie et al., 1978, 1985). These programs computed estimates under a small number of models and computed associated goodness-of-fit and likelihood ratio test statistics for use in model selection and testing of assumptions. Program BAND2 was de-
G.2. Estimation of Abundance and Demographic Parameters for Open Populations veloped by Wilson et al. (1989) to compute sample sizes needed to achieve specific design objectives based on the models of ESTIMATE and BROWNIE. A primary reason for limiting ESTIMATE and BROWNIE to a fixed set of specified models involved the numerical methods used to obtain estimates. The numerical methods for maximizing the likelihoods were based on a Newton-type approach requiring analytic expressions for the elements of the information matrix [the second partial derivatives of the likelihood function with respect to the parameters; see Brownie et al. (1978)]. White (1983) and Conroy and Williams (1984) developed methods and software that provided maximum likelihood estimates using methods that were entirely numerical (they did not require analytic expressions for the first and second partial derivatives of the likelihood function with respect to parameters), thus opening the door to very flexible software. SURVIV (White, 1983) is extremely flexible, permitting maximum likelihood estimation under virtually any user-defined model that fits within a multinomial framework. Program MULT (Conroy et al., 1989b) is more user friendly, yet less flexible than SURVIV, permitting models incorporating linear covariates, models for unequal time intervals between banding periods, and models for the analysis of reward band studies. The preferred software for most band recovery analyses today is MARK (White and Burnham, 1999). MARK is capable of performing analyses on a wide range of data structures involving marked animals, including capture-recapture, band recovery, analyses with covariates, combined recovery and recapture data, known-fate data (e.g., radiotelemetry and nest success; Chapter 15), and multistate capture-recapture data. MARK is particularly powerful when data are stratified by a multiple classification procedure, in which case construction and comparison of a potentially huge number of models is facilitated by a model output screen that is automatically sorted by AIC values; individual model output, parameter estimates, and graphical analysis of residuals are all easily obtained by "clicking" on appropriate graphical objects. Program MARK permits the indexing of parameters by any strata identified at the time of marking and release (e.g., sex, areas, treatment groups) as well as the usual dimensions of time and age. The strategy is to construct the models that represent all the potentially relevant sources of variation (and combinations thereof) for the problem at hand, and to use goodnessof-fit and AIC statistics to select the model (or subset of models) that is best supported by the data. Because of its flexibility, SURVIV is still needed to carry out estimation under certain special classes of models. Fi-
741
nally, it should be noted that band recovery models can also be implemented in general software packages such as MATLAB (Catchpole, 1995).
G.2.2. Capture-Recapture Models for Open Populations Modern capture-recapture models for open animal populations were first developed in the mid-1960s by Cormack (1964), Jolly (1965), and Seber (1965) (also see Chapters 17 and 18). The estimators derived by these authors for the Cormack-Jolly-Seber (CJS) model represented closed-form solutions, and analytic expressions were derived for the variance and covariance estimators. Computations were sufficiently complicated, especially for variance estimators, that workers such as White (1971) and Arnason and Kreger (1973) developed programs to carry them out for the Jolly-Seber (JS) model (unconditional likelihood, Chapter 18). A. N. Arnason, like D. R. Anderson, has been an influential pioneer in software development for animal population estimation problems. Arnason has continued to support his JS program POPAN and has released improved versions over the years (Arnason and Baniuk, 1978, 1980; Arnason and Schwarz, 1987, 1995, 1999). Buckland (1980,1982) developed program RECAPCO to implement a modified version of the JS model that constrained estimates of survival probabilities to the interval [0,1] and dealt with known deaths that occurred between sampling occasions. In a manner that paralleled developments in band recovery models, Jolly and Dickson (1980; also see Jolly, 1982) and Brownie et al. (1986) developed software that used numerical methods with analytic derivatives to obtain estimates under reduced-parameter single-age JS models. Brownie et al. (1986) and Pollock et al. (1990) released programs JOLLY and JOLLYAGE, written by J. E. Hines to compute estimates, fit statistics, and between-model test statistics for a small set of one- and two-age open-population capture-recapture models. Hines also wrote a companion program, CAPQUOTA, to compute expected precision of JS estimates based on various characteristics of study design (e.g., capture probability). However, as was the case with band recovery models, in the 1980s the use of numerical methods that did not require analytic derivatives led to the development of more flexible capture-recapture models for open populations. Crosbie and Manly (1981, 1985) developed program CMR to produce estimates under reduced-parameter JS models using their superpopulation approach (Section 18.3). CMR also included the ability to model conditional recruitment parameters, but was not widely used because it was developed on a
742
Appendix G
TABLE G.2 Site location/institution
Estimation Software
Public Sites for Obtaining Estimation Software
Contact
Programs
World wide web site/ftp
University of Manitoba
A. Neil Arnason ([email protected]), Carl J. Schwarz ([email protected])
EAGLES POPAN SMOLT SPAS
http ://www.cs.umanitoba.ca /~popan /
St. Andrews University, UK
S.T. Buckland ([email protected])
DISTANCE
http ://www.ruwpa. st-and, ac.uk / distance /
Colorado State University
Gary C. White ([email protected])
BROWNIE CAPTURE DISTANCE ESTIMATE MARK NOREMARK RADIOTRK RELEASE SURVIV
http ://www.cnr.colostate.edu /~gwhite / software.html
Cornell University
Evan Cooch ([email protected])
CAPTURE CONTRAST DISTANCE ESTIMATE JOLLY JOLLYAGE MARK NOREMARK POPAN MSSURVW RELEASE SURGE SURPH SURVIV
http:# www.phidot.org./software /
Illinois Natural History Survey
Robert Diehl ([email protected]), Ronald Larkin ([email protected])
BROWNIE CAPTURE CONTRAST DISTANCE ESTIMATE JOLLY JOLLYAGE RELEASE SURGE TRANSECT
h ttp ://detritus.inhs.uiuc.edu / wes /
University of Connecticut
Robert K. Colwell ([email protected])
EstimateS
http:# viceroy, eeb.uconn.edu/EstimateS
Columbia Basin Research, University of Washington
J.R. Skalski ([email protected])
SURPH
http://www.cqs.washington.edu / surph / surph.html
(continues)
G.2. Estimation of Abundance and Demographic Parameters for Open Populations TABLE G.1 Site location/institution
Patuxent Wildlife Research Center
Contact
James E. Hines ([email protected])
Burroughs computing platform. This superpopulation approach was later adopted by Schwarz and Arnason (1996) and incorporated into POPAN-4 (Arnason and Schwarz, 1995) and POPAN-5 (Arnason and Schwarz, 1999). Both CMR and POPAN were developed to compute abundance estimates as well as estimates of survival and capture probability. Flexible, numerical methods were also adopted by those interested in obtaining estimates of survival probability using the conditional approach of Chapter 17. Program SURGE was developed by Lebreton and Clobert (1986; also see Clobert and Lebreton, 1985; Clobert et al., 1985,1987) for this purpose. Using logistic and other link functions, SURGE placed the modeling of survival and capture parameters in a general linear models framework (Lebreton et al., 1992), permitting ultrastructural modeling of survival and capture parameters as functions of covariates, as well as modeling with additive effects (so-called parallelism models). SURGE was revised and became a very powerful and flexible piece of software (Pradel and Lebreton, 1991). Program RELEASE (Burnham et al., 1987) was also developed for conditional modeling of survival and capture probabilities using flexible numerical methods. However, the program was developed for a specific kind of experimental design, so was not nearly as flexible as SURGE, for example. Nevertheless, the goodness-of-fit test implemented in RELEASE was such an
743
(Continued) Programs
BAND2 BROWNIE CAPQUOTA CAPTURE CENTROID COMDYN CONTRAST DISTANCE DOBSERV DOSECOMP ESOR ESTIMATE JOLLY JOLLYAGE MAYFIELD MSSURVW MULT NOREMARK ORDSURVIV POPAN5 RD-SURVIV RELEASE SPECRICH TM-SURVW
World wide web site/ftp
http://www.mbr-pwrc.usgs.gov/ software.html ftp://ftp.pandion.er.usgs.gov
important development that RELEASE was (and is) frequently used to provide fit statistics for the general CJS model. The developers of SURGE produced a software package, CR (Cezilly et al., 1992; Pradel et al., 1995), that provided access to RELEASE, JOLLY, and JOLLYAGE, as well as SURGE. It should also be noted that Cormack (1979, 1985, 1993) has had success fitting some kinds of open models using the general software package GLIM. Despite the flexibility of SURGE for CJS modeling of survival and capture probabilities conditional on releases, there was still room for additional development. Skalski et al. (1993) reported the development of SURPH (Smith et al., 1994), a package that permits modeling of survival and capture probabilities as a function of individual covariates and permits proportional hazards modeling (see Section 15.4.4; proportional hazards also can be modeled in MARK). Hines (1994) developed MSSURVIV, a front-end program for SURVIV (White, 1983) designed to write cell probabilities under first- and second-order Markov process multistate models (see Section 17.3) and compute estimates of survival, capture, and movement/transition probabilities. Similarly, TMSURVIV was written by J. E. Hines to implement the so-called transient parameterization of the Brownie-Robson (1983) model (Pradel et al., 1997; also see Section 17.1.6). Hines also wrote RDSURVIV to compute estimates under likelihood-
744
Appendix G Estimation Software
based robust design models (Kendall and Hines, 1995; also see Sections 19.3 and 19.4.1). As with band recovery models, the most useful software for open-population capture-recapture models is MARK (White and Burnham, 1999). As briefly described above, it is both extremely flexible and very user friendly. In addition to carrying out virtually all of the kinds of analyses of SURGE, MARK also handles multistate models, individual covariates, temporal symmetry models (Sections 17.4 and 18.4), and the robust design, as well as models for which previous software was not available or not well-developed (e.g., some of the models of Section 17.5). In addition, MARK is very well maintained and White is continually adding new models and capabilities as they are developed. Although MARK does deal with some parameterizations of unconditional JS models, emphasis is on conditional modeling. Arnason and Schwarz (1999) have implemented many modeling capabilities in POPAN-5 centered on their superpopulation modeling approach (Section 18.3). So our basic recommendation for those interested in capture-recapture modeling for open populations is to look first at MARK for conditional modeling and the robust design, and at POPAN-5 if primary interest is in unconditional modeling and abundance estimation. Then if the model(s) of interest cannot be implemented in these excellent packages, the web sites in Table G.1 can be searched for the relevant model structure. If appropriate models cannot be found, then they may have to be developed using either SURVIV or a general package such as MATLAB (e.g., see Caswell et al., 1999; Catchpole, 1995; Caswell, 2001).
G.2.3. Telemetry and Nest Survival G. C. White has written a set of programs (RADIOTRK) for conducting analyses of radiotelemetry data (Table G.1; see also White and Garrott, 1990). Otherwise, relatively little software has been written spe-
cifically for the estimation of survival from telemetry and nest survival data for which detection probabilities are assumed to be 1 (e.g., MAYFIELD, by J. E. Hines). The primary reason for this is the fact that general biomedical survival analysis software is often appropriate for such data (e.g., see Pollock et al., 1989a,b). MARK (White and Burnham, 1999) and SURVIV (White, 1983) also provide an extremely flexible modeling framework for use with such data (see example in Miller, 1999).
G.3. E S T I M A T I O N OF
COMMUNITY PARAMETERS Species richness can be estimated from closedpopulation capture-recapture models (Chapter 20), so program CAPTURE (Rexstad and Burnham, 1991) is especially useful for this purpose. A special limiting form of the jackknife estimator for abundance under heterogeneity model M h was developed by Burnham and Overton (1979) for the purpose of estimating richness from an empirical species abundance distribution (see Section 20.2.3). Hines et al. (1999) developed program SPECRICH for the purpose of computing estimates using this estimator. The robust design approach to estimation of community-level vital rates (Section 20.3) is implemented in program COMDYN, also written by Hines (Hines et al., 1999). Program EstimateS was developed by Robert K. Colwell to estimate species richness, diversity, and related community parameters (Table G.1). The software contains some algorithms that assume perfect detectability and some algorithms that allow for detection probability < 1.
G.4. SOFTWARE AVAILABILITY Nearly all of the software described above can be obtained from web-based sources free of charge. Information on software sources is presented in Table G.1.
A P P E N D I X
H The Mathematics of Optimization
H.1. U N C O N S T R A I N E D OPTIMIZATION
The generic optimization problem is to choose values for a set of decision instruments, so as to maximize some objective that is expressed in terms of the instruments. In natural resource management, the decision instruments can consist of system parameters, system state variables, state variable trajectories, control trajectories, a n d / o r combinations of these. Optimization objectives incorporate values that are based on system states at specific times, or aggregates of these values across time, or functions of system controls over time, or elements of the time frame, or factors that are linked to system behaviors, etc. Some optimization approaches distinguish between state and control variables as functions of time; others focus on the selection of decision instruments without explicit reference to time. The existence and sufficiency of optimal solutions d e p e n d on convexity of the set of potential solutions and concavity of the objective functions. Set convexity is expressed mathematically as follows: the set _X is convex if ax 1 + (1 - a)x 2 ~ X for all X1 (~ Xr X2 E Xl and a e [0, 1]. Thus, there can be no "depressions" along the surface of a convex set, for then it w o u l d be possible to exit the set in moving along a line from one side of the depression to the other (Fig. 22.6). A function F(x) is concave over a convex set X if
Here we describe optimization procedures for problems in which a vector of decision variables is chosen to maximize an objective function, given that the set of allowable values for the decision variables is not constrained. We assume in w h a t follows that the objective function is twice differentiable with continuous derivatives.
H.1.1. Univariate D e c i s i o n Problem Consider first a single decision variable x, where the problem is to identify a local optimum, i.e., a value x* such that F(x* + &x) <- F(x*) for "small" values Ax. From Taylor's theorem we m a y express F(x* + Ax) by
dF d2F F(x* + Ax) = F(x*) + ~xx(x*)Ax + -~x2(X* + OAx)Ax2/2, with 0 < 0 < 1. Because F(x* + &x) <- F(x*) for all values Ax, we have
dF (x*)Ax + d2F dx -d~x2(x* § OAx)Ax2/2 ~ 0
F[ax 1 + (1 - a)x 2] ~ aF(x 1) + (1 - a)F(x 2)
for positive as well as negative &x. Division by arbitrarily small but positive Ax shows that (dF/dx)(x*) -< 0. On the other hand, division by arbitrarily small negative &x shows that (dF/dx)(x*) >- O. It follows that
for all X1 ~ Xr X_.2 (~ Xr and a e [0, 1] (Fig. 22.7). In mathematical p r o g r a m m i n g problems, concavity in the objective function over convex X is sufficient to guarantee that a local m a x i m u m is also global.
(dF/dx)(x*) = 0,
745
(H.1)
746
Appendix H
The Mathematics of Optimization
and the Taylor expansion reduces to
and 0 < h < 1. This equation can be expressed in matrix notation as
d2F
F(x* + Ax) = F(x*) + -~x2(X* + OAx)Ax2/2. F(x* + hax) = F(x*) + haF (x*)ax + -
-
-
O X -
-
h2
,32F
,
ax ~x2(X + hOax)ax,
- 2
-
-
From F(x* + Ax) <_ F(x*) we therefore have d2F d x 2 ( X * 4- O A x ) A x 2 / 2
where (aF/Ox)(x) = [(aF/3Xl)(X), (aF/ax2)(x)] is the gradient of F at x and
G 0
32F
or 32F
(x) =
3x2-
d2F (x* + OAx) < O. dx 2
02_____~F(x)
~ax~(X) 32F
(x)
OXlOX2 -32F
Ox23xl -
Because 0 is of arbitrary m a g n i t u d e and Ax is of arbitrary sign, this in turn implies
d2F dx 2 (x*) ~ 0.
is the Hessian matrix of F at x. As in the univariate case, F(x* + h&x) <- F(x*)yields
hO-~F(x*),~x +
(H.2)
OX --
--
h2 .
,32F
~(x*
- ~ a X OX 2 _
+ hOAx)Ax < O. _
_ --
m
This second-order condition (H.2) (modified for strict inequality) in combination with (dF/dx)(x*) = 0 guarantees that x* is a local m a x i m u m . Note that (dF/dx)(x*) = 0 is not by itself sufficient to ensure a m a x i m u m , because the derivative vanishes for local minima and inflection points as well as local maxima. Nor is a local m a x i m u m that is identified by first- and second-order optimality conditions guaranteed to be a global maximum. However, concavity in the objective function at every point x is sufficient to guarantee a global maxim u m at x*.
H.1.2. Bivariate Decision Problem N o w suppose the problem involves a twice differentiable objective function in two decision variables x 1 and x2. Again, we wish to identify a value x* = (x~, x~)' such that F(x* + hAx) <_ F(x*) for "small" variations h&x in an arbitrary direction Ax = (&Xl, Ax2)'. Applying Taylor's Theorem, we have
Dividing this expression by h and taking the limit as h approaches zero leads to OF(x*) = 0 OX-
and A ,O2F
x 7x2(X*)ax _< 0.
Sufficient conditions for a local m a x i m u m are firstorder stationarity [(OF/Ox)(x*) = _0'] and a negativedefinite Hessian matrix [i.e., A x ' ( O 2 F / O x 2 ) ( x * ) A x < 0 for arbitrary Ax]. Note that these are analogous to optimality conditions, Eqs. (H.1) and (H.2), for k = 1. For notational convenience, in w h a t follows the gradient of F at point x* is denoted by
OF(x,) = VF(x*), F(x* + hAx) = F(x*) + h~" (x*)Axa + h _ _ _ OX1 7 0 x 2 ( X * ) A X 2
3F
_ )2 + -~ ~x21(x* + Ohax)(aXl
OX-
and the Hessian at x* is denoted by
h232F
02F (x*) -- H_H_H_~(x*). OX2 -
h 2 O2F.
+ ~2 ~"2(X*2ox -
+
0hAx)(Ax2)2
32F + h2
(x* + O h A x ) ( A X l A X 2 ) 2 OXlOX2 -_ ,
where OF/OXi and 32F/cOXiOXj a r e first and second partial derivatives of F with respect to x i and xj, respectively,
H.1.3. Multivariate Decision Problem In the general multivariate case, a vector _x* = (x~, ..., x*)' is sought for which F(x*_ + hAx) <_ F(x*) for "small" variations hAx in an arbitrary direction Ax
H.1. Unconstrained Optimization - - ( A X l , ..., AXn)'. As in the bivariate case, the secondorder Taylor expansion about x* is
F(x* + hAx) = F(x*) + h~oF _ _ _ ( x * ) a x
= F(x*) + h[VF(x*)]Ax
h2 +-~-Ax'[HF(X* + h0Ax)]Ax, where again
] a2F
32F
CgXlOX2 (X)
a2F
-
...
OXlCgX-----~n(X)
...
32F c~X2(X__)
=
32F (x) _c~Xne~xl
32F (x) C~XnC~X2
Derivative-free methods involve a systematic search for a value x* that satisfies OF/~x = 0', while avoiding the sometimes difficult task of differentiation of F(x). The search begins with repeated evaluations of F(x) in multiple directions &x_in a neighborhood around some starting value x 0, followed by movement in the direction of m a x i m u m change. A value x I thus identified, x 0 is replaced by x I and another search centered on x 1 ensues. This process can be repeated until the optimal value x* is found, or until the difference between successive approximations of x* [or successive approximations of F(x*)] no longer exceeds a m i n i m u m stopping criterion. n
H.1.4.1.1. Simplex Search
is the gradient of F at x, and -
H.1.4.1. Nondifferential Methods
m
h2, ,a2F + ~-ax 7x2(X* + h0ax)ax
747
is the Hessian matrix at x. As before, necessary conditions for a m a x i m u m at x* are VF(x*) = 0'
(H.3)
and &x'[HF(x*)]Ax --< 0.
(H.4)
Sufficient conditions for a local m a x i m u m are firstorder stationarity [VF(x*)_ = 0'] and a negative-definite Hessian matrix {&x'[HF(x*)]&x < 0 for arbitrary &x}. Again, conditions (H.3) and (H.4) reduce to conditions (H.1) and (H.2) for a univariate problem.
H.1.4. Solution Approaches Procedures for identifying x* often can take advantage of the fact that any optimal value must satisfy the stationarity condition (H.3), with a search for optimal values reduced to a search for zeros of the system of equations OF/Ox = 0'. In the majority of cases, iterative search procedures must be used, in which the direction and size of each step in the procedure are determined from results of the previous step. Here we describe some approaches that utilize either differential or nondifferential methods.
A simplex search begins with the specification of an equilateral simplex (in 2-space, a triangle in the opportunity set _X with equal sides), at each vertex of which the objective function is evaluated. A direction line then is projected from the minimizing vertex x 0, bisecting the side of the simplex opposite x 0. This line establishes the direction of search for a new candidate for x*. Once a choice X__1 in that direction is made, it becomes a vertex of a new simplex along with the vertices of the bisected side, and the process is repeated. Variants of this approach include the use of nonequilateral simplices and different algorithms for determining the step size for each iteration.
H.1.4.1.2. Sequential Univariate Searches Here the approach consists of a search in the direction of each axis of the opportunity set X, using a univariate search algorithm to find an optimizing value of the corresponding decision variable. Thus, from a starting value x 6 = (x01, x02, ..., X0n), one identifies a new value (x11, x02, ..., Xon) by optimizing in the direction (1, 0, ..., 0). From ( X l l , X 0 2 , ..., XOn)one then identifies yet another new value (x11, x12, ..., x0n) by optimizing in the direction (0,1,..., 0), and so on until optimizations have occurred in each direction of dimensionality in the opportunity set. This process can be repeated until the optimal value x* is found or until the difference between successive approximations of F(x*) no longer exceeds a m i n i m u m stopping criterion. For a two-dimensional optimization, the method produces a "zig-zag" pattern in the sequence of iterations, wherein movement in the direction of (1, 0) is followed by movement in the direction of (0, 1), which again is followed by movement in the direction of (1, 0), and so on.
H.1.4.1.3. Conjugate Direction Method The idea with this approach is to choose successive directions to avoid "undoing" the optimality gains of
748
Appendix H
The Mathematics of Optimization
previously chosen directions, as often happens with sequential univariate searches. One w a y is to modify the search directions in successive iterations so that they conjugate, i.e., point in directions that are mathematically independent. An iterative conjugate direction m e t h o d consists of the following steps:
recognizing that the gradient on a contour F(x) = c points in the direction of steepest ascent for F(x) (Fig. 22.2). Starting at a value x 0 in X, the approach is essentially to choose a value
9 Start with the set _DO = {D~ "", n D~ of line directions as defined by _ID~= (1, 0, ..., 0), 2D O = (0, 1, ..., 0), ..., and D O = (0, ..., 0, 1). Search along each of these directions in sequence, starting at an initial point x 0. Let P0 be the point located at the end of these searches, with direction A 0 = P0 - x0 from x 0. Find a maximizing point x I in the direction of A 0. 9 Let D 1 = {D~ .... , D n1} be a new set of directions obtained from the set D O by {D], "", __n-lD1, ranD1} __ {D O..... D ~ A0}. Search along each of these directions in sequence, starting at x 1. Let P l be the point located at the end of these searches, with direction A 1 = P1 x I from x 1. Find a maximizing point x 2 in the direction
as an approximation of x*, with 80 the step size maximizing F(x 1) in the direction of VF(x0). Then a second approximation of x* is chosen by x 2 = X 1 -F 81 VF(Xl)' , with 81 again chosen to maximize the value F(x 2) in the direction of VF(Xl). This sequence can be repeated until the optimal value _x* is found, or until the difference between successive approximations of x* [or successive approximations of F(x*)] no longer exceeds a m i n i m u m stopping criterion. With the gradient VF(Xk) specifying the direction of m o v e m e n t at each step k, the challenge in the m e t h o d of steepest descent is to determine the step size 8k. One procedure substitutes
o f a 1.
9 For step k, let D k = {D k, ..., D k} be a new set of directions obtained from the se~ D k-1 by {D1k, ..., D nk- 1 , _Dk} = 21Dk-1 t , ..., nDk-1 , a k - 1}" Search along each of these directions in sequence, starting at x k. Let P k be the point located at the end of these searches, with direction Ak = P k -- Xk f r o m Xk. Find a maximizing point Xk+l in the direction of Ak. 9 Continue until a stopping criterion is satisfied.
Xk+ 1 -- __Xk = 8 V F ( X k ) '
from the u p d a t i n g algorithm into the quadratic approximation of F about x k, to get V F k + 1 = F ( X k + 1) --
q- 0 . 5 ( X k + 1 --
q-
~.. zd
OXiOy.(Xk) (X i -- X k) (Xj --
82 + ~ [VF(Xk)][H-H-~(xk)][vF(xk)]'' The idea is to choose 8 so as to maximize the difference VFk+I, which is accomplished by differentiating the above expression with respect to 8 and setting the result to zero:
d(VFk+l)
= F(Xk) + VF(Xk)(X -- Xk) q-0.5(X-
Xk)'[HF(X__k)](Xk+ 1 -- Xk)
= 8[VF(xk)l[VF(Xk)]'
A f r a m e w o r k for iterative differential approaches is based on the quadratic approximation
I
F(Xk)
= VF(Xk)(Xk+ 1 -- Xk)
H.1.4.2. Differential Methods
F(x-) = F(Xk) + ~-J [3~xi(Xk)](Xi--
80VF(x0)'
X1 -- X0 q-
d~
= [VF(Xk)] [VF(Xk)]'
+ 8[VF(Xk)][H_H_H_H__~(Xk)] [VF(Xk)]'
X_k)'[H___F(Xk)](X -- X k)
=0, of F about a value x k = (Xlk, ..., Xkn)', where VF(x k) and Hr(x k) are the gradient vector and Hessian matrix. For a quadratic function F(x), this approximation is exact; otherwise, it is more or less representative of F(x), depending on the nonquadratic nature of the function and the difference between x and x k.
H.1.4.2.1. Method of Steepest Ascent This approach utilizes the gradient VF(x) to determine a direction of search for an optimal value x*,
or
[VF(Xk)] [VF(Xk)]' 8k --
[VF(Xk)] [Hr(Xk)] [VF(xk)I'
The resulting algorithm chooses Xk+1 according to X k + l --- Xk -- {[VF(Xk) ]
[HF(Xk)]
[VF(Xk)]'} -1
x {[VF(xk)] [VF(xk)l'} VF(Xk)'.
H.2. Classical Programming This algorithm is effective when the Hessian is positive-definite over the operating range of values Xk. However, with more irregular functions it is less likely to be useful, except for short segments of the steps in some of the iterations, and directions other than the gradient may lead to larger increases in the objective function for the same step size.
749
gate, i.e., point in directions that are mathematically independent. Mathematically, conjugation of vectors D k and Dk+ 1 with respect to the matrix A is defined b y D ~ A D k + 1 = 0.
The mathematical logic of conjugate gradients is rather complicated, but the description of an algorithm is fairly straightforward. Algorithm steps include the following:
H.1.4.2.2. Newton's Method
At each iteration, the method of steepest ascent assumes a search direction given by the gradient VF(xk), with step size chosen to maximize F(x) in that direction. An alternative approach is Newton's method, which uses the quadratic approximation
1. For a given a starting value x 0, find D o = VF(x0). 2. Find a maximizing value X__1 in the direction of D O and determine rE(x1). 3. For iteration k + 1, use the previous direction D k along with successive gradients VF(x k) and VF(Xk+1) to determine the new search direction
F(x) = F(X k) + [VF(Xk)](x - Xk)
[VF(X k+I)VF(X k+_I)' ] + 0.5(x- xk)'[H_;(xk)](x- xk)
D k + 1 = VF(X_k+l) + L
to identify simultaneously both the search direction and step size that maximizes F(x). The approach uses optimality condition (H.3) in combination with the quadratic approximation of F(x) to find X__k+1 directly, based on Xk and the values of the gradient and Hessian at x k. Condition (H.3) is satisfied by differentiating both sides of the approximation and equating the result to zero: dF d-~(x) = VF(xk) + ( x - Xk)'[H___y(Xk)] =
O F"
For nonsingular H__F(Xk),this yields the iterative formula Xk+ 1
= X k -- [H___F(Xk)]-IVF(Xk) '
VF(Xk)VF(Xk)'
Dk"
4. Find a maximizing value X__k+2 in the direction of D k + 1 and determine VF(Xk+2). 5. Continue as above until a stopping criterion is satisfied. It can be shown that every pair of search directions D i and Dj in the sequence {Dk} is conjugate and every pair of the corresponding gradients VF(xi) and VF(xj) is orthogonal. An important advantage of conjugate gradient methods is that the search direction at each iteration can be found without the need to calculate the Hessian (or any other state-specific scaling matrix). The advantages as to computational efficiency should be obvious.
(H.5)
that defines Newton's method. On condition that H__F(xk) is positive-definite over the operating range of values Xk, the sequence of values {Xk}generated from repeated application of Eq. (H.5) converges to x*. A comparison of Newton's method and the method of steepest ascent shows that the former is a more computer-intensive algorithm, in that it not only relies on the computation of first and second partial derivatives at each iteration, but also requires the inverse of HF(Xk). This additional computing burden is compensated by much more rapid convergence in a neighborhood of x* (Luenberger, 1989).
H.2. C L A S S I C A L PROGRAMMING Classical programming extends the unconstrained optimization problem, by allowing for equality constraints of the form gi(x) = a i. A full expression of the classical programming problem is maximize
F(x)
subject to gl(X) = a I
H.1.4.2.3. Conjugate Gradient Methods
These methods combine the advantages of classical gradient methods such as steepest ascent with those of nonderivative "conjugate" methods (see Section H.1.4.1.3). The idea is to modify the search directions at successive iterations so as to ensure that they conju-
!
kgm(X)'= a m with m < k.
750
Appendix H The Mathematics of Optimization
H.2.2. Multivariate Classical Programming
H.2.1. Bivariate Classical Programming Consider an optimization problem involving two decision variables Xl and x 2, along with a single equality constraint g(x) = a. The only feasible values for an o p t i m u m are along the curve defined by the constraint, and an optimum is found at a point of tangency of the constraint curve and a contour of F(x) (Fig. 21.4). On condition that F(x) and g(x) are differentiable, it is relatively straightforward to show that this condition is expressed mathematically by c3g/Ox 1
3F/ax I
3g/ax 2
3F/Ox 2"
(H.6)
Condition (H.6) also can be obtained by the method of Lagrangian multipliers, by incorporating the constraint directly into the objective function: L(x, k) = F(x) + k[a - g(x)]. where k is the Lagrangian multiplier for the constraint and x 1, x 2, and k are treated as independent (unconstrained) variables in the expanded "Lagrangian function" L(x, k). Differentiating L(x, k) with respect to x and k yields aL = O F cOX1
The general multivariate problem with x' Xk) and m equality constraints gl(X)
aX 2
...,
a
(m < k) can be handled in much the same way as the bivariate problem with a single constraint. Thus, the only feasible candidates for an o p t i m u m are on the hypersurface defined by the constraint equations, and an o p t i m u m is found at a point of tangency of the hypersurface and a contour of F(x). The mathematical argument proceeds as before, with modifications for the multidimensional character of the problem. We assume that the variables in x can be reordered into vectors x I and x__2 of dimension n - m and m, respectively, with x 2 - - h(x 1) a local solution. In addition, the Jacobian matrix =
ag
3gl ( X ) OK--n-
"3gl
cOX1'
GK1
eL = a__r - x ag OX2
(Xl,
a1
Lgmix)
kag
OX1
=
' 9
.
.
OX2' 9
and
~
g~m a (x)
o
...
ag___~m (x)
_ OX1 --
COXn -- _
aL/0k = a - g(x), Vgl(X)
so that z
OF = kag 3x Ox
(H.7)
and
Vgm(X)J is assumed to be full rank at x* (see Appendix B). After some rather complicated mathematics, it can be shown that a first-order condition for optimization is m
g(x) = a
(H.8)
at an o p t i m u m x*. Eliminating X from cOL/ax I = 0 and a L / a x 2 = 0 in Eq. (H.7) produces the optimality condition (H.6). Of course, differentiation of the Lagrangian function by X simply restates the constraint equation g(x) = a. Equations (H.7) and (H.8) establish necessary conditions for a maximizing value x*. A m a x i m u m is guaranteed by a negative-definite Hessian of F in Xl and x 2, subject to certain conditions on the constraint gradient.
OF 3x
ag
X-- = 0' -3x -
at x*, where _Xis given by -1
_X = Ox__2kax2/ with the existence of [(3g/Ox__2)(x*)] -1 guaranteed by the full rank of the Jacobian matrix.
H.3. Nonlinear Programming First-order stationarity also can be derived by the method of Lagrangian multipliers, by incorporating the constraints directly into the objective function by means of m Lagrangian multipliers _k = (Xl, ..., )~m): L(x, X) = F(x) + X[a - g(x)],
OF
ag
Ox
Ox
- Ox
be negative-definite, provided x* satisfies the gradient condition Vg(x*)d(x) = I~x_(X*)ldx.
(H.9)
where Xi is the Lagrangian multiplier for the constraint gi(x) -- a i and the variables in _xare treated as independent (unconstrained) variables in L(x, X). Differentiation of L(x, X) with respect to x and k_ yields aL
751
and aL
a-x = [a_ - 3 (x)].
Necessary conditions for a local maximum x* are therefore
Negative-definiteness in turn is guaranteed if the last n - m principal minors of HL(X, x) alternate in sign, starting with (--1) m+l. Because the Hessian Hr(x) of the original objective function is bordered by the Jacobian Jz(x) in HL( ~, x), the latter is often referred to as a bordered Hessian. The optimal Lagrangian multipliers M, ..., X* provide useful information about the marginal influence of the corresponding constraints. Heuristically, we may think of the decision variables and Lagrangian multipliers as functions of the constraint constants in a. Then the Lagrangian in Eq. (H.9) can be expressed as L(a) = F[x(a)l + X(a){a- g[x(a)]}, E
and differentiation with respect to a yields
ag
~
3x-
= x*=(x*) - 3x-
(H.10)
and g (x*) = a.
(H. 11)
=
--ox!
+
But Eqs. (H.9) and (H.11) ensure the first and second terms in this expression go to zero at x*, so that OL (x*, X *)
Expressions (H.10) and (H.11) represent n and m necessary conditions for optimization, and in combination they constitute a system of n + m equations in the n + m variables in x and k__.Equation (H.11) asserts that the equality constraints are satisfied at an optimal solution x*, and these equalities in turn establish that maximization of L(X, x) is equivalent to maximization of F(x) at x*. Geometrically, Eq. (H.11)says that an optimal solution x* must be on the hypersurface defined by the constraint g(x*) = a, and Eq. (H.10) says that the gradient OF/Ox of the objective function at x* is a linear combination of the contour gradients (3gi/3x)(x*) (with weights given in k_*). Along with the necessary conditions (H.10) and (H.11) for optimization, sufficiency can be expressed in terms of the Hessian
o
Ig(x) ]
[a - g(x)l + x.
-
OF =
-- )t*.
H.3. N O N L I N E A R PROGRAMMING Nonlinear programming extends the classical programming problem by allowing for inequality constraints gi(x) ~ b i on allowable values of the decision variables. The problem is stated thus: maximize
F(x)
subject to gl(X) G b I
I
of the Lagrangian, where HF(x) is the Hessian of the original objective function, Jz(x) is the Jacobian matrix for the constraints, and 0 is the null matrix of dimension m. Thus, for a point x* satisfying Eqs. (H.10) and (H.11) to be locally maximizing, it is sufficient that HL(_k*, x*)
Lg,,(x) <- bn x>_O. B
Both equality and inequality constraints can be accommodated in the formulation, because the equality con-
752
Appendix H
The Mathematics of Optimization
straint gi(x) = b i can be expressed as the pair of inequality constraints gi(x) < b i and -gi(x_.) < - b i. Note that it is possible to express a free (unconstrained) variable, say x 1, as the difference Xll - - X12 of t w o nonnegative variables. The nonnegativity conditions x -> 0 restrict the feasible solutions in nonlinear programming to the nonnegative orthant of the n-dimensional Euclidean space E n. Feasible solutions are restricted further to a subset of the orthant by each constraint gi(x) ~ b i. The opportunity set X consists of values x in the intersection of these subsets.
to zero (for x~ = 0). Then ..., n, so that ar (x,)x
= ~,
ax_--
j
(aF/Oxj)(x*)xj = OF
~(x*)xj
for j - 1,
= o.
-
It follows that a maximizing value x* must satisfy OF re(x*) - 0, Ox-
OF
E. G(x,)x
(H.13)
=o
(H.14)
]
H.3.1. Nonnegative Constraints Only
and
A special case of the nonlinear p r o g r a m m i n g problem restricts the constraints to nonnegativity conditions only: maximize
0
x* - 0.
(H. 15)
F(x)
H.3.2. General Inequality Constraints subject to
In this case, the optimization problem is x_>0.
maximize
For a local m a x i m u m x*, F(x* + hAx) <_ F(x*) for all x* + Ax in a neighborhood of x*. As with unconstrained optimization, we utilize a Taylor series expansion about x* to get F(x* + h 7OF _ + hAx) _ = F(x*) _ x (x*)ax h2 ,, ,32F + ~ - a x 7x2(X*) + h 0 a x ) a x ,
where hAx is an arbitrary "small" deviation from x*, (OF/Ox)(x) is the gradient of F at x, (32F/Ox2)(x) is the Hessian matrix for F at x, and 0 -< 0 <- 1. This leads to
F(x)
subject to g(x) <-- b,
x__>O. The approach here is to introduce "slack variables" s' = (s 1, ..., sin), one for each inequality constraint, so that the problem can be described in terms of equality constraints and nonnegativity conditions. Defining the nonnegative vector s by s -= b - g(x), we can express the optimization problem as
m
h2,, ,32F h aa xf (x*)ax + - ~ a X T x 2 (x* + h O a _x ) a x_ - < 0. _
maximize
F(x)
subject to g(x) + s = b,
(H.12) X_>0, m
If x* is interior to X, i.e., x > 0, the same conditions for optimality obtain as for the unconstrained optimization problem, and in particular, all first-order partial derivatives vanish at x*. Assume, however, that the optimal solution includes a decision variable with optimizing value of zero: x~ = 0. Because first-order partial derivatives for the remaining decision variables vanish, from condition (H.12) we have (3F/Oxj)(x*)Axj 0. Because Axj >_ 0 by virtue of xj = 0 and the nonnegativity condition x -> 0, it follows that (OF/Oxj)(x*) ~ O. From this we conclude that the partial derivatives of F either vanish (for x~ > 0) or are less than or equal
s_>0. m
Disregarding for the m o m e n t the nonnegativity conditions x >- 0 and s -> 0, this is a classical p r o g r a m m i n g problem that can be handled with Lagrangian multipliers: L = F(x) + k[b - g(x) - s], w
m
where k includes n Lagrangian multipliers corresponding to the equality constraints. As before, firstorder necessary conditions for an o p t i m u m are that the partial derivatives of L with respect to x, s, and __k
H.3. Nonlinear Programming vanish. However, the nonnegativity constraints on x and s require that these conditions be expanded as in expressions (H.13)-(H.15). The expanded conditions can be written as 0L,,
,)
tx
OF =
0g (x*) -
0x (x*)_< o ' ,
m
(x* x*)x* = .=[~ (x*) _
x* = o, _
_
n decision variables and m constraints, restricting the search to the opportunity set means that the dimension of the decision space (and thus the number of independent decision variables) is essentially n - m. Examples are gradient projection, reduced gradient methods, and the method of feasible directions. H.3.3.1.1. Gradient Projection
3L'(x* k*) = b - g(x*) > O, Ok ' --
OL
753
_
(H.16)
h*OL(x * K*) = K*[b - g(x*)] = 0, . . .
- 3k . m
X* --~ 0, k*>_O. These are the well known Kuhn-Tucker conditions of nonlinear programming. The direction of the inequalities in the Kuhn-Tucker conditions indicates that (x*, _K*)is a saddle point of the Lagrangian, in that L(x, ~) is maximized with respect to the decision variables x and minimized with respect to the Lagrangian multipliers _h. As with classical programming, the optimal Lagrangian multipliers can be interpreted in terms of a marginal change in the objective function with respect to the constraint coefficients:
The idea with gradient projection is to search in the direction of the gradient, but with suitable reorientation of the search direction hs needed to account for the constraints. It starts with an initial value x 0 in the opportunity set X and moves at each step in the direction of the gradient of F, provided that direction remains in X. A formula describing each step in the algorithm is Xk+l -- Xk q- ~kVF(Xk )'
with step size
m
OF (x*)= ~* 0b -"
(H.17)
Note that the Kuhn-Tucker conditions reduce to conditions (H.13)-(H.15) in the absence of inequality constraints.
H.3.3. Solution
Approaches
[VF(xk)l [VF(xk)l' [VF(xk)] [H__p(Xk)] [VF(xk)]" provided Y_k+1 remains in the opportunity set X._ If at some step in the iteration the gradient direction is infeasible [i.e., if x k is on a boundary of X and VF(x k) points away from X], the direction of movement is altered to follow the projection of the gradient vector on the tangent to the boundary of X. The corresponding step size is chosen to increase the value of the objective function while remaining in the opportunity set. The sequence {Xk}can be shown to converge to x* provided the objective function is concave and the opportunity set is convex. H.3.3.1.2. Reduced Gradient
The overall logic with iterative methods for constrained optimization is to move at each step in a "best" (unconstrained) direction unless a constraint forces a change in that direction. Approaches to constrained optimization can be grouped into four broad classes, roughly corresponding to a focus on (1) the decision variables constituting the objective function, (2) a subset of those variables as specified by the constraints, (3) the Lagrangian multipliers, or (4) the combination of both decision variables and Lagrangian multipliers.
This approach is a variant of gradient projection, in that the gradient VF(xk) again is adjusted at each iteration to ensure that feasible approximations of x* are generated. The added feature here is that one takes advantage of the constraints to reduce the dimensionality over which the search is conducted. To illustrate, consider a constrained optimization problem with n decision variables and m constraints"
H.3.3.1. P r i m a l M e t h o d s
subject to
Primal methods involve the search for an optimal value x* via procedures for unconstrained optimization, as adapted to ensure the search remains within the opportunity set X. For classical programming with
maximize
F(x)
A x = b, m
where the constraints are linear and m < n. The decision variables can be grouped into vectors x I and X__2
754
Appendix H
The Mathematics of Optimization
of dimension m and n - m, respectively, with x' = (x~, x~). Then A x can be partitioned as ay
= a l x 1 q- a 2 x 2 ,
where the matrix a I is of dimension m • m. To simplify notation we assume that the constraints are linearly independent, so that A is full rank and therefore nonsingular (see Appendix B). Thus, the constraints can be rewritten as a l x I q- a2x__2 -- b or Y1 - - a l I [ b
-
-
a2x__2].
This allows us to rewrite the optimization problem absent the constraints, in terms of the n - m decision variables in x 2 only: maximize
direction of vector Dk is chosen to ensure that a search from x k in that direction at least initially yields candidates for Xk+l in the opportunity set. Each step in the algorithm thus consists of a constrained line search in a feasible direction, with the selection along the line of an optimal value Xk+1 via an optimizing choice of 8k. A useful criterion for selection of the direction vector D k is that it be aligned with the gradient dF(xk)/dx = VF(xk)' as closely as possible and still remain feasible. We illustrate with a problem that has linear constraints. Assuming D k is normalized by requiring its elements to sum to unity, the optimizing choice of D k can be obtained as the solution of the linear programming problem: minimize
VF(xk)D
subject to
F(a11[b -- A2x2] , X2).
AD<-b, m
Gradient projection then can be used to search an opportunity space that is of dimension n - m. This approach can be generalized to allow for inequality constraints, through the incorporation of slack variables (see Section 22.5) to transform the inequalities into equality constraints. Another generalization that addresses nonlinear constraints h(x) <- 0 is via a similar partitioning of the decision variables so that the m • m matrix a h / a x 2 is nonsingular at (x 1, x2). In the latter case, an iterative search involves movement along a tangent to the constraint surface, followed by a correction as in gradient projection to return to the constraint surface (Luenberger, 1989). H.3.3.1.3. Feasible Direction
This approach involves choosing a direction D k to deviate as little as possible from VF(Xk), while ensuring that at least some movement in this direction is possible. With linear constraints, under some rather mild conditions on the normalization of candidate directions D k, a feasible direction can be found at each step via linear programming. The corresponding step size in the direction of Dk typically is determined by the nearer of (1) the point where the direction vector leaves the opportunity set X, or (2) the point at which F(x) reaches a m a x i m u m in the direction of D k. Mathematically, the method of feasible directions is described as a search of the opportunity set with steps of the form Xk+ 1 = X__k -}- ~)kDk,
with D k a direction vector at step k, and 8k an appropriately chosen nonnegative step size. At each step the
]Di]-
1,
i
where the inequality constraints correspond to the constraints that define the opportunity set X. In some sense this describes the locally "best" feasible direction in which to search, with the step size chosen to maximize the objective function in that direction. The problem is of course more complicated for nonlinear constraints. For certain kinds of nonlinearities, it may be necessary to relax the requirement of feasibility by allowing the points x k to deviate somewhat from the constraint surface, or otherwise allow the search to move along curves rather than straight lines (Luenberger, 1989). Given their general applicability, primal methods constitute an important class of iterative solution procedures for constrained optimization. They have certain advantages over other approaches to nonlinear optimization. First, all the candidates in an iterative sequence {xk} generated by primal methods are feasible, so that termination of the sequence at any iteration is guaranteed to produce a feasible approximation of x*. Second, the sequence will converge to a local optim u m for reasonably well-behaved problems. Third, primal methods are applicable to general programming problems, whether they include linear or nonlinear objective functions a n d / o r linear and nonlinear constraints. But primal methods also have certain disadvantages. For example, one must identify a starting point in the opportunity set, which sometimes can be difficult to find. In addition, difficulties arise in trying to remain within the opportunity set when the constraints are nonlinear, and indeed, nonlinearity can lead to a failure of the iterative search to converge.
H.3. Nonlinear Programming H.3.3.2. P e n a l t y and Barrier M e t h o d s
These approaches involve the approximation of a constrained optimization problem by an unconstrained problem, which then can be solved with procedures that are appropriate for unconstrained optimization. The approximation can be effected in either of two ways: (1) in the case of penalty methods, by adding to the objective function a term prescribing a high cost for violation of the constraints, or (2) for barrier methods, by adding a term that favors points in the interior of the opportunity set rather than its boundary. Associated with these added terms is a weighting factor that determines the severity of the penalty or barrier and therefore the degree to which the modified optimization problem approximates the original problem. As the weighting factor increases in magnitude, the approximation becomes an increasingly accurate representation of the constrained problem, and the solution of the approximating problem typically converges to x* (Luenberger, 1989). Penalty and barrier approaches involve the full suite of n decision variables, in contrast to primal methods, which are restricted by the space defined by the opportunity set. Penalty methods can be described for the generic constrained optimization problem maximize
F(x)
subject to xeX
by introducing a continuous function P such that (1) P(x) -> 0 for all x e E n and (2) P(x) = 0 if and only if x ~ X. For positive constant c, the optimization problem can be approximated by minimize
- F ( x ) + cP(x).
The larger the value of c, the more sensitive is the optimization to the second term in this objective function. Barrier methods also use an approximating objective function, which includes a barrier function that serves as a barrier on the boundary of the opportunity set that prevents a search procedure from leaving the set. A continuous barrier function B(x) is defined by (1) B(x) --- 0 and (2) B(x)~oo as x approaches the boundary of X. For positive constant c, the optimization problem can be approximated by minimize
- F ( x ) + B(x)/c.
755
Two issues are especially important in the application of penalty and barrier methods. One concerns the degree to which the unconstrained problem actually approximates the constrained problem, and thus how well the unconstrained solution approximates the constrained solution. Let {Ck}be a sequence of constants with Ck+l ~ Ck, such that Ck----)oo.Then the corresponding sequence {xk} of minimizing solutions for both penalty and barrier approximations can be shown to converge to the solution of the original optimization problem (Luenberger, 1989). The other is how to devise efficient solution algorithms for problems containing a penalty or barrier term. The difficulty here is that convergence rates of many standard algorithms decrease dramatically as the weighting factor is increased to improve the approximation. It thus is necessary to devise specialized procedures for accelerating convergence to an optimizing solution. It often is useful to apply a modified Newton's method, or, if there are only a few active constraints, a conjugate gradient method. One often can greatly accelerate convergence simply by a careful choice of the penalty and barrier functions. H.3.3.3. Dual Methods
Dual methods focus on constraint weights (the Lagrangian multipliers) as the fundamental variables to be optimized, with the idea that determination of optimizing values for the Lagrangian multipliers is tantamount (at least in some cases) to finding the optimal solution x*. Thus, the original constrained optimization problem involving n decision variables and m constraints is replaced with a dual problem involving m decision variables. Provided the number of constraints is substantially lower than the number of decision variables, this can result in substantial efficiencies in finding an optimal solution. To illustrate, consider the constrained optimization problem maximize
F(x)
subject to h(x) <- O,
x~O. Under certain convexity assumptions, a maximizing value x* for this problem can be found as a solution of an unconstrained problem, with the Lagrangian L(x, k) = F(x) - kh(x)
In this case, the smaller the value of c, the more sensitive is the optimization to the second term in this objective function.
as an objective function in the n decision variables in x and m Lagrangian multipliers in _k = (k 1, ..., k m)
756
Appendix H The Mathematics of Optimization
(see Section 22.3). Because the partial gradient of the Lagrangian with respect to the variables in _Kis simply VL~(x, ~) = -h(x)', the original optimization problem can be written as
L(x, ~)
maximize
[VL~(x, ~)] ~'
-
x
H.3.3.4.
Lagrangian
Methods
It also is possible to address the optimization of constrained problems via the Lagrangian, by simultaneously solving for the optimizing values of the decision variables and Lagrangian multipliers in the Lagrangian function. To illustrate, consider the classical programming problem
F(x)
maximize
subject to -VLdx, K) -~ 0',
subject to
h(x) = O,
x~O, where the maximization is with respect to x. This suggests a symmetric or "dual" problem, namely, minimize
L(x, )0
-
[VLx(x, ~)]
with x an n-dimensional vector of decision variables and h(x) = 0 a set of m constraints. A solution approach involves the use of m Lagrangian multipliers _h = (~1, .... Xm) to find optimizing values x* and _h* for the Lagrangian
subject to
L(x, X) = F(x) - Xh(x). -VLx(x, k) >- 0',
The gradient of L(x, ~) is
m
h~O,
VL(x, )0 = [VLx, VL~]
where the minimization is now with respect to _h. But the partial gradient VLx(x, ~.) of the Lagrangian L(x, ~) can be written as
= [VF(x) - )~J, h (x)'], u
where
VL x = VF(x) - h_J_h(X),
Vh 1
where J_h(X) is the Jacobian matrix
l= Vhl(X)
LVhmJ
=
is the Jacobian matrix for h(x). Necessary conditions for optimization are therefore
LVh2(/
VF(x) + XV/= 0'
Thus, the dual programming problem is minimize
F(x)- kh(x)-
[ V F ( x ) - kJh(X)]X
and
h(x) = O.
subject to
m
lh(x) -
VF(x) _> 0',
D
This constitutes a system of m + n equations in x and ~. As with the other classes of constrained optimization methods, there are several methods based on the Lagrangian. Among others, these include first-order methods, conjugate directions, Newton's method, and modified Newton's method, which are extensions of methods described above to accommodate searches in (n + m)-dimensional space. An example is the differential gradient method, which uses the gradient and Hessian of the Lagrangian in a manner analogous to the method of steepest ascent for unconstrained optimization (see Section H.1.4). m
K&0. In essence, the dual method of solving constrained optimization problems is to focus on the latter formulation, with a goal of finding a solution ~*. It sometimes is possible to describe the dual problem in terms of the Lagrangian multipliers only, based on a prior conditional optimization with respect to x. Then the solution of the dual problem corresponds to that of the original "primal" problem, with the solution of the latter obtained directly from the solution of the former. m
H.4. Linear Programming It should be clear from the above discussion that there are strong interconnections between and among the methods in the different classes of iterative approaches, both in terms of implementation and performance. In fact, many of the procedures for constrained optimization were adapted from procedures for unconstrained problems. Much as the rates of convergence for unconstrained problems are determined by the Hessian of the objective function, so are the rates of convergence for constrained optimization determined in most cases by the structure of the Hessian matrix of the Lagrangian (Luenberger, 1989).
757
H.4.1. Kuhn-Tucker Conditions for Linear Programming As a special case of nonlinear programming, the linear programming problem is amenable to the use of Lagrangian multipliers for derivation of the Kuhn-Tucker conditions [Eq. (H.16)]. The Lagrangian function for linear programming is L(x,
h) =
cx +
h(b -
Ax),
and the corresponding Kuhn-Tucker conditions at (x*, h*) are OL/Ox = c -
hA
<- 0 ' ,
OL/Oh = b -
Ax
>- O,
H.4. LINEAR P R O G R A M M I N G m
The linear programming problem is a special case of nonlinear programming, in which the objective function is a linear form F(x) = c x with c = (cI .... , Cn), and the opportunity set is defined by linear constraints A x -< b with
w
(OL/Ox)x = [cK(OL'/Oh)
hAlx
= 0,
Ax]
= 0,
= k[b -
x>_O,
k_>0'. all a21 a
a12 a22
... ...
aln a2n
.__
H.4.2. Dual Linear Programming m
and b = (bl,
..., bin)'.
am1
am2
...
amn
. .
A statement of the problem is
maximize
F(x) = c x
subject to Ax
An important consequence of linearity in the objective function and inequality constraints is that a second, or dual, linear programming problem can be defined by switching the role of the Lagrangian multipliers and the decision variables. The dual problem involves the minimization of an objective function in the Lagrangian multipliers:
<- b,
minimize
hb
x>_0. subject to As before, the nonnegativity constraints x _ 0 restrict feasible solutions to the nonnegative orthant of E n. Additional restrictions are imposed by the linear constraints a i l x I + ai2x 2 + ... + a , x n <_ b i
in A x -< b, each of which defines a set of points in the n-dimensional Euclidean space E n on one side of a corresponding hyperplane a i l x I q- ai2x 2 q- ... + a , x n -- bi"
The opportunity set X consists of values x in the intersection of these subsets (Fig. 21.5). m
hA >-- c, h>O' where hA >- c specifies n linear constraints on the decision variables in h. Note that the constraint constants of the original (or primal) problem are used here as objective constants, the objective constants in the primal problem are now the constraint constants, and the direction of the inequalities has changed in the dual problem. The Lagrangian of the dual problem is w
L(h,
x) = hb + (cD
hA)x,
758
Appendix H The Mathematics of Optimization
with x now representing m Lagrangian multipliers corresponding to the m constraints in k A >- c. The corresponding Kuhn-Tucker conditions at (_k*, x*) are
ation ~ + 8~ on the system rate of change. Thus, the objective functional can be expressed as J = J(8)
fttfo I(x + 8~I, Yc + 841, t)dt,
=
03L ' / 03)~ = b - A x >_ O, aL/ax = c-
;~A <_ 0',
X(aL'/aM = x[b - Ax] = O, (aL/ax)x = [ c -
highlighting the fact that J is a function of 8. A necessary condition for {x(t)} to be optimal is that the derivative of J must vanish at 8 = 0:
k A l x = O,
=
I(x + 8~1, Yc + a~h, t ) d t
k ~> 0',
= to
x_>0. But these are the same conditions as for the primal linear programming problem, though the roles of x and k are reversed. The identity of the Kuhn-Tucker conditions for both problems confirms the remarkable result that the primal and dual optimization problems yield the same values x* and k_*. Thus, if a solution x* to the primal problem and a solution _k* to its dual problem can be found, taken together the pair (x*, k_*) is guaranteed to meet the Lagrangian conditions and thus to solve the extended Lagrangian optimization problem. In this way the dual problem provides an efficient means of identifying the sensitivities (03F/03b)(x*) = _k* as in Eq. (H.17).
n+
=0.
By integrating the second term in brackets by parts, this equation can be written as f
[+i
to
"q - ~ x - ~
maximize
f ttfoI(x, Yr, t) dt
subject to x (t 0) = x 0,
x(t? = x , where x = x(t) = [xl(t) .... , xk(t)]'.
H.5.1. Univariate Calculus of Variations Following the differential approach of classical optimization, a solution approach here is to consider a variation x(t) + 8Tl(t) on the state trajectory, where ~(t) is an arbitrary trajectory with ~(t 0) = Tl(tf) = 0 to accommodate the initial and terminal time conditions. Then 8~q(t) represents a "small" deviation from x(t) over T, and this variation induces a corresponding vari-
+i]+
-~x
dt +~q-~x
to
=0,
from which it follows that Euler's equation +I 03x
dt a-~ = 0
(H.18)
and the transversality condition = 0
H.5. C A L C U L U S OF V A R I A T I O N S The problem addressed by the calculus of variations is to choose a piecewise differentiable function x(t) that maximizes an integral objective of the function, its time rate of change _~, and time. The problem statement is
n at
(H.19)
to
must be satisfied for an optimal trajectory. Differentiating 031/03Ycwith respect to t allows Euler's equation to be rewritten as
(0321~d2x
-d7 +
( o32I ~ dx ox
-d-t +
( 0321 o7-&
0
Thus the search for an optimal trajectory for the calculus of variations problem reduces to a search for solutions of a second-order differential equation in x, 2, and t, with boundary conditions given by the initial and terminal conditions and x(t o) = Xo and x(tf) = xf [the transversality conditions are met automatically, because ~(t 0) = +q(tf) - 0]. There are other conditions besides Euler's Eq. (H.18) and the transversality conditions (H.19) that also must be satisfied by an optimal solution. For example, the Legendre condition 0321/03X2~ 0 must be met at all points in the time frame between t o and tf. In addition, the Weierstrass condition imposes "concavity" requirements on I with respect to 2, and the WeierstrassErdmann condition requires that aI/03Yc and I - (03I/aYc)Yc be continuous over the time frame.
H.6. Pontryagin's Maximum Principle
H.5.2. Multivariate Calculus of Variations Optimality conditions for the multivariate problem are completely analogous to those for the univariate problem. In particular, the multivariate version of Euler's equation is
759
each of the equality constraints. The augmented objective functional is l' = l +
f
t~ _x(t) [[(x, u, t) - x_]dt to
(I(x, U, t) + _X(t) [fix, U, t) - x_'l)dt + Fl[X(tf)l,
ax
dtkaYcJ = 0',
(H.20)
involving n equations, one for each of the state variables. For fixed initial and terminal times, the corresponding transversality conditions are
and the search for an optimal trajectory {U(t)} becomes a search for optimal trajectories of both U(t) and _X(t) for the expanded problem. The augmented objective function can be written in terms of the Hamiltonian
H(t) = I(x, U, t) + X_(t) [(x, U, t),
(H.21)
so that
for t = t 0, tf. Extremal trajectories satisfying condition (H.20) play an analogous role to that of stationary points satisfying aF/Ox = 0' in mathematical programming.
J' =
[S(x, U, X, t ) -
X21 dt + fl[x(tf)l.
to
Integration of the second term in the integrand by parts produces
l' =
[H(x, U, X, t) + X x] dt to
H.6. P O N T R Y A G I N ' S
+ {Fl[X(tf)] -
_X(qx(9} +
_x(t0)x(t0).
MAXIMUM PRINCIPLE The maximum principle generalizes the calculus of variations by including complex constraints on the control variables in U(t). The control problem is maximize U(t) ~ U
m
n
f tf I(x, U, t) dt + Fl[X(tf)]
Taking a variation 8U in controls induces a variation 8x in the system state and thus in J', so that 8J'=
ft {
(~xH+X_)Sx+(aa---~)SU
to
}
dt
-
to
F1
+ WIx(9 subject to
- x_(9
}
9
m
x_"= [(x, U, t), x(t0) = x0,
A necessary condition for optimization is that 8J' must vanish for arbitrary variations 8U and 8x, which in turn gives, as conditions for optimality,
OH/OU_ = _0,
9 = x, -
where x = x(t) = [xl(t) , ..., Xn(t)]'. The controls in {U(t)} are constrained to the control set U, and system change x_"is influenced but not directly controlled by U(t).
H.6.1. Unconstrained Optimal Control Consider a special case of the control problem in which there are no limits on the control set, i.e., the control trajectory {U(t)} can be any piecewise continuous function of t. A solution can be obtained by introducing "costate" variables _X(t) = [Xl(t) .... , hn(t)] for
aHlax
[K__ -- c~Fl l O x l a x
t o <_ t <_ tf;
= _~.,
t o <_ t <- tf;
= o,
t = tf.
Thus, an optimal solution {U*(t)} maximizes the Hamiltonian at each time t in the time frame, and the search for an optimal trajectory is limited to the investigation of trajectories meeting this condition. This result depends, of course, on the fact that there are no constraints on U at each point in time, so that a value of U(t) can be found at a point where the derivative a H / a U vanishes. The differential equations involving the derivatives in "_Xdescribe the required time rate of
760
Appendix H The Mathematics of Optimization
change for the costate variables in K, and the transversality condition, 0 = [~-
OF1/cgX__I~x,
t=
tf,
J*[U*(t)]- ][U*(t) + 8U(t)]
requires that the components of 8x or_X - 3F 1/3x vanish at the endpoint tf. A terminal time constraint x(tf) = Xf forces the variation 8x_ to vanish at tf, thereby allowing X_(tf) to be free. On the other hand, the absence of a terminal time constraint allows the variation 8x to be free at tf, so that _X(tf) = (aF 1/0X)( 9. The net effect in the latter case is to replace a boundary condition on x(t) with one on _X(t). Then the optimal control is given by the solution of the equations
Yc = OH~Ok (H.22) = f(x, u, t)
and =
-OH/Ox,
augmented objective functional by J*[U*(t)] and J[U*(t) + 8U(t)] for U*(t) and U*(t) + 8U(t), respectively, then
(H.23)
m
where U is given in terms of state and costate variables by 3H/OU = 0' and x(t 0) = x 0, X_(tf) = OF1/3x(tf). This is a two-point boundary value problem, wherein integration of the state equation proceeds forward in time from t o to tf and integration of the costate equation proceeds backward in time from tf to t 0. These integrations are complicated by the fact that x(t) appears in the costate equation, and _X(t)appears in the state equation [through U(t), a function of _k and _x]" Sufficient conditions for a control strategy to be optimal, at least in a neighborhood of the strategy, require that the Euler-Lagrange conditions be satisfied at each point in the time frame, and also require that the Hessian matrix for H with respect to the controls be negative definite at each point in the time frame. These are local criteria that ensure neighborhood-optimal control strategies, but they do not guarantee that a local solution is also a global optimum over E n. For highly nonlinear systems, this can mean that convergence of a search procedure to an optimal trajectory may require that the search be initiated with a trajectory that already is close to optimal.
H.6.2. Constraints on the Control Trajectory In this case, the feasible control trajectories are constrained to be in some bounded control set U, and the constraints translate into bounds on U(t) at each point in time. An approach to optimization under these conditions is based on variations 8U(t) about an optimal control strategy U*(t). If we designate the value of the
=
f
tf
{H[x*(t), U*(t), _X*(t), t] - H[x*(t), U*(t) to
+ 8U(t), k*(t), t]} dt >- O. This condition must hold for arbitrary 8U(t), which means that
H[x*(t), U*(t), k*(t), t] - H[x*(t), U*(t) + 8U(t), k*(t), t] _>0 for t o ~- t ~ tf. In words, an optimal control strategy must maximize the Hamiltonian at each point t in the time frame. This "maximum principle" is consistent with a solution approach for unconstrained problems based on stationarity and concavity conditions, because it is satisfied under the latter conditions. For constrained optimization problems with ~-~trepresenting the constraints at time t, the Hamiltonian is to be maximized by choosing the appropriate value U(t) in 12t. The optimization problem then is to maximize
H(x, U, t)
U(t) * ~t
for all t ~ T. An optimal value is obtained either at an interior point of 12t, in which case OH/OU vanishes, or at a boundary point of f~t. Irrespective of the nature of the constraints, it is easy to see from Eq. (H.21) that the partial derivative OH~Ok_ of the Hamiltonian reproduces the time rate of change of the state variables. Thus, a general solution of the optimal control problem consists of trajectories x(t), U(t), and _X(t) for which the Hamiltonian is maximized over ~'~t at each point in the time frame, and the canonical equations are satisfied:
Yc = OH~OK_, ~_ = -OH/Ox,
x(t 0) = x0;
X_(tf) = OF 1/o3Xf.
Maximizing the Hamiltonian with respect to U(t) typically allows one to identify the optimal control variables as functions of the state and costate variables, so that the canonical equations can be expressed in terms of 2n state and costate variables without reference to controls. Thus, the optimal state and costate trajectories solve a system of 2n ordinary differential equations with split boundary values. The optimal control trajectory subsequently is identified by using the optimal state and costate trajectories in the functional relationship derived from the Hamiltonian.
H.6. Pontryagin's Maximum Principle
H.6.3. Discrete-Time M a x i m u m
Principle
It is possible to derive a version of the canonical equations for problems in which the time frame is discrete. The relevant optimization problem in discrete time is
constrained, this in turn requires that oJ'/oe = 0 and o2J'/O82 < 0. The stationary condition oJ'/O8 = 0 gives
aH
t=to
tf-1 maximize
~
{ u ~ u}
761
Ox(ti 3(t) + OU(t)E(t)- ~_(t + 1)3(t + 1) OF1
I[x(t), U(t), t]
+
~(t0)3(t 0) + ~
Fl[X(tf) ]
t=to
subject to
= x(t + 1) = x(t) + fix, U, t),
t=to
Ox(t)'q(t) - h_(t)3(t)+ Oa~iI((t)~(t)] --
OF x_-~f)3 (tf) - _h(tf)3(tf) ].
x(t0) = x0. Lagrangian multipliers can be used to incorporate the transition equations and initial conditions into the objective functional, so that
At each point in time, the elements of this equation must vanish, which means
OH ~(t) = 0,
tf - 1
J'=
~
]
~I[x(t), U(t), t ] -
h_(t + 1){x(t + 1)
au(t)
-
t=to
- / [ x ( t ) , U(t), t]}~ + _K(to)[Xo - x(to)]
+
I OH Ox(t)
Fl[X__(tf)]
K(t)]~(t) = 0,
-
-
tf--1
= ~
{H[x(t), U(t), _h(t + 1), t] - _K(t + 1)x(t + 1)}
and
t=to
El
where
H[x(t), U(t), )~(t + 1), t] = I[x(t), U(t), t)]
H.6.3.1. Discrete-Time Optimization in the Absence of Control Constraints First we consider a situation in which there are no constraints on the control trajectory, i.e., the vector U(t) of control variables can be anywhere in E n. A perturbation argument can be used as before, in which small perturbations {e~_(t)} and ~(t) about trajectories x(t) and U(t) give tf--1
{H[x(t) + ~3(t), U(t) + ~(t), _h(t + 1), t]
=0
over the time frame. Because the perturbation trajectories {_~(t)} and {~(t)} are arbitrary, an optimal trajectory therefore must satisfy
+ ~_(t + 1)/[x(t), U(t), t].
/'(8) = ~
_ ]~(tf)
x_-~f)- K(tf)
+ _k(to)[Xo - x(to) ] + Fl[X(tf)],
OH/aU(t) = 0', K(t) = OH/Ox(t), and
x(t) = OH~OK(t),
m
along with the transversality condition
I El
_
x_~f)- k(tf)
] ~(tf)
=0
and initial condition x(t0) = x 0.
t=to
- K_(t + 1)[x(t + 1) + 83(t + 1)]} + K_(t0){[x0 - [ x ( t 0 ) + ~q(to) ]}
+ Fl[X(t f)
q-
8~(tf)].
Conditions for {x(t)} and {U(t)} to be maximal are that J'(0) > J'(~) for "small" values of e and for arbitrary values in {~(t)} and {E(t)}. Because U(t) is unm
Note that these optimality conditions are essentially identical to the Euler-Lagrange Eqs. (H.22) and (H.23) for continuous problems. In both cases the optimization problem reduces to a two-point boundary value problem and typically requires the solution of a system of nonlinear transition equations in state and costate variables. In general, both discrete-time and continuous-time problems must be solved by iterative tech-
762
Appendix H The Mathematics of Optimization
niques. In most instances, the solution of a discretetime problem converges to its continuous-time analog as the partitioning of the time frame becomes increasingly fine. H.6.3.2. Discrete-Time O p t i m i z a t i o n w i t h C o n s t r a i n t s on Controls
In this case, there are constraints on the vector U(t), i.e., U(t) ~ f~t. On assumption that the optimality index, transition functions, and the terminal value function are continuously differentiable, a maximum principle can be derived much as before for continuous systems. Thus, the Hamiltonian
Time differentiation of _k then gives
But differentiation of the Hamiltonian with respect to x also produces _k, with = -aH/ax = -aI/ax.
Eliminating "_kfrom the last two equations yields
ax
H[x(t), U(t), k ( t + 1), t] = I[x(t), U(t), t]
+ k.(t + 1)/[x(t), U(t), t]
H(x, U, t)
U ( t ) ~~ t
for all t e T. An optimal value is obtained either at an interior point of f~t, in which case a H / a U vanishes, or at a boundary point of ~t. A general solution of the optimal control problem consists of trajectories {x(t)}, {U(t)}, and {k(t)} for which the Hamiltonian is maximized over 1~t at each point in the time frame, and the discrete canonical equations are satisfied:
Ox
x(to) = Xo;
~_(t) = aH/ax(t),
~_(tf) = aF1/axf.
H = I(x, 2, t) + k2, _
_
m
_
= O.
H.7. D Y N A M I C P R O G R A M M I N G Dynamic programming seeks a control strategy {U(t)} from some constrained set U that maximizes an objective functional of system states, controls, and possibly time. For deterministic systems with continuous time frames, the problem is expressed as maximize
H.6.4. The Maximum Principle and the Calculus of Variations It is possible to use the maximum principle to derive the optimality conditions of the calculus of variations. Consider the standard calculus of variations problem, with unconstrained control exercised directly through the time rate of change: x_"= U and U ~ E k. The Hamiltonian is
dt
Thus, Euler's equation follows directly from the maxim u m principle. Similar examinations show that the Legendre, Weierstrass, and Weierstrass-Erdmann conditions of the calculus of variations also can be derived from the maximum principle.
m
x(t) = aH/ak_(t),
-~x
or
is to be maximized by choosing the appropriate value U(t) in f~t: maximize
dt
I(x, U, t) dt + fl[x(tf)]
U(t) ~ U
to
subject to 2_ = [(x, U, t),
x(t0) = x0, x( 9
-
An analogous statement for discrete-time systems utilizes
and differentiation with respect to the control gives
tf - - 1
l = ~ , I(x, U, t) + Fl[X(tf)]
OH~a2 = aI/ O2 + k = 0
t = to
so that
and k = -aI/a2.
x(t + 1) = x(t) + F(x, U, t).
H.7. Dynamic Programming Stochastic effects are handled via expectation in the objective functional:
(H.24)
763
H.7.1.1. Linear-Quadratic Control in Continuous Time
An important application in continuous time involves a quadratic objective functional and linear system transitions. A formal statement is l = 1/2 f tr [x'Qx + U'R U] dt
maximize
to
where the expectation is with respect to random elements Z(t) that influence system behaviors by
subject to Yc=Ax
x(t + 1) = m x(t) + F(x, U, Z, t),
with {Z(t)} a time series stochastic process. The solution approach in all cases is to embed the particular problem in a larger class of problems and then seek a global solution by means of the Principle of Optimality (Bellman, 1957), which can be used to generate a recurrence relation for the extended class and from which is derived a partial differential or difference equation. Its solution holds for all problems in the class and is applied to the original problem as a special case.
H.7.1. Deterministic Dynamic Programming In applying dynamic programming, it is useful to write the optimal value of the objective function as J*[x(t), t] to emphasize its dependence on time and the starting point of the state trajectory. Application of the Principle of Optimality yields the fundamental recurrence relation
:(to) = x0,
:(9
- ol*lat = max[I(x, U, t) + (ol*lox)f(x, u, t)],
U,U
which, along with the boundary condition
provides the analytic framework for solving the optimal control problem for continuous systems.
:
x,
with Q and R negative-definite matrices. The HJB equation for this system is oJ* _ max at Eu
U, t)]
~ I(x, U, t) + -~x
= max I(xtQx + UtR U)/2 + OJ~C(Ax + B U)I. u
....
Ox
Because it is unconstrained, the maximization requires the derivative of the expression in brackets to vanish: 0 I (x'Qx + U'RLD/2 +-~x ,91" (Ax + BLD ] = U' R + OJ--~*B OU Ox_=
0'I
from which we get
U* = - R - I B ' ( O ] * ~ '. -
-
-
\Oxl
Substituting this expression back into the HJB equation gives
J*[x(t), t] = max [I(x, u, t)&t + J*(x + Ax_, t + At)l, U,U a form that is appropriate for solution of control problems with discrete states and time frames. Additional smoothing assumptions ensuring the continuous differentiability of J[x(t)] yields the Hamilton-Jacobi-Bellman (HJB) equation
+ BU,
oJ" at
l x , Q x + 1 (oJ" ea_le,OJ*'~ -2 \ Ox - Ox I + al-- A x ax
-
(03]~ B R -1 B, oJ~P~ \ ax ax I
or
oJ* _ l x, Q x + __ OJ* A x _ 1(a J* B R -1 B' oJ*') . at -2 Ox -2 \ ox - ox
If we also assume that tf = oo, then oJ*/Ot = 0 at any particular time t and state x(t), so that the HJB equation becomes 'Qx + Ol___~*A x - 1 (0J* BR _ l e ' 0J*') = 0 .
~x
- - -
ox
-2 \ o x
-
ox
764
Appendix H The Mathematics of Optimization
This is an ordinary differential equation in x, which can be shown to have a solution of the form J* = x ' P x / 2 . Then oJ*/Ox = x'P and the HJB equation becomes
which in turn requires that Q + P A + A ' P - P B R - 1 B ' p = O.
On solving this system of k(k + 1) / 2 equations for P, we then can utilize 3J*/Ox. = x'P . and . U. = - R - 1 B ' (3J*/ox)' to identify optimal controls by
where Z(t) usually is modeled as an uncorrelated white-noise process with a stationary, time-invariant distribution. The corresponding control systems are known as Markov decision processes, and the appropriate formulation of the HJB equation for systems with unit time step is /*[x(t), t] = max E{I(x, U, t) + / * [ x ( t + 1), t + 1]}, U,U
-R-1B'Px(t),
thus describing a linear feedback strategy in x(t). Substituting U*(t) back into the transition equations produces Yc = A x + B U* BR-1B'P]x
= [A -
System transitions can include a stochastic element Z(t) to account for such random factors: x(t + 1) = x(t) + fix, U, Z, t),
x' [Q + P A + A ' P - P B R - 1 B ' P ] x = O,
U*(t) =
H.7.2. Stochastic Dynamic Programming
with a solution k
m
where the expectation refers to the stochastic structure of the exogenous environment or the control trajectory or both. If the optimality index in Eq. (H.24) is time discounted with discount factors that are less than unity, the objective functional is necessarily finite. General conditions that guarantee finite values for the objective functional include, but are not limited to, the discounting of optimality indices (Williams, 1988). The time average
Ci(vieait),
x(t) = ~ i=1
] = lim ( r +
where )~1, ..., )~k are the eigenvalues of A - B R - 1 B ' p and v 1, ..., v k are the corresponding right eigenvectors (see Appendix B). H.7.1.2. A p p l i c a t i o n s in Discrete Time
The HJB equation for discrete-time dynamic programming is J*[x(t), t] = max {I(x, U, t)~t + J*[x(t + At), t + At]},
T~ t
1) - 1 E
~
I(x, U, Z, t)
t= 0
avoids problems with potentially infinite objective functionals, and in particular is appropriate for problems with undiscounted optimality indices. An optimal control strategy for a time-averaged objective functional maximizes the average single-step system gain. It also identifies state-specific optimal values corresponding to the optimal system gain.
U,U
in which the time increment over which system change occurs is At. As a matter of convenience, the time interval in most applications is taken to be 1, so that the HJB equation is expressed by /*[x(t), t] = max {I(x, U, t) +/*[x(t + 1), t + 1]}.
(H.25)
U,U m
Equation (H.25) can be used to identify a solution by means of backward interation from the terminal time tf. At each step in the iteration, one need only choose the time-specific action that maximizes the sum I(x, U, t) + l*[x(t + 1), t + 1] m
of present and future values. Of course, this requires that the future optimal values J*[x(t + 1), t + 1] are available; hence the motivation for backward iteration.
H.7.3. Dynamic Programming and Variational Mathematics Both dynamic programming and variational mathematics share a crucial limitation with respect to the number of state and control variables that can be addressed. Thus, only under special conditions can either approach be used for systems of more than a few variables. This "curse of dimensionality" is especially formidable in analysis of continuous-time systems, with dynamic programming requiring the solution of a partial differential equation and variational mathematics requiring the solution of 2k ordinary differential equations with split boundary values. In either case, the computational difficulties can be severe. In order for these techniques to be applicable, large systems must either be simplified or decomposed into separately an-
H.7. Dynamic Programming alyzed subsystems. Though the tremendous growth in computing power in recent years has dramatically improved our ability to apply these techniques, optimization with dynamic programming and variational approaches still can easily exceed computing capacity. It remains a challenge to construct realistic optimization models of animal populations with feasible computational requirements. It is worthwhile to point out some operational differences between dynamic programming and variational mathematics. An important difference concerns the nature of the solution that is produced. As indicated above, the variational approach results in optimal control strategies in terms of state and costate variables. From this relationship, one sometimes can make generalizations about the form of the optimal strategy without solving the system of difference or differential equations. The iterative procedures in dynamic programming, on the other hand, typically result in statespecific optimal decisions at each point in time, decisions that are appropriate only for the particular problem for which they are derived. It thus is more difficult to characterize mathematically the solutions of dynamic programming problems, or to identify general patterns in the solutions, than it is for a variational approach. On the other hand, dynamic programming offers certain advantages, even in the case of continuous systems. For example, the dynamic programming solution is in the form of feedback control, with the optimal action identified as a function of the system state and
765
time. This is extremely useful in applying the results of dynamic programming. There also can be strong computational advantages in the stage-wise approach, because the optimization can proceed through a series of single-stage optimizations, wherein additional constraints on controls and states can be used to advantage to limit the amount of computation. Furthermore, because dynamic programming is primarily used in the context of multistage optimization, considerable flexibility is allowable in the form of the resource model. The method is applicable to systems with inequality constraints, discontinuities, nonlinearities in both state and control variables, stochastic influences, and other effects that are analytically intractable in a variational approach. On the other hand, a chief advantage of the variational approach results from the decomposition of the control problem into two parts, wherein first the Hamiltonian is used to identify optimal controls as a function of state and costate variables, and then the resultant system of ordinary differential equations is solved. This "stepwise" approach effectively eliminates the control variables from the differential equations, thereby simplifying computations. Furthermore, optimization of the Hamiltonian often exposes patterns of the optimal solutions without the need to solve the equations explicitly. Finally, for sufficiently simple systems it sometimes is possible to determine optimal strategies under parametrically specified boundary conditions, without resorting to a new formulation and solution of the problem for each case.
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Index
A Abundance, see also Density capture-recapture-based estimation, see Capture-recapture methods conservation and extinction models, 231-233 birth-death processes, 232 birth processes, 231-232 minimum viable populations, 233 persistence time, 232-233 count-based estimation canonical population estimator, 243-245 detectability, 244 spatial sampling, 244-245 complete detectability on equal area samples, 245-247 abundance estimators, 245 assumptions, 245-246 sample size effects, 247 shape of unit effects, 246-247 size of unit effects, 246-247 stratification effects, 247 survey designs, 246-247 complete detectability on unequal area samples, 247-250 ratio estimators, 248-249 stratified design-based estimators, 249-250 weighted estimators, 249 description, 241-243, 261 partial detectability, estimation based on a subset of sample units, 250-253 aerial-ground survey comparisons, 250-252 incomplete ground counts, 252-253 partial detectability estimation,
with all sample units, 253-257 bounded counts, 256-257 detection probability estimation, 254-256 marked subpopulations, 255-256 multiple dependent observers, 254-255 multiple independent observers, 254 sampling proportional to size, 253-254 sighting probability models, 256 simple random sampling, 253 population censuses, 245 population indices, 257-261 abundance relationship, 257-259 constant slope linear relationship, 258-259 counts as indices, 257 design, 260 interpretation, 260-261 noncount indices, 260 nonhomogeneous slope linear relationship, 259 nonlinear relationships, 259 definition, 3 distance-based estimation issues, 286-287 cue counting, 286 g(0) less than 1,286 trapping webs, 286-287 line transect sampling, 265-278 assumptions, 266-267 density estimation with clusters, 275-276 distance function estimation, 270-273 fourier series approximations, 271, 273
793
interval estimation, 277-278 key function approach, 271-272 maximum likelihood estimation, 273-274 modeling approach, 265-266 model selection, 276-277 random sampling, 267 sample size determination, 282-283 sampling scheme, 265-266 statistical models, 267-270 variance of D estimation, 274-275 overview, 263, 287 point sampling, 278-281 assumptions, 280-281 data structure, 278-279 estimators, 279-280 models, 279-280 sample size determination, 283-284 sampling scheme, 278-279 point-to-object methods, 263-265 data structure, 264 estimators, 264-265 models, 264-265 sampling scheme, 264 study design, 281-286 experimental design, 285-286 field procedures, 281-282 sample size determination, 282-284 stratified sampling, 284-285 study population replication, 285-286 model identification correlated estimates, 178 time function, 178-180 multiple-dimensional models, 180
794 Abundance (continued) one-dimensional models, 179 two-dimensional models, 179-180 population management, see Population management population reconstruction, 342-343 species detection, 557 Accuracy, estimator accuracy in parameter estimation, 45-46 Adaptive optimization generalizations, 656-658 learning, 653-654 passive optimization, 658-660 Adaptive resource management harvested population impact determination, 230-231 relationship to adaptive optimization, 653 waterfowl sport harvest case study, 671-672, 679-680 Adaptive sampling, population parameters, 71-74 Additive mortality hypothesis, description, 227-230 Aerial surveys, ground surveys compared, 250-252 Age determination, in band-recovery studies, 408 Age models age frequency analysis, demographic parameter estimation, 337343 age stability unknown, 341 known stable age distribution, 340-341 life tables, 337-339 model likelihoods, 339-340 population reconstruction, 342-343 sampling assumptions, 341-342 survival estimation, 339-342 age-structured projection models continuous-time models, 155-157 discrete-time models, 144-155 demographic relationships, 152-154 harvest, 154-155 rate of growth, 147-151 sensitivity analysis, 151-152 stable age distribution, 147-151 life tables, 143-144 overview, 143 population characterization by age, 157-158 capture-recapture methods age-0 cohort models, 444M47 alternative modeling, 446 estimation, 445-446
Index estimator robustness, 445447 model assumptions, 445447 model selection, 445-447 model structure, 445 sampling design, 444M45 age-specific breeding models, 447-454 alternative modeling, 451-454 estimation, 451 model assumptions, 450-451 model structure, 448-450 sampling design, 447-448 capture history effects, 428-430 computer software, 740 data structure, 439-440 multiple ages, 439-454, 543-550 parameterization for transient individuals, 428-430 Pollock's model, 440-444 alternative modeling, 443-444 estimation, 442--443 estimator robustness, 444 model assumptions, 442, 444 model selection, 444 model structure, 440-442, 543 sampling design, 440 probabilistic modeling, 419-426 time-specific covariates, 373-374, 426 trap response, 428 survival rate estimation multiple-age models, 383-391 covariate effects, 389-390 data structure, 383 group effects, 389-390 model selection, 390--391 multiple bandings per year, 390 probabilistic modeling, 383-385 reduced-parameter models, 385-387 robust models, 543-550 temporary banding effect, 387 unrecognizable subadult cohorts, 387-389 Akaike's information criterion, model selection, 55-57, 432-434 ARMA process description, 219 Autoregressive processes, stationary time series, 217-218
B BAND2 program, 411-413, 742 Band-recovery combined with capture-recapture methods, 476-480 alternative modeling, 480
computer software, 740-741 data structure, 476--478 estimation, 480 estimator robustness, 480 model assumptions, 480 model selection, 480 model structure, 478-480 multiple-age models, 383-391 covariate effects, 389-390 data structure, 383 group effects, 389-390 model selection, 390-391 multiple bandings per year, 390 probabilistic modeling, 383-385 reduced-parameter models, 385-387 temporary banding effect, 387 unrecognizable subadult cohorts, 387-389 nonharvested species analysis, 398-402 cohort band recovery models, 400-401 data structure, 398 juvenile only banding, 400-401 probabilistic models, 398-400 unknown number of banded birds, 401-402 overview, 365-366, 414-415 poststratification of recoveries, 402-406 areas coincide, 405-406 areas differ, 402-405 reward studies for reporting rates, 391-398 data structure, 391-393 direct recoveries, 393-398 indirect recoveries, 393-396 spatial variation modeling, 396-398 single-age models, 366-383 covariates, 373-374 data structure, 367-368 estimator robustness, 377-378 goodness of fit, 376 individual covariates, 374, 430-431, 552 model selection, 376-377 multiple bandings per year, 375-376 multiple groups, 373 probabilistic modeling, 368-371 reduced-parameter models, 371-372 sampling correlation interpretation, 378-380 sampling scheme, 367-368 temporary banding effect, 372-373
Index time-specific covariates, 373-374, 426 survival rate estimation, 365-415 banding study design, 406-414 age determination, 408 banding station, 409 capture methods, 407-408 difference detection studies, 413-414 level of precision, 412 marking methods, 407-408 parameter of interest, 411-412 recovery rate enhancement, 409-410 recovery rate expectations, 413 sample size determination, 410-414 sex determination, 408 study duration, 409, 411 survival rate expectations, 412-413 targeted precision levels, 410-411 time of year effects, 408-409 two-age analyses, 413 Barrier methods, nonlinear programming, 755 Bayesian extension of likelihood theory, description, 57-58 Bayes' theorem, conditional probability, 685-686 Behavioral response model, capture-recapture-based abundance estimation, 299-300, 302-304 Bernoulli counting processes Bernoulli distribution, 35, 47, 723 number of successes, 189-190 waiting times, 190-192 Beta distribution, description, 730 Bias, estimator bias in parameter estimation, 43-44 Binomial distribution description, 35, 723 discrete survival analysis, 343-345 Birth processes conservation and extinction models, 231-232 continuous Markov processes, 203-204 Bivariate decision problem, unconstrained optimization, 589-590, 746 Bivariate programming, classical programming, 593-594, 750 Bootstrap estimation, variation estimation, 737-738 Branching processes, description, 213-215
Brownian motion, 210-213 description, 21 0-212 extensions, 212-213 absorbed motion, 212 geometric motion, 212-213 integrated motion, 212-213 reflected motion, 212 BROWNIE program description, 740-742 multiple-age models, 386, 389 BUGS program, 361
C Canonical population estimator capture-recapture-based abundance estimation, Jolly-Seber approach, 501-504 count-based abundance estimation, 243-245 detectability, 244 spatial sampling, 244-245 CAPQUOTA program, description, 741 Capture history closed models, 289-290, 296-297 multiple-age models, 439-441 multistate models, 455-456 recruitment and abundance estimation, Jolly-Seber approach, 497, 501-502 robust design, 524-529 single-age models, 419, 428-430 CAPTURE program capture-recapture model analysis, 298, 303-308, 311, 313 community parameters, 558-560 robust design, 530, 534-535 description, 739-740, 743-744 Capture-recapture methods, see also Band-recovery methods auxiliary data use, 476-489 band recoveries, 476-480 alternative modeling, 480 computer software, 740-741 data structure, 476-478 estimation, 480 estimator robustness, 480 model assumptions, 480 model selection, 480 model structure, 478-480 radiotelemetry, combined with capture-recapture data, 485-489 alternative modeling, 488-489 data structure, 486-487 estimation, 488 estimator robustness, 489 model assumptions, 488-489 model selection, 489
795 model structure, 487-488 resightings between sampling occasions, 481-485 alternative modeling, 485 data structure, 481-482 estimation, 484--485 estimator robustness, 485 model assumptions, 484--485 model selection, 485 model structure, 482-484 closed populations change-in-ratio methods, 325-330 data structure, 326 estimators, 326-329 model assumption violation, 329-330 models, 326-329 sampling scheme, 326 study design, 330 density estimation, 314-319 geometric analysis, 319 gradient designs, 317-319 grid trapping, 314-317 movement distances, 314-315 nested grid approach, 315-317 trapping webs, 317-319 uniform sampling effort, 314-317 K-sample models, 296-314 behavioral response models, 299-300, 302-304 between-model tests, 307 closure, 305-306 confidence interval estimation, 304-305 constant capture probability-model M o, 298-299 data structure, 296 diagnostic statistics, 308-310 discriminant analysis, 307-308 estimator robustness, 310-311 goodness of fit, 307 individual heterogeneity-model Mh, 300-302 model assumption testing, 305-306 modeling approach, 296-298 model selection, 306-310 population size estimation, 298-304 sampling scheme, 296 study design, 311-313 tag loss, 306 temporal variation-model Mt, 299 Lincoln-Petersen two-sample estimator, 290-296 closure, 293-294 data structure, 290-291
796 Capture-recapture methods (continued) equal capture probability, 294 estimators, 291-293 model assumption violation, 293-295 models, 243, 291-293 sampling scheme, 290-291 study design, 295-296 tag loss, 294-295 overview, 289-290, 331 removal methods, 319-325 data structure, 320-321 Delury approach, 321 estimators, 321-322 Leslie-Davis approach, 321 maximum likelihood approach, 322 model assumption violations, 322-323 models, 321-322 Ricker approach, 321 sampling scheme, 320-321 study design, 323-324 three least-squares approaches, 321-322 combined closed and open models, 523-554 ad hoc approach, 529-535 alternative modeling, 532-533 closed only estimation models, 533-535 estimation, 531-532 estimator robustness, 533 model assumptions, 531,533 models, 529-531 model selection, 533 data structure, 524-529 recruitment components, 543-550 reverse-time approach, 545-550 special estimation problems, 538-552 alternative data sources, 552 capture frequency data as covariates, 552 catch-effect studies, 550-551 multiple ages, 543-550 standard-time approach, 544-545 temporary emigration, 538-543 unconditional closed-population modeling, 551-552 future research directions, 551-552 likelihood-based approach, 535-538 alternative modeling, 537-538 estimation, 537 estimator robustness, 538 model assumptions, 537-538
Index models, 535-537 model selection, 538 overview, 523-524, 553-554 study design, 552-553 computer software, 740 open populations multiple-age models, 439-454 age-0 cohort models, 444--447 alternative modeling, 446 estimation, 445-446 estimator robustness, 445-447 model assumptions, 445-447 model selection, 445-447 model structure, 445 sampling design, 444M45 age-specific breeding models, 447-454 alternative modeling, 451-454 estimation, 451 model assumptions, 450-451 model structure, 448-450 sampling design, 447-448 data structure, 439-440 Pollock's model, 440-444 alternative modeling, 443-444 estimation, 442-443 estimator robustness, 444 model assumptions, 442, 444 model selection, 444 model structure, 440-442, 543 sampling design, 440 multistate models, 454--468 Markovian models, 454-464 alternative modeling, 459-460 data structure, 454-456 estimation, 459 estimator robustness, 460-464 model assumptions, 458-464 model selection, 460-464 model structure, 456-458 memory models, 464-468 alternative modeling, 466-467 data structure, 464-465 estimation, 466 estimator robustness, 467-468 model assumptions, 465-468 model selection, 467-468 model structure, 465 recruitment and abundance estimation, 495-554 approach relationships, 518-520 combined closed and open robust models, 543-550 reverse-time approach, 545-550 standard-time approach, 544-545 data structure, 496M97 Jolly-Seber approach, 497-508 alternative modeling, 499-508
canonical estimator use, 501 capture history dependence, 501-502 estimation, 498-499 estimator robustness, 504-507 group-specific covariates, 501 individual covariates, 504 model assumptions, 498, 504-507 model selection, 504-507, 518-520 model structure, 497-498 multiple-age models, 502-503 multistate models, 503-504 partially open models, 499-500 reduced-parameter models, 500-501 time-specific covariates, 501 overview, 495-496, 518-520, 522 Pradel's temporal symmetry model, 512-518 alternative modeling, 515 estimation, 514-515 estimator robustness, 515-517 model assumptions, 514-517 model selection, 515-520 model structure, 512-514 study design, 520-522 estimator precision, 522 instantaneous sampling, 521 model assumptions, 520-522 parameters, 520 rate parameter homogeneity, 520-521 tag retention, 521 temporary emigration, 521522 superpopulation approach, 508-512 alternative modeling, 510-511 estimation, 510 estimator robustness, 511 model assumptions, 509-511 model selection, 511,518-520 model structure, 508-509 reverse-time models, 468-476 multistate models, 472-476 alternative modeling, 475 data structure, 473 estimation, 474--475 estimator robustness, 475 model assumptions, 475 model selection, 475 model structure, 473-474 robust models, 545-550 single-state models, 468-472 alternative modeling, 472 data structure, 468-469
Index estimation, 470-472 estimator robustness, 472 model assumptions, 470, 472 model selection, 472 model structure, 469-470 single-age models, 418-439 capture history effects, 428-430 Cormack-Jolly-Seber model, 419-426 estimation, 423-425 estimator robustness, 434-436 model assumptions, 422-423 reduced-parameter models, 425-426 structure, 419-422 data structure, 419 estimator robustness, 434-436 capture history independence, 436 homogeneous capture probabilities, 434-435 homogeneous survival probabilities, 435 instantaneous sampling, 435 permanent emigration, 435-436 tag loss absence, 435 individual covariates, 374, 430-431,552 model selection, 431-434 multiple groups, 427 parameterization for transient individuals, 428-430 probabilistic modeling, 419-426 time-specific covariates, 373-374, 426 trap response, 428 study design, 489-492 estimator precision, 492 model assumptions, 490-492 instantaneous sampling, 491 permanent emigration, 491 rate homogeneity, 490-491 tag retention, 491 model parameters, 489-490 sampling designs, 489-490 Case studies, s e e Waterfowl sport harvest case study Catch-effort studies band-recovery methods multiple-age models covariate effects, 389-390 group effects, 389-390 temporary banding effect, 387 single-age models, temporary banding effect, 372-373 time of year effects, 408-409 removal models, 319-325 robust capture-recapture methods, 550-551
Causation in animal ecology necessary causation, 12, 81 study approaches, 12-13 sufficient causation, 12, 81 experimental control issues, 81 survey sampling misinterpretations, 75-76 Cell means model blocking, 90 experimental design, 85-86 Census, s e e Counts CENTROID program, description, 743 Change-in-ratio methods, 325-330 data structure, 326 estimators, 326-329 model assumption violation, 329-330 models, 326-329 sampling scheme, 326 study design, 330 Chi-square distribution, description, 728 Classical programming, optimal decision analysis, 593-597, 749-751 bivariate programming, 593-594, 750 multivariate programming, 594-596, 750-751 sensitivity analysis, 596 Closed populations capture-recapture-based abundance estimation computer software, 739 density estimation, 314-319 distance sampling, 317-319 geometric analysis, 319 gradient designs, 317-319 grid trapping, 314-317 movement distances, 314-315 nested grid approach, 315-317 trapping webs, 317-319 uniform sampling effort, 314-317 K-sample models, 296-314 behavioral response--models, 299-300, 302-304 between-model tests, 307 closure, 305-306 confidence interval estimation, 304-305 constant capture probability-model M o, 298-299 data structure, 296 diagnostic statistics, 308-310 discriminant analysis, 307-308 estimator robustness, 310-311 goodness of fit, 307
797 individual heterogeneity-model Mh, 300-302 model assumption testing, 305-306 modeling approach, 296-298 model selection, 306-310 population size estimation, 298-304 sampling scheme, 296 study design, 311-313 tag loss, 306 temporal variation-model M t, 299 Lincoln-Petersen two-sample estimator, 290-296 closure, 293-294 data structure, 290-291 equal capture probability, 294 estimators, 291-293 model assumption violation, 293-295 models, 243, 291-293 sampling scheme, 290-291 study design, 295-296 tag loss, 294-295 overview, 289-290, 331 removal methods, 319-325 data structure, 320-321 Delury approach, 321 estimators, 321-322 Leslie-Davis approach, 321 maximum likelihood approach, 322 model assumption violations, 322-323 models, 321-322 Ricker approach, 321 sampling scheme, 320-321 study design, 323-324 three least-squares approaches, 321-322 computer software, 739-740 capture-recapture models, 740 observation-based methods, 739-740 CLOSTEST program, capture-recapture model analysis, 306 Cluster sampling line transect sampling, 275-276 full likelihood estimation, 276 size dependent on distance, 275-276 size independent of distance, 275 survey sampling, 67-69, 73-74 CMR program, description, 741-742 Cohort models age cohorts band-recovery methods, 383-391
798 Cohort models (continued) covariates, 389-390 data structure, 383 juvenile only banding, 400-401 model selection, 390-391 multiple bandings per year, 390 probabilistic modeling, 383-385 reduced-parameter models, 382-387 temporary banding effect, 387 capture-recapture methods, 444--447 alternative modeling, 446 estimation, 445-446 estimator robustness, 445-447 model assumptions, 445-447 model selection, 445--447 model structure, 445 sampling design, 444M45 continuous-time models, 155-157 demographic parameter estimation, 337-343 age stability unknown, 341 known stable age distribution, 340-341 life tables, 337-339 model likelihoods, 339-340 population reconstruction, 342-343 sampling assumptions, 341-342 survival estimation, 339-342 discrete-time models, 144-155 demographic relationships, 152-154 harvest, 154-155 rate of growth, 147-151 sensitivity analysis, 151-152 stable age distribution, 147-151 cohort transitions, 143 independent cohort populations, 141-143 overview, 141 size cohorts continuous-time models, 159 discrete-time models, 158-159 Colonization temporal variation at a single location local colonizing species, 566 recolonization probabilities, 566-567 time and space variation, 571 COMDYN program community parameter analysis, 567 description, 743-744 Community definition, 9 parameter estimation, 555-573 assumptions, 572
Index computer software, 744 geographic variation at a single time, 569-571 species cooccurrence, 569-570 time and space variation, 571-572 unique species, 570-571 overview, 555-556, 561-562, 573 Pollock's robust design, 562 population-community relationship, 556-557 abundance and species detection, 557 count statistics, 556-557 detection probabilities, 556-557 inferences, 556-557 species detection, 557 state variables, 556 vital rates, 556 species richness estimation, 557-561 empirical species abundance distributions, 560-561 field sampling recommendations, 561 multiple sampling occasions, 559-560 quadrat sampling, 557-559 temporal variation at a single location, 563-568 annual extinction probabilities, 566-567 local colonizing species, 566 local extinction probability, 564-565 local species turnover, 565-566 rate of change, 564 recolonization probabilities, 566-567 time and space variation, 571-572 time and space variation, 571-572 colonization rate, 571 relative change, 571-572 variance estimation, 563 Compensatory mortality hypothesis, description, 227-230 Competition models general model for interacting species, 170-171 Lotka-Volterra equations multiple competing species, 168-169 two competing species, 165-167 competitive exclusion, 166 stable coexistence, 166, 703-704 unstable population equilibrium, 166-167 resource competition, 169-170
Computer software, see also specific programs
availability, 744 band recovery models, 740-741 closed populations, 739-740 capture-recapture models, 740 observation-based methods, 739-740 community parameter estimation, 744 nest survival, 744 open populations, capture-recapture models, 741-744 overview, 739 simulations, 114-115 telemetry, 744 Conceptual models, description, 22 Conditional distributions Bayes' theorem, 685--686 description, 40-42 Confidence intervals parameter estimation, 48-50 Conjugate direction method, description, 747-748 Conjugate gradient methods, description, 749 Conservation models, see also Population management metapopulations, 233-235 overview, 223, 237-238 patch dynamics, 235-237 individual-based models, 236-237 source-sink models, 235-236 spatially explicit models, 236-237 population size effects, 231-233 birth-death processes, 232 birth processes, 231-232 minimum viable populations, 233 persistence time, 232-233 Constant capture probability model, capture-recapture-based abundance estimation, 298-299 Continuous distributions beta distribution, 730 chi-square distribution, 728 exponential distributions, 729-730 F distribution, 731-732 gamma distributions, 729-730 normal distribution, 36-38, 725-728 Student's t distribution, 730-731 variation estimation, 733-734 Continuous-time models age cohorts, 155-157 continuous Markov processes, 202-205 birth processes, 203-204 death processes, 203-204
Index Kolmogorov differential equations, 204-205 differential equations, s e e Differential equations discrete-time models compared, 115-117 optimal decision analysis, dynamic programming, 628-630 size cohorts, 159 CONTRAST program band recovery estimates, 380 description, 743 Cormack-Jolly-Seber model overview, 419-422, 492--493 single-age survival rate estimation, 419-426 estimation, 423--425 estimator robustness, 434-436 capture history independence, 436 homogeneous capture probabilities, 434-435 homogeneous survival probabilities, 435 instantaneous sampling, 435 permanent emigration, 435-436 probabilistic models, 422-423 tag loss absence, 435 model assumptions, 422-423, 434 reduced-parameter models, 425-426 structure, 419-422 Correlation, in statistical distributions, 42 Counts all sample units-based detectability estimation, 253-257 bounded counts, 256-257 detection probability estimation, 254-256 marked subpopulations, 255-256 multiple dependent observers, 254-255 multiple independent observers, 254 sampling proportional to size, 253-254 sighting probability models, 256 simple random sampling, 253 canonical population estimator, 243-245 detectability, 244 spatial sampling, 244-245 complete detectability on equal area samples, 245-247 abundance estimators, 245 assumptions, 245-246 sample size effects, 247 shape of unit effects, 246-247
size of unit effects, 246-247 stratification effects, 247 survey designs, 246-247 complete detectability on unequal area samples, 247-250 ratio estimators, 248-249 stratified design-based estimators, 249-250 weighted estimators, 249 overview, 241-243, 261 partial detectability, 250-257 aerial-ground survey comparisons, 250-252 incomplete ground counts, 252-253 subset-based estimation, 250-253 population censuses, 245 population-community relationship, 556-557 population indices, 257-261 abundance relationship, 257-259 constant slope linear relationship, 258-259 design, 260 interpretation, 260-261 noncount indices, 260 nonhomogeneous slope linear relationship, 259 nonlinear relationships, 259 survey counts, 257 Covariates experimental design role, 91-92 parameter estimation, 92 statistical models, 91-92 testing, 92 individual covariates, capture-recapture methods capture frequency data use, 552 computer software, 740-744 Jolly-Seber approach, 504 single-age models, 430-431 recruitment and abundance estimation, Jolly-Seber approach, time-specific covariates, 501 in statistical distributions, 42 survival rate estimation using bandrecovery and capture-recapture methods multiple-age models, 389-390 single-age models individual covariates, 374, 430-431 time-specific covariates, 373-374, 426 Crossover designs, description, 84, 94-96, 100 Cox proportional hazards model, s e e Proportional hazards model, failure time analysis
799
Cue counting, distance-based abundance estimation issues, 286
D Data analysis, data-prediction comparison, 14-16 /) estimator, variance estimation, 274-275 Decision analysis 643-666 adaptive optimization and learning, 653-654 general framework, 645-650 generalization of adaptive optimization, 656-600 overview, 643-644, 660-661 uncertainty and optimal control, 650-653 value of information, 654-655 Decision-theoretic models, s e e Optimal decision analysis Deductive logic in science, 16-17 9 Delta method, variation estimation, 736-737 Delury approach, removal methods in closed populations, 321 Demographics age-structured matrix model relationships, 152-154 parameter estimation, 333-363 age frequency analysis, 337-343 age stability unknown, 341 known stable age distribution, 340-341 life tables, 337-339 model likelihoods, 339-340 population reconstruction, 342-343 sampling assumptions, 341-342 survival estimation, 339-342 detectability, 334-337 movement probabilities, 335-336 reproductive rates, 336-337 survival rates, 335 discrete survival analysis, 343-351 binomial survival model, 343-345 movement studies, 350-351 nest success estimation models, 345-350 radiotelemetry survival, 350-351 failure time analysis, 351-361 explanatory variable incorporation, 357-359 Kaplan-Meier method, 354357
800 Demographics (continued) model assumptions, 359 nonparametric survival estimation, 354-357 parametric survival estimation, 353-354 proportional hazards model, 357-359 radiotelemetry study design, 359-361 statistical models, 352-353 survival distribution comparison tests, 355-357 known-fate data, 361-362 nest success estimation models, 345-350 binomial modeling alternatives, 346-347 historical perspectives, 345 Mayfield method, 345-346 model assumptions, 346-347 nest encounter parameters, 347-348 random effects model, 348 study design, 349-350 overview, 333-334, 362-363 random effects, 361-362 stochasticity, stationary time series, 219-220 Density, see also Abundance capture-recapture-based estimation, 314-319 computer software, 740-744 geometric analysis, 319 gradient designs, 317-319 grid trapping, 314-317 movement distances, 314-315 nested grid approach, 315-317 trapping webs, 317-319 uniform sampling effort, 314-317 definition, 3 distance-based estimation line transect sampling with clusters, 275-276 point-to-object methods, 263-265 Density dependence density-dependent growth, logistic model, 139-141 harvest incorporation, 140-141 time lag incorporation, 141 description, 5-6 Density independence density-independent growth, exponential models, 136-139 habitat effects, 138 harvest effects, 138-139 perturbations, 139 description, 5-6 Detectability, in survey samples, 61
Index Deterministic models dynamic programming, 627-635 continuous-time applications, 628-630 linear-quadratic control, 629-630, 763-764 stochastic models compared, 116117 Difference equations first-order linear homogeneous equations, 710-716 population dynamics estimation multiple species, 715-716 one species, 710 two species, 710-715 nonlinear homogeneous equations, stability analysis, 716-719 multiple species, 719 one species, 716-718 two species, 718-719 overview, 709-710 Differential equations first-order linear homogeneous equations, 694-700 multiple species, 699-700 one species, 694-695 two species, 695-699 matrix approach, 697-699 second-order equations, 695-697 graphical methods, 705-708 stability assessment with null clines, 706-708 Kolmogorov differential equations, 204-205 matrix algebra, 692 nonlinear homogeneous equations, stability analysis, 700-705 multiple species, 704-705 one species, 700-701 two species, 701-704 overview, 693-694 translation into dynamic models, 125-126 Discrete distributions binomial distribution, 35, 723 difference equations, see Difference equations geometric distribution, 723 hypergeometric distributions, 724-725 Markov processes decision processes, 207 Markov chains description, 197-198 stationary distributions, 201-202 state classification, 198-201 absorbing states, 200-201 communicating states, 198-199 ergodic states, 201
recurrent states, 199-200 transient states, 199-200 multinomial distribution, 35-36, 721 negative binomial distributions, 723 overview, 721 Poisson distribution, 192-193, 721-722 survival analysis, 343-351 binomial survival model, 343-345 movement studies, 350-351 nest success estimation models, 345-350 binomial modeling alternatives, 346-347 historical perspectives, 345 Mayfield method, 345-346 model assumptions, 346-347 nest encounter parameters, 347-348 random effects model, 348 study design, 349-350 radiotelemetry survival, 350-351 variance estimation, 733-734 Discrete-time models age cohorts, 144-155 demographic relationships, 152-154 harvest, 154-155 rate of growth, 147-151 sensitivity analysis, 151-152 stable age distribution, 147-151 continuous-time models compared, 115-117 size cohorts, 158-159 Distance-based estimation, abundance estimation issues, 286-287 cue counting, 286 g(0) less than 1,286 trapping webs, 286-287 line transect sampling, 265-278 assumptions, 266-267 density estimation with clusters, 275-276 distance function estimation, 270-273 fourier-based series approximations, 271,273 interval estimation, 277-278 key function approach, 271-272 likelihood estimation, 276 maximum likelihood estimation, 273-274 modeling approach, 265-266 model selection, 276-277 random sampling, 267 sample size determination, 282-283 sampling scheme, 265-266
Index statistical models, 267-270 variance of/~ estimation, 274-275 overview, 263, 287 point sampling, 278-281 assumptions, 280-281 data structure, 278-279 estimators, 279-280 models, 279-280 sample size determination, 283-284 sampling scheme, 278-279 point-to-object methods, 263-265 data structure, 264 estimators, 264-265 models, 264-265 sampling scheme, 264 study design, 281-286 experimental design, 285-286 field procedures, 281-282 sample size determination, 282-284 stratified sampling, 284-285 study population replication, 285-286 DISTANCE program description, 740 line transect sampling, 274, 277-278, 280, 287 Distributions, see Population dynamics; Statistical distributions; spe-
temporary emigration, 521-522, 538-543 Markovian emigration model, 541-543 random migration model, 539-541 Empirical models, 24-27 description, 22 multiple-hypothesis approach, 2427 Popper's natural selection of hypotheses, 26 single-hypothesis approach, 24 Environmental variation optimal decision analysis, 644-646 waterfowl sport harvest case study, 674-675 Ergodic states classification, 201 model assessment, 130-131 Error experimental error, 82-83 in hypothesis testing type I errors, 50-51 type II errors, 50-51 ESTIMATE program band recovery estimates, 399 description, 740-741, 743 Estimation, see Computer software; spe-
cific distributions
Euler-Lagrange equation, description, 618-619, 624, 761 Euler-Lotka equation, description, 148-149, 157 Euler's equation, variation analysis, 609-610 Expected value of perfect information, 654-655 identical policies for models, 655 identical values for models, 655 Experimental studies completely randomized designs, 83-89 hypothesis testing, 88-89 model associations, 86-87 multifactor randomized designs, 85-86 cell means model, 85-86, 90 fully parameterized model, 86-87, 90-91 restricted parameterization model, 86-87 single-factor randomized designs, 83-85 testable hypothesis, 87-88 average main effects, 88 interactions, 88 constrained experimental design, 102-106
DOBSERV program, description, 739 Double sampling, population parameters, 69-71 ratio estimator, 71 regression estimator, 70-71 Dynamic programming, optimal decision analysis advantages, 639-641, 762 continuous time applications, 628-630 deterministic programming, 627-635, 763-764 discrete time applications, 630-635, 764 linear-quadratic control, 629-630, 763-764 stochastic dynamic programming, 635-638, 764 uncertainty effects, 650-651 variational mathematics, 764-765
E
Eigenvectors, matrix algebra, 690-692 Emigration, in capture-recapture models permanent emigration, 435-436, 476-480, 491
cific methods; specific parameters
801 observational studies, 105-106 covariance, 91-92 statistical models, 91-92 covariates parameter estimation, 92 testing, 92 harvested population impact determination, 230 hierarchical designs, 92-97 crossover designs, 84, 94-96, 100 nested design associations, 99-100 population model design strategies, 122-124 repeated-measures designs, 96-97 split-plot designs, 93-94 estimation, 93-94 statistical models, 93 testing, 93-94 nested designs, 97-100 estimation, 98-99 fixed effects, 98-99 hierarchical design associations, 99-100 random effects, 99 statistical models, 98 testing, 98-99 overview, 79-80, 106-107 principles, 80-83 control, 80-81 experimental error, 82-83 randomization, 82 replication, 81-82 random effects, 97-100 randomized block designs, 89-91 estimation, 91 restricted randomization, 89-90 statistical models, 90-91 testing, 91 sample size determination, 101-102 statistical power, 100-101 Exponential models exponential distributions, 729-730 population dynamics modeling, 136-139 habitat effects, 138 harvest effects, 138-139 perturbations, 139 Extinction models community dynamics, temporal variation at a single location annual extinction probabilities, 566-567 local extinction probability, 564-565 metapopulations, 233-235 overview, 223, 237-238 patch dynamics, 235-237 individual-based models, 236-237
802
Extinction models ( c o n t i n u e d ) source-sink models, 235-236 spatially explicit models, 236-237 population size effects, 231-233 birth-death processes, 232 birth processes, 231-232 minimum viable populations, 233 persistence time, 232-233
F Failure time analysis, 351-361 explanatory variable incorporation, 357-359 Kaplan-Meier method, 354-357 model assumptions, 359 nonparametric survival estimation, 354-357 parametric survival estimation, 353-354 proportional hazards model, 357-359 radiotelemetry study design, 359-361 statistical models, 352-353 survival distribution comparison tests, 355-357 F distribution, description, 731-732 Feasible direction approach, description, 754 Finite rate of population growth, definition, 136 Fitness, individuals, 7-9 FORTRAN program, description, 740 Fourier series, distance-based abundance estimation, line transect sampling, 271, 273 Fully parameterized model blocking, 90-91 experimental design, 86 restricted parameterization model compared, 86-87
G Gamma distributions, description, 729-730 Geographic-structured models, description, 159-161 Geographic variation, community dynamics at a single time, 569-571 species cooccurrence, 569-570 time and space variation, 571-572 unique species, 570-571 Geometric distribution, description, 723 GLIM program, description, 740 Goodness-of-fit tests community dynamics models, 567, 572
Index estimation with mark-recapture Jolly-Seber model, 504 superpopulation approach, 511 temporal symmetry model, 516 in hypothesis testing, 52-54 K-sample capture-recapture models, 307 line transect models, 277 removal methods, 323 robust designs, 537 survival rate estimation using band-recovery methods, single-age models, 376 using capture-recapture band recoveries as auxiliary data, 481 multiple-age models, 444 multistate models, 460 radiotelemetry as auxiliary data, 489 reverse-time models, 472 resightings between caputre occasions, 485 single-age models, 431-433 Gradient designs capture-recapture-based density estimation, 317-319 gradient projection, 753 reduced gradient, 753-754 Graphical analysis, differential equations, 705-708 stability assessment with null clines, 706-708 Grid trapping, capture-recapturebased density estimation, 314-317 Growth, see Population growth rate
H Habitat, exponential modeling, 138 Harvested populations, see also Bandrecovery methods conservation and management dynamic models additive mortality, 227-230 case studies, see Waterfowl sport harvest case study compensatory mortality, 227-230 partial compensation model with variable thresholds, 229-230 impact determination, 230-231 adaptive resource management, 230-231 experimental studies, 230 observational studies, 230 overview, 223, 237-238 sustained yield
concept, 223-224 maximum sustained yield, 224-227 traditional population dynamics models age-structured models, 154-155 density-dependent growth, 140-141 density-independent growth, 138-139 exponential model, 138-139 logistic model, 140-141 waterfowl sport harvest case study, 663-684 adaptive harvest management, 671-672, 679-680 harvest objectives, 676-677 ongoing issues, 681-684 goal setting, 681-683 management, 683-684 scale, 683-684 overview, 663-667, 684 population dynamics modeling environmental variation, 674-675 model development, 665-667 partial management control, 675-676 recruitment, 673-674 structural uncertainty, 672-674 survival, 673 regulatory process alternatives, 677-679 biological monitoring, 668-670 harvest regulations, 664-665, 667-668 optimal regulation identification, 679-680 outcome prediction, 670-671, 677-679 Hazard rates, failure time analysis proportional hazards model, 357359 statistical models, 352-353 Heuristic approaches to decision analysis, 638-639 computer simulation, 638 genetic algorithms, 639 simulated annealing, 639 Hierarchical designs crossover designs, 84, 94-96, 100 nested design associations, 99-100 population model design strategies, 122-124 split-plot designs, 93-94 estimation, 93-94 statistical models, 93 testing, 93-94
Index Hypergeometric distribution, description, 724-725 Hypothesis testing complementary hypothesis, 18-19 completely randomized designs, 88-89 goodness-of-fit tests, 52-54 likelihood ratio tests, model comparisons, 53-55 models relationship empirical uses multiple-hypothesis approach, 24-27 Popper's natural selection of hypotheses, 26 single-hypothesis approach, 24 scientific process, 14, 30-31 null hypothesis, 19 overview, 50 scientific method confirmation, 16-17 hypothesis formation, 14, 30-31 statistical inference, 18 statistical power, 51-52 type I errors, 50-51 type II errors, 50-51
I
Impact studies, constrained experimental design, 103-105 Individual heterogeneity model, capture-recapture-based abundance estimation, 300-304 Inductive logic, scientific process, 17-18 Information theory approaches, description, 55-57 Interval estimation, 48-50 closed population capture-recaputre models, 304-306 distance-based abundance estimation, line transect sampling, 277-278
J Jackknife estimators, 737 Joint distributions, description, 38, 40 JOLLYAGE program, description, 741-743 JOLLY program, description, 741-743 Jolly-Seber approach, recruitment and abundance estimation, 497-508 alternative modeling, 499-508 canonical estimator use, 501 capture history dependence, 501-502 estimation, 498-499 estimator robustness, 504-507
group-specific covariates, 501 individual covariates, 504 model assumptions, 498, 504-507 model selection, 504-507, 518-520 model structure, 497-498 multiple-age models, 502-503 multistate models, 503--504 partially open models, 499-500 reduced-parameter models, 500-501 time-specific covariates, 501
K
Kaplan-Meier method, failure time analysis, 354-357 Key function estimation, distancebased abundance estimation, line transect sampling, 271-272 Known-fate model, random effects, 361-362 Kolmogorov differential equations, description, 204-205 K-sample models, capture-recapturebased abundance estimation, 296-314 behavioral response-models Mb, 299-300, 302-304 between-model tests, 307 closure, 305-306 computer software, 740-744 confidence interval estimation, 304-305 constant capture probability-model M o, 298-299 data structure, 296 diagnostic statistics, 308--310 discriminant analysis, 307-308 estimator robustness, 310-311 goodness of fit, 307 individual heterogeneity-model M h, 300-302 model assumption testing, 305-306 modeling approach, 296-298 model selection, 306-310 population size estimation, 298-304 sampling scheme, 296 study design, 311-313 tag loss, 306 temporal variation-models M t, 299 Kuhn-Tucker conditions, optimal decision analysis using linear programming, 602-603, 606, 757
L Lagrangian multipliers decision analysis, 614-615
803 dual methods, 755-756 model identification with time series data, 180-181 nonlinear programming, 756-757 Learning, adaptive optimization, 653--654 Least-squares approach model identification with time series data, 174--176 removal methods in closed populations, 321-322 Leslie-Davis approach, removal methods in closed populations, 321 Leslie matrix models, sensitivity analysis, 151-152 Life tables age frequency analysis, 337-339 individual life history characteristics, 7-9 structure, 143-144 Likelihood-based models Bayesian extension, 57-58 estimation from age-structure data, 339-342 maximum likelihood estimation, 46-49, 273-274 model comparisons, ratio tests, 53-55 Limitation, description, 6 Lincoln-Petersen two-sample estimator, capture-recapturebased abundance estimation, 290-296 closure, 293-294 data structure, 290-291 equal capture probability, 294 estimators, 291-293 model assumption violation, 293-295 models, 243, 291-293 sampling scheme, 290-291 study design, 295-296 tag loss, 294-295 Linear equations, first-order linear homogeneous equations difference equations multiple species, 715-716 one species, 710 two species, 710-715 differential equations, 694-700 multiple species, 699-700 one species, 694-695 two species, 695-699 Linear programming, optimal decision analysis, 601-606, 757-758 advantages, 639-641 dual linear programming, 603, 757-758 Kuhn-Tucker conditions, 602-603, 606, 757
804 Linear programming, optimal decision analysis (continued) linear-quadratic control, 629-630 nonlinear problem solutions, 603605 simplex solution algorithm, 605, 747 LINETRAN program, description, 739 Line transect sampling, abundance estimation, 265-278 assumptions, 266-267 density estimation with clusters, 275-276 distance function estimation, 270-273 fourier-based series approximations, 271,273 interval estimation, 277-278 key function approach, 271-272 maximum likelihood estimation, 273-274, 276 modeling approach, 265-266 model selection, 276-277 random sampling, 267 sampling scheme, 265-266 statistical models, 267-270 study design, 281-286 field procedures, 281-282 sample size determination, 282-283 stratified sampling, 284-285 variance of D estimation, 274-275 LINLOGN program, description, 740 Logistic models, population dynamics modeling harvest incorporation, 140-141 overview, 139-140 predator-prey models, 163-164 time lag incorporation, 141 Lotka-Volterra models continuous-time models, 161-164 logistic effects, 163-164 oscillation size and shape, 162-163 discrete-time models, 164 equations decision analysis, 622-623 multiple competing species, 168-169 two competing species, 165-167 competitive exclusion, 166 stable coexistence, 166, 703-704 unstable population equilibrium, 166-167
M M o model, capture-recapture-based abundance estimation, 298-299
Index Management, see Population management Marginal distributions, description, 40 Markov processes continuous processes, 202-205 birth processes, 203-204 death processes, 203-204 Kolmogorov differential equations, 204-205 decision processes, 207-210 discrete-time processes, 207 objective functionals, 207-208 stationary policies, 208-210 finite processes, 208 infinite processes, 209-210 semi-Markov processes, 210 discrete processes decision processes, 207 Markov chains description, 197-198 stationary distributions, 201-202 state classification, 198-201 absorbing states, 200-201 communicating states, 198-199 ergodic states, 201 recurrent states, 199-200 transient states, 199-200 multistate capture-recapture models, 454-464 alternative modeling, 459-460 data structure, 454-456 estimation, 459 estimator robustness, 460-464 model assumptions, 458-464 model selection, 460-464 model structure, 456-458 semi-Markov processes, 205-207 stationary limiting distributions, 206-207 stationary policies, 210 MARK program k estimation, 514, 517 band recovery estimates, 380, 386, 408-409 capture-recapture model analysis, 301,347 description, 740-743 likelihood-based approach, 537, 540 multiple-age models, 386, 389 multistate models, 459, 480 single-age models, 371,373-374, 379 state transition estimation, 424, 438, 443 Mark-recapture methods, see Capture-recapture methods Martingales, stochastic processes, 216 Mathematical models, description, 22 MATLAB program, description, 741 Matrix algebra methods
addition, 688 definitions, 687 determinants, 689 eigenvalues, 690-692 eigenvectors, 690-692 inverse of a matrix, 689-690 linear forms, 692, 713-715 matrix differentiation, 692 matrix multiplication, 688 orthogonal matrices, 690 orthonormal matrices, 690 overview, 687 positive-definite matrices, 692 quadratic forms, 692 scalar multiplication, 688 semidefinite matrices, 692 trace of a matrix, 690 Maximum likelihood estimators description, 49, 54 removal methods in closed populations, 322 Mayfield method, nest success estimation, 345-346 MAYFIELD program, description, 743-744 M b model, capture-recapture-based abundance estimation, 299-300 Mean absolute error method, model identification with time series data, 176-177 Mean estimation, jackknife estimators, 737 Mechanical models, description, 22 Memory models, multistate capture-recapture methods, 464--468 alternative modeling, 466-467 computer software, 740-744 data structure, 464-465 estimation, 466 estimator robustness, 467-468 model assumptions, 465-468 model selection, 467-468 model structure, 465 Metapopulations, extinction models, 233-235 M h model, capture-recapture-based abundance estimation, 300-302 Minimum viable populations, conservation and extinction models, 233 Models, see also specific models abundance capture-recapture-based estimation change-in-ratio methods, 326-329 K-sample models, 296-314
Index Lincoln-Petersen two-sample estimator, 243, 291-293 removal methods, 321-322 count-based estimation, sighting probability models, 256 distance-based estimation line transect sampling, 265-270, 276-277 point sampling, 279-280 point-to-object methods, 264-265 K-sample models, 296-314 behavioral response models, 299-300, 302-304 between-model tests, 307 closure, 305-306 confidence interval estimation, 304-305 constant capture probability-model Mo, 298-299 data structure, 296 diagnostic statistics, 308-310 discriminant analysis, 307-308 estimator robustness, 310-311 goodness of fit, 307 individual heterogeneity-model M h, 300-302 model assumption testing, 305-306 modeling approach, 296-298 model selection, 306-310 population size estimation, 298-304 sampling scheme, 296 study design, 311-313 tag loss, 306 temporal variation-model Mt, 299 model identification with time series data correlated estimates, 178 time function, 178--180 assessment equilibria identification, 128-129 model behavior stability, 129130 model ergodicity, 130-131 sensitivity analysis, 127-128 validation, 126-127 verification, 126-127 comparisons, likelihood ratio tests, 53-55 conservation models metapopulations, 233-235 overview, 223, 237-238 patch dynamics, 235-237 individual-based models, 236-237 source-sink models, 235-236
spatially explicit models, 236-237 population size effects, 231-233 birth-death processes, 232 birth processes, 231-232 minimum viable populations, 233 persistence time, 232-233 demographic parameter estimation age frequency analysis, 339-340 discrete survival analysis binomial survival model, 343-345 nest success estimation models, 345-350 failure time analysis model assumptions, 359 proportional hazards model, 357-359 statistical models, 352-353 nest success estimation models, 345-350 binomial modeling alternatives, 346-347 historical perspectives, 345 Mayfield method, 345-346 model assumptions, 346-347 nest encounter parameters, 347-348 random effects model, 348 study design, 349-350 development attributes, 114-117 continuous models, 115-117 deterministic models, 116-117 discrete models, 115-117 quantitative approaches, 114-115 stochastic models, 116-117 goals, 113-114 overview, 111-113 population model characteristics, 117-126 canonical processes, 120-122 components, 119-120 construction, 122-126 differential equation translation into dynamic models, 125-126 hierarchical, 122-124 mathematical formulation, 118-119 notation, 118-119 stochastic factor incorporation, 125 variable encoding relationships, 124 extinction models metapopulations, 233-235
805 overview, 223, 237-238 patch dynamics, 235-237 individual-based models, 236-237 source-sink models, 235-236 spatially explicit models, 236-237 population size effects, 231-233 birth-death processes, 232 birth processes, 231-232 minimum viable populations, 233 persistence time, 232-233 harvested population dynamics additive mortality, 227-230 case studies, s e e Waterfowl sport harvest case study compensatory mortality, 227-230 partial compensation model with variable thresholds, 229-230 impact determination, 230-231 adaptive resource management, 230-231 experimental studies, 230 observational studies, 230, 739-740 overview, 223, 237-238 sustained yield concept, 223-224 maximum sustained yield, 224-227 hypotheses relationship, 30-31, 53-55 identification, time series data, 173-185 Lagrangian multipliers, 180-181, 755-757 least squares method, 174-176 mean absolute error method, 176-177 multiple-dimensional models, 180 one-dimensional models, 179 optimal identification, 178 overview, 173-174 parameter estimate stability, 181-182 population size correlated estimates, 178 system property identification in non-modeled situations, 182-184 two-dimensional models, 179-180 overview, 21-22, 31 selection of estimation models, s e e Information theory approaches, description successful use, 22-23 systematic approach, 131-134 boundaries, 131-132
8{}6 Models (continued) features identification, 131-132 goal establishment, 131 mathematical model development, 132 population management, 133 sensitivity analysis, 132 stability analysis, 133 validation, 133 verification, 132-133 traditional population dynamics models, 135-172 age-structured models, 143-158 continuous-time models with age cohorts, 155-157 demographic relationships, 152-154 discrete-time models with age cohorts, 144-155 harvest, 154-155 life tables, 143-144 population characterization by age, 157-158 sensitivity analysis, 151-152 stable age distribution, 147-151 stable rate of growth, 147-151 survival estimation, 339-342 cohort models, 141-143 continuous-time models with age cohorts, 155-157, 159 discrete-time models, 144-155, 158-159 independent cohort populations, 141-143 transitions among cohorts, 143 competing populations models, 165-170 competitive exclusion, 166 Lotka-Volterra equations, 165-169 multiple competing species, 168-169 resource competition models, 169-170 stable coexistence, 166 unstable population equilibrium, 166-167 density-dependent growth, 139-141 harvest incorporation, 140-141 time lag incorporation, 141 density-independent growth, 136-139 habitat effects, 138 harvest effects, 138-139 perturbations, 139 exponential model, 136-139 habitat effects, 138 harvest effects, 138-139
Index perturbations, 139 general interacting species model, 170-171 geographic-structured models, 159-161 logistic model, 139-141 harvest incorporation, 140-141 time lag incorporation, 141 Lotka-Volterra models continuous-time models, 161-164 discrete-time models, 164 logistic effects, 163-164 oscillation size and shape, 162-163 overview, 135-136, 171-172 predator-prey models continuous-time models, 161-164 discrete-time models, 164 logistic effects, 163-164 oscillation size and shape, 162-163 size-structured models, 158-159 continuous-time models with size cohorts, 159 discrete-time models with size cohorts, 158-159 types, 22 uses, 23-28 empirical uses, 24-27 multiple-hypothesis approach, 24-27 Popper's natural selection of hypotheses, 26 single-hypothesis approach, 24 theoretical uses, 23-24 utility determinants, 28-30 mechanistic versus descriptive models, 29-30 more versus less integrated model parameters, 30, 86 simple versus complex models, 28-29 Mortality, models, harvested population dynamics additive mortality, 227-230 compensatory mortality, 227-230 partial compensation model with variable thresholds, 229-230 Movement studies band recoveries, 476-480 demographic parameter estimation detectability, 335-336 discrete survival analysis, 350-351 multistate models, 454-468 Markovian models, 454-464 alternative modeling, 459-460 data structure, 454-456
estimation, 459 estimator robustness, 460-464 model assumptions, 458-464 model selection, 460-464 model structure, 456-458 memory models, 464-468 alternative modeling, 466-467 data structure, 464-465 estimation, 466 estimator robustness, 467-468 model assumptions, 465-468 model selection, 467-468 model structure, 465 overview, 417-418, 492-493 radiotelemetry, 485-489 alternative modeling, 488-489 data structure, 486-487 estimation, 488 estimator robustness, 489 model assumptions, 488-489 model selection, 489 model structure, 487-488 resightings between sampling occasions, 481-485 alternative modeling, 485 data structure, 481-482 estimation, 484-485 estimator robustness, 485 model assumptions, 484-485 model selection, 485 model structure, 482-484 reverse-time models, 468-476 multistate models, 472-476 alternative modeling, 475 data structure, 473 estimation, 474-475 estimator robustness, 475 model assumptions, 475 model selection, 475 model structure, 473-474 robust models, 545-550 single-state models, 468-472 alternative modeling, 472 data structure, 468-469 estimation, 470-472 estimator robustness, 472 model assumptions, 470, 472 model selection, 472 model structure, 469-470 study design, 489-492 estimator precision, 492 model assumptions, 490-492 instantaneous sampling, 491 permanent emigration, 491 rate homogeneity, 490-491 tag retention, 491 model parameters, 489-490 sampling designs, 489-490
Index Moving-average processes, stationary time series, 218-219 MSSURVIV program description, 742-743 multistate models, 459-460, 466-467, 548 M t model, capture-recapture-based abundance estimation, 299 Multinomial distribution, description, 35-36, 721 Multiple-age and multistate models band-recovery methods, 383-391 covariate effects, 389-390 data structure, 383 group effects, 389-390 model selection, 390-391 multiple bandings per year, 390 probabilistic modeling, 383-385 reduced-parameter models, 385-387 temporary banding effect, 387 unrecognizable subadult cohorts, 387-389 capture-recapture methods, 439-454 age-0 cohort models, 444M47 alternative modeling, 446 estimation, 445-446 estimator robustness, 445-447 model assumptions, 445--447 model selection, 445-447 model structure, 445 sampling design, 444M45 age-specific breeding models, 447-454 alternative modeling, 451-454 estimation, 451 model assumptions, 450-451 model structure, 448-450 sampling design, 447-448 Markovian models, 454--464 alternative modeling, 459-460 data structure, 454-456 estimation, 459 estimator robustness, 460-464 model assumptions, 458-464 model selection, 460-464 model structure, 456-458 memory models, 464-468 alternative modeling, 466-467 data structure, 464-465 estimation, 466 estimator robustness, 467-468 model assumptions, 465-468 model selection, 467-468 model structure, 465 Pollock's model, 440--444 alternative modeling, 443-444 estimation, 442-443 estimator robustness, 444
model assumptions, 442, 444 model selection, 444 model structure, 440-442, 543 sampling design, 440 reverse-time models alternative modeling, 475 data structure, 473 estimation, 474-475 estimator robustness, 475 model assumptions, 475 model selection, 475 model structure, 473-474 multistate models, 472-476 robust models, 545-550 computer software, 740-744 data structure, 439--440 Multiple-factor design models completely randomized designs, 85-86 cell means model, 85-86, 90 fully parameterized model, 86-87, 90 restricted parameterization model, 86 Multivariate decision problem calculus of variation, 613--614, 759 unconstrained optimization, 590-591, 764-747 Multivariate programming, description, 594-596, 750-751 MULT program band recovery estimates, 380, 393, 396-397 description, 741, 743 single-age models, 375-376
N Negative binomial distribution, description, 35, 723 Nested designs capture-recapture-based density estimation, 315-317 estimation methods, 98-99 fixed effects, 98-99 hierarchical design associations, 99-100 random effects, 99 statistical models, 98 testing, 98-99 Nest success, estimation models, 345-350 binomial modeling alternatives, 346-347 computer software, 744 historical perspectives, 345 Mayfield method, 345-346 model assumptions, 346-347 nest encounter parameters, 347-348
807 random effects model, 348 study design, 349-350 Newton's method, description, 749 Nonlinear homogeneous equations, stability analysis difference equations, 716-719 multiple species, 719 one species, 716-718 two species, 718-719 differential equations, 700-705 multiple species, 704-705 one species, 700-701 two species, 701-704 Nonlinear programming, optimal decision analysis, 597-601, 751-757 advantages, 639-641 inequality constraints, 598-599, 752-753 linear programming solutions, 603-605 nonnegative constraints only, 597-598, 752 solution algorithms, 599-600, 753-757 barrier methods, 755 dual methods, 755-756 feasible direction, 754-755 gradient projection, 753 Lagrangian methods, 180-181, 755-757 penalty methods, 755 primal methods, 753-754 reduced gradient, 753-754 NOREMARK program, description, 739, 743 Normal distribution, description, 36-38, 725-728 Null hypothesis, definition, 19
O Observational studies computer software, 739-740 constrained experimental design, 105-106 harvested population impact determination, 230 Optimal decision analysis adaptive optimization generalizations, 656-658 learning, 653-654 passive optimization, 658-660 calculus of variation, 608-618, 758-759 equality constraints, 614-615 Euler's equation, 609-610 inequality constraints, 615-617 integral constraints, 617
808
Optimal decision analysis ( c o n t i n u e d ) multivariate problem, 613-614, 759 optimality index forms, 612-613 transversality conditions, 610-612 univariate problem, 758 classical programming, 593-597, 749-751 bivariate programming, 593-594, 750 multivariate programming, 594-596, 750-751 sensitivity analysis, 596 dynamic programming, 627-638, 762-765 continuous time applications, 628-630 deterministic programming, 627-635, 763-764 discrete time applications, 630--635 linear-quadratic control, 629-630, 763-764 stochastic dynamic programming, 635-638, 764 variational mathematics, 764-765 dynamic resource control, 650-651 at equilibrium conditions, 579-580 expected value of perfect information, 654--655 identical policies for models, 655 identical values for models, 655 geometry of optimization, 584-585 convexity requirements, 585 heuristic approaches, 638-639 learning through management, 28 linear programming, 601-606, 757-758 dual linear programming, 603, 757-758 Kuhn-Tucker conditions, 602-603, 606, 757 nonlinear problem solutions, 603--605 simplex solution algorithm, 605, 747 multiple-model approach control, 652-653 decision times, 649-650 description, 27-28 outcomes, 649 natural resource conservation overview, 643-644 process uncertainty incorporation, 646-648 at nonequilibrium conditions, 580-581 nonlinear programming, 597-601, 751-757 inequality constraints, 598-599, 752-753
Index linear programming solutions, 603--605 nonnegative constraints only, 597-598, 752 solution algorithms, 599-600, 753-757 barrier methods, 755 dual methods, 755-756 feasible direction, 754-755 gradient projection, 753 Lagrangian methods, 180-181, 755-757 penalty methods, 755 primal methods, 753-754 reduced gradient, 753-754 objective functions, 579 overview general framework, 648-650, 660-662 mathematics, 745 modern approaches, 607-608 natural resource conservation, 643--644 optimization approaches compared, 639-641 traditional approaches, 583-584, 606 uses, 577-578, 581 partial observability, 655-656 Pontryagin's maximum principle, 618-627, 759-762 autonomous problems, 621 calculus of variation, 762 control trajectory constraints, 620-621, 760 discrete-time maximum principle, 624-627, 761-762 linear systems control, 621-622 sensitivity analysis, 623--624 singular controls, 622-623 unconstrained optimal control, 618-620, 759-760 single best model approach, 27 single model control, 651-652 stationary optimization, 580-581 uncertainty accounting for sources, 658 effects, 650-651 incorporation, 646-648 unconstrained optimization bivariate decision problem, 589-590, 746 differential approaches, 591-593 discrete-time maximum principle, 624-625, 761-762 multivariate decision problem, 590-591, 764-747 nondifferential approaches, 591 solution algorithms, 591-593
solution approaches, 747-749 differential methods, 748-749 nondifferential methods, 747-748 univariate decision problem, 586-589, 745-746 waterfowl sport harvest case study, regulatory process, 679-680 Optimization, see Optimal decision analysis
P Parameter estimation, see Population parameters Partial compensation hypothesis description, 229-230 waterfowl sport harvest case study, management control, 675-676 Partial observability, optimal decision strategies, 655-656 Patch dynamics, conservation and extinction models, 235-237 individual-based models, 236-237 source-sink models, 235-236 spatially explicit models, 236-237 Penalty methods, nonlinear programming, 755 Persistence time, conservation and extinction models, population size effects, 232-233 Physical models, description, 22 Point sampling, distance-based abundance estimation assumptions, 280-281 data structure, 278-279 estimators, 279-280 models, 279-280 sample size determination, 283-284 sampling scheme, 278-279 study design, 281-286 field procedures, 281-282 sample size determination, 282-283 stratified sampling, 284--285 Point-to-object estimators, abundance estimation, 263-265 data structure, 264 estimators, 264-265 models, 264-265 sampling scheme, 264 Poisson counting processes, 192-197 description, 192-193, 721-722 extensions, 194-196 compound processes, 194-195 interarrival times, 196-197 nonstationary processes, 195-196 superposition, 194
Index Poisson distribution, description, 192-193, 721-722 Pollock's model multiple-age capture-recapture methods, 440-444 alternative modeling, 443-444 estimation, 442-443 estimator robustness, 444 model assumptions, 442, 444 model selection, 444 model structure, 440-442, 543 sampling design, 440 Pontryagin's maximum principle, in optimal decision analysis, 618-627, 759-762 autonomous problems, 621 calculus of variation, 762 control trajectory constraints, 620-621, 760 discrete-time maximum principle, 624-627, 761-762 linear systems control, 621-622 sensitivity analysis, 623-624 singular controls, 622-623 unconstrained optimal control, 618-620, 759-760 POPAN-5 program abundance estimation, 501, 504, 510-511 description, 741-743 Popper's natural selection of hypotheses, description, 26 Population, definition, 3 Population censuses, count-based abundance estimation, 245 Population density, s e e Abundance; Density Population dynamics, s e e a l s o Population parameters abundance, s e e Abundance affecting factors, 4-5 case studies, s e e Waterfowl sport harvest case study community-population relationship, 556-557 abundance and species detection, 557 count statistics, 556-557 detection probabilities, 556-557 inferences, 556-557 species detection, 557 state variables, 556 vital rates, 556 conservation models metapopulations, 233-235 population size effects, 231-233 birth-death processes, 232 birth processes, 231-232
minimum viable populations, 233 persistence time, 232-233 description, 4 extinction models metapopulations, 233-235 population size effects, 231-233 birth-death processes, 232 birth processes, 231-232 minimum viable populations, 233 persistence time, 232-233 first-order linear homogeneous equations difference equations multiple species, 715-716 one species, 710 two species, 710-715 differential equations multiple species, 699-700 single species, 694-695 two species, 695-699 harvested population models additive mortality, 227-230 compensatory mortality, 227-230 partial compensation model with variable thresholds, 229-230 impact determination, 230-231 adaptive resource management, 230-231 experimental studies, 230 observational studies, 230 overview, 223, 237-238 sustained yield concept, 223-224 maximum sustained yield, 224-227 nonlinear homogeneous equations, stability analysis multiple species, 704-705 single species, 700-701 two species, 701-704 optimization, 578-579 traditional models, 135-172 age-structured models, 143-158 continuous-time models with age cohorts, 155-157 demographic relationships, 152-154 discrete-time models with age cohorts, 144-155 harvest, 154-155 life tables, 143-144 population characterization by age, 157-158 sensitivity analysis, 151-152 stable age distribution, 147-151 stable rate of growth, 147-151
809 survival estimation, 339-342 cohort models, 141-143 continuous-time models with age cohorts, 155-157, 159 discrete-time models, 144-155, 158-159 independent cohort populations, 141-143 transitions among cohorts, 143 competing populations models, 165-170 competitive exclusion, 166 Lotka-Volterra equations, 165-169 multiple competing species, 168-169 resource competition models, 169-170 stable coexistence, 166 unstable population equilibrium, 166-167 density-dependent growth, 139-141 harvest incorporation, 140-141 time lag incorporation, 141 density-independent growth, 136-139 habitat effects, 138 harvest effects, 138-139 perturbations, 139 exponential model, 136-139 habitat effects, 138 harvest effects, 138-139 perturbations, 139 general interacting species model, 170-171 geographic-structured models, 159-161 logistic model, 139-141 harvest incorporation, 140-141 time lag incorporation, 141 Lotka-Volterra models continuous-time models, 161-164 discrete-time models, 164 logistic effects, 163-164 oscillation size and shape, 162-163 overview, 135-136, 171-172 predator-prey models continuous-time models, 161-164 discrete-time models, 164 logistic effects, 163-164 oscillation size and shape, 162-163 size-structured models, 158-159 continuous-time models with size cohorts, 159
810 Population dynamics ( c o n t i n u e d ) discrete-time models with size cohorts, 158-159 Population ecology affecting factors, 4-7 density dependence, 5-6 density independence, 5-6 individual characteristics, 7-9 management, 6-7 population limitation, 6 regulation, 5 community dynamics, see Community definitions, 3-4 difference equations, see Difference equations population dynamics, s e e Population dynamics Population growth rate demographic parameter detectability, 334-335 density-independent growth, exponential models, 136-139 habitat effects, 138 harvest effects, 138-139 perturbations, 139 Population indices count-based abundance estimation, 257-261 abundance relationship, 257-259 constant slope linear relationship, 258-259 design, 260 interpretation, 260-261 noncount indices, 260 nonhomogeneous slope linear relationship, 259 nonlinear relationships, 259 survey counts, 257 optimality index, 612-613 Population management description, 6-7 limiting factors, see Limitation models decision-theoretic model use, 27-28 learning through management, 28 multiple-model approach, 27-28 single best model approach, 27 demographic stochasticity, 219-220 systematic approach, 133 optimal decision making, see Optimal decision analysis regulation, see Population regulation waterfowl sport harvest case study, 663-684
Index adaptive harvest management, 671-672, 679-680 harvest objectives, 676--677 ongoing issues, 681-684 goal setting, 681-683 management, 683-684 scale, 683-684 overview, 663-667, 684 population dynamics modeling environmental variation, 674-675 model development, 665-667 partial management control, 675--676 recruitment, 673-674 structural uncertainty, 672-674 survival, 673 regulatory process alternatives, 677-679 biological monitoring, 668-670 harvest regulations, 664-665, 667-668 optimal regulation identification, 679-680 outcome prediction, 670-671, 677-679 Population models, see Models Population parameters, see a l s o Population dynamics, Abundance, Recruitment estimation, Survival rates demographic parameter estimation, 333-363 age frequency analysis, 337-343 age stability unknown, 341 known stable age distribution, 340-341 life tables, 337-339 model likelihoods, 339-340 population reconstruction, 342-343 sampling assumptions, 341-342 survival estimation, 339-342 detectability, 334-337 movement probabilities, 335-336 population growth rates, 334-335 reproductive rates, 336--337 survival rates, 335 discrete survival analysis, 343-351 binomial survival model, 343-345 movement studies, 350-351 nest success estimation models, 345-350 radiotelemetry survival, 350-351 failure time analysis, 351-361
explanatory variable incorporation, 357-359 Kaplan-Meier method, 354-357 model assumptions, 359 nonparametric survival estimation, 354-357 parametric survival estimation, 353-354 proportional hazards model, 357-359 radiotelemetry study design, 359-361 statistical models, 352-353 survival distribution comparison tests, 355-357 known-fate data, 361-362 nest success estimation models, 345-350 binomial modeling alternatives, 346-347 historical perspectives, 345 Mayfield method, 345-346 model assumptions, 346-347 nest encounter parameters, 347-348 random effects model, 348 study design, 349-350 overview, 333-334, 362-363 random effects, 361-362 estimation approaches confidence intervals, 48-50 double sampling ratio estimator, 71 regression estimator, 70-71 estimator accuracy, 45-46 estimator bias, 43-44 estimator precision, 44--45 experimental design covariates, 92 nested designs, 98-99 randomized block designs, 91 split-plot designs, 93-94 interval estimation, 48-50 overview, 33-34, 42-43, 59-60 procedures, 46-48 maximum likelihood estimation, 46-49 method of moments, 46 simple random sampling, 63-64 stability, 181-182 stratified random sampling, 65-66 model utility, more versus less integrated model parameters, 30, 86 overview, 33-34 statistical distributions, 38-39 survey sampling adaptive sampling, 71-74
Index cluster sampling, 67-69, 73-74, 275-276 design features aerial-ground survey comparisons, 250-252 complete detectability on equal area samples, 246-247 partial individual detectability, 250-252 population indices, 257 randomization, 62 replication, 61-62 variation control, 62-63 design problems, 74-76 cause and effect misinterpretations, 75-76 pattern misinterpretation, 75-76 population definition, 74-75 pseudoreplication, 75 target definition, 74-75 detectability, 61 double sampling, 69-71 ratio estimator, 71 regression estimator, 70-71 issues, 60-61 overview, 59-60, 76-77 simple random sampling all sample units-based detectability estimation, 253 estimation, 63-64 sample size determination, 64-65 spatial heterogeneity, 60 stratified random sampling, 65-67 estimation, 65-66 sample size determination, 66--67 systematic sampling, 69 temporal heterogeneity, 60 variability, 60-61 Population reconstruction, age frequency analysis, 342-343 Population regulation, description, 5 Population size, s e e Abundance; Density Pradel's temporal symmetry model, recruitment and abundance estimation, 512-518 alternative modeling, 515 estimation, 514-515 estimator robustness, 515-517 model assumptions, 514-517 model selection, 515-520 model structure, 512-514 Precision, in parameter estimation, 44-45 Predator-prey models continuous-time models, 161-164 logistic effects, 163-164
oscillation size and shape, 162-163 general model for interacting species, 170-171 Prediction data-prediction comparison, 14-16 scientific process, 14 Probability density function, description, 35 Probability distributions, s c e Continuous distributions; Discrete distributions Proportional hazards model, failure time analysis, 357-359
Q Quadrat sampling, species richness estimation, 557-559
R
Radiotelemetry capture-recapture methods, 485-489 alternative modeling, 488-489 computer software, 744 data structure, 486-487 estimation, 488 estimator robustness, 489 model assumptions, 488-489 model selection, 489 model structure, 487-488 demographic parameter estimation discrete survival analysis, 350-351 failure time analysis study design capture and attachment procedure, 359-360 censoring, 360-361 fate determination, 360-361 monitoring frequency, 360 radio effects, 361 sample selection, 360 study area, 360 Randomization in experimental design completely randomized designs, 83-89 average main effects, 88 cell means model, 85-86, 90 fully parameterized model, 86-87, 90-91 hypothesis testing, 88-89 interactions, 88 model associations, 86-87 multifactor designs, 85-86 restricted parameterization model, 86 single-factor randomized designs, 83-85 testable hypothesis, 87-88
811 nested designs, 99 principles, 82 random effects, 97-100 randomized block designs, 89-91 estimation, 91 restricted randomization, 89-90 statistical models, 90-91 testing, 91 in failure time models, 359 in survey sampling all sample units-based detectability estimation, 253 design features, 62 simple random sampling all sample units-based detectability estimation, 253 estimation, 63-64 line transect sampling, 267 sample size determination, 64-65 stratified random sampling, 65-67 estimation, 65-66 sample size determination, 66--67 Randomized block designs, 89-91 estimation, 91 restricted randomization, 89-90 statistical models, 90-91 testing, 91 Ratio estimators complete detectability on unequal area samples, 248-249 double sampling, 71 RDSURVIV program description, 742-743 likelihood-based approach, 537, 540 Recruitment estimation capture-recapture methods, 495-522 approach relationships, 518-520 computer software, 740-744 data structure, 496-497 Jolly-Seber approach, 497-508 alternative modeling, 499-508 canonical estimator use, 501 capture history dependence, 501-502 estimation, 498-499 estimator robustness, 504-507 group-specific covariates, 501 individual covariates, 504 model assumptions, 498, 504-507 model selection, 504-507, 518-520 model structure, 497-498 multiple-age models, 502-503 multistate models, 503-504 partially open models, 499-500
812 Recruitment estimation (continued) reduced-parameter models, 500-501 time-specific covariates, 501 overview, 495-496, 518-520, 522 Pradel's temporal symmetry model, 512-518 alternative modeling, 515 estimation, 514-515 estimator robustness, 515-517 model assumptions, 514-517 model selection, 515-520 model structure, 512-514 study design, 520-522 estimator precision, 522 instantaneous sampling, 521 model assumptions, 520-522 parameters, 520 rate parameter homogeneity, 520-521 tag retention, 521 temporary emigration, 521-522, 538 superpopulation approach, 508-512 alternative modeling, 510-511 estimation, 510 estimator robustness, 511 model assumptions, 509-511 model selection, 511,518-520 model structure, 508-509 waterfowl sport harvest case study, 673-674 Reduced-parameter models recruitment and abundance estimation, Jolly-Seber approach, 500-501 survival rate estimation using bandrecovery methods multiple-age models, 385-387 single-age models, 371-372 Regression, estimation, double sampling, 70-71 RELEASE program description, 742-743 goodness-of-fit test, 431 Removal methods, closed populations, 319-325 data structure, 320-321 Delury approach, 321 estimators, 321-322 Leslie-Davis approach, 321 maximum likelihood approach, 322 model assumption violations, 322-323 models, 321-322 Ricker approach, 321 sampling scheme, 320-321 study design, 323-324
Index three least-squares approaches, 321-322 Renewal processes, description, 215 Repeated-measures designs, description, 96-97 Replication in experimental design, 81-82 in survey sampling design features, 61-62 problems, 75 Reproductive rates definition, 8 detectability, 336-337 Restricted parameterization model experimental design, 86 fully parameterized model compared, 86-87, 90 Retrospective study, constrained experimental design, 105-106 Reverse-time models, capture-recapture methods, 468-476 computer software, 740-744 multistate models, 472-476 alternative modeling, 475 data structure, 473 estimation, 474-475 estimator robustness, 475 model assumptions, 475 model selection, 475 model structure, 473--474 robust design models, 545-550 single-state models, 468-472 alternative modeling, 472 data structure, 468-469 estimation, 470-472 estimator robustness, 472 model assumptions, 470, 472 model selection, 472 model structure, 469-470 Reward studies, reporting rate using band recoveries, 391-398 data structure, 391-393 direct recoveries, 393-398 indirect recoveries, 393-396 spatial variation modeling, 396-398 Ricker approach, removal methods in closed populations, 321 Ring recovery, see Band-recovery combined with capture-recapture methods Robust design, combined closed and open capture-recapture models, 523-554 ad hoc approach, 529-535 alternative modeling, 532-533 closed only estimation models, 533-535 estimation, 531-532 estimator robustness, 533
model assumptions, 531,533 models, 529-531 model selection, 533 computer software, 740-744 data structure, 524-529 estimation concerns, 538-552 alternative data sources, 552 capture frequency data as covariates, 552 catch-effect studies, 550-551 multiple ages, 543-550 recruitment components, 543-550 reverse-time approach, 545-550 standard-time approach, 544-545 temporary emigration, 521, 538-543 unconditional closed-population modeling, 551-552 future research directions, 551-552 likelihood-based approach, 535538 alternative modeling, 537-538 estimation, 537 estimator robustness, 538 model assumptions, 537-538 models, 535-537 model selection, 538 overview, 523-524, 553-554 study design, 552-553 Robustness of estimators band-recovery-based estimation, single-age models, 377-378 capture-recapture-based estimation K-sample closed population models, 310-311 open population models, 434-436, 504-507, 511,515-517
S Sample size count-based abundance estimation, complete detectability on equal area samples, 247 determination band-recovery studies, 410-414 difference detecting studies, 413-414 parameter of interest, 411-412 precision level desired, 412 recovery rate expectation, 413 survival rate expectation, 412-413 targeted precision level studies, 410-411 two-age analyses, 413 years of study, 412 change-in-ratio mehtods, 330 distance sampling studies, 282-284
Index line transect sampling, 282-283 point sampling, 283-284 K-sample mark-recapture, 312 nest success studies, 349 open-population mark-recapture studies, 492, 522 abundance and recruitment, 522 survival, recruitment, and state transition, 492 power-based determination, 101-102 radiotelemetry studies, 360 removal methods, 323-324 simple random sampling, 64-65 stratified random sampling, 66-67 Sampling methods abundance estimation aerial-ground survey comparisons, 250-252 capture-recapture-based estimation computer software, 740-744 distance sampling, 317-319 K-sample models, 296 Lincoln-Petersen two-sample estimator, 243, 290-291 removal methods, 320-321 uniform sampling effort, 314-317 change-in-ratio methods, 326-327 complete detectability on equal area samples, 246-247 count-based estimation all sample units-based detectability estimation, 253-254 canonical population estimator, 244-245 line transect sampling, 265-278 assumptions, 266-267 density estimation with clusters, 275-276 distance function estimation, 270-273 fourier series approximations, 271,273 interval estimation, 277-278 key function approach, 271-272 maximum likelihood estimation, 273-274, 276 modeling approach, 265-266 model selection, 276-277 random sampling, 267 sample size determination, 282-283 sampling scheme, 265-266 statistical models, 267-270 variance of/) estimation, 274-275
partial individual detectability, 250-252 point sampling, 278-281 assumptions, 280-281 data structure, 278-279 estimators, 279-280 models, 279-280 sample size determination, 283-284 sampling scheme, 278-279 point-to-object methods, 264 population indices, 257 study design, 284-285 adaptive sampling, 71-74 cluster sampling, 67-69, 73-74, 275-276 counts, s e e Counts design features, 61-63 randomization, 62 replication, 61-62 variation control, 62-63 design problems, 74-76 cause and effect misinterpretations, 75--76 pattern misinterpretation, 75-76 population definition, 74-75 pseudoreplication, 75 target definition, 74-75 detectability, 61 double sampling, 69-71 ratio estimator, 71 regression estimator, 70-71 issues, 60-61 overview, 59-60, 76-77 simple random sampling, 63-65 estimation, 63-64 sample size determination, 64-65 spatial heterogeneity, 60 stratified random sampling, 65-67 estimation, 65-66 sample size determination, 66-67 systematic sampling, 69 temporal heterogeneity, 60 variability, 60-61 Schnabel census, s e e K-sample models, capture-recapture-based abundance estimation Scientific process causation necessary causation, 12 study approaches, 12-13 sufficient causation, 12 complementary hypothesis, 18-19 inductive logic, 17-18 overview, 11, 19-20 scientific method data, 14-16 hypotheses, 14 hypothesis confirmation, 16-17
813 observations, 14, 739-740 theory, 13-14 statistical inference, 18 Sensitivity analysis of age-structured models, 151-152 in model assessment, 127-128, 132 optimal decision making, 596, 623-624 Pontryagin's maximum principle, 623-624 Sequential univariate searches, description, 747 Sex determination, in band-recovery studies, 408 Sighting probability models count-based abundance estimation, 256 resightings between capture-recapture sampling occasions, 481-485 alternative modeling, 485 computer software, 740-744 data structure, 481--482 estimation, 484-485 estimator robustness, 485 model assumptions, 484-485 model selection, 485 model structure, 482-484 Simple random sampling, population parameters, 63-65 estimation, 63-64 sample size determination, 64-65 Simplex solution algorithm, optimal decision analysis using linear programming, 605, 747 Single-factor models completely randomized designs, 83-85 decision-theoretic models in population management, 27, 651 empirical models, 24 optimal control, 651-652 survival rate estimation using band recoveries, 366-383 computer software, 740-741 covariates, 373-374 data structure, 367-368 estimator robustness, 377-378 goodness of fit, 376 individual covariates, 374 model selection, 376-377 multiple bandings per year, 375-376 multiple groups, 373 probabilistic modeling, 368-371 reduced-parameter models, 371-372 sampling correlation interpretation, 378-380
814 Single-factor models (continued) sampling scheme, 367-368 temporary banding effect, 372-373 time-specific covariates, 373-374 Size-structured models continuous-time models, 159 discrete-time models, 158-159 SIZETRAN program, description, 740, 743 Software, see Computer software Source-sink models, conservation and extinction dynamics, 235-236 Spatially explicit models, conservation and extinction dynamics, 236-237 Spatial sampling canonical population estimator, 244-245 direct recovery reporting rate variation, 396-398 heterogeneity, 60 poststratification of band recoveries, 402-406 multistate models, 454-468 Markovian models, 454-464 memory models, 464-468 reverse-time models, 472-476 Species diversity, see Community SPECRICH program community parameter analysis, 561 description, 743-744 Split-plot designs, 93-94 estimation, 93-94 statistical models, 93 testing, 93-94 Stability assessment graphical methods with null clines, 706-708 model behavior stability, 129-130 model identification, 181-182 nonlinear homogeneous equations difference equations, 716-719 multiple species, 719 one species, 716-718 two species, 718-719 differential equations, 700-705 multiple species, 704-705 one species, 700-701 two species, 701-704 systematic approach, 133 Stable age distribution age frequency analysis age stability unknown, 341 known stable age distribution, 340-341 description, 147-151 Stationary processes
Index discrete Markov chains, stationary distributions, 201-202 Markov decision processes, stationary policies, 208-210 finite processes, 208 infinite processes, 209-210 semi-Markov processes, 210 semi-Markov processes stationary limiting distributions, 206-207 stationary policies, 210 stationary time series, 216-220 autoregressive processes, 217-218 demographic stochasticity, 219-220 moving-average processes, 218-219 population projection, 219-220 Statistical analysis, see specific m e t h o d s Statistical distributions conditional distributions, 40--42 continuous distributions, see Continuous distributions correlation, 42 covariance, 42 discrete distributions, see Discrete distributions joint distributions, 38, 40 marginal distributions, 40 overview, 34-35 parameters, 38-39 replication, 39-40 statistical independence, 39-40 Statistical expectation, in distribution parameters, 38 Statistical independence, in statistical distributions, 39-40 Statistical models, see also Models distance-based abundance estimation, line transect sampling, 267-270 experimental studies covariance, 91-92 hierarchical designs, 93 nested designs, 98 randomized block designs, 90-91 split-plot designs, 93 failure time analysis, 352-353 hazard rate, 352-353 Statistical power experimental design relationship, 100-101 in hypothesis testing, 51-52 Steepest ascent method, description, 748 Stochastic processes Bernoulli counting processes, 189-192 Bernoulli distribution, 35, 47, 723
number of successes, 189-190 waiting times, 190-192 branching processes, 213-215 Brownian motion, 210-213 description, 210-212 extensions, 212-213 absorbed motion, 212 geometric motion, 212-213 integrated motion, 212-213 reflected motion, 212 deterministic models compared, 116-117 discrete Markov processes decision processes, 207 Markov chains description, 197-198 stationary distributions, 201-202 state classification, 198-201 absorbing states, 200-201 communicating states, 198-199 ergodic states, 201 recurrent states, 199-200 transient states, 199-200 dynamic programming, 635-638, 764 Markov decision processes, 207-210 discrete-time processes, 207 objective functionals, 207-208 stationary policies, 208-210 finite processes, 208 infinite processes, 209-210 semi-Markov processes, 210 martingales, 216 model design factors, 125 overview, 187-189, 220-221 Poisson counting processes, 192197 description, 192-193, 721-722 extensions, 194-196 compound processes, 194-195 interarrival times, 196-197 nonstationary processes, 195-196 superposition, 194 renewal processes, 215 semi-Markov processes, 205-207 stationary limiting distributions, 206-207 stationary policies, 210 single-age models in band-recovery estimation, 378 stationary time series, 216-220 autoregressive processes, 217-218 demographic stochasticity, 219-220 moving-average processes, 218-219 population projection, 219-220 Stratified sampling count-based abundance estimation
Index complete detectability on equal area samples, 247 complete detectability on unequal area samples, 249-250 distance-based abundance estimation, 284-285 population parameters, 65-67 estimation, 65-66 sample size determination, 66--67 Student's t distribution, description, 730-731 Superpopulation approach, recruitment and abundance estimation, 508-512 alternative modeling, 510-511 estimation, 510 estimator robustness, 511 model assumptions, 509-511 model selection, 511,518-520 model structure, 508-509 SURGE program, description, 742-743 SURPH program, description, 743 Survey sampling, s e e Sampling methods Survival rates, 476-489 band recoveries, 365-415, 476-480 alternative modeling, 480 computer software, 740-741 data structure, 476--478 estimation, 480 estimator robustness, 377-378, 480 model assumptions, 480 model selection, 480 model structure, 478-480 movement analysis, 402-406 areas coincide, 405-406 areas differ, 402-405 multiple-age models, 383-391 covariate effects, 389-390 data structure, 383 group effects, 389-390 model selection, 390-391 multiple bandings per year, 390 probabilistic modeling, 383-385 reduced-parameter models, 385-387 temporary banding effect, 387 unrecognizable subadult cohorts, 387-389 nonharvested species, 398-402 cohort band recovery models, 400--401 data structure, 398 juvenile only banding, 400-401 probabilistic models, 398-400 unknown number of banded birds, 401--402 overview, 365-366, 414--415
poststratification of recoveries, 402-406 areas coincide, 405--406 areas differ, 402-405 reward studies for reporting rates, 391-398 data structure, 391-393 direct recoveries, 393-398 indirect recoveries, 393-396 spatial variation modeling, 396-398 single-age models, 366-383 covariates, 373-374 data structure, 367-368 estimator robustness, 377-378 goodness of fit, 376 individual covariates, 374, 430-431,552 model selection, 376-377 multiple bandings per year, 375-376 multiple groups, 373 probabilistic modeling, 368-371 reduced-parameter models, 371-372 sampling correlation interpretation, 378-380 sampling scheme, 367-368 temporary banding effect, 372-373 time-specific covariates, 373-374, 426 study design, 406-414 age determination, 408 banding station, 409 capture methods, 407-408 difference detection studies, 413-414 level of precision, 412 marking methods, 407-408 parameter of interest, 411-412 recovery rate enhancement, 409-410 recovery rate expectations, 413 sample size determination, 410-414 sex determination, 408 study duration, 409, 411 survival rate expectations, 412-413 targeted precision levels, 410-411 time of year effects, 408-409 two-age analyses, 413 definition, 8 demographic parameter estimation age frequency analysis, 339-342 detectability, 335 discrete survival analysis, 343-351
815 binomial survival model, 343-345 movement studies, 350-351 nest success estimation models, 345-350 radiotelemetry survival, 350-351 failure time analysis nonparametric survival estimation, 354-357 parametric survival estimation, 353-354 statistical models, 352-353 survival distribution comparison tests, 355-357 multiple-age mark-recapture models, 439-454 age-0 cohort models, 444-447 alternative modeling, 446 estimation, 445-446 estimator robustness, 445-447 model assumptions, 445-447 model selection, 445-447 model structure, 445 sampling design, 444-445 age-specific breeding models, 447-454 alternative modeling, 451-454 estimation, 451 model assumptions, 450-451 model structure, 448-450 sampling design, 447-448 data structure, 439-440 Pollock's model, 440-444 alternative modeling, 443-444 estimation, 442-443 estimator robustness, 444 model assumptions, 442, 444 model selection, 444 model structure, 440-442, 543 sampling design, 440 multistate models, 454-468 Markovian models, 454-464 alternative modeling, 459-460 data structure, 454-456 estimation, 459 estimator robustness, 460-464 model assumptions, 458-464 model selection, 460-464 model structure, 456-458 memory models, 464-468 alternative modeling, 466-467 data structure, 464-465 estimation, 466 estimator robustness, 467-468 model assumptions, 465-468 model selection, 467-468 model structure, 465 overview, 417-418, 492-493 radiotelemetry, 485-489
816 Survival rates ( c o n t i n u e d ) alternative modeling, 488-489 data structure, 486-487 estimation, 488 estimator robustness, 489 model assumptions, 488-489 model selection, 489 model structure, 487-488 resightings between sampling occasions, 481-485 alternative modeling, 485 data structure, 481-482 estimation, 484-485 estimator robustness, 485 model assumptions, 484-485 model selection, 485 model structure, 482-484 reverse-time models, 468-476 multistate models, 472-476 alternative modeling, 475 data structure, 473 estimation, 474M75 estimator robustness, 475 model assumptions, 475 model selection, 475 model structure, 473-474 robust models, 545-550 single-state models, 468-472 alternative modeling, 472 data structure, 468-469 estimation, 470-472 estimator robustness, 472 model assumptions, 470, 472 model selection, 472 model structure, 469-470 single-age models, 418-439 capture history effects, 428-430 Cormack-Jolly-Seber model, 419-426 estimation, 423-425 estimator robustness, 434-336 model assumptions, 422-423 reduced-parameter models, 425-426 structure, 419-422 data structure, 419 estimator robustness, 434-436 capture history independence, 436 homogeneous capture probabilities, 434-435 homogeneous survival probabilities, 435 instantaneous sampling, 435 permanent emigration, 435-436 probabilistic models, 422-423 tag loss absence, 435
Index individual covariates, 374, 430-431, 552 model selection, 431-434 multiple groups, 427 parameterization for transient individuals, 428-430 probabilistic modeling, 419-426 time-specific covariates, 426 trap response, 428 study design, 489-492 estimator precision, 492 model assumptions, 490-492 instantaneous sampling, 491 permanent emigration, 491 rate homogeneity, 490-491 tag retention, 491 model parameters, 489-490 sampling designs, 489-490 waterfowl sport harvest case study, 673 SURVW program description, 347, 376, 393-397, 406, 739-744 multistate models, 459-460, 467, 480, 548 state transition estimation, 424 Sustained yield, models, harvested population dynamics concept, 223-224 maximum sustained yield, 224-227 Systematic sampling, population parameters, 69
T Tagging methods, see Band-recovery methods; Capture-recapture methods; Radiotelemetry t Distribution, description, 730-731 Telemetry, see Radiotelemetry Temporal factors repeated-measures designs, 96--97 survey sampling heterogeneity, 60 survival rate estimation, banding effects multiple-age models, 387 single-age models, 372-373 Temporal symmetry model, recruitment and abundance estimation, 512-518 alternative modeling, 515 estimation, 514-515 estimator robustness, 515-517 model assumptions, 514-517 model selection, 515-520 model structure, 512-514 Temporal variation model
capture--recapture-based abundance estimation, 299, 302-304 community dynamics at a single location, 563-568 annual extinction probabilities, 566-567 local colonizing species, 566 local extinction probability, 564-565 local species turnover, 565-566 parameter analysis, 567-567 rate of change, 564 recolonization probabilities, 566-567 time and space variation, 571-572 Temporary emigration, in capture-recapture models, 521-522, 538-543 Markovian emigration model, 541-543 random migration model, 539-541 Theoretical models description, 23-24 information-theoretic approaches, 55-57 population management, 27-28 learning through management, 28 multiple-model approach, 27-28 single best model approach, 27 Time series data models identification methods, 173-185 Lagrangian multipliers, 180-181, 755-757 least squares method, 174-176 mean absolute error method, 176-177 multiple-dimensional models, 180 one-dimensional models, 179 optimal identification, 178 overview, 173-174 parameter estimate stability, 181-182 population size correlated estimates, 178 system property identification in non-modeled situations, 182-184 two-dimensional models, 179-180 reverse-time capture-recapture models, 468-476 computer software, 740-744 multistate models, 472-476 alternative modeling, 475 data structure, 473 estimation, 474-475 estimator robustness, 475 model assumptions, 475
817
Index model selection, 475 model structure, 473-474 robust models, 545-550 single-state models, 468-472 alternative modeling, 472 data structure, 468-469 estimation, 470-472 estimator robustness, 472 model assumptions, 470, 472 model selection, 472 model structure, 469-470 survival rate estimation using bandrecovery methods, singleage models, 373-374, 426 Time-specific covariates, capture-recapture methods recruitment and abundance estimation, Jolly-Seber approach, 501 single-age models, 373-374, 426 TMSURVIV program, description, 742-743 TRANSECT program description, 739-740 distance-based abundance estimation, line transect sampling, 277-278 Transient individuals, parameterization methods, 428-430 Transversality conditions, optimal decision analysis, 610-612 Trapping webs capture-recapture-based density estimation, 317-319 distance-based abundance estimation issues, 286-287 Trap response, in single-age models capture probabilities, 428 survival probabilities, 428 Tribolium models, description, 22 Type I errors, in hypothesis testing, 50-51 Type II errors in hypothesis testing, 50-51 statistical power relationship, 100-101
U Uncertainty, in optimal decision analysis accounting for sources, 658 effects, 650-651 incorporation, 646-648
u Validation, model assessment systematic approach, 133 verification compared, 126--127 Variability estimation methods, 733-738 bootstrap estimation, 737-738 distribution-based estimation, 733-734 empirical-based estimation, 734 information matrix methods, 734-736 jackknife estimators, 737 overview, 733 geographic variation, community dynamics at a single time, 569-571 species cooccurrence, 569-570 time and space variation, 571-572 unique species, 570-571 optimal decision analysis calculus of variation, 608-618, 758-759 equality constraints, 614-615 Euler's equation, 609-610 inequality constraints, 615-617 integral constraints, 617 multivariate problem, 613-614, 759 optimality index forms, 612-613 transversality conditions, 610-612 univariate problem, 758 classical programming bivariate programming, 593-594, 750 multivariate programming, 594-596, 750-751 dynamic programming, 764-765 environmental variation, 644-646 unconstrained optimization bivariate decision problem, 589-590, 746 multivariate decision problem, 590-591, 764-747 univariate decision problem, 586-589, 745-746 in survey sampling control, 62-63 description, 60-61 temporal variation capture-recapture-based abundance estimation, 299, 302-304
community dynamics at a single location, 5~3-568 annual extinction probabilities, 566-567 local colonizing species, 566 local extinction probability, 564-565 local species turnover, 565566 parameter analysis, 567-567 rate of change, 564 recolonization probabilities, 566-567 time and space variation, 571-572 waterfowl sport harvest case study, 674--675 Verbal models, description, 22
W
Waterfowl sport harvest case study, 663--684 adaptive harvest management, 671-672, 679--680 harvest objectives, 676-677 ongoing issues, 681-684 goal setting, 681-683 management, 683-684 scale, 683-684 overview, 663-667, 684 population dynamics modeling environmental variation, 674-675 model development, 665-667 partial management control, 675-676 recruitment, 673-674 structural uncertainty, 672-674 survival, 673 regulatory process alternatives, 677-679 biological monitoring, 668670 harvest regulations, 664-665, 667-668 optimal regulation identification, 679-680 outcome prediction, 670-671, 677-679 Weighted estimators, complete detectability on unequal area sampies, 249
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