ECOLOGY, ECO LOGY, GENETICS, G EN ETICS, AND EVOLUTION AND EVOLUTION OF OF METAPOPULATIONS M ETAPOPULATIO N S
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COLOGY, CS, E ECO LOGY, GENETI G EN ETICS, AND EVOLUTION AND EVOLUTION OF OF METAPOPULATIONS M ETAPO PU LATIO N S
Edited Edited by by
IIkka Hanski Ilkka Hanski
Metapopulation Research Group Department of Ecology Ecology and Systematics University of Helsinki, Finland
Oscar Oscar E. E. Gaggiotti Gaggiotti
Metapopulation Research Group Department of Ecology Ecology and Systematics University of Helsinki, Finland Finland
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C ONTENTS CONTENTS
CONTRIBUTORS PREFACE PREFACE
ix
xiii
ACKNOWLEDGMENTS xv
PERSPECTIVES PERSPECTIVES ON ON SPATIAL SPATIAL DYNAMICS DYNAMICS 1 1
11
METAPOPULATION METAPOPULATION BIOLOGY: BIOLOGY PAST, PRESENT, PRESENT, AND FUTURE 3 3 ANDFUTURE by Ilkka Hanski and Oscar E.Gaggiotti E.Gaggiotti
2 2
METAPOPULATION METAPOPULATION DYNAMICS: DYNAMICS: PERSPECTIVES PERSPECTIVES FROM FROM LANDSCAPE LANDSCAPE ECOLOGY ECOLOGY
23 23
by Kimberly A. With With
3 3
CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS FOR FOR POPULATION POPULATION DYNAMICS DYNAMICS
45 45
by by Benjamin M. M . Balker Bolker
II METAPOPULATION METAPOPULATION ECOLOGY ECOLOGY 4 4
71 71
METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
73 73
by by Otso Otso Ovaskainen and Ilkka Ilkka Hanski v V
vi
CONTENTS CONTENTS
5 5
APPLICATION OF STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS TO REAL METAPOPULATIONS METAPOPULATIONS 105 105 by Rampal S. Etienne, Cajo].F. Caj0J.F. ter Braak, and Claire C. Vos by
6 6
FROM METAPOPULATIONS METAPOPULATIONS TO METACOMMUNITIES METACOMMUNITIES FROM
133 133
by Mathew A. Leibold and Thomas E. Miller by
III METAPOPULATION METAPOPULATION GENETICS GENETICS
1151 51
7 SELECTION SELECTION AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS 153 153 7 by Michael C. C. Whitlock by 8 8
METAPOPULATIONS METAPOPULATIONS AND COALESCENT COALESCENT THEORY
175 175
by John Wakeley by John Wakeley 9 9
METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS: GENETICS: THE METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS OF POPULATION POPULATION
DIFFERENTIATION DIFFERENTIATION
199 199
by Charles Charles J. Goodnight by
Part IV
EVOLUTIONARY DYNAMICS DYNAMICS EVOLUTIONARY IN METAPOPULATIONS METAPOPULATIONS 225 225 IN 10 10
LIFE HISTORY HISTORY EVOLUTION EVOLUTION IN IN METAPOPULATIONS METAPOPULATIONS 227 227 LIFE by Ophelie Ophe'lie Ronce Rome and Isabelle Olivieri by
11 11
IN METAPOPULATIONS: METAPOPULATIONS: THE COEVOLUTION COEVOLUTION SELECTION IN SELECTION OF PHENOTYPE PHENOTYPE AND CONTEXT CONTEXT 259 259 by Michael JJ.. Wade Wade by
12 12
SPECIATION SPECIATION IN IN METAPOPULATIONS METAPOPULATIONS 275 275 by by Sergey Sergey Gavrilets Gavrilets
Partt V
INTEGRATION AND APPLICATIONS INTEGRATION APPLICATIONS 13 13
305 305
CAUSES, MECHANISMS MECHANISMS AND CONSEQUENCES CONSEQUENCES DISPERSAL 307 307 OF DISPERSAL by Jean Clobert, Clobert, Rolf Rolf Anker Ims, and Franfois FranGois Rousset by
CONTENTS CONTENTS
14 14
vii vII MECHANISMS MECHANISMS OF POPULATION POPULATION EXTINCTION EXTINCTION
by Oscar E. Gaggiotti and Ilkka Hanski
15 15
337 337
MULTILOCUS MULTILOCUS GENOTYPE GENOTYPE METHODS METHODS FOR FOR THE STUDY STUDY OF OF METAPOPULATION METAPOPULATION PROCESSES PROCESSES 367 367
by Oscar E. Gaggiotti
16 16
ECOLOGICAL AND EVOLUTIONARY OF ECOLOGICAL AND EVOLUTIONARY CONSEQUENCES CONSEQUENCES OF SOURCE-SINK SOURCE-SINK POPULATION POPULATION DYNAMICS DYNAMICS
by Tadeusz J. Kawecki
17 17
387 387
METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS OF INFECTIOUS INFECTIOUS DISEASES DISEASES 415 415
by Matt J. Keeling, Bjornstad, and Bryan T. Keeling, Ottar N. Bj~rnstad, T. Grenfell
18 18
TOWARD A METAPOPULATION METAPOPULATION CONCEPT CONCEPT FOR FOR PLANTS PLANTS 447 447 by N.J. Ouborg and O. Eriksson
19 19
LONG-TERM STUDY OF OF A A PLANT-PATHOGEN LONG-TERM STUDY PLANT-PATHOGEN METAPOPULATION METAPOPULATION
by Janis Antonovics
20 20
471 471
METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN CHANGING CHANGING ENVIRONMENTS: ENVIRONMENTS" BUTTERFLY BUTTERFLYRESPONSES RESPONSES TO HABITAT HABITAT AND CLIMATE AND CLIMATE CHANGE CHANGE 489 489 by Chris Chris D. Thomas and Ilkka Ilkka Hanski
21 21
INFERRING AND PROCESS INFERRING PATTERN PATTERN AND PROCESS IN IN SMALL SMALL MAMMAL MAMMAL METAPOPULATIONS: METAPOPULATIONS" INSIGHTS INSIGHTS FROM FROM ECOLOGICAL ECOLOGICAL AND GENETIC AND GENETIC DATA DATA 515 515
by Xavier Lambin, Jon Aars, Aars, Stuart B. Piertney, Piertney, and Sandra Sandra Telfer
22 22
METAPOPULATION AND RESERVE METAPOPULATION DYNAMICS DYNAMICS AND RESERVE NETWORK NETWORK DESIGN DESIGN 541 541
by Mar Cabeza, Cabeza, Atte Moilanen, and Hugh P. P. Possingham
23 23
VIABILITY ANALYSIS ANALYSIS FOR VIABILITY FOR ENDANGERED ENDANGERED METAPOPULATIONS: METAPOPULATIONS" APPROXIMATION APPROACH APPROACH 565 A DIFFUSION DIFFUSION APPROXIMATION 565 by E.E. Holmes and B.X. B.X. Semmens REFERENCES REFERENCES 599 599
INDEX
683 683
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C ONTRIBUTORS CONTRIBUTORS
Numbers Numbers in parentheses parentheses indicate indicate the pages on which which the authors' authors" contributions contributions begin. Jon Jon Aars Aars (515) (515) NERC Molecular Genetics Genetics in Ecology Ecology Initiative, Aberdeen Aberdeen
Population Ecology Ecology Research Unit, School School of Biological Biological Sciences, Sciences, University University of Aberdeen, Aberdeen, Aberdeen Aberdeen AB24 AB24 2TZ, Scotland; Scotland; current address: address: Norwegian Polar Institute, Troms0, Norway Institute, Polarmilj0senteret, Polarmiljosenteret, N-9296 Tromso, Janis Antonovics (471) Biology Department, University of Virginia, Charlottesville, Virginia Virginia 22904 Ottar N. Bjornstad Bj0rnstad (415) Departments of Entomology and Biology, Biology, Penn State University, University Park, Pennsylvania Pennsylvania 16802 University, University Benjamin Zoology Department, University of Florida, Benjamin M. Bolker (45) (45) Zoology Florida, Gainesville, Florida 32611 J.F. ter Braak (105) Biometrics, Cajo J.F. Biometrics, Wageningen Wageningen University University and Research Centre, Box 100, NL-6700 AC Wageningen, The Netherlands Mar Mar Cabeza (541) Metapopulation Research Group, Department of Ecology Ecology and Systematics, Systematics, University University of Helsinki, FIN-00014 Helsinki, Finland Jean Clobert (307) Laboratoire d'Ecologie, Universite Universit~ Pierre Pierre et Marie Curie, Batiment B~timent A, 75252 75252 Parix cedex OS, 05, France O. Eriksson (447) Department of Botany, Botany, Stockholm University, University, SE-I06 SE-106 91 Stockholm, Sweden Rampal S. Etienne Etienne (105) (105) Community and Conservation Ecology Ecology Group, University University of Groningen, Box 14, NL-9750 AA Haren, The Netherlands Oscar E. Gaggiotti Gaggiotti (3, 337, 367) Genomique G~nomique de Populations Populations et Biodiversite, Biodiversit~, LECA-CNRS LECA-CNRS UMR 5553 Universite Universit~ Joseph Fourier, Fourier, F-38041 F-38041 Grenoble Cedex 9, France Sergey Sergey Gavrilets (275) (275) Department of Ecology Ecology and Evolutionary Biology, Biology, University of Tennessee, Department of Mathematics, University Tennessee, Knoxville, Knoxville, Tennessee Tennessee 37996 37996 Ix ix
CONTRIBUTORS CONTRIBUTORS
xx
Charles J. Goodnight (201) Department of Biology, Biology, University University of Vermont,
Marsh Life Life Sciences Sciences Building, Building, Burlington, Vermont 05405 05405
Bryan T. Grenfell Grenfell (415) Department of Zoology, Zoology, University University of Cambridge,
Downing Street, Street, Cambridge CB2 3EJ, 3EJ, England
I1kka 337, 489) Metapopulation Research Group, Department Ilkka Hanski (3, 73, 73,337,
of Ecology University of Helsinki, FIN-00014 Ecology and Systematics, Systematics, University FIN-00014 Helsinki, Finland E.E. Holmes Holmes (565) (565) National Marine Fisheries Fisheries Service, Service, Northwest Fisheries Fisheries Science Science Center, Center, Seattle, Seattle, Washington 98112 Rolf Anker Anker Ims (307) (307) Institute of Biology, Biology, University University of Troms0, Tromso, N-9037 Troms0, Tromso, Norway Tadeusz J. Kawecki Kawecki (387) Unit Unit for Ecology Ecology and Evolution, Department of Biology, University Biology, University of Fribourg, CH-1700 Fribourg, Switzerland Matt Matt J. Keeling (415) Maths Institute and Department of Biological Biological Sciences, Sciences, University AL, England University of Warwick, Coventry, Coventry, CV4 77AL, Xavier Lambin Aberdeen Population Ecology Lambin (515) Aberdeen Ecology Research Unit, Unit, School School of Biological Sciences, Sciences, University University of Aberdeen, Aberdeen, Aberdeen Aberdeen AB24 2TZ, 2TZ, Scotland Mathew Mathew A. Leibold Leibold (133) Department of Ecology Ecology and Evolution, University University of Chicago, Chicago, Illinois Illinois 60637 (133) Department of Biological Science, Florida State Thomas E. Miller (133) Biological Science, State University, University, Tallahassee, Florida 32306 Atte Atte Moilanen Moilanen (541) Metapopulation Research Research Group, Department of Ecology Ecology and Systematics, Helsinki, FIN-00014 Helsinki, Helsinki, Finland Systematics, University University of Helsinki, Isabelle (229) Institut des Sciences IsabeUe Olivieri (229) Sciences de l'Evolution UMR5554, Universite Universit~ Montpellier II, Place Place Eugene Eugene Bataillon, Bataillon, 34095 Montpellier cedex 5, France N.J. Ouborg (447) Department of Molecular Ecology, Ecology, University University of Nijmegen, Nijmegen, The Netherlands Toernooiveld 1, 6525 Ed Nijmegen, Otso Otso Ovaskainen (73) (73) Metapopulation Research Research Group, Department of Ecology Helsinki, Ecology and Systematics, Systematics, University University of Helsinki, Helsinki, FIN-00014 Helsinki, Finland Stuart B. Piertney Piertney (515) (515) NERC Molecular Genetics Genetics in Ecology Ecology Initiative, Aberdeen Population Population Ecology Ecology Research Unit, Unit, School School of Biological Biological Sciences, University University of Aberdeen, Aberdeen Aberdeen AB24 AB24 2TZ, Scotland Hugh Hugh P. P. Possingham (541) Departments of Zoology Zoology and Mathematics, The University University of Queensland, Queensland, St Lucia Lucia QLD 4072, 4072, Australia Ophelie (229) Institut des Sciences Oph~lie Ronce (229) Sciences de l'Evolution UMR5554, Universite Universit~ Montpellier II, Place Place Eugene Eugene Bataillon, Bataillon, 34095 34095 Montpellier cedex 5, France Fran�ois Francois Rousset Rousset (307) (307) Institut des Sciences Sciences de l'Evolution, l'Evolution, UMR5554 UMR5554 Universite Universit~ Montpellier, 34095 Montpellier cedex cedex 5, France B. Semmens Semmens (565) (565) Zoology Department, University University of Washington, Seattle, Washington 98195 98195 Sandra Animal Infectious Disease Sandra Telfer Telfer (515) (515) Small Small Animal Disease Group, Leahurst, University University of Liverpool, Neston CH64 7TE, 7TE, England Chris Chris D. Thomas Thomas (489) Department of Biology, Biology, University University of Leeds, Leeds, Leeds Leeds LS2 9JT, England Claire C. Vos (105) (105) Alterra Green Green World Research, Research, Box 47, NL-6700 NL-6700 AA Wageningen, The Netherlands
CONTRIBUTORS CONTRIBUTORS
Michael J.
xl xi
Wade (259) Department of Biology, Biology, Indiana University, Bloomington, Indiana 47405 47405 John John Wakeley Wakeley (175) Biological Biological Laboratories, Harvard University, University, Cambridge, Massachusetts Massachusetts 02138 02138 Michael Michael C. Whitlock (153) (153) Department of Zoology, University University of British Columbia, Vancouver, Vancouver, BC V6T 1 1 Z4, Z4, Canada Kimberly A. With With (23) (23) Division of Biology, Biology, Kansas State University, Manhattan, Kansas Kansas 66506 66506
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CE PREFA PREFACE
Over past 15 years, metapopulation metapopulation biology has Over the the past has developed from from a set of ideas, simple models, models, and and a limited number number of case studies studies to to an an essential part of population population biology. Some areas population biology continue to part areas of meta metapopulation continue to flourish flourish with bold new new visions and and attempts attempts to to clarify them with models, models, but but other areas areas have already become consolidated consolidated into into a solid body of theory and and have have been thoroughly investigated empirically. Progress Progress has has been so great great that that contents of this this volume bear the contents bear only superficial resemblance resemblance to to the contents contents Metapopulation Biology of the predecessor, predecessor, Metapopulation Biology (Hanski (Hanski and and Gilpin, Gilpin, 1997), 1997), to to say Metapopulation Dynamics nothing nothing about about the the first edited edited volume in this series, Metapopulation Dynamics (Gilpin and Hanski, 1991). and Hanski, 1991 ). In this volume volume we have achieved, for the first time, an equal equal coverage of metapopulation metapopulation ecology, meta population genetics, metapopulation genetics, and and evolutionary meta population biology. There is no parity, however. metapopulation biology. There no complete parity, however. Metapopulation which was at the stage of conceptual Metapopulation ecology, which conceptual development and turned to and budding budding empirical empirical studies studies 15 years ago, has has by now now turned to a well wellestablished established discipline with with substantial substantial impact on on practical practical conservation. conservation. In contrast, metapopulation meta population genetic and contrast, and evolutionary studies studies are at an an earlier stage, stage, with with less well-developed integration integration of theoretical and and empirical work. work. But such integration integration is undoubtedly undoubtedly coming, coming, and and it is hoped hoped that that this volume will stimulate further further development in this direction. chapters in this volume are entirely new, All the chapters new, nothing nothing has has been copied copied from from Metapopulation Metapopulation Biology Biology.. The The previous previous volume includes contributions contributions that are well worth that worth reading reading even today, today, but but we we did not not include them them here in the the interest of giving space space to to a new set of authors authors and and chapters, chapters, and and also because because the previous volume is still available. available. One One important important similarity remains. This is an remains. This an edited edited volume volume in which which we have not not forced forced the same app approach roach in treatment of the subject matter matter in all the the chapters. Some chapters are primarily or or even entirely theoretical, theoretical, whereas whereas others others are based based on on empirical research. present an metapopulation research. Some chapters chapters present an overview of one slice of metapopulation xlii xiii
xxlv lv
PREFACE PREFACE
biology, whereas others are focused more narrowly on new developments. There are up-to-date reviews of all areas of metapopulation biology. We are confident that there is not a single population biologist in this world world who would find nothing new in this volume, nor are there many who would find all the chapters easy bed-time reading. reading. But we trust that that most of our readers will appreciate the diversity and the challenge, and will be inspired by at least some of the visions, comprehensive empirical studies, and modeling efforts described in this volume. volume. Our Our aim was to produce a volume that serves serves both both as a reference for researchers and as a text for advanced students in ecology, genetics, evolutionary biology, and conservation biology. The emphasis is on integration across disciplines. disciplines. Several chapters are relevant for conservation conservationists in setting the stage for new applications. It is hoped that graduate students will find material in this volume for innovative Ph.D. Ph.D. projects. We are grateful to a large number of colleagues who provided truly helpful Araujo, Frederic Austerlitz, Hans reviews of particular particular chapters: Miguel Arafijo, Baveco, Peter Beerli, Beerli, Thomas Thomas Berendonck, Ben Bolker, Bolker, Cajo ter Braak, Mark Mark Burgman, Burgman, Jeremy Burdon, Burdon, Dennis Couvet, Michael Michael Doebeli, Stephen Ellner, Rampal Rampal Etienne, Patrick Foley, Foley, Robert Freckleton, Sylvain Gandon, Gandon, Gisela Garcia, Nicholas Gotelli, Mikko Mikko Heino, Jessica Hellmann, Eric Imbert, Imbert, Rolf Ims, Par P~r K. K. Ingvarsson, Kevin Kevin Laland, Xavier Lambin, Russ Lande, Martin Martin Lascoux, Richard Law, Michel Lareau, Loreau, Michael McCarthy, Juha Merila, Meril~i, Atte Moilanen, Moilanen, Allen J. Moore, Moore, Isabelle Olivieri, Olivieri, Otso Otso Ovaskainen, Ovaskainen, John Pannell, Craig Primmer, Jonathan Pritchard, Pritchard, Chris Ray, Ray, Steven Riley, Riley, Ilik Saccheri, Saccheri, Mikko Jonathan Silvertown, Peter Smouse, Mikko J. Sillanpaa, Sillanp~i~i, Jonathan Smouse, Per Sj6gren-Gulve, Chris Thomas, Thomas, Xavier Vekemans, Jana Jana Verboom, and Franjo Weissing. We thank thank Marjo Marjo Saastamoinen and Tapio Gustafsson for indispensable secretarial help. Chuck Crumly from Academic Press had trust in this volume from our very first correspondence, correspondence, and Kelly Kelly Sonnack, Angela Dooley, Michael Sugarman and and Eric DeCicco at Academic Press made our task as editors as easy as possible. Finally, our thanks to all the authors authors for showing great enthu enthusiasm and keeping deadlines. deadlines. Ilkka Hanski Oscar Oscar Gaggiotti April 2003, Helsinki
A CKNOWLEDG MENTS ACKNOWLEDGMENTS
CHAPTER CHAPTER 1 1
We thank thank Rolf Ims and Chris Thomas for comments on the chapter. Supported by the Academy of Finland (Centre of Excellence Programme 2000-2005). 2000-2005).
CHAPTER CHAPTER 2 2
I thank thank Hans Baveco, Ilkka Hanski, Greg Schrott, Per Sjogren-Gulve, Sj6gren-Gulve, and Jana Verboom for their comments on the chapter. chapter. My research on the effects of landscape structure and and dynamics on extinction risk for spatially structured populations populations has been supported by past grants from the National Science Science Foundation and, most recently, by the u.S. U.S. Environmental Protection Agency (R829090).
CHAPTER CHAPTER 3 3
I thank thank the Isaac Newton Newton Institute for supporting a workshop on scaling in biological systems where some of these ideas were developed and Toshinori Okuyama and Graeme Cumming for useful discussions.
CHAPTER CHAPTER 44
We thank thank Ben Bolker, Bolker, Cajo van ter Braak, Rampal Etienne, and Karin Karin Frank for comments on the chapter. chapter. Supported by the Academy of Finland (Centre of Excellence Programme Programme 2000-2005). 2000-2005). XV xv
Part I Perspectives on Spatial Dynamics
sdfsdf
1
META PO PULATION M ETAPO PU LATIO N BIOLOGY: BIOLOGY: PAST, PAST! PRESENT, AND FUTURE PRESENT, AND FUTURE Ilkka Ilkka Hanski Hanski and and Oscar Oscar E. Gaggiotti
11.1 .1
INTRODUCTION INTRODUCTION The The term term metapopulation metapopulation stems stems from from the the general general notion notion of of the the hierarchical hierarchical structure structure of of nature. nature. Just Just like like the the term term population population is is needed needed to to describe describe an an assemblage assemblage of of interacting interacting individuals, individuals, it it seems seems apt apt to to have have aa term term for for an an assem assemblage blage of of spatially spatially delimited delimited local local populations populations that that are are coupled coupled by by some some degree degree of 970) . It of migration m i g r a t i o n- the the metapopulation metapopulation (Levins, (Levins, 11970). It is is conceptually conceptually attract attractive, and explicitly consider ive, and helpful helpful for for the the study study of of population population biology, biology, to to explicitly consider the the sequence of entities from populations. sequence of entities from individuals individuals to to local local populations populations to to meta metapopulations. Theoretical Theoretical studies studies are are greatly greatly facilitated facilitated by by the the view view of of landscapes landscapes as as networks networks of inhabited by local populations. just theory: of habitat habitat patches patches inhabited by local populations. And And it it is is not not just theory: there there are are innumerable innumerable species species that that definitely definitely have have such such aa spatial spatial population population structure structure in in some some landscapes, landscapes, and and continuing continuing habitat habitat loss loss and and fragmentation fragmentation force species to population structure. force ever ever greater greater numbers numbers of of species to conform conform to to aa meta metapopulation structure. Other continuous spatial spatial distributions distributions in Other species species have have more more continuous in less less distinctly distinctly patchy some purposes patchy environments, environments, but but even even for for these these species species and and for for some purposes the the meta population view metapopulation view of of nature nature can can be be helpful. helpful. A metapopulation metapopulation approach approach refers refers to to research research or or management management that, that, in in one one form form or or another, another, adopts adopts the the view view that that local local populations, populations, which which the the metapopula metapopulations (or relatively tions consist consist of, of, are are discrete discrete (or relatively discrete) discrete) entities entities in in space space and and that that these these local populations population local populations interact interact via via migration migration and and gene gene flow. flow. Classic Classic meta metapopulation
Ecology, Ecology, Genetics, Genetics, and Evolution of Metapopulations
3 3
Copyright 2004, Elsevier, Elsevier,Inc. 0-12-323448-4
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ILKKA ILKKA HANSKI HANSKIAND AND OSCAR OSCAR E. E. GAGGIOTII GAGGIOTTI
dynamics 1969, 11970) 970) focus dynamics in in the the sense sense pioneered pioneered by by Levins Levins ((1969, focus on on the the processes processes of local local extinction extinction and and recolonization recolonization in in the the same same manner as as population population dynam dynamof ics are are concerned concerned with with births births and and deaths deaths of of individuals. individuals. However, However, such such popula populaics tion turnover turnover is is not not aa necessary necessary condition condition for for the the meta metapopulation approach to to tion population approach be be useful, useful, nor nor aa characteristic characteristic feature feature of of all all species species that that are are structured, structured, in in some some landscapes, landscapes, into into spatially spatially discontinuous discontinuous local local populations. populations. Important Important questions questions need to be asked asked about the the interaction interaction of permanent local populations, for context of source-sink source-sink dynamics (Chapter (Chapter 116). instance in the context 6). Metapopulation biology represents one one way of of explicitly explicitly putting population biology biology into into aa spatial spatial context. context. The The basic basic tenet tenet of of spatial spatialecology, ecology,which which includes includes metapopulation ecology as well as other approximations approximations (see (see alter) alter),, is that the individuals and populations matter, in the sense of influ influspatial positions of individuals encing the growth populations and growth rate and dynamics of populations and meta metapopulations their competitive, predator-prey and other interactions. Likewise, we may use the term spatial spatial population population biology biology to emphasize the influence of the spatial positions of individuals and populations on their genetic and evolutionary dynamics as well as their ecological dynamics. That That spatial spatial positions positions matter matter is is aa trivial trivial observation observation for for biologists biologists working working on plants and other sessile organisms. Thus Harper ((1977) 1 977) entitled one of the Population Biology of Plants as "The effects of neigh neighfive main sections of his Population bours. bours."" It has been less obvious that spatial positions of individuals matter in the case of mobile animals, which may form more or less less random-mating (panmictic) populations. However, from the point of view of of ecological ecological interactions, spatial positions often do matter even in mobile animals. One example example is is the the large large number number of of insect insect species species with with mobile mobile adults adults but but immobile immobile larval stages. Larvae do most of the interactions and so the spatial distribution of larvae matters to single-species single-species (de (de long, 979), competitive of larvae matters greatly greatly to Jong, 11979), competitive ((Hanski, Hanski, 11981, 9 8 1 , 11990a), 990a), and and predator-prey predator-prey dynamics (Hassell, 11978, 978, 2000). 2000). dynamics (Hassell, Indeed, from the 1970s 1 970s onward, indi onward, the spatial aggregation of interacting individuals has been one of the most important important themes in population dynamics. within-population spatial structures also have evolutionary evolutionary These types of within-population ( 1970), Boorman Boorman and and consequences, which which have been investigated by Levins (1970), ( 1 973 ), Cohen and Eshel (1976), ( 1 976) , Wilson Wilson (Wilson, 1980; 1 980; Wilson aI., Levitt (1973), Wilson et al., 1 992; Mitteldorf Mitteldorf and population 1992; and Wilson, 2000), and others. Interestingly, the the population genetic modeling populations initiated by Wright modeling of continuously continuously distributed distributed populations ( 1 940, 1943, 1 943, 1946) 1 946) and and Malecot Malecot (1948) ( 1 94 8 ) faced faced difficulties difficulties precisely precisely because because of of (1940, the spatial spatial aggregation of of individuals (Felsenstein, (Felsenstein, 1975). 1 975). Much Much progress progress the has been made made in this area in the last decade decade using Monte Monte Carlo Carlo simulations, simulations, has autocorrelation methods, methods, and and lattice lattice models (Eppenson and and Allard, Allard, spatial autocorrelation 1989, 1 989, 1993a,b, 1 993a,b, 1995; 1 995; Rousset, Rousset, 2000). 2000). Taking the population population structure structure in in which reproduction reproduction is panmictic panmictic but but Taking ecological interactions interactions are are localized, localized, as described earlier, as the the starting starting point, point, ecological there are are two two ways ways of of moving moving to to the the domain domain of of metapopulation metapopulation dynamics. dynamics. First, First, there widespread dispersal may may not not occur occur in in every every generation, generation, in which which case patches patches widespread of microhabitat microhabitat harbor harbor not not just just single-generation assemblages assemblages of of interacting interacting of individuals, but but multigeneration multigeneration local local populations. populations. Insects Insects living in decaying individuals, wood provide provide good good examples, examples, ranging ranging from from those those that that disperse disperse completely completely in in wood each each generation generation to to species species that that form form local local populations populations in in particular particular (large) (large) trunks for for tens tens or or even even hundreds hundreds of of generations generations (Fig. 1.1). 1 . 1 ) . The The decisive factor factor is trunks
11.. METAPOPULATION METAPOPULATIONBIOLOGY BIOLOGY
5 5
.1 Oak woodland in Sweden population Fig. 11.1 Sweden where a long-term long,term study has examined the meta metapopulation biology of the beetle Osmoderma Osmodermoeremita, eremita, with long-lasting local local populations populations inhabiting individual oak trees trees (Ranius, (Ranius, 2000; Ranius Ranius and Jansson, Jansson, 2000). Photograph by Jonas Jonas Hedin.
simply simply the the longevity longevity of of the the microhabitat microhabitat in in relation relation to to the the life life span span of of individu individuals, underscoring the more generally valid point that that metapopulation dynamics are are typically typically determined determined as as much, much, or or more, more, by by the the structure structure and and dynamics dynamics of of the the physical environment as by the properties of the species. In the population population genet genetics literature, the sort of situation represented by insect populations inhabiting long-lasting microhabitats microhabitats has been examined under under the rubric of the haystack model (Maynard Smith, 11964; 964; Bulmer and Taylor, 11981). 981). The population domain The second second way way of of moving moving ttoo the the meta metapopulation domain from from panmictic panmictic local populations is simply by expanding the spatial scale: most organisms have limited dispersal powers, hence there is a spatial scale at which most inter interactions, including mating, occur "within populations," populations," whereas at larger spatial scales, scales, these these local local populations populations are are connected connected by by migration migration and and gene gene flow. flow. It It is is especially population approach especially natural natural to to turn turn to to the the meta metapopulation approach if if the the environment environment is is physically fragmented into pieces of habitat habitat that that may support support local populations. Metapopulation Metapopulation biology recognizes that that many, if not not most, ecological, genetic, and evolutionary processes occur at spatial scales that that are greater greater than than the scale within within which which most individuals disperse. Hence there is spatial structure at the meta population scale to still metapopulation scale that that should should not not be be ignored. ignored. Moving Moving to still larger larger spatial spatial scales, to the geographical ranges of species, brings in other processes that that are beyond population concept beyond the the meta metapopulation concept and and domain. domain. We population processes We emphasize emphasize the the significance significance of of meta metapopulation processes rather rather than than spatial structures. It is tempting to attempt attempt to classify different kinds of spatial population 994), and some terminology is needed population structures (Harrison, (Harrison, 1991, 11994), for communication, communication, but the danger is that that we impose an order order to nature nature that that is not not there. Landscapes are all different, hence there must be a huge diversity of "metapopulation - migration, gene "metapopulation structures." structures." Focusing Focusing on on the the processes processes--migration, gene
6 6
ILKKA HANSKI OSCAR E. ILKKA HANSKI AND AND OSCAR E. GAGGIOTTI GAGGIOTTI
flow, flow, spatially spatially correlated correlated dynamics, dynamics, local local extinction, extinction, genetic genetic drift, drift, local local adapta adaptation, tion, and and so so forth forth ~ circumvents circumvents the the need need to to infer infer processes processes from from patterns patterns where necessary ((in in many cases there where this this is is not not necessary many cases there is, is, however, however, valuable valuable informa information in patterns that should not ignored; see tion in patterns that should not be be ignored; see Wiegand Wiegand et et aI., al., 2003). 2003). By By emphasizing emphasizing the the metapopulation metapopulation approach, approach, we we also also underscore underscore the the point point that that this .3). this is is only only one one approach approach and and not not always always the the most most appropriate appropriate one one (Section (Section 11.3). There There is is little little doubt doubt that that spatially spatially localized localized interactions interactions and and movements movements influ influence genetic, and majority of ence the the ecological, ecological, genetic, and evolutionary evolutionary dynamics dynamics of of the the vast vast majority of species. particular approach species. It It is is another another question question which which particular approach is is the the most most effective effective in in uncovering the interactions and uncovering the biological biological consequences consequences of of spatially spatially localized localized interactions and movements for research and movements for both both research and management. management.
11.2 .2
METAPOPULATION IN THE METAPOPULATION BIOLOGY: BIOLOGY: PAST PAST TRENDS TRENDS IN THE LITERATURE LITERATURE The population biology The history history of of research research in in meta metapopulation biology has has been been narrated narrated by by Hanski 1 997) and 1999b). Rather Hanski and and Simberloff Simberloff ((1997) and Hanski Hanski ((1999b). Rather than than repeating repeating it it here, examine that here, we we will will examine that history history in in light light of of the the number number of of citations citations to to relevant relevant key key words. words. Such Such aa systematically systematically "documented "documented history" history" of of metapopulation metapopulation biol biology 970s. We ogy goes goes back back to to the the 11970s. We used used the the BIOSIS BIOSIS database, database, which which yielded yielded 1087 1087 citations citations to to the the key key word word metapopulation metapopulationin in the the title title of of aa paper paper or or in in its its abstract abstract (years 970-2001 ) . To (years 11970-2001). To get get aa fair fair idea idea of of the the temporal temporal patterns patterns in in the the number number of of citations, pooled number citations, we we divided divided the the yearly yearly totals totals by by the the pooled number of of citations citations in in the the database total volume volume of database in in that that year, year, aa measure measure of of the the total of the the literature. literature. Thus measured, the number of citations to Thus measured, the number of citations to metapopulation metapopulation has has increased increased more less linearly linearly since 990 (Fig. (Fig. 11.2), .2), with more or or less since 11990 with only only aa few few earlier earlier citations, citations, even even if already in 970 (Levins, if the the metapopulation metapopulation concept concept itself itself was was introduced introduced already in 11970 (Levins, 11970). 970). Some Some inaccuracy inaccuracy is is due due to to less less thorough thorough coverage coverage of of the the literature literature in in the the database database in in the the 1970s 1970s than than later later on, on, but but this this does does not not change change the the broad broad picture. picture. One 20-yr time One can can think think about about several several reasons reasons for for the the 20-yr time lag lag in in the the wider wider use use of of the the metapopulation metapopulation concept, concept, which which is is in in sharp sharp contrast contrast to to the the early early success success of of the island biogeographic 1963, 1967), the island biogeographic theory theory of of MacArthur MacArthur and and Wilson Wilson ((1963, 1967), pub published 1 969, 11970) 970) metapopulation lished only only aa few few years years prior prior to to Levins's Levins's ((1969, metapopulation idea idea and and model 996). First, model (Hanski, (Hanski, 11996). First, MacArthur MacArthur and and Wilson Wilson published published their their theory theory in in leading journal for population population biology and as a high-profile monograph, a leading whereas papers were published in illustrious journals. whereas Levins's Levins's papers were published in less less illustrious journals. Second, Second, MacArthur MacArthur and and Wilson Wilson were were purposely purposely in in the the business business of of turning turning aa page page in in the the history 1 969) immediate immediate goal history of of biogeography, biogeography, whereas whereas Levins's Levins's ((1969) goal was was more more modest, to model to examine alternative alternative strategies eradica modest, to construct construct aa model to examine strategies of of pest pest eradication. Third, Third, MacArthur MacArthur and Wilson were widely respected scientists, whereas Levins was was aa hero hero for for aa more more limited limited number number of of people. people. Fourth, Fourth, and and what what may may Levins be be really really important, important, the the island island theory theory became became associated associated with with the the species-area species-area relationship, enhancing the ecologists could relationship, enhancing the theory'S theory's popularity popularity because because ecologists could use use it it in research (whether made aa lasting in their their research (whether this this application application of of the the theory theory made lasting contri contribution similar opportunity bution is is another another matter). matter). There There was was no no similar opportunity to to do do empirical empirical work linked with models work that that would would be be similarly similarly linked with Levins's Levins's models ~ aa situation situation that that was 990s with was to to change change only only in in the the 11990s with further further development development of of the the theory theory ((Section Section 1.3). awareness of biological conse 1.3). Finally, Finally, the the heightened heightened awareness of the the dire dire biological consequences 980s onward quences of of habitat habitat loss loss and and fragmentation fragmentation from from the the late late 11980s onward has has
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Fig. .2 Number Number of of citations citations in in the the BIOSIS database to the the key words words indicated indicated in in the the panels panels Fig. 11.2 BIOSIS database divided of citations citations in a particular particular year year (to (to control control for divided by by the the total total number number of for the the increasing increasing total total volume of of literature literature over the years). Note Note that that the the scale on the the vertical vertical axis is different different in different different volume rows of panels. See text for discussion. rows of text for
practically forced interest in metapopulation practically forced an an interest metapopulation biology, making making the the rediscovrediscov ery of of Levins's early work work inevitable. inevitable. The top top row row in Fig. 1.2 1 .2 gives the the number of citations to the the key words land The number of citations to words landand island island biogeograph* hiogeograph* as well as to scape ecology ecology and scape to the the key word word metapopulametapopula as tion ( biogeograph" includes all words words starting starting with with "biogeograph," "biogeograph," such such as tion (biogeograph* "biogeography" "biogeography" and and "biogeographic"). "biogeographic"). The The temporal temporal patterns patterns show show intriguing intriguing differences. Landscape ecology ecology was was established established in the the literature literature in the the beginning beginning differences. Landscape of of the the period period considered, considered, in in 1970, 1 970, but but for for the the next next 15 1 5 years years the the frequency frequency of of citations citations remained remained at at a constantly constantly low low level. A distinct distinct growth growth phase phase began began around around 1985, 1 985, and and definitely definitely earlier earlier than than in the the case case of of metapopulation. metapopulation. At At present, metapopulation is cited cited somewhat somewhat more more frequently frequently than than landscape landscape ecolecol present, metapopulation ogy. Island Island biogeograph* hiogeograph * has has appeared appeared in in the the literature literature since since the the mid-1970s mid-1 970s and and ogy. the the frequency frequency has has remained remained high high until until the the present, present, with with ups ups and and downs. downs. Perhaps Perhaps
8 8
ILKKA AND OSCAR OSCAR E. ILKKA HANSKI HANSK! AND E. GAGGIOTII GAGGIOTTI
surprisingly, surprisingly, the the standardized standardized number number of of citations citations to to island island biogeograph'� biogeograph* was was higher in in 2001 2001 than than ever ever before before m 34 34 years years since since the the classic classic monograph monograph by by higher MacArthur 1967) established MacArthur and and Wilson Wilson ((1967) established the the modern modern era era in in ecological ecological biogeography. biogeography. It It is is noteworthy noteworthy that that the the peaks peaks in in the the time time series series for for landscape landscape 980s, ecology ecology and and island island biogeograph': biogeograph*' agree agree rather rather closely closely since since the the late late 11980s, suggesting suggesting that that many many papers papers refer refer to to both both key key words. words. Next examined combinations including metapopulation Next we we examined combinations of of key key words, words, including metapopulation and something else. second row Fig. 11.2 .2 compares and something else. The The second row in in Fig. compares the the three three subdiscip subdiscip* , and * , genetic * . In cases, the lines lines ecolog ecolog*, genetic*, and evolution evolution*. In all all cases, the first first papers papers were were published 978 and and 1985. Most of these papers listed in in all published in in 11978 1985. Most of these papers were were in in fact fact listed all three Gill's ((1978a,b) 1 978a,b) papers papers on three searches searches and and include include Gill's on the the metapopulation metapopulation ecol ecology ogy of of the the red-spotted red-spotted newt, newt, its its migration migration rate, rate, and and effective effective population population size; size; Couvet 1 985) study Couvet et et al.'s al.'s ((1985) study on on the the population population genetics genetics in in spatially spatially structured structured populations; Fix's ((1985) 1985) theoretical populations; and and Fix's theoretical study study of of the the evolution evolution of of altruism. altruism. Since 990, ecolog has accumulated Since 11990, ecolog** has accumulated many many more more citations citations than than genetic genetic** or or * . The evolution appear to evolution*. The temporal temporal patterns patterns appear to indicate indicate that that while while ecolog ecolog** has has not been growing 994, genetic* not been growing systematically systematically since since 11994, genetic* has has been been growing growing until until the 990s and the late late 11990s and the the number number of of citations citations to to evolution" evolution': appears appears still still to to be be growing. These These trends trends are growing. are consistent consistent with with our our general general perception perception of of shifting shifting research research interests, interests, as as well well as as with with the the change change in in the the contents contents of of the the three three volumes on biology (Gilpin volumes on metapopulation metapopulation biology (Gilpin and and Hanski, Hanski, 1991; 1991; Hanski Hanski and and Gilpin, 11997; 997; present Gilpin, present volume) volume).. A A somewhat somewhat different different interpretation interpretation of of the the fig figures for 997 ures for ecolog* ecolog* associates associates the the peak peak in in the the number number of of citations citations in in 1996-1 1996-1997 to population volume, to the the publication publication of of the the previous previous meta metapopulation volume, which which appeared appeared in in the 996 (Hanski 997). In the year year 11996 (Hanski and and Gilpin, Gilpin, 11997). In any any case, case, it it is is apparent apparent that that the the has increased again since 996. number number of of citations citations to to ecolog': ecolog*' has increased again since 11996. The .2 gives The next next row row in in Fig. Fig. 11.2 gives some some further further comparisons. comparisons. Theory Theory has has main maintained tained its its position position well well over over the the years years (key (key word word model), model), although although evidence evidence also also indicates indicates that that empirical empirical work work has has been been catching catching up up to to theoretical theoretical studies studies in in recent recent years. years. This This is is shown shown by by aa significant significant declining declining trend trend in in the the ratio ratio of of cita cita990 tions tions to to metapopulation metapopulation + model model over over metapopulation metapopulation (yearly (yearly counts counts for for 11990 until 990 count until 2001, 2001, the the 11990 count also also including including all all the the previous previous papers; papers; linear linear regres regression, F course, many sion, F= = 7.76, 7.76, P P = 0.02). 0.02). Of Of course, many of of the the papers papers referring referring to to model model might might not not be be theoretical theoretical papers, papers, and and part part of of the the continuing continuing increase increase in in model model papers population stud papers is is due due to to an an increase increase in in genetic genetic and and evolutionary evolutionary meta metapopulation studies. Conservation ies. Conservation combined combined with with metapopulation metapopulation has has increased increased steadily steadily for for the the past 995-1996, paralleling past decade, decade, with with the the exception exception of of aa striking striking peak peak in in 11995-1996, paralleling (although peak for (although not not exactly exactly matching) matching) the the corresponding corresponding peak for ecolog ecolog*. The very very " . The low frequency frequency of of citations citations to to metapopulation metapopulation + landscape landscape ecology ecology is is not not sur surlow prising disciplines that prising in in the the light light of of the the continuing continuing separation separation of of these these two two disciplines that seemingly (more about seemingly have have so so much much in in common common (more about this this in in the the next next section). section). Let Let us number of citations to us hope hope that that the the relatively relatively large large number of citations to metapopulation metapopulation + + land landscape scape ecology ecology scored scored for for 2001 2001 represents represents the the beginning beginning of of aa new new era! era! Finally, .2 examines Finally, the the last last row row in in Fig. Fig. 11.2 examines three three taxa, taxa, plants, plants, fishes, fishes, and and but butterflies, terflies, all all of of which which show show the the same same increasing increasing trend trend as as metapopulation metapopulation itself. itself. The number of citations to The pooled pooled number of citations to metapopulation metapopulation + "taxon" "taxon" for for the the years years 1996 22; mammal, 1996 to to 2000 2000 is is as as follows follows for for the the following following taxa: taxa: bird, bird, 22; mammal, 85; 85; fish, fish, 38; butterfly, 38; butterfly, 49; 49; and and plant, plant, 94. 94. These These overall overall figures figures are are somewhat somewhat misleading, misleading, however. however. For For instance, instance, there there are are many many more more "hard "hard core" core" metapopulation metapopulation
11.. METAPOPULATION METAPOPULATION BIOLOGY BIOLOGY
9 9
papers on papers on butterflies butterflies than than on on birds birds and and mammals, mammals, undoubtedly undoubtedly because because the the metapopulation many butterflies metapopulation approach approach is particularly particularly applicable applicable to many butterflies (Chapter 20; Hanski, Hanski, 11999, 999, Ehrlich Hanski, 2004). (Chapter 20; Ehrlich and and Hanski, 2004). This This is is also also reflected reflected in in the database for .2. the type type of of the the very very first first papers papers in in the the database for the the taxa taxa shown shown in in Fig. Fig. 11.2. For 1 988) on For butterflies butterflies the the pioneering pioneering study study is is Harrison Harrison et et al. al. ((1988) on the the mainland mainlandisland population structure island meta metapopulation structure in in the the Bay Bay checkerspot checkerspot butterfly butterfly (Euphydryas (Euphydryas editha) in California, fishes and and plants editha) in California, whereas whereas for for fishes plants the the first first papers papers are, are, respect respectively, Hanzelova and 1 992) essentially ively, Hanzelova and Spakulova's Spakulova's ((1992) essentially biometric biometric study study and and Ellstrand et al.'s 1 984) notion Ellstrand et al.'s ((1984) notion of of an an inflorescence inflorescence as as aa metapopulation. metapopulation.
11.3 .3
AN OVERVIEW OVERVIEW OF CURRENT RESEARCH RESEARCH AN OF CURRENT This This section section outlines outlines some some noteworthy noteworthy recent recent developments developments in in metapopula metapopulation tion ecology, genetics, and and evolutionary evolutionary studies studies as well as their their integration. integration. This This section section refers refers extensively extensively to to the the remaining remaining chapters chapters in in this this volume. volume. Although Although the motivation motivation for for research typically stems from past past scientific dis discoveries coveries and and perceived perceived opportunities opportunities for for further further discoveries, discoveries, the the ongoing ongoing loss, loss, alteration, alteration, and and fragmentation fragmentation of of natural natural habitats habitats are are widely widely viewed viewed as as other other important reasons for population biology. important reasons for conducting conducting research research in in meta metapopulation biology.
Ecology Ecology The population approach The meta metapopulation approach is is conceptually conceptually closely closely related related to to the the dynamic dynamic theory island biogeography 1967). Most theory of of island biogeography of of MacArthur MacArthur and and Wilson Wilson ((1967). Most import importantly, islands antly, both both theories theories advocate advocate the the same same "island "island perspective," perspective," whether whether the the islands habitat islands, and both both theories are concerned with local are true islands or habitat extinctions extinctions and and recolonizations, recolonizations, although although this this is is not not an an exclusive exclusive interest interest in in meta population biology, metapopulation biology, as as pointed pointed out out earlier. earlier. The The apparent apparent difference difference in in the the focus population theories focus of of the the island island theory theory on on communities communities and and of of meta metapopulation theories on on single assumes independindepend single species species is is not not aa fundamental fundamental difference, difference, as as long long as as one one assumes ent (as the ent dynamics dynamics in in the the species species that that comprise comprise the the community community (as the basic basic island island model model does). does). The The similarity similarity between between the the island island biogeographic biogeographic model model and and the the classic population model pop classic meta metapopulation model is is underscored underscored by by the the spatially spatially realistic realistic meta metapopulation a; Hanski ulation theory theory (Hanski, (Hanski, 2001 2001a; Hanski and and Ovaskainen, Ovaskainen, 2003; 2003; Chapter Chapter 4; 4; see see later), later), which which adds adds the the effects effects of of habitat habitat patch patch area area and and isolation isolation on on extinctions extinctions and classic meta population theory. and colonizations colonizations into into the the classic metapopulation theory. In In fact, fact, we we can can now now see population model see that that Levins's Levins's meta metapopulation model and and MacArthur MacArthur and and Wilson's Wilson's island island two special cases of a more comprehensive comprehensive model (Hanski, 2001a). model are two One One advantage advantage of of the the metapopulation metapopulation theory theory over over the the island island theory theory is is that that the the former former but not the latter allows each species to have its own patch network network in the the same same landscape, landscape, reflecting reflecting differences differences in in the the habitat habitat selection selection of of the the species. species. In any case, it is intriguing that population theory that the island theory theory and meta metapopulation have been widely have been widely considered considered as as representing representing two two different different paradigms paradigms in in conser conservation (see discussion 997). vation biology biology (see discussion in in Hanski Hanski and and Simberloff, Simberloff, 11997). The island theory and and metapopulation metapopulation theory are are not not the only approaches approaches to spatial ecology. . 3 gives simple classification spatial ecology. Figure Figure 11.3 gives aa simple classification of of three three main main approaches. approaches. The The key key issue issue is is what what is is assumed assumed about about the the structure structure of of the the environment. environment. In In one one extreme, extreme, labeled labeled as as the the theoretical theoretical ecology ecology approach, approach, the the
1100
ILKKA OSCAR E. ILKKA HANSKI HANSKI AND OSCAR E. GAGGIOTTI GAGGIOTTI Metapopulation ecology
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common assumption assumption is that the environment is completely homogeneous. Here the primary aim of research is to elucidate the consequences consequences of spatially restricted interactions and/or and/or migration of individuals to the dynamics and spatial structures of populations. Chapter 3 describes at length this approach approach to spatial ecology. The mathematical tools commonly employed include lattice latticebased models, such as interacting particle systems, cellular automata automata and coupled-map coupled-map lattices, spatial moment equations, and partial differential equa equations, as well as simulations. Recent work on "neutral" "neutral" theories of community structure ((Bell, Bell, 2000; Hubbell, 200 1 ) also fit in this category, although these 2001) Chapter 6). The models deal with evolutionary as well as ecological dynamics ((Chapter assumption assumption of homogeneous space facilitates the study of population population processes as opposed to the heterogeneous heterogeneous landscape in creating and maintaining spatial variation in population densities, but this assumption assumption also practically Chapter 3, eliminates the possibility of testing model predictions. As suggested in Chapter the models studied by theoretical ecologists are strategic models designed to investigate general principles rather than tactical models designed to answer specific questions about about specific populations. populations. Nonetheless, even the general theory has to be related to the real world. It is hence important important that that recent modeling studies in this framework framework have attempted to relax the assumption of homogeneous homogeneous space. For instance, Murrell and Law (2000) have used the method method of moments to model the dynamics of carabid beetles in heterogeneous
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landscapes with three different classes of land type, woodland, woodland, agricultural land, and urban areas, and Keeling (2000b) has applied the method of moments to single-species and predator-prey predator-prey dynamics in coupled local populations in a meta population (for further metapopulation further discussion, see Chapter 33).) . IInn the other extreme depicted iinn Fig. 11.3, .3, which iiss represented bbyy much of landscape ecology, the starting point is just the opposite, a detailed description of of the the often often complex structure structure of of real real landscapes. landscapes. Chapter Chapter 22 presents presents an an overview of landscape ecology as far as it is concerned with population processes. processes. Given Given the the complex complex description description of of the the landscape landscape structure structure and and the the emphasis on individual movements in much of landscape ecology (Schippers et aI., 996; Pither and Taylor, 11998; 998; Haddad, 11999a; 999a; Bunn et aI., al., 11996; al., 2000; Jop.sen Jopsen and Taylor, 2000; Byers, 200 1 ), it is not surprising that 2001), that the prevalent modeling tool has been individual-based simulation (With and Crist, 11995; 995; With With and King, 11999b; 999b; Hill and Caswell, 11999; 999; Fahrig, 2002; Chapter .3, Chapter 2). As seen from Fig. 11.3, we we view view the the metapopulation metapopulation approach approach as as occupying occupying the the middle middle ground ground in in this this classification: the environment is assumed to consist of discrete patches of suit suitable habitat for the focal species, usually ignoring the shape of these patches, sur surrounded by the landscape matrix that that is not suitable for reproduction reproduction but through through which individuals may migrate. These assumptions can be somewhat somewhat relaxed population the relaxed without without compromising compromising the the possibility possibility of of developing developing meta metapopulation theory. For example, one may allow for matrix heterogeneity by calculating effec effective patch connectivities, and one may replace real patch areas by effective areas allowing for spatial variation in habitat patch quality (Moilanen and Hanski, 11998; 998; Hanski, 11999b). 999b). What What still remains intact is the core assumption of dis discrete local populations inhabiting discrete patches of habitat. metapopulation perIn terms of theory in meta population ecology, our admittedly partial per spective spective inclines inclines us us to to emphasize emphasize the the significance significance of of the the spatially spatially realistic realistic metapopulation metapopulation theory theory (SMT). The core mathematical mathematical models in this theory are are stochastic stochastic patch patch occupancy occupancy models models (SPOM). (SPOM). SPOMs assume assume aa network network of of habitat habitat patches, which which have only two two possible states, occupied by the focal species or empty. If there are n n patches in the network, the metapopulation metapopulation has 2 2 nn possible states, which is such a large number for large n n that that a rigorous mathematical analysis is not not possible and some simplification is called for. One simplification is to assume a homogeneous homogeneous SPOM, SPOM, with identical habitat habitat patches, which allows a rigorous analysis of even the stochastic stochastic model (this is the the familiar familiar "island "island model"). model"). Another Another simplification simplification is is to to resort resort to to determin deterministic models that that ignore spatial correlations in the pattern pattern of patch patch occupancy and variability due to a finite number number of patches in the network. network. The Levins model model makes makes both both simplifying simplifying assumptions assumptions at at the the same same time time ~ it it is is aa deter deterministic approximation approximation of a homogeneous homogeneous SPOM. What What we now now know know is that that rigorous just one rigorous theory theory can can be be constructed constructed by by making making just one of of the the simplifying simplifying assumptions. SMT is obtained by combining a heterogeneous SPOM, in which heterogeneous SPOM, patches patches have have different different extinction extinction and and colonization colonization probabilities, probabilities, with with assump assumptions tions as as to to how how the the structure structure of of the the landscape landscape influences influences these these probabilities probabilities (or (or rates in the case of continuous-time continuous-time models). Chapter 4 describes SPOMs and the theory in the spatially spatially realistic realistic metapopulation metapopulation theory in detail. detail. The population theory The spatially spatially realistic realistic meta metapopulation theory makes makes aa contribution contribution toward toward aa unification of research in population population biology in several fronts (Hanski and Ovaskainen, 2003). First, as already pointed out, the island theory and the
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classic metapopulation a). metapopulation theory are two two special cases of SMT (Hanski, 2001 2001a). Second, SMT contributes to to the unification of metapopulation metapopulation ecology and landscape landscape ecology with with its explicit focus on the influence of the structural fea features tures of of the the landscape on on population processes. processes. As As Chapter Chapter 2 2 shows, shows, some some land landscape scape ecologists ecologists have have worked worked toward toward the the same same goal goal from from their their own own tradition. tradition. These developments suggest that the merging of the two fields of metapopulametapopula Third, SMT tion ecology and landscape ecology is finally starting to take place. Third, is mathematically closely related to matrix population models (Caswell, 1) (Caswell, 200 2001) for age-structured and size-structured populations, which have also been employed in the study of source-sink metapopulations Chapter 16). Fourth, metapopulations ((Chapter SMT shares common theoretical underpinnings with with epidemiological theory (Grenfell and Harwood, 997; Ovaskainen and Grenfell, 2003). Fifth, a great Harwood, 11997; advantage of the models stemming from from SMT is that that they can be parameterized rigorously with data on the dynamics and pattern pattern of habitat patch occupancy. Chapter 5 presents a review of the methods of parameter estimation and how the models can be applied to real metapopulations. metapopulations. Chapter 22 employs SMT to combine spatial dynamics with reserve site selection algorithms to incorporate the concept of population persistence into reserve selection procedures. The close linking of theory to empirical research that SMT facilitates is somewhat analogous to the link between the dynamic theory of island biogeography and empirical research on the species-area relationship in the 1970s. The difference, however, is that population models can be parameterized rigorously with that meta metapopulation with empirical data, whereas just documenting the species-area relationship is not sufficient to parameterize, nor to test, the island biogeographic model. The rea reason for the success of the meta population models in this respect is that metapopulation that they are typically applied to meta populations with metapopulations with many and often small local popula populations with with a measurable rate of population turnover. Data available hence relate to spatial dynamics as well as to the consequent spatial patterns patterns of habitat habitat occu occupancy. This is in contrast with with past research on the island theory and species-area relationship, which was largely restricted, due to a low rate of popu population turnover on large islands, to analyses of spatial patterns rather than of processes. Spatially realistic meta population theory is focused on the actual spatial metapopulation structure of metapopulations, metapopulations, in the sense of specifying the probabilities with which particular habitat patches in a fragmented landscape are occupied. Another class of structured meta population models considers the distribution metapopulation of local population sizes but ignores the actual spatial structure by assuming that all local populations are equally equaUy connected (Hanski, 11985; 985; Hastings and Wolin, 11989; 989; Hastings, 1991; GyUenberg 992; GyUenberg Gyllenberg and Hanski, 11992; Gyllenberg et aI., al., 11997). 997). These models are particularly concerned with the the influence of emigra emigration and immigration on local dynamics in the meta population context and are, metapopulation in this respect, akin to source-sink models ((Chapter Chapter 116). 6) . The aforementioned modeling studies assume an infinite number of local populations with deter deterministic local dynamics. Lande et al. ((1998)developed 1 998) developed another class of models structured by local population populations with stochastic population size for finite meta metapopulations local dynamics. The most interesting new phenomenon phenomenon predicted by population size-structured meta population models is the possibility of alternative stable metapopulation equilibria in meta population size, one of which corresponds to meta population metapopulation metapopulation extinction, the other one to a positive and possibly large meta population size metapopulation
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(Hanski, 985; Gyllenberg 992; Hanski 993). (Hanski, 11985; Gyllenberg and and Hanski, Hanski, 11992; Hanski and and Gyllenberg, Gyllenberg, 11993). The The processes processes that that lead lead to to alternative alternative stable stable equilibria equilibria are are the the rescue rescue effect, effect, extinction due to immigration, immigration, and the Allee effect, which reduced rate of local extinction increases per capita increases the the rate rate of of successful successful colonization colonization per capita with with increasing increasing immi immigration gration rate. rate. These These processes processes can can be be added added to to SPOMs SPOMs only only in in aa nonmechanis nonmechanistic manner Hanski, 2001 manner (Ovaskainen and Hanski, 2001),) , as SPOMs are concerned concerned with habitat habitat patch occupancy, not not with numbers numbers of individuals. Another great advantage population models models structured advantage of the meta metapopulation structured by the actual size size of local populations populations is the possibility to extend extend the analysis to evolutionary evolutionary issues. For instance, 1 ), and instance, Ronce Ronce et et al. al. (2000b), (2000b), Metz Metz and and Gyllenberg Gyllenberg (200 (2001), and Gyllenberg Gyllenberg et al. (2002) population size et al. (2002) studied studied the the evolution evolution of of migration migration rate rate with with population sizestructured structured metapopulation metapopulation models models [see [see also Heino Heino and Hanski Hanski (2001 (2001)) for for a spatially model and spatially realistic realistic model and Chapter Chapter 1100 for for comprehensive comprehensive discussion]. discussion]. Ecological population dynamics Ecological models models of of meta metapopulation dynamics tend tend to to make make simple simple assumptions assumptions about about migration. migration. Emigration Emigration is is typically typically assumed assumed to to be be density density independent, and individuals are independent, and migrating migrating individuals are assumed assumed to to follow follow aa correlated correlated random about the random walk walk or or some some less less mechanistic mechanistic simple simple assumption assumption is is made made about the behavior behavior of migrants. Chapter Chapter 1133 presents presents a thorough thorough review of what what is known known about about migration migration at the level of individual individual behavior. Not Not surprisingly, there no strong the simple in most most models. there is is no strong support support for for the simple assumptions assumptions made made in models. contrast, migration migration is seen as a complex complex behavior behavior involving a series of In contrast, decisions decisions that that often often depend depend on on the the state state (condition) (condition) of of individuals individuals and and their their interactions interactions with with other other individuals. individuals. In In particular, particular, migration migration is is often often density density dependent, dependence is dependent, although although both both positive positive and and negative negative density density dependence is commonly reported Chapter 113). 3 ) . Positively commonly reported ((Chapter Positively density-dependent density-dependent emigration emigration and negatively density-dependent density-dependent immigration immigration are expected to enhance enhance the and growth population, increasing growth rate rate of of the the meta metapopulation, increasing the the range range of of conditions conditions under under which Saether et aI., 999). These effects occur which the metapopulation metapopulation is viable ((Saether al., 11999). occur because because the the pattern pattern of of migration migration will will influence influence the the strength strength of of the the rescue rescue effect effect and the probability brief, it is clear that and probability of successful colonization. colonization. In brief, that migrants migrants in behavior than assumed by most in most most species species have have more more sophisticated sophisticated behavior than assumed by most models. would it models. What What is is not not clear, clear, however, however, is is when when would it be be necessary necessary to to (greatly) (greatly) complicate complicate the the models models by by including including many many behavioral behavioral details, details, and and indeed indeed to to what what extent extent should should the the models models be be modified. modified. Turning Turning from from rigorous rigorous mathe mathematical just for matical models models to to simulations simulations just for the the sake sake of of adding adding some some "realism" "realism" is is not not necessarily necessarily warranted. warranted. What What is is needed needed is is aa family family of of models models incorporating incorporating different different amounts amounts of of detail. detail. No No systematic systematic study study of of this this type type has has yet yet been been conducted conducted on migration migration and metapopulation metapopulation dynamics. dynamics. Ecologists Ecologists working working with with population population viability viability analysis analysis tend tend to to prefer prefer individ individual-based (Possingham and Noble, 1991; Ak�akaya Ferson, 11992; 992; Lacy, Akqakaya and Ferson, 11993, 993, 2000; 2000; Ak�akaya, Akqakaya, 2000a) 2000a) or or population-based population-based (Sjogren-Gulve (Sj6gren-Gulve and and Ray, Ray, 11996) 996) simulation these models that any simulation models. models. The The advantage advantage of of these models is is that any processes processes and and mechanisms mechanisms that that the the researcher researcher may may wish wish to to add add to to the the model model can can be be added added readily. readily. The The disadvantage disadvantage is is that that general general insights insights are are difficult difficult to to extract extract from from complex simulations. Furthermore, it is practically impossible to estimate rigor rigorously ously the the often often large large number number of of parameters parameters and and to to test test the the structural structural model model assumptions; assumptions; the the modeling modeling results results are are thus thus of of questionable questionable value value for for manage management. ment. The The best best use use of of these these models, models, as as perhaps perhaps of of any any population population models, models, for for conservation that differ conservation and and management management is is to to contrast contrast alternative alternative scenarios scenarios that differ in in
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only a small number of factors (Hanski, 11997a; 997a; Ralls and Taylor, 997; Beissinger Taylor, 11997; and Westphal, 11998; 998; Ak�akaya Akqakaya and Sjogren-Gulve, Sj6gren-Gulve, 2000). One would hope that the result of such comparisons is relatively insensitive to the many uncertainties uncertainties in parameter values and even even in the structure of the model itself. itself. The interested interested reader is referred to many chapters in two edited volumes (Sjogren-Gulve (Sj6gren-Gulve and Ebenhard, 2000; Beissinger 1 ) . Our emphasis in this vol Beissinger and McCullough, 200 2001). volume is on SPOMs for the reasons that much progress has been made in recent years in developing both the theory ((Chapter Chapter 4) and applications to real metapopulations ((Chapters Chapters 5, 20, and 22; see also Dreschler aI., 2003). Dreschler et al., To To present present aa balanced balanced view view about about the the standing standing of of the the metapopulation metapopulation approach in ecology, it is appropriate to acknowledge the critical opinions that have been voiced about its general significance. Harrison ((1991, 1 99 1 , 11994; 994; Harrison, 11994; Hastings and Harrison, 994; Harrison and Taylor, 11997; 997; Harrison and Bruna, 11999) 999) has suggested repeatedly that that the occurrence of species "in the balance between the extinction and recolonization of populations is an improbable 994, p. 15 ) . To improbable condition" condition" (Harrison, (Harrison, 11994, p. 1115). To some some extent, extent, Harrison's Harrison's population theory, which concerns are answered by the spatially realistic meta metapopulation relaxes many of the simplifying assumptions of the nonspatial homogeneous patch occupancy models, such as the Levins model, and which shows how realistic variation in habitat patch areas and connectivities can be incorporated into models. Another line of response is provided by the scores of empirical studies that that demonstrate the operation, in practice, of metapopulation dynam dynamics with a frequent turnover that lack large and turnover of local populations in systems that permanent "mainland" "mainland" populations. Chapter 20 assesses the performance of the metapopulations metapopulations approach approach in dynamic (nonequilibrium) landscapes, where Harrison's criticisms initially seem most relevant. In fact, the models perform well in the situations examined and can be used to gain valuable insights about the long-term behavior of metapopulations. metapopulations. Research on European butterflies, in particular, has produced much empir empirical evidence for metapopulation processes in shaping not only the ecologiecologi cal dynamics (Thomas, 1994b; Thomas and Hanski, 11997; 997; Hanski, 11999b; 999b; CD. Saccheri et aI., 998; Nieminen C.D. Thomas et aI., al., 2002), but but also genetic ((Saccheri al., 11998; et al., aI., 2001; Scmitt and Seitz, 2002) and evolutionary dynamics (Kuussaari et aI., al., 1 ; Heino and Hanski, 2001 2000; Hanski and Singer, Singer, 200 2001; 2001;; Thomas et aI., al., 2001; al., 2002) of butterflies. Chapter 20 in this volume and a volume 200 1 ; Hill et aI., checkerspot on the biology of checkers pot butterflies (Ehrlich and Hanski, 2004) present two overviews covering much of this research. Butterflies possess several traits that make them a convenient model group of species for metapopulation research: specific host plant and habitat requirements, meaning that that many landscapes are highly fragmented for butterflies; small body size size allowing the presence of local breeding populations in relatively small habitat patches; and high population growth rate but also great sensitivity to environmental condi conditions, leading to high population turnover 990). Additional turnover (Murphy et aI., al., 11990). advantages that that butterflies offer include the facility of estimating population sizes and migration rates with mark-release-recapture mark-release-recapture methods methods and the often great between the and the matrix. It may great distinction distinction between the suitable suitable habitat habitat and the landscape landscape matrix. It may remain a matter of opinion as to how representative, and representative of what, the many butterfly studies are, but minimally we expect that butterflies fairly represent a large number of specialized insect species.
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Chapters 8 , 119, 9, and 21 discuss plant, plant-pathogen, Chapters 118, plant-pathogen, and small mammal metapopulation metapopulation dynamics, respectively. The metapopulation metapopulation dynamics of many plants are influenced by the seed bank and very long-lived adult indi individuals, which complicate empirical empirical studies studies greatly but also mean that that certain phenomena, phenomena, such as long transients in the dynamics of species in changing environments, Chapter 118). 8 ) . The basic issue environments, are especially important important in plants ((Chapter habitat is often often very difficult in the case of of delimiting suitable but occupied habitat plants because plant species typically compete for space and hence a single singlespecies approach approach is likely to be inadequate. Chapter Chapter 1199 on plant-pathogen plant-pathogen metapopulation metapopulation dynamics is focused on a two-species interaction, interaction, which add additionally involves a coupling of ecological and evolutionary dynamics that that may be responsible population sizes. Metapopulation responsible for long-term trends trends in meta metapopulation Metapopulation dynamics in small mammals ((Chapter Chapter 2 1 ) may also often 21) often involve more than than one species, for instance, a specialist predator predator driving some of the population population turnover turnover in the prey species, potentially potentially leading to spatially correlated correlated patterns patterns of habitat occupancy. More More generally, there is a clear need for more studies on meta communities metacommunities m assemblages of interacting metapopulations. metapopulations. Chapter 6 reviews the conceptual conceptual framework framework and current research on metacommunities. metacommunities. Not Not surprisingly, webs of direct direct and indirect indirect interactions interactions in communities, com combined with with webs of spatially connected connected populations, populations, complicate complicate matters greatly, and we may need several different theoretical frameworks frameworks to cover the full range of possibilities that that arise in metacommunity metacommunity dynamics. Returning Returning to the criticism against the general significance of the metapopu metapopuapproach, Fahrig ((1997, 2001,, 2002) has suggested repeatedly lation approach, 1 997, 11998, 998, 2001 that the persistence of species in (increasingly) fragmented landscapes is little affected by habitat fragmentation as such, but rather what what matters is the total area of the (remaining) habitat. In other words, in Fahrig's opinion, opinion, the spatial configuration of the habitat makes little difference. If this were generally the case, much of the contents contents of this volume would would be superfluous. superfluous. We, however, consider that that Fahrig's conclusions are too far-fetched. Considering the plane depicted .4, defined depicted in in Fig. Fig. 11.4, defined by by the the proportion proportion of of the the suitable suitable habitat habitat in in the the landscape landscape and the migration range of the focal species, habitat fragmentation may indeed be of little significance in most parts of this plane. However, a huge number of species/landscape combinations combinations crowd the lower-left corner of Fig. 11.4: .4: highly fragmented landscapes, landscapes, in which which only a small fraction of the total total area is covered covered by the suitable habitat; habitat; and relatively poorly dispersing species at the scale of interest. Furthermore, Furthermore, as we all know, know, human-caused human-caused habitat loss and fragmentation continuously push further combinations combinations of species and landscapes .4, where landscapes to this corner corner in Fig. 11.4, where the spatial configura configuration of the remaining habitat should not be ignored. The metapopulation metapopulation theory ((Chapter Chapter 4) is helpful in delineating the parts of the shaded square in Fig. 11.4 .4 that allow long-term metapopulation metapopulation persistence from those parts that that lead to meta population extinction. metapopulation
Genetics Genetics Metapopulation Metapopulation genetic studies have their roots in Sewall Wright's island model of population 93 1 ), which assumes distinct local population structure structure (Wright, 11931), populations (colonies, demes) connected by migration and gene flow. In this
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0/
Total amount of habitat in the landscape
Fig. 11.4 .4 Habitat fragmentation (the spatial configuration Fig. fragmentation (the configuration of the remaining remaining habitat) habitat) matters on the the abundance in landscapes where the suitable habitat on abundance and persistence persistence of species in landscapes where habitat covers only a small fraction fraction of the total landscape area and and the migration the focal species species is migration range of the limited.
classic model, model, all all local populations populations are identical identical (same (same size) classic size) and equally connected (constant migration rate), which are also the assumptions assumptions of the ecological Levins model (a (a deterministic approximation approximation of a homogeneous homogeneous SPOM, see SPOM, see earlier earlier discussion) discussion).. However, However, while while the the latter latter was was focused focused on on popu population turnover, Wright's island model assumed permanent permanent local populations. populations. The The first first formal formal application application of of the the classic classic ecological ecological metapopulation metapopulation concept concept in the domain domain of population population genetics was a generalization of the island model to to cover cover the the case case where where local local populations populations would would go go extinct extinct and and new new ones ones were were established Slatkin, 11977). 977). The ideas to established ((Slatkin, The extension extension of of these these ideas to stepping-stone stepping-stone models followed shortly afterward (Maruyama and Kimura, 1980). The pion pioneering 1 977) was eering work work of of Slatkin Slatkin ((1977) was followed followed by by studies studies by by Wade Wade and and McCauley 1 9 8 8 ) and McCauley ((1990). 1 990). The McCauley ((1988) and Whitlock Whitlock and and McCauley The aim aim of of all all these these investigations was extinctions and investigations was to to clarify clarify the the effects effects that that extinctions and recolonizations recolonizations have populations, that have on on the the genetic genetic structure structure of of meta metapopulations, that is, is, the the partitioning partitioning of of genetic variability between local like in genetic variability within within and and between local populations. populations. Just Just like in classic classic ecological ecological metapopulation metapopulation models, models, the the effect effect of of local local dynamics dynamics on on genetic genetic structure structure was was ignored ignored to to facilitate facilitate the the study study of of factors factors such such as as the the extinction extinction rate individuals that rate and and the the genetic genetic composition composition of of the the groups groups of of individuals that establish establish new populations. populations. The population genetic The effect effect of of local local dynamics dynamics on on meta metapopulation genetic structure structure has has been been addressed 1992a), Gaggiotti addressed in in aa series series of of papers papers published published by by Whitlock Whitlock ((1992a), Gaggiotti and and Smouse 1 996), Gaggiotti 1 996), and 1 997). These Smouse ((1996), Gaggiotti ((1996), and Ingvarsson Ingvarsson ((1997). These studies studies demonstrate interaction between local dynamics demonstrate that that the the interaction between local dynamics and and migration migration pat patterns consequences for terns can can have have important important consequences for the the genetic genetic structure structure of of metapopu metapopulations. In metapopulations of populations lations. In metapopulations of the the Levins Levins type, type, with with all all local local populations having population size having the the same same carrying carrying capacity, capacity, fluctuations fluctuations in in local local population size and/or and/or migration migration rate rate increase increase genetic genetic differentiation differentiation among among populations populations (Whitlock, (Whitlock, 11992a). 992a). Slow population growth similar effect Slow population growth following following colonization colonization has has aa similar effect when migration rate 997). In when the the migration rate is is constant constant (Ingvarsson, (Ingvarsson, 11997). In the the case case of of
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source-sink populations, the source-sink meta metapopulations, the degree degree of of genetic genetic differentiation differentiation among among sources sources and and sinks, sinks, among among sinks, sinks, and and the the level level of of genetic genetic variability variability maintained maintained by by sink sink populations populations is is largely largely determined determined by by the the variance variance in in propagule propagule size. size. The lower the variance, the higher higher the degree of genetic differentiation differentiation and the lower Gaggiotti lower the the level level of of genetic genetic variability variability maintained maintained by by sink sink populations populations ((Gaggiotti and 996; Gaggiotti, 996). and Smouse, Smouse, 11996; Gaggiotti, 11996). All All the the theoretical theoretical studies studies mentioned mentioned so so far far have have been been concerned concerned with with the the genetic structure neutral genes, most thoroughly genetic structure of of selectively selectively neutral genes, which which is is the the most thoroughly studied population genetics. studied subject subject in in meta metapopulation genetics. Following Following the the publication publication of of comprehensive 1 997) and Rousset ((1999a,b), 1999a,b), comprehensive analyses by Whitlock Whitlock and Barton ((1997) which which extended extended the the results results of of the the previous previous studies studies to to models models that that cover cover aa wide wide variety population scenarios, scenarios, theoretical theoretical research variety of of meta metapopulation research into into the the genetic genetic structure diminished. Presently, structure of of metapopulations metapopulations has has diminished. Presently, the the most most active active area area in in meta population genetics is concerned concerned with selected genes and metapopulation and quantitative quantitative genetic Chapters 77 and genetic variation variation ((Chapters and 9). 9). This This recent recent work work has has added added important important new processes such heterosis, and new processes such as as inbreeding, inbreeding, heterosis, and mutation mutation accumulation accumulation into population approach into the the meta metapopulation approach and and its its application application to to conservation conservation and and management of endangered endangered species (see Chapter Chapter 7). To some extent, extent, further further management advance area is lack of advance in in this this area is hampered hampered by by the the substantial substantial lack of knowledge knowledge that that exists spontaneous mutations exists about about the the rates rates and and effects effects of of spontaneous mutations (discussed (discussed in in Chapter point where Chapter 14). 14). Indeed, Indeed, the the fact fact that that we we have have reached reached aa point where further further progress area of meta population biology progress in in aa specific specific area of metapopulation biology requires requires the the resolution resolution of a fundamental fundamental issue in such an established established discipline as genetics is an indication indication of of how how fast fast the the field field has has progressed. progressed. Studies classic quantitative Studies reviewed reviewed in in Chapter Chapter 99 have have extended extended classic quantitative genetics genetics theory populations. The theory to to meta metapopulations. The classic classic theory theory was was concerned concerned with with measur measuring response to selection and ing the the response to selection and largely largely ignored ignored epistatic epistatic interactions, interactions, whereas population quantitative whereas the the more more recent recent meta metapopulation quantitative genetics genetics theory theory is is concerned concerned with with measuring measuring differentiation differentiation among among populations populations and and empha emphasizes Chapter 99).) . This sizes the the importance importance of of epistatic epistatic interactions interactions ((Chapter This shift shift in in emphasis emphasis has has uncovered uncovered new new mechanisms mechanisms for for speciation speciation and and is is aa good good example populations can example of of how how aa focus focus on on meta metapopulations can shed shed new new light light onto onto key key evolutionary problems. evolutionary problems. Another population biology Another important important recent recent development development in in meta metapopulation biology is is the the extension of the coalescent approach 982a; reviewed by Fu and Li, approach (Kingman, 11982a; 11997) 997) to to cover cover metapopulation metapopulation scenarios scenarios (Wakeley (Wakeley and and Aliacar, Aliacar, 2001 2001).). The The coalescent coalescent approach approach represented represented aa big big leap leap forward forward for for population population genetics genetics because about past because it it provides provides aa theoretical theoretical framework framework to to make make inferences inferences about past events sample representing events based based on on aa genetic genetic sample representing the the present present population. population. The The essence theory is move back essence of of the the coalescent coalescent theory is to to start start with with aa sample sample and and to to move backward ward in in time time to to identify identify events events that that occurred occurred in in the the past past since since the the most most recent recent common common ancestor ancestor of of the the sample. sample. Chapter Chapter 88 provides provides an an overview overview of of the the coa coalescent population context lescent process process in in the the meta metapopulation context and and describes describes ways ways in in which which it it can can be be used used to to make make statistical statistical inferences. inferences. Although Although current current work work in in this this area area is lead to is highly highly theoretical, theoretical, it it will will lead to useful useful applications applications such such as as the the development development of of statistical statistical approaches approaches for for the the analysis analysis of of molecular molecular data data aimed aimed at at making making inferences inferences about about metapopulation metapopulation processes. processes. This This in in turn turn will will facilitate facilitate the the inte integration gration of of theoretical theoretical and and empirical empirical work work as as well well as as the the demographic demographic and and genetic population biology. genetic approaches approaches to to meta metapopulation biology.
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Evolution Evolution Application of the meta population approach metapopulation approach in the domain of evolutionary biology broad issues: issues: the biology has has been been motivated motivated mainly mainly by by three three broad the shifting shifting balance balance theory SBT) (Wright, 9 3 1 , 11940), 940), the theory ((SBT) (Wright, 11931, the evolution evolution of of migration migration rate, rate, and and the the evolution of species' ranges. An important controversy in evolutionary biology e.g., Coyne et aI., 997, 2000; deals with two opposing views of adaptation ((e.g., al., 11997, Wade and Goodnight, 11998; 998; Goodnight and Wade, 2000) 2000).. One view, view, called the Fisherian view, view, advocates that the bulk of adaptive evolution results from Darwinian mass selection. The other view maintains that adaptation cannot be explained and that such as as genetic explained by by selection selection alone alone and that stochastic stochastic processes processes such genetic drift drift often has been been the often play play an an important important role. role. Sewall SewaU Wright Wright has the main main advocate advocate of of this this latter view view and and he shifting balance latter he formalized formalized it it in in his his shifting balance theory. theory. The shifting balance theory is based on the idea that species are subdivided into many local populations (demes) that are weakly connected by migration. The small size allow genetic The small size of of the the local local populations populations would would allow genetic drift drift to to overwhelm overwhelm the effects of natural selection and take the populations to the domain of attraction of new adaptive peaks (phase I). Individual selection could then move the population toward the new peak itself (phase 11), II), at which point selection among the local populations would act to pull the entire species (meta population) toward peak (phase (phase III). (metapopulation) toward the the new new adaptive adaptive peak III). At At the the time time of of the the publication volume (Hanski 997), the publication of of the the predecessor predecessor to to this this volume (Hanski and and Gilpin, Gilpin, 11997), the SBT (Barton and SBT was was imperfectly imperfectly understood understood and and largely largely untested untested (Barton and Whitlock, Whitlock, 11997). 997). However, a large number of theoretical studies have provided new insight into the feasibility of the genetic mechanisms underlying the SBT. SBT. These studies have also uncovered many alternative forms of evolution in "adaptive landscapes" that are theoretically and empirically better supported than the SBT (Whitlock and Phillips, Phillips, 2000). Much of this work was influenced by or even population paradigm. even based based on on the the meta metapopulation paradigm. This This body body of of literature literature and and its its connection 9, 111, 1, and connection to to some some recent recent theories theories are are discussed discussed in in Chapters Chapters 9, and 12. 12. A populations differs A particularly particularly brilliant brilliant example example of of how how evolution evolution in in meta metapopulations differs from large panmictic provided by recent stud from evolution evolution in in large panmictic populations populations is is provided by the the recent studies Chapter 111). 1 ) . IGEs based ies of of indirect indirect genetic genetic effects effects (IGEs, (IGEs, Chapter IGEs are are genetically genetically based environmental influences that are generated whenever the phenotype of one 997). IGEs cre individual acts as an environment for another (Moore et aI., al., 11997). create causal pathways phenotypes of ate causal pathways between between the the genes genes on on individuals individuals and and the the phenotypes of other permitting the other related related or or unrelated unrelated individuals individuals permitting the coevolution coevolution of of phenotype phenotype and Chapter 111). 1 ). Another and context context that that is is unique unique to to metapopulations metapopulations ((Chapter Another import important advance in the evolutionary studies of meta populations is the recently metapopulations developed theory Chapter 12). theory developed theory of of "holey "holey adaptive adaptive landscapes" landscapes" ((Chapter 12). This This theory provides genetically explicit explicit approach the dynamics provides aa genetically approach for for the the study study of of the dynamics of of speciation and diversification in spatially explicit systems. Evolution Evolution of of the the migration migration rate rate is is aa well-studied well-studied topic in in evolutionary evolutionary ecology, metapopulation paradigm paradigm has ecology, but but use use of of the the metapopulation has shed shed new new light light onto onto the the selective pressures created by population turnover (Olivieri and Gouyon, 11997). 997). For 1 0 have shown For example, several several studies studies reviewed reviewed in in Chapter Chapter 10 shown that that under some circumstances, under some circumstances, migration migration is is aa nonmonotonic nonmonotonic function function of of the the extinction extinction rate, rate, with with high high extinction extinction rates rates leading leading to to reduced reduced migration migration propensity, contrary to the prevailing view. now is to find out view. The challenge now
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what 01) what actually actually happens happens iinn real real metapopulations. metapopulations. Heino Heino and and Hanski's Hanski's (20 (2001) modeling study ooff evolution ooff the migration rate iinn checkerspot butterflies demonstrated the possibility of a reduced migration rate with an increasing extinction rate, but they concluded that that this would not occur under conditions met in natural metapopulations of the butterflies. Other studies have made a start in developing a more general framework of life life history evolution in metapopulations, metapopulations, including traits other than migration rate and interactions among different traits. This research is reviewed in Chapter 10. Another Another evolutionary evolutionary problem problem that that has has benefited benefited from from application application of of the the metapopulation approach is the evolution of species' ranges. The basic ques question being populations at tion being asked asked here here is: is: Why Why do do populations at the the range range margin margin not not adapt adapt to their local conditions and then spread outward outward (Kirkpatrick and Barton, 11997)? 997) ? One populations receive One answer answer to to this this question question is is that that peripheral peripheral populations receive migrants migrants from from the the center center of of the the species' species' range. range. These These immigrants immigrants will will be be well well adapted adapted to to the the conditions conditions at at the the range range center center but but not not to to conditions conditions at at the the periphery and, therefore, the genes that that they bring hinder adaptation at the periphery (Mayr, 11963). 963). Thus, peripheral populations are forced into the role of demographic sinks, preventing the range from expanding outward outward ((Kirkpatrick Kirkpatrick and Barton, 11997). 997). An appropriate conceptual framework used to study the interplay between migration and selection in peripheral popula populations is the source-sink metapopulation framework framework (e.g., Holt and Gaines, 11992). 992). The usual approach approach in this context has been to consider the conditions that fit that would allow the increase of a rare allele with antagonistic effects on fitness and Gomulkiewicz, 1 997; Gomulkiewicz ness in in two two habitats habitats (Holt (Holt and Gomulkiewicz, 1997; Gomulkiewicz et et aI., al., 11999; 999; Kawecki, Kawecki, 2000; 2000; Kawecki Kawecki and and Holt, Holt, 2002 2002).) . Use Use of of the the source-sink source-sink meta metapopulation approach has led to an important important general conclusion about sink populations: the parameter that that governs the rate of spread of the beneficial mutation not its relative fitness, as is the mutation is the absolute fitness fitness of the mutant, mutant, not case in Gomulkiewicz, 11997). 997). Use of case in populations populations of of constant constant size size (Holt (Holt and and Gomulkiewicz, Use of the source-sink metapopulation concept has also shed new light on the evolu evolutionary consequences of asymmetric migration in heterogeneous heterogeneous landscapes ((Ronce Ronce and Kirkpatrick, Kirkpatrick, 2001 2001;; Kawecki and Holt, 2002). These studies are described in detail in Chapter 116. 6.
Integration Integration across across Disciplines Disciplines and and Applications Applications A population biology A clear clear indication indication of of the the maturity maturity that that the the field field of of meta metapopulation biology has has reached reached is is the the appearance appearance of of increasing increasing numbers numbers of of studies studies that that attempt attempt to to integrate many or even all of the main subdisciplines covered by the broader field of population biology. The integration of ecology and genetics has been in 93 1 , Sewall in the the minds minds of of population population biologists biologists for for aa long long time. time. As As early early as as 11931, Sewall Wright attempted the integration of ecological and population genetic processes processes through through his his shifting shifting balance balance theory, theory, as as described described earlier, earlier, with with the the aim aim of demonstrating that that evolution could proceed rapidly in spatially structured populations. years that most of both populations. In In the the years that followed, followed, most of the the work work that that included included both ecological and genetic considerations was empirical and did not explicitly attempt such integration. However, the importance importance of such integration was widely paper published published in in 11960 960 by widely accepted accepted as as attested attested by by the the conceptual conceptual paper by
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L. C. Birch. Although using slightly different terms, Birch ((1960) 1 960) referred to many of the problems that are the current focus of meta population biology, metapopulation such as feedback between population dynamics and genetic variability, the importance of sink populations, and so forth, and provided numerous refer references to the empirical work available at that time. As an aside, it is worth not notcontribution to early development of ing that L. C. Birch also made a lasting contribution the ecological metapopulation ideas by his textbook with H. G. Andrewartha (Andrewartha and Birch, 11954). 954). The first step toward toward a more formal integra integraMacArthur ((1962), anation of the two disciplines can be traced back to MacArthur 1 962), who ana lyzed a selection model in which population population regulation plays a central role. Subsequent studies continued continued to explore the way in which which population dynam dynamics affects natural selection (e.g., Anderson, 1971; Asmussen, 11979, 979, 11983a,b), 983a,b), but left most other questions unexplored. but New New impetus for the integration of the two two disciplines came with the realization that human impact is the primary cause of species extinctions in many landscapes and that that extinctions are taking place at an alarming rate. Just Just known about about the interaction of demographic, over a decade ago, little was known 1 988) urged population ecologic, and genetic factors factors in extinction, and Lande ((1988) population biologists to address this fundamental fundamental but difficult problem. Much Much progress contribution, and despite its short history, has occurred since Lande's key contribution, metapopulation metapopulation biology has facilitated substantial progress in this area. Chapters population Chapters 13 to 16 cover many of the key contributions contributions of meta metapopulation biology toward toward the integration integration of population population biology. For this integration to be truly successful, we need to extend it also to the domain domain of empirical research. Current Current developments in the field of statistical genetics provide new tools that Of particular develthat will help accomplish accomplish this goal. Of particular importance importance is the devel opment multilocus genotype to make inferences about about opment of of powerful powerful multilocus genotype methods methods to the origin (natal populations) migrating individuals populations) of migrating individuals (e.g., Smouse et al., aI., 1 990a; Rannala Rannala and Mountain, 1997; 1 997; Pritchard Pritchard et al., aI., 2000; Dawson and and 1990a; and Mountain, 2000; Dawson 200 1 ) . These methods, methods, when when implemented implemented under under the hierarchical Belkhir, 2001). framework, can be used to combine genetic, demographic, and Bayesian framework, demographic, and environmental data model (e.g., (e.g., Gaggiotti aI., 2002). 2002). environmental data in a single statistical model Gaggiotti et al., turn, provides a way way of testing hypotheses about This approach, approach, in turn, about the demo demographic and and environmental factors that control graphic factors that control metapopulation metapopulation processes. These very recent recent developments developments are covered in Chapter Chapter 15. There There are already good examples examples of studies studies that that have have employed employed the the metapopulation metapopulation approach approach good to integrate integrate ecology, ecology, genetics, and/or and/or evolution, evolution, including including studies studies on on to host-pathogen interaction interaction (Chapter ( Chapter 19), 1 9 ), butterflies butterflies (Chapter ( Chapter 20), 20), and and small small host-pathogen mammals (Chapter mammals ( Chapter 21). 21). mentioned earlier, the renewed renewed interest interest in the the metapopulation metapopulation concept concept As mentioned was was fostered fostered by its potential potential application application to to the the field of of conservation conservation biology, and it it is now now clear clear that that the the initial expectations expectations were well founded. founded. The The design of of reserve networks networks (Chapter ( Chapter 22) is a good good example of of a problem problem that that needs to be addressed addressed using the the metapopulation metapopulation approach. approach. Another Another important important to example is the the extension extension of of population population viability analysis (PVA) to to fragmented fragmented populations. In In the the past, past, most most PVA PYA methodologies methodologies either either took took no no account account of of populations. spatial structure structure or or did did so in in ways ways that that have have unrealistic unrealistic data data requirements. requirements. spatial Chapter 23 23 presents presents a practical practical approach approach that that considers considers spatial spatial population population Chapter structure and and can can be be parameterized parameterized using using available available data. data. This This chapter chapter structure
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describes how such a model can be used in the management management of endangered species Columbia basin species using using the the contentious contentious Columbia basin salmon salmon stocks stocks as as an an example. example. Practical applications of the meta population concept metapopulation concept have gone beyond the domain domain of of conservation conservation biology biology and and now now include include epidemiological epidemiological studies studies of of infectious infectious diseases diseases in in humans humans and and domestic domestic animals. animals. Chapter Chapter 1177 explores explores how how meta population theory metapopulation theory at a variety of scales can help understand understand epidemio epidemiological dynamics and how how this newly gained insight can be used in the design of efficient vaccination vaccination programs. programs.
11.4 .4
CHALLENGES FOR THE CHALLENGES FOR THE NEAR NEAR FUTURE FUTURE It is generally difficult ((and and often unnecessary) to try try to predict the the course that research in a particular field will take even over a short short period of a few that years. A truly novel discovery may radically change the way we think about about a particular particular issue; new modeling tools are introduced, introduced, allowing researchers to tackle previously could cumbersome simu tackle questions questions that that previously could be be studied studied only only via via cumbersome simulations; and new methods of field study may open up possibilities that that we could only dream about about in the past. One good example is the study of migra migration and gene flow, which which has benefited greatly from from new statistical models of both both demographic and genetic data data and the combination combination of the two, two, as well as of the the high-resolution genetic markers that that have become recently available. Chapters 15 and 21 illustrate Chapters 15 and 21 illustrate the the power power of of these these new new tools. tools. We anticipate that that the integration of ecological, genetic, and evolutionary studies studies will will continue continue in in the the near near future. future. Metapopulation Metapopulation biology biology is is well well placed placed to make ground-breaking contributions contributions here. Theoretical challenges start from the need need to to combine combine currently currently distinct distinct ecological ecological modeling modeling approaches, approaches, such as as stochastic patch occupancy models, spatial moment moment equations, and metapopu metapopulation models structured by local population population size. Adding realistic description of landscape structure structure into genetic and evolutionary models is another another chal challenge. The new statistical methods that that integrate genetic, demographic, and environmental data ((Chapter Chapter 115) 5 ) offer a route to merging ecology and genetics but also the possibility of linking theory ever more closely with empirical research. Few of these methods are currently widely available, but we expect that many will be developed further in the near feature. feature. Somewhat Somewhat more specific specific research research tasks tasks include include the the need need to to better better understand understand the the interactive interactive effects spatial structure effects of of populations' populations' age/stage-structure age/stage-structure and and their their spatial structure on on the the maintenance clines, inbreeding maintenance of of genetic genetic variability, variability, genetic genetic clines, inbreeding depression, depression, and and so so forth (Mills and Smouse, 11994; 994; Gaggiotti et al., aI., 11997; 997; Gaggiotti and Vetter, 11999). 999). To what extent can the meta population approach metapopulation approach be developed to address such large-scale issues as determination of species' range boundaries and their responses to global changes ((Chapter Chapter 20), and indeed the global extinction risk of species? We have already commented on the relative lack of studies on metacommunities. metacommunities. population context is needed Combining ecology and genetics in the meta metapopulation for conservation and epidemiology. Chapter Chapter 22 takes an important important step forward forward in adding spatial dynamics to existing reserve site selection procedures. procedures. We imagine that that including genetics in the same package package would would be worth population dynamics worth the effort. Research on plant-pathogen plant-pathogen meta metapopulation
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((Chapter Chapter 119) 9 ) shows the way forward for epidemiology ((Chapter Chapter 117) 7) in includ includframework. Finally, ing genetic and evolutionary issues into the demographic framework. ever since Lande's ((1988) 19 8 8 ) key contribution, ecology and genetics have been integral parts of conservation biology. Opinions have shifted over the years on their relative importance ((Chapter Chapter 115). 5 ) . The coming years may demonstrate that asking about the "relative importance" has been a somewhat misleading (although necessary) question, as often the real question is about interactions. That That being being said, said, we we should should not not lose lose perspective perspective on on the the kinds kinds of of threats threats that that operate at present, of which habitat loss and fragmentation are the most important ones. The immediate adverse effects of habitat loss and fragmenta fragmentametapopulation are largely ecological, and it remains a major challenge for metapopula tion biologists to develop predictive models and robust understanding of this key issue to be able to provide solid scientific advice to the society.
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META PO PULATION M ETAPO PU LATIO N DYNAMICS DYNAMICS:: PERS PECTIVES FROM PERSPECTIVES FROM LANDS CA PE ECOLOGY LANDSCAPE ECOLOGY Kimberly A. With
2.1 2.1
INTRODUCTION INTRODUCTION It It is is no no coincidence coincidence that that the the current current biodiversity biodiversity crisis crisis occurs occurs at at aa time time when when landscapes are landscapes are being being transformed transformed faster faster than than ever ever before before in in human human history history (With, 2004). Many conservation issues are ultimately human human land-use issues (Wiens, become (Wiens, 2002), 2002), which which is is why why the the discipline discipline of of landscape landscape ecology ecology has has become increasingly biological diver increasingly relevant relevant for for the the management management and and conservation conservation of of biological diversity (e.g., Gutzwiller, 2002 2002).) . Processes that that operate at broader broader spatial scales likely influence the occurrence and persistence of an organism at a local scale, and ultimately required assessing species' and thus thus aa landscape landscape perspective perspective is is ultimately required for for assessing species' extinction extinction risk. risk. Such Such acknowledgment acknowledgment of of the the importance importance of of landscape landscape ecology ecology for for conser conservation reinforces the common common misconception misconception that that landscape ecology is con concerned cerned solely solely with with broad broad spatial spatial scales, scales, however. however. In In the the present present context, context, this this would entail understanding meta population dynamics at a "landscape metapopulation "landscape scale" (e.g., Rushton et al., 11997). 997). Apart Apart from the usual broad-scale anthropocentric anthropocentric definition landscape, aa landscape landscape is definition of of landscape, is defined defined more more appropriately appropriately as as aa "spa "spatially heterogeneous area" 99 1 ) that area" (Turner and Gardner, 11991) that is scaled relative
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to the process or organism of interest (Wiens, 1989). By this rendering, meta population dynamics can then be studied in fragmented landscapes that that metapopulation range in scale from that encompassing encompassing bacteria and protozoan communities ((Burkey, Burkey, 11997) 997) to spotted owls (Strix (Strix occidentalis; occidentalis; Gutierrez Guti~rrez and Harrison, Harrison, 11996). 996). The landscape thus provides a spatial context context for understanding understanding population dynamics and persistence in frag processes contributing contributing to meta metapopulation fragmented landscapes. Although meta population theory metapopulation theory is the current current paradigm paradigm for the conserva conservation of spatially structured structured populations in fragmented fragmented landscapes (Hanski and Simberloff, 11997), 997), landscape ecology provides an additional perspective and suite of approaches that that can complement complement metapopulation metapopulation theory, particularly in applications that that are not not handled well by existing theory, such as those involving continuous habitat habitat distributions distributions or recently fragmented fragmented landscapes. Metapopulation Metapopulation theory is not applicable to species in landscapes landscapes in which the habitat habitat is not not distinctly patchy or already fragmented extensively (Moilanen and Hanski, 200 1 ) . Nor 2001). Nor is the application application of metapopulation metapopulation theory theory necessar necessarily appropriate appropriate for species in recently fragmented systems, given the assump assumption of equilibrium colonization-extinction colonization-extinction dynamics that that underlies much of the theory [but see Ovaskainen and Hanski (2002) and Chapter Chapter 4 for advances in metapopulation metapopulation theory involving transient transient dynamics] dynamics].. In particu particular, landscapes fragmented by human human land-use activities may represent transient population transient nonequilibrium nonequilibrium dynamics in which a formerly continuous population has become subdivided into smaller, more isolated populations. populations. Dispersal among populations populations is disrupted disrupted such that that a functional metapopulation metapopulation is not not created; created; local extinctions extinctions are not not balanced by recolonization, recolonization, and conse consequently, all populations populations slowly decline to extinction extinction (Hanski and Simberloff, 11997). 997). Thus, a declining population population may superficially resemble a metapopula metapopulation in structure, but not not function function like one. Spatial subdivision is a necessary, necessary, but not not sufficient, condition condition for metapopulation metapopulation dynamics. Metapopulation Metapopulation theory has nevertheless drawn drawn attention attention to the importance importance of landscape landscape structure and dispersal for maintaining population population persistence 996). Indeed, the effect of patch structure on dispersal and coloniza (Wiens, 11996). colonization success is a unifying theme in both metapopulation metapopulation theory and landscape ecology (Wiens, 11997). 997). Colonization success is not simply a function function of the distance distance between between patches, but also depends depends on the nature nature of the intervening habitat or land-use matrix matrix through which organisms disperse, which deter deterIncorporation of mines the "effective isolation" isolation" of patches (Ricketts, 2001 2001).). Incorporation the more complex mosaic structure population structure of real landscapes landscapes into meta metapopulation models has been viewed as the main promise of landscape landscape ecology for metapopulation e.g., Hanski and Simberloff, 11997; 997; Wiens, 11997). 997). metapopulation theory ((e.g., In 1 996, 11997) 997) identified In aa couple couple of of earlier earlier reviews, reviews, Wiens Wiens ((1996, identified several several land landscape ecological concepts metapopulation ecology and concepts that that are relevant to metapopulation which emphasize the dual importance of dispersal dispersal and heterogeneous heterogeneous land landscape mosaics for understanding 1 ) landscape understanding metapopulation metapopulation dynamics: ((1) connectivity, which emerges as the interaction interaction of individual movement with with landscape pattern, pattern, is important important for metapopulation metapopulation persistence; persistence; (2) the land landscape matrix matters for meta population dynamics because it affects dispersal metapopulation and thus colonization success; ((3) 3 ) landscapes landscapes are heterogeneous mosaics of habitats and land uses, such that that habitat quality varies across the landscape,
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setting the stage for source-sink population dynamics ((Chapter Chapter 16); and (4) landscape dynamics may affect, or even generate, meta population dynamics. metapopulation The latter represents landscape heterogeneity in time as well as space. In In addition addition to to these these potential potential contributions contributions of of landscape landscape ecology ecology to to meta population ecology, a more fundamental principle emerges from the metapopulation definition of landscape ecology itself. Landscape ecology is the study of the effect of spatial pattern on ecological process (Turner, 11989). 989). From this it fol follows that that adopting a landscape ecological perspective to metapopulation dynamics entails understanding understanding how spatial pattern, pattern, such as habitat fragmen fragmentation or heterogeneity, affects processes that contribute to the dynamics of spatially structured populations. This might involve, for example, under understanding the relative effects of habitat fragmentation on dispersal (coloniza (colonization) and demography demography on metapopulation metapopulation persistence. This expanded expanded perspective of landscape ecology is adopted adopted in this chapter. The 1 ) demonstrate The objectives objectives in in this chapter are are thus thus to to ((1) demonstrate what what aa landscape landscape ecological perspective can contribute population contribute toward toward understanding understanding meta metapopulation dynamics, beyond the usual suggestions that that landscape ecology offers a broader scale perspective or more spatially complex rendering of landscape structure; (2) discuss how landscape structure is expected, or has been demon demonstrated, to affect various processes (dispersal, demography) that affect meta population persistence and thus extinction risk; ((3) 3 ) assess the implica metapopulation implications of adopting a landscape ecological perspective for management and con conservation; and (4) identify theoretical and empirical research needs that that would help contribute contribute to the further development of this "exciting scientific synthe synthesis" between meta population biology and landscape ecology (see metapopulation (see Hanski and 991). Gilpin, 11991).
2.2 2.2
CONTRIBUTIONS CONTRIBUTIONS OF OF LANDSCAPE LANDSCAPE ECOLOGY ECOLOGY TO TO UNDERSTANDING ETAPOPULATION DYNAMICS UNDERSTANDING M METAPOPULATION DYNAMICS This section addresses how how landscape structure affects, or is expected to affect, the dynamics of meta populations. This includes a discussion on issues metapopulations. pertaining to landscape connectivity, landscape connectivity and dispersal thresholds, the relative importance importance of dispersal for metapopulation persist persistence, landscape effects on demography and extinction risk, the source-sink potential of landscapes, extinction risk in dynamic landscapes, and the relative effects of habitat loss and fragmentation on meta population persistence. metapopulation
Landscape Landscape Connectivity Connectivity Issues: Issues: Patch-Based Patch-Based vs vs Landscape-Based Landscape-Based Measures Measures Habitat Habitat connectivity is a central theme in both landscape ecology and metapopulation 999a; Tischendorf and Fahrig, 2000a). metapopulation ecology (Hanski, 11999a; Connectivity refers to the ability of organisms to to access habitat, which affects colonization rates and thus metapopulation persistence on the landscape (e.g., Gonzalez et aI., 998; Kindvall, 11999). 999). The emphasis in meta population ecol al., 11998; metapopulation ecology, however, has been on deriving patch-based patch-based measures measures related to the prox proximity and area of neighboring patches, which quantify the accessibility of
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999a; Moilanen habitat patches to an individual on the landscape (Hanski, 11999a; and Hanski, 200 1 ) . An and Hanski, 2001). An overall overall measure measure of of patch patch connectivity connectivity for for the the landscape landscape can be obtained as a weighted average of patch patch isolation, which then gives the amount amount of habitat accessible to a random random individual on the landscape (Hanski, 11999a). 999a). Overall patch connectivity may give an indication indication of landscape con connectivity, nectivity, but but the the latter latter is is not not formally formally derived derived mathematically mathematically from from such such patch-based Fahrig, 2001 connect patch-based measures measures (Tischendorf (Tischendorf and and Fahrig, 2001).). Patch-based Patch-based connectivity measures are best applied to extensively fragmented or distinctly patchy landscapes and are less applicable to more continuous continuous habitat habitat distributions (Moilanen 1 ). (Moilanen and and Hanski, Hanski, 200 2001). In contrast, landscape ecologists have focused on deriving measures of overall landscape landscape connectivity. Habitat Habitat connectivity is thus being assessed at different different scales scales ~ patch patch based based vs vs landscape landscape based based ~ in in these these two two disciplines. disciplines. Landscape connectivity is defined as the degree to which various habitat habitat types 993; With facilitate movement across the landscape (Taylor et aI., al., 11993; With et aI., al., 11997; 997; Tischendorf Tischendorf and and Fahrig, 2000a) 2000a) and and can can thus be assessed for continu continuous habitat distributions heterogeneous landscapes Schippers et ous habitat distributions and and heterogeneous landscapes ((Schippers et aI., al., 11996; 996; With et aI., al., 1997). Landscape connectivity can be quantified in a number of ways, such as by the use of percolation theory and its neutral neutral landscape derivatives ((Gardner Gardner et aI., 987; With, 997), al., 11987; With, 1997, 1997, 2002; With With and and King, 11997), graph theory 1 ) , and theory (Urban and and Keitt, 200 2001), and various other other approaches approaches (e.g., Schumaker, 996; Tischendorf Schumaker, 11996; Tischendorf and and Fahrig, Fahrig, 2000a,b). 2000a,b). Although Although aa full full render rendering lies beyond beyond the ing of of how how landscape landscape connectivity connectivity can can be be quantified quantified lies the scope scope of of this common theme this chapter, chapter, the the common theme underlying underlying all all of of these these approaches approaches is is how how the the movement movement behavior behavior of of organisms organisms interacts interacts with with the the patch patch structure structure of of land landscapes. Landscape connectivity thus emerges as a species-specific response to landscape habitat affinities, landscape structure structure based based on on factors factors such such as as the the species' species' habitat affinities, gap-crossing abilities, movement movement rates, response to patch boundaries, boundaries, and differential mortality through 993; through elements of the landscape (Wiens et a!., al., 11993; Dale et aI., 994; With, 11997; 997; With et aI., 997; Tischendorf al., 11994; al., 11997; Tischendorf and Fahrig, 200 1 ; Vos et aI., 1). 2001; al., 200 2001). Landscape connectivity is important important for understanding understanding the emergence of spatial structure in populations, populations, which in turn turn is expected to have implications for the persistence and dynamics of meta populations. As an example of how metapopulations. how species-specific species-specific responses responses to to heterogeneity heterogeneity affect affect landscape landscape connectivity connectivity and and population 1 995) used habitat-specific rates of population distributions, distributions, With and Crist ((1995) movement in an individual-based simulation model inspired by percolation theory to predict the distributional patterns of two two acridid grasshopper species in a heterogeneous landscape within the shortgrass steppe of the North North American Great Plains. The largest species (Xanthippus (Xanthippus corallipes) corallipes) moved rapidly through through the grass matrix matrix (65% of the landscape), suggesting that that the overall of this overall landscape landscape was was highly highly connected connected from from the the standpoint standpoint of this species. species. Its Its reduced rate of movement in the remaining third of the landscape resulted in the observed patchy population population distribution, distribution, consistent with model expecta expectations that that good dispersers should exhibit patchy distributions distributions when the landscape contained :::; 3 5 % preferred habitat because their high mobility -<35% allows individuals to locate and allows individuals to locate and aggregate aggregate within within the the preferred preferred habitat habitat (assum (assuming that that individuals reduce their rates of movement and exhibit greater resi residence dence times times in in preferred preferred habitats). habitats). In In contrast, contrast, the the lower lower mobility mobility of of the the
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smaller smaller species species (Psoloessa (Psoloessa delicatula) delicatula) prevented prevented large large numbers numbers of of individuals individuals from locating and aggregating within its preferred preferred habitat, habitat, which constituted aa minor ((8%) 8 % ) component component of of the the landscape. landscape. Because Because of of its its relatively relatively greater rates of movement through species was rates of movement through other other grassland grassland habitats, habitats, this this species was expected expected to landscape, which to be be distributed distributed randomly randomly across across the the landscape, which was was consistent consistent with with its its observed distribution distribution in the field.
Landscape Connectivity Issues: Issues: Data Requirements Patch-based population Patch-based connectivity connectivity measures measures that that form form the the basis basis of of meta metapopulation theory Hanski, theory have have the the distinct distinct advantage advantage of of ease ease of of model model parameterization parameterization ((Hanski, 11999b; 999b; Hanski Hanski et aI., al., 2000; Chapter Chapter 5). Consistent Consistent with with metapopulation metapopulation the theory's abstraction abstraction of landscape structure as discrete patches embedded in an ecologically neutral matrix, matrix, patchwise patchwise measures of connectivity have often been been based based on on simplistic simplistic measures measures such such as as nearest-neighbor nearest-neighbor distances distances (Moilanen and Nieminen, Nieminen, 2002 2002).) . More More sophisticated measures of patch patch connectivity connectivity have have been been developed, developed, however, however, which which incorporate incorporate patch-area patch-area effects effects on on emigration emigration and and immigration immigration rates rates and and species-specific species-specific dispersal dispersal distances 1 ; Chapter distances (Moilanen and Hanski, Hanski, 2001 2001;; Vos et aI., al., 200 2001; Chapter 4). As the aforementioned aforementioned grasshopper grasshopper example example illustrates, illustrates, however, however, the the connectivity connectivity of of habitat habitat patches is not just a simple function function of the distance between between patches. Because the intervening matrix matrix may determine determine the effective isolation of patches (Ricketts, 2001 aI., 2001),) , and thus thus overall overall landscape connectivity (With et al., 11997), 997), explicit consideration consideration of how how the complex complex mosaic mosaic structure of hetero heterogeneous landscapes affects population extinction geneous landscapes affects colonization colonization success success and and population extinction risk risk has typically been viewed as one of the most most important important contributions contributions that that landscape has to population ecology ecology ((Hanski Hanski and and landscape ecology ecology has to offer offer meta metapopulation Simberloff, 11997; 997; Wiens, 11997). 997). The The connectivity of heterogeneous landscapes is not easily captured captured by a simple index or landscape metric unfortunately, unfortunately, but but is commonly tackled with an data on an individual-based individual-based simulation simulation modeling modeling approach. approach. Empirical Empirical data on habitat habitatspecific specific movement movement parameters parameters or or residence residence times times within within different different elements elements of of the the landscape landscape are are used used to to parameterize parameterize aa rulerule- or or vector-based vector-based movement movement model to simulate dispersal across a heterogeneous landscape map. Landscape connectivity is then inferred by extrapolating extrapolating habitat-specific rates of move movement, and perhaps perhaps other information information (if available) about about behavior at habitat edges (Lidicker and Koenig, 11996) 996) or mortality mortality risk while dispersing through through the different elements of the landscape, to determine determine whether whether individuals are able to able to colonize colonize aa suitable suitable habitat habitat successfully. successfully. Some Some presumed presumed correlate correlate of of aa connected landscape, such as dispersal success, degree of population population aggrega aggregation 995; With 997), or tion (With (With and and Crist, Crist, 11995; With et et aI., al., 11997), or population population connectivity connectivity (Schippers et aI., 996), is then used as an indirect measure of landscape al., 11996), connectivity. Admittedly, quantifying the resistance of different different habitat habitat types to move movement is a challenge in practice. practice. Direct Direct observation observation of individual movement movement responses to landscape structure structure is time intensive and is necessarily limited in temporal and therefore bound to change temporal ((and therefore spatial) extent, extent, although although this is bound with with the the increasing increasing availability availability of of satellite-tracking satellite-tracking devices devices that that permit permit the the near-continuous near-continuous monitoring monitoring of individuals. As an alternative, alternative, investigators
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typically rely upon typically rely upon mark-recapture mark-recapture techniques techniques to to derive derive estimates estimates of of inter interpatch patch movements movements in in different different matrix matrix types types (e.g., (e.g., Pither Pither and and Taylor, Taylor, 1998; 1998; Ricketts, 1 ) or Ricketts, 200 2001) or make make inferences inferences about about matrix matrix resistance resistance based based on on observed e.g., observed patterns patterns of of patch patch occupancy occupancy in in different different landscape landscape contexts contexts ((e.g., Moilanen 99 8 ) . For Moilanen and and Hanski, Hanski, 11998). For example, example, mark-recapture mark-recapture data data were were used used to to quantify quantify the the resistance resistance of of different different matrix matrix types types to to butterfly butterfly movement movement within naturally heterogeneous heterogeneous landscape within a naturally landscape located located in an alpine alpine valley of the Colorado 1 ). Movement Colorado Rocky Rocky Mountains Mountains (Ricketts, (Ricketts, 200 2001). Movement through through coniferous coniferous forests likely than forests was was 3-12 3-12 times times less less likely than movement movement through through willow willow thickets thickets for for the these meadow meadow butterfly the majority majority of of these butterfly species. species. Meadows Meadows separated separated by by coniferous coniferous forest forest are thus thus effectively more more isolated isolated than than meadows meadows separated separated by by aa similar similar distance, distance, but but embedded embedded in in aa willow willow matrix. matrix. Because Because of of the the diffi difficulty and and cost cost of culty of attempting attempting to to reconnect reconnect habitat habitat fragments fragments with with corridors corridors or stepping stepping stones stones to to enhance enhance dispersal dispersal and and thus thus colonization colonization success, it might practical to isolation of of patches patches by alter might be be more more practical to reduce reduce the the effective effective isolation by altering management management practices practices in the surrounding surrounding matrix Ricketts, 200 1). matrix ((Ricketts, 2001). Corridors Corridors need need not not bbee linear linear features features ooff the the landscape, landscape, but but can can occur occur or or be be created created through through the the juxtaposition juxtaposition of of certain certain matrix matrix types, types, such such as as different different habitats funnel individuals habitats or land land uses, which which serve to funnel individuals among among habitat habitat patches Gustafson and 996; Vos et al., aI., 2002) patches ((Gustafson and Gardner, Gardner, 11996; 2002)..
landscape Landscape Connectivity Connectivity Issues: Issues: Asymmetrical Asymmetrical Connectivity Connectivity Among Among Populations Populations Connectivity Connectivity in in heterogeneous heterogeneous landscapes landscapes may may thus thus be be difficult difficult to to identify identify from a simple analysis of landscape structure. Nor landscape structure. Nor is the transfer transfer of individu individuals among patches als among patches necessarily necessarily symmetrical symmetrical in in aa heterogeneous heterogeneous landscape landscape matrix. 1 996) developed an individual-based matrix. Gustafson Gustafson and Gardner Gardner ((1996) individual-based simula simulation tion model model to to explore explore how how altering altering landscape landscape heterogeneity heterogeneity affected affected dispersal dispersal success success for for aa generic generic organism organism among among fragments fragments of of deciduous deciduous forest forest in in several several agricultural agricultural landscapes landscapes located located in in the the midwestern midwestern United United States. States. Emigration Emigration and among forest and immigration immigration rates rates among forest fragments fragments often often were were not not symmetrical, symmetrical, leading Gustafson 1 996) to Gustafson and and Gardner Gardner ((1996) to speculate that that such asymmetrical asymmetrical transfers rule rather transfers may may be be the the rule rather than than the the exception exception in in heterogeneous heterogeneous land landscapes. If true, true, this presents presents a problem problem for for patchwise patchwise connectivity connectivity measures measures in which which colonization colonization probabilities probabilities are are based based on on interpatch interpatch distances distances that that ignore ignore matrix problem, distances would matrix effects. effects. To To overcome overcome this this problem, distances between between patches patches would need inter need to to be be specified specified by by direction direction and and weighted weighted by by the the resistance resistance of of the the interUrban and 1 ). vening habitat to vening matrix matrix habitat to movement movement (i.e., (i.e., dij dii *' 4: dji; dii; see see Urban and Keitt, Keitt, 200 2001). The potential The potential for for asymmetrical asymmetrical connectivity connectivity among among populations populations of of European European badgers (Meles (Meles meles) meles) was indicated indicated by a GIS-based random random walk walk model model applied applied to to aa landscape landscape in in the the central central part part of of The The Netherlands Netherlands (Schippers (Schippers et 996). Urban areas, canals, et aI., al., 11996). Urban areas, canals, and and motorways motorways either either created created barriers barriers to to movement movement or or increased increased mortality mortality such such that that asymmetries asymmetries in in the the connectivity connectivity of of populations populations emerged. emerged. Asymmetrical Asymmetrical connectivity connectivity has has also also been been found found among among populations populations of of the the critically critically endangered endangered Iberian Iberian lynx lynx (Lynx (Lynx pardinus) pardinus) in in aa human-dominated region in southwest human-dominated and and therefore therefore extensively extensively fragmented fragmented region in southwestern Spain (Ferreras, 200 1 ). Although emi 2001). Although two two source populations populations had had similar emigration rates, they differed in their connectivity connectivity to outlying populations populations
2. 2.
M ETAPOPULATION DYNAMICS DYNAMICS METAPOPULATION
29
because of of differences differences in in the the matrix type surrounding surrounding each each source. source. One One of of the the because matrix type sources was embedded in an an agricultural agricultural matrix, matrix, which which lynx lynx avoided, avoided, and and sources was embedded in were instead instead funneled funneled along along aa narrow narrow corridor corridor of of aa more more suitable suitable habitat habitat to to were populations in in the the south south and and west. west. The The other other source source population population was was located located in in populations the southwestern southwestern region, which consisted consisted mainly mainly of of Mediterranean Mediterranean scrubland scrubland the region, which and tree tree plantations, plantations, and and most most dispersing dispersing individuals individuals from from this this population population and settled in this this region region instead instead of of dispersing dispersing toward toward the the northeast. northeast. Thus, Thus, there there is is settled in an asymmetrical asymmetrical transfer transfer of of individuals individuals that occurs among among populations: indi an that occurs populations: individuals to disperse disperse from from the the northeast northeast to to the the southwest, southwest, but but not not in in the the viduals tend tend to opposite direction. direction. Such Such asymmetries asymmetries could could lead lead to to aa reduction reduction in in the the effective effective opposite connectivity, and and thus thus metapopulation metapopulation capacity capacity (see (see Hanski Hanski and and Ovaskainen, Ovaskainen, connectivity, 2000), of of the the landscape landscape for for the the species. species. 2000),
Landscape Connectivity Connectivity Issues: Issues: Thresholds Thresholds in in Connectivity Connectivity Landscape Because of landscape connectivity the Because of the the importance importance of of landscape connectivity for for evaluating evaluating the structure metapopulations, it it would advantageous to to structure and and dynamics dynamics of of metapopulations, would be be advantageous identify when when landscapes become disconnected disconnected and and thus thus when when metapopulametapopula identify landscapes become tion likely to to be be disrupted. disrupted. Both Both patch tion processes processes such such as as colonization colonization rates rates are are likely patch connectivity (Hanski, 1999a) 1 999a) and and measures measures of of landscape landscape connectivconnectiv connectivity measures measures (Hanski, ity predict critical critical thresholds thresholds in habitat connectivity, where the the ity (With, (With, 2002) 2002) predict in habitat connectivity, where habitat network becomes disconnected at at a critical level of remaining habitat network becomes abruptly abruptly disconnected of remaining habitat. percolation-based approaches, approaches, for landscape connectivity habitat. In In percolation-based for example, example, landscape connectivity cluster) spans spans whether a single habitat habitat cluster cluster (the percolating cluster) is assessed by whether the landscape. when the the critical critical habitat habitat the landscape. Landscape Landscape connectivity connectivity is is disrupted disrupted when nodes nodes forming forming the the "backbone" "backbone" of of the the percolating percolating cluster cluster are are destroyed, destroyed, which which abruptly abruptly breaks breaks the the percolating percolating cluster cluster into into two two or or more more fragments. fragments. The The critical critical level level of of habitat habitat at at which which landscape landscape connectivity connectivity becomes becomes disrupted disrupted (percolation threshold) depends depends on on aa number number of of assumptions assumptions regarding regarding species-specific (gap-crossing abilities, species-specific movement movement attributes attributes (gap-crossing abilities, movement movement rates rates through habitat types, mortality) and through different different habitat types, matrix matrix mortality) and the the representation representation and landscape itself habitat and configuration configuration of of the the landscape itself (grid (grid geometry, geometry, degree degree of of habitat fragmentation) issue, however, fragmentation) (for (for aa review, review, see see With, With, 2002). 2002). The The issue, however, is is whether whether thresholds thresholds in in landscape landscape connectivity, connectivity, or or measures measures of of landscape landscape connectivity connectivity more generally, success and more generally, relate relate to to processes processes such such as as colonization colonization success and local local extinction extinction rates, rates, which which are are important important for for predicting predicting metapopulation metapopulation persistence persistence on on landscapes. landscapes. In In other other words, words, is is landscape landscape connectivity connectivity both both aa necessary population persistence e.g., necessary and and aa sufficient sufficient condition condition for for meta metapopulation persistence ((e.g., With, With, 1999)? 1999) ?
Landscape Landscape Connectivity Connectivity Thresholds Thresholds and and Dispersal Dispersal Success Success Dispersal populations together Dispersal is is the the "glue" "glue" that that keeps keeps meta metapopulations together (Hansson, (Hansson, 11991), 99 1 ) , and and thus thus colonization colonization success success is is deemed deemed crucial crucial to to metapopulation metapopulation persistence. persistence. Clearly Clearly there there should should be be some some relationship relationship between between landscape landscape connectivity connectivity and and dispersal dispersal (colonization) (colonization) success: success: dispersal dispersal success success is is expected expected to to be be higher higher in in landscapes landscapes with with aa high high degree degree of of connectivity. connectivity. What What is is less less clear, clear, however, however, is is whether whether thresholds thresholds in in landscape landscape connectivity connectivity should should neces necessarily sarily coincide coincide with with thresholds thresholds in in dispersal dispersal or or colonization colonization success. success.
WITH KIMBERLY A. WITH
30 30
To this, With With and and King King (1999a) ( l 999a) quantified quantified dispersal dispersal success success on on a To address this, of landscapes landscapes with with complex complex (fractal) (fractal) habitat habitat distributions distributions that that reprerepre series of sented a gradient gradient of of fragmentation fragmentation severity severity (Fig. 2.1a). 2.la). Dispersal Dispersal success success was was sented defined as the the proportion proportion of of independent independent dispersers dispersers that that successfully successfully located located a defined suitable suitable habitat habitat patch patch (cell). (cell) . Consider Consider that that if if dispersal dispersal is truly random, random, such such that dispersal dispersal occurs occurs to to a random random point point on on the the landscape, landscape, then then the the underlying underlying that spatial spatial pattern pattern of of the the landscape landscape is unimportant unimportant for for predicting predicting dispersal success success
a
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120 100 .S 80 2!.� 60 40 j 20 x
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1 .0 0.9 s: 0.8 or. 0.7 ti: 0.6 Q; 0.5 'E - Lacunarity 0.4 C Fragmented 0.3 � 0.2 :E Clumped 0.1 e , 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0
(f', (f', a: c..:
(f', c (f',
Cl: .c a..
C
g.
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� 0.9
� 0.8 -al 0.7 13 0.6 Clumped g? 0.5 Fragmented 8 0.4 2!- 0.3 15 0.2 0.1 0:: O.O-t-�-T--'-----;r--.--�--r---.,..---, 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0 Proportion suitab le habitat c
I'
Fig. Fig. 2.1 (a) Examples of fractal landscape patterns illustrating extremes in fragmentation fragmentation severity severity ((H, success declines precipitously below H, spatial autocorrelation of habitat). (b) Dispersal success 20% habitat (dispersal threshold), coinciding with the lacunarity threshold of landscape struc structure, which isis a landscape-wide measure of interpatch distances. The lacunarity index isis not affected by the landscape pattern for analyses conducted at the finest scale scale (1 (1 x x 11 grid cell), shown here for clarity of of presentation. The lacunarity curve at other scales scales isis qualitatively simi similar, but lacunarity indices tend to be higher in clumped fractal landscapes than in fragmented lar, ones due to the greater variability in gap sizes. sizes. (c) Percolation thresholds, aa patch-based assess assessment 2-cell ment of of landscape landscape connectivity, connectivity, do not coincide coincide with dispersal thresholds (assuming aa 112-cell dispersal neighborhood). Modified 999a). Modified from from With With and King (1 (1999a).
31 31
2. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS 2.
and only only the the fraction fraction of of habitat habitat (h) (h) and and number number of of dispersal dispersal steps steps (m; (m; equivaequiva and lently the the "dispersal "dispersal neighborhood" neighborhood" or or "dispersal "dispersal ability" ability" of of the the species) species) affect affect lently dispersal success success as as dispersal Pr( success ) = Pr(success) = 1 1 -- (1 ( 1 -- h)m. h)m.
(2.1) (2. 1 )
Equation (2.1) (2. 1 ) represents represents the the mean-field mean-field approximation. approximation. If If dispersal dispersal is is instead instead Equation constrained to to movement movement through through adjacent adjacent cells cells (but ( but still still random random in in direction) direction) constrained to force force individuals individuals to to interact interact with with the patch structure structure of of the landscape, then then to the patch the landscape, dispersal on fractal fractal landscapes landscapes can can no no longer longer be be derived first dispersal success success on derived from from first principles and and may may lack lack aa closed-form closed-form solution. solution. Thus, Thus, dispersal dispersal success success on on principles fractal landscapes landscapes had to be be obtained obtained through through numerical numerical simulations simulations (With (With fractal had to and King, King, 1999a). 1 999a). and As expected, dispersal dispersal success success declined declined with with decreasing decreasing habitat habitat and and As expected, increasing fragmentation fragmentation of of the the landscape, landscape, but but the the rate rate of of this this decline decline accelaccel increasing erated 1 0-20% (Fig. 2 . 1 b ) . In In other erated once once the the amount amount of of habitat habitat fell fell below below 10-20% (Fig. 2.1b). other words, response to habitat amount. amount. words, dispersal dispersal success success exhibited exhibited aa threshold threshold response to habitat This with percolation to This dispersal dispersal threshold threshold did did not not coincide coincide with percolation thresholds thresholds used used to quantify landscape connectivity, allowing for dispersal quantify landscape connectivity, even even after after allowing for aa larger larger dispersal neighborhood in which neighborhood to to define define habitat habitat connectivity, connectivity, in which individuals individuals could could move move through cells nonhabitat in their search habitat site site ((as as in in the the through cells of of nonhabitat in their search for for aa suitable suitable habitat simulation; 2.1c). Intuitively, Intuitively, dispersal to decline as simulation; Fig. Fig. 2.1c). dispersal success success is is expected expected to decline as patches become more isolated because the up patches become smaller smaller and and more isolated because the disperser disperser ends ends up spending of its its time in the the matrix be greater. spending much much of time in matrix where where mortality mortality may may be greater. Lacunarity analysis the "gap landscapes and is related related Lacunarity analysis quantifies quantifies the "gap structure" structure" of of landscapes and is to variance-to-mean ratio to the the variance-to-mean ratio of of the the distances distances among among patches patches on on the the land landscape 993). The scape (Plotnick (Plotnick et et aI., al., 11993). The higher higher the the lacunarity lacunarity index, index, the the greater greater the the variability variability in in distances distances among among patches. patches. Lacunarity Lacunarity is is not not merely merely the the inverse inverse of of some some measure measure of of patch patch structure, structure, such such as as the the fractal fractal dimension dimension of of the the land landscape, scape, however; however; it it can can resolve resolve differences differences in in landscape landscape pattern pattern that that may may be be obscured obscured by by patch-based patch-based measures measures (Plotnick (Plotnick et et aI., al., 1993 1993).) . Landscape Landscape lacu lacunarity narity exhibited exhibited aa strong strong threshold threshold effect effect around around 20% 20% habitat; habitat; interpatch interpatch dis distances tances became became greater greater and and more more variable variable when when habitat habitat fell fell below below this this critical critical level 999a; Fig. level (With (With and and King, King, 11999a; Fig. 2.1b). 2.1b). Thresholds Thresholds in in dispersal dispersal success success thus thus coincide percolation thresholds coincide with with lacunarity lacunarity thresholds thresholds rather rather than than percolation thresholds of of landscape landscape connectivity. connectivity. Empirical Empirical tests tests of of percolation percolation theory theory have have been been performed performed in in the the field field with with insects insects moving moving across across experimental experimental "microlandscapes" "microlandscapes" in in which which habitat habitat (grass (grass sod) 997; sod) was was arrayed arrayed as as either either aa random random or or aa fractal fractal distribution distribution (Wiens (Wiens et et aI., al., 11997; 999; With 999). Although McIntyre Mclntyre and and Wiens, Wiens, 11999; With et et aI., al., 11999). Although it it is is not not clear clear whether whether parameters parameters that that describe describe movement movement pathways pathways should should exhibit exhibit threshold threshold behav behavior, ior, let let alone alone coincide coincide with with percolation percolation thresholds thresholds in in landscape landscape connectivity connectivity (With (With et et aI., al., 1999), 1999), tenebrionid tenebrionid beetles beetles (Eleodes (Eleodes obsoleta) obsoleta) nevertheless nevertheless exhibited exhibited threshold threshold behavior behavior in in several several movement movement parameters parameters when when grass grass cover cover fell fell below below 20% 997). This 20% (Wiens (Wiens et et aI., al., 11997). This is is in in the the domain domain of of lacunarity lacunarity thresholds thresholds (With (With and 999), suggesting and King, King, 11999), suggesting that that landscape landscape measures measures of of gap gap structure structure may may be better better predictors predictors of of dispersal dispersal success success than than landscape landscape measures measures of of ultimately be patch population theory, patch structure. structure. This This reiterates reiterates one one of of the the main main tenets tenets of of meta metapopulation theory, that that patch patch isolation isolation measures measures (and (and therefore therefore patch-based patch-based connectivity connectivity measures measures
32 32
KIMBERLY KIMBERLYA. A. WITH WITH
that that incorporate incorporate interpatch interpatch distances) distances) are are aa strong strong correlate correlate of of colonization colonization suc success, population cess, at at least least in in extensively extensively fragmented fragmented landscapes landscapes that that meet meet the the meta metapopulation ideal ideal of of habitat habitat patches patches embedded embedded in in an an ecologically ecologically neutral neutral matrix. matrix. As As dis discussed cussed previously, previously, the the relationship relationship between between landscape landscape structure structure and and colonization colonization success landscapes, where success is is more more complicated complicated in in heterogeneous heterogeneous landscapes, where patch patch isolation isolation may may be be less less important important than than the the quality quality of of the the matrix matrix habitat habitat through through which which the the organism organism disperses. disperses. For For example, example, large-scale large-scale forestry forestry in in Sweden Sweden resulted resulted in in extensive extensive ditching ditching to to drain drain clear-cut clear-cut areas, areas, which which created created an an inhospitable inhospitable matrix matrix colonizing breeding that that prevented prevented pool pool frogs frogs (Rana (Rana lessonae) lessonae) from from colonizing breeding ponds, ponds, irre irrespective 996). spective of of their their proximity proximity to to an an occupied occupied pond pond (Sjogren-Gulve (Sj6gren-Gulve and and Ray, Ray, 11996). The The lack lack of of concordance concordance between between percolation percolation thresholds thresholds and and dispersal dispersal suc success has led investigators to invent other, seemingly more-relevant more-relevant measures of landscape landscape structure structure for for predicting predicting dispersal dispersal or or colonization colonization success success (e.g., (e.g., Schumaker, 996; Tischendorf Schumaker, 11996; Tischendorf and and Fahrig, Fahrig, 2000a) 2000a).. The The problem problem is is not not the the measure used to measure used to quantify quantify landscape landscape connectivity, connectivity, however, however, but but with with the the scale scale at at which which habitat habitat connectivity connectivity is is assessed assessed relative relative to to the the scale scale of of dispersal. dispersal. Landscape Landscape connectivity connectivity relates relates to to the the potential potential of of organisms organisms to to traverse traverse the the entire entire landscape, landscape, whereas whereas dispersal dispersal or or colonization colonization success success pertains pertains only only to to the the likelihood organism will will successfully suitable habitat likelihood that that aa dispersing dispersing organism successfully find find aa suitable habitat patch patch (or (or cell cell in in aa grid-based grid-based landscape). landscape). Although Although the the two two are are related, related, assess assessments ments of of landscape landscape connectivity connectivity and and dispersal dispersal success success are are ultimately ultimately per performed at different scales. Individual movement movement is constrained constrained in the latter formed assessment individuals that locate suitable suitable habitat assessment (success (success is is scored scored for for individuals that locate habitat within within aa dispersal dispersal neighborhood, point colonization neighborhood, at at which which point colonization occurs occurs and and individuals individuals are are assumed assumed to to stop stop moving), moving), but but not not in in the the former former where where the the emphasis emphasis is is on individuals to landscape (whether on the the ability ability of of individuals to move move across across the the entire entire landscape (whether the the organism organism actually actually does does or or not). not). Thus, Thus, the the grain grain of of movement movement may may be be the the same individuals move same ~ how how individuals move within within or or between between habitat habitat types types or or cells cells ~ but but the the spatial spatial extent extent of of movement movement is is different. different. Habitat Habitat connectivity connectivity is is obviously obviously important important for for colonization colonization success success at at some some scale. scale. The The challenge challenge is is to to identify identify what what scale scale is is appropriate appropriate for for predicting predicting colo colonization nization success success in in aa given given species, species, however. however. This This involves involves adopting adopting aa species' species' perspective perspective of of habitat habitat connectivity connectivity (Wiens (Wiens and and Milne, Milne, 1989; 1989; With, With, 1994; 1994; Pearson Pearson et 996; Vos 1 ) . Although et aI., al., 11996; Vos et et aI., al., 200 2001). Although this this has has been been done done using using percolation percolationbased neutral related approach based neutral landscape landscape models models (see (see With, With, 2002), 2002), aa related approach involves involves the the use use of of graph graph theory. theory. In In graph graph theory, theory, the the grid grid structure structure of of the the landscape landscape is is represented represented as as aa graph graph in in which which habitat habitat patches patches (vertices (vertices or or nodes) nodes) are are con con(lines or nected nected across across varying varying distances distances (lines or edges) edges) (Urban (Urban and and Keitt, Keitt, 2001 2001).) . The The graph representation permits measure of graph representation permits aa process-based process-based measure of connectivity connectivity for for indi individual patches Overall connectivity vidual patches as as well well as as the the entire entire landscape. landscape. Overall connectivity of of the the graph graph (i.e., assessed in (i.e., landscape) landscape) is is simply simply assessed in terms terms of of whether whether each each node node is is connected connected to node. Although several ways to some some other other node. Although there there might might be be several ways to to connect connect the the various various nodes nodes of of the the graph graph to to form form aa spanning spanning tree, tree, the the one one with with the the shortest shortest minimum spanning length length is is termed termed the the minimum spanning tree. tree. There There is is aa critical critical threshold threshold distance disconnected, reminiscent reminiscent of distance at at which which the the graph graph becomes becomes disconnected, of the the percola percolation tion threshold threshold of of landscape landscape connectivity connectivity for for grid-based grid-based landscapes landscapes (Urban (Urban and and Keitt, Keitt, 2001 2001).) . Using Using aa graph-theoretic graph-theoretic approach, approach, van van Langevelde Langevelde (2000) (2000) identi identified fied different different scales scales of os connectivity connectivity and and related related this this to to colonization colonization patterns patterns of os the the European European nuthatch nuthatch (Sitta (Sirra europaea) europaea) occupying occupying woodlots woodlots within within fragmented fragmented
METAPOPULATIONDYNAMICS 2. METAPOPULATION
33 33
Netherlands (Fig. 2.2a). Patch Patch occupancy patterns of landscapes of The Netherlands nuthatches were correlated with with a critical threshold distance distance of 2.4-3 2.4-3 km nuthatches that woodlots woodlots located located >3 >3 km from a neighboring neighboring forest patch (Fig. 2.2b), such that were unlikely to be colonized by dispersing nuthatches (van Langevelde, 2000). nuthatch populations populations is related to to both both the connectivity The extinction of local nuthatch size of forested patches (Verboom et aI., al., 11991) and the size 99 1 ) and underscores again habitat connectivity ~ at some scale ~ for population population per perthe importance of habitat Merriam, 11985). metapopulation dynamics dynamics of sistence (e.g., Fahrig and Merriam, 985). The metapopulation nuthatches within this fragmented landscape have also been assessed using an incidence function model (Ter Braak et aI., al., 11998). incidence 998).
a
Above threshold (4 km)
Below threshold (2 km)
b � "5
U Q)
c c 0 u
'0
� Q)
"0
Ol Q)
�
Q) >
0
1 2,000 1 0,000 8,000 6,000 4,000 2,000 0 0
2
3
4
5
6
7
8
9
10
Threshold distance (km) Fig. 2.2 2.2 (a) A A graph graph theoretic of habitat habitat connectivity connectivity for for European nuthatches nuthatches (Sitta (Sitta Fig. theoretic analysis of europaea) in an agricultural agricultural landscape based based on on an analysis assuming assuming a dispersal distance distance of of 2 km km (just Gust below below the the landscape connectivity connectivity threshold; threshold; b) b) and and a dispersal distance distance of of 44 km km (above (above the the connectivity threshold; threshold; b). Gray areas are habitat habitat fragments, fragments, and and black black lines indicate indicate landscape connectivity connections connections among among patches based on on the the indicated dispersal distance. distance. (b) (b) Landscape Landscape connectivity connectivity exhibits a threshold threshold in in this landscape landscape at at about about 2.5 km. Patches farther farther than than 3 km km apart apart were were unlikely to to be colonized colonized by nuthatches. nuthatches. Modified Modified from from van van Langevelde Langevelde (2000). (2000).
KIMBERLY A. A. WITH WITH KIMBERLY
334 4
Relative Importance Importance of of Dispersal Dispersal for for Metapopulation Metapopulation Persistence Persistence Relative Studies on how how landscape landscape structure structure and and matrix matrix heterogeneity heterogeneity affect affect dispersal dispersal Studies on (colonization) assume assume that that the the fine-scale fine-scale movements movements of of individuals individuals translate translate (colonization) into broader broader patterns patterns of of population population distribution distribution (e.g., (e.g., Turchin, Turchin, 1991; 1 99 1 ; With With into and Crist, Crist, 1996; 1 996; With With et et al., al., 1997), 1 997), which which in in turn turn may may have have consequences consequences for for and metapopulation persistence persistence on on the the landscape. landscape. How How important important is is dispersal dispersal for for metapopulation predicting metapopulation metapopulation persistence? persistence? Dispersal Dispersal is is aa key key component component of of most most predicting spatially explicit explicit population population models, models, which which serve serve as as the the main main arsenal arsenal in in the the spatially landscape ecological ecological approach approach to to predicting predicting metapopulation metapopulation persistence persistence in in landscape fragmented landscapes, landscapes, particularly particularly in in evaluating evaluating the the consequences consequences of of differdiffer fragmented ent scenarios scenarios of of land-use land-use change change on on extinction extinction risk risk for for species species of of conservation conservation ent concern (Dunning (Dunning et et al., al., 1995). 1 995). As As mentioned mentioned previously, previously, it it is is difficult difficult to to concern obtain speciesspecies- and and habitat-specific habitat-specific information information on on dispersal, dispersal, which which may may result result obtain in errors errors in in the the estimation estimation of of dispersal dispersal success. success. Such Such estimation estimation errors may in errors may propagate in spatially affect estimates of species' species' extincextinc propagate in spatially explicit explicit models models and and affect estimates of tion risk (Ruckleshaus (Ruckleshaus et et al., al., 1997), 1 997), although although the magnitude of of these these errors tion risk the magnitude errors may have been been overestimated overestimated (Mooij (Mooij and and DeAngeles, DeAngeles, 1999). 1 999). may initially initially have Landscape structure not always always important important for predicting dispersal dispersal sucsuc Landscape structure is is not for predicting cess, however. Using an an individual-based individual-based model model of dispersal on on neutral neutral land cess, however. Using of dispersal landscape King and and With that the the mean-field mean-field approximation approximation scape models, models, King With (2002) (2002) found found that [Eq. (2. 1 )] was was sufficient sufficient for for predicting predicting dispersal dispersal success when >40% >40% of of the the [Eq. (2.1)] success when landscape below this this level, level, specifics related to to dispersal dispersal behavior behavior landscape was was suitable; suitable; below specifics related and landscape pattern became more important. Given and landscape pattern became more important. Given that that many many species species of of conservation in landscapes <40% suitable conservation concern concern occur occur in landscapes with with substantially substantially <40% suitable habitat, however, however, it is likely that landscape structure the configuration and habitat, it is likely that landscape structure m the configuration and heterogeneity generally be heterogeneity of of land-cover land-cover types types - - will will generally be important important for for predicting predicting dispersal dispersal success. success. Although Although dispersal dispersal (colonization) (colonization) success success is is considered considered an an important important process process necessary population persistence, necessary for for meta metapopulation persistence, demographic demographic factors factors that that affect affect extinction may actually more important some species 999; extinction risk risk may actually be be more important for for some species (South, (South, 11999; With 999b). This With and and King, King, 11999b). This is is especially especially true true for for good good dispersers, dispersers, such such as as birds, birds, where landscape landscape structure has a greater effect effect on reproductive reproductive output output through through edge 995b; Dooley Dooley and edge effects effects than than on on immigration immigration rates rates (e.g., (e.g., Donovan Donovan et et al., al., 11995b; and Bowers, 998). The landscape structure Bowers, 11998). The effect effect of of landscape structure on on demography demography and and extinction extinction risk risk on on metapopulation metapopulation persistence persistence is is explored explored in in the the next next section. section.
Landscape Landscape Effects Effects on on Demography Demography and and Extinction Extinction Risk Risk Habitat Habitat loss and fragmentation fragmentation pose the greatest threats to biodiversity (Wilcove 998) and (Wilcove et et al., al., 11998) and are are the the inevitable inevitable consequence consequence of of the the transforma transformation tion of of landscapes landscapes by by humans humans (With, (With, 2004). 2004). Beyond Beyond the the sheer sheer magnitude magnitude and and rate problem: the rate of of this this transformation transformation lies lies aa more more insidious insidious problem: the effects effects of of habi habitat tat loss loss and and fragmentation fragmentation on on population population viability viability are are not not linear. linear. Habitat Habitat loss loss may may precipitate precipitate aa sudden sudden and and rapid rapid decline decline in in the the probability probability of of metapopula metapopulation tion persistence persistence (i.e., (i.e., aa threshold). threshold). Using Using aa demographic demographic model model founded founded on on Levins' 1 969) classic 1987) first Levins' ((1969) classic metapopulation metapopulation model, model, Lande Lande ((1987) first defined defined extinction extinction thresholds thresholds for for territorial territorial vertebrates vertebrates as as aa function function of of their their demo demopotentials (k, (k, a composite composite parameter parameter derived algebraically from graphic potentials
2. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS 2.
335 5
life-history parameters parameters such such as as net net lifetime lifetime reproductive reproductive output, output, R'o, R' 0' and and disdis life-history persal ability, ability, m). m). The The extinction extinction threshold threshold is is the the critical critical level level of of habitat habitat (h~) (hcl persal at which which the the population population no no longer longer occurs occurs on on the the landscape landscape (patch (patch occupancy, occupancy, at p'~ p * == 0) 0) and and is is defined defined mathematically mathematically as as hhec == 11 -- k.k. The The decline decline in in patch patch occupancy accelerates accelerates past past aa certain certain reduction reduction in in habitat, habitat, such such that that the the occupancy approach to to the the extinction extinction threshold threshold (h,) (he) is is usually usually nonlinear. nonlinear. approach Lande's (1987) ( 1 987) model model was was in in the the tradition tradition of of the the classical classical metapopulation metapopulation Lande's model, which which only only assumes assumes the the existence existence of of habitat habitat patch patch structure structure (i.e., (i.e., itit is is model, spatially implicit). implicit). Habitat Habitat is is assumed assumed to to be be distributed distributed randomly randomly across across the the spatially landscape or, or, alternatively, alternatively, is is randomly randomly accessible accessible by by dispersing dispersing individuals. individuals. landscape Bascompte and and Sold Sole (1996) ( 1 996) developed developed aa spatially spatially explicit explicit realization realization of of this this Bascompte model using using grid-based grid-based landscapes landscapes with with random random habitat habitat distributions distributions and and model found that that extinction thresholds generally generally occurred occurred at at about about the the same same level level found extinction thresholds (he) as as in in Lande's Lande's (1987) ( 1 987) spatially spatially implicit model, but but the the decline decline in in patch patch (h,) implicit model, occupancy occurred occurred faster, faster, resulting resulting in in steeper steeper thresholds. thresholds. occupancy The effect effect of of habitat habitat fragmentation fragmentation on on extinction thresholds has has been been The extinction thresholds explored using using fractal fractal landscapes, landscapes, which which generate generate complex complex landscape landscape patterns patterns explored across aa gradient gradient of of fragmentation fragmentation severity severity (e.g., (e.g., Fig. Fig. 2.1a; 2. 1a; With With and and King, King, across 1 999b; Hill and and Caswell, Caswell, 1999). 1999). Because of of the the complexities complexities of of how how species 1999b; interact with with fractal fractal landscape landscape patterns, patterns, itit was was necessary necessary to to parse parse the the demodemo interact (R' 0 and m) to to evaluate evaluate the the graphic potential (k) graphic potential (k) into into its its constituent constituent parameters parameters (R'o and m) these life-history life-history parameters on extinction. Different combin relative effects effects of of these parameters on extinction. Different combinrelative R' 0 and and rn m may may give give rise rise to to the the same potential (k), (k), but but ations ations of of R'o same demographic demographic potential have population persistence persistence on on the the landland have very very different different consequences consequences for for meta metapopulation scape extinction thresholds. thresholds. On On fractal fractal landscapes, reproduc scape in in terms terms of of their their extinction landscapes, reproductive 0 ) had population persistence tive output output (R' (R'o) had aa much much greater greater effect effect on on population persistence (he) (h~) than than dispersal ability (m), (m), which which is the opposite opposite of what what was found found in Lande's ((1987) 1 987) model, (in essence) essence) aa random model, which which assumes assumes (in random landscape landscape (i.e., (i.e., compare compare the the rate rate at at which which he hc declines declines as as aa function function of of increasing increasing R' R'o0 as as opposed opposed to to increasing increasing m m in in fractal fractal landscapes, landscapes, relative relative to to the the rate rate at at which which those those same same parameters parameters decline decline in in the the random random landscape, landscape, Fig. Fig. 2.3). 2.3). Enhancing Enhancing reproductive reproductive output, such as through through the conservation conservation of high-quality habitats habitats or supple supplementation mentation of of nesting nesting habitat, habitat, may may thus thus have have aa greater greater effect effect than than enhancing enhancing dispersal dispersal success, success, by by the the maintenance maintenance or or restoration restoration of of habitat habitat connectivity, connectivity, on on mitigating mitigating extinction extinction risk. risk. This This is is not not to to say say that that landscape landscape structure structure had had no no effect effect on on population population per persistence, .0) sistence, however. however. Populations Populations in in landscapes landscapes that that were were not not fragmented fragmented (H = = 11.0) were were generally generally able able to to persist persist throughout throughout almost almost the the entire entire range range of of habitat habitat avail availability .0, Fig. ability (i.e., (i.e., he h~ :::; --- 0.1; 0.1; H H = = 11.0, Fig. 2.3). 2.3). Reducing Reducing fragmentation fragmentation and and maintain maintaining habitat connectivity thus mitigate extinction risk, as expected. In fact, fact, species with with low low demographic demographic potentials, potentials, due due to to aa combination combination of of low low reproductive reproductive out output 0 ) and put (R' (R'o) and poor poor dispersal dispersal ability ability (m), (m), generally generally went went extinct extinct sooner sooner on on frag frag1987) model. mented mented landscapes landscapes (H = = 0.0) 0.0) than than predicted predicted by by Lande's Lande's ((1987) model. Because Because many many species species of of conservation conservation concern concern have have these these combined combined traits traits of of low low fecund fecundity ity and and poor poor dispersal dispersal ability, ability, such such species species may may be at at a greater risk of extinction from from habitat habitat loss loss and and fragmentation fragmentation than than previously previously suspected. suspected. The The problem problem of of how how habitat habitat fragmentation fragmentation affects affects extinction extinction thresholds thresholds has has also also been been tackled tackled by by evaluating evaluating the the metapopulation m e t a p o p u l a t i o n capacity capacity of of the the landscape. landscape. The population capacity The meta metapopulation capacity is is basically basically the the sum sum of of the the relative relative contribution contribution of of
KIMBERLY KIMBERLY A. WITH
36 36 Fractal H = O.O
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Fig. Fig. 2.3 2.3 Fragmentation effects effects on extinction thresholds for species species with with different demo demographic potentials (the combined effects of reproductive output, output, R'o, Ro, and dispersal ability, m). Random landscapes are the most extensively fragmented; .0 are the fragmented; fractal landscapes, H H= = 11.0 least fragmented (i.e., (i.e., clumped; Fig. Fig. 2.1 a). Data from and King (1 999b). 2.1a). from With and (1999b).
individual individual patches patches (their (their "value" "value")to ) to metapopulation metapopulation persistence persistence based based on their size and degree of connectivity connectivity to other other habitat habitat fragments fragments in the landscape landscape (Hanski and Ovaskainen, 2000; (Hanski 2000; Chapter Chapter 4). This is a spatially realistic exten extension 1 969) model landscape structure sion of of Levin's Levin's ((1969) model in in which which landscape structure (patch (patch area area and and iso isolation) is allowed to affect the metapopulation metapopulation processes processes of colonization colonization and and extinction. extinction. For For example, example, the the probability probability of of extinction extinction is is calculated calculated as as aa function function of of the the inverse inverse of of patch patch area area because because extinction extinction is is more more likely likely in in small small patches patches than (see Chapter than in in large large ones ones (see Chapter 4 4 for for the the mathematical mathematical details details of of this this model). model). For population capacity For aa given given landscape, landscape, the the meta metapopulation capacity increases increases with with the the disper dispersal enhances connectivity sal range range of of the the species species because because dispersal dispersal enhances connectivity and and thus thus patch patch colonization colonization rates. rates. For For aa given given species, species, landscapes landscapes can can be be ranked ranked according according to to their capacity populations. A their capacity to to support support viable viable meta metapopulations. A landscape landscape is is capable capable of of supporting supporting a viable metapopulation metapopulation if its metapopulation metapopulation capacity (analogous to model) exceeds by to the the fraction fraction of of habitat habitat in in Lande's Lande's model) exceeds aa threshold threshold determined determined by the potential" of the species' the "metapopulation "metapopulation potential" of the the species species (the (the ratio ratio of of the species' extinc extinction rates, analogous analogous to potential in tion to to colonization colonization rates, to the the demographic demographic potential in Lande's Lande's model). Landscape fragmentation, fragmentation, created model). Landscape created by by the the random random destruction destruction of of habi habitat, tat, resulted resulted in in aa decline decline in in the the metapopulation metapopulation capacity capacity of of the the landscape landscape that that was habitat was roughly roughly proportional proportional to to the the amount amount of of habitat habitat lost. lost. Destruction Destruction of of habitat in population capacity in large large blocks blocks caused caused the the meta metapopulation capacity to to decline decline slower slower than than the the loss loss of of habitat, habitat, and and therefore therefore was was less less detrimental detrimental to to metapopulation metapopulation persist persistence. ence. Landscape Landscape structure structure thus thus affects affects extinction extinction thresholds thresholds and and metapopula metapopulation tion persistence, persistence, which which is is consistent consistent with with the the findings findings of of other other spatially spatially realistic realistic metapopulation models (Hill and 999; With 999b). metapopulation models (Hill and Caswell, Caswell, 11999; With and and King, King, 11999b). Although Although these these theoretical theoretical investigations investigations demonstrate demonstrate that that some some critical critical level level of of habitat habitat is is required required for for metapopulation metapopulation persistence, persistence, what what empirical empirical evidence evidence is is there support the extinction thresholds? there to to support the existence existence of of extinction thresholds? Although Although extinction extinction thresholds thresholds have have been been quantified quantified mathematically mathematically for for various various species species based based on on available available demographic demographic information information and and estimates estimates of of the the fraction fraction of of suitable suitable habi habitat tat in in the the landscape landscape (e.g., (e.g., Lande, Lande, 1988; 1988; Carlson, Carlson, 2000), 2000), extinction extinction thresholds thresholds have have been been identified identified empirically empirically as as an an abrupt abrupt decline decline in in the the occupancy occupancy of of habitat habitat patches patches across across aa series series of of landscapes landscapes that that vary vary in in the the amount amount of of habitat. habitat. For For example, (Melitaea cinxia) cinxia) exhibited exhibited example, the the endangered endangered Glanville Glanville fritillary fritillary butterfly butterfly (Melitaea
2. METAPOPULATION METAPOPULATIONDYNAMICS DYNAMICS 2.
37 37
metapopulation capacity of patch net neta threshold response to declines in the metapopulation works distributed among the A land Islands in southwest Finland (Hanski and ]kland Ovaskainen, Ovaskainen, 2000; 2000; Chapter Chapter 4). 4). Many Many birds birds may may not not exhibit exhibit aa threshold threshold response response to the amount of habitat, however, particularly if they are migratory and exist regionally due to coupled source and sink landscape dynamics (With and King, 200 1 ). Such species may be able to occupy all remaining habitat fragments even 2001). found for most neotropical migra migrain extensively fragmented landscapes, as was found tory songbirds across a landscape gradient of increasing agricultural dominance al., 1999). Threshold Threshold responses to a in southern Ontario, Canada (Villard et aI., reduction reduction in in forest forest cover cover were were generally generally absent absent for for most most species, species, except except for for two two species (ovenbird, Seiurus Seiurus aurocapillus; aurocapillus; black-and-white warbler, Mniotilta Mniotilta varia) varia) <10% that were not found in landscapes with < 1 0% mature forest. For For many many species, species, such such as as neotropical neotropical migrants, migrants, the the efficiency efficiency of of patch patch occupancy occupancy does does not not decline decline with with habitat habitat loss loss and and fragmentation. fragmentation. Many Many plant plant species may also maintain maintain constant constant patch occupancy occupancy despite a reduction reduction in i.e., suitable habitat, habitat, due to life-history strategies such as seed dormancy dormancy ((i.e., species escape in time as well as space; Chapter 8 ) . Although Chapter 118). Although the extinction extinction threshold threshold is defined as the fraction fraction of all sites that that are suitable but but not not occu occulandscape, Eriksson and and Kiviniemi ((1999) performed a modified modified cal calpied on a landscape, 1999) performed culation in which which the "quasi-equilibrium" "quasi-equilibrium" threshold threshold was obtained obtained as only the culation fraction of suitable sites that that were not not occupied occupied (hi (h'c). 44% fraction c ) . By this measure, 44% ((8/18) 8/1 8 ) of plants they Sweden were were of the the grassland grassland plants they evaluated evaluated in in southeastern southeastern Sweden extinction threshold, threshold, which which means that that although such existing below the extinction species were currently found found in these landscapes, landscapes, the amount amount of suitable habi habitat tat was was not not sufficient sufficient to to permit permit the the long-term long-term persistence persistence of of these these species; species; species response to habitat loss. loss. Thus, Thus, the the identification identification of of species exhibited exhibited aa lagged lagged response to habitat extinction and availability availability of suitable suitable extinction thresholds, thresholds, based based on site occupancy occupancy and habitat, may not always be sufficient population persistence and habitat, may not always be sufficient for for evaluating evaluating population persistence and extinction an analysis of population responses responses to to extinction risk. In such cases, an of lagged lagged population Extinction Risk in Dynamic Dynamic Landscapes). Landscapes). landscape landscape change change is required required (see Extinction Risk in
The Source-Sink Source-Sink Potential Potential of of Landscapes The Landscapes Spatial resulting from from differences in in the size, shape, and quality quality Spatial heterogeneity, resulting the size, shape, and of habitats habitats comprising the landscape, affects species' demographic demographic rates. of Reproductive success may be be maximized, maximized, or or survivorship survivorship may may be in Reproductive success may be minimized, minimized, in habitat. Habitat-specific Habitat-specific survivorship survivorship and and reproductive reproductive success set aa particular particular habitat. success set the stage source-sink dynamics dynamics (Pulliam, (Pulliam, 1988) 1988) in in which population growth growth the stage for for source-sink which population rates are are positive positive (birth ( birth rates rates exceed exceed death death rates) rates) in in some some patches patches (sources) (sources) but but rates are negative negative in in others others (sinks). (sinks). The The relative relative amount amount of of source source and and sink sink habitat habitat on on are landscape may may thus thus affect affect persistence persistence of of the the metapopulation metapopulation at at the the landscape landscape a landscape scale (Pulliam (Pulliam and and Danielson, Danielson, 1991; 1 99 1 ; Donovan Donovan et et al., aI., 1995b; 1 995b; Chapter Chapter 16). 1 6) . landscape perspective perspective is ultimately ultimately required required to to assess assess source-sink source-sink dynamdynam A landscape and to to evaluate evaluate how how changes changes in landscape landscape structure, structure, such such as from habitat ics and from habitat fragmentation or fragmentation or land-use land-use change, change, may may affect affect these these dynamics dynamics and and thus thus meta population persistence. Unfortunately, Unfortunately, most most of of the the previous previous efforts efforts to to metapopulation model source-sink source-sink dynamics dynamics have have been been spatially spatially implicit implicit (e.g., (e.g., Pulliam Pulliam model and and Danielson, Danielson, 1991), 1 99 1 ), including including those those that that have have attempted attempted to to determine determine the the effects of of habitat habitat fragmentation fragmentation on on the the source-sink source-sink status status of of populations populations effects
KIMBERLY WITH
KIMBERLY A. A. WITH
38 38
( e.g., Donovan Donovan et et al., aI., 1995a; 1 995a; but but see see Ritchie, Ritchie, 1997). 1 997). Although Although demographic demographic (e.g., rates vary vary spatially spatially in in such such models, models, they they are are usually usually fixed fixed input input parameters parameters rates that are are independent independent of of landscape landscape structure. structure. In In birds, birds, for for example, example, landscape landscape that structure is is known known to to affect affect reproductive reproductive output output in in many many species species due due to to higher higher structure edge effects effects in in fragmented fragmented landscapes landscapes in in which which nesting nesting success success is is lower lower in in habihabi edge tat fragments fragments because because of of greater greater nest nest predation predation or or brood brood parasitism parasitism (such (such as as tat Moluthrus in North America ) along frag by the the brown-headed by brown-headed cowbird, cowbird, M o l u t h r u s ater, ater, in North America) along fragedges (Donovan (Donovan et et al., aI., 1995b, 1 995b, 1997). 1 997). Thus, Thus, reproductive reproductive output output (a (a ment edges ment demographic rate) rate) is spatially dependent dependent and and varies varies as as aa function function of patch size size demographic is spatially of patch and shape, shape, being being reduced reduced in in fragments fragments dominated dominated by by edge edge and and maximized maximized in in and large patches patches of of contiguous contiguous habitat. habitat. large With and King King (2001) (20 0 1 ) devised devised aa functional functional relationship patch With and relationship between between patch structure and and reproductive reproductive success success for for neotropical neotropical migratory migratory songbirds songbirds as as part part structure of aa spatially spatially structured structured demographic demographic model model developed developed to to assess assess the the of source-sink potential potential of of fragmented landscapes. Reproductive Reproductive success success declines declines source-sink fragmented landscapes. as effects) . For For example, example, some some as aa function function of of increasing increasing edge edge (i.e., (i.e., negative negative edge edge effects). species were "edge sensitive" and decline in in reproductive reproductive out species were "edge sensitive" and exhibited exhibited aa steep steep decline output in small irregularly shaped shaped patches were dominated put in small or or irregularly patches that that were dominated by by edge edge (edge (edge index � .0; Fig. Fig. 2.4a). 2Aa). The The demographic of landscape landscape structure structure index --~ 11.0; demographic consequences consequences of were assessed the expected expected number of female female offspring offspring produced produced per per were assessed as as the number of female, per per patch, patch, for for all across the the entire entire landscape landscape (bL). ( bLl. A A simple simple female, all individuals individuals across and two-stage table combining combining fecundity fecundity (bL) (bLl and survivorship (juvenile, (juvenile, So two-stage life life table and survivorship So and adult, s) was for each 2Aa) in in aa given given landscape adult, s) was then then constructed constructed for each species species (Fig. (Fig. 2.4a) landscape (e.g., Fig. Fig. 2.1a). 2.1a). From From the the life table, we we calculated calculated the finite rate rate of of increase increase for for (e.g., life table, the finite the entire entire landscape landscape population population (A.Ll as the the solution solution to to the the characteristic characteristic equa the (XL) as equation 988): tion (Lande, (Lande, 11988): kL,~ _ skL,~-1
-
-
bLl~ = 0
(2.2)
for for A.k � - 11 and and 00 < < ss < < 1, 1, where where lex Is is is survivorship survivorship at at the the age age of of first first breeding, breeding, ss isis the (> 11 yr), yr), and the annual annual probability probability of of survivorship survivorship for for breeding breeding adults adults (> and bL bE is is
derived derived from from the the population population across across the the entire entire landscape. landscape. The The landscape landscape popu population was .0, declining lation was stable stable when when A.L XL = 11.0, declining when when A.L XL < <11 and and increasing increasing when when A.L 1 . The annual rate change in %/yr) is XL > >1. The annual rate of of change in the the size size of of the the metapopulation metapopulation ((%/yr) is (A.L .0) >,* 1100. 00. Thus, (XL -- 11.0) Thus, this this modeling modeling approach approach treats treats aa demographic demographic rate rate (b) (b) as as aa spatially spatially dependent dependent variable; variable; it it is is aa model model output output rather rather than than aa fixed fixed model model parameter parameter as as in in traditional traditional demographic demographic models. models. Furthermore, Furthermore, this this approach approach extends extends the the concept concept of of source-sink source-sink populations populations from from the the scale scale of of patches to patches to the the entire entire landscape, landscape, such such that that the the potential potential of of aa given given landscape landscape to to function function as as aa population population source source or or sink sink is is ultimately ultimately assessed. assessed. For For species species with with low low edge-sensitivity edge-sensitivity (Fig. (Fig. 2Aa), 2.4a), landscapes landscapes supported supported viable viable metapopulations metapopulations and and had had the the potential potential to to function function as as sources sources across across aa wide 1 ) . Fragmented wide range range of of available available habitat habitat (Fig. (Fig. 2Ab; 2.4b; With With and and King, King, 200 2001). Fragmented landscapes populations of landscapes (random) (random) could could not not support support viable viable meta metapopulations of this this species, species, however, .0) when however, and and functioned functioned as as sinks sinks (A.L (XL < < 11.0) when habitat habitat fell fell below below 30% 30% (Fig. (Fig. 2Ab). 2.4b). The The situation situation was was bleaker bleaker for for species species with with high high edge-sensitivity, edge-sensitivity, which which had had aa difficult difficult time time persisting persisting in in landscapes landscapes with with <50% <50% habitat habitat even even when when the the landscape landscape was was managed managed to to preserve preserve large large tracts tracts of of contiguous contiguous habi habi.0, Fig. tat tat (H (H = = 11.0, Fig. 2Ac). 2.4c).
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Fig. decline in in reproductive Fig. 2.4 2..4 (a) (a) Degree Degree of of edge edge sensitivity s e n s i t i v i t y- the the decline reproductive success success as as aa function function of increasing edge songbirds. (b) of increasing edge m for for aa couple couple of of generic generic migratory migratory songbirds. (b) Effect Effect of of fragmentation fragmentation (random (random = = maximum maximum fragmentation) fragmentation) on on aa species species with with low low edge-sensitivity. edge-sensitivity. (c) (c) Effect Effect of of frag fragmentation high edge-sensitivity. King (2001 mentation on on aa species species with with high edge-sensitivity. Modified Modified from from With With and and King (2001).).
This This spatially spatially structured structured avian avian demographic demographic model model was was parameterized parameterized for for the the Henslow's Henslow's sparrow sparrow (Ammodramus (Ammodramus henslowii) henslowii) in in aa heavily heavily managed managed land landscape of north-central aI., 2000). north-central Kentucky in the eastern eastern United United States (King et al., Henslow's Henslow's sparrow sparrow is is an an area-sensitive area-sensitive species species that that requires requires large large tracts tracts of of dense dense 994) and tallgrass tallgrass prairie prairie for for nesting nesting (patch (patch sizes sizes �30 ->30 ha; ha; Herkert, Herkert, 11994) and is is thus thus aa species of concern given 1 % of species of conservation conservation concern given that that < <1% of the the historical historical tallgrass tallgrass prairie Plains of (Knopf and prairie remains remains throughout throughout the the Great Great Plains of North North America America (Knopf and Sampson, 11994). 994). Only Only 0.6% suitable breeding breeding Sampson, 0.6% of of the the managed managed landscape landscape was was suitable habitat, which barely habitat, which was was fragmented fragmented and and consisted consisted of of many many small small patches patches barely large area requirements large enough enough to to support support aa breeding breeding pair, pair, let let alone alone meet meet the the area requirements of this species (the largest patch was 5511 ha ha).l . The finite rate of increase for for the metapopulation in the Henslow's Henslow's sparrow sparrow metapopulation in this this landscape landscape was was "-L kL = - 0.86. 0.86. Thus, Thus, the Henslow's sparrow sparrow was declining at an annual annual rate of 14%/yr 14 %/yr such that that the
KIMBERLY KIMBERLY A. A. WITH WITH
40 40
landscape output, b, be increased landscape was was aa sink sink for for this this species. species. Reproductive Reproductive output, b, must must be increased by about 1.5 by about 1.5 times times its its current current level level to to restore restore the the Henslow's Henslow's sparrow sparrow to to aa stable stable or population, which or increasing increasing meta metapopulation, which would would require require an an increase increase in in the the land landscape-wide 39 to 8 % . To scape-wide nesting nesting success success from from 39 to 558%. To reverse reverse the the current current decline decline in in this population, aa land minimize disturbances this meta metapopulation, land manager manager should should thus thus minimize disturbances that that contribute habitat fragmentation contribute to to habitat fragmentation of of grassland grassland habitat, habitat, thereby thereby increasing increasing reproductive output. output. patch sizes and minimizing minimizing edge effects that decrease reproductive Although Although this this spatially spatially structured structured demographic demographic model model focuses focuses on on how how patch patch geometry geometry for for aa single single habitat habitat type type affects affects reproductive reproductive success, success, this this approach approach can easily to landscapes in can be be extended extended easily to heterogeneous heterogeneous landscapes in which which reproductive reproductive success or varies as success or survivorship survivorship additionally additionally varies as aa function function of of habitat habitat quality. quality.
Extinction Extinction Risk Risk in in Dynamic Dynamic Landscapes Landscapes The population processes The effects effects of of landscape landscape structure structure on on the the meta metapopulation processes discussed discussed thus assumed aa static landscape in thus far far have have assumed static landscape in which which the the amount, amount, suitability, suitability, and and configuration of patches remain remain unchanged configuration of habitat habitat patches unchanged on on the the landscape. landscape. Even Even studies that habitat loss studies that have have explored explored the the effects effects of of habitat loss and and fragmentation fragmentation on on meta population persistence persistence have been conducted metapopulation have been conducted on on aa series series of of static static land landscapes habitat availability scapes representing representing aa gradient gradient of of habitat availability and and fragmentation fragmentation sever severity 9 96; Hill Caswell, 11999; 999; With ity (e.g., (e.g., Bascompte Bascompte and and Sole, Sol~, 11996; Hill and and Caswell, With and and King, King, 11999b, 999b, 2001 2001),) , which which assumes assumes that that these these landscapes landscapes all all lie lie on on aa particular particular tra trajjectory ectory of of landscape landscape change. change. Real Real landscapes landscapes are are not not static, static, however, however, especially especially given given the the current current rate rate at at which which most most landscapes landscapes are are being being transformed transformed by by human human land-use land-use activities. activities. Different Different trajectories trajectories of of land-use land-use change change could could gen generate consequences for erate similar similar landscape landscape patterns, patterns, but but have have very very different different consequences for the the dynamics persistence of dynamics and and persistence of metapopulations metapopulations on on these these landscapes. landscapes. The The rate rate of of landscape landscape change change is is an an important important component component of of landscape landscape structure 9 92; Keymer structure that that affects affects extinction extinction risk risk (e.g., (e.g., Fahrig, Fahrig, 11992; Keymer et et aI., al., 2000) 2000) and and which which may may be be responsible responsible for for generating generating metapopulation metapopulation dynamics, dynamics, particularly successional habitat Hanski, 11999a; 999a; Johnson, particularly in in ephemeral ephemeral or or successional habitat ((Hanski, Johnson, 2000). 2000). Patch Patch demographics, demographics, such such as as the the life life span span of of aa patch, patch, drive drive the the dynamics dynamics of population in critical of the the meta metapopulation in these these systems. systems. For For aa given given species, species, there there is is aa critical rate changes too relative to rate of of patch patch turnover turnover in in which which the the landscape landscape changes too fast fast relative to the the scale population persist scale of of the the extinction-colonization extinction-colonization process process to to permit permit meta metapopulation persistence ence (Keymer (Keymer et et aI., al., 2000). 2000). Metapopulation Metapopulation extinction extinction is is thus thus predicted predicted to to occur more frequently than in static landscape occur more frequently in in dynamic dynamic than in static landscape scenarios. scenarios. These These recent recent theoretical theoretical treatments treatments of of metapopulations metapopulations on on dynamic dynamic land landscapes scapes have have been been concerned concerned primarily primarily with with systems systems in in which which there there is is aa constant constant rate habitat turnover (but see Hanski and rate of of habitat turnover (but see Hanski and Ovaskainen, Ovaskainen, 2002; 2002; Ovaskainen Ovaskainen and and Hanski, Hanski, 2002; 2002; Chapter Chapter 4). 4). Many Many landscapes landscapes are are subjected subjected to to chronic chronic habi habitat tat loss loss and and fragmentation, fragmentation, however, however, in in which which habitat habitat that that has has been been destroyed destroyed is restored. To this latter is not not restored. To address address this latter scenario, scenario, Schrott, Schrott, With With and and King King (unpub (unpublished) extended the spatially structured structured avian demographic model of With and King (200 1 ) to dynamic landscape King (2001) to aa dynamic landscape context context in in which which habitat habitat was was destroyed destroyed at at various %/yr) until denuded. The various rates rates (0.5, (0.5, 11 and and 55%/yr) until the the landscape landscape was was entirely entirely denuded. The most surprising result result of population appeared appeared to most surprising of this this analysis analysis was was that that the the meta metapopulation to persist across aa greater range of lost rap persist across greater range of habitat habitat destruction destruction when when habitat habitat was was lost rapthan when it was destroyed slowly (0.5%/yr); in words, idly ((5%/yr) 5%/yr) than in other words,
2.
METAPOPULATION DYNAMICS DYNAMICS METAPOPULATION
41 41
extinction extinction appeared appeared to to occur occur sooner sooner in in landscapes landscapes subjected subjected to to lower lower rates rates of of disturbance disturbance (Fig. (Fig. 2.5a) 2.5a).. This This paradox paradox is is resolved resolved by by considering considering the the decline decline in in metapopulation metapopulation growth growth rates rates as as aa function function of of time time (Fig. (Fig. 2.5b), 2.5b), which which demon demonstrates strates that that populations populations on on landscapes landscapes subjected subjected to to rapid rapid rates rates of of habitat habitat loss loss ((5%/yr) 5%/yr) will will go go extinct extinct within within 20 20 yr yr (the (the time time to to total total landscape landscape denudation), denudation), whereas whereas populations populations in in landscapes landscapes subjected subjected to to lower lower rates rates of of habitat habitat loss loss can can apparent prolonged prolonged persistence of persist for up to three times as long. The apparent meta populations in metapopulations in landscapes landscapes undergoing undergoing rapid rapid change change results results from from aa lagged lagged response response by by the the species. species. The The generic generic migratory migratory songbird songbird being being modeled modeled in in this this study %/yr, total study had had aa life life span span of of 88 yr, yr, such such that that at at aa habitat habitat loss loss rate rate of of 55%/yr, total denudation denudation of of the the landscape landscape would would occur occur in in aa little little over over two two generations. generations. The The landscape landscape is is changing changing more more rapidly rapidly than than the the demographic demographic potential potential of of the the species, species, and and thus thus declines declines in in the the metapopulation metapopulation growth growth rate rate (ALJ (kL) lag lag behind behind the rate of habitat loss. This "extinction debt" has been demonstrated to be
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Years Years % habitat habitat destroyed/yr) destroyed/yr) on on metapopumetapopu Fig. 2.5 2.5 (a) Effect of the the rate of of habitat habitat destruction destruction (r, Fig. Effect of (r, % lation persistence for aa species species with with intermediate intermediate edge edge sensitivity sensitivity in in moderately fragmented lation persistence for moderately fragmented 0.5) dynamic dynamic landscapes. landscapes. Because Because of of the the initial initial conditions conditions of of the the model, model, in in which which the the (H == 0.5) (H (AL == 1.0), 1 .0), the the landscape landscape population population can can only only decline decline metapopulation growth rates are are stabilized stabilized (~kL metapopulation growth rates as habitat is lost. lost. (b) (b) Time TIme to to extinction extinction for for the the same same species species in in dynamic dynamic landscapes landscapes undergoing undergoing as habitat is moderate fragmentation fragmentation (H (H -= 0.5) 0.5) at at different different rates. From From Schrott, Schrott, With With and and King King (unpublished). (unpublished). moderate
KIMBERLY WITH KIMBERLY A. A. WITH
42 42
especially great for meta populations that especially great for metapopulations that are are close close to to their their extinction extinction thresholds thresholds following habitat loss following habitat loss and and fragmentation fragmentation (Hanski (Hanski and and Ovaskainen, Ovaskainen, 2002). 2002). These These results results emphasize emphasize the the importance importance of of understanding understanding the the historical historical landscape patterns, habiforces shaping landscape patterns, such as the rate of land-use change or habi tat tat destruction, destruction, for for assessing assessing species' species' extinction extinction risk. risk. Two Two landscapes landscapes with with iden identical tical metrics metrics could could have have achieved achieved their their present present state state via via vastly vastly different different trajectories change, with trajectories of of landscape landscape change, with different different consequences consequences for for metapopula metapopulation (Fig. 2.5a). landscape metrics tion persistence persistence (Fig. 2.5a). Conventional Conventional landscape metrics thus thus cannot cannot be be used populations in used to to evaluate evaluate extinction extinction risk risk for for meta metapopulations in dynamic dynamic landscapes. landscapes. Time Time lags lags in in species' species' responses responses to to landscape landscape change change may may also also create create relict relict dis distributions species' occurrence explained by tributions in in which which the the species' occurrence is is better better explained by aa historical historical landscape configuration "ghosts of landscape past"; landscape configuration than than the the current current one one (("ghosts of landscape past"; Nagelkerke Nagelkerke et et aI., al., 2002). 2002). For For example, example, the the distribution distribution of of carabid carabid beetles beetles (Abax (Abax parallelepipedus) within an parallelepipedus) in in aa hedgerow hedgerow network network within an agriculturally agriculturally dominated dominated landscape France was landscape of of France was better better explained explained by by the the well-connected well-connected hedgerow hedgerow distribution from Burel, 1998). distribution from 40 40 yr yr ago ago than than the the current current network network (Petit (Petit and and Burel, 1998). From From aa conservation conservation and and land-management land-management standpoint, standpoint, the the potential potential for for lagged lagged population responses is disquieting because population responses is especially especially disquieting because the the effects effects of of landscape landscape change change may may go go unnoticed unnoticed for for long long periods periods of of time, time, such such that that the the window window of of opportunity opportunity for for affecting affecting aa recovery recovery may may close close before before the the problem problem is is realized realized and action is taken. Alternatively, such lagged effects may buy the necessary time measures before time in in which which to to implement implement conservation conservation and and restoration restoration measures before the the species goes extinct. This assumes that that the problem problem can be recognized in time, which argues for the importance importance of performing performing theoretical theoretical and empirical analy analyses landscape change ses of of the the effect effect of of dynamic dynamic landscape change on on extinction extinction risk. risk.
Relative Relative Effects Effects of of Habitat Habitat Loss Loss and and Fragmentation Fragmentation on on Metapopulation Metapopulation Persistence Persistence A A number number of of empirical empirical and and theoretical theoretical investigations investigations have have attempted attempted to to the relative relative importance importance of of the the amount amount of of habitat habitat versus versus the the degree degree of of assess the habitat habitat fragmentation fragmentation on species occurrence occurrence and and extinction extinction thresholds thresholds (e.g., McGarigal 995; Trzcinski et aI., 999; Villard et aI., 999; McGarigal and and McComb, McComb, 11995; al., 11999; al., 11999; Fahrig, 997, 2002; Fahrig, 11997, 2002; Flather Flather and and Bevers, Bevers, 2002). 2002). Part Part of of the the difficulty difficulty in in evalu evaluating the relative relative effects of these two two components components of landscape landscape structure, structure, however, thresholds are however, has has been been differences differences in in how how extinction extinction thresholds are defined, defined, which which is measure of is aa consequence consequence of of the the specific specific modeling modeling approach approach or or measure of population population viability viability used used to to assess assess extinction extinction risk. risk. Depending Depending on on the the modeling modeling construct, construct, extinction thresholds thresholds have been defined variously as extinction have been defined variously as the the critical critical level level of of habi habitat 1 ) is tat at at which which the the population population ((1) is unlikely unlikely to to persist persist for for (or (or after) after) some some speci specified amount individual-based simulation amount of time (e.g., duration duration of run in individual-based simulation models; Fahrig, 997; Flather 1 ), (2) models; Fahrig, 11997; Flather and and Bevers, Bevers, 200 2001), (2) no no longer longer occupies occupies any any of the available habitat habitat (metapopulation (metapopulation models, where p'; p* = 0; Lande, 1987; Bascompte 996; With 999b), or 3 ) is Bascompte and and Sole, Sold, 11996; With and and King, King, 11999b), or ((3) is no no longer longer stable, stable, as as assessed assessed by by some some demographic demographic index, index, such such as as net net reproductive reproductive rate rate (Ra) (Ro) or or the population growth the finite finite rate rate of of population growth (A) (k) (spatially (spatially structured structured demographic demographic models, .0; With measures of pop models, where where Ra Ro or or Ak < < 11.0; With and and King, King, 2001 2001).). Some Some measures of population viability may be more more sensitive to fragmentation fragmentation effects, however however ((Flather Flather and Bevers, Bevers, 2002); measures of population 2002); for example, example, patch-based patch-based measures population
2.
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viability (patch occupancy, p" p*,, of metapopulation metapopulation models) may be more sen sensitive to fragmentation fragmentation effects than than probabilities probabilities of population population persistence derived from individual-based simulation models. Thus, the debate over the relative effects of habitat population persist habitat loss and fragmentation fragmentation on meta metapopulation persistence is muddled muddled by the specific modeling approach approach employed in theoretical studies (Flather and Bevers, 2002), 2002), the corresponding corresponding measure of population population viability used, and the taxonomic taxonomic group group being assessed assessed in the case of empirical studies. For example, birds may not be the best test of the relative importance importance of fragmentation effects, effects, as many species are efficient at occupying available habitat 999). habitat even when habitat habitat is rare (e.g., Villard Villard et ai., al., 11999). Nevertheless, population Nevertheless, habitat amount is generally the best predictor of meta metapopulation persistence, although fragmentation m the explicit arrangement of patches in space m becomes increasingly important below the extinction threshold (Fahrig, 11997; 997; Flather and Bevers, 2002). As pointed out in the previous section, however, the discussion of the relative effects of habitat loss and fragmentation on metapop metapopinformation on landscape dynamics. ulation persistence is moot in the absence of information The rate of patch turnover turnover or habitat destruction destruction has been shown to have a more profound profound effect on extinction thresholds than either the amount amount or the fragmen fragmentation of habitat (Keymer et ai., al., 2000; Schrott, With and King (unpublished). (unpublished).
2.3 2.3
SUMMARY SUMMARY Although landscape population ecology share a common landscape ecology and meta metapopulation common goal in predicting predicting the persistence of spatially structured structured populations populations in frag fragmented landscapes, they differ in scope and academic tradition, tradition, which is reflected in the different approaches approaches typically employed by ecologists from the two two disciplines. Landscape Landscape ecology is an interdisciplinary field that that arose from European European traditions traditions of regional geography and vegetation science, science, and which combined combined the spatial approach of the geographer with the functional functional approach approach of the ecologist (Turner et ai., 1 ) . As a consequence, there has been greater al., 200 2001). emphasis placed on remotely sensed data, geographical information information systems, and spatial statistics to generate, display, and analyze complex landscape landscape pat patterns within within landscape landscape ecology, as opposed opposed to the more abstract abstract representation representation of landscapes inspired by the patch-based ecological theory adopted adopted by meta population ecology. In North metapopulation North America, the systems ecology background background of many "first-generation" "first-generation" landscape ecologists contributed contributed to the widespread widespread use of computer computer simulation models to tackle problems related to the effect of land-use change on resource resource management, management, an application application that that additionally could take advantage advantage of GIS. GIS. Subsequently, this led to the development development of spa spatially explicit population population models, the main tool of the landscape landscape ecologist for assessing population population viability in fragmented landscapes. In contrast, contrast, metapopu metapopulation ecologists hail from a background population ecology that background in population that is rich in mathematical mathematical theory and thus tend to approach approach the problem of population population persistence persistence in fragmented landscapes analytically rather rather than than numerically. Despite Despite its its diverse diverse disciplinary disciplinary breadth, breadth, landscape landscape ecology is is fundamentally fundamentally concerned with the effects effects of spatial pattern on ecological processes (Turner, 11989), 989), at whatever scale spatial heterogeneity emerges. In the context context metapopulation dynamics, any study that that incorporates incorporates the effect of spatial of metapopulation
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KIMBERLY KIMBERLYA. WITH
pattern, such as habitat fragmentation or matrix heterogeneity, on processes con contributing to metapopulation persistence is, by definition, adopting a landscape ecological perspective. Thus, it could be argued that the "exciting scientific synthesis" between landscape ecology and metapopulation theory that was envienvi sioned by Hanski and Gilpin ((1991) 1 99 1 ) a decade ago is well underway. This is evident in current metapopulation metapopulation theory that that utilizes spatially explicit (or realis realistic) models (e.g., Hanski, 11999b) 999b) and by the use of analytical approaches from metapopulation 999b). metapopulation theory in landscape ecology (e.g., With and King, 11999b). The goal of this chapter has has been to demonstrate demonstrate that landscape ecology has more to offer meta population ecology than metapopulation than just the incorporation incorporation of a broader spatial scale or landscape heterogeneity to existing meta population theory by metapopulation providing a spatial context for understanding processes contributing to the populations. Nevertheless, there are still sev dynamics and persistence of meta metapopulations. several areas at this nexus of landscape ecology and meta population biology that metapopulation that are in need of further research: First, spatially structured demographic models require further development and testing. Demographic factors, such as repro reproductive output output and survivorship, may be spatially dependent. Estimates of meta population viability that ignore this spatial dependency may give erroneous metapopulation and overly optimistic assessments of a species' status on a landscape (With and King, 200 1 ) . Second, matrix effects on dispersal success and extinction risk need 2001). importance of to be evaluated further. Research has demonstrated the potential importance managing the matrix for enhancing colonization success and reducing extinction risk, and thus we need to move beyond assessment of mere patch size size and iso isolation effects on metapopulation persistence (e.g., Fleishman et aI., al., 2002), par particularly in understanding understanding asymmetrical flows among populations populations and evaluating the source-sink potential of landscapes for species of conservation concern. Third, the effect of landscape dynamics on extinction thresholds deserves greater attention (e.g., Keymer et aI., al., 2000; Schrott, With and King (unpublished)). Chronic habitat loss and fragmentation increase the potential for lagged responses to landscape change, which may produce an extinction debt (Hanski and Ovaskainen, 2002), such that the status and future viability of meta populations may not be well predicted by current landscape patterns. metapopulations Fourth, meta population viability analysis needs to be extended to a broader, metapopulation approach" is particularly important important regional scale. A "metalandscape modeling approach" for assessing dynamics among among source-sink source-sink landscape populations populations (e.g., Donovan 995a). Finally, empirical work to address these issues, analo Donovan et aI., al., 11995a). analogous to how population theory has been applied to real meta populations in how meta metapopulation metapopulations metapopulation metapopulation ecology (Chapters 4 and 5), should be a research priority in landscape ecology. Otherwise, Otherwise, continued continued progress progress toward toward this developing syn synthesis between landscape ecology and metapopulation metapopulation ecology will slow.
3
CONTINUOUS-S PA CE CONTINUOUS-SPACE MODELS MODELS FOR PO PULATION POPULATION DYNAMICS DYNAMICS Benjamin M. Bolker
3.1 3.1
INTRODUCTION INTRODUCTION Metapopulation Metapopulation ecologists have explored the dynamics of biological popu populations that because they live in patchy that are discontinuous discontinuous both spatially ((because patchy habi habitats) and temporally ((because because local populations .The populations frequently go extinct) extinct).The success of meta population theory comes both from metapopulation from the ecological importance of such populations (e.g., populations populations in fragmented fragmented landscapes) landscapes) and and from the simplicity of meta population theory. Dealing with population dynamics in metapopulation more general continuous continuous landscapes m those that that are neither neither completely patchy nor completely hhomogeneous o m o g e n e o u s- requires some compromise between generality and tractability, which has in turn turn divided ecologists into two two camps. Landscape Landscape ecologists ecologists have focused on large-scale patterns patterns of popula population distributions distributions in heterogeneous heterogeneous environments, environments, generally considering the effects of exogenous environ exogenous heterogeneity imposed by the abiotic or biotic environment with less less concern for the endogenous endogenous heterogeneity caused by interac interactions within the populations. populations. In contrast, contrast, spatial spatial ecologists ecologists have focused on smaller scale patterns patterns of population population distribution or expansion expansion and have been more interested in endogenous endogenous than in exogenous heterogeneity. Landscape Landscape
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BENJAMIN BENJAMIN M. BOlKER BOLKER
ecologists 982), designed ecologists favor favor tactical tactical models models (Nisbet (Nisbet and and Gurney, Gurney, 11982), designed to to answer questions about populations (e.g., answer specific specific questions about specific specific populations (e.g., distributions distributions of of extinction extinction times times under under different different management management scenarios), scenarios), whereas whereas spatial spatial ecolo ecologists "toy" models, gists generally generally prefer prefer strategic strategic or or "toy" models, which which are are more more abstract abstract and and are designed to explore general principles of spatial dynamics (see Chapter ). dynamics (see Chapter 11). Landscape Landscape ecologists ecologists use use computational computational tools tools such such aass geographic geographic information information systems GIS) and models, which flexible enough systems ((GIS) and individual-based individual-based models, which are are flexible enough to to incorporate realistic patterns incorporate realistic patterns of environmental environmental heterogeneity heterogeneity and individual individual behavior, behavior, whereas whereas spatial spatial ecologists ecologists use use simpler simpler models, models, such such as as partial partial differ differential equations equations (PDEs) and interacting interacting particle systems (IPSs), which may lead to to more more general general results results and and are are easier easier to to analyze. analyze. (As (As discussed discussed in in Chapters Chapters 11 and population models models are used in entire range tactical to and 4, 4, meta metapopulation are used in the the entire range from from tactical to strategic modeling.) This chapter brief! ) overview of models for ecological dynamics chapter gives a ((brief!) dynamics in continuous continuous space, space, focusing focusing on on spatial spatial rather rather than than landscape landscape ecology. ecology. In In partic particular, ular, it it explores explores spatial spatial moment moment eequations, relatively new new framework framework for for ana anaquations, aa relatively lyzing population densities lyzing spatial spatial dynamics dynamics in in terms terms of of mean mean population densities and and spatial spatial covariances. covariances. The The chapter chapter gives gives aa sample sample derivation derivation of of aa set set of of spatial spatial moment moment equations; equations; summarizes summarizes various various applications applications of of moment moment equations equations for for single singlespecies species and and community community dynamics; dynamics; contrasts contrasts the the strengths strengths and and weaknesses weaknesses of of the the IPSs; and discusses future approach with other frameworks, frameworks, such as PDEs and IPSs; directions moment equations. directions and and potential potential of of spatial spatial moment equations. Although Although this this chapter chapter concentrates endogenous rather concentrates on on the the effects effects of of endogenous rather than than exogenous exogenous variability, variability, it it also also describes describes some some strategies strategies for for incorporating incorporating both both kinds kinds of of heterogeneity heterogeneity and and bridging the gap between spatial and landscape ecology. Spatial moment moment equa equations are a powerful tool for this task; other other advantages advantages (not to leave the reader reader in suspense for too long) include preservation preservation of the spatial and and stochastic character analytical tractability; character of of ecological ecological systems; systems; analytical tractability; and and simple simple connections connections to individual dispersal to field field data data on on individual dispersal and and performance performance and and to to well-established well-established spatial spatial statistical statistical measures. measures.
3.2 3.2
OVERVIEW OVERVIEW OF OF CONTINUOUS-SPACE CONTINUOUS-SPACE MODELS MODELS The The full full range range of of continuous-space continuous-space models models for for ecological ecological systems systems is is far far too too large to review properly properly in a short short book book chapter, but but this section gives abbrevi abbreviated ated descriptions descriptions that that serve serve at at least least to to put put spatial spatial moment moment equations equations in in perspective. For each category discussed earlier, the the text text describes the the basic approach; approach; gives gives some some examples examples of of how how it it has has been been used used to to study study ecological ecological dynamics; dynamics; discusses discusses how how models models of of this this type type have have been been used used to to explore explore the the combination of endogenous combination endogenous and exogenous exogenous variability; and and gives some starting points in in the literature literature for further points further exploration. exploration. Table Table 3.1 3.1 categorizes categorizes mathematical mathematical models models for for population population dynamics dynamics in in con continuous tinuous space. space. Most Most of of these these models models are are discussed discussed in in the the following following sections. sections. Any Any I) can model model with with discrete discrete individuals individuals (type (type = = I) can be be considered considered an an individual-based individual-based model model (although (although the the following following section section focuses focuses on on relatively relatively complex, complex, flexible flexible IBMs), IBMs), whereas whereas models models with with continuous continuous individuals individuals are are covered covered in in the the section section on on continuum continuum models. models. All All of of the the discrete-space discrete-space models models in in Table Table 3.1 3.1 have have regularly regularly L) arranged arranged sites sites or or patches patches and and so so fall fall under under the the rubric rubric of of lattice lattice models models (type (type = = L)
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TABLE TABLE 3.1 3.1
i
Partial Enumeration of Continuou s-Space Models Continuous-Space Models
Space
Time
Population
Random Random
Discrete Discrete Discrete Discrete
Discrete Discrete Discrete Discrete
Discrete Discrete Discrete Discrete
Discrete Discrete Discrete Discrete
Discrete Continuous Discrete Continuous Continuous Continuous Discrete Discrete
Continuous Continuous Continuous Continuous
Discrete Discrete Either Either
Continuous Continuous
Continuous Continuous Continuous Continuous
Continuous Continuous
Continuous Continuous Continuous Continuous
Deterministic Deterministic Cellular Cellular automaton (CA) (CA) Stochastic Stochastic Probabilistic Probabilistic CA CA = = stochastic stochastic CA CA Either Either Coupled-map Coupled-map lattice Stochastic Stochastic Interacting Interacting particle particle system system =pair ~pair approximations Either Either Integrodifference equation equation Stochastic Spatial point Stochastic Spatial point process process =spatial ~spatial moment moment equations Deterministic Integrodifferential Deterministic Integrodifferential equation (IDE), (IDE), equation (PDE) partial differential differential equation (PDE) = = reaction-diffusion reaction-diffusion equation Stochastic Stochastic IDE, PDE Stochastic StochasticIDE, PDE
Continuous Continuous Discrete Discrete
Model
Type Type"" I,L I,L L I,L I,L C I C
C C
aa "Type" model; L, lattice model; C, continuum continuum model) "Type" (I, (I, individual-based individual-based model; lattice model; model) gives gives a reference reference to the section(s} section(s) ill
i
i
i
i
that describe describe different different classes classes of models models -- some some models models fall fall in more than one class. class. As As discussed discussed in the text, "continuum" refers refers to continuous continuous population densities densities rather than continuous space. space.
(as opposed (as opposed to to the the "continuous-space" "continuous-space" models, models, where where continuity continuity is is used used in in the the narrow narrow mathematical mathematical sense sense rather rather than than the the broader broader sense sense used used elsewhere elsewhere in in the the chapter). equations, discussed chapter). Pair Pair approximations approximations and and spatial spatial moment moment equations, discussed in in some some detail approximations to models; both detail later, later, are are two two classes classes of of approximations to stochastic stochastic spatial spatial models; both eliminate the eliminate the explicitly explicitly stochastic, stochastic, discrete-individual discrete-individual nature nature of of their their parent parent mod models els but but preserve preserve some some of of the the important important properties properties of of discrete discrete stochastic stochastic spatial spatial dynamics 994). Other lead dynamics (Durrett (Durrett and and Levin, Levin, 11994). Other approximations approximations and and limits limits also also lead to models: for to connections connections between between different different classes classes of of models: for example, example, interacting interacting par particle ticle systems systems converge converge to to PDEs PDEs in in the the "hydrodynamic" "hydrodynamic" limit limit where where individual individual movement movement is is on on aa rapid rapid timescale timescale relative relative to to intraintra- and and interspecific interspecific interactions interactions (Durrett 994). (Durrett and and Neuhauser, Neuhauser, 11994).
Individual-Based I n d i v i d u a l - B a s e d Models Models By By definition, definition, an an individual-based individual-based model model tracks tracks the the fates fates of of all all of of the the indi individual vidual organisms organisms within within an an ecological ecological community. community. IBMs IBMs need need not not be be spatial, spatial, but but their their flexibility flexibility appeals appeals to to spatial spatial and and landscape landscape ecologists. ecologists. In In addition, addition, keeping keeping track track of of unique unique individuals individuals is is often often the the simplest simplest way way to to model model spatial spatial pattern pattern in in populations. populations. Spatial Spatial IBMs IBMs are are one one kind kind of of spatially spatially explicit explicit popula populamodels that individuals as tion tion models models (SEPMs): (SEPMs): models that take take individuals as discrete discrete individuals individuals occupying occupying aa continuous, continuous, usually usually two-dimensional two-dimensional landscape. landscape. If If individual individual locations locations are are described described as as discrete, discrete, infinitesimal infinitesimal points points (rather (rather than than disks disks with with aa finite finite radius, radius, for for example) example),, then then the the model model is is aa spatial spatial point point process process model, model, which which will will form form the the basis basis for for the the spatial spatial moment moment approximations approximations later later in in this this chapter. chapter. Spatial Spatial IBMs IBMs allow allow dynamics dynamics to to be be stochastic stochastic and and are are usually usually compu computational tational rather rather than than analytical. analytical. IBMs IBMs are are widely widely used used in in wildlife wildlife and and conserva conservation about organisms tion ecology, ecology, where where researchers researchers have have questions questions about organisms moving moving and and interacting aI., 1998). interacting on on very very specific specific landscapes landscapes (Wiegand (Wiegand et etal., 1998). They They are are convenient convenient because because they they can can incorporate incorporate aa wide wide range range of of individual individual behaviors behaviors
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and and landscape landscape structures. structures. The The downside downside of of these these models models is is that that they they may may be be computationally computationally intensive, data data hungry, and relatively hard hard to generalize (Wennergren ai., 11995; 995; Ruckelshaus ai., 11997; 997; Mooij (Wennergren et et al., Ruckelshaus et et al., Mooij and and Deangelis, Deangelis, 11999; 999; South, 999). South, 11999). In In addition addition to to aa large large variety variety of of tactical tactical simulation simulation models models (Turner (Turner et et ai., al., 11994; 994; Deangelis 998; Johnson 998; Werner Deangelis et et ai., al., 11998; Johnson et et ai., al., 11998; Werner et et ai., al., 2001; 2001; Mullon Mullon et et ai., al., 2002; 2002; Woodbury Woodbury et et ai., al., 2002), 2002), spatial spatial IBMs IBMs have have been been used used to to study study the the endogenous endogenous spatial spatial dynamics dynamics of of predator-prey predator-prey and and larger larger trophic communities and trophic communities and to to compare compare their their results results with with those those from from other other model model formulations (McCauley et 993; Wilson ai., 11993; 993; Keitt, formulations (McCauley et ai., al., 11993; Wilson et et al., Keitt, 1997; 1997; Schmitz Schmitz and 997; Wilson, 998; Donalson 999). Forest and Booth, Booth, 11997; Wilson, 11998; Donalson and and Nisbet, Nisbet, 11999). Forest models models are are aa well-developed well-developed subset subset of of spatial spatial IBMs IBMs that that are are individually individually based based but but only indi only sometimes sometimes fully fully spatially spatially explicit; explicit; they they usually usually track track the the locations locations of of individual 00 m vidual trees trees only only to to within within the the nearest nearest "gap," "gap," an an area area of of 1100 m 22 or or so so that that rep represents resents the the crown crown size size of of an an adult adult canopy canopy tree, tree, although although more more recent recent models models have localized localized individuals 984; Urban individuals to points points in the plane plane (Shugart, 11984; Urban et ai., al., 1991; 993; Chave, 999; Phillips 2003). Spatial 1991; Pacala Pacala et et ai., al., 11993; Chave, 11999; Phillips et et ai., al., 2003). Spatial forest forest models are tactically and models are used used both both tactically and strategically. strategically. The flexibility the effects effects on on popu The flexibility of of spatial spatial IBMs IBMs lends lends itself itself to to modeling modeling the population lation movement movement and and dynamics dynamics of of different different configurations configurations of of barriers barriers and and habitat habitat types types in in the the landscape landscape (Jonsen (Jonsen and and Taylor, Taylor, 2000; 2000; Cumming, Cumming, 2002). 2002). While While strategic strategic spatial spatial IBMs IBMs focus focus on on endogenous endogenous heterogeneity, heterogeneity, tactical tactical spatial spatial IBMs IBMs often often incorporate incorporate exogenous exogenous heterogeneity heterogeneity ~ frequently frequently derived derived from from aerial or satellite images images of a particular Caspersen et ai., 999; particular landscape landscape ((Caspersen al., 11999; Murrell Murrell and and Law, Law, 2000; 2000; Mooij Mooij et et ai., al., 2002). 2002). Further 1 992) gives Further reading. The The classic classic book book by by DeAngelis DeAngelis and and Gross Gross ((1992) gives aa good good introduction introduction to to IBMs IBMs in in general general (although (although most most of of the the material material does does not not empha emphasize spatial models), whereas Grimm and co-workers co-workers give more recent reviews and major directions directions in research (Grimm, 999; and highlight highlight some some of of the the major in IBM IBM research (Grimm, 11999; Grimm Grimm et et ai., al., 1999). 1999).
Lattice Lattice Models Models Lattice models models represent represent a continuous continuous landscape landscape as a regular (usually) square square lattice. lattice. Each Each lattice lattice cell cell may may contain contain aa single single individual; individual; aa population; population; or individuals or populations of or individuals or populations of multiple multiple species, species, depending depending on on the the model. model. Lattice individuals are Lattice cells cells change change their their state state ((individuals are born, born, grow, grow, or or die, die, or or are are eaten eaten by populations of by predators, predators, or or sites sites are are taken taken over over by by populations of other other species) species) deter deterministically or ministically or stochastically stochastically according according to to rules rules based based on on the the local local density density of of prey, prey, predators, predators, or or competitors. competitors. Interacting Interacting particle particle systems systems are are stochastic stochastic lat lattice tice models models that that run run in in continuous continuous time. time. Stochastic Stochastic or or probabilistic probabilistic cellular cellular automata model stochastic site occupancy automata model stochastic changes changes in in individual individual site occupancy in in discrete discrete model continuous continuous (deterministic time, whereas whereas coupled-map time, coupled-map lattices lattices model (deterministic or or stochastic) populations stochastic) populations in in discrete discrete time. time. Lattice Lattice models models are are another another form form of of SEPM spatial IBMs. IBMs. A individu SEPM that that overlaps overlaps with with spatial A lattice lattice model model where where single single individuals equivalent to spatial IBM within-species variation als occupy occupy cells cells is is equivalent to aa spatial IBM with with no no within-species variation (so that (so that the the only only unique unique property property of of an an individual individual is is its its spatial spatial location) location).. However, and IPSs However, lattice lattice models models ((and IPSs iinn particular) particular) have have their their own own literature, literature, which general theoretical theoretical questions questions such which focuses focuses on on general such as as competitive competitive coexistence coexistence
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACE MODELS MODELS
49 49
and and the connections connections between IPS and partial differential questions. In contrast contrast to the situation for complex spatial IBMs, there is a well-developed mathe mathematical theory that that can be used to construct construct formal proofs of persistence or extinction in IPS ((Durrett, Durrett, 1988; Bramson ai., 11991; 9 9 1 ; Durrett, 992; Durrett Bramson et al., Durrett, 11992; Durrett and Neuhauser, 11994). 994). IPS and stochastic stochastic cellular automata automata have been used to Sole, 11997); 997); explore persistence of single-species populations populations (Bascompte and So16, competitive interactions (Harada 994; Schwinning and (Harada and and Iwasa, 11994; and Parsons, 11996; 996; Takenaka et ai., 997); predator-prey al., 11997); predator-prey and parasitoid-host parasitoid-host interactions interactions ((Hassell Hassell et ai., 9 9 1 ; Comins et ai., 992; Wilson et ai., 993; Cuddington al., 11991; al., 11992; al., 11993; Cuddington of epidemics and Yodzis, 2000; 2000; Hosseini, 2003); 2003); and dynamics and evolution evolution of ((Sat6 Sat6 et ai., 994; Rand 995; Boots and Sasaki, 2000, al., 11994; Rand et ai., al., 11995; 2000, 2002). 2002). Some researchers researchers have used lattice models to to explore the effects of envir environmental heterogeneity, particularly the viability of single-species populations populations degraded or fragmented habitats. habitats. Bascompte and Sole Sol~ ((1997) in degraded 1 997) and Hiebeler (2000) 1 ) understand (2000) have have both both used used IPS IPS to to ((1) understand how how population population densities densities vary vary in in response to the amount amount and pattern pattern of habitat habitat destruction destruction and (2) compare the results to the predictions of patch occupancy models under under similar scenarios al., 11994; (Gotelli, 11991; 99 1 ; Tilman et ai., 994; Lavorel and Chesson, 11995). 995). Further reading. Durrett and Levin's classic paper paper on on "The Importance Importance of Being Discrete (and Spatial) 1 994) lucidly compares Spatial)"" ((1994) compares the dynamics and coexis coexistence properties of nonspatial models, continuum (see below), and IPS. IPS. continuum models (see
Pair Approximation
Pair Pair approximation, approximation, an approximate approximate method method for analyzing IPS that that focuses on the joint occupancy probability probability of neighboring pairs of cells, has been an important important complement complement to more rigorous analytical techniques techniques (Tainaka, 1988, 11994; 9 94; Harada 994; Harada 995; Nakamaru 997; Harada and Iwasa, 11994; Harada et ai., al., 11995; Nakamaru et ai., al., 11997; Takenaka ai., 11997; 997; Iwasa, 2000; Sat6 Takenaka et al., Sat~ and Iwasa, Iwasa, 2000; 2000; Boots and Sasaki, 2000 2000).) . Pair approximation approximation makes the assumption assumption of conditional conditional independ independence. Pairs of neighbors are assumed to be independent independent (the conditioning is on the presence of a focal individual) so that that the probability of a particular particular con configuration of a focal individual (e.g., a predator) predator) and and two two neighbors neighbors (e.g., two different prey individuals) is the product product of the probabilities probabilities of each pair of neighbors. neighbors. Although a great deal of foundational foundational work in IPSs IPSs has been done on simple models, such as the contact contact process and the biased voter voter model (the IPS analogues of logistic growth growth and and simple competition, respectively), pair approximations approximations allow rapid rapid construction construction and analysis of models dealing with larger communities or more complex complex biological rules. In addition addition to basic pair pair approximation approximation on the lattice lattice with a single, fixed interaction interaction scale, variants of pair approximation approximation have been developed to handle handle more general spaces, such as networks 997; van Baalen and Rand, 998; Rand, 1 999; networks (Keeling et ai., al., 11997; Rand, 11998; Rand, 1999; Keeling, 11999a,b; 999a,b; van van Baalen, 2000); to to approximate approximate the spatial dynamics of expanding 99 8 ); and expanding populations populations (Ellner et ai., al., 11998); and to model systems with multi multiple scales of movement and interaction interaction (Ellner, (Ellner, 2001 2001).) . Further reading. Chapters 1 999) give the Chapters bbyy van Baalen (2000) and Rand ((1999) clearest general explanation of pair approximation approximation derivations (they both dis discuss network network models, which are more general than than lattices); chapters by Sat6 Sat~ and and Iwasa (2000) and Iwasa (2000) are also usefui. useful.
BENJAMIN BENJAMINM. M. BOlKER BOLKER
50 50
C o n t i n u u m Models Models Continuum
The The "continuum" "continuum" in in the the definition definition of of continuum continuum models models refers refers primarily primarily to to the the representation representation of of local local populations populations as as continuous continuous densities densities rather rather than than dis discrete crete numbers, although although the the most most common continuum continuum models models (partial (partial differ differential equation and and integrodifferential equations) also operate operate in continuous continuous space and time. Continuum Continuum models are are more common in strategic strategic than in in tactical tactical applications. applications. The The simplest simplest reaction-diffusion reaction-diffusion models models allow allow only only completely completely local local reaction reaction terms terms (predator-prey (predator-prey interactions, interactions, competition, competition, etc.) etc.) that are are formulated similarly to to their nonspatial analogues analogues coupled with move movement by local diffusion; more detailed integrodifferential models allow interacindividuals to move or interact nonlocally according to movement or interac tion kernels. Continuum models can also be defined in discrete time, in which case they become integrodifference equations. The continuous-population assumption eliminates demographic demographic stochas stochasticity from continuum models, although the effects of demographic stochas stochasticity can be reintroduced either by assuming Poisson or multinomial statistical variation in the local population ((Durrett Durrett and Levin, 11994; 994; Griinbaum, 11994) 994) or by adding a stochastic term to create a stochastic POE PDE or integral equation (Lande et aI., 999; Engen et aI., al., 11999; al., 2002) 2002).. Continuum models models have have aa proud proud history history in in ecology; ecology; before before computational computational models models were were practical, they were the only way to model dynamics in continuous space, foundation. Continuum mod modand they still benefit from a strong theoretical foundation. wavefront dynamics of els are particularly powerful for modeling spread and wavefront genes, diseases, and populations (Metz 995), but genes, diseases, and populations (Metz and and van van den den Bosch, Bosch, 11995), but they they are also useful for studying equilibrium states and complex spatiotemporal dynamics (Holmes et ai., 1 994; Klausmeier, 1999; 1 999; Briscoe et al., ai., 2002). al., 1994; Although many ecologists perceive them them as oversimplified, PDEs equations Although incorporate fairly sophisticated models of behavior (Kareiva and Odell, can incorporate 1 9 8 7; Griinbaum, 1 9 94; Turchin, 1 998; Griinbaum, 1 998; Moorcroft et al., aI., 1987; Griinbaum, 1994; Turchin, 1998; Griinbaum, 1998; Moorcroft et 1999). 1 99 9 ) . Historically, most most continuum continuum models models have have explored explored the the effects effects ooff endogenendogen Historically, of work work on the distribudistribu ous variability only. One exception is the large body of tion of animals following different different foraging and and dispersal strategies tion strategies in heterogeneous environments (Griinbaum, (Griinbaum, 1994); 1 994); classical work on population population heterogeneous work on and competition in the presence of of environmental environmental features, persistence and features, such as resource patches patches and and gradients gradients (Kierstead and and Slobodkin, Slobodkin, 1953; 1 953; Pacala Pacala and and resource Roughgarden, 1982); 1 982); and and a series series of of papers papers by Roughgarden Roughgarden on on the the shape shape and and Roughgarden, of spatial distributions distributions of of organisms organisms in the presence of of particular particular envirenvir scale of (Roughgarden, 1974, 1974, 1977, 1 9 77, 1978; 1 978; Sasaki, 1997). 1 997). onmental heterogeneities (Roughgarden, [Klausmeier (1999) ( 1 999) shows shows how how the the combination combination of of endogenous endogenous dynamics with with [Klausmeier in smooth slope slope can generate generate the the heterogeneous heterogeneous pattern pattern of of "tiger "tiger bush" bush" in a smooth semi-arid landscapes.] landscapes.] More More recently, recently, Lande Lande et et al., ai., (1999) ( 1 999) and and Engen Engen et et al., ai., semi-arid (2002) (2002) have have developed developed a series of of approximate approximate and and exact exact stochastic stochastic PDE PDE modmod els for for the the spatial spatial synchrony synchrony of of population population fluctuation fluctuation that that extend extend els Roughgarden's work. work. Results Results from from this this approach approach of of modeling modeling noise-driven noise-driven Roughgarden's correlations on on an an exogenously exogenously variable variable landscape landscape (also (also used used by by Snyder Snyder and and correlations Chesson, 2002) 2002) have have begun begun to to converge converge on on some some of of the the results results of of spatial spatial Chesson, moment moment equations equations described described later. later.
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS
51 51
Further reading. Continuum Continuum models models are are presented presented in in Okubo's Okubo's book book on on dif diffusion Okubo, 11980) 980) and, and, more more recently, recently, by fusion models models ((Okubo, by Shigesada Shigesada and and Kawasaki Kawasaki ((1997) 1 997) [the [the edited edited volume volume by by Okubo Okubo and and Levin Levin (2002) (2002) is is more more up up to to date date than than Okubo ((1980), 1 980), but Okubo but gives gives more more of of an an overview overview of of developments developments in in ecological ecological diffusion 1 998) does diffusion models models than than an an introduction]; introduction]; Turchin Turchin ((1998) does aa particularly particularly good job connecting good job connecting continuum continuum models models with with data data on on animal animal movement. movement. Books Books by 1 990) and 1 988) cover by Murray Murray ((1990) and Edelstein-Keshet Edelstein-Keshet ((1988) cover the the use use of of PDEs PDEs in in math mathematical ematical biology biology more more generally, generally, but but do do have have useful useful ecological ecological material. material.
3.3 3.3
SPATIAL MOMENT MOMENT EQUATIONS EQUATIONS SPATIAL How SMEs) fit How do do spatial spatial moment moment equations equations ((SMEs) fit into into this this assemblage assemblage of of mod models tools? The els and and tools? The basic basic structure structure behind behind spatial spatial moment moment equations equations is is that that of of spatial (possibly heterogeneous) spatial point point processes processes in in aa (possibly heterogeneous) two-dimensional two-dimensional space. space. As As pair pair approximations approximations do do for for lattice lattice models, models, SMEs SMEs attempt attempt to to reduce reduce aa spatial den spatial point point process process to to aa set set of of equations equations for for the the dynamics dynamics of of population population densities and and spatial spatial pattern pattern in in the the system. SMEs capture the spatial spatial properties properties of of sities system. SMEs capture the an tracking the mean dens an ecological ecological system system by by tracking the first first two two spatial spatial moments, moments, the the mean densdifferent species and and the spatial correlations among among individuals of all ities of different species, species, which which are are analogous analogous to to the the mean mean and and variance variance of of aa nonspatial nonspatial distri distribution. Like models, the representations of spatial pattern bution. Like continuum continuum models, the actual actual representations of spatial pattern are continuous, differentiable are continuous, differentiable functions. functions. While While moment moment equations equations are are not not technically technically stochastic stochastic m they they represent represent average average behavior behavior over over space space or or across across ensembles similar ecological ensembles of of similar ecological arenas arenas - - they they do do preserve preserve some some of of the the import important ant effects effects of of stochasticity stochasticity in in finite finite populations populations that that are are lost lost from from typical typical con continuum tinuum models. models. Many Many nonspatial nonspatial models models use use similar similar methods methods to to track track the the properties properties of of population 964) can population distributions. distributions. Moment Moment generating generating functions functions (Bailey, (Bailey, 11964) can be be used used to to analyze analyze the the entire entire infinite infinite series series of of moments moments of of aa distribution, distribution, but but are are difficult nonlinear processes, difficult to to apply apply to to nonlinear processes, let let alone alone nonlinear nonlinear spatial spatial processes. processes. Nonspatial Nonspatial moment moment equations equations that that use use some some rule rule to to approximate approximate higher higher moments (including constant, or common in moments (including setting setting them them constant, or to to zero) zero) are are common in ecology, ecology, epidemiology, 992; Turelli epidemiology, and and genetics genetics (Dobson (Dobson and and Hudson, Hudson, 11992; Turelli and and Barton, Barton, 11994; 994; Grenfell 995; Dushoff, 999; Cornell Grenfell et et aI., al., 11995; Dushoff, 11999; Cornell et et aI., al., 2000). 2000). For For example, example, Anderson 1 985) modeled Anderson and and May May ((1985) modeled macroparasite macroparasite dynamics dynamics by by assuming assuming that that the binomial distribution the number number of of parasites parasites per per host host has has aa negative negative binomial distribution with with aa fixed fixed overdispersion overdispersion parameter parameter so so that that the the variance variance can can be be calculated calculated for for any any given value of given value of the the mean. mean. The The next next section section derives derives the the spatial spatial moment moment equations equations for for the the simplest simplest par parasitoid-prey asitoid-prey system, system, equivalent equivalent to to Lotka-Volterra Lotka-Volterra predator-prey predator-prey equations equations with with aa predator predator efficiency efficiency of of one one (predator (predator bith bith rate rate equal equal to to predation predation rate). rate). The The parasitoid parasitoid moves moves by by random random jumps, jumps, while while the the prey prey is is sessile sessile and and disperses disperses only close enough only as as aa seed seed or or propagule. propagule. When When the the parasitoid parasitoid is is close enough to to the the prey, prey, it prey into into aa single This system, rather than than the it turns turns the the prey single new new parasitoid. parasitoid. This system, rather the analogous been chosen chosen for analogous equations equations for for competition competition or or epidemics, epidemics, has has been for novelty; novelty; derivations derivations of of competitive competitive and and epidemic epidemic systems systems appear appear elsewhere elsewhere (Bolker (Bolker and 999; Bolker, 999). Moment and Pacala, Pacala, 11999; Bolker, 11999). Moment equations equations have have been been used used to to model model
52 52
M. BOLKER BOlKER BENJAMIN M.
predator-prey metapopulations metapopulations (Keeling, (Keeling, 2000; 2000; Keeling Keeling et et al., aI., 2002) 2002) and and spaspa predator-prey tial epidemics epidemics without replenishment (Bolker, (Bolker, 1999; 1 999; Brown, Brown, 2001), 200 1 ), and and pair pair tial without replenishment approximations approximations have have been been used used for for more more general general epidemics epidemics (Sato (Saw et et al., aI., 1994), 1 994), spatial moment moment equations equations for for predator-prey predator-prey systems systems have have not not been been published published spatial (M. Desai, Desai, unpublished unpublished manuscript). manuscript). The The derivation derivation is is sketched, leaving out out the the (M. sketched, leaving straightforward but but scary-looking algebra; this this approach approach makes makes things slightly straightforward scary-looking algebra; things slightly the serious serious reader reader who who wants wants to to follow follow the the algebra algebra line line by by line, line, but but harder for the harder for it it allows allows the the discussion discussion to to focus focus on on the the important important steps steps in in the the derivation, derivation, pointing out out where where alternative alternative assumptions assumptions or or procedures procedures are are possible. possible. [Similar [Similar pointing derivations can can be be found found in in papers papers by by Bolker Bolker and and Pacala Pacala (1999) ( 1 999) and and by by derivations Dieckmann and and Law Law (2000).] (2000).] Dieckmann Further reading. reading. Other Other introductions introductions to to SMEs SMEs can be found found in in chapters chapters by by Further can be Bolker et et al. al. and and Dieckmann Dieckmann et et al. ai. in in Dieckmann Dieckmann et et al. ai. (2000) (2000) and and in in other other papers papers Bolker by Law, Law, Dieckmann, Dieckmann, and and Murrell Murrell (Law (Law and and Dieckmann, Dieckmann, 2000; 2000; Law Law et et al., aI., 2001, 2001, by 2003). 2003).
Write Rules W r i t e down down Stochastic Stochastic Rules We by writing writing down change in in the of aa small We start start by down the the expected expected change the occupancy occupancy of small patch of w located in aa landscape landscape of patch of size size 00 located at at position position x x over over aa small small time time M, At, in of O. In In this ( V) reproduce reproduce at at aa constant constant per capita rate patches D.. patches this system, system, prey prey (V) per capita rate t fvv (leading to aa Poisson-distributed number of of offspring per unit unit time) time) and and disperse disperse (leading to Poisson-distributed number offspring per them in aa random with aa distance given by by the kernel 73v. Vv. them in random direction direction with distance given the dispersal dispersal kernel [Kernels are that determine the rate rate or probability with [Kernels are continuous continuous functions functions that determine the or probability with which will interact interact with with another another site site or or individual individual at given disdis which an an individual individual will at aa given tance. kernels represent probability distributions tance. Dispersal Dispersal kernels represent probability distributions and and so so must must integrate integrate to all possible also scale scale interacinterac to l1 over over all possible locations; locations; for for analytical analytical convenience, convenience, we we also tion kernels Kernels are tion kernels so so they they integrate integrate to to 11.. Kernels are typically typically symmetric, symmetric, and and to to avoid avoid interactions at infinite distances, distances, kernels distance becomes becomes interactions at infinite kernels must must approach approach zero zero as as distance large. We typically typically use use simple simple decreasing decreasing functions functions such such as as the the Laplacian Laplacian (back (backlarge. We to-back exponential) Gaussian distribution, distribution, but to-back exponential) or or Gaussian but other other shapes shapes are are feasible.] feasible.] Prey Prey die die at at aa density-independent density-independent per per capita capita rate rate !.Lv, ~v, and and intraspecific intraspecific competition leads leads to additional mortality neigh competition to additional mortality at at aa rate rate proportional proportional to to the the neigh- xl) all conspecific borhood borhood density, density, measured measured as as the the sum sum of of aUYY{ly ~Uvv(lYxl) for for all conspecific neighbors, overall strength neighbors, where where a c~ is is the the overall strength of of competition competition and and Uw b/w (p) (P) is is the the rel relative ative strength strength of of competition competition at at aa distance distance p. p. [In [In the the standard standard nonspatial nonspatial Lotka-Volterra Lotka-Volterra model, model, specifying specifying birth birth and and density-independent density-independent death death rates rates separately separately would would be be redundant, redundant, but but in in stochastic stochastic and and spatial spatial models models the the ratio ratio of of birth birth to to death death rates rates as as well well as as the the difference difference between between birth birth and and death death rates rates is 999).] Self-limitation is important important (Bolker (Bolker and and Pacala Pacala 11999).] Self-limitation of of the the prey prey may may seem seem to to be be an an unnecessary unnecessary complication, complication, but but it it is is actually actually required required to to keep keep the the spa spatial Desai, unpublished tial moment moment equations equations from from blowing blowing up up (M. (M. Desai, unpublished manuscript). manuscript). Parasitoids Parasitoids (P) (P) die die at at aa density-independent density-independent per per capita capita rate rate !.Lp, ~p, and and at at rate rate mp ump from mp they they jjump from their their current current location location in in aa random random direction direction with with aa dis dispersal persal kernel kernel Vp• De. Parasitism Parasitism occurs occurs as as aa function function of of parasitoid-prey parasitoid-prey distance, distance, with is the with aa rate rate proportional proportional to t o 'YUpY{ly ~/b/PV(ly - xl) xl) [again, [again, 'Y ~/is the overall overall rate rate and and Upv btpv(p) (p) predation rate at different distances]. When When predation governs the relative predation occurs, occurs, the the prey prey turn turn into into newborn newborn parasitoids. parasitoids. The The processes processes described described here here can can be be written written formally formally as as stochastic stochastic rates: rates:
3. 3.
CONTINUOUS-SPACE MODELS MODELS CONTINUOUS-SPACE
53 53
Event Event
Change Change
Rate Rate
Prey birth birth Prey
V(x)-; V(x) ++ 11 V(x)--->V(x)
( ly -- xl)~176 ~L yEf/fvV(y)Dv yE n fv V(y )D v (lY xl )w
Prey death death Prey (density independent) independent) (density
V(x)-;V(x) V(x) -- 11 V(x)--->
ILVV(X) ~vV(x)
Prey death death Prey (density dependent) dependent) (density
V(x)->V(x) -- 11 V(x)-+V(x)
V(x) ~ aV(y) Uvv( l y -- xxI) V(x) ] y~a L Y E n oLV(y)Uw(ly l)
Parasitoid movement Parasitoid movement
P(x)->P(x) -- 11 P(x)-+P(x) P(y)->P(y) + + 11 P(y)-->P(y)
mpp P(x) P(x) D/, xl )w ( ly -- xl)~ m Dp (lY
Parasitoid death death Parasitoid
P(x)-;P(x) -- 11 P(x)-+P(x)
f.LpP(x) ~/,P(x)
V(x)-; V(x) -- 11 V(x)-+V(x)
xl ) V(x) ~ yP(y) UepvY( (l yly -- xl) V ( x ) ~L yyEn ),P (y) U
Parasitism Parasitism
P(x) -;P(x) + + 11 P(x)-+P(x)
The limits represent represent the locations of patches in the landland The summation summation limits the set set of of locations of all all patches in the scape. All All events involve the the probability probability of of aa propagule or an an individual scape. events that that involve propagule or individual landing in in aa particular particular patch prey birth birth rate rate and movement landing patch (the (the prey and parasitoid parasitoid movement rates in this this case) patch size size (00) (w) term term because because the the probability probability of an rates in case) contain contain aa patch of an individual patch is to patch patch area. individual landing landing in in aa patch is proportional proportional to area. Interactions Interactions that that involve the parasitism parasitism term, do not not because because the the corcor involve two two individuals, individuals, such such as as the term, do responding expressions individuals rather rather than than areas. areas. When When responding expressions sum sum over over discrete discrete individuals -7 0), 0), itit will will be be important important to to have have we the continuous-space continuous-space limit limit later (w -+ w e take take the later (00 these terms correct. these terms correct. Alternatives Alternatives
This small, discrete in This derivation derivation is is heuristic, heuristic, starting starting from from small, discrete patches patches evolving evolving in discrete discrete time time and and then then taking taking straightforward straightforward limits limits to to reach reach the the continuous continuousspace, space, continuous-time continuous-time equations. equations. This This procedure procedure makes makes it it easier easier to to see see when when and (removable) singularities possible to and why why (removable) singularities occur occur in in the the equations. equations. It It is is also also possible to construct construct aa rigorous rigorous derivation derivation starting starting from from infinitesimal infinitesimal space space and and time time increments increments rather rather than than small small but but finite finite ones ones and and using using probability probability measures, measures, but but we we have have always always found found it it easier easier and and clearer clearer to to proceed proceed heuristically. heuristically. more rigor Mathematically can see see Barton Mathematically inclined inclined readers readers can Barton et et al. al. (2002) for for aa more rigorous continuous space ous approach approach to to stochastic stochastic population population dynamics dynamics in in continuous space in in aa pop population-genetic ulation-genetic context. context. The The predation predation process process just just described, described, where where predators predators turn turn their their prey prey into into aa single single newborn newborn predator, predator, models models aa parasitoid-host parasitoid-host interaction interaction where where only only one one egg egg can can emerge emerge from from aa host. host. This This kind kind of of parasitoid parasitoid model model is is the the simplest simplest to to analyze analyze because because only only two two spatial spatial locations locations are are involved involved in in the the interaction. interaction. If If we we wanted wanted to to allow allow more more than than one one newborn newborn predator predator per per host, host, we we could could define define aa natal natal dispersal dispersal kernel kernel according according to to which which the the newborn newborn predators predators jumped jumped away away from from the the host, host, but but we we would would then then have have to to keep keep track track of of three three spatial spatial locations locations (predator, (predator, prey, prey, and and newborn) newborn) at at aa time; time; to to calculate calculate the the expected expected change change in in predator-predator predator-predator covariances, covariances, we we would would need need to to estimate estimate the the probability probability of of aa four-point newborn, and four-point configuration configuration (predator, (predator, prey, prey, newborn, and neighbor neighbor predator). predator). We We could could mitigate mitigate this this problem problem by by turning turning the the prey prey into into an an incubating incubating prey, prey, which which would would be be represented represented as as aa different different class class of of individuals individuals in in the the equations equations and and which which could could then then be be immune immune from from superinfection. superinfection. At At aa constant constant rate, rate, the the
BENJAMIN M. BOlKER BENJAMIN M. BOLKER
54 54
incubating incubating prey prey transforms transforms into into one one or or more more newborn newborn predators predators ~ separating separating the time of birth from the time of predation eliminates the need to keep track of many locations locations (Fig. .1). of as as many (Fig. 33.1). Similar problems, problems, and similar solutions, arise iinn predator-prey predator-prey models. If newborn newborn predators predators disperse disperse away away either either from from the the location location of of the the prey prey or or from from the location of their parent, several spatial locations have to be tracked. Another analogous to Another possibility, possibility, analogous to incubating incubating prey, prey, is is to to say say that that predation predation turns turns aa hungry hungry predator predator into into aa sated sated predator. predator. The The sated sated predator predator would would "decay" "decay" at at aa constant constant rate rate to to aa pair pair of of predators, predators, aa parent parent and and aa newborn; newborn; if if sated cannot kill kill prey, model structure would also also give sated predators predators cannot prey, this this model structure would give rise rise to to aa handling time. Multiple stages of satiation could make the time lag between predation predation and predator reproduction reproduction gamma distributed distributed (Keeling and and Grenfell, 11998); 998); predator predator efficiency less than than one could could be handled handled by requir requiring experience multiple ing that that predators predators experience multiple prey prey events events or or by by assuming assuming there there is is aa probability less than one that satiated predator probability less than one that aa satiated predator reproduces reproduces when when it it becomes becomes hungry again. These problems problems of definition are implicit in all ecological models, from sim simnonspatial models to SEPMs. Lattice models often make slightly artificial ple nonspatial Durrett and rules to avoid having multiple individuals in the same lattice cell ((Durrett Levin, Levin, 2000); 2000); in in nonspatial nonspatial models, models, these these details details can can often often be be accounted accounted for for by by setting up proper proper functional and numerical responses (Keeling et a!., al., 2000; Cuddington and Yodzis, 2002). The parasitoids parasitoids in this model are stupid, stupid, neglecting any local cues that that could guide them to higher prey densities. We could introduce introduce adaptive adaptive for foraging rules by letting the probability of movement, or the length of moves, Upv
4,
Upv
(~
4,
G
(~
4, t o + At
o
to + At
to+At
II
4,
t o +t=
to + i
I
(a) Standard interaction: interaction: predator predator (parasitoid) immediately immediately converts converts prey into a single offspring. offspring.
o
+4,
(b) (b) Parasitism with with incubation: incubation: predator nto an predator converts converts prey prey iinto Incubating Incubating prey, which which later (after t,) tl) becomes predator predator offspring. offspring.
tG
i
(c) Predation: prey disappears, disappears, prey becomes Sated Sated (and pregnant), later (after gestation tG) tc) gives gives birth to a single single off offspring.
Fig. Predator-prey Fig. 3.1 3.1 Predator-prey transition transition rules. Predators are Po, P0, P1; P1; prey are V; V; incubating incubating prey (e.g., containing containing parasitoid eggs or larvae) are I; and sated, pregnant pregnant predators are S. S. At At denotes an instantaneous change, whereas whereas other time time lags t" tl, tG tG denote events that that occur after an exponentially distributed distributed lag time.
3. 3. CONTINUOUS-SPACE CONTINUOUS-SPACE MODELS MODELS
$$ 55
decrease decrease in in response response to to increasing increasing local local prey prey density. density. This This behavioral behavioral response response would result result in in an an aggregation aggregation of of parasitoids in areas areas of of high high prey prey density, density, would parasitoids in mitigating the prey in parasitoid mitigating the prey depletion depletion that that occurs occurs in parasitoid neighborhoods. neighborhoods. Another Another common common foraging foraging rule, rule, allowing allowing the the directions directions of of successive successive moves moves to 99 8 ) , to become become less less correlated correlated in in response response to to favorable favorable habitat habitat (Turchin, (Turchin, 11998), could could bbee incorporated incorporated iinn the the point point process process model model but but iiss harder harder ttoo preserve preserve in in moment moment equations. equations. We We could could also also add add effects effects of of environmental environmental heterogeneity heterogeneity simply simply by by making making some some of of the the parameters parameters be be functions functions of of location. location. Making Making death death or or movement movement rates although perhaps rates spatially spatially heterogeneous heterogeneous is is simpler simpler ((although perhaps less less interesting) interesting) than than making making predation predation or or birth birth rate rate spatially spatially heterogeneous heterogeneous because because making making the the predation predation rate rate heterogeneous heterogeneous induces induces aa triple triple spatial spatial interaction interaction in in the the process, which interaction when process, which will will turn turn into into aa four-way four-way interaction when the the covariance covariance is is calculated. calculated.
Compute Changes in and Pair Pair Densities Compute Expected Expected Changes in Singlet Singlet and Densities For For each each mean mean density, density, the the derivation derivation of of the the spatial spatial moment moment equations equations requires requires two two averages averages or or expectations. expectations. The The first first average average is is the the expected expected change change in in occupancy occupancy over over aa short short time time M At in in aa patch patch of of size size w ~o for for aa given given starting starting con configuration figuration of of neighbors; neighbors; this this is is the the standard standard expected expected change change calculated calculated in in deriving deriving most most ecological ecological models models and and is is denoted denoted by by an an overbar. overbar. In In general, general, this this expected expected rate rate of of change change is is �F(x) = i) [change in AF(x) = k(rate E(rate of of event event/)[change in F(x) F(x) if if event event ii occurs]�t occurs]At
(throughout (throughout this this derivation, derivation, E F, G, G, and and H H refer refer to to arbitrary arbitrary state state variables; variables; w, w, x, x, y, y, and and zz are are arbitrary arbitrary locations). locations). Once Once we we have have taken taken this this first first average average to to cal calculate culate the the expected expected amount amount of of change change that that will will occur occur at at aa particular particular location location starting from from aa particular particular configuration, configuration, we we need need to to average average all all possible possible con constarting figurations figurations across across the the ensemble ensemble to to find find the the expected expected amount amount of of change change that that will will occur occur at at the the location location starting starting from from all all configurations configurations with with particular particular mean mean densities densities and and correlations. correlations. This This second second expectation, expectation, denoted denoted by by angle angle brackets brackets 0, (.), reduces reduces the the information information on on the the right-hand right-hand side side of of the the equation equation from from all all of of the ust the the information information about about the the configuration configuration to to jjust the expected expected patch patch densities densities and and joint joint patch patch densities densities (doublets). (doublets). The The same same procedure procedure is is applied applied to to calculate calculate the the average average expected expected rate rate of of change change of of doublets, doublets, e.g., e.g., <�F(x)G(y) (AF(x)G(y)),, the the rate rate of of change change of of pairs pairs with with prey prey in in aa patch patch at at point point x x and and parasitoid parasitoid in in aa patch patch at at point point y; y; these these rates rates will will involve involve doublets doublets and and triplets. triplets. Alternatives
For point that can drop For simplicity, simplicity, we we assume assume at at this this point that �t At is is short short so so that that we we can drop terms happen with terms where where two two events events occur occur at at once once [which [which happen with probability probability pro pro2 ] . Retaining portional possible [leading portional to to (�t) (At)2]. Retaining finite finite time time steps steps is is also also possible [leading even eventually tually to to an an integrodifference integrodifference equation equation for for the the changes changes in in spatial spatial covariances covariances ((Lewis, Lewis, 2000)], models, it 2000)], although although as as usual usual with with discrete-time discrete-time models, it requires requires more more care care in in defining defining the the possible possible orders orders of of events. events. In In addition, addition, retaining retaining double double events events will will lead lead to to fourth-moment fourth-moment terms terms in in expressions expressions for for covariances, covariances,
BENJAMIN M. BOLKER BENJAMINM. BOLKER
56
which which will will have have to to be be dealt dealt with with by by some some extended extended form form of of moment moment closure closure (see (see below). below).
Assume Assume Spatial Spatial Homogeneity Homogeneity By isotropy or By assuming assuming spatial spatial homogeneity homogeneity (specifically (specifically homogeneity homogeneity and and isotropy or translational make all translational and and rotational rotational invariance), invariance), we we can can make all singlet singlet expectations expectations (F(x) = joint products F(x) G (y) ) depend into global averages, into global averages, (F(x)) - (F). (F). Similarly, Similarly, joint products (((F(x)G(y))) depend only points: (FG) (FG) ((Ix only on on the the distance distance between between two two points: - yl). Yl). Assuming Assuming that that space space Ix is algebra, allowing is homogeneous homogeneous considerably considerably simplifies simplifies the the algebra, allowing us us to to collapse collapse terms (e.g., «(AF(x)) ((AF(x)) F(y) F(x)(AF(y))) = 2 (AF(x)F(y) (AF(x)F(y))).). In addition, various terms F(y) + F(x) (AF(y))) = it expectations across it simplifies simplifies the the interpretation interpretation of of the the expectations across ensembles ensembles introduced introduced above, above, which which we we can can now now take take as as averages averages across across space space in in aa single single realization realization rather rather than than as as averages averages across across ensembles. ensembles. Alternatives Alternatives
Certain easier to Certain kinds kinds of of spatial spatial heterogeneity heterogeneity are are easier to incorporate incorporate than than others. others. As assume that point As long long as as we we assume that the the probability probability of of events events occurring occurring at at aa point depends only only on depends on the the occupancy occupancy of of that that point point and and of of its its neighborhood, neighborhood, rather rather than absolute position, position, we express the than on on its its absolute we can can still still express the dynamics dynamics in in terms terms of of moments - covariances between moments~covariances between population population densities densities and and environmental environmental parameter distributions predation rate. parameter distributions such such as as mortality mortality or or predation rate. At price of some algebraic relax the At the the price of some algebraic complexity, complexity, we we can can relax the assumption assumption of of spatial spatial homogeneity homogeneity still still further. further. For For example, example, Lewis Lewis (2000) (2000) and and Lewis Lewis and ala (2000) mean and invad and Pac Pacala (2000) analyzed analyzed the the mean and covariance covariance structure structure of of an an invading ing population population by by preserving preserving information information about about the the distance distance of of points points from isotropy (directionality) from the the wave wave front. front. As As another another example, example, non nonisotropy (directionality) in in some some processes processes could could be be studied studied by by allowing allowing joint joint product product expectations expectations to to depend G (y) = (FG) depend on on direction: direction: (F(x) (F(x)G(y))= (FG) ((IxI x -- YI, y[, 60).). Allowing Allowing for for some some degree rather than degree of of absolute absolute ((rather than relative) relative) spatial spatial heterogeneity heterogeneity also also opens opens the the possibility possibility of of modeling modeling covariance covariance dynamics dynamics as as aa function function of of position position along Edge effects along aa gradient gradient or or distance distance from from an an edge. edge. ((Edge effects are are not not easy easy to to incorporate dynamics in incorporate in in moment moment equations, equations, which which typically typically explore explore dynamics in aa hypothetical hypothetical infinite infinite landscape. landscape. Modeling Modeling dynamics dynamics as as aa function function of of dis distance one-dimensional habitat tance from from edge edge in in aa one-dimensional habitat or or from from an an infinitely infinitely long, long, straight straight edge edge in in aa two-dimensional two-dimensional habitat habitat might might be be feasible, feasible, but but edge edge effects landscape would would probably effects in in aa finite finite two-dimensional two-dimensional landscape probably only only be be cal calculable culable numerically. numerically.))
Product Product Expansion/Convert E x p a n s i o n / C o n v e r t Joint Joint Products Products to to Covarlances Covarlances This This step step first first expands expands joint joint and and triple triple products products in in terms terms of of means, means, covari covariances, and ances, and third third moments moments and and then then subtracts subtracts A(F)(G) A(F)(G) (the (the expected expected change change in in the mean densities) densities) from the product product of of the the mean from joint-product joint-product equations equations to to turn turn them them into covariance equations instead of into covariance equations instead of joint-product joint-product equations. equations. It It is is useful useful to to explicitly explicitly write write out out singular singular or or "delta "delta function" function" terms terms that that take take into into account account
3. CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS 3.
57 57
the the possibility possibility that that the the two two individuals individuals share share the the same same patch patch (when (when we we take take the the continuous-space continuous-space limit limit below, below, this this means means that that there there is is really really only only one one individ individual present). present). The The definition definition of of the the covariance covariance is is thus thus ual (FG)([x-
Yl)=
(F)(G)
CFG(IX-
+
Yl) + ~xyO'FG"
((3.1) 3.1 )
We We can can expand expand the the triple triple product product (FGH) (FGH) (x, (x, y, y, z) z) similarly, similarly, arriving arriving at at aa (fairly F), (G), (fairly ugly) ugly) formula formula containing containing products products of of the the mean mean densities densities (((F), (G), (H») (H)) and all all of of the the covariances, covariances, as as well well as as the the third third moment moment M3pCH(x, M3F~H(x, y, y, z). [We and z). [We specify third central moments M3 by all three locations involved in specify third central moments M3 by all three locations involved in the the triple: triple: with with spatial spatial homogeneity homogeneity and and isotropy isotropy they they are are fully fully determined determined once once we know know two two sides sides and and an an angle angle (so we could could quote quote them them as as M3 M3pcH(]x - Yy],I , we (so we pCH( l x Ix zj, 8x -xz)), but this notation is simpler.] In addition, the expansion Ix - z], 0xy-xz)), but this notation is simpler.] In addition, the expansion con cony tains aa series series of of singular singular third-moment third-moment terms terms that that result result when when two two or or more more tains points in in the the triangle triangle coincide. coincide. For For example, example, aa term term (H)apc (H)crFc + + CH,pd CH.Fc(]x -- zz]) lx points l) results when P and G are in same patch: results when F and G are in the the same patch: apc O'FG denotes denotes the the covariance covariance of of P F and G G within within the the same distance zero) zero),, and and CH,pc CH.FC denotes denotes the and same patch patch ((at at distance the covariance patch containing both P and G When we covariance of of H H with with aa patch containing both F and G.. When we scale scale to to continuous too small more than continuous space space below, below, and and patches patches become become too small to to contain contain more than one one individual, individual, the the singular singular terms terms will will express express the the case case where where one one individual individual plays plays two two roles roles at at once, once, e.g., e.g., as as both both neighbor neighbor and and competitor. competitor. The The within withinpatch covariance covariance apc crF~ will zero for for F F i:~ G G and and (F) (F) for for F F = = G, G, which which we we will equal equal zero patch can become can express express by by writing writing it it as as o8F~ (F) (or (or o8Fc (G)); CF,CH CF.cH will will similarly similarly become Pc (G»); Pc (F) 0CHC 8cHCFc. One can can sometimes sometimes justify justify dropping dropping the the third-moment third-moment singular singular PC. One terms for for analytical analytical simplicity simplicity in in the the limit limit of of long-range long-range interactions interactions ((Bolker terms Bolker and Pacala, 1999). and Pacala, 1 999). The that we to The only only delicate delicate part part of of these these expansion expansion is is that we have have to to remember remember to drop nothing really really happens: happens: for drop cases cases where where nothing for example, example, if if the the event event "parasitoid "parasitoid moves to y the same points xx == y, y, then then no no move move really really takes takes moves from from x x to y"" involves involves the same points place and we should term. place and we should omit omit the the singular singular term. Alternatives Alternatives
The covariance is is only one way way of of expressing dependence. We We The covariance only one expressing spatial spatial dependence. (FG) or could also leave all all expressions in terms terms of of joint joint densities could also leave expressions written written in densities (FG) or densities, scaled scaled we could could write write the the spatial spatial dynamics dynamics in in terms terms of of conditional we conditional densities, or muhiplicative multiplicative covariances covariances (neglecting ( neglecting singular singular covariances, correlations, or covariances, correlations, terms): terms) : Conditional Conditional density density Scaled Scaled covariance covariance Correlation Correlation Multiplicative Multiplicative covariance covariance
(F I G) G) = = ~Fc) � (FI (C) f.:EG - (FC) CoFG __ (FG) -- (F) (F) (C) (G) SCPC = (P I G) G) -- (F) (F) SC;G ~c) = - (c) - (FI (C) -(C) (FC) ...il'.QL (F) (C) _ CFG CFG = ( F G ) (F)(G) __ (FG) rC~A PC - (F) (C) 1 (C) - - 1 ~ F G - - (F) (G) (F) (F) (G) (F) (G) (C) - (F) _
(FG) ...il'.QL (F) (G) (C) (F)
_
-
'
= CCP + 11 F GC + --
These forms forms are are all all equivalent, equivalent, but but have have advantages advantages in in different different situations: situations: These for for example, example, using using conditional conditional densities densities makes makes itit easy easy to to take take the the limit limit where where one one species species is is invading invading at at low low densities. densities.
BENJAMIN BENJAMIN M. M. BOLKER BOlKER
58 58
Moment Closure M o m e n t Closure Moment closure closure is is the the key key step step in in the the derivation, derivation, but but also also (potentially) (potentially) Moment one of of the the simplest. simplest. One One way way or or another, another, we we have have to to deal deal with with the the presence presence of of one third-moment terms, terms, which which incorporate incorporate the the information information about about the the spatial spatial pattern pattern third-moment that cannot cannot be be fully fully captured captured by by pairwise pairwise approximations. approximations. The The simplest simplest solution solution that is power-1 power-l closure, closure, which which assumes assumes that that the the probability probability of of aa given given triangular triangular concon at location location x, x, type type G G at at location location y, y, type type H H at at location location z) z) figuration (e.g., (e.g., type type FF at figuration is described described adequately adequately by by taking taking the the probability probability of of pairs pairs and and assuming assuming that that the the is third point point in in the the triangle triangle is is independent. independent. This This corresponds corresponds to to the the relationship relationship third (FGI-I) (FGH) = = (FG) (FG) (t-t) (H) + + (GH) (GH) (F) (F) + + (FH) (FH) (G) (G) -- 22 (F) (F> (G) (G) (t-I). (H). If If we we expand expand this this relationship in in terms terms of of central central moments, moments, we we find find happily happily that that the the power-1 power-1 relationship assumption corresponds corresponds to to M3FcH(x, M3FCH(X, y, y, z) z) -= O, 0, so so we we simply simply drop drop third-moment third-moment assumption terms. terms. Alternatives Alternatives
Different moment moment closures closures are are possible possible (see (see Box Box 3.1). 3 . 1 ) . In In some some cases, cases, it it may may Different actually actually be be simpler simpler to to substitute substitute the the relationships relationships between between noncentral noncentral moments earlier earlier in in the the derivation: derivation: the the author's author's habit habit of of waiting waiting until until this this point point moments comes from from using using power-1 power-1 closures, closures, which which are are simplest to apply apply at at this this step step by by comes simplest to setting moments to setting central central moments to zero. zero. Criteria for choosing aa moment closure are are analytical Criteria for choosing moment closure analytical simplicity, simplicity, stability, stability, and and numerical accuracy. Different better for for different kinds of Different closures closures may may work work better of numerical accuracy. problems, for for example, in competitive competitive communities communities vs vs predator-prey predator-prey or epidemic problems, example, in or epidemic systems. There is even even aa difference difference between between different regions of of phase phase space space systems. There is different regions within within the the same same model. model. For For example, example, power-1 power-1 closure closure is is unstable unstable for for calculating calculating invasion rates in in aa simple simple epidemic model m in in the the limit limit of of small small densities densities of of invasion rates epidemic model infectives, infectives, the the equations equations simply simply blow blow up up m but but actually actually predicts predicts equilibrium equilibrium den densities sities more more accurately accurately than than the the power-2 power-2 closure closure (Brown, (Brown, 2001 2001).). Power-1 Power-1 closures closures are unstable, but 999); power-2 are unstable, but analytically analytically tractable tractable (Bolker (Bolker and and Pacala, Pacala, 11999); power-2 clo closures sures may may be be the the best best overall overall choice choice for for stability, stability, although although power-3 power-3 closures closures per perform form better better under under some some conditions conditions (Law (Law et et aI., al., 2003). 2003). Filipe Filipe has has discussed discussed aa variety of of more more complex, complex, more more accurate accurate closure closure schemes; schemes; while while these these are are dis disvariety cussed cussed in in aa lattice lattice context, context, they they could could easily easily be be adapted adapted to to continuous-space continuous-space mod models 999; Filipe 1 ; Filipe Maule, 2003). els (Filipe, (Filipe, 11999; Filipe and and Gibson, Gibson, 200 2001; Filipe and and Maule, 2003). (These (These closures closures are are intended, intended, and and are are probably probably more more useful, useful, for for cases cases where where accurate accurate numeric solutions solutions are are required required rather rather than than for for cases cases where where one one will will attempt attempt to to numeric analyze analyze the the moment moment equations.) equations.) Hybrid Hybrid closures, closures, which which scale scale between between two two dif different ferent closures closures in in different different regions regions of of phase phase space space (e.g., (e.g., when when infection infection is is rare rare or or common), 1999). The common), have have been been used used in in aa nonspatial nonspatial context context by by Dushoff Dushoff ((1999). The choice choice of of closures closures is is still still more more of of an an art art than than aa science. science. At At present, present, recommendations recommendations are are ((1) 1 ) use use power-1 power-1 closures closures for for analytically analytically tractable tractable results, results, except except possibly possibly in in inva invasion sion cases; cases; (2) (2) use use power-2 power-2 closures closures otherwise; otherwise; and and (3) (3) explore explore the the accuracy accuracy of of aa particular particular closure closure with with simulations simulations and and always always check check key key results. results.
Take Take Continuum C o n t i n u u m limits Limits Now Now we we let let the the patch patch size size w 03 and and the the time time step step !::. Att both both become become infinitesi infinitesimally small. small. Prior Prior to to taking taking the the limits limits we we rescale rescale the the mean mean densities densities (F) (F) to to den denmally sities clw2• sities ff = = (F)/w, (F)/03, and and the the covariances covariances CFC CFc to to covariance covariance densities densities CFC cFc == =- CF CFd03 2.
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS
59 59
BOX 3.1 Verbal, Analytic, and Graphical Descriptions of Different Moment Closures
Each moment closure breaks the probability of a triangular configuration (FGI-I) down in a different way: while power-l assumes that each point is independent of the remain ing pairs, power-2 and power-3 deal with pairs only. Power-2 considers pairs (sides) two at a time, while power-3 considers all three sides simultaneously. For each geometric assumption, we can write out the relationship between the triple and the pairs and points (noncentral moments), expand it in terms of central moments, and solve it for M3. Dieckmann and Law (2000) give more details on the consistency conditions that must be met by any moment closure rule.
Closure Power-l Power-2 (asymmetric) Power-3
Noncentral moments
Central moments
(FG) (H) + (GH) (F) + (FH) (H) - 2 (F) (G) (H) (FG)(FH)
o
«FGH»
(M3)
(F)
(eFG (Iy - xl) eFH (Ix - zl) eGH (Iy - zl)
(FG)(FH)(GH) (F)(G)(H)
+
+
+
eFG (Ix - y l) eFH (Ix - Zl) (G)(H) eFc (lx - y l) eGH (ly - zl) (F)(H)
eFH (Ix - zl) eGH (Iy - zl) (F)(G»
I « F)(G)(H»
The power-2 asymmetric closure shown here multiplies only one pair of sides «FG) and (FI-I) . It can be made symmetric by including terms for the other two pairs of sides, but it is useful in the asymmetric form when a triangle naturally contains a focal indi vidual, which breaks the symmetry (Murrell and Law, 2000). Symmetric power-2 clos ures require a slightly counterintuitive correction term in order to make them consistent (Dieckmann and Law, 2000). Power-2 closure is analogous to the usual pair approxi mation used in lattice models; it represents an assumption of independence between neighbors of a focal individual. I n the graphical representation below, lines between circles indicate dependence; juxtaposition indicates multiplication (combining independent probabilities). • •
10 :.0 -0 -2 0 +
+.
•
.0
power-1 power-2
power-3
BENJAMIN M. BOlKER BENIAMIN M. BOLKER
60 60
In In this this limit, limit, summations summations become become integrals integrals and and delta delta functions functions become become Dirac Dirac delta delta functions, functions, but but all all Dirac Dirac delta delta functions functions are are found found inside inside integrals integrals and and can can be be extracted extracted by by the the rule rule that that Jf 3x ~x d dxx = - 11.. We We can can also also simplify simplify variance variance terms terms (0" there can only be be zero zero or patch, (FF) (or;2)) because because when when there can only or one one individual individual per per patch, (FF) = = (F) (this is (F) = = 00 or or 11 and and (0" (or;2)) = = (F) (F) (this is equivalent equivalent to to assuming assuming Bernoulli Bernoulli statistics statistics within within each each patch) patch).. Alternative Alternative
If preferred, we can retain sizes, although although in this case case we may If preferred, we can retain finite finite patch patch sizes, in this we may have time evolution within have to to write write out out separate separate equations equations for for the the time evolution of of the the withinpatch variances variances 0" cr;2 ((Bjornstad Bolker, 2000) 2000) and and the the within-patch within-patch Bjornstad and and Bolker, patch covanances covariances O"pc. O'FG.
Simplify Delta Functions and Convolutions Simplify Delta Functions and Convolutions The The last last step step is is just just algebraic algebraic tidying tidying and and involves involves no no new new assumptions assumptions or or conceptual leaps. leaps. We values out conceptual We take take delta delta functions functions and and average average values out of of integrals integrals (y) dy ); write ((all all kernels kernels K normalized so /C are are normalized so that that JK f/C(y) dy = 11); write [y ] y- x[ x I = r; r; and y[) dz and write write terms terms of of the the form form Jf K([z /C([z-- x[)cPc x])cFo ([z ( ] z- y]) dz as as convolutions, convolutions, (K· (r) (Bolker also rewrite (/C*cFc)(I x - Yy[) l ) == (K*cpc) (/C*cFG)(r) (Bolker et et ai., al., 2000). 2000). We We can can also rewrite the the · cPc)([x average ( [y - x[)cpc([y a v e r a g e covariances c o v a r i a n c e s that that appear appear in in the the mean mean equations, equations, JK f/C(]yx])cFo(]y(we can K omit in the subscript when it is clear from context x[)dy, x[)dy, as as CK,P e~,FG (we can /C omit in the subscript when it is clear from context C which kernel is weighting term). which kernel is being being used used as as aa weighting term). All All the the algebra algebra results results in in the the following following mean mean equations equations for for the the parasitoid parasitoid model: model:
��dv = fvv - f..LVV - a(v2 + dt
@ dp dt dt
- fw
-
~w
- ~ ( v 2 + cuvv' CU~v, vv) vv) - "'( ~ ((vp vp + + cUpv, -~u~, vp) ~p)
= (vp ++ Cc-upv, ..Lpp - "'( ~l(vp v p )- f ~PP Um vp)
(3.2) (3.2)
_
These These are are the the covariance covariance equations: equations:
ac a c vyy v ((r) r) Dvy( (r)v = 2[f 2 [ f vyD r)v + + f f vy((D D vy**ccvyy v ) ()(r) r)-'--'-- = at 8t
- f..Ll~vCvv(r) yc yy(r)
2Uyy(r) ~((g4vv -- a(( Uyy * cCvv)(r) yy)(r) ++ vv2Ltvv(r) --~l(v(blnv*Cnv)(r) Pcvv(r)) "'( (v(Upy *cpy)(r) ++ pc yy(r)) *
ac yp(r) Ocvn(r) 8t at
+ + Uyy b l v v ((r)c r ) c vyy v + + cCvv(r)) yy(r))
(3.3) (3.3)
(pcyp(r) ++ v( Cpp)(r) + + Up Unv(r)vp) yp)( r ) -- f..LI~vCvn(r) Upy ** cpp)(r) - f fv(Z)v v(Ltnv v(Dy ** cCvp)(r) ---'--'- = y(r)vp) vc yp(r) -- "'(~l(pcvn(r) pc yp(r) + (v( Up y ** cvp)(r) + "'( ~l(v(blnv Cvn)(r) + + pc Pcvv(r)) - f..L I~nCvp(r) yy(r)) -- mC m c vynp(r) (r) + + m(Dp m(g)n * cCvn)(r) yp)(r) -- a(vc a(VCvn(r) Uyy ** cCvn)(r)) yp)(r)) yp(r) ++ ((blvv
(3.4) (3.4)
acpp(r) Ocnn(r) (pc p(r) ++ v( = 2 2[~l(pcvp(r) v(blpv Cpp)(r) + + U b lpeyv(r)vp) ( r ) v p ) -- f p~pCpp(r) Upy "* cpp)(r) -..Lpcpp(r) aOtt--'- = ["'( y mcpe(r) + + m(Dp m(7)p '*:. cpp)(r)] Cpe)(r)] -- mcpp(r)
(3.5) (3.5)
•.
3. 3.
CONTINUOUS-SPACE MODELS MODELS CONTINUOUS-SPACE
3.4 3.4
61 61
NUMERICAL N U M E R I C A L RESULTS RESULTS We can evaluate the spatial dynamics of of the parasitoid-prey parasitoid-prey system stochas stochastically, by running an individual-based individual-based simulation, or deterministically (and approximately), approximately), by integrating the moment equations numerically. As suggested previously, it is important when beginning to work equations to when beginning work with moment moment equations check check their their accuracy accuracy (possibly (possibly for for aa variety variety of of different different closures) closures) over over aa wide wide range of chapter focuses range of conditions. conditions. Because Because this this chapter focuses on on the the derivation derivation and and meaning meaning of of moment moment equations, equations, the the following following section section shows shows only only aa brief brief overview overview of of the the numerical numerical results results from from the the parasitoid-prey parasitoid-prey system. system. Starting Starting from from the the monocul monoculture ture equilibrium equilibrium of of the the prey prey (running (running only only the the equations equations for for V V and and Cvv Cvv to to equi equilibrium) and then introducing librium) introducing a small density of of parasitoids, the system shows shows (In fact, damped damped oscillations oscillations to to an an equilibrium equilibrium (Fig. (Fig. 3.2). 3.2). (In fact, because because moment moment equa equations tions represent represent the the dynamics dynamics of of the the average average neighborhood neighborhood across across an an infinitely infinitely
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N umerical solutions Numerical solutions of of parasitoid-prey parasitoid-prey moment m o m e n t equations. equations 9 Parameters Parameters (nonspatial): (nonspatial): tv fv = = 2.0, 29
fv .0, a .0. All All kernels fv = = 11.0, c~ = = 0.1 0.1,, fLp l~p = = 0.2, 0.2, 'Y 7 = = 0.1 0.1.. Parasitoid Parasitoid movement movement rate rate m mpp = = 11.0. kernels exponential exponential (sym (symmetric Wm), with and competition = 0.2, metric exponential), exponential), K /C = = 221m / m exp(exp(-Irl/m), with prey prey dispersal dispersal and competition scale scale mDv mDv = = muvv muvv-0.2,
.0. Solutions parasitoid parasitoid dispersal dispersal and and parasitism parasitism scale scale mupv mupv = = mvp m~p = = 11.0. Solutions by by the the Euler Euler method method with with total total spatial spatial length 25, 25, Ax Ax = = 0.25, 0.25, total total time time = 80, 80, At At = = 0.025. 0.025. length =
62 62
BENJAMIN BENJAMIN M. M. BOLKER BOLKER
extended extended stochastic stochastic system, system, they they should should always always converge converge to to an an equilibrium: equilibrium: without without global global interactions, interactions, there there is is no no way way that that parasitoid-prey parasitoid-prey populations populations can can stay stay in in synchrony synchrony across across an an infinite infinite landscape.) landscape.) For For the the parameters parameters chosen chosen in Fig. 3.2, the dynamics (Fig. 3.2a) dynamics of the mean mean densities in the spatial model (Fig. are are nearly nearly identical identical to to the the dynamics dynamics obtained obtained from from the the analogous analogous nonspatial nonspatial model (Fig. .2c and 3.2d) show (Fig. 3.2b). However, the covariances (Fig. (Fig. 33.2c show interest interesting patterns: the parasitoid-prey parasitoid-prey covariance at short short spatial lags quickly becomes negative itoids deplete deplete the becomes negative as as paras parasitoids the prey prey in in their their immediate immediate neigh neighborhood, driving the average parasitoid prey covariance covariance negative (Fig. (Fig. 3.2d). This This zone zone of of depletion depletion grows grows rapidly rapidly up up to to about about tt = = 55 with with aa maximum maximum corre correlation 12 units scale of lation length length of of 12 units and and then then shrinks shrinks back back to to aa length length scale of approxi approximately 33 units. prey mately units. The The average average covariance covariance between between parasitoids parasitoids and and prey (cvp (Cvp = =f fUvp(r)cvp(r)dr), which describes describes the the net net effect effect of of spatial spatial segregation segregation on on Uvp(r)cvp(r)dr), which the (Fig. 3.2d): the parasitism parasitism rate, rate, is is always always negative negative (Fig. 3.2d): the the spatial spatial structure structure always always reduces rate. The reduces the the effective effective parasitism parasitism rate. The average average intraspecific intraspecific covariance covariance of cvv = fflgvv(r) cvv(r) dr) of prey( prey(evv dr) is is always always positive positive ~ prey prey experience experience intraspe intraspeUvv(r) Cvv(r) cific raises their - but cific crowding crowding that that raises their effective effective density density~ but drops drops as as parasitoids parasitoids reduce also experience experience intraspecific reduce the the prey prey density. density. Parasitoids Parasitoids also intraspecific aggregation aggregation (cpp (epp = ffblvp(r) Cpp(r) dr dr > > 0), 0), but but this this aggregation aggregation has has no no direct direct effect effect on on the the Uvp(r) cpp(r) dynamics mean densities. dynamics of of the the mean densities. What What about about the the effects effects of of (for (for example) example) changing changing parasitoid parasitoid scales? scales? Figure Figure 3.3 3.3 compares compares the the results results of of simulations simulations with with numerical numerical integration integration of of the parasitoid the moment moment equations equations for for aa broad broad range range of of parasitoid parasitoid scales scales ((parasitoid dispersal scale were equal in dispersal scale scale and and parasitism parasitism scale were set set equal in all all cases) cases).. For For large large parasitoid relatively accurate; parasitoid scales, the moment moment equations equations are relatively accurate; the prey den density spatial case; para sity converges converges on on the the expected expected density density in in the the non nonspatial case; and and the the parasitoid density is slightly below the nonspatial case, the only nonspatial density. In this case, effect enhance intraspecific competition among effect of space is to enhance intraspecific competition among the prey. As in many many simple simple parasitoid-prey parasitoid-prey models, models, the the lowered lowered productivity productivity of of the the prey prey base base is is reflected reflected in in aa lowered lowered density density of of the the parasitoids parasitoids rather rather than than in in the the prey prey density density itself. itself. As As the the parasitoid parasitoid scale scale decreases, decreases, the the intraspecific intraspecific aggregation aggregation of of the the prey prey increases ) . The increases (from (from Cvv evv ~ 0.2 0.2 to to = ~ 11). The much much larger larger effect, effect, however, however, is is the the large large increase increase in in magnitude magnitude of of the the spatial spatial segregation segregation between between parasitoids parasitoids and and prey prey (cv p). p). In (ev In effect, effect, this this spatial spatial segregation segregation lowers lowers the the efficiency efficiency of of the the parasitoids parasitoids by simultaneous increase by making making it it harder harder for for them them to to find find prey prey nearby. nearby. The The simultaneous increase in in both both parasitoid parasitoid and and prey prey densities densities with with increasing increasing spatial spatial segregation segregation may may seem seem surprising. surprising. However, However, we we can can understand understand it it by by seeing seeing that that in in the the nonspatial nonspatial ana analogue logue of of the the current current model, model, decreasing decreasing parasitoid parasitoid effectiveness effectiveness increases increases both both parasitoid and prey density; limiting parasitoid parasitoid effectiveness increases the pro productivity ductivity of of the the prey prey base, base, which which counterintuitively counterintuitively feeds feeds through through to to increase increase the the parasitoid parasitoid density density as as well. well. (If (If we we were were modeling modeling the the evolutionary evolutionary dynamics dynamics of probably see of this this system, system, we we would would probably see that that the the evolutionary evolutionary stable stable strategy strategy for for parasitoids and interaction parasitoids of of unlimited unlimited movement movement and interaction lowers lowers the the average average density density of looking at Eq. (3.2), of the the population.) population.) By By looking at Eq. (3.2), we we see see that that we we can can encapsulate encapsulate all all of of the the effects effects of of spatial spatial structure structure by by rescaling rescaling the the parasitoid parasitoid effectiveness effectiveness to to 2 in gen 'V [1 ~'' = = 'V ~ [[11 - cvp/(vp)] evp/(Vp)] and and the the prey prey competition competition to to a c~'' = = a ~ [1 - cvv/(v evv/(v2)]; )]; in general, eral, 'V y'' < < 'V y and and a c~'' > > a. c~. The The mean mean equations equations tell tell us us that that parasitoid parasitoid density density will will begin begin to to drop drop if if 'V ~/'' falls falls below below aa critical critical value value (fv ( f v -- fLv)/(2afLp), ~v)/(2oL~p), but but for for the the =
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