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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Fig. 3.3 Comparisons Comparisons of moment equations. in Fig. 3.3 of simulations simulations with with numerical numerical integration integration of of moment equations. Parameters Parameters as as in Fig. 3.2, 3.2, but but with and parasitoid Fig. with varying varying (equal) (equal) scales scales of of parasitism parasitism (mupv) (mupv) and parasitoid dispersal dispersal (mDp)' (m~p).
parameter ",' < - the parameter values values given given earlier earlier that that would would require require ~/' < 0.4", 0.4~/m the effects effects of of spatial seen here here decrease spatial structure structure seen decrease ",' ~/' only only to to aa minimum minimum of of 0.84",. 0.84~/. How How accurate accurate are are the the moment moment equations equations at at capturing capturing the the observed observed changes changes in in the the covariances? covariances? In In this this case, case, the the simple simple power-1 power-1 closure closure that that we we have have captures captures the the qualitative qualitative changes changes in in the the means means and and covariances, covariances, although although it it underpredicts underpredicts all of spatial structure. all of the the effects effects of of spatial structure. It It does does the the worst worst job job predicting predicting parasitoid parasitoid clustering clustering (luckily, (luckily, parasitoid-parasitoid parasitoid-parasitoid covariance covariance is is the the one one covariance covariance term term that that has has no no direct direct effects effects on on mean mean density). density). Other Other symmetric symmetric closures closures (power-2 (power-2 and and power-3; power-3; not not shown) shown) give give very very similar similar results results to to the the power-1 power-1 closure, closure, but but an an asymmetric asymmetric power-2 power-2 closure closure fails fails badly, badly, predicting predicting zero zero parasitoid-prey parasitoid-prey covari covariance. (but spatial ance. In In general, general, we we expect expect the the moment moment equations equations to to work work well well (but spatial struc structure ture to to be be irrelevant) irrelevant) for for large large scales scales of of movement movement or or interaction interaction and and to to fail fail at at extremely scales. We based on kernel extremely short short scales. We can can define define aa neighborhood neighborhood size size based on the the kernel shape exponential kernel shape and and the the scale scale parameter parameter (Bolker, (Bolker, 1999). 1999). For For the the exponential kernel used used here, here, the the neighborhood neighborhood size size equals equals 4m. 4m. In In the the past, past, we we have have found found that that moment moment equations equations give give accurate accurate predictions predictions of of mean mean densities densities when when individuals individuals average average = 1 0-100 discrete 10-100 discrete neighbors neighbors within within their their interaction interaction neighborhood neighborhood (Bolker (Bolker and and Pacala, 997), although Pacala, 11997), although qualitative qualitative results results may may apply apply for for lower lower numbers. numbers.
BENJAMIN BENJAMIN M. BOlKER BOLKER
64
Predictions 3b) for Predictions of of prey prey density density are are reasonably reasonably accurate accurate (Fig. (Fig. 3b) for scales scales in in the the range 2, this range of scales corresponds corresponds to range 0.5-2. 0.5-2. Because Because prey prey density density is is = ~ 2, this range of scales to aa neighborhood population of 4 6, slightly below the expected range. Similarly, 4 to 116, for parasitoid density scales between 0.1 and 11.0 .0 work, which for a density of = ~88 gives gives aa range range of of 3.2 3.2 to to 32 32 neighbors. neighbors. A full analysis of this system would other changes in would explore the effects of other relative interaction interaction scales; for example, does changing parasitoid parasitoid movement and parasitism scales and parasitism scales independently independently have have any any interesting interesting effects? effects? [If [If one one sets sets all (prey dispersal, dispersal, parasitoid parasitoid movement, equal and all scales scales (prey movement, parasitism) parasitism) equal and varies varies them them simultaneously, simultaneously, the the effect effect of of prey prey aggregation aggregation dominates dominates by by lowering lowering prey parasitoid densities.] One prey productivity productivity and and hence hence lowering lowering equilibrium equilibrium parasitoid densities.] One could also look for simplified cases where some analysis is possible; as shown 1 999), one can find the equilibria of moment moment equations in Bolker and Pacala ((1999), analytically if the problem is restricted restricted to one or two two different scales or if some processes are given infinite (global) scales. In the parasitoid-prey parasitoid-prey case, one one might might get get quite quite simple simple results results if if prey prey competition competition were were global global and and para parasitoids did not not move, being able only to eat prey that that settled within within their their par parasitism neighborhood. neighborhood. Ultimately, strategies for simplifying simplifying and analyzing the equations, equations, and and connecting connecting the the results results to to the the more more general general results results from from numer numerical solutions and simulations, depend on the ecological questions of interest.
3.5 3.5
APPLICATIONS APPLICATIONS Spatial Spatial moment moment equations equations have have yet yet to to see see widespread widespread use use in in ecology, ecology, although a few groups of researchers have used them to answer a variety of questions section reviews questions in in spatial spatial ecology. ecology. This This section reviews briefly briefly the the (small) (small) literature literature of applications of spatial moment equations, making connections to nonspa nonspatial or noncontinuous noncontinuous models where appropriate. appropriate.
Single-Species Single-Species Dynamics Dynamics The The simplest simplest possible possible application application of of moment moment equations equations is is to to populations populations of of a single species; this system is called the spatial logistic model and is analogous nonspatial contact process on to both the classical non spatial logistic equation and the contact the lattice (Law et al., aI., 2003) 2003).. The model defines the fecundity and density densityindependent mortality rates of individuals, the strength of density dependence, and the kernels for dispersal and competition. competition. Competition can lower the fecundity of parents with increasing neighborhood neighborhood density; lower establishment probability of offspring with increasing density of the neighborhood where they land increase mortality land after after dispersing; dispersing; or or increase mortality probability probability of of adults. adults. The The results results are are simple: spatial structure structure can be either aggregated (positive average covariance) or even (negative average covariance). Aggregation occurs when individuals have short dispersal scales relative to competition scales and low intrinsic reproductive numbers (fecundity numbers (fecundity divided divided by by density-independent density-independent mortality mortality or or expected expected lifetime reproduction in the absence of competition). The overall strength of spatial effects is largest when neighborhood size, size, the number of neighbors with which an individual interacts, is small: the neighborhood size size is proportional to population spatial carrying population density density (determined (determined approximately approximately by by the the non nonspatial carrying
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS
65 65
capacity). In the absence of interspecific competition or environmental hetero heterogeneity, the intrinsic rate of reproduction (fecundity minus density-independent mortality) and the spatial scale of dispersal merely set the temporal and spatial scales of the population without without having any qualitative effect on population dynamics ((Bolker Bolker and Pacala, 11997). 997). Heterogeneous environments widen the range of possibilities. Hetero Heterogeneity could affect fecundity, mortality, competition, or the probability of dispersal (or the scale of dispersal and competition, although this would lead to some thorny technical problems). Models Models with heterogeneous fecundity and mortality connect directly with applied questions of population viability in degraded and fragmented landscapes, as well as with basic questions about the evolution of dispersal (Rousset and Gandon, 2002). Models with variation about foraging in dispersal are connected more closely to classical questions about and aggregation in heterogeneous landscapes, but are also relevant to conser conservation. Either heterogeneous mortality or heterogeneous movement probabilities can lead to individuals aggregating in a good (low-mortality, low-movement) habitat (Murrell and Law, 2000; Bolker, Bolker, 2003). This habitat association can shield individuals from the worst effects effects of habitat degradation, but with sufficiently short dispersal scales the negative effects effects of intraspecific competition overwhelm the advantages of habitat association [and the moment equations break down; they do not capture capture the percolation phenomena that can lead to sudden sudden extinction in some continuous-space models (Bascompte and Sole, 997)]. Sol~, 11997)]. Finally, one can use single-species moment equations to predict invasions and to begin to understand the feedbacks among environmental heterogeneity, endogenous heterogeneity, and invasion speed at the edge of a spreading wave. Various pieces of the puzzle are present, but they have not been put together into a single picture. The only work with spatial moment equations analyzes the patchiness of invading populations and shows that patchiness and local crowd crowding slow invasion speeds below what would be expected from a continuous population (Lewis, 2000). Other work has been done with lattice models (Ellner et aI., 998) and diffusion equations (Shigesada and Kawasaki, 11997) 997) to explore al., 11998) the the effects effects of of endogenous endogenous and and exogenous exogenous variability, variability, respectively, respectively, but but these these phe phenomena have not been fully integrated into the elegant literature on dispersal and invasion speeds (Skellam, 11951; 95 1 ; Mollison, 1991; Clark et al., aI., 11998). 998).
Interspecific Interspecific Competition Spatial dynamics and spatial models have been explored extensively in research on competitive coexistence, particularly in plant communities (Tilman, 11994). 994) . The natural history of weedy and early successional species clearly suggests that that variations in spatial strategies could maintain diversity in environments where there is insufficient variation in resource availability for 993). Spatial niche separation to maintain diversity (Tilman and Pacala, 11993). coexistence can be interpreted as the exploitation of endogenous endogenous and exoge exogenous spatial covariance in resource availability. The basic mechanisms of spa spametapopulation tial coexistence have been explored in meta population ((Levins Levins and Culver, 1971; Tilman, 11994; 994; Pac ala and Rees, 11998; 998; Chesson, 2000a), lattice Pacala (Silvertown et aI., 992; Holmes and Wilson, 1998), and continuous-space al., 11992;
BENJAMIN BENJAMIN M. M. BOLKER BOLKER
66
models Gopasalmy, 11977a,b); 977a,b); more models ((Gopasalmy, more recent recent work work has has applied applied the the tools tools of of pair pair (Harada 994) and spatial moment (Harada and Iwasa, 11994) moment equations equations (Bolker and Pacala, 11999; 999; Bolker aI., 2003). Bolker et et aI., al., 2000; 2000; Dieckmann Dieckmann and and Law, Law, 2000; 2000; Bolker et et al., 2003). Chesson (2000a,b) (2000a,b) has developed an analytic framework framework to to quantify and test the processes by which coexisting species partition exogenous temporal and spatial variability. His framework framework largely neglects the dynamics of endogenous variation, although although it has recently begun to incorporate incorporate continuous space; the results on equilibrium equilibrium correlations correlations developed by Snyder and Chesson (2003) correspond correspond to the results one one would would get by deriving moment moment equations equations but neglecting the singular terms in the covariances. Finally, the evolutionary dynamics of dispersal and spatial coexistence remain largely unexplored unexplored (Hovestadt 1 ; Rousset and Gandon, 2002). A full accounting (Hovestadt et aI., al., 200 2001; accounting of spa spatial coexistence will have to incorporate incorporate both endogenous and exogenous vari variation and to recognize that that spatial traits such as dispersal represent just a subset of all plant plant competitive and life life history traits and that that the coevolution of these traits needs to be considered as a whole (Rees, 11996). 996).
Predator-Prey Predator-Prey The The spatial spatial dynamics dynamics of of predator-prey predator-prey interactions interactions are are currently currently an an area area of of active research, with many theoreticians theoreticians trying to understand understand whether differ differences between spatial and nonspatial models in stability and equilibrium densities are caused by diffusion limitation or other other spatial modifications modifications of the predation 993; Cuddington predation process (McCauley et aI., al., 11993; Cuddington and Yodzis, 2002; Hosseini, 2003). Keeling et al. (2002) used moment equations (for a metapopu metapopulation system) to understand understand the broad broad conditions under which spatial structure stabilizes or destabilizes a predator-prey predator-prey model. The The usual usual approaches approaches to to understanding understanding predator-prey predator-prey and and other other complex complex ecologies in continuous continuous space make use of simulation models and a variety of 995; Pascual and Levin, 11999) 999) to identify the scaling rules (Rand and Wilson, 11995; relevant scales at which these ecologies are best understood. understood. Looking at the problem from a very different perspective, spatial moment equations have a different set of advantages and disadvantages. Moment equations equations typically look at the average densities and covariance covariance densities of an entire system and, as a result, have some difficulty capturing local stability or instability. Capturing Capturing the effects of finite system size, which might be another another way to address the problem, problem, is similarly difficult because of the algebraic complexity of moment equations with spatial inhomogeneity. In addition, addition, some of the nonlinearities nonlinearities and resulting coherent coherent structures (e.g., spirals) that that arise in typical predator-prey predator-prey models may be poorly represented represented by second-order second-order moment 994). moment closures (Tainaka, 11994). However, spatial spatial moment moment equations equations also offer unique unique advantages. advantages. Other Other approaches approaches have explained the changes in predator-prey predator-prey dynamics from nonspatial nonspatial expectations expectations as a result of endogenous endogenous patchiness, patchiness, but have not not tried tried to predict predict the endogenous endogenous patchiness patchiness itself. By calculating calculating the shape and scale of equilibrium equilibrium covariances covariances within within and and between between species, we can understand understand how how different spatial and demographic demographic parameters parameters determine the the endogenous endogenous structure structure of of predator-prey predator-prey systems systems in in continuous continuous space space and, by extension, extension, the spatial dynamics of predator-prey predator-prey systems (M. Desai,
3. CONTINUOUS-SPACE CONTINUOUS-SPACEMODELS MODELS
67 67
unpublished moment equations can natnat unpublished manuscript). manuscript). Furthermore, Furthermore, spatial spatial moment equations can urally be include exogenous urally be extended extended to to include exogenous variation variation in in predation predation risk risk in in aa more general form than the patchy structure structure implied by metapopulation metapopulation models. models.
Epidemics Epidemics Epidemics are a subtype of a predator-prey system that that is of obvious prac practical and theoretical interest. Spatial models of epidemics go back to the panpan demic models of Kermack and McKendrick ( 1 927); since then, epidemics have demic models of Kermack and McKendrick (1927); since then, epidemics have been studied in metapopulation, lattice, and continuum models. Stochastic spatial models have always presented a challenge, however, and until recently, computational simulation models have been the main method of exploration (Duryea 999). Moment (Duryea et et al., al., 11999). Moment equations equations and and their their discrete discrete pair pair approxima approximation analogues have proved to be powerful tools for analyzing spatial epidemic models (SaW et 994; Keeling al., 11997; 997; Rand, 999; Keeling, 999a; models (Sato et al., al., 11994; Keeling et et al., Rand, 11999; Keeling, 11999a; Boots and Sasaki, 2000, 2002). Many of the analyses have used social contact Boots and Sasaki, 2000, 2002). Many of the analyses have used social contact networks spatial structure, networks as as the the underlying underlying spatial structure, which which is is probably probably more more appro appropriate for human diseases than a two-dimensional plane. For diseases of plants and wildlife, however, lattice or continuum models may be best. Some of the remaining challenges in using spatial moment equations for epidemiology are adapting some of the sophisticated lattice closures developed by Filipe and Gibson 1 ) to Gibson (200 (2001) to the the continuous-space continuous-space case; case; incorporating incorporating exogenous exogenous spatial spatial heterogeneity in suceptibility and infectivity; and combining disease transmis transmission with other ecological processes such as competition to explore the effects of disease in a broader ecological setting.
3.6 OUTLOOK 3.6 OUTLOOK Continuous-space Continuous-space models models represent represent aa different different approach approach to to spatial spatial ecology ecology than population models than the the meta metapopulation models discussed discussed throughout throughout most most of of this this book; book; they they are relevant for populations living in different habitats from the spatially dis disconnected, connected, temporally temporally persistent persistent patches patches that that are are the the main main concern concern of of meta population ecology. This chapter has reviewed continuous-space models metapopulation with with a strong focus on spatial moment equations, a relatively new analytical framework for reducing simple individual-based models to a set of integral equations for the densities and spatial covariances of interacting populations. Why Why would would one one choose choose continuous-space continuous-space models models (continuum, (continuum, lattice, lattice, or or spatial IBM) over patch occupancy or meta population models? There metapopulation There are two two basic reasons: first, one might be studying an organism or community that that inhabits a continuous habitat, where the patchy-habitat assumption is simply unrealistic. Second, one might be asking explicit questions about the scale and pattern of spatial structure structure that have no meaning in a discontinuous spatial model. Tactical questions and landscapes incorporating specific features such as barriers and complex boundaries between habitat types would favor spatial IBMs, IBMs, whereas whereas strategic strategic questions questions and and simple simple landscapes landscapes with with more more conti continuous nuous variation variation would would favor favor lattices lattices or or continuum continuum models. models.
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BENJAMIN M. BOLKER BENJAMIN M. BOLKER
Given Given aa continuous-space continuous-space model, model, how how does does one one choose choose among among the the different different options? options? As As described described earlier, earlier, spatial spatial IBMs IBMs are are flexible flexible and and realistic realistic but but hard hard to to generalize, generalize, whereas whereas lattice lattice and and continuum continuum models models are are more more general general and and tractable tractable but but make make different different kinds kinds of of unrealistic unrealistic assumptions. assumptions. Lattice Lattice models models assume assume aa particular, particular, unrealistic unrealistic spatial spatial structure structure that that complicates complicates the the connec connection tion with with measured, measured, individual-level individual-level data data [except [except in in cases cases where where data data are are gathered 1 994)]. Continuum gathered at at the the level level of of lattice lattice cells; cells; Silvertown Silvertown et et al. al. ((1994)]. Continuum mod models although it els eliminate eliminate demographic demographic stochasticity stochasticity ((although it can can be be reintroduced reintroduced by by adding adding noise noise with with the the appropriate appropriate scaling scaling properties) properties) and and tend tend to to reduce reduce the the complexity complexity of of individual individual movement movement rules rules to to diffusion diffusion and and advection advection terms. terms. Nevertheless, be Nevertheless, both both of of these these model model types types can can be be extremely extremely useful. useful. Both Both can can be solved numerically by simpler solved numerically by standard standard packages packages (lattice (lattice models models are are much much simpler computationally) computationally) and and both both are are some some what what tractable tractable analytically, analytically, although although most analyzed approximately approximately most nonlinear nonlinear spatial spatial models models of of any any type type can can only only be be analyzed or special cases invasion conditions. or for for special cases such such as as invasion conditions. The spatial moment equations emphasized emphasized in The spatial moment equations in this this chapter chapter represent represent aa dif different, relatively set of ferent, relatively new new set of compromises. compromises. Spatial Spatial moment moment equations equations have have two major disadvantages. two major disadvantages. First, First, because because they they depend depend on on aa heuristic heuristic approx approximation imation of of the the full full spatial spatial configuration configuration of of the the system, system, they they may may fail fail under under certain spatial certain conditions conditions ~ at at present, present, one one must must always always verify verify the the results results of of spatial moment equations with moment equations with simulations simulations for for at at least least aa few few sets sets of of parameters. parameters. Second, equations can be Second, the the detailed detailed derivation derivation and and analysis analysis of of moment moment equations can be daunting daunting for for the the average average ecologist ecologist or or evolutionary evolutionary biologist, biologist, although although some some automated automated tools tools have have been been developed developed to to help help the the process. process. Spatial Spatial moment moment equations equations do do have have strong strong counterbalancing counterbalancing advantages, advantages, however. however. Because Because their their parameters parameters are are expressed expressed directly directly in in terms terms of of individual individual behavior, behavior, moment moment equations equations are are useful useful for for studying studying the the connections connections between between individ individual Because their variables (spatial ual behavior behavior and and spatial spatial dynamics. dynamics. Because their state state variables (spatial cor correlations) preserve preserve information information about about the the shape shape and and scale scale of of spatial spatial pattern, pattern, relations) moment causes and spa moment equations equations are are useful useful for for understanding understanding the the causes and effects effects of of spatial pattern. Spatial moment equations equations can tial pattern. Spatial moment can preserve preserve some some aspects aspects of of demo demographic landscape heterogeneity, graphic stochasticity stochasticity and and of of landscape heterogeneity, although although they they cannot cannot typically typically handle handle movement movement orientation orientation by by individuals individuals or or landscape landscape struc structures, such flexibility than tures, such as as edges edges and and barriers, barriers, and and they they have have less less flexibility than spatial spatial IBMs IBMs for for incorporating incorporating details details of of individual individual state; state; at at present, present, these these state state dynamics (Fig. 33.1). .1 ). dynamics can can only only be be included included by by adding adding new new types types to to the the model model (Fig. Spatial Spatial moment moment equations equations can can bbee analyzed, analyzed, although although often often only only by by simplify simplifying models considerably. ing the the models considerably. Nevertheless, Nevertheless, spatial spatial moment moment equations equations can can form form aa bridge bridge between between complex complex spatial spatial IBMs IBMs and and simple, simple, analytically analytically tractable tractable models. models. Like Like metapopulation metapopulation models, models, moment moment equation equation approaches approaches can can be be used used to to explore population explore aa broad broad range range of of ecological ecological interactions. interactions. Also, Also, like like meta metapopulation models, models, moment moment equations equations work work best best in in communities communities with with aa particular particular kind kind of equations, these of underlying underlying spatial spatial geometry. geometry. In In the the case case of of moment moment equations, these are are communities communities where where both both population population densities densities and and environmental environmental characteristics characteristics change individual is predictable (at change relatively relatively smoothly; smoothly; where where the the fate fate of of an an individual is predictable (at least least on on average) average) from from the the characteristics characteristics of of its its neighborhood; neighborhood; and and where where that that neighborhood neighborhood is is large large enough enough to to encompass encompass at at least least aa dozen dozen or or so so interacting interacting organisms. Moment equations can shapes and scales of organisms. Moment equations can predict predict the the shapes and scales of population population
3. 3.
CONTINUOUS-SPACE CONTINUOUS-SPACE MODELS MODELS
69 69
patterns patterns generated generated by by the the interaction interaction between between environmental environmental variability variability and and endogenous endogenous population population interactions, interactions, and these predictions predictions can sometimes sometimes be handled handled analytically. analytically. Equally Equally important, important, moment moment equations equations have have aa natural natural con connection nection with data data that that are collected at an individual individual scale, with individual-based individual-based computational computational models, models, and and ultimately ultimately with with the the life life and and death death of of individuals individuals in in ecological ecological communities. communities.
sdfsdf
Part II Metapopulation Ecology
sdfsdf
M FTAPO PU LATIO N PO PULATION 4 � META 4
DYN AMICS IN IG H LY DYNAMICS IN H HIGHLY FRAGMENTED FRAG M ENTED LANDSC A PES LANDSCAPES
Otso Ovaskainen and Ilkka Hanski
4.1 4.1
INTRODUCTION INTRODUCTION The ecological dynamics of meta populations have many facets, which can metapopulations hardly be all studied in any single investigation, nor analyzed with with a single metapopulation metapopulation model. Ecologists have constructed constructed particular models to 995; Smith et aI., investigate source-sink dynamics (Pulliam, 1988; 1988; Paradis, 11995; al., 11996; 996; Walters, 2001; Hels, 2002; Waiters, 2001; 2002; Chapter 16), 16), the the influence of immigration and emigration on the type of local population population dynamics (Gyllenberg et aI., al., 11993; 993; Rohani et aI., 11996; 996; Ruxton aI., 11997; 997; Saether et aI., 11999; 999; etal., Ruxton et etal., etal., Nachman, Nachman, 2000), 2000), models of habitat habitat selection influencing metapopulation metapopulation dynamics (Ray et al., aI., 1991; 1991; Doncaster, 2000; 2000; Etienne, 2000; 2000; Danchin et aI., al., 200 1 ), and 2001), and so forth. Models of individual movement behavior in heterogen heterogeneous landscapes (Hanski et aI., al., 2000; 2000; Ricketts, 2001; 2001; Gobeil and Villard, 2002; 2002; Ovaskainen and Cornell, 2003) 2003) provide building blocks for metapopu metapopulation models, and single-species meta population models in turn can be metapopulation extended to model the dynamics of metacommunities 992; metacommunities (Wilson, 11992; Holyoak, 2000; 2000; Klausmeier, Klausmeier, 2001; 2001; Holt, 2002). 2002). The purpose of the models ranges from attempts to clarify general principles, as described by Bolker in
Ecology, Ecology, Genetics, and Evolution of Metapopulations Metapopulations
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Copyright 2 004, Elsevier, 2004, Elsevier, Inc.
2-323448-4 0-12-323448-4 0-1
OTSO OVASKAIN EN AND ILKKA OVASKAINEN ILKKA HANSKI
74 14
Chapter Chapter 33 for spatial spatial continuum continuum models, to spatially extended population population viability analysis (Sjogren-Gulve (Sj6gren-Gulve and Hanski, Hanski, 2000; 2000; Akcakaya, 2000; 2000; Lindenmayer et aI., al., 2001; 2001; Ferreras et aI., al., 2001). 2001). This chapter chapter focuses on models that that are iinn the hard hard core ooff ecological metapopulation SPOM) and their metapopulation theory: stochastic patch patch occupancy occupancy models ((SPOM) approximations, including the mother mother of all ecological metapopu metapopudeterministic approximations, lation models, the Levins model. There are several reasons for devoting an entire chapter chapter to these models. The classic metapopulation metapopulation idea of a "popula "population" tion" of local populations populations is captured, captured, in its bare bare essence, by SPOMs. In a broader broader biological perspective, SPOMs belong to "island models," models," which have played an important important role not not only in population population and community ecology (MacArthur (MacArthur and Wilson, 1963, 1963, 1967), 1967), but also in population population genetics (Wright, 1931; 1931; Slatkin, 1977; 1977; Wade and McCauley, 1988; 1988; Whitlock and McCauley, 1990) and evolutionary studies (Wright, 1931, 1931, 1940). 1940). These models often 1990) allow a rigorous mathematical analysis, an obvious advantage for theory. However, there is yet another another reason for focusing on SPOMs in this chapter, a reason reason that that complements the advantages of the models for theoreticians: theoreticians: SPOMs are good good models for real metapopulations metapopulations living in highly fragmented landscapes, landscapes, to the extent extent that that they can be parameterized with empirical data (Chapter 5) 5) and turned turned into tools that that hold substantial promise for conserva conservation, landscape landscape planning, and management. Figure 4.1 4.1 gives an example of a highly fragmented landscape, landscape, in fact two two representations representations of it. The map on the left shows the 56 habitat habitat patches (dry meadows) suitable for reproduction reproduction by the Glanville fritillary butterfly (Melitaea land Islands in Southwest Finland (Melitaea cinxia) cinxia) in one part of the A .~land (Hanski, 11999b). 999b). The long-term project on the Glanville fritillary has played an
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A network of 56 an example throughout this chapter. This 56 habitat patches used used as an throughout this patch network is part of a larger network of habitat patches inhabited by the Glanville fritillary butterfly (Melitaea land Islands 999b; Nieminen (Melitaea cinxia) in the A ~,land Islands in Southwest Finland (Hanski, 11999b; et aI., al., 2004). 2004). Panel A shows a map of of the real real landscape, whereas panel BB shows the simplified view (network of circular habitat patches) assumed by the spatially realistic metapopulation theory. (network The areas areas of the the circles in BB are proportional to the the sizes sizes of the habitat patches, but but they have have circles in not been drawn to scale. The patch indicated by an arrow is analyzed in Fig. Fig. 4.7. 4.7.
Fig. 4.1
4.. 4
M ETAPOPULATION DYNAMICS METAPOPULATION DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
75 75
important important role in stimulating stimulating research on SPOMs, and and we will use the the results of empirical studies example in of our our empirical studies of of this this butterfly butterfly as as aa focal focal example in this this chapter. chapter. The The rest of the landscape landscape in Fig. 4.1 mostly consists of cultivated cultivated fields and and forests. The "highly fragmented The characteristic characteristic feature feature of of "highly fragmented landscapes" landscapes" is is that that the the pooled pooled area of suitable suitable habitat habitat for for the species of interest interest is limited limited to to say less than than 110% 0 % of the total total area of the landscape, landscape, and and the habitat habitat occurs in discrete 1 ) shows patches, none of which is very large. The map map on the right (Fig. 4. 4.1) shows the simplified view of the same patch patch network network in which which we only distinguish distinguish between between the the breeding breeding habitat habitat and and the the remaining remaining matrix matrix and and in in which which the the actual approximated by actual shapes shapes of of the the patches patches are are approximated by circles. circles. How How common common are are "highly "highly fragmented fragmented landscapes? landscapes?"" Nobody Nobody knows knows the the answer species, but answer for for aa comprehensive comprehensive sample sample of of species, but there there are are reasons reasons to to suspect suspect that that aa large large fraction fraction of of invertebrates invertebrates and and increasing increasing numbers numbers of of other other species species occur occur in highly fragmented fragmented landscapes. landscapes. Community Community ecologists ecologists have docu documented mented repeatedly repeatedly that that most most species species in in most most communities communities are are locally locally rare rare (Whittaker, 975; May, 975; Gaston, 994) . While While there there are reasons (Whittaker, 11975; May, 11975; Gaston, 11994). are many many reasons for of habitat. habitat. Furthermore, for local local rarity, rarity, one one likely likely reason reason is is aa limited limited amount amount of Furthermore, locally rare species tend tend to be rare also in the the surrounding surrounding landscape (Hanski, 11998b), 998b), suggesting frag suggesting that that for for such such species species the the landscape landscape is is often often highly highly fragmented. For the land Islands Islands it it has has been mented. For the Glanville Glanville fritillary fritillary in in the the A Aland been relatively relatively straightforward straightforward to document document the degree of fragmentation fragmentation empirically empirically ~ ca. 4000 0.6% of 4000 fragments fragments covering covering 0.6% of the the total total landscape landscape (Nieminen (Nieminen et et aI., al., 2004) 2004) - but 1 990) have argued but for most most other other species it is not. Murphy Murphy et aI. al. ((1990) argued that that species species with with small small body body sizes, sizes, high high rates rates of of population population increase, increase, short short genera generation tion times, times, and and specific specific resource resource requirements requirements are are predisposed predisposed to to have have aa meta population structure. structure. A fraction of insects possess metapopulation A large large fraction of insects possess these these attributes. attributes. There There are are also also very very large large numbers numbers of of species species that that live live in in discrete discrete "minor "minor habi habitats" 949), such but tats" (Elton, (Elton, 11949), such as as decaying decaying tree tree trunks, trunks, which which are are not not permanent permanent but change change in in time. time. Furthermore, Furthermore, large large numbers numbers of of parasites parasites inhabit inhabit aa highly highly frag fragmented once we that, from their perspective, host individu mented landscape landscape once we realize realize that, from their perspective, host individuals are Grenfell and 1 997). The metapopulation als are habitat habitat patches patches ((Grenfell and Harwood, Harwood, 1997). The metapopulation theory with due to the theory developed developed here here applies applies also also to to these these systems, systems, with due attention attention to the dynamics dynamics of of the the landscape. landscape. Finally, Finally, as as we we all all know, know, the the expanding expanding human human enterprise fragmentation of many habitat enterprise typically typically leads leads to to the the loss loss and and fragmentation of many habitat types types in of species highly in most most landscapes, landscapes, increasing increasing the the frequency frequency of species whose whose habitat habitat is is highly fragmented. fragmented.
Stochastic Stochastic Patch Patch Occupancy Occupancy Models Models Stochastic Stochastic patch patch occupancy occupancy models models are are based based on on two two major major simplifying simplifying assumptions. relates to assumptions. The The first first assumption assumption relates to the the structure structure of of the the landscape, landscape, which which is is assumed assumed to to consist consist of of discrete discrete patches patches of of breeding breeding habitat habitat surrounded surrounded by Fig. 4 .1 landscape is by the the matrix matrix as as shown shown in in Fig. 4.1 ~ the the landscape is highly highly fragmented. fragmented. The The second major assumption second major assumption concerns concerns the the description description of of population population dynamics; dynamics; SPOMs possible states SPOMs recognize recognize only only two two possible states for for each each habitat habitat patch, patch, which which can can be be either either occupied occupied by by the the focal focal species species or or unoccupied. unoccupied. Thus Thus the the sizes sizes and and struc structures explicitly accounted accounted for. tures of of local local populations populations are are not not explicitly for. While While these these simpli simplifications fications restrict restrict the the range range of of problems problems to to which which SPOMs SPOMs may may be be applied, applied, they they make make the the models models both both tractable tractable for for rigorous rigorous mathematical mathematical analysis analysis
76 16
OTSO OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HANSKI HANSKI
((Ovaskainen Ovaskainen and 2002, 2003a; and Hanski, Hanski, 2001 2001,, 2002, 2003a; Ovaskainen, Ovaskainen, 2003) 2003) and and turn turn them tools that used increasingly empirical studies them into into effective effective tools that are are used increasingly in in empirical studies (Moilanen, 999, 2000; 999b; Hanski (Moilanen, 11999, 2000; Hanski, Hanski, 11999b; Hanski and and Ovaskainen, Ovaskainen, 2000; 2000; Ter Ter Braak Braak and and Etienne, Etienne, 2003; 2003; Chapter Chapter 55).) . The The original original "island "island model" model" approach approach to to modeling modeling spatially spatially structured structured dynamics in population population biology assumes that that all the patches are identical. In the the case case of of SPOMs, SPOMs, this this assumption assumption also also means means that that all all the the patches patches and and all all the the local local populations populations occupying occupying these these patches patches are are equally equally connected connected to to each each other. We call models making this assumption assumption homogeneous SPOMs, which are patches obviously are described described in in Section Section 4.2. 4.2. Assuming Assuming identical identical patches obviously simplifies simplifies the models greatly, the models greatly, which which has has been been the the overriding overriding justification justification for for this this assumption. One might also ask what what could be gained by assuming spatial assumption. variation variation in in the the properties properties of of the the patches patches - - would would such such more more complex complex mod models provide any novel insight to the processes processes that SPOMs are used to study? that SPOMs The is. The The answer answer depends depends on on what what our our aim aim is. The greatest greatest advantage advantage of of employ employSPOMs with ing ing heterogeneous heterogeneous SPOMs with variation variation in in patch patch properties properties stems stems from from the the opportunity apply them opportunity to to meaningfully meaningfully apply them to to real real metapopulations metapopulations living living in in highly fragmented landscapes, either for the purposes purposes of management management and conservation or for the purpose of critically testing model predictions predictions for advancing understanding. A advancing scientific scientific understanding. A core core idea idea in in this this research research is is to to make make structural structural model assumptions assumptions about how the key landscape features, such as habitat habitat patch areas, qualities, and spatial locations, locations, will influence the two two key processes of classic classic metapopulation dynamics; local local extinction and recolon processes metapopulation dynamics; recolonization. We have called the combination combination of heterogeneous SPOMs (assuming dissimilar patches patches and hence patch-specific dissimilar and hence patch-specific transition transition probabilities) probabilities) with with these these structural structural model model assumptions assumptions as as the the spatially spatially realistic realistic metapopulation metapopulation theory ((SMT; SMT; Hanski, 1 b; Hanski Hanski and Hanski, 200 2001b; and Ovaskainen, Ovaskainen, 2003 2003).). The The assump assumppopulation processes to landscape structure are discussed in tions relating population Section models based Section 4.3, 4.3, where where we we also also describe describe the the two two most-studied most-studied models based on on this this theory: theory: the the spatially spatially realistic realistic Levins Levins model model and and the the incidence incidence function function model. model. The The stochastic stochastic theory theory of of heterogenous heterogenous SPOMs SPOMs poses poses mathematical mathematical prob probthat a meta metapopulation lems, which relate to the fact that population living in a large patch described by presence or network network with with n n patches patches and and described by the the presence or absence absence of of the the number of possible states, 2 nn.. We describe species in these patches has a huge number two two approaches to overcome these problems. First, for many purposes it is sufficient approximation of sufficient to to use use aa deterministic deterministic approximation of the the stochastic stochastic model; model; the the deterministic theory is described in Section 4.4. Second, Ovaskainen (2002a) deterministic theory has has described described an an effective effective approach approach for for analyzing analyzing the the stochastic stochastic model model for for aa heterogeneous patch network with the help help of an appropriate heterogeneous patch appropriate stochastic model model for for aa homogeneous homogeneous network, network, which which latter latter model model can can be be analyzed analyzed mathematically. mathematically. This This approach, approach, described described in in Section Section 4.5, 4.5, is is akin akin to to the the idea idea of of effective effective population population size size in population population genetics: construct an unstructured unstructured model that that behaves similarly to a structured model with respect to some model features of interest. Following the description description of deterministic deterministic and sto stochastic chastic theories theories of of SPOMs, SPOMs, Section Section 4.6. 4.6. presents presents some some comparisons comparisons with with other approaches other approaches to to modeling modeling metapopulation metapopulation dynamics, dynamics, particularly particularly focused focused on key issues issues in on one one of of the the key in ecological ecological metapopulation metapopulation dynamics, dynamics, the the extinc extinction Section 4.7 tion threshold threshold for for long-term long-term persistence. persistence. Section 4.7 discusses discusses the the current current
4.4.
METAPOPULATIONDYNAMICS DYNAMICS IIN HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION N HIGHLY
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state of the theory and its applications, and Section 4.8 outlines the broader significance significance of of the the spatially spatially realistic realistic metapopulation metapopulation theory, theory, emphasizing emphasizing the the contributions contributions that that it it makes makes toward toward greater greater unification unification of of research research in in popula population biology. This This chapter chapter is is mostly mostly about about theory theory and and models, models, although although our our purpose purpose is is not not just to describe a mathematically rigorous body of theory but also to extract that stem extract biologically biologically relevant relevant messages messages that stem from from SMT. SMT. We We illustrate illustrate many many of of the the results results with with examples examples based based on on the the habitat habitat patch patch network network for for the the Glanville . 1 . We Glanville fritillary fritillary butterfly butterfly depicted depicted in in Fig. Fig. 44.1. We will will not not review review the the litera literature on on the the application application of of the the models models to to real real meta metapopulations, which is is cov covture populations, which 999b, in Chapter Chapter 55 (for (for more more ecologically ecologically oriented oriented reviews, reviews, see see Hanski Hanski 11999b, ered in 200 l b) . The 2001b). The kind kind of of models models examined examined here here are are employed employed in in some some of of the the analy analy20-22 in this volume. ses presented in Chapters 20-22
4.2 4.2
PATCH OCCUPANCY OCCUPANCY MODELS: MODELS: HOMOGENEOUS HOMOGENEOUS PATCH PATCH NETWORKS NETWORKS PATCH Mathematically, Mathematically, SPOMs SPOMs are are defined defined as as Markov Markov chains chains (discrete-time (discrete-time models) models) or Markov Markov processes processes (continuous-time (continuous-time models). models). However, However, most most of of the the existing existing or theory models, but theory does does not not relate relate to to the the full full stochastic stochastic models, but to to their their deterministic deterministic approximations, speaking, account approximations, which, which, mathematically mathematically speaking, account only only for for the the drift drift in in the dynamics dynamics while while ignoring ignoring the the variance variance due due to to stochastic stochastic fluctuations. fluctuations. This This the section considers considers models models that assume identical identical habitat habitat patches. section that assume patches. We We start start with with the the familiar model, which familiar Levins Levins model, which is is helpful helpful in in highlighting highlighting the the basic basic concepts concepts and and qualitative theory behind SPOMs. qualitative theory
Levins Metapopulation Metapopulation Model Model Levins Levins 1 969, 11970) 970) assumed habitat patches patches in patch Levins ((1969, assumed that that the the number number of of habitat in aa patch network is is infinite, network infinite, which which allowed allowed him him to to formulate formulate aa patch patch occupancy occupancy model model directly from the deterministic viewpoint. viewpoint. However, However, we we will will view directly from the deterministic view here here the the Levins the deterministic deterministic mean-field mean-field approximation of the the stochastic Levins model model as as the approximation of stochastic identical and and logistic (Box 4. 1 ) , which assumes aa finite network of of nn identical logistic model model (Box 4.1), which assumes finite network equally connected of which may be be either equally connected patches, patches, each each one one of which may either occupied occupied or or empty. If patch is is occupied, occupied, it it is is assumed assumed to to go go extinct extinct at at aa fixed fixed rate empty. If aa patch rate E E == e.e. The colonization colonization rate rate of of an an empty is assumed to depend on the the fraction fraction The empty patch patch is assumed to depend on of occupied occupied patches patches as as C C == ck/n, ck/n, where where cc is is aa colonization colonization rate rate parameter parameter and and of o ::; k-< k ::; nn isis the the number number of of occupied occupied patches. patches. The The reasoning reasoning behind behind this this 0-< expression for for the the colonization colonization rate rate is is that that kk occupied occupied patches patches are are assumed assumed to to expression produce emigrants emigrants that that disperse disperse randomly randomly to to any any of of the the habitat habitat patches patches (both ( both produce occupied and and empty). empty). As As described described in in Box Box 4.1, 4 . 1 , these these transition transition rates rates from from occupied empty to to occupied occupied patches patches and and vice vice versa versa define define the the stochastic stochastic logistic logistic model model empty as a Markov Markov process. process. If the the number number of of patches patches is is large, large, the the stochastic stochastic logistic logistic model model may may be be If approximated To do p(t) == k(t)/n k(t)/n approximated by by aa diffusion diffusion process. process. To do this, this, we we denote denote by by p(t) the fraction fraction of of occupied occupied patches patches at at time time t.t. The The diffusion diffusion process process is is determined determined the by E[dp 2] /dt of of by the the mean mean (or (or drift) drift) ~(p) j.L(p) == E[dp]/dt E[dp]/dt and and the the variance variance ~r2(p) (j2 (p) == E[dp2]/dt the the infinitesimal infinitesimal rate rate of of change change (Karlin (Karlin and and Taylor, Taylor, 1981), 1 9 8 1 ), which which are are given, given,
OTSO OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HANSKI
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BOX 4.1
The Stochastic logistic Model
The stochastic logistic model has been studied in a variety of contexts, including pop ulation biology, chemistry, and sociology, and the dynamic behavior of the model is well understood (Kryscio and Lefevre, 1 989; Jacquez and Simon, 1 99 3; Ovaskainen, 2001 ). We will describe the model here as a SPOM describing the dynamics of a species inhab iting a network of n identical habitat patches. The stochastic logistic model is defined as a Markov p rocess, which assumes that an occupied patch turns empty at a fixed rate E e and that an empty patch turns occu pied at a rate C ck/n, where c is a colonization rate parameter and 0 s k s n is the number of occupied patches. To put the model into a mathematical setting, k(t) is the random variable giving the number of occupied patches at time t, and q(t) is the prob ability distribution describing the state of the metapopulation, with the component qlt) being the probability that the system is in state k(t) = i (i 0, ... ,n) at time t. By stand ard theory of Markov processes (e.g., Grimmet and Stirzaker, 2001 ), the probability dis tribution q evolves according to the forward equation dq(t)/dt q(t)P, where P is the (n + 1 )*(n + 1 ) matrix =
=
=
=
(1 )
In Eq. (1 ), the diagonal elements are defined as di ei + Ci, and Ci ci(n - i)/ n and ei give total (instead of patch-specific) colonization and extinction rates, assuming that presently i of the n patches are occupied. In the stochastic logistic model, eventual metapopulation extinction is certain, and the biologically most fundamental quantity that may be derived from the model is the time that the meta population is expected to persist. Assuming that the initial state of the metapopulation is drawn from the quasistationary distribution (Box 4.2), the mean time to extinction may be approximated by (Andersson and Djehiche, 1 998; Ovaskainen, 2001 ) ei
=
=
=
*
T
(b- exp( - np* ) -;; \j n ( 1 p* )n-l p*2' 1
=
(2)
_
where p is the deterministic equilibrium value for the fraction of occupied patches * given by p = 1 - e/c. Equation (2) shows that the time to extinction increases expo * nentially with the number of habitat patches (assuming that p is kept fixed). Before extinction, the process approaches a quasistationary distribution 'IT (Box 4.2), which * may be approximated by a normal distribution with mean p and variance (1 -p*)/n (Ovaskainen, 2001 ).
for the for the stochastic stochastic logistic logistic model, model, as as (Saether ( Saether et et al., aI., 1999; 1 999; Ovaskainen, Ovaskainen, 2002a) 2002a) ~(p) ep, p) -- ep, cp ( l -- p) � (p) == cp(1
cr2(p) a2 (p) ==
cp(1 + e p+ cp( l --p )p) nn
(4.1) (4. 1 )
ep .
.
(4.2) (4.2)
4 4..
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The Levins model model is The Levins is obtained obtained from from the the diffusion diffusion approximation approximation by by assum assuming ing that that the the number number of of patches patches is is so so large large that that stochastic stochastic fluctuations fluctuations may may be be ignored. ignored. In In this this case, case, the the model model reduces reduces to to the the familiar familiar deterministic deterministic differen differential equation equation
@ dp dt = = f.1(p) I~(p) dt
=
cp( 1 -- pp))
= cp(1
-
-
e ep..
p
(4.3) (4.3)
Figure 4.2 compares the behavior of the stochastic logistic model with the deterministic deterministic Levins model. Realizations of the stochastic model fluctuate fluctuate around around the deterministic prediction, and the deterministic equilibrium state P" p* corresponds corresponds to to the the peak peak in in the the quasistationary quasistationary distribution distribution of of the the stochastic stochastic model model (for (for the the definition definition of of quasistationarity quasistationarity see see Box Box 4.2). 4.2). A A more more throughout throughout analysis relationship between analysis of of the the relationship between the the deterministic deterministic Levins Levins model model and and the the stochastic logistic model stochastic logistic model has has been been presented presented by by Ovaskainen Ovaskainen (2001 (2001),) , Etienne Etienne (2002b), (2002b), and and Etienne Etienne and and Nagelkerke Nagelkerke (2002). (2002). Lande 1987, 1988b) Lande ((1987, 1988b) extended extended the the Levins Levins model model to to account account for for habitat habitat loss loss by by assuming assuming that that aa fraction fraction 11 - hh of of the the habitat habitat patches patches are are permanently permanently colonization. We destroyed, destroyed, and and thus thus only only the the fraction fraction h h remains remains suitable suitable for for colonization. We assume to discriminate assume that that the the migrating migrating individuals individuals are are unable unable to discriminate between between suit suitable able and and unsuitable unsuitable patches, patches, hence hence they they attempt attempt to to colonize colonize the the latter latter in in pro prois portion portion to to their their number. number. The The colonization colonization rate rate of of empty empty patches patches ((11 - p p) ) is thereby c p h , and and the the model model is is given given as as thereby reduced reduced to to cph, -
-
@ dp
--:it dt = - cph( cph(11 - p) p)
-
-
e e p ,,
p
(4.4) (4.4)
where where p p is is the the fraction fraction of of occupied occupied patches patches out out of of the the suitable suitable ones. ones. The The key key prediction population will prediction made made by by Lande's Lande's model model is is that that the the meta metapopulation will persist persist (there is a nontrivial nontrivial equilibrium state) if and only if the threshold threshold condition condition e e
hh > > - cc
(4.5)
B
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ij' c �
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0.05 0.04 0.03 0.02 0.01 0.2
0.4
0.6
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Occupancy state p
Fig. 4.2 model and and the Fig. 4 . 2 Comparison Comparison between between the the stochastic stochastic logistic logistic model the Levins Levins model. model. (A) (A) A A sin single gle simulation simulation realization realization of of the the stochastic stochastic logistic logistic model model compared compared to to the the deterministic deterministic predic prediction and its tion of of the the Levins Levins model. model. (B) (B) The The exact exact quasistationary quasistationary distribution distribution 'IT ~r and its normal normal distribution distribution [mean 00. [mean p*, p*, variance variance (1 (1 - p*)/n] approximation. approximation. Parameter Parameter values values ee = = 11,, c = = 2, 2, n n = 1100. =
OTSO OTSO OVASKAINEN OVASKAINENAND ILKKA ILKKAHANSKI HANSKI
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BOX 4.2 4.2
Quasistationary Quasistationary Distributions Distributions and and Extinction Extinction nmes Times
Mathematically, SPOMs SPOMs are defined as as Markov chains or Markov processes, processes, for which amount of general theory is available (e.g., Grimmet Grimmet and Stirzaker, Stirzaker, 2001 2001). ). We a large amount describe briefly here part of the theory that is relevant in the analysis analysis of SPOMS. SPOMS. As As described in Section 4.3, the state of of a general heterogeneous SPOM SPOM is specified =1, where the component by the random variable 0 O = {0/}7 {Oi}n=l, component 0/ Oi specifies specifies whether patch ii ). We assume is empty (0/ (Oi = 0) or occupied (O{ (Oi = 11). assume that these 22nn states states are ordered in some manner with with metapopulation extinction (0/ (Oi = = 0 0 for all i) being the first state. let (t) :s Let 11 :s _< Q Q(t) _< 2n 2 n denote the random variable describing the state of the metapopula metapopulaq� t)};;: , denote the probability distribution defined tion at time tt and let q(t) q ( t ) = {{q~[t)}/2=~l by q,(t) q,(t) = P[ P[Q(t) = i]. i]. By standard theory, the probability distribution q q evolves accord accordQ(t) = ing to the forward forward equation either as dq(t)ldt dq(t)/dt = q(t)P (for Markov processes) processes) or as q(t + 11)) = q(t = q(t)P q(t)P (for Markov chains). The 2n 2 n** 22nn matrix P P is called the generator matrix (Markov processes) processes) or the transition matrix (Markov chains) and is composed of transi transition rates rates (or probabilities) from any occupancy state to any other occupancy state (Grimmet 1 ). (Grimmet and Stirzaker, 200 2001). If a metapopulation happens to to go extinct, there are no occupied patches to to produce If migrants, and thus empty patches cannot be recolonized. Mathematically, metapopumetapopu lation extinction is an absorbing state, which the process process will eventually reach with with istribution of the Markov process probability 11,, and thus the stationary d distribution process (or chain) is concentrated on q, ql.. Although the stationary distribution is uninformative, it is still mean meaningful to SPOMs. This may be done in terms of to study the limiting behavior of SPOMs. of the quasistationary distribution 'IT, ~r, which is defined as the limiting distribution conditioned on nonextinction. In practical terms, a metapopulation that succeeds succeeds in persisting for a long time converges toward the quasistationary distribution. More precisely, we first condition the probability distribution q(t) q(t) on nonextinction as =
=
=
=
=
=
(t) m/(t) mj
= Pr( P r (Q Q (t) (t) = =
=
. t) '" ilQ(O ~ 1I )) IIQ(
= =
�(O
11 (t) - q, q1(t)
(1 (I))
after which the quasistationary distribution is defined as the limiting distribution, ~r = ilim ~ met). re(t). The quasistationary distribution 'IT ~r exists exists and is unique, provided that the dln pat etwork is irreducible, meaning that any patch is able to colonize (possibly patch network through through intermediate colonizations) any other patch. Technically, the quasistationary distribution 'IT ~r may be derived as the left leading eigenvector of matrix Po, P0, which is obtained by deleting the first row row and the first column from P. P. Furthermore, drawing the initial state from the quasistationary distribution, the expected time to metapopula metapopulation extinction is given by 'IT =
P
T
=
T=
{-liP 1 /(1 -
- 1/13 for Markov processes, processes, 1/(1 - p) p) for Markov Markovchains, chains,
(2) (2)
where p is the leading eigenvalue of Po 965, 11967). 967). P0 (Darroch and Seneta, Seneta, 11965,
is is satisfied. satisfied. In In other other words, words, the the species species is is expected expected to to persist persist only only if if the the amount amount (� = of of habitat habitat (h) (h) exceeds exceeds aa threshold threshold value value (8 = e/c), e/c), which which is is set set by by the the proper properties model, the ties of of the the species. species. In In the the original original Levins Levins model, the threshold threshold condition condition is is given given by by cc > > e, e, as as in in that that model model h h = = 11.. Based Based on on Eq. Eq. (4.5), (4.5), we we may may con conclude clude that that the the long-term long-term persistence persistence of of aa species species in in aa fragmented fragmented landscape landscape is is
4. 4.
METAPOPULATION DYNAMICS DYNAMICS IN I N HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
81 81
facilitated by by increasing increasing amount amount of of suitable suitable habitat habitat (large (large h), h), aa small small risk risk of of local local facilitated e), and and aa good good colonization colonization ability ability (large (large c). c). extinction (small (small e), extinction The The most most fundamental fundamental message message from from these these simple simple models models for for ecology ecology and and conservation is is that that aa species species may may go go deterministically deterministically extinct extinct even even though though some some conservation suitable habitat habitat remains remains at at the the landscape landscape level, level, but but exactly exactly how how much much habitat habitat is is suitable needed for long-term persistence, persistence, and and how how could could we we estimate estimate this this value? value? The The needed for long-term p * the the fracfrac answer given given by by Lande's Lande's model model is is as as follows. follows. First, First, let let us us denote denote by by p* answer tion of of occupied in an an initial initial situation situation in in which which aa fraction fraction h0 ho of of the the landland tion occupied patches patches in scape consists consists of of suitable suitable habitat. habitat. Assuming Assuming that that the the species species is is at at its its scape population-dynamic equilibrium, equilibrium, we we may may solve solve the the value value of of the the species species paramparam population-dynamic eter 80 = eter e/c from from Eq. = e/c Eq. (4.4) (4.4) as as 80 == h0(1 ho( 1 -- p*), p". ), after after which which the the threshold threshold condition condition [Eq. (4.5)] This observation observation leads leads to to the the (4.5)] may may be be rewritten rewritten as as hh >> h0(1 ho( l -- p*). p * ). This [Eq. seemingly very very useful useful result, result, dubbed dubbed the the Levins Levins rule rule (Hanski (Hanski et et al., aI., 1996), 1 996), that that the the seemingly minimum minimum amount amount of of habitat habitat necessary necessary for for long-term long-term persistence persistence can can be be estimated estimated by just just recording recording the the amount amount of of empty empty habitat habitat while while the the species species is is still still common. common. by Carlson the Levins rule to to an an endangered endangered bird bird species in Carlson (2000) (2000) applied applied the Levins rule species in Sweden and Finland, leucotos). Sweden and Finland, the the white-backed white-backed woodpecker woodpecker (Dendrocopos (Dendrocopos leucotos). Using from the National Park Park in Poland, where where the the species species is Using data data from the Bialowieza Bialowieza National in Poland, is still still = 0.66 p* == 0.81, 0.81, from which the the extinction common, he estimated common, he estimated that that ho = 0.66 and and p* from which extinction This result result is is consistent consistent with with the of suitsuit threshold is estimated threshold is estimated as as 0.13. 0.13. This the amount amount of able habitat 0.12) and and in in Finland Finland (h (h = = 0.08), remaining in Sweden (h (h < < 0.12) able habitat remaining in Sweden 0.08), where where the populations have is encouraging, but we we consider the populations have declined declined severely. severely. The The result result is encouraging, but consider that and simplified the land that it it is is based based on on such such aa simplified simplified theory theory and simplified description description of of the landscape that the the Levins has primarily value. There are several factors scape that Levins rule rule has primarily pedagogic pedagogic value. There are several factors that are are not not included included in the model model but but which which are are likely likely to to influence influence metapopulametapopula that in the tion tion dynamics dynamics and and hence hence the the threshold threshold condition condition in in practice, practice, including including the the rescue rescue effect 996; Gyllenberg Hanski, 11997). 997). Furthermore, effect (Hanski (Hanski et et aI., al., 11996; Gyllenberg and and Hanski, Furthermore, as as the the models do take into account variation models do not not take into account variation in in the the properties properties of of the the patches, patches, any any best. The estimates estimates of of h h and and P p':" are are rough rough approximations approximations at at best. The same same comments comments and Lande models in genapply to the stochastic logistic model and the Levins and in gen eral: population dynamics, eral: they they provide provide qualitative qualitative insight insight to to classic classic meta metapopulation dynamics, but but they they have have limited limited value value for for aa quantitative quantitative metapopulation metapopulation analysis. analysis. To To be be fair, fair, they they were were not not meant meant to to do do that. that. To To follow follow our our interest interest in in developing developing predictive predictive metapopulation models landscapes, we metapopulation models for for highly highly fragmented fragmented landscapes, we next next turn turn to to heterogeneous SPOMs, SPOMs, which which may may be be used used more more readily readily in in the the study study of of real real heterogeneous metapopulations in in real real landscapes. landscapes. It It will will turn turn out, out, however, however, that that the the behavior behavior metapopulations of of the the spatially spatially heterogeneous heterogeneous models models may may often often be be best best understood understood by by studying studying their their homogeneous homogeneous counterparts, counterparts, which which leads leads to to resurrection resurrection of of the the Levins Levins model model via via aa new new interpretation interpretation of of the the model model parameters. parameters.
4.3 4.3
PATCH PATCH OCCUPANCY OCCUPANCY MODELS: MODELS: HETEROGENEOUS HETEROGENEOUS PATCH PATCH NETWORKS NETWORKS fundamental difference difference between homogeneous and heterogeneous heterogeneous The fundamental homogeneous and SPOMs SPOMs is is that that in in the the latter latter the the colonization colonization and and extinction extinction probabilities probabilities ((or or rates) rates) are are different different for for different different habitat habitat patches. patches. The The state state of of aa meta metapopulation living living in a heterogeneous heterogeneous network network of nn patches, patches, such as shown shown in population Fig. 1 , is Fig. 4. 4.1, is described described by by the the vector vector 0 O = {OJ}i=l' {Oi}in=l, where where the the component component OJ Oi
OTSO SKI OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HAN HANSKI
82 82
specifies OJ = specifies whether whether patch patch ii is is empty empty ((Oi = 0) 0) or or occupied occupied ((OJ 0 i= - - 11) ).. The The metapopulation metapopulation model model may may be be specified specified either either in in continuous continuous time, time, in in which which case case the the model model is is mathematically mathematically aa Markov Markov process, process, or or in in discrete discrete time, time, in in which which case case the the model model becomes becomes aa Markov Markov chain. chain. We We denote denote the the colonization colonization rate (for (for Markov Markov processes) processes) or or the the colonization colonization probability probability (for (for Markov Markov rate chains) of an empty patch i by Cj = Cj(O) and the extinction rate (or chains) of an empty patch i by Ci = Ci(O) and the extinction rate (or prob probpatch by Ej Ei = = Ej(O). Ei(O). ability) of an occupied patch Before Before turning turning to to specific specific SPOMs SPOMs for for heterogeneous heterogeneous networks, networks, it it is is worth worth noting that the qualitative theory presented in Box 4.1 for the stochastic logisnoting that the qualitative theory presented in Box 4.1 for the stochastic logis tic tic model model applies applies almost almost independently independently of of the the functional functional forms forms for for the the coloniza colonization tion and and extinction extinction processes processes Cj Ci and and Ej• Ei. Metapopulation Metapopulation extinction extinction is is an an absorbing absorbing state, state, which which the the process process will will eventually eventually reach reach with with probability probability one. one. Before Before the the inevitable inevitable extinction, extinction, the the process process converges converges toward toward aa quasistationary quasistationary distribution 'IT, which is given as the subdominant eigenvector distribution ~r, which is given as the subdominant eigenvector of of aa stochastic stochastic transition matrix transition matrix P (see (see Box Box 4.2 4.2 for for details). details). This This quasistationary quasistationary distribution, distribution, referred to as ""(stochastic) often referred (stochastic) metapopulation equilibrium" (Hanski, 11999b) 999b) or population ((Frank Frank and or as as the the "established "established phase" phase" of of the the meta metapopulation and Wissell, Wissell, 2002; 2002; Grimm 2004), is Grimm and and Wissel, Wissel, 2004), is of of great great importance importance for for ecological ecological applications applications of of the population and the theory. theory. It It relates relates directly directly to to the the average average size size of of the the meta metapopulation and is is needed population that needed for for determining determining the the extinction extinction risk risk of of aa meta metapopulation that has has already already persisted persisted for for some some time. time. As As the the quasistationary quasistationary distribution distribution does does not not account account for for transient not sufficient dynam transient dynamics, dynamics, however, however, it it is is not sufficient for for the the description description of of the the dynamics of of aa newly newly established established metapopulation metapopulation nor nor aa metapopulation metapopulation that that has has been been ics perturbed perturbed recently. recently. In In such such cases, cases, one one needs needs to to find find out out also also the the probability probability of of reaching reaching the the quasistationary quasistationary distribution, distribution, which which influences influences the the extinction extinction risk risk of the metapopulation al., 1991a; of the metapopulation within within a given given time time horizon horizon (Verboom et et aI., 1 99 1a; Stephan, 1 993; Ovaskainen Hanski, 2002; 2002; Grimm Grimm and and Wissel, Wissel, 2004). 2004). Stephan, 1993; Ovaskainen and and Hanski, We will will next formulate deterministic approximations of the full full stochastic stochastic We next formulate deterministic approximations of the = {pi}n_ {Pi}7= 11 aa vector vector with with the the component component ii givgiv SPOM. To this, we we denote by pp = SPOM. To do do this, denote by is occupied. occupied. A A deterministic deterministic version version of ing the probability that that patch ing the probability patch ii is of the the by the the SPOM may may be be obtained by replacing replacing the the vector vector of patch occupancies occupancies 0 SPOM obtained by of patch O by vector of of occupancy probabilities probabilities p ((Ovaskainen Ovaskainen and and Hanski, Hanski, 2001), 200 1 ),
dpi(t) dt pi(t + 1) - pi(t)
= Ci(p(t))(1 - p i ( t ) ) -
Ei(p(t))pi(t).
(4.6)
In this this equation, equation, the the upper upper formula formula relates relates to to continuous-time continuous-time models models and and the the In lower one one to to discrete-time discrete-time models. models. If If the the colonization colonization and and extinction extinction rates rates lower (probabilities) depend depend on on the the occupancy occupancy state state in in aa linear linear fashion, fashion, Eq. Eq. (4.6) (4.6) may may (probabilities) be derived derived from from the the stochastic stochastic model model in in the the same same way way as as the the Levins Levins model model was was be derived from from the the stochastic stochastic logistic logistic model. model. Note Note that that for for nonlinear nonlinear models, models, derived Eq. (4.6) (4.6) does does not not necessarily necessarily correspond to the the drift drift term, term, although although in in Eq. correspond exactly exactly to practice practice it often often gives a good good approximation. approximation. We will will put put flesh flesh to to the the skeleton skeleton of of SPOMs SPOMs by by introducing introducing two two examples examples of of We biologically reasonable reasonable models, models, which which have have been been dubbed dubbed the the spatially spatially realistic realistic biologically Levins model model (Hanski (Hanski and and Ovaskainen, Ovaskainen, 2000) 2000) and and the the incidence incidence function function Levins
4. 4.
83 83
M ETAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
model (Hanski, (Hanski, 1994a). 1 994a). Before Before doing we have have to to add add aa critical critical compocompo model doing that, that, we nent to to the the models, models, the the set set of of assumptions assumptions that that relate relate the the colonization colonization and and nent extinction rates rates to to the the structure structure of of the the fragmented fragmented landscape. landscape. extinction
Metapopulation Theory Spatially Realistic Metapopulation A particular particular SPOM SPOM is is defined defined by by describing describing how colonization and and extinction extinction A how colonization rates (or (or probabilities) probabilities) depend depend on on the the structure structure of of the the landscape landscape and and on on the the prespres rates ent occupancy occupancy pattern. pattern. The The dependence dependence of of population population turnover turnover on on habitat habitat fragfrag ent ment areas areas and and spatial spatial locations, locations, and and possibly possibly on on other other quantities, quantities, such such as as habitat habitat ment quality, places the theory explicitly explicitly in a spatial context. context. As one the metapopulation metapopulation theory one may use use here here landscape landscape measures measures that that describe describe the the structure structure of of real real fragmented fragmented may landscapes, we we have have called called the the combination combination of of heterogeneous heterogeneous SPOMs SPOMs and and the the landscapes, assumptions mapping mapping population population turnover turnover to to landscape landscape structure structure the the spatially spatially assumptions realistic metapopulation metapopulation theory theory (Hanski, (Hanski, 2001b; 2001 b; Hanski Hanski and and Ovaskainen, Ovaskainen, 2003). 2003). realistic The starting point description of the landscape landscape structure structure as depicted in The starting point is is aa description of the as depicted in Fig. 4.1. 4.1. Thus Thus the the network network is is assumed assumed to Fig. to consist consist of of n circular circular habitat habitat patches. patches. We area of of patch patch i, by dij di; the the distance cen We denote denote by by AAii the the area i, and and by distance between between the the centroids of extinction and and colonization rates are are now now defined troids of patches patches ii and and j. j. The The extinction colonization rates defined as functions of di;, using using some specific arguments, arguments, of of which which the the followfollow as functions of AAii and and dij, some specific ing sections give two examples. examples. In most cases cases it it is is sensible to assume assume that that aa ing sections give two In most sensible to decreasing patch area increases the extinction the contribu decreasing patch area increases the extinction rate rate and and decreases decreases the contribution of the the respective population to to connectivity populations because because tion of respective population connectivity of of other other populations population sizes tend decreasing patch patch area. Likewise it tend to decrease with with decreasing local population sensible to to assume assume that that the the probability probability of of an an empty empty patch patch becoming colonis sensible becoming colon ized ized increases increases with with increasing increasing connectivity connectivity to to existing existing populations. populations. Figure Figure 4.3 4.3 gives gives an an example example for for the the well-studied well-studied Glanville Glanville fritillary fritillary butterfly. butterfly. We unites the metapopulation theory We will will note note in in passing passing that that SMT SMT unites the classic classic metapopulation theory ((CMT) CMT) based 1 969, 11970) 970) and based on on the the pioneering pioneering models models by by Levins Levins ((1969, and the the MacArthur and Wilson dynamic theory of island biogeography (DTIB) of MacArthur ((1963, 1 963, 11967) 967) (Hanski, because (Hanski, 2001b). 2001b). CMT CMT and and DTIB DTIB are are obviously obviously related related because the the expected expected number number of of species species on on an an island island or or in in aa habitat habitat fragment fragment (the (the
A 1 .?;:3 .Q 0.8 0.8
.m .0
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<1l .c . 0
a.
§ U O
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o. 0.6 ._o 0.4 .N c 0.2 t--
0.4 0.4
to
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Connectivity Iogl0S
4.3 The dependence dependence of extinction extinction probability on patch size (A) and the dependence dependence of Fig. 4.3 (B). Dots represent data collected for nine successive colonization probability on connectivity (S). network of ca. 4000 4000 habitat patches in the A Aland generations of the Glanville fritillary from a network land al., 2004). 2004). Lines Lines depict depict maximum maximum likelihood likelihood estimates based on the entire Islands (Nieminen et aI., 7, �em data data set. set. Parameter Parameter values values e = 0.38, 0.38, c = 0.083, Ci ~ = = 0.84, 0.84, �ex ~ex 0.1 0.17, ~em = 0.07 0.07 and and �im ~im = 0.30. 0.30. = --
--
--
OT50 OTSO OVA5KAINEN OVASKAINEN AND AND ILKKA ILKKA HAN5KI HANSKI
84 84
dynamic dynamic variable variable in in DTIB) DTIB) is is given given by by the the sum sum of of probabilities probabilities of of different different species species occurring occurring in in that that fragment fragment (the (the dynamic dynamic variable variable in in CMT). CMT). Assuming Assuming aa mainland identical and species, the basic DTIB mainland pool pool of of R R identical and independent independent species, the basic DTIB is is obtained by obtained by multiplying multiplying both both sides sides of of Eq. Eq. (4.6) (4.6) by by R R and and assuming assuming the the same same colonization species. However, colonization and extinction extinction rate parameters parameters for for all the species. and equally connected connected habitat habitat patches as in CMT, assuming a set of identical and Eq. the rate of change in the the probability of occupancy for each Eq. (4.6) (4.6) gives gives the rate of change in probability of occupancy for each frag fragment, which also the change in ment, which is is also the rate rate of of change in the the fraction fraction of of occupied occupied fragments. fragments.
Spatially Levins Model Spatially Realistic Realistic Levins Model (SRLM) (SRLM) Our Our first example example is the the spatially spatially realistic Levins model, model, which which is defined by the assumptions about the following following general general assumptions about the the colonization colonization and and extinction extinction rates: rates: Ci = Z cijOj , j4:i Ei = 8i~
(4.7) (4.7)
where where Cciiij denotes denotes the the contribution contribution that that patch patch jj (when (when occupied) occupied) makes makes to to the the words, the colonization colonization colonization rate of patch patch i.i. In other words, colonization rate of patch patch ii is aa linear occupancies of linear function function of of the the occupancies of potential potential source source patches, patches, whereas whereas the the extinction extinction rate rate is is independent independent of of the the occupancies occupancies of of the the other other patches. patches. Equation Equation (4.7) (4.7) is is structurally structurally similar similar to to the the stochastic stochastic logistic logistic model. model. If If the the patches are identical SRLM reduces patches are identical and and equally equally connected, connected, SRLM reduces to to the the stochastic stochastic logistic model. Our next task is to make specific assumptions assumptions as to how how colon colonization ization and and extinction extinction rates rates depend depend on on the the structure structure of of the the patch patch network. network. Local Local Extinction
Assuming Assuming that that all all the the patches patches are are of of equal equal quality, quality, it it is is reasonable reasonable to to assume assume that that the the carrying carrying capacity capacity of of aa patch patch is is proportional proportional to to its its area. area. Thus Thus number of individuals in local population Ki K i =k =k A A ii gives gives the the number of individuals in patch patch ii when when the the local population is is at at its its carrying carrying capacity, capacity, and and kk denotes denotes the the density density of of individuals. individuals. If If there there is is variation variation in in density density from from patch patch to to patch patch due due to to differences differences in in patch patch quality, quality, this in the as one this can can be be straightforwardly straightforwardly taken taken into into account account in the model, model, as as long long as one has has estimates estimates of of patch-specific patch-specific density. density. In In order order to to estimate estimate the the extinction extinction rate rate of assumptions on of aa local local population, population, we we need need to to make make assumptions on the the type type of of local local dynamics. dynamics. One possibility would would be to assume that that the local population population behaves according to the stochastic stochastic logistic model, model, now now viewed as a model for the number in aa local this case, the rate of local the number of of individuals individuals in local population. population. In In this case, the rate of local extinction extinction would would essentially essentially decrease decrease exponentially exponentially with with local local carrying carrying cap capK; (Box (Box 4. 1 ). The acity, acity, Exti E x t i ex: ~ r e -Ki 4.1). The exponential exponential scaling scaling arises arises from from demographic demographic stochasticity, stochasticity, which which is is of of major major concern concern in in very very small small populations populations but but not not in in larger ones. all but is larger ones. Typically, Typically, the the extinction extinction risk risk of of all but the the smallest smallest populations populations is dominated Chapter 14). Adding Adding environmen dominated by environmental environmental stochasticity stochasticity ((Chapter environmental stochasticity population dynamics stochasticity to models for for local population dynamics leads to the much much milder scaling (Lande, 993; Foley, 994) milder power-law power-law scaling (Lande, 11993; Foley, 11994)
Extj Exti = =
e e e = A!ex, K ~{e ex x - A {ex ' K 1 1
(4.8) (4.8)
4.
METAPOPULATION IN HIGHLY METAPOPULATION DYNAMICS DYNAMICS IN HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
85 85
where where ee is is an an extinction extinction rate rate parameter. parameter. The The scaling scaling factor factor �~ex may be be inter interex may preted preted as as aa measure measure of of the the strength strength of of environmental environmental stochasticity, stochasticity, with with aa low low value x indicating that environmental stochasticity is of major importance value of of �Gx indicating that environmental stochasticity is of major importance e determining the extinction extinction risk (Hanski, 11998a). in determining 998a). Colonization Colonization
To To estimate estimate the the colonization colonization rate rate of of an an empty empty patch, patch, we we make make the the simplify simplifying assumption assumption that that all the occupied occupied patches patches are at their local carrying carrying capac capacity (or that that the sizes of the local populations populations are proportional proportional to the respective individuals produces carrying carrying capacities). capacities). We We assume assume that that each each of of the the Ki K i individuals produces propagules propagules in in aa continuous continuous fashion fashion with with aa rate rate ss and and that that the the propagules propagules are are randomly (and independently randomly (and independently of of each each other) other) redistributed redistributed according according to to aa radi radially symmetric Box 4.3) ally symmetric dispersal dispersal kernel kernel f(r) fir) ((Box 4.3).. Assuming Assuming that that each each propagule propagule has probability qq of colonization, it col has aa constant constant probability of successful successful colonization, it follows follows that that the the colonization onization rate of an empty empty patch patch ii is given by Ci(O) = ~a KjsqgqOj = c a i ~ -- Ajfldq)Oj j:/:i j~i
= ~ cqOj, j~i
(4.9)
where = f(dij)Ai where 'Vij ~///= f(dij)Ai is is the the fraction fraction of of propagules propagules produced produced by by an an individual individual in in patch patch jj that that are are expected expected to to migrate migrate to to patch patch i,i, and and the the colonization colonization rate rate parameter collate the parameter cc is is defined defined as as cc = = ks ksq. What this this formula formula does does is is to to collate the con conq . What tributions all source tributions of of all source populations populations (currently (currently occupied occupied patches) patches) to to the the colo colonization nization rate rate of of the the focal focal patch; patch; the the contribution contribution of of source source patch patch jj depends depends on on the the size size of of patch patch jj and and its its distance distance from from the the focal focal patch patch ii as as specified specified by by the the dis diskernel. persal kernei. Generally, Generally, both both the the source source and and the the target target patch patch areas areas may may influence influence indi individual vidual movements movements and and hence hence the the contribution contribution of of patch patch jj to to the the colonization colonization of of patch patch ii in in aa nonlinear nonlinear manner manner (Hanski (Hanski et et ai., al., 2000; 2000; Moilanen Moilanen and and Nieminen, Nieminen, 2002; Ovaskainen 2002; Ovaskainen and and Cornell, Cornell, 2003). 2003). Assuming Assuming aa power-law power-law relationship relationship for for both and immigration, may be be generalized to the both emigration emigration and immigration, Eq. Eq. (4.9) (4.9) may generalized to the form form Cij and Cij = - cA c A fi ~m iAm~emAf q'dt 1emf(dij) I~ ij!~ (Ovaskainen, (Ovaskainen, 2002b). 2002b). Values of the scaling factors factors �~im im and �em depend on ~em depend on the the biological biological processes processes determining determining immigration immigration and and emigra emigration empirical data tion rates and and may may either be estimated estimated from empirical data or be derived from from submodels for for immigration immigration and emigration. emigration. submodels The model is The deterministic deterministic mean-field mean-field model is obtained obtained by by replacing replacing the the vector vector of of patch occupancy probabilities Gyllenberg and patch occupancies occupancies by by the the vector vector of of occupancy probabilities ((Gyllenberg and Hanski, 997). For example, Hanski, 11997). example, Hanski Hanski and and Ovaskainen Ovaskainen (2000) (2000) assumed assumed a sim simple continuous-time e/Aii ple continuous-time deterministic deterministic SRLM SRLM given given by by Eq. Eq. (4.6) (4.6) with with Ei Ei = ---e/A where we have assumed that �ex and C IAjexp( - adij)pj, where and Cii = = ccEAjexp(-oLdij)pj, we have assumed that ~ex = = �em ~em = = 1 1,, an an exponential exponential dispersal dispersal kernel kernel (the (the normalization normalization constant constant is is included included in in the the � im = parameter parameter c), c), and and no no effect effect of of target target patch patch size size on on colonization colonization ((i~im = 00).) . The The exponential exponential dispersal dispersal kernel kernel used used here here iiss phenomenological phenomenological (does (does not not have have aa mechanistic mechanistic explanation) explanation) but but it it typically typically fits fits well well with with observa observations 999; Sutherland tions (e.g., (e.g., Conrad Conrad et et ai., al., 11999; Sutherland et et ai., al., 2000; 2000; Byrom, Byrom, 2002; 2002; Griffith Griffith and and Forseth, Forseth, 2002) 2002).. Note Note that that it differs differs from from the the dispersal dispersal kernel kernel introduced 1 ) in 4.3 by introduced in in Eq. Eq. ((1) in Box Box 4.3 by the the factor factor r. r. As As seen seen in in Section Section 4.4, 4.4, the the deterministic deterministic model model yields yields many many useful useful results results about about metapopulation metapopulation dynamics. dynamics.
86 8 6
OTSO OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HANSKI HAN SKI
BOX
4.3
Dispersal Kernels
In order to construct a SPOM, one needs to know the fraction 9ij of propagules pro duced by an individual in patch j that are expected to migrate to patch i. The fraction 9ij is often described with the help of a dispersal kernel 0, which specifies how propag ules are redistributed in a landscape. More precisely, assuming that the propagules are produced in the origin, the fraction of propagules that disperse to a region X k R2 is assumed to be given by fxO(x,y)dA, where dA denotes integration with respect to area. As the total number of propagules should be conserved, we always assume that the dis persal kernel is normalized as fR2 O(x,y)dA = 1 . In many cases, it is reasonable to assume that the dispersal kernel is radially symmetric, in which case it may be denoted by f(r) O(r,O), and the fraction of propagules that disperse to distance r with r, < r < r2 is given by 21rf��f(r)rdr. For example, if one assumes that propagules move toward a fixed (but random) direction u ntil they settle with a fixed rate, one has =
f(r)
=
271'r ex
(1 )
e-C1r,
where the parameter ex depends on the ratio of the speed of the individuals and the rate at which they settle. For another example, if one assumes that propagules move accord ing to random walk u ntil they settle with a fixed rate 8, one has f(r)
=
2!/O(lfr),
(2)
where Ko is a modified Bessel function of the second kind and a is the diffusion coeffi cient that may be calculated from the parameters of the random walk (Turchin, 1 998). Assuming that the landscape is highly fragmented, i.e., that the patches are small with respect to the dispersal distances, it follows that 9ij may by approximated by
(3) Deriving 9ij from a dispersal kernel 0 is best justified for passive propagule dispersal, whereas dispersal strategies with active search behavior may not correspond exactly to any dispersal kernel. For example, if the propagules move according to random walk but bias their movement toward the habitat patches, the patches are, in a sense, competing for the migrants. In such a case, the dispersal rate between any two patches does not depend just on the areas of and the distance between the two patches, but also on the spatial config uration of the entire patch network (Ovaskainen and Cornell, 2003). Further, strong het erogeneity of the dispersal habitat (e.g., dispersal corridors or barriers) may prevent the use of dispersal kernels, although in general they provide a reasonable first approximation.
The SRLM may also be formulated as a discrete-time model by assuming that that the the time time step step of of the the model model is is sufficiently sufficiently small. small. In In this this case, case, general general expres expresextinction probabilities are given by sions for the colonization and extinction
{
-- exp ( - � CiPj) , exp(-.j~/cijOj), Ei -e ) E i = 11 - exp( exp(-ei). Ci Ci = = 1l
i
.
(4.10) (4.10)
ETAPOPULATION DYNAMICS N HIGHLY 4. 4. M METAPOPULATION DYNAMICS IIN HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
87 87
As As continuous-time continuous-time SRLM SRLM is is structurally structurally simple simple and and thus thus analytically analytically tractable, tractable, we we use use it it later later for for developing developing much much of of the the mathematical mathematical theory. theory. However, connect the using the as However, we we will will connect the theory theory to to data data using the discrete-time discrete-time SRLM, SRLM, as the rest the latter latter is is more more appropriate appropriate for for the the Glanville Glanville fritillary fritillary butterfly. butterfly. For For the the rest of model 4 . 1 0 with the of this this chapter, chapter, we we will will call call the the model 4.10 with parameters parameters estimated estimated for for the Glanville fritillary butterfly model, whereas Glanville fritillary butterfly (Fig. (Fig. 4.3) 4.3) as as the the Glanville Glanville fritillary fritillary model, whereas SRLM realistic SRLM will will always always mean mean the the continuous-time continuous-time version version of of the the spatially spatially realistic Levins model. model. Levins
Incidence Incidence Function Function Model Model (IFM) (IFM) The The incidence incidence function function model model is is aa discrete-time discrete-time SPOM SPOM that that was was introduced introduced by Hanski ((1994a) 1994a) and been applied empirical studies by Hanski and has has since since been applied widely widely in in empirical studies Ovaskainen, 2002a) (Hanski, 999b, 2001b; (Hanski, 11999b, 2001b; Chapter Chapter 5). 5). An An extended extended version version ((Ovaskainen, 2002a) of of the the incidence incidence function function model model is is defined defined as as Ci =
z Sz S i + l/c'
Ei -- m l "n ( ~ e x ] ) ( 1
(4. 11) (4.11) -- Ci' r
i
where where
m ·�(' d··II ) OI· s SiI = A ~�im i m xX � E A �e gem = A ..:::.. A Ii 1/if(dij)Oj I ij:/=i *i
(4.1 ( 4 . 1 22) )
iiss the the metapopulation metapopulation dynamic dynamic connectivity connectivity ooff patch patch ii to to extant extant local local popula populations. tions. In In comparison comparison with with the the linear linear SRLM, SRLM, the the IFM IFM is is more more complex, complex, as as it it has has two two structural structural parameters. parameters. First, First, parameter parameter zz relates relates to to the the assumptions assumptions behind behind the the colonization colonization process, process, values values greater greater than than zz = = 11 reflecting reflecting the the presence presence of of an 1994a) assumed an Allee Allee effect. effect. In In the the original original version version of of the the IFM, IFM, Hanski Hanski ((1994a) assumed aa could follow relatively strong Allee effect relatively strong Allee effect (z (z = = 2), 2), which which could follow from from the the interaction interaction of immigrants at colonization in a sexually reproducing species. of immigrants at colonization in a sexually reproducing species. Second, Second, parameter parameter rr measures measures the the strength strength of of aa rescue rescue effect, effect, describing describing aa reduced reduced extinction size. Hanski extinction risk risk due due to to immigrants immigrants enhancing enhancing the the local local population population size. Hanski ((1994a) 1994a) assumed that r = 1 . As with the SRLM, the deterministic assumed that r = 1. As with the SRLM, the deterministic approxima approximation tion is is obtained obtained by by replacing replacing the the vector vector of of patch patch occupancies occupancies with with aa vector vector of of occupancy probabilities in Eq. (4. 12). occupancy probabilities in Eq. (4.12).
4.4 4.4
DETERMINISTIC DETERMINISTIC THEORY THEORY This be This section section focuses focuses on on three three ecologically ecologically interesting interesting issues issues that that may may be addressed First, we kind of addressed by by the the deterministic deterministic theory. theory. First, we examine examine what what kind of thresh threshold SPOMs predict old conditions conditions SPOMs predict for for metapopulation metapopulation persistence persistence and and use use these these conditions SPOMs into classes. Second, conditions to to classify classify SPOMs into three three qualitatively qualitatively distinct distinct classes. Second, we we ask ask about about the the "values" "values" of of individual individual habitat habitat patches patches in in the the sense sense of of the the con contributions population dynamics tributions that that the the patches patches make make to to meta metapopulation dynamics and and persistence. persistence.
88 88
OTSO OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HANSKI HANSKI
Third, Third, we we explore explore transient transient dynamics, dynamics, the the specific specific aim aim being being to to quantify quantify the the length length of of time time it it takes takes for for aa metapopulation metapopulation to to return return to to equilibrium equilibrium follow following a perturbation.
Metapopulation Capacity Capacity Metapopulation The question that The most most fundamental fundamental question that may may be be addressed addressed by by aa patch patch occu occupancy model model is is whether whether aa given given species species is is expected expected to to persist persist in in the the long long term term pancy in aa given given fragmented fragmented landscape. landscape. In In stochastic stochastic models, models, where where eventual eventual extinc extincin tion tion is is always always certain, certain, it it is is natural natural to to assess assess the the risk risk of of extinction extinction by by analyzing analyzing the the time time it it takes takes for for the the metapopulation metapopulation to to go go extinct. extinct. In In the the deterministic deterministic framework, framework, aa metapopulation metapopulation may may be be classified classified as as viable viable if if the the model model pos possesses �. , toward sesses aa stable stable nontrivial nontrivial equilibrium equilibrium state state pp*, toward which which the the deterministic deterministic model model converges converges asymptotically. asymptotically. The The deterministic deterministic perspective perspective is is useful useful espe espemetapopulations for for which which the role of stochastic fluctuations is cially for large metapopulations not as pronounced pronounced as for for small metapopulations. metapopulations. not In In the the Levins-Lande Levins-Lande model, model, the the deterministic deterministic threshold threshold condition condition for for persist persistis given given by by hh > > 8, 8, where where h h is is the the amount amount of of suitable suitable habitat habitat and and 88 = -- e/c e/c is is ence is parameter defined defined by the properties properties of the species (Section 4.2). 4.2). Hanski and a parameter Ovaskainen Ovaskainen (2000) (2000) extended extended this this result result to to the the spatially spatially realistic realistic Levins Levins model, model, in in threshold condition for for deterministic persistence is given by which the threshold
(4.13) (4.13)
~.M > g
Here Here A.M ~M is is called called the the metapopulation metapopulation capacity capacity ooff the the fragmented fragmented landscape, landscape, and and 88 == e/c is aa species species parameter in the the original Levins model. model. Comparing Comparing e/c is parameter as as in original Levins we note that in in the spatially realistic realistic model, model, the the metapopumetapopu Eqs. Eqs. (4.5) (4.5) and and (4.13), (4.13), we note that the spatially role of of the the amount amount of of habitat habitat (h) in the nonspatial lation capacity lation capacity (A.M) (~M) plays plays the the role (h) in the nonspatial model. A.M also takes into into account how well well the the patches patches are are connected model. However, However, ~.M also takes account how connected A.M is given as the to each the meta population capacity to each other. other. Mathematically, Mathematically, the metapopulation capacity X M is given as the leading eigenvalue eigenvalue of matrix M M with elements mii and leading of the the n*n n*n matrix with elements mii = 00 and =
+ �i~imA~emf(d#) mAi�em/t:(' d1/ ) .· mijII-- = 8cij/ei - - il�ex + A~ex = A m = 8c II-le-I =
(4.14) (4.14)
The gives the that patch patch jj makes makes to the colonization colonization The element element mij mij gives the contribution contribution that to the rate multiplied by the expected lifetime mij thus thus measthe expected lifetime of of patch patch i.i. mij meas rate of of patch patch i,i, multiplied ures fraction of were the would be be occupied occupied if if patch patch jj were the only only ures the the fraction of time time that that patch patch ii would source of of immigrants immigrants (Hanski (Hanski and and Ovaskainen, Ovaskainen, 2000; 2000; Ovaskainen Ovaskainen and and source Hanski, Hanski, 2001). 2001). The analogy analogy between between ~M A.M and and hh extends extends beyond beyond the the threshold threshold condition. condition. In In The the the Levins-Lande Levins-Lande model, model, the the equilibrium equilibrium fraction fraction of of occupied occupied patches patches is is given given by p* p * == 11 - ~/h, where pp t 8/h, whereas whereas in in the the SRLM SRLM itit is is given given by by ppt~ == 11 - ~/XM, 8/A.M' where by the weights being i, = ki iiss aa weighted weighted fraction fraction ooff occupied occupied patches, patches, Px ~iWiPi, the weights Wi Wi being Wi Pi.. P defined by patch for details). defined patch values (see later later for details). Figure 4.4 gives an empirical example example of the extinction extinction threshold threshold using using the the Figure 4.4 gives an empirical of the Glanville structure of Glanville fritillary fritillary model. model. Due Due to to the the nonlinear nonlinear structure of the the discrete-time discrete-time from the the other other model model model, itit is is not not possible possible to to extract extract aa species species parameter parameter ~8 from model, parameters, and and consequently consequently the the threshold threshold condition condition for for persistence persistence is is given given parameters, A.M >> 11 (Ovaskainen (Ovaskainen and and Hanski, Hanski, 2001). 2001). Although Although there there is is aa in this this model model as as kM in -
-
=
4. 4.
889 9
METAPOPULATION DYNAMICS DYNAMICS IN I N HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
B
A
or{'
'* (D iii
0.8 0.8
>- 0.6 0 .6 >,
� 0.4
r<.l t-
o13
cc. ~
0o o<.l
9
0.2
9 .
..
_O _ . . . L
~o .....
0C;:
i
, . o. "..o: .. . 9: o ~ I 9 9� : . .
. ' . "" , .. : . . ooO . .. ~ , oe 9 .el 9 , .
.
.
.
.
0.8 0.8
•• go • • O0 O0 " 9 '
oo
>, >- 0.6
� 0.4 <.l
e-
o8
c. �
0o o
.
-0.25 0 0.25 0.5 -1 .25 -1 -1 -0.75-0.5 -0. 75 -0.5 -0.25 -1.25
Metapopulation capacity capacity Ioglo 109,0 ;~M Metapopulation AM
.
9
• 9• ,.. • •• • • 9 O'Oo0 • •• 9
o•
• 9 9•
0.2
•
-1 -1
.O
~.,_
.L
.
~,
-0.5 -0.5
9
-~
-!
0
0.5
~-
9 . ..... . 9.
g~O0
.•o•• . •
•
9
•
- - =_ . . . . .
1
1 .5 1.5
Total area area IogloA 1091QA Total
Fig. 4.4 4.4 An empirical empirical example example of of the the treshold condition condition for for persistence persistence using the the Glanville Glanville Fig. fritillary model parameters from Fig. 4.3. Each dot dot depicts depicts 11 of of 59 59 semi-independent semi-independent patch patch fritillary model with with parameters from Fig. networks like the the one one illustrated illustrated in in Fig. Fig. 4.1. 4.1 . The The horizontal horizontal axis axis shows shows the the metapopulation metapopulation capacapa networks like city (A) or the the total area (B) (B) of of the the network, network, and and the the vertical vertical axis axis shows shows the the weighted weighted average of city (A) or total area average of the observed observed probability probability of of patch patch occupancy occupancy during during the the 9-year 9-year study study (weights (weights patch patch values values in in A A the and patch areas areas in in B). B). The The dashed dashed line line in in A A depicts depicts the the threshold threshold for for persistence persistence (kM (AM = 1). 1 ). and patch =
variation in in patch patch quality quality and to regional sto lot of lot of scatter, scatter, much much of of it it due due to to aa variation and to regional stochasticity, it is is evident metapopulation capacity is able rank the the netnet chasticity, it evident that that the the metapopulation capacity is able to to rank works in in terms terms of presence or the species more accurately works of the the presence or absence absence of of the species more accurately than than the total area area of of habitat habitat in the the patch network ((Fig. the total patch network Fig. 4.4B). 4.4B). Note Note that that many many of of AM == 11 have, have, howhow the the threshold the patch patch networks networks that that fall fall below below the threshold condition condition aM ever, had substantial substantial metapopulations metapopulations during the study study period. period. The ever, had during the The explan explanation is twofold. twofold. First, First, the the different different networks networks are are not not completely completely ation for for this this result result is independent of thus aa large large metapopulation meta population may may keep keep aa nearby nearby independent of each each other, other, and and thus small small patch patch network network occupied, occupied, although although the the small small network network would would alone alone be be below below the the threshold threshold condition condition for for long-term long-term persistence. persistence. Second, Second, the the parameter parameter estimates estimates (Fig. (Fig. 4.3) 4.3) are are based based solely solely on on the the observed observed annual annual transi transitions, tions, and and they they thus thus ignore ignore the the occupancy occupancy state state observed observed in in the the first first year year ((see see Chapter Chapter 5). 5). As As there there has has been been aa general general decline decline in in the the occurrence occurrence of of the the butterfly I. butterfly during during the the study study period period (which (which may, may, however, however, be be only only temporary; temporary; I. Hanski, Hanski, unpublished unpublished results), results), the the parameter parameter estimates estimates are are likely likely to to be be biased biased in in the the direction direction of of overestimating overestimating the the critical critical amount amount of of habitat habitat needed needed for for meta population persistence metapopulation persistence (Moilanen, (Moilanen, 2000). 2000). Correcting Correcting for for the the bias bias would would move move the the vertical vertical line line in in Fig. Fig. 4.4A 4.4A somewhat somewhat toward toward the the left. left.
Classification of Classification o f SPOMs SPOMs The The most most fundamental fundamental classification classification of of metapopulations metapopulations in in the the determinis deterministic tic context context is is into into those those that that are are beyond beyond the the extinction extinction threshold threshold and and persist persist (have (have aa stable stable nontrivial nontrivial equilibrium equilibrium state state p p*" > > 0) 0) and and those those that that are are below below the the extinction extinction threshold threshold and and go go extinct. extinct. A A more more refined refined classification classification consid considers ers in in addition addition whether whether the the trivial trivial equilibrium equilibrium state state p p*':' = 00 corresponding corresponding to to meta population extinction unstable, aa small metapopulation extinction is is stable stable or or unstable. unstable. If If it it is is unstable, small meta population has metapopulation has aa deterministic deterministic tendency tendency to to grow, grow, and and thus thus the the metapopu metapopulation lation may may be be expected expected to to be be able able to to invade invade an an empty empty patch patch network network success successfully. fully. The The threshold threshold condition condition for for successful successful invasion invasion may may be be written written as as AI k1 > > 8, 8, where where AI kI is is called called the the invasion invasion capacity capacity of of the the network network (Ovaskainen (Ovaskainen and and Hanski, Hanski, 2001). 2001). In In the the SRLM, SRLM, the the metapopulation metapopulation capacity capacity AM kM coincides coincides with with
90 90
OTSO SKI OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKAHAN HANSKI
the the invasion invasion capacity capacity }.J, ~.I, but but this this is is not not the the case case in in general. general. Three Three types types of of sit situations uations may may be be distinguished distinguished based based on on the the relationship relationship between between metapopula metapopulation 1 ). Models tion and and invasion invasion capacities capacities (Fig. (Fig. 4.5; 4.5; Ovaskainen Ovaskainen and and Hanski, Hanski, 200 2001). Models for for which which }.[ X1= = }.M XMare are called called Levins-type Levins-type models. models. Models Models for for which which }.[ ]kI < < }.M )kM possess population level possess aa meta metapopulation level Allee Allee effect, effect, meaning meaning that that although although aa metapopu metapopulation lation could could persist persist in in aa network, network, aa single single small small local local population population cannot cannot invade invade an population level an empty empty network. network. A A meta metapopulation level Allee Allee effect effect leads leads to to multiple multiple equi equilibria population dynamics. libria in in meta metapopulation dynamics. In In mechanistic mechanistic terms, terms, this this may may be be caused caused by by an an Allee Allee effect effect in in the the colonization colonization process, process, by by aa rescue rescue effect effect in in the the extinc extinction process, process, or or by by aa combination combination of of the the two two ((Ovaskainen and Hanski, Hanski, 2001 2001).). tion Ovaskainen and While While the the SRLM SRLM belongs belongs to to Levins-type Levins-type models, models, the the IFM IFM is is an an example example of of aa model with a strong metapopulation level Allee effect ((Fig. Fig. 4.5). Multiple equilibria equilibria in in meta metapopulation dynamics are are difficult difficult to to test test because because Multiple population dynamics this this requires requires data data for for several several independent independent patch patch networks. networks. The The long-term long-term study land Islands study of of the the Glanville Glanville fritillary fritillary in in the the A Aland Islands has has produced produced the the most most convincing 995a), including convincing example example so so far far (Hanski (Hanski et et aI., al., 11995a), including aa demonstration demonstration of Hanski, 11999b). 999b). A of the the rescue rescue effect effect on on local local extinction extinction ((Hanski, A signature signature of of multi multiple ple equilibria equilibria is is aa bimodal bimodal distribution distribution of of patch patch occupancy occupancy frequencies frequencies (or, (or, more more properly, properly, of of p}... px in in the the case case of of heterogeneous heterogeneous networks) networks).. Putative Putative exam examples Hanski, 11982) 982) have ples of of bimodal bimodal "core-satellite" "core-satellite" distributions distributions ((Hanski, have been been described 999b). The described for for aa wide wide range range of of taxa taxa (Hanski, (Hanski, 11999b). The fundamental fundamental message message
:is
0.8 1l 0.8 .13
25 0.8
K~. 0.6 0.6 O
._o 0.4 0.4 ,g e.c:
0.2 .�E 0.2 O o O -0 o()
B B
A
�
"no Allee effect /
~
weak Allee effect
s 0.6 o. 0.4
strong Allee effect
._~ 0.2
N
. . . . . . . . . .
"~
0.2 0.4 0.6 0'.8' ' :1 Incidences the other Incidences inin the other patches patches 88
AI = AM
a. 6 o_ 6
t~ a: "Oi~ 4 a:
FI
�
22
0.2 0.4 0.6 0.8 0.4 0.6 0.2 0.8 1 the other other patches patches Incidences inin the Incidences
Levins-type model Levi ns-typem odel
~ "~~~---;q /
rescue rescue effect effect
c C
~M
4
uJ
no no rescue rescue effect effect
weak Allee effect effect weak Allee
~ ~ . ~ A l l e e estron f f e cgt Allee effect
0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 0.2 Incidences inin the the other other patches patches Incidences Fig. 44.5 One-dimensional illustration illustration of of three three qualitatively qualitatively different different metapopulation metapopulation models. models. Fig. . 5 One-dimensional (A) The The colonization colonization probability probability of of patch patch ii with with no no Allee Allee effect, effect, with with a weak weak Allee Allee effect, effect, and and with with (A) with and and without without aa rescue effect. effect. strong Allee Allee effect. effect. (B) (6) Extinction Extinction probability probability of of patch patch ii with a strong p)g(p)/p for for Levins-type Levins-type models models and and for for models models possessing possessing aa weak weak (C) The The principal principal map map (1 (1 - p)g(p)/p (C) (2001) for for discussion discussion and and for for the the definition definition strong Allee Allee effect. effect. See See Ovaskainen Ovaskainen and and Hanski Hanski (2001) or or aa strong of of the the principal principal map map g9 (modified (modified from from Ovaskainen Ovaskainen and and Hanski, Hanski, 2001). 2001). -
4. 4.
91 91
METAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
for conservation conservation is is that that aa metapopulation metapopulation may may crash crash unexpectedly unexpectedly to to extincextinc for tion from from aa state state of of commonness. commonness. This This happens happens when when the the metapopulation metapopulation tion crosses the the unstable unstable internal internal equilibrium equilibrium due due to to perturbation perturbation or or when when the the pospos crosses itive internal internal equilibrium equilibrium is is lost lost due due to to an an environmental environmental change. change. itive
Patch Values Values Patch
While AM characterizes characterizes the the capacity capacity of of an an entire network to to support support While ~.M entire patch patch network viable metapopulation, metapopulation, one one might might often wish to to assess assess the the contributions contributions that that aa viable often wish particular patches patches make make to to the the persistence persistence of of aa metapopulation. metapopulation. For For example, example, particular in metapopulation metapopulation management, management, limited may force one to to decide decide in limited resources resources may force one which of of two two patches patches is is more more valuable valuable and and should should be be conserved conserved (Lindenmayer (Lindenmayer which and Possingham, Possingham, 1996a,b; 1 996a,b; Moilanen Moilanen and and Cabeza, Cabeza, 2002; 2002; Chapter Chapter 22). 22). In In the the and context of of metapopulation meta population dynamics, dynamics, the the value value of of aa habitat habitat patch patch depends depends not not context only on on the the size size and and the the quality quality of of the the patch, patch, but but also also on on its its connectivity connectivity to to the the only remaining network. Hanski (2003a) examined the remaining network. Ovaskainen Ovaskainen and and Hanski (2003a) examined the concept concept of of "patch value" by particular patches patches make make "patch value" by quantifying quantifying the the contributions contributions that that particular to population dynamics. the value to meta metapopulation dynamics. They They concluded concluded that that the value of of aa particular particular patch assessed properly without specifying what exactly is meant meant by by patch cannot cannot be be assessed properly without specifying what exactly is aa "contribution Table 4.1 four biologically biologically "contribution to to metapopulation metapopulation dynamics. dynamics."" Table 4.1 lists lists four meaningful alternatives. meaningful alternatives. the dynamic value highlight one Wi, termed We will We will here here highlight one particular particular measure, measure, Wi, termed the dynamic value of More precisely, precisely, Wi Wi is is defined the long-term that of the the patch. patch. More defined as as the long-term contribution contribution that patch makes to to the colonization events events in in the the network. network. To define W W= = {Wi}n= patch ii makes the colonization To define l, { Wi}i= 1, we into account that patch patch jj makes makes to to aa colcol we first first take take into account b bii, the direct direct contribution contribution that ij, the onization population is onization event event in in patch patch i.i. Assuming Assuming that that the the meta metapopulation is at at equilibrium, equilibrium, it it is is natural natural to to define define bb~iji as as (Ovaskainen, (Ovaskainen, 2002a) 2002a)
bij = kiPj* TABLE TABLE 4. 4.11
dCol(p * + 8ej) de.
(4. 15) (4.15)
Four Used to Patch Val ues Four Measures Measures Used to Characterize Characterize Patch Values i
Target Target quantity quantity
Perturbation Perturbation measures
Metapopulation capacity Metapopulation capacity �M kM
dAM vSi- dAi
Colonization Colonization events events Metapopulation Metapopulation size size SS
dS US = � U~ = dA; dAi u~== Ss -- Ssii UL d d logT logT t1 = ~- � dAi t7 = log(TIT) t~= log(T/T)
Dynamic Dynamic measures
Appropriate for for Appropriate Rare species
Wi Wi (see (see text) text)
Common species Common species
1
Time Time to to extinction extinction T
1
Rare species
aa Superscripts Superscripts SS and and L L refer refer to to small small and and large large perturbations, perturbations, respectively. respectively. Metapopulation Metapopulation size size SS is is defined defined as as population SS = = 2,iS • i s i piP * ,j, where where Si si isis the the weight weight given given to to patch patch i.i. The The quantities quantities �k, k~, Si, Si, and and T T denote denote the the meta metapopulation i
i
capacity, population size, capacity, the the meta metapopulation size, and and the the time time to to extinction extinction in in aa network network from from which which patch patch i has has been been removed removed (modified (modified from from Ovaskainen Ovaskainen and and Hanski, Hanski, 2003a). 2003a).
92 92
OTSO SKI OTSO OVASKAINEN OVASKAINEN AND AND ILKKA ILKKA HAN HANSKI
at at e~ = = 0, 0, where where eeji is is the the jth jth unit unit vector vector and and the the scaling scaling factor factor kkii is is chosen chosen so so 5), the 'j measures that that '!. Eij bbiiij = = 1. 1. In In Eq. Eq. (4.1 (4.15), the term term pp')" measures the the fraction fraction of of time time that that patch jj may may possibly possibly contribute contribute to to the the colonization colonization rate rate of of patch patch i,i, whereas whereas patch the remaining remaining term term measures measures how how sensitive sensitive the the colonization colonization rate rate of of patch patch ii is is the to the the contribution contribution that that patch patch jj makes. makes. For For example, example, in in the the SRLM, SRLM, bb ij0 is is to given as b//=
Picij
(4.16) (4.16)
.
E kP ~gik While While bbiiij measures measures the the direct direct contribution contribution that that patch patch jj makes makes to to aa coloniza colonizapatch i,i, we would would ultimately like to measure measure the long-term con contion event in patch tribution tribution by by including including the the full full chain chain of of colonization colonization events events through through the the network. network. This This may may be be done done by by raising raising the the matrix matrix B to to an an infinite infinite power, power, which is which is equivalent equivalent to to defining defining W W as as the the left left leading leading eigenvector eigenvector of of matrix matrix B = 7, = ( Ovaskainen and Hanski, 2003a). Doing so, W measures = {b {bii}in, i=l (Ovaskainen and Hanski, 2003a). Doing so, W i measures j ij} j l the colonization events the long-term long-term contribution contribution of of patch patch jj to to colonization events in in the the entire entire network network and and is is thus thus independent independent of of the the target target patch patch i.i. Furthermore, Furthermore, as as '!. = 1 , W represents the relative value of patch j. Figure 4.6 illustrates E/W/= 1, W/represents the relative value of patch j. Figure 4.6 illustrates j jWj the behavior of Glanville fritillary the behavior of the the measure measure W W using using the the Glanville ffitillary model model in in the the net network shown shown in in Fig. Fig. 4.1. 4.1. Note Note that, that, in in this this example, the patch patch values values are are dis disexample, the work tributed Fig. 4. 1B). This tributed more more evenly evenly than than patch patch areas areas ((Fig. 4.1B). This is is not not the the case case in in general, general, but but it it happens happens in in the the Glanville Glanville fritillary fritillary model model as as the the patch patch area area scal scal2002b; Ovaskainen ing ing factor factor �~ = �ex ~ex + + �e ~em "at-�~im is less less than than 11 (Ovaskainen, (Ovaskainen, 2002b; Ovaskainen m+ im is and Hanski, and Hanski, 2003a). 2003a). =
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Fig. 4 4.6 An illustration illustration of of the the patch patch value value measure measure W W in in the the patch patch network network shown shown in in Fig. Fig. 4.1. 4. 1 . Fig. . 6 An (A) (A) Sizes Sizes of of the the dots dots are are proportional proportional to to the the values values of of the the habitat habitat patches, patches, and and contour contour lines lines indiindi cate the the relative relative value hypothetical patch would attain attain ifif added added to to aa particular particular location location cate value that that aa hypothetical patch would within within the the network. network. (B) (6) Values Values of of the the habitat habitat patches patches with with respect respect to to patch patch areas. areas. The The slope slope of of the the fitted fitted regression regression line line is is 0.31. 0.31 . The The figure figure is is based based on on the the Glanville Glanville fritillary fritillary model model with with paramparam eters e --= 0.30 0.30 and and cc == 0.13 0.1 3 estimated estimated from from data data restricted restricted to to the the network network shown shown in in Fig. Fig. 4.1 4.1 and and eters the the remaining remaining structural structural parameters parameters estimated estimated from from the the entire entire metapopulation metapopulation (Fig. (Fig. 4.3). 4 .3).
4. 4.
93 93
METAPOPULATION IN HIGHLY FRAGMENTED LANDSCAPES METAPOPULATION DYNAMICS IN LANDSCAPES
Transient Dynamics
Metapopulation Metapopulation capacity capacity A.M ~M and and patch patch values values refer refer to to the the equilibrium equilibrium state state
p" relate to p*,, and and they they thus thus relate to the the long-term long-term behavior behavior of of the the metapopulation metapopulation with with-
out population happens out any any reference reference to to its its present present state. state. If If the the meta metapopulation happens to to be be far far away from away from its its population population dynamic dynamic equilibrium, equilibrium, it it may may be be of of great great importance importance to about the to be be able able to to say say something something about the transient transient dynamics. dynamics. For example, consider For example, consider aa species species that that persists persists initially initially very very well well so so that that most most of of the the habitat habitat patches patches would would be be occupied occupied most most of of the the time. time. Assume Assume then then that that due habitat loss, loss, the due to to habitat the situation situation changes changes rapidly rapidly so so that that many many of of the the patches patches are metapopulation capacity are lost lost from from the the network. network. As As the the metapopulation capacity of of the the network network declines, the population is declines, the meta metapopulation is expected expected to to move move to to aa lower lower occupancy occupancy state, state, or or it it may may even even go go extinct. extinct. However, However, this this does does not not happen happen instantaneously, instantaneously, and and the the length length of of the the transient transient period period is is often often of of great great interest. interest. A A transient transient may may also also occur occur in in the the opposite opposite direction, direction, as as is is the the case case if if aa species species invades invades an an initially initially empty empty network. network. Figure Figure 4.7 4.7 illustrates illustrates that that the the length length of of such such transient transient periods periods may may well well be be up up to to 5-1 5-100 generations generations in in the the Glanville Glanville fritillary fritillary model. model. Ovaskainen Ovaskainen and and Hanski Hanski (2002) (2002) investigated investigated the the transient transient time time in in the the SRLM SRLM by model is able to approximate by first first demonstrating demonstrating that that the the original original Levins Levins model is able to approximate the the transient transient behavior behavior of of the the SRLM SRLM and and then then calculating calculating the the transient transient time time for for the task, we the former. former. To To accomplish accomplish the the first first task, we denote denote the the parameters parameters of of the the Levins Levins model model by by c T and and Ii ~" so so that that the the model model is is defined defined as as
ap dp c p ( 1 -- pp)) -- � ep. dt = =� 2"p(1 ~p. dt
(4.1 7) (4.17)
This model approximates following trans This model approximates the the behavior behavior of of the the SRLM SRLM given given the the following transformations formations (Ovaskainen (Ovaskainen and and Hanski, Hanski, 2002). 2002). First, First, one one has has to to interpret interpret the the Pi, the Eq. (4.1 7) as variable variable p p in in Eq. (4.17) as p" px = = kjWj ]~iWiPi, the weighted weighted fraction fraction of of occupied occupied patches. patches. Second, Second, the the parameter parameter Ii ~ is is interpreted interpreted as as the the effective effective extinction extinction rate, rate,
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in the model in in the Fig. 4.7 4.7 Transient Transient behavior behavior in the Glanville Glanville fritillary fritillary model the network network shown shown in in Fig. Fig. 4.1 4.1 with with parameter values as in Fig. Fig. 4.6. In panel A all the patches are initially assumed to to be occupied, whereas in panel B, only the patch indicated by an arrow 4.1 is initially occupied. The thick arrow in Fig. Fig. 4.1 line shows Thin lines mean of 000 simu line shows the the prediction prediction of of the the deterministic deterministic model. model. Thin lines show show the the mean of 11000 simulations, lations, the the lower lower line line derived derived from from all all replicates replicates and and the the upper upper one one ignoring ignoring such such replicates, replicates, which extinct by that time. which have have gone gone extinct by that time. Dashed Dashed lines lines show show 95% 95% confidence confidence intervals intervals derived derived from all simulation from all simulation replicates.
94 94
OTSO LKKA HAN SKI OTSO OVASKAINEN OVASKAINEN AND AND IILKKA HANSKI
defined defined by by e ~ = = 1/(ki 1/(EiWi/Exti). Third, the the parameter parameter c T is is interpreted interpreted as as the the W/Ext;). Third, "AM/a. effective colonization effective colonization rate, rate, defined defined by by c 7" = =e ~" kM/8. Using the SRLM to Using the aforementioned aforementioned transformation transformation of of SRLM to the the Levins Levins model, model, the the length period, defined defined as time it length of of the the transient transient period, as the the time it takes takes for for the the metapopu metapopulation close the lation to to return return from from its its initial initial state state to to aa state state close the new new equilibrium, equilibrium, may may be Ovaskainen and be written written as as the the product product of of four four factors factors ((Ovaskainen and Hanski, Hanski, 2002). 2002). First, First, the the length length of of the the transient transient period period increases increases with with the the distance distance between between the the present state. Second, transient is present state state and and the the equilibrium equilibrium state. Second, the the length length of of the the transient is longer span) than longer for for species species that that have have slow slow dynamics dynamics (e.g., (e.g., due due to to aa long long life life span) than for species with dynamics. Third, transient period for species with fast fast dynamics. Third, the the length length of of the the transient period is is longer longer in in aa patch patch network network that that has has few few large large patches patches than than in in aa network network with with many many small small patches, patches, as as the the turnover turnover rate rate is is expected expected to to be be slower slower for for larger larger patches. transient period patches. Fourth Fourth and and most most important, important, the the length length of of the the transient period is is expected to species that close to their extinction expected to be be especially especially long long for for species that are are close to their extinction threshold threshold following following perturbation. perturbation. The The fourth fourth conclusion conclusion has has the the important important implication implication for for conservation conservation that that many many rare rare species species living living in in recently recently deteriorated deteriorated landscapes landscapes may may be be "doomed" "doomed" to to extinction. extinction. They They still still exist exist because because they they have have not not had had time time to to go go extinct extinct yet, yet, and and the the time time it it takes takes to to go go extinct extinct is is especially especially long long in in the the case case of of species species whose whose long-term number of long-term persistence persistence is is most most precarious. precarious. The The number of species species that that are are pre predicted habitat loss loss and dicted to to ultimately ultimately go go extinct extinct due due to to past past habitat and fragmentation fragmentation rep represents 994; Hanski resents the the extinction extinction debt debt in in the the community community (Tilman (Tilman et et a!., al., 11994; Hanski and and Ovaskainen, Ovaskainen, 2002). 2002). The The extinction extinction debt debt is is paid paid either either by by letting letting the the species species go go extinct extinct or or by by improving improving the the quality quality of of the the landscape landscape sufficiently sufficiently for for the the species species that debt. Hanski Hanski and that constitute constitute the the extinction extinction debt. and Ovaskainen Ovaskainen (2002) (2002) discuss discuss an an example example of of extinction extinction debt debt in in beetle beetle species species living living in in boreal boreal forests forests in in Finland. Finland.
4.5 4.5
STOCHASTIC STOCHASTIC THEORY THEORY Recall analysis of occupancy models models is Recall that that the the analysis of stochastic stochastic patch patch occupancy is difficult difficult because space is heterogeneous network because the the size size of of the the state state space is 2n 2 n for for aa heterogeneous network of of n n patches. This develops approximation approximation methods patches. This section section develops methods that that take take advantage advantage of of possible possible links links between between deterministic deterministic and and stochastic stochastic frameworks. frameworks. In In particular, particular, this section shows values may heterogeneous this section shows that that patch patch values may be be used used to to transform transform aa heterogeneous meta population to metapopulation to its its homogeneous homogeneous ("ideal" ("ideal")) counterpart, counterpart, which which behaves, behaves, in in some some relevant relevant respects, respects, similarly similarly as as the the original original heterogeneous heterogeneous metapopulation. metapopulation.
Effective Effective Metapopulation Metapopulation Size Size Let Let us us start start with with the the SRLM. SRLM. As As stated stated in in the the previous previous section, section, both both the the equi equilibrium librium state state and and the the transient transient behavior behavior of of the the deterministic deterministic SRLM SRLM can can be be approximated approximated by by the the one-dimensional one-dimensional Levins Levins model model by by replacing replacing the the original original model parameters parameters with with the the effective effective colonization rate c F~ and and the the effective effective model colonization rate extinction extinction rate rate e. T. These These factors factors account account for for the the deterministic deterministic drift drift (growth) (growth) in in the model. The the model. The main main difference difference between between deterministic deterministic and and stochastic stochastic models models is is that that the the latter latter account account for for stochastic stochastic fluctuations fluctuations around around the the mean mean dynamics, dynamics, which which arise arise due due to to the the finite finite size size of of the the network. network. We We may may ask ask whether whether one one
4. 4.
995 5
METAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES METAPOPULATION
could also also transform transform the the size size of of aa heterogeneous heterogeneous metapopulation meta population into into an an effeceffec could tive metapopulation metapopulation size size ~', ii, which which would would control control for for the the amount amount of of stochastic stochastic tive fluctuations. To To incorporate incorporate the the latter latter into into the the model, model, we we require require that that the the fluctuations. (T22 in in Eq. Eq. (4.2) (4.2) be be the the same same for for the the weighted weighted fraction fraction of of infinitesimal variance variance (r infinitesimal occupied patches patches Px p" in in the the SRLM SRLM and and for for the the simple simple fraction fraction of of occupied occupied occupied patches pp in in the the one-dimensional one-dimensional Levins Levins model. model. Equating Equating the the two two variances variances at at patches the equilibrium leads to to the equilibrium state state p': p ':- leads n' n~ = =
cp '� ( l - p t )
ep '� Exti( P':- )P/' ] ZL W2[C~ W;[ Co[i(p" )(l -- Pi':) Pi', ) + Exti(p*)pi*] i +
Tp'~(1 - p'~) + ~p'~ ----- .
(4.18) (4. 1 8 )
t
As expected, expected, the the effective effective number number of of habitat habitat patches patches increases increases with with the the real real As number of of patches patches and and with with decreasing decreasing variance in the values of of these patches. number variance in the values these patches. If all all patches are identical, the effective effective number number coincides coincides with the actual actual numnum If patches are identical, the with the ber. the SRLM SRLM is in Fig. 4.8. The line corcor ber. Transformation Transformation of of the is illustrated illustrated in Fig. 4.8. The dashed dashed line responds to to the transformed model, model, whereas whereas the responds the analytically analytically transformed the continuous continuous line line corresponds fitted model. corresponds to to aa numerically numerically fitted model. In nonlinear models, such such as fritillary model, model, or IFM, it In nonlinear models, as the the Glanville Glanville fritillary or the the IFM, it may not be be possible possible to to derive derive analytical expressions for the effective coloniza may not analytical expressions for the effective colonization and extinction rates rates or for for the effective metapopulation tion and extinction metapopulation size. size. These quanquan tities may still still be be determined fitting aa structurally tities may determined numerically numerically by by fitting stcucturally similar similar homogeneous model to of the the heterogeneous model homogeneous model to the the drift drift and and the the variance variance of heterogeneous model (Ovaskainen, 2002a). Figure example for the Glanville Glanville fritillary fritillary (Ovaskainen, 2002a). Figure 4.9 4.9 gives gives an an example for the model. The homogeneous model model consists 50 patches, which is is somewhat model. The homogeneous consists of of 50 patches, which somewhat less 56 patches. less than than the the actual actual number number of of 56 patches. The The difference difference between between the the effective effective number number of of habitat habitat patches patches and and the the actual actual number number of of habitat habitat patches patches is is explained (Fig. 4.6). example, the explained by by aa variation variation in in patch patch values values (Fig. 4.6). In In this this example, the distri distribution bution of of patch patch values values is is relatively relatively even even and and hence hence the the difference difference is is not not very very great. great.
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Fig. model in Fig. 4.8 4 . 8 Transformation Transformation of of the the SRLM SRLM to to the the stochastic stochastic logistic logistic model in the the network network shown shown in sample of 00 randomized in Fig. Fig. 4.1 4.1.. Dots Dots represent represent aa sample of 1100 randomized occupancy occupancy states states from from the the quasista quasistationary tionary distribution distribution obtained obtained by by simulation. simulation. Lines Lines represent represent fitted fitted models; models; the the continuous continuous line line isis based based on on numerical numerical fitting, fitting, and and the the dashed dashed line line is is based based on on the the analytical analytical transformation transformation given 1 8). The given by by Eq. Eq. (4. (4.18). The two two panels panels show show (A) (A) infinitesimal infinitesimal mean mean f.L I~ and and (B) (B) infinitesimal infinitesimal variance variance a2. ~2. Parameter Parameter values values as as in in Fig. Fig. 4.6 4.6 except except ee = = 11,, Cc = = 0.5. 0.5.
96 96
OTSO OVASKAINEN AND ILKKA ILKKA HANSKI B B
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Fig. 4.9 in Fig. Fig. 4.8 but for model with values as 4.9 As As in 4.8 but for the the Glanville Glanville fritillary fritillary model with parameter parameter values as in in Fig. Fig. 4.6. 4.6. Single realizations of of the the full heterogeneous model model (C) and of of the transformed transformed homogeneous homogeneous model (D).
Spatially Spatially Correlated Correlated and and Temporally Temporally Varying Varying Environmental Environmental Conditions Conditions Metapopulation Metapopulation models models typically assume assume that the dynamics of local popula populations are independent independent of each other other and that that the environmental environmental conditions remain constant. In reality, both assumptions are violated to a smaller or greater extent ((Baars Baars and Van Dijk, 11984; 984; Ims and Steen, 11990; 990; Hanski, 1999b; Lundberg Lundberg et et aI., al., 2000; 2000; Nieminen Nieminen et et aI., al., 2004). 2004). Spatially Spatially correlated correlated and and tempo tempoenvironmental conditions present a challenge for for meta metapopulation rally varying environmental population theory from the viewpoint of both model analysis (Heino et al., aI., 11997; 997; Engen et aI., al., 2002; Ovaskainen, 2002a) and parameter parameter estimation (Thomas, 1991; Hanski and Woiwod, 11993; 993; Bjornstad et aI., al., 1999, Williams and Liebhold, al., 2002). Viewing the metapopulation metapopulation as a population population of 2000; Peltonen et aI., populations, populations, stochasticity stochasticity in in patch patch occupancy occupancy dynamics dynamics in in aa constant constant environ environment ment is is analogous analogous to to demographic demographic stochasticity, stochasticity, whereas whereas temporal temporal variability variability in dynamics leads to variability that that is analogous to environmental stochastic stochasticity. Hanski ((1991) 1991 ) has termed these two population-level stosto two forms of meta metapopulation-level chasticities regional stochasticity, chasticities as as extinction-colonization extinction-colonization stochasticity stochasticity and and regional stochasticity, respectively. In a finite network network of habitat habitat patches, regional stochasticity leads to both spatially correlated correlated and temporally temporally varying parameter parameter values; these two two phenomena phenomena actually actually represent represent the the two two sides sides of of the the same same coin. coin. The The effective effective metapopulation metapopulation size size approach approach described described earlier earlier can can be be extended extended to situations in allowing one include to situations in which which the the parameters parameters vary vary temporally, temporally, allowing one to to include regional model. This regional stochasticity stochasticity into into the the model. This is is illustrated illustrated in in Fig. Fig. 4.10, 4.10, which which is is otherwise identical to Fig. 4.9 but now now with the parameter parameter values for for the Glanville fritillary butterfly estimated separately for the eight annual transitions temporal variation to the model present in our dataset. As expected, adding temporal increases the variance Var[Llp;.J (Figs. 4.9B and 4.10B) and thus the amplitude Var[APx] (Figs.
4.
97 91
METAPOPULATION METAPOPULATION DYNAMICS IN HIGHLY HIGHLY FRAGMENTED LANDSCAPES LANDSCAPES E[Ap~] E[l1p J
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f-----------""'-<3 00 O 3000 4 400 00 30 00 5500 260 300 360 400 460 500 $60 00 2200 1lo0 00 200 1100 Time Time tt Time t Fig. 4.9 but with Fig. 4.1 4 . 1 00 As in Fig. 4.9 with temporally temporally varying varying parameters. parameters. The parameter parameter distributions distributions e = (0.24, 0048, 34, 0.1 7, 0.30, 0.53, 0.04) and c = 1 4, 0.1 0, 0. 1 0, 0.33, 0.08, 0.08, 0.48, 0.33, 0. 0.34, 0.17, 0.04)and = (0. (0.14, 0.10, 0.10, 0.06, 0.22) have been estimated separately from 3), from the eight eight annual transitions transitions (see (see Fig. Fig. 4. 4.3), whereas the remaining remaining parameters have been kept fixed. o o
of stochastic fluctuations (Figs. 0C). Figure 4. 1 1 compares (Figs. 4.9C and 4.1 4.10C). 4.11 compares the statistical properties of the two two models and their homogeneous homogeneous counterparts. counterparts. As variation flattens As expected, expected, temporal temporal variation flattens the the quasistationary quasistationary distribution distribution (Fig. 4.1 1A), but note that (Fig. 4.11A), that it also adds especially low-frequency fluctuations fluctuations to the behavior of the model (Fig. (Fig. 4.1 1B). As illustrated by Figs. 4.9, 4.10, and 4.11B). 4.1 1, the effective metapopulation 4.11, metapopulation size approach approach gives a very accurate description of the Glanville fritillary model in all the senses investigated here. The example in Fig. 44.10 . 1 0 demonstrates cor demonstrates that that temporal temporal variation (spatial correlation) in parameter values increases the extinction risk of a meta population metapopulation by amplifying stochastic fluctuations, fluctuations, which which may be restated restated by observing that that temporal variation decreases the effective number of habitat patches. B
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0.1
98 98
OTsO LKKA HANSKI HAN SKI OTSO OVAsKAINEN OVASKAINEN AND AND IILKKA
Ovaskainen Ovaskainen and and Hanski Hanski (2003b) (2003b) made made this this statement statement explicit explicit in in the the context context of of the the stochastic stochastic logistic logistic model. model. They They showed showed that that if if extinctions extinctions and and coloniza colonizations tions are are correlated, correlated, with with aa correlation correlation coefficient coefficient p, p, the the effective effective number number of of patches patches is is reduced reduced to to
nF/ nnee = = ----- 9 ((nn -- 11 )p ) p ++ 11
(4. 19) (4.19)
Such changes the model in Such aa correlation correlation changes the qualitative qualitative behavior behavior of of the the model in several several ways. Most metapopulation extinc ways. Most importantly, importantly, for for pp > > 00 the the mean mean time time to to metapopulation extinction increase exponentially habitat patches, patches, as tion does does not not increase exponentially with with the the number number of of habitat as predicted . 1 , but predicted by by Eq. Eq. (2) (2) in in Box Box 4 4.1, but it it now now grows grows according according to to the the power-law power-law T Ovaskainen and T IX oc nlfp nl/~ ((Ovaskainen and Hanski, Hanski, 2003b). 2003b).
4.6 4.6
COMPARISON COMPARISON WITH W I T H OTHER OTHER MODELING MODELING APPROACHES APPROACHES This This section section compares compares stochastic stochastic patch patch occupancy occupancy models models with with other other mod modeling eling approaches approaches that that have have been been used used in in metapopulation metapopulation studies. studies. The The other other approaches include individual-based simulation models, models structured approaches include individual-based simulation models, models structured by by population size, lattice models, and spatial moment moment equations, population size, lattice models, and spatial equations, which which were were described Chapters 11 and 3. This described briefly briefly in in Chapters and 3. This section section compares compares the the different different modeling modeling approaches approaches in in the the context context of of extinction extinction thresholds thresholds and and their their relation relation to landscape structure, issues in population to landscape structure, which which is is one one of of the the key key issues in meta metapopulation dynamics dynamics for for both both research research and and management. management. To To facilitate facilitate this this comparison, comparison, we we start start by by summarizing summarizing three three ecologically ecologically significant significant messages messages that that have have been been discussed discussed in in this this chapter chapter in in the the context context of of SPOMs. SPOMs. 9 First First and and most most important, important, there there is is aa critical critical amount amount of of habitat habitat below below which which aa species species is is expected expected to to go go deterministically deterministically extinct. extinct. In In highly highly frag fragmented landscapes, extinction threshold depends not mented landscapes, the the extinction threshold depends not only only on on the the total also on configuration of total amount amount of of habitat, habitat, but but also on the the spatial spatial configuration of the the habi habitat population capacity, tat patch patch network. network. As As characterized characterized by by meta metapopulation capacity, metapopulation by increasing metapopulation persistence persistence is is facilitated facilitated by increasing connectivity connectivity among the habitat patches. among the habitat patches. • population 9 Second, Second, the the contribution contribution that that individual individual patches patches make make to to meta metapopulation persistence persistence depends depends not not only only on on their their area area and and quality, quality, but but also also on on their their position network. Well-connected patches make make generally position within within the the network. Well-connected patches generally aa greater greater contribution contribution than than isolated isolated patches. patches. • population extinction 9 Third, Third, meta metapopulation extinction is is aa stochastic stochastic event, event, which which depends depends not population dynamics, not just just on on the the deterministic deterministic drift drift in in meta metapopulation dynamics, but but also also on on fluctuations fluctuations around around the the drift. drift. Stochastic Stochastic fluctuations fluctuations increase increase with with aa decreasing decreasing number number of of habitat habitat patches, patches, and and thus thus extinction-colonization extinction-colonization stochasticity stochasticity increases increases the the extinction extinction risk, risk, especially especially in in small small metapopu metapopuFurthermore, spatially correlated local dynamics or or temporally lations. Furthermore, varying varying environmental environmental conditions conditions amplify amplify stochastic stochastic fluctuations fluctuations and and thus thus increase increase the the risk risk of of metapopulation metapopulation extinction. extinction. •
The The first first message message has has been been studied studied extensively extensively in in the the literature. literature. Fahrig Fahrig (2002) (2002) reviewed reviewed aa number number of of lattice-based lattice-based modeling modeling studies studies attempting attempting to to disentangle disentangle
LANDSCAPES D SCAPES 4. METAPOPULATION DYNAMICS IN HIGHLY FRAGMENTED LAN
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fragmentathe effects of habitat loss (fraction of destroyed habitat) and habitat fragmenta tion (spatial configuration of the remaining habitat). She categorized the studies modinto two classes based on the type of models used, either patch occupancy mod (colonization-extinction models; Hill and Caswell, 11999; With and King, els (colonization-extinction 999; With 11999a) 999a) or individual-based simulation models ((birth-immigration-death-birth-immigration-death- emigration Bevers, 2002; emigration models; models; Flather Flather and and Bevers, 2002; Fahrig, Fahrig, 2001 2001).). Fahrig Fahrig (2002) (2002) drew drew three major conclusions, which are all well in line with the results derived from SPOMs. SPOMs. First, First, all all of of the the modeling modeling studies studies she she considered considered predicted predicted aa threshold threshold amount metapopulation persistence. Second, a smaller amount of available habitat for metapopulation total population persistence total amount amount of of habitat habitat was was required required for for meta metapopulation persistence if if the the habi habiThird, the spatial configuration configuration of the remaining habi habitat occurred in clusters. Third, remained tat had the strongest effect if only a small fraction of the landscape remained suitable for the species. Similar conclusions were reached by Hiebeler (2000) (2000) and and Ovaskainen et al. (2002) (2002) using a lattice-based patch occupancy model. distinction between patch patch occupancy occupancy models and individual-based individual-based A major distinction simulation simulation models models reviewed reviewed by by Fahrig Fahrig (2002) (2002) was was that that the the latter latter predicted predicted only only habitat fragmentation fragmentation on extinction extinction threshold, threshold, which prediction prediction a mild effect of habitat with much much of the empirical literature literature (Harrison and Bruna, 11999; is in line with 999; Fahrig, 2002). 2002). Fahrig (2002) (2002) argued that that the difference between the two two types metapopulation of models occurs because the underlying reasons decreasing meta population persistence persistence with with increasing increasing amount amount of of habitat habitat loss loss are are different. different. In In patch patch occu occupancy pancy models, models, the the colonization colonization rate rate of of empty empty patches patches decreases decreases with with increasing increasing habitat loss, whereas in individual-based individual-based simulation simulation models, an increasing habitat number number of individuals individuals spend their time in the landscape landscape matrix, matrix, where repro reproduction duction is is not not possible possible and and mortality mortality is is assumed assumed to to be be high. high. However, However, as as migra migration mortality mortality is one of the major reasons why why a reduced reduced colonization colonization rate with tion major reasons rate with distance is assumed assumed in extinction-colonization extinction-colonization dynamics dynamics (Hanski, 11999b), distance 999b), the difference between the the two causes for extinction threshold threshold is is somewhat somewhat difference between two causes for the the extinction superficial. parameterize a patch superficial. Indeed, one could parameterize patch occupancy model using data from simulation simulation of an individual-based which case the the two two moddata individual-based model, in which mod should give more more or less identical identical predictions. eling approaches approaches should predictions. We suggest that that the difference difference between the two approaches is not two modeling approaches not fundamental fundamental and and on quantitative quantitative that the differences differences observed by Fahrig (2002) that (2002) were were based on model assumptions. Most Most importantly, to model assumptions. importantly, the lattice-based lattice-based studies are sensitive to spatial (instead of "patches" "patches" consisting consisting of of spatial scale, especially if single lattice cells (instead a cluster cluster of assumed to support independent independent local populations. The of cells) are assumed to support populations. The (2002) did did not not include include environmental environmental simulation reviewed by Fahrig simulation studies studies reviewed Fahrig (2002) stochasticity at at the the level of of habitat habitat patches, patches, which, which, in the the case of of lattice lattice models, models, stochasticity would amount to to spatially correlated stochasticity stochasticity in in clusters clusters of of lattice lattice cells cells would amount spatially correlated comprising a single habitat patch (Gu ( Gu et al., aI., 2002). 2002). In SPOMs SPOMs and and in reality, comprising habitat patch the patch patch area-dependent area-dependent extinction extinction rate rate leading leading to to aa high high rate rate of of extinction extinction of of the small populations populations in in small small patches patches comprises comprises an an important important reason reason why why the the spaspa small configuration of of the the habitat habitat may may greatly influence the the extinction threshold tial configuration greatly influence extinction threshold in highly fragmented fragmented landscapes. landscapes. Concerning our our second second message, which which relates relates to to the the value value of of individual individual Concerning habitat patches, patches, most most of of the the previous previous literature literature has has been been based based on on simulation simulation habitat studies. For For example, example, Lindenmayer Lindenmayer and and Possingham Possingham (1996a,b) ( 1 996a,b) developed developed aa studies. simulation simulation model model to to assess assess the the persistence persistence of of Leadbeater's Leadbeater's possum possum in in southsouth eastern Australia. Australia. In In line line with with our our results, results, they they concluded concluded that that extinction extinction eastern
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probability probability was was influenced influenced both both by by the the size size and and by by the the spatial spatial arrangement arrangement of of habitat patches. They found that in some circumstances circumstances the probability probability that that metapopulation is extant extant may be a good meas measa patch is occupied while the metapopulation for metapopulation metapopulation viability. The habitat habitat patch networks used ure of its value for from a single 300-ha reserve reserve to twelve twelve 25-ha reserves. in their studies varied from As As expected, expected, metapopulations metapopulations in in networks networks of of many many small small reserves reserves were were vul vulnerable to extinction extinction due to demographic demographic and environmental environmental stochasticities. stochasticities. Conversely, a metapopulation metapopulation inhabiting inhabiting the the single large reserve was suscep susceptible to extinction extinction in a catastrophic catastrophic wildfire, highlighting highlighting the need for for several scattered reserves. more scattered Other 1 997), Other studies studies that that have have addressed addressed patch patch values values include include Keitt Keitt et et al. al. ((1997), who used percolation percolation theory to quantify quantify connectivity at multiple scales and and assigned conservation conservation priorities to habitat habitat patches based on their their contribu contributions to connectivity. In addition, 1 997) addition, using sensitivity analysis, Keitt et al. ((1997) identified identified critical critical "stepping "stepping stone" stone" patches patches that, that, when when removed removed from from the the land landscape, would 19 9 8 ) would cause large changes changes in connectivity. connectivity. Moilanen Moilanen et al. ((1998) constructed constructed aa somewhat somewhat aadd hoc hoc measure measure ooff patch patch values values while while analyzing analyzing dynamics dynamics in a classic metapopulation metapopulation of the American American pika using the incidence function function model. Their measure measure was an attempt attempt to define the patch patch value in the the sense Moilanen et 1998 ) considered sense of of our our measure measure W, W, although although Moilanen et al. al. ((1998) considered only only aa two-generation two-generation impact of habitat habitat patches patches on colonizations. colonizations. Verboom et al. (200 1 ) emphasized (2001) emphasized the the importance importance of of large large "key "key patches" patches" with with aa stabilizing stabilizing role role in in habitat habitat patch patch networks. networks. Patch Patch values values have have also also been been addressed addressed analyt analytpopulation models ically in the context context of meta metapopulation models structured structured by the sizes of local populations. 1 994) and Gyllenberg and 1 997) noted populations. Lawton Lawton et al. ((1994) and Hanski Hanski ((1997) noted that down due due to that the the Levins rule (Section ( Section 4.2) may may break break down to a variation variation in habihabi tat destruction of high-quality patch has a greater impact tat patch patch quality, as destruction of a high-quality patch has impact destruction of low-quality patch. on metapopulation metapopulation size than than destruction of a low-quality patch. Our which is concerned interplay between Our third third message, which concerned with with the the interplay between stochas stochastic and dynamics, has been studied studied by Casagrandi and Gatto Gatto and deterministic deterministic dynamics, Casagrandi and ((1999, 1999, 2002a,b) 2002a,b) using using metapopulation metapopulation models models structured structured by by local local population population size. They They assumed of local populations negative-binomially assumed that that the sizes of populations are negative-binomially distributed to with our results, Casagrandi Casagrandi distributed to facilitate facilitate the the model model analysis. analysis. In line with our results, and parameter space into and Gatto Gatto classified the the parameter into four four regions, of which which one extreme the other to extreme corresponded corresponded to deterministic deterministic extinction extinction and and the other one to metapopulation persistence. persistence. Stochasticity Stochasticity was was shown shown to to create create intermediate intermediate metapopulation which either either demographic demographic stochasticity stochasticity or or environmental environmental catastrocatastro zones, in which phes imposed imposed a substantial substantial risk risk of of extinction, extinction, even though though the the metapopulation metapopulation phes was was predicted predicted to to persist persist in a deterministic deterministic model. model. Many Many studies studies have have concon cluded that that the the risk risk of of metapopulation metapopulation extinction increases with with increasing increasing cluded extinction increases regional stochasticity, stochasticity, that that is, increasing increasing the the scale of of environmental environmental variability. variability. regional For For example, example, Palmqvist Palmqvist and and Lundberg Lundberg (1998) ( 1 998) used used a coupled coupled map map lattice lattice model density-dependent dynamics dynamics and model consisting consisting of of local local populations populations with with density-dependent and density-independent migration. migration. They They found found that that the the major major determinant determinant of of the the density-independent risk of between local of metapopulation metapopulation extinction extinction is the the balance balance between local population population variability synchrony in local extent variability and and synchrony local population population fluctuations. fluctuations. To what what extent regional stochasticity stochasticity increases increases the the extinction extinction risk risk of of real real metapopulations metapopulations is a regional quantitative question, question, which which cannot cannot be be answered answered by by just just demonstrating demonstrating that that quantitative there there is some some degree degree of of spatial spatial correlation correlation in local dynamics. dynamics.
METAPOPULATION DYNAMICS DYNAMICS IN I N HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES 4. METAPOPULATION 4.
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CURRENT STATE STATE OF OF THE THE THEORY THEORY A AND APPLICATIONS 4.7 CURRENT 4.7 N D APPLICATIONS Metapopulation theory theory based based on on stochastic stochastic patch patch occupancy occupancy models models has has its its Metapopulation roots in in the the Levins Levins model model (Levins, (Levins, 1969), 1 969), which which was was originally originally developed developed to to roots study alternative alternative pest pest eradication eradication strategies. strategies. In In the the 1970s, 1 970s, models models stemming stemming study from the the Levins Levins model model were were constructed constructed to to study study group group selection selection (Levins, (Levins, from 1 970; Levitt, Levitt, 1978) 1 978) and and competitive competitive (Levins (Levins and and Culver, Culver, 1971; 1971; Horn Horn and and 1970; MacArthur, 1972; 1 972; Slatkin, Slatkin, 1974; 1 974; Hanski, Hanski, 1983) 1 98 3 ) and and predator-prey predator-prey interinter MacArthur, actions (Maynard (Maynard Smith, Smith, 1974). 1 974) . Patch Patch occupancy occupancy models models for for two two or or more more actions species were were investigated investigated further further in in the the 1990s 1 990s (Nee (Nee and and May, May, 1992; 1 992; Tilman Tilman et et species al., 1994; 1 994; Lei Lei and and Hanski, Hanski, 1997; 1 997; Nee Nee et et al., al., 1997; 1 997; Holt, Holt, 1997). 1 997). These These studies studies al., have largely remained remained at at the the theory theory level, level, although although Hanski Hanski and and Ranta Ranta (1983) ( 1983) have applied the the Levins model model extended extended to to three three competitors competitors to to study study the the dynamics dynamics applied water fleas fleas in in rock rock pools. A few few other other studies studies have have employed employed of Daphnia of Daphnia water homogeneous SPOMs SPOMs in the the study study of of actual actual metapopulations metapopulations (Lande, 1988b; 1 9 88b; homogeneous As et al., 1997; Kindvall, 1996; 1 996; As 1 997; Doncaster Doncaster and Gustafsson, 1999; 1 999; Carlson, Carlson, Kindvall, and Gustafsson, 2000). However, given the the typically marked marked heterogeneities heterogeneities in real habitat habitat 2000). patch networks, these applications applications are best best interpreted interpreted as attempts attempts to to increase increase patch networks, these qualitative about metapopulation metapopulation dynamics. Quantitative predictions predictions qualitative insight insight about dynamics. Quantitative are best best based based on on heterogeneous heterogeneous SPOMs SPOMs and and the the spatially spatially realistic realistic metapopumetapopu are lation lation theory theory that that has been developed developed since the first first incidence function function models (Hanski, 1 992, 1994a). 1 994a). (Hanski, 1992, advantage of of SMT in comparison of the other The great advantage comparison with with much much of other theory in spatial ecology is that applied to real metapopulations metapopulations for that the models can be applied purposes research, management, management, and conservation. purposes of research, conservation. The key here is the set of assumptions assumptions that that these models make on the influence of landscape structure on population population processes. SMT prescribes research tasks for ecologists engaged in empirical work work in terms of testing model assumptions assumptions and predictions. The incidence 994a) has been employed widely in the incidence function model (Hanski, 11994a) study of classic metapopulations metapopulations of insects, frogs, small mammals, and birds (reviews in Hanski, 11999b, 999b, 2001 b ) . Statistically rigorous methods have been 2001b). developed to estimate model parameters of both SPOMs and their mean-field approximations, including maximum likelihood (Moilanen, 11999, 999, 2000) and Bayesian estimation estimation (O'Hara (O'Hara et al., 2002; Ter Braak and Etienne, 2003; Chapter Chapter 5). 5). Although metapopulation theory based oonn SPOMs has advanced greatly in recent years, years, the theory is still still in a state of active development and major steps forward forward can be expected to be taken in the near future. We list here some of the challenges for for further research. First, it would be interesting to integrate SPOMs SPOMs with with metapopulation metapopulation models models structured structured by by the the sizes sizes of of local local popula populations Gyllenberg and 992; Gyllenberg 997; Casagrandi tions ((Gyllenberg and Hanski, Hanski, 11992; Gyllenberg et et al., al., 11997; Casagrandi and and Gatto, 11999). 999). Among other other things, things, this would allow allow one one to extend extend mechanis mechanistic Etienne, 2000, 2002b) tic models models for the rescue effect ((Etienne, 2002b) to heterogeneous heterogeneous patch patch networks. networks. Second, and related related to the former, not accounting accounting for for individuals largely prevents prevents the extension extension of of SPOMs to address population genetic and evolutionary questions. questions. However, independently of of the ecological ecological research on SPOMs, 1 997) have SPOMs, Whitlock Whitlock and and Barton Barton ((1997) have started started to to develop develop population population genetic genetic theory theory that that takes takes into into account account the the influence influence of of spatial spatial heterogeneity heterogeneity at at the the landscape landscape level, level, and and Heino Heino and and Hanski Hanski (2001 (2001)) have have modeled modeled the the influence influence
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of of landscape landscape structure structure on on the the evolution evolution of of migration migration rate rate (see (see also also Parvinen, Parvinen, 2002; 2002; Parvinen et aI., al., 2003 2003).). There There is much potential in these directions. Third, Third, SPOMs apply best to highly fragmented fragmented landscape, landscape, although although the theory theory has also been applied to has also been applied to landscapes landscapes described described as as lattices lattices by by interpreting interpreting the the regularly spaced lattice cells as habitat patches ((Ovaskainen Ovaskainen et aI., al., 2002; 2002; Pakkala Pakkala et aI., al., 2002). 2002). In this context context in particular, the the standard standard assumption assumption of independent independent local dynamics dynamics becomes becomes very questionable, questionable, and ways have to be devised of incorporating incorporating spatially correlated correlated dynamics dynamics into the model (for one approach aI., 2002). approach with with potential potential for for some some practical practical applications, applications, see see Gu Gu et et al., 2002). Finally, it may be possible to to combine combine SPOMs with with other other techniques techniques used in spatial Chapter 3). spatial ecology, such such as spatial spatial moment moment equations equations ((Chapter
4.8 4.8
LINKS WITH LINKS W I T H LANDSCAPE LANDSCAPE ECOLOGY, ECOLOGY, EPIDEMIOLOGY, EPIDEMIOLOGY, AND AND MATRIX MATRIX POPULATION POPULATION MODELS MODELS Incorporating population Incorporating the the influence influence of of landscape landscape structure structure on on meta metapopulation processes in the manner manner that that this is done in SMT has led to the realization realization that that more substantial substantial links can be forged forged between between meta metapopulation and more population ecology and than has been perceived previously (Hanski, 2001 2001b; related disciplines than b; Hanski Ovaskainen, 2003) 2003).. In the first place, SMT contributes contributes to the conceptual conceptual and Ovaskainen, unification metapopulation ecology landscape ecology. unification of of metapopulation ecology and and landscape ecology. Merging Merging of of these been anticipated these two two fields fields has has been anticipated for for aa long long time time (Hanski (Hanski and and Gilpin, Gilpin, but not not much much has has happened happened until until now, now, as as can can be be seen seen from from aa lack lack of of 11991), 99 1 ) , but Chapter 11,, Fig. 11.2). .2). Metapopulation papers that that refer to both disciplines ((Chapter Metapopulation ecology ecology and and landscape landscape ecology ecology have have largely largely adhered adhered to to their their own own research research tra traditions, to the extent extent that that shared key concepts concepts such as connectivity connectivity are used in aa different different manner manner (Tischendorf (Tischendorf and and Fahrig, Fahrig, 2000a; 2000a; Moilanen Moilanen and and Hanski, Hanski, 200 1 ) . Nonetheless, 2001). Nonetheless, many landscape landscape ecologists ecologists have the same goal in their research spatial structure research as as we we do, do, to to take take account account of of the the influence influence of of the the spatial structure of of the other processes processes (Turner 1 ) . So the landscape landscape on on population population and and other (Turner et et aI., al., 200 2001). So far, far, most of these on simulations Chapter 2). SMT rep most of these studies studies have have been been based based on simulations ((Chapter 2). SMT represents aa coherent coherent body resents body of of theory theory that that should should go go aa long long way way in in answering answering the the needs of those landscape needs of those landscape ecologists ecologists who who are are interested interested in in population population and and metapopulation processes. metapopulation Second, spatially spatially realistic meta population models metapopulation models are closely related to matrix matrix population population models models for for age-structured age-structured and and size-structured size-structured populations populations ((Caswell, Caswell, 200 1 ) . As 2001). As aa matter matter of of fact, fact, much much of of the the mathematical mathematical theory theory is is the the same Ovaskainen and 1 ) . Just same ((Ovaskainen and Hanski, Hanski, 200 2001). Just as as traditional traditional matrix matrix models models divide divide populations populations into age classes, SMT SMT divides metapopulations metapopulations into individual individual habitat habitat patches. patches. Important Important mathematical mathematical similarities similarities include include the the role role of of the the dominant dominant eigenvalue eigenvalue and and eigenvector eigenvector of of the the respective respective population population matrices matrices in in determining population growth persistence. There determining population growth rate rate and and persistence. There are are also also significant significant differences. differences. For For example, example, in in SMT, SMT, transitions transitions are are possible possible between between any any pairs pairs of of "classes", "classes", whereas whereas in in age-structured age-structured models, models, individuals individuals move move from from one one class class to to another another in in aa predictable predictable manner. manner. Another Another difference difference is is in in the the manner manner in in which which density density dependence dependence typically typically enters enters into into the the models. models. Age-structured Age-structured models drives intraspecific models assume assume that that the the effective effective population population density density that that drives intraspecific competition competition (e.g., by increasing mortality) is the overall density, calculated calculated as
4. 4. METAPOPULATION METAPOPULATION DYNAMICS DYNAMICS IN IN HIGHLY HIGHLY FRAGMENTED FRAGMENTED LANDSCAPES LANDSCAPES
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the weighted sum of individuals in the different age classes. This simplification does models, where does not not apply apply to to spatially spatially structured structured models, where one one needs needs to to know know exactly which patches are occupied (local as opposed to global density) to cal calculate culate the colonization colonization rate rate of an empty patch. Third, SPOMs are closely related to epidemiological models (see (see Chapter 20 20 and references therein). The archetypal metapopulation metapopulation model, the Levins model, is identical to a basic epidemiological model, the susceptible-infected-susceptible (SIS) (SIS) model. model. Ovaskainen Ovaskainen and and Grenfell Grenfell (2003) (2003) utilized utilized the the correspondence correspondence between metapopulation and between metapopulation and epidemiological epidemiological models models in in using using the the patch patch 4.4 in the study of the effectiveness of various value measures discussed in Section 4.4 intervention intervention scenarios for sexually transmitted transmitted diseases. Further cross crossfertilization of these two populations two fields has much much potential, potential, including meta metapopulations inhabiting dynamic landscapes versus disease transmissions in dynamically changing changing contact contact networks, networks, the the evolution evolution of of migration migration rate rate and and other other life life history history traits versus the evolution of virulence, and predator-prey population mod predator-prey meta metapopulation models versus SIR models. With such links as described earlier developing among meta population metapopulation ecology, landscape ecology, epidemiology, and general population population ecology, a new era of exciting research is in the horizon.
sdfsdf
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A PPLICATION OF APPLICATION OF STOCH ASTIC PATCH STOCHASTI C PATC H OCCUPANCY OCCUPANCY MODELS TO RE AL MODELS TO REAL META PO PULATIONS M ETAPO PU LATIO N S Rampal S. Etienne, Cajo J.F. J.F. ter Braak, and Claire C. Vos
5 .1 5.1
INTRODUCTION INTRODUCTION Consensus has emerged about about the generally detrimental effects of habitat loss populations (Hanski, loss and and habitat habitat fragmentation fragmentation on on the the viability viability of of meta metapopulations (Hanski, 11999). 999). Habitat Habitat loss and fragmentation both affect the balance between extinc extinction populations and recolonization of tion of of local local populations and recolonization of empty empty patches: patches: patch patch occupancy occupancy decreases and the meta population is more prone to extinction. Simple metapopulation Simple statistical models models for for patch patch occupancy occupancy have have been been used used to to ascertain ascertain the the existence existence of of these these detrimental detrimental effects effects in in real real metapopulations metapopulations (Van (Van Dorp Dorp and and Opdam, Opdam, 1987; 1987; Merriam, 11988), 988), but their predictive power to show the magnitude of these effects limited, as incorporate the effects is is very very limited, as they they do do not not incorporate the mechanisms mechanisms underlying underlying metapopulation dynamics ((local local extinction and colonization) colonization).. The Levins ((1969, 1969, 11970) 970) model, the prototype metapopulation model that is based on these these mechanisms, mechanisms, is is biologically biologically too too unrealistic, unrealistic, whereas whereas size-structured size-structured and and individual-based metapopulation models are mostly too complex to be para parameterized with with available data. These data almost never consist of accurate
Ecology, Ecology, Genetics, Genetics, and and Evolution Evolution of of Metapopulations Metapopulations
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estimates of sizes. At best, they comprise observations estimates of population population sizes. At best, they comprise observations of of patches patches in in aa patch being occupied occupied or single year patch network network being or empty empty for for aa single year (snapshot (snapshot data; data; consecutive, years (which give Hanski, Hanski, 1994) 1994) or or for for several, several, not not always always consecutive, years (which give infor information mation of of the the population population turnover). turnover). There There is, is, however, however, one one class class of of metapopulation metapopulation models models that that can can both both capture biological detail easily parameterized capture sufficient sufficient biological detail and and be be relatively relatively easily parameterized with with these data: data: the patch occupancy see Chapter these the stochastic stochastic patch occupancy models models (SPOMs, (SPOMs, see Chapter 4). 4). Given Given the the current current occupancy occupancy states states of of all all patches patches in in aa patch patch network, network, SPOMs SPOMs predict predict for for each each patch patch the the probability probability that that it it will will be be occupied occupied at at any any time time in in the local extinction the future. future. These These probabilities probabilities depend depend on on the the probabilities probabilities of of local extinction colonization, which, in turn, turn, may be constant constant (resulting in a stochastic and colonization, Levins depend in variety of Levins model) model) or or depend in aa variety of ways ways on on the the quantities quantities that that are are con considered relevant, relevant, such quality and (or spatial sidered such as as patch patch quality and connectivity connectivity (or spatial cohesion; cohesion; Opdam Opdam et et aI., al., 2003 2003).). Perhaps Perhaps the the simplest simplest such such quantities quantities are are the the area area of of the the patch patch and and the the distance distance between between patches, patches, but but other other quantities quantities are are sometimes sometimes preferable because because area preferable area and and interpatch interpatch distance distance are are not not always always the the best best pre predictor 1b). This dictor variables variables (Thomas (Thomas et et aI., al., 200 2001b). This chapter chapter uses uses as as an an example example aa model model for for aa tree tree frog frog (Hyla (Hyla arborea arborea)l metapopulation metapopulation with with area area and and interpatch interpatch distance as well two predictor variables variables (Vos distance as well two other other predictor (Vos et et aI., al., 2000) 2000).. For For another another 1996). example, ai. ((1996). example, see see Hanski Hanski et et al. Application Application of of aa SPOM SPOM to to aa real real metapopulation metapopulation consists consists of of four four parts. parts. First, First, patch SPOM must patch occupancy occupancy data data must must be be collected. collected. Second, Second, the the SPOM must be be formulated. formulated. Here Here questions questions such such as as "what "what processes processes do do II want want to to describe describe and and what what vari variables do ables do II need need and and what what processes processes and and variables variables can can II dispense dispense with?" with?" must must be be answered. mathematical representation answered. Third, Third, given given the the mathematical representation of of the the SPOM SPOM that that results model must parameterized using results from from the the second second step, step, the the model must be be parameterized using the the dataset dataset at parameters. Fourth, at hand hand and/or and/or independent independent information information about about these these parameters. Fourth, pre predictions model by scenarios. The reli dictions can can be be made made with with the the model by considering considering different different scenarios. The reliability of ability of these these predictions predictions must must be be assessed assessed through through uncertainty uncertainty analysis. analysis. This assumes the SPOM to This chapter chapter assumes the dataset dataset and and the the SPOM to be be given given and and concen concentrates trates on on the the third third part part by by reviewing reviewing the the various various methods methods that that have have been been developed SPOMs using developed to to parameterize parameterize SPOMs using snapshot snapshot and/or and/or turnover turnover data. data. It It then focuses on some of still remaining, their consequences then focuses on some of the the problems problems still remaining, and and their consequences for populations; these for the the fourth fourth part, part, making making model model predictions predictions for for real real meta metapopulations; these problems are frog example. example. We We problems are illustrated illustrated with with the the afore-mentioned afore-mentioned tree tree frog show to determine show that that the the model model predictions predictions are are useful useful to determine the the best best conservation conservation strategies metapopulation. For strategies for for the the real real metapopulation. For this this we we compare compare the the distribution distribution pattern 2002 with pattern in in 2002 with the the distribution distribution pattern pattern predicted predicted from from the the model model cali cali1-1983 and brated brated with with data data from from 198 1981-1983 and 1986. 1986. The The chapter chapter ends ends with with aa dis discussion application of cussion of of the the four four parts parts in in the the application of SPOMs SPOMs and and an an overview overview of of the the insights application of insights gained gained by by the the application of SPOMs SPOMs as as reported reported in in the the literature. literature.
5.2 5.2
STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS Stochastic Stochastic patch patch occupancy occupancy models models have have been been formulated formulated in in discrete discrete time time 994) and ((Day Day and Possingham, 1995; and Possingham, 1995; Hanski, Hanski, 11994) and continuous continuous time time (Verboom (Verboom 998; Etienne, et 9 9 1 ; Frank Frank and et aI., al., 11991; and Wissel, Wissel, 11998; Etienne, 2002; 2002; Etienne Etienne and and
5. 5.
11 007 1
STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS STOCHASTIC
Nagelkerke, 2002; 2002; Chapter Chapter 4). 4). Because Because census census data data are are usually usually separated separated by by Nagelkerke, or several several years, years, itit seems seems natural natural to to use use aa discrete-time discrete-time model model with with aa time time 11 or step of of i1 year; year; in in this this way way difficulties difficulties with with extinction extinction and and colonization colonization probprob step abilities not not being being constant constant during during the the year year can can be be avoided. avoided. Consider Consider now now aa abilities patch network network of of N N patches. patches. If If aa patch patch ii contains contains aa population, population, this this popupopu patch lation can go go extinct extinct in in one one time time step probability Ei, Ei, and and if if the the patch patch lation can step with with probability is empty, empty, it it can can be be colonized colonized with with probability probability Ci Ci (Ei (Ei and and Ci Ci are are denoted denoted by by is Exti and and COli in Chapter Chapter 4, 4, but but we we adhere adhere to to the the symbols symbols introduced introduced here here Exti Coli in for better comparison comparison with with the the literature literature cited). cited) . Let Let us us denote denote the the state state of of for better at time time tt by by X/(t), Xi(t), which is aa binary binary variable: variable: we we have have for for an an occuoccu patch patch ii at which is = 1 1 and and for an empty empty patch patch Xi(t) Xi(t) == 00 (Xi (Xi is is denoted denoted by by Oi 0i pied patch patch Xi(t) Xi(t) = pied for an in total state state of in Chapter Chapter 4). 4 ) . The The total of the the metapopulation metapopulation at at time time t,t, X(t), X(t), can can then then N ones ones and and zeros. zeros. The The colonization colonization be described described by by aa vector vector containing containing N be at time time tt + + 11 usually usually depends depends on on the the state state X(t) X(t) probability of of patch patch ii at probability because this this state state determines determines the the number number of of dispersers dispersers being being sent sent out that because out that may such as patch i,i, but may end end up up in in patch but not not on on earlier earlier states states such as X(t X(t - 11). ) . The The extincextinc tion probability probability is considered to be independent independent of X(t), but but when when tion is usually usually considered to be of X(t), there is Brown and and Kodric-Brown, Kodric-Brown, 1977) 1 977) populations on on there is aa rescue rescue effect effect ((Brown - - populations the brink brink of are rescued extinction by by immigrants immigrants i- the the the of extinction extinction are rescued from from extinction extinction on the probability and thereby extinction probability probability depends depends on the colonization colonization probability and thereby on X(t). on X(t). The the SPOM by aa single single formula, formula, valid valid The dynamics dynamics of of the SPOM can can thus thus be be described described by for each each patch patch i, i, that that specifies specifies the the probability the state of patch patch ii at time for probability of of the state of at time tt + the meta population at at time time t: + 11 conditional conditional on on the the state state of of the metapopulation t: - Ei
P[Xi(t + 1)IX(t)]
=
li i
i,ci
Ci
if if Xi(t) Xi(t)== if if Xi(t) Xi(t) = = = if if Xi(t) Xi(t) = if if Xi!t) Xi( t) = =
11 and Xi(t + and Xi(t + 11 and and Xi(t Xi(t + + 00 and Xi(t + and Xi(t + 00 and and Xi(t Xi( t +
11 )) == 11)) = = 11)) = = 11)) = =
11 00 00 11
((s.1) 5. 1 )
in in which which Ei E i and and Ci C i are a r e shorthand shorthand for for EM) Ei(t ) and and Ci(t), Ci(t), the the extinction extinction and and colonization colonization probabilities probabilities for for the the transition transition from from time time tt to to time time tt + + 11.. These These may may depend depend on on species species and and landscape landscape characteristics characteristics that that can can be be treated treated either either as as observed observed variables variables or or known known or or unknown unknown parameters parameters in in the the model. model. Because Because SPOMs SPOMs possess possess the the Markov Markov property property that that the the probability probability distri distribution bution for for the the state state at at time time tt + + 11 is is completely completely determined determined by by the the state state at at time time t, t, Markovian Markovian theory theory can can be be invoked invoked to to state state various various characteristics characteristics of of the the metapopulation. metapopulation. One One such such characteristic characteristic is is that that the the metapopulation, metapopulation, left left undisturbed population not undisturbed and and conditional conditional on on the the meta metapopulation not having having gone gone extinct extinct untimely, untimely, will will settle settle in in aa pseudoequilibrium pseudoequilibrium in in which which the the probability probability that that it it is in in aa certain certain state state (e.g., (e.g., all all patches patches occupied) occupied) no no longer longer changes changes in in time. time. This This is pseudoequilibrium pseudoequilibrium is is also also called called the the quasistationary quasistationary state state (see (see further further Darroch Darroch and 965; Gyllenberg 994; Gosselin, 998; and Seneta, Seneta, 11965; Gyllenberg and and Silvestrov, Silvestrov, 11994; Gosselin, 11998; Ovaskainen, Ovaskainen, 2001; 2001; Chapter Chapter 4). 4). This This is is aa property property used used frequently frequently in in param parameterization eterization methods, methods, as as discussed discussed later. later. We We will will first first give give aa few few examples examples of of SPOMs. SPOMs.
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
08 1 08
Hanskl's Hanski's Incidence Incidence Function Function Model Model In 1 994) and In probably probably the the best-known best-known SPOM, SPOM, described described by by Hanski Hanski ((1994) and better better known incidence function known as as the the incidence function model model (IFM), (IFM), the the extinction extinction probability probability is is set set proportional to proportional to the the inverse inverse of of patch patch area: area:
)
(5.2) (5.2)
ex Ei = min 1, A-~.
where where ee and and x x are are parameters parameters (e is is sometimes sometimes written written as as A5 A~ where where Ao A0 is is the the minimum minimum possible possible patch patch area area in in which which aa viable viable population population can can exist, exist, x x is is 4) and denoted denoted by by �~ex in Chapter Chapter 4) and the the colonization colonization probability probability is is aa saturating saturating ex in function measure S: function of of connectivity connectivity measure S: (5.3) (5.3)
Ci-- $2i + y2
where where y y is is aa parameter parameter and and Si Si is is given given by by aa sum sum of of contributions contributions from from all all occupied occupied patches patches weighted weighted by by their their distance distance to to patch patch i:i: Si(t) = ~.~ j ~ i X j ( t ) A i e x p ( - ~ d i
((5.4) 5 .4)
i)
where where ex e~ is is aa parameter parameter that that can can be be interpreted interpreted as as the the inverse inverse of of the the mean mean dispersal 19 94) dispersal distance distance of of the the species species under under consideration. consideration. This This is is Hanski's Hanski's ((1994) model model without without the the rescue rescue effect. effect. Incorporating Incorporating the the rescue rescue effect effect means means multiplying multiplying the the extinction extinction probability probability by by 11 - Ci• Ci. In In later later work, work, Hanski Hanski and and co-workers co-workers added added the the area area of of the the destination destination patch patch Ai A i to to the the connectivity connectivity and and parameters parameters that that determine determine the the strength strength of of the the contribution contribution of of the the patches patches of of origin 996; Hanski, 998a,b; Moilanen origin and and destination destination (e.g., (e.g., Wahlberg Wahlberg et et aI., al., 11996; Hanski, 11998a,b; Moilanen and 998; Chapter Moilanen and and Hanski, Hanski, 11998; Chapter 4). 4). See See Moilanen and Nieminen Nieminen (2002) (2002) for for aa discussion Formulas for extinction discussion of of different different measures measures of of connectivity. connectivity. Formulas for the the extinction and some mechanistic basis as and colonization colonization probabilities probabilities have have some mechanistic basis as pointed pointed out out by by Hanski 1 998a,b). Hanski ((1998a,b). The 1 994) derived The model model is is called called IFM IFM because because Hanski Hanski ((1994) derived an an incidence incidence function colonization probabilities. function from from the the extinction extinction and and colonization probabilities. For For this, this, he he first first defined defined the the incidence incidence ]; Ji of of patch patch ii as as the the probability probability that that it it is is occupied. occupied. He He then then assumed population is quasistationary state, assumed that that the the meta metapopulation is in in the the quasistationary state, which which he he interpreted time. This interpreted as as ]; Ji being being independent independent of of time. This leads leads to to -
J.1 == J·(l Ji Ji(1 - E Ei)1 ) + + ((11 - J J·)G i )1 C i1 ==> ~ JJ.i1 == 1 -
-
c Ci 1
Ci Ci + + E Eii
(5.5) (,.5.:5)
For For example, example, for for the the IFM IFM with with rescue rescue effect effect [but [but without without the the cutoff cutoff at at 11 of of Ei Ei in Eq. Eq. (5.2)], (5.2)], the the incidence incidence function function becomes becomes in 1
Ji =
ey2 1+
Axs 2
(5.6) (5.6)
5. STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS 5.
1109 09
The The Extended Extended IFM IFM for f o r Tree Tree Frog Frog Data Data Vos et al. ai. (2000) extended the IFM to model a tree frog metapopulation metapopulation in aa pond southwest The pond network network in in southwest The Netherlands, Netherlands, for for which which occupancy occupancy data data have 98 1-1983 and 986 (see (see Box . 1 ) . In model, the have been been collected collected in in 11981-1983 and 11986 Box 55.1). In their their model, the form replacing the form of of the the function function for for colonization colonization probability probability is is relaxed relaxed by by replacing the exponent (as already 994) and exponent of of 2 2 by by parameter parameter Zz (as already suggested suggested by by Hanski, Hanski, 11994) and by by allowing allowing the the contribution contribution of of patch patch area area in in the the connectivity connectivity measure measure to to be 996; bb is be determined determined by by parameter parameter bb (Wahlberg (Wahlberg et et aI., al., 11996; is denoted denoted by by �~em in em in
BOX 5.1 The Tree Frog Metapopulation in Zealand Flanders, The Netherlands The tree frog metapopulation exists in an agricultural landscape in southwest The Netherlands (Fig. B5. 1 ). The suitable habitat forms approximately 1 .5% of the total landscape cover and is separated by intensively used agricultural fields that are unsuit able for the species. (Semi)natural vegetation can be found on dikes, in coastal sand dunes, and in meadows with cattle-drinking ponds, the aquatic habitat used by the tree frog. The terrestrial habitat of the tree frog consists of shrubs, bushes, and vegetation of high herbs. The tree frog distribution has declined sharply in The Netherlands since the 1 950s as a result of intensification of agriculture. Most (semi)natural elements such as cattle-drinking ponds and hedgerows have been cleared. Zealand Flanders
ftAst
•
o
4
IO km
L' _ --'_ ---' ,
Fig. 85.1
(2000).
The position of the tree frog study area. Reprinted with permission from Vos et al.
11 11 00
RAM PAL S. RAMPAL S. ETIENNE ETIENNE ET ET Al. AL.
From 1 98 1 until 1 986 the distribution pattern of the tree frog in Zealand Flanders was monitored. Statistical analysis showed that the occupation probability of a pond increased with the number of other ponds and. the a rea of high herbs and bushes in the surrounding (Vos and Stumpel, 1 996). Analysis of turnover data from 1 98 1 to 1 983 showed that local extinctions were related to spatial features of the landscape: extinction probability decreases with pond size (Vos et a I ., 2000). Pond size encom passes both pond area and suitable terrestrial habitat within a radius of 250 m around the pond . Also, colonization events were correlated to connectivity as well as habitat quality factors. Comparing dispersal distances with distances between ponds shows that the habitat network is still connected by dispersing individuals (Fig. B5 .2). Analysis of dispersal events showed that occupied ponds were preferred over empty
.-
, '.
"
t
, , . .. "
"
� " '.
": : ":-
, ,
,
"
,
,
"
"
" ,
, .'
, ..
,
-
:
"
.
• •
#
.
. . i::,:
# .' -.
, .. '
.
.
.
.. ... "
roT; , -;,!..!; ,:;-"� "': 4...A .•
-:' , --,.
pond
dispersal between ponds
land habitat
•
4 km
»
Fig, 85,2 Overview of dispersal events registered by capture-recapture techniques in Zealand Flanders during the period 1 98 1 -1 989. Reprinted with permission from Vos et al. (2000).
5. 5. STOCHASTIC STOCHASTICPATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
1111 11
ponds, although many empty ponds were present at close ranges. This implies the importance of conspecific attraction and a rescue effect in this metapopulation pond system. In the next step toward spatially explicit connectivity measures the permeability of the landscape matrix is incorporated. Results of a radio telemetry study 0/os, 1 999), show that moving tree frogs have a high preference for hedgerows, that they avoid arable land, and that pasture takes an intermediate position. Incorporating these char acteristics in the connectivity measure of the extended IFM model will improve the model further. The results seem to justify the application of the metapopulation concept for guide lines for optimal spatial habitat configuration for this species. The protection policy plan for the tree frog in The Netherlands is actually based on meta population concepts. Apart from the necessary improvements of habitat quality, the size and connectivity of the habitat networks of the tree frog are being improved (Crombaghs and Lenders, 2001 ). Results for a tree frog meta population in Twente, a region in the east of The Netherlands, already show that the species reacts positively to these restoration meas ures (Braad, 2000).
Chapter Chapter 4). 4). Furthermore, Furthermore, three three variables variables were were introduced: introduced: B;j Bii,' H Hi,i, and H H2, 2,ii.' I ,;, and describes the the quality of the the matrix matrix habitat habitat between and j, j, and describes quality of between patches patches ii and and its value was all combinations = 00 for for aa total total barrier, barrier, its value was set set for for all combinations of of ii and and jj (e.g., (e.g., Bij = water Bij Bii == 11 for for no no barrier, barrier, and and B;j Bii == 0.5 0.5 for for aa semibarrier). semibarrier). HI,i Hl,i isis the the water conductivity, percentage cover cover of both conductivity, and and H H2,i the percentage of the the water water vegetation; vegetation; both 2,; isis the were each pond. deter were measured measured for for each pond. Two Two additional additional parameters, parameters, q qll and and q q2, 2 , deterextinction and mine Hl,i and H2,i respectively. respectively. The The extinction and colonization colonization mine the the effect effect of of H I ,i and probabilities probabilities are are now now given given by by
Bij
(
)
eH 1,i ql) eHt 1
E 1 ,, ax , Ei1· = = ((11 - C)min Ci)min/1 1 AX 1
__
(5.7) (5.7)
and and
sf CCii == -----'-- y Szi ++ HYq2 sT q2,i H22,i
(5.8) (5.8)
with with
SSi(i (t)t) = L ~ Xj(t)A Xj( t)A~Bq e x p (-ad - (xd/j) fB ij exp( ij) jjri=- i
(5.9) (5.9)
(Note (Note that that y y in in this this model model is is not not the the same same as as y y in in Hanski's Hanski's IFM.) IFM.) We We mention mention this this model, model, not not only only as as an an example example of of aa SPOM SPOM with with other other variables variables than than patch patch area area and and interpatch interpatch distance, distance, but but also also because because it it is is used used as as an an illustration illustration of of some some of of the the problems problems in in parameter parameter estimation. estimation.
11 11 22
5.3 5.3
RAMPAL S. ETIENNE RAMPAL S. ETIENNE ET ET AL. AL.
METHODS METHODS TO TO ESTIMATE ESTIMATE PARAMETERS PARAMETERS OF OF STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS Because Because SPOMs SPOMs contain contain few few parameters, parameters, it it is is attractive attractive to to attempt attempt to to esti estimetapopulation data, that that is, data data on the occupancy occupancy of some mate them using meta population data, or years. Several or all all of of the the patches patches in in aa network network for for 11 or or several several years. Several methods methods have have for this purpose. They use only data data on transitions been developed for transitions between occupancies colonizations, noncolonizations), occupancies (extinctions, (extinctions, nonextinctions, nonextinctions, colonizations, noncolonizations), data data from a snapshot year (which typically shows spatial clusters of snapshot of a single year shows spatial occupied occupied patches patches as as the the result result of of the the history history of of extinctions extinctions and and colonizations colonizations patch network), network), or both. This section section reviews these methods. methods. Table 55.1 in the patch .1 summarizes summarizes their their main main features and and shortcomings. shortcomings.
Hanski 1 994): Regression Hanski ((1994): Regression of of Snapshot Snapshot Data Data In his classic paper, Hanski ((1994) 1 994) not not only introduced introduced the IFM, but but also showed showed how how it it can can be be parameterized parameterized in in aa simple simple way way using using snapshot snapshot data data of of patch presented by Hanski ((1991) 1991 ) patch occupancies, occupancies, elaborating elaborating ideas ideas presented by Peltonen Peltonen and and Hanski and Hanski Hanski ((1992). 1 992). He proposed proposed nonlinear regression to obtain obtain point point estimates nonlinear regression parameters, with with standard errors, by maximizing maximizing the logarithmically logarithmically of the parameters, transformed likelihood, i.e., the loglikelihood, loglikelihood, transformed
E[Xi lnJi +
(1 -
Xi)ln(1 - Ji)]
(5.10) (5.10)
l
Note Note from Eq. (5.6) that that in the IFM without without rescue effect, e and and y form a single composite parameter so that they cannot be estimated separately composite parameter that cannot separately from snapshot data. Independent Independent information information on the minimum minimum sustainable patch snapshot size separate ee and size Ao A0 can can help help to to estimate estimate ee and and thus thus separate and y. y. out that that Eq. (5.6) ( 5 . 6 )- the IFM with with rescue Ter Braak et al. ((1998) 1 998) pointed out effect linearized by logit effect and and with with all all patches patches larger larger than than Ao A 0- can can be be linearized by the the logitIi giving transformation of transformation of Ji giving 10g log
0g( Si ) Ji)) == �o [30 + �l ~311og(Ai)+ 132 log( Ai ) + � (� 2 1log(S/) li~ji -j;
(5.11) (5.1 1)
with - log(ey2 ), �131 1= with �o 130 = =-log(ey2), = x, x, and and � ~ 22 = z 2 2.. Therefore, Therefore, available available data data being being binary values, point point estimates parameters (with approximate standard binary values, estimates for for the the parameters (with approximate standard errors) can be obtained obtained easily by using logistic regression (McCullagh (McCullagh and NeIder, - at Nelder, 1989) 1989) of of the the occupancies occupancies Xi on on 10g(Ai) log(Ai) with with offset offset 210g(Si) 21og(Si)at least least when when the the values values of of a cx and and bb in in the the definition definition of of Si Si are are given. given. (An (An offset offset is is aa . ) Although predictor with a regression coefficient coefficient of 11.) Although this is mathematically mathematically equivalent equivalent to Hanski's nonlinear nonlinear regression, it is computationally computationally much much more more efficient. It is historically 1 994) already historically interesting interesting that that Eber and Brandl ((1994) performed occupancies without performed such such logistic logistic regression regression of of patch patch occupancies without knowing knowing the link to the incidence function function model (which had had not not been published yet); they they just just used used aa statistical statistical model. model. The The extended extended IFM IFM can can be be fitted fitted to to snapshot snapshot data data using using aa logistic logistic regression regression of of occupancies occupancies Xi on on 10g(Ai), log(A/), 10g(Si), log(S/), log( log(Hi,i) H1,i) a and b by logistic regression by calculating and ,i)' We can estimate and log(H log(H2,i). We can estimate oL and b by logistic regression by calculating 2
>,
e-
0
0 E
2~ 0
0
0
Turnover only
No (ignored)
Yes
Logistic regression
Likelihood optimization (non-linear regression)
Likelihood optimization (MC simulation)
<
Approximations of standard errors
Approximations of standard errors
Approximations of Full uncertainty confidence intervals distribution
Full uncertainty distribution
Z
No
No
No
Z
Yes
Yes
Fast / very fast
Fast
Fast
Slow
Very slow
Slow
gO
Ter Braak and Etienne (2003) i_
Not at all (turnover only) Yes
o~
Spatial structure and turnover Correctly Yes
~.j r
. ~0
0..~
0
Bayesian MCMC (MetropolisHastings) 9 ,.~
.
~
r
r
c,.) .~,
0 .,,~
9- ~ 9- ~
0 .,.~
.~
9
9-0
Bayesian MCMC (Gibbs sampling)
9
0 0
O'Hara et al. (2002) s_
O~
0
- - ~ o~
Z
z-'~
~
Not at all (turnover only) No
0
r
Turnover only
m
O~
o l
Z
~
r ~.~
r
o
.,..~
0""
~ ~
0
r
~
0
No (ignored/ considered empty) Likelihood optimization (nonlinear regression)/ with rescue effect: logistic regression Approximations of standard errors No
oO O~ O~
Spatial structure and turnover Correctly
Spatial structure only Approximation
r
o
Is uncertainty analysis of model predictions possible? Computational efficiency?
E
Spatial structure and turnover Approximation
What type of information is used? How is quasistationarity incorporated? Are missing data taken into account?
How is uncertainty in estimates specified?
E
Moilanen (1 999)
Hanskl (1994)
Method (reference)
What method are parameter estimates based on?
,1,,,o
Ter Braak et al. (1 998)/ Vos et al. (2000)
~
o9~ =
Verboom et al. (l99l)/SjtigrenGulve and Ray (1996)
. m
0
,i,,,o
0 c-
4-o
. m
x
4-o
0
i1
0
0
o~
Overview o f Features o f Existing Methods t o Estimate Parameters o f SPOMs from Occupancy Data
u~
...1
TABLE 5.1
9
o0
0
0
113
. . IN
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
4 1 114
the regression over values of locating where the regression over aa grid grid of of values of ex 0~ and and bb and and by by locating where the the log log likelihood [Eg. (5.1 0)] is likelihood [Eq. (5.10)] is maximum. maximum. There 998; Ter There are are five five caveats caveats in in the the method method (Ter (Ter Braak Braak et et aI., al., 11998; Ter Braak Braak and and Etienne, Etienne, 2003 2003).). First, First, the the condition condition of of nonextinction nonextinction of of the the metapopulation metapopulation (required (required for for quasistationarity) quasistationarity) is is omitted omitted in in Eq. Eq. (5.5 (5.5).) . This This is is perhaps perhaps aa valid valid approximation approximation for for large large networks networks that that are are unlikely unlikely to to go go extinct extinct quickly, quickly, but but because population models because meta metapopulation models are are often often used used to to explore explore scenarios scenarios intended intended to threatened meta populations, this need not to conserve conserve threatened metapopulations, this need not always always be be the the case. case. Second, Second, it it is is assumed assumed in in Eq. Eq. (5.5) that that the the extinction extinction and and colonization colonization prob probabilities abilities of of patch patch ii are are constant constant in in time. time. This This is is not not true, true, for for they they both both depend depend on on the the evolving evolving states states of of the the other other patches. patches. Third, Third, the the connectivity connectivity S S is is calculated calculated using using the the patch patch occupancies occupancies of of the the same same year year and and not not using using the the patch patch occupancies occupancies of of the the previous previous year year as as it it should should according according to to the the model model [Eq. reason is data. [Eq. (5.4)]; (5.4)]; the the reason is of of course course that that these these are are unknown unknown for for snapshot snapshot data. Fourth, likelihood [Eq. [Eq. ((5.10)], 5 . 10)], patch Fourth, in in the the log log likelihood patch occupancies occupancies are are assumed assumed to to be be statistically statistically independent. independent. However, However, this this assumption assumption is is not not met met because because the the patch patch occupancies occupancies are are regressed regressed on on the the connectivity connectivity SS that that is is calculated calculated from from the same data. 5 . 1 0 ) is best aa pseudo-log the same data. Therefore, Therefore, Eq. Eq. ((5.10) is at at best pseudo-log likelihood. likelihood. Fifth, Fifth, in in the the derivation derivation of of the the incidence incidence function function Eq. Eq. (5.6), the the cutoff cutoff at at 1I of of Ei Ei in Ao. in Eq. Eq. (5.2 (5.2)) is is neglected. neglected. This This is is warranted warranted only only if if all all areas areas are are larger larger than than A0. Missing Missing data data (occupancies (occupancies are are unknown unknown for for some some patches) patches) cannot cannot be be dealt dealt with with properly properly in in this this method. method. They They are are basically basically ignored. ignored. It It is is apparent apparent from from Eq. Eq. (5.4) (5.4) that that this this may may affect affect the the connectivity. connectivity. By By ignoring ignoring patches patches of of which which no no data data are are available, available, one one effectively effectively assumes assumes them them to to be be empty empty because because they they do contribute to do not not contribute to connectivity. connectivity. If If the the patches patches were were actually actually occupied, occupied, the the colonization colonization probability probability will will be be underestimated. underestimated.
Verboom al. (1 991 ), Sjogren-Gulve and Ray 996), Verboom et et al. (1991), Sj6gren-Gulve and Ray (1 (1996), and Eber Eber and and Brandl 996): Regression and Brandl (1 (1996): Regression of of Turnover Turnover Data Data While While Hanski Hanski ((11994) 9 9 4 ) oonly n l y considered considered snapshot snapshot data, data, Verboom Verboom et et ai. al. ((1991), 1 991 ), Sjogren-Gulve 1 996), and, Sj6gren-Gulve and and Ray Ray ((1996), and, less less commonly commonly cited, cited, Eber Eber and and Brandl 1 996) looked 1 99 1 ) Brandl ((1996) looked only only at at turnover turnover data data [actually, [actually, Verboom Verboom et et ai. al. ((1991) also also looked looked at at the the frequency frequency that that aa patch patch is is found found occupied, occupied, see see later]. later]. They They realized that 5 . 1 ) are realized that Ei and and Ci in in Eq. Eq. ((5.1) are conditional conditional probabilities probabilities that that can can be be fit fitted logistic regression regression (McCullagh ted to to data data by by logistic (McCullagh and and Nelder, Nelder, 1989). 1989). To To parame parame)] with terize possible pairs Xj(t + 11)] terize Ei Ei,, they they created created aa dataset dataset with with all all possible pairs [X;(t), [Xi(t), Xi(t with and then then applied applied logistic logistic regression regression with with Xi(t Xi(t + 11)) as as the the response response Xi(t) = 11 and variable. Similarly, variable. Similarly, Cj Ci was was parameterized parameterized by by applying applying another another logistic logistic regres regression all possible )] with Xj(t + 11)) as sion to to all possible pairs pairs [Xj(t), [Xi(t), Xi(t + 11)] with X;(t) Xi(t) = = 00 and and with with Xi(t as the the response response variable. variable. They They thus thus assumed assumed that that the the probabilities probabilities of of extinction extinction and and colonization colonization behave behave logistically: logistically: Ei Ei = --
1
----1 + exp( - Ue,i )
1 + exp(-ue,i) 11 Cj Ci = -- ---11 + + eexp( xp(-u cUc,i , i))
(5.12) (5.12)
5. STOCHASTIC STOCHASTICPATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS 5.
11 11 55
where where UUee and and UUcc are are linear linear functions functions of of the the variables variables of of importance, importance, such such as as patch area area and and connectivity, connectivity, the the logarithm logarithm of of which which are are treated treated as as explanatory explanatory patch variables variables in in logistic logistic regressions. regressions. The The parameters parameters in in Ue Ue and and UUcc are are fitted fitted from from the the ""extinction extinction dataset" dataset" and and the the "colonization "colonization dataset," dataset," respectively, respectively, giving giving point point estimates estimates and and standard standard errors. errors. They They found found that that extinction extinction is is significantly significantly related related to to patch patch area area (and (and not not connectivity) connectivity) and and colonization colonization to to connectivity connectivity area).. ((and and not to area) Sjogren-Gulve 1 996) used Sj6gren-Gulve and and Ray Ray ((1996) used point point estimates estimates in in subsequent subsequent computer computer simulations simulations of of the the discrete-time discrete-time SPOM SPOM to to predict predict the the long-term long-term trend trend in in 1991 ) had occupancy, occupancy, whereas whereas 55 years years earlier earlier Verboom Verboom et et al. al. ((1991) had missed missed that that opportunity, opportunity, partly partly because because they they adhered adhered to to their their continuous-time continuous-time SPOM. SPOM. Note Note that that extinction extinction probabilities probabilities do do not not have have the the logistic logistic form form in in the the IFM IFM [Eq. [Eq. (5.2)] (5.2)] and and they they are are even even linked linked to to the the colonization colonization probabilities probabilities in in the the ((extended) extended) IFM IFM with with aa rescue rescue effect effect [see [see Eq. Eq. (5.7)]. (5.7)]. We We must must therefore therefore resort resort to nonlinear nonlinear regression regression (Vos (Vos et et aI., al., 2000) 2000) where where the the log log likelihood likelihood to to be be to maximized is
2: t+l 1 )))) InE ~ [X [ X ii((t)(1 t ) ( 1 -X - X i (it( + lnEii + + X X ii(( tt)X ) X ii(( tt+ + l )1l)nIn( ( 11 - E Ei)] i)]
i, (5.13) (5.~31 ++ � ~ [ ([(1 1 - X X ii(( tt))X ) ) X ii((tt + + l1)lnCi Xi(t))(1 X ii(( tt+ + l 1) )))l nln(1 (1 - Ci)] Ci)] ) lnCi ++ ((11 -- X i( t))(l -- X tI
Ter Ter Braak Braak et et al. al. (1998) (1998) showed showed that that this this is is equivalent equivalent to to maximizing maximizing the the extinction extinction and and colonization colonization parts parts of of the the log log likelihood likelihood separately, separately, provided provided Ei Ei and independent parameters, parameters, so and Ci Ci have have different, different, independent so giving giving aa formal formal justification justification approach by Verboom et al. Sj6gren-Gulve and Ray (1996). (1996). 1991 ) and Sjogren-Gulve of the approach al. ((1991) Although this method is theoretically sound are no technical difficulties Although this method is theoretically sound (there (there are no technical difficulties as using only turnover data has some some shortcomings. shortcomings. First, First, as with with snapshot snapshot data), data), using only turnover data has it requires data collection in at at least least 2, 2, but but preferably years. it requires data collection in preferably several several consecutive consecutive years. Second, while snapshot data many extinctions extinctions and coloniza Second, while snapshot data are are the the result result of of many and colonizations in tions in the the history history of of the the metapopulation metapopulation and and are are therefore therefore considered considered to to con contain information, turnover turnover data data provide provide little tain a lot of of information, little information information if if turnover turnover that data data show only extinctions and and colonizations colonizations [most inforis slow so ::hat only a few extinctions [most infor mation would be provided provided if the number number of mation would be if the of extinctions extinctions (cq. (cq. colonizations) colonizations) and and the were equal]. the number number of of nonextinctions nonextinctions (cq. (cq. noncolonizations) noncolonizations) were equal]. Third, Third, miss missing resulting in the same bias as ing data data are are again again ignored, ignored, resulting in the same bias as for for snapshot snapshot data. data.
Ter Ter Braak Braak et et al. al. (1998) (1998) and and Vos Vos et et al. al. (2000): (2000): Combining Combining the the Previous Previous Approaches Approaches To make make full full use use of of data, data, both both the the historical information contained To historical turnover turnover information contained in dataset and the turnover in the the first first year year of of aa dataset and the turnover information information in in the the following following (1991) not not only only applied applied years should should be be extracted. extracted. Interestingly, Interestingly, Verboom Verboom et et al. al. (1991) years logistic regression regression to to turnover turnover events events as as discussed discussed earlier, earlier, but but also also applied applied logistic logistic regression to to the the frequency frequency that that aa patch is found found occupied the logistic regression patch is occupied during during the years of of survey. survey. For For aa dataset dataset of of only only 11 year, year, the the latter latter is is formally formally equivalent equivalent years (1994). For For data data of of several several years, years, the the to the the snapshot snapshot data data analysis analysis of of Hanski Hanski (1994). to method also also takes takes turnover turnover events events into into account, account, but but extinction extinction and and colonizacoloniza method tion considered equal 1 1 1 000 sequence sequence is is considered equal to to 101010). 101010). tion are are not not separated separated (a (a 111000
11 11 66
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
Verboom 1991 ) must Verboom et et al. al. ((1991) must have have realized realized that that there there is is more more information information in in 1 994, 11996) 996) certainly aa dataset dataset than than only only turnover. turnover. Eber Eber and and Brandl Brandl ((1994, certainly realized realized this, approaches. this, but but they they did did not not combine combine the the two two approaches. It 1998) who It was was Ter Ter Braak Braak et et al. al. ((1998) who mentioned mentioned aa pragmatic pragmatic method method to to combine likelihood of combine the the two two previous previous approaches. approaches. For For this, this, the the likelihood of the the dataset dataset is incidence dataset" is partitioned partitioned in in three three parts: parts: the the ""incidence dataset" from from the the first first year year of of aa dataset, dataset, the the "extinction "extinction dataset," dataset," and and the the "colonization "colonization dataset" dataset" from from the the following (see also following years. years. The The likelihood likelihood to to be be maximized maximized (see also later) later) is is the the sum sum of 5 . 1 3 ), so of Eq. Eq. (5.10) and and Eq. Eq. ((5.13), so Hanski's Hanski's incidence incidence function function model model [Eq. [Eq. (5.6)] is used used for for the the first first year. year. This This combined combined approach, approach, first first applied applied in in Vos Vos et et al. al. is (2000), suffers suffers of of course course from from the the same same problems problems as as its its constituents: constituents: it it uses uses aa pseudo-likelihood pseudo-likelihood for for the the first first year year data, data, it it yields yields only only point point estimates estimates of of the the mean mean and and standard standard error error of of the the parameters, parameters, and and cannot cannot handle handle missing missing data data properly. properly.
Moilanen 999) : Monte Moilanen (1 (1999): Monte Carlo Carlo Simulation Simulation Moilanen 1 999) provided Moilanen ((1999) provided the the first first solution solution to to problems problems involving involving the the pseudo-likelihood based on pseudo-likelihood and and missing missing data data in in aa new new approach approach based on maximum maximum likelihood estimation using Carlo simulations. Because of likelihood estimation using Monte Monte Carlo simulations. Because of the the Markov Markov property SPOM, the probability of values property of of the the SPOM, the probability of aa dataset dataset X of of T years, years, given given values for for the the parameters parameters 0 O,, can can be be written written as as
P[X P[XO]=P[X(1)]P[X(2)IX(1)]...P[X(t+I)X(t)]...P[X(T)X(T-1) (5.14) . 14) [ 0 ] = P[X( 1 )]P[X( 2 ) [X( 1 )] . . . P[X( t + 1 ) [X( t ) ] . . . P[X( T) [ X( T- l ) ]] (5 with, with, for for each each year year t of of this this sequence, sequence,
PP[X(t+l)X(t)]=i-iP[Xi(t+l)X(t) [X( t + 1 ) [X( t ) ] = IT P [Xi( t + 1 ) [ X( t)]]
. 15) (5 (5.15)
I
because because the the states states are are independent independent conditional conditional on on the the state state of of the the system system in in the the previous between spatial in previous year. year. Note Note the the separation separation between spatial information, information, contained contained in P[X(1)], P[X(l )], and the turnover turnover information, contained in the remaining conditional probabilities, al. ((1998). 1 998). Equation probabilities, as as noted noted by by Ter Ter Braak Braak et et al. Equation (5.14) (5.14) is is the the true true likelihood that likelihood that needs needs to to be be maximized. maximized. Instead 1 999) approximated 1 )] by Instead of of using using Eq. Eq. (5.6), (5.6), Moilanen Moilanen ((1999) approximated P[X( P[X(1)] by Monte Monte Carlo Carlo simulation. simulation. From From an an arbitrary arbitrary state, state, the the IFM IFM is is simulated simulated until until the the quasi quasistationary stationary equilibrium equilibrium is is considered considered to to be be reached reached and and then then for for another another L L time time steps steps to to obtain obtain simulated simulated states states X Xuu ((uu =- L.L). 1...L). The The approximation approximation is is then then
1 u~lP[X ( 1 )]Xu]
(5.1 6) (5.16)
_
for some large large number for some number L. L. This This Monte Monte Carlo Carlo approximation approximation derives derives from from the the equation (Ter equation (Ter Braak Braak and and Etienne, Etienne, 2003 2003)) K K
= k2~,P[Yk]P[X(1)IYk]= 2: P[Yk ]P[X(l ) [ Yk ] Pe[x(1)] [X( l )] = k=l
(5.17)
5. 5. STOCHASTIC STOCHASTICPATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
1 11 77
where the summation summation is over all possible states Y Yk. probabilities P[Y P[Yk] drop k] drop k • The probabilities out simulation series out of of Eq. Eq. (5.17) because because the the simulation series X Xuu is is self-weighing, self-weighing, that that is, is, each each Xuu is generated by the simulation simulation with probability probability proportional proportional to P[X P[Xu]. X u]' The usually huge reason for Carlo simulation simulation [Eq. [Eq. (5.16)] is reason for the the Monte Monte Carlo is that that K K is is usually huge so so that summation in cannot be that the the summation in Eq. Eq. (5.17) cannot be carried carried out out in in practice. practice. Missing years in data data can be handled handled by simulation simulation as well. If, for for example, example, data data for for year year 2 are are missing, missing, we we require require in in the the likelihood likelihood [Eq. (5.14)] the the term term
(1 )]P[X(3)IXk(2)] X(1)])] = = k~ X(1)]P[X(3)Xk(2)] PP[X(3) [X(3)IX(1 P[Xk(2) IX ±]=~P[xk(2) K
1i = 1
(5. 18) (5.18)
where where the the summation summation iiss over over all all possible possible states states X Xk(2). To approximate approximate k (2). To Eq. (5.18), the (M being large) from the state state in in year year 2 is is simulated simulated M M times times (M being large) from the the state state in Carlo approximation in year year 1. The The Monte Monte Carlo approximation is is then then [analogously [analogously to to Eq. (5.16)] e[x(3)lx(1)] ~
1
M
-c;~,I'[X(3)Xu(2)] IVl u=~-I
(5.19)
Although 1 999) does Although Moilanen Moilanen ((1999) does not not mention mention it it explicitly, explicitly, the the same same procedure procedure can can be be applied applied if, if, instead instead of of aa complete complete year, year, only only aa few few occupancies occupancies are are miss missing in a single year. Evidently, only the missing data data are simulated. simulated. Being 1 999) approach Being aa maximum maximum likelihood likelihood method, method, Moilanen's Moilanen's ((1999) approach only only produces produces point point estimates, estimates, although although estimates estimates of of confidence confidence limits limits can can be be com computed but require puted ((but require aa lot lot of of computing computing time). time).
O'Hara O'Hara et et al. al. (2002): ( 2 0 0 2 ) : Bayesian Bayesian MCMC M C M C on on Turnover Turnover Data Data To To obtain obtain aa full full joint joint probability probability distribution distribution of of the the model model parameters parameters instead instead of of mere mere point point estimates, estimates, the the maximum maximum likelihood likelihood method method must must be be abandoned. abandoned. O'Hara et al. (2002) were O'Hara et al. were the the first first to to adopt adopt aa Bayesian Bayesian approach approach to to parameter parameter estimation. estimation. The The central central idea idea in in Bayesian Bayesian theory theory (e.g., (e.g., Gelman Gelman et et aI., al., 1995) is is that that our value of our knowledge knowledge of of the the value of aa parameter parameter can can be be represented represented by by aa probability probability distribution containing new information about this parameter distribution and and that that data data containing information about parameter can be be used used to to adjust adjust this this probability probability distribution. distribution. The The probability probability distributions distributions before been used called prior before and and after after data data have have been used to to update update our our knowledge knowledge are are called prior and posterior probability distributions. Bayes' formula posterior probability distributions. Bayes' formula describes how how the prob probability distribution distribution of model model parameter parameter E> | is adjusted adjusted using data data X:
P[E>IX] /'[olx] = =
P[XIE>]P[E>] e[xlo]e[o] e[x] P[X]
(5.20) (5.20)
The can The posterior posterior probability probability distribution distribution P[E>IX] P[| can often often be be approximated approximated through Markov Markov chain Monte Carlo through chain Monte Carlo (MCMC) (MCMC) simulation simulation with with the the Metropolis-Hastings Metropolis-Hastings algorithm. algorithm. The The Metropolis-Hastings Metropolis-Hastings algorithm algorithm consists consists of of the following steps. First, arbitrary values values of the model parameters parameters are chosen. New New values values of of the the model model parameters parameters are are then then drawn drawn from from aa probability probability distri distribution called The form of this jumping distribution " bution called the the jumping jumping distribution distribution ]Ju. The form of this jumping distribution U is is arbitrary, arbitrary, but but aa smart smart choice choice will will facilitate facilitate calculations calculations and and convergence convergence of of the the
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
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simulation simulation to to the the posterior posterior distribution. distribution. These These new new values, values, denoted denoted collectively collectively by by e | , are are now now accepted accepted with with probability probability "
, P[|
X]Ju[| u-110.]
r = min(1 p - [ ~ - i ~ 7 ~ ) u - - - 1 ] )
(5.2 1) (5.21)
where where e | u -1 represent represent the the previous previous values values of of the the model model parameters. parameters. This This pro procedure cedure of of drawing drawing new new values values and and accepting accepting or or rejecting rejecting them them is is iterated iterated many many (u denotes times times (u denotes the the iteration iteration number) number) and and creates creates aa Markov Markov chain; chain; acceptance acceptance parameter values u only depends depends on the values accepted in of new parameter values in iteration iteration u iteration (a sam iteration u u - 11.. The The set set of of values values e | u generated generated in in this this way way constitutes constitutes (a samThe ple from) from) the the posterior posterior distribution distribution p[eIX]. P[| The first first half half of of the the iterations iterations is is ple often needs some some time often discarded discarded because because the the simulation simulation needs time (called (called the the burn-in burn-in period) to period) to converge converge to to the the stationary stationary distribution distribution of of the the Markov Markov chain chain (which (which has has nothing nothing to to do do with with the the Markov Markov property property of of the the SPOM). SPOM). 1 ), the When When Eq. Eq. (5.20) is is inserted inserted into into Eq. Eq. (5.2 (5.21), the probabilities probabilities P[X] P[X] cancel cancel and and only only the the prior prior probability probability distribution distribution pre] P[| and and the the probability probability of of data data remain. Note conditional on conditional on the the model model parameters parameters p[Xle] P[XI| remain. Note that that p[Xle] P[XI| is is the the likelihood. prior can based on prior knowledge knowledge of model likelihood. The The prior can be be chosen chosen based on prior of the the model parameters, given by SPOM itself, is, parameters, and and the the likelihood likelihood p[Xle] P[XI| is is given by the the SPOM itself, that that is, Eq. and (5.1). O'Hara O'Hara et al. (2002) used only turnover turnover Eq. (5.14) with Eqs. Eqs. (5.15) and data case the data in in which which case the state state in in the the first first year year is is considered considered to to be be given given so so that that P[X(1)] P[X( l)] drops out of Eq. (5.14). For the jumping distribution they used a nornor mal mean e matrix k (to be chosen arbi mal distribution distribution with with mean | u and and covariance covariance matrix E (to be chosen arbitrarily, trarily, but but aa smart smart choice choice speeds speeds up up convergence). convergence). As 1 999) approach, As in in Moilanen's Moilanen's ((1999) approach, the the problem problem of of missing missing data data can can be be tack tackled led by by simulating simulating them, them, but but in in aa different different way. way. In In the the Bayesian Bayesian context, context, missing missing data are are in in fact fact treated treated as as parameters; parameters; the the MCMC thus also also yields yields posterior posterior data MCMC thus probability probability distributions distributions for for these these missing missing data. data. The The Metropolis-Hastings Metropolis-Hastings algor algorithm alternate sampling ithm allows allows alternate sampling of of (sets (sets of) of) parameters, parameters, that that is, is, they they do do not not need need to joint jumping to be be drawn drawn from from aa single single joint jumping distribution distribution simultaneously. simultaneously. It It is is most most convenient to sample the parameters and convenient to sample the set set of of model model parameters and the the set set of of missing missing data data chose to sample missing missing data in al. (2002) chose in turn. turn. O'Hara O'Hara et et al. to sample data for for each each patch patch in in 5.2 on each each year year separately separately (see (see Box Box 5.2 on Gibbs Gibbs sampling). sampling). As As O'Hara O'Hara et et al. al. (2002) only only considered considered turnover turnover data, data, they they could could not not use use all all information information in in aa dataset. dataset. This This problem problem was was solved solved by by Ter Ter Braak Braak and and Etienne Etienne (2003). -
Ter Ter Braak Braak and and Etienne Etienne (2003): (2003): Bayesian Bayesian MCMC MCMC o onn the the Full Full Dataset Dataset While al. (2002) were analysis of While O'Hara O'Hara et et al. were working working on on their their Bayesian Bayesian analysis of also developing turnover data, Ter Braak and Etienne (2003) were turnover data, Ter Braak and Etienne were also developing aa Bayesian Bayesian method. method. This This method method turned turned out out to to generalize generalize the the approach approach by by O'Hara O'Hara et et al. al. (2002) on on two two main main points. points. First, First, Ter Ter Braak Braak and and Etienne Etienne (2003 (2003)) were were able able to to also also exploit exploit the the informa informamissing preyears, tion tion in in the the first first year. year. Their Their idea idea was was to to extend extend data data with with L L missing preyears, with with L L aa large large number, number, and and to to choose choose arbitrary arbitrary fixed fixed states states for for the the year y e a r --L. L. The data, given chosen states The likelihood likelihood of of extended extended data, given the the chosen states in in year y e a r --L, L , is is simply simply aa product with the product of of L L + T T - 1 transition transition probabilities probabilities [compare [compare Eq. Eq. (5.14) (5.14)with the -
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5. 5. STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
BOX 5.2 Simulating Missing Patch Data One at a Time by Gibbs Sampling When some patch states are unknown (Le., with missing data), the expression for the likelihood [Eq. (5 . 1 4)] cannot be calculated, even if P[X(l )] were known, because some transition probabilities of Eq. (5. 1 ) are then unknown. The trick is to fill in missing data from the correct conditional distribution. O'Hara et al. (2002) achieved this by simulat ing each missing patch state in turn, starting from initial guessed states and initial model parameters. After each missing state is simulated once (or more than once), new model parameters are proposed [and accepted with the acceptance probability of Eq. (5.21 )] . With the then current model parameters, each missing state is simulated again. Next, new model parameters are proposed and so on until convergence (i.e., when the dis tribution of the model parameters and simulated states does not change any more). To simulate a single missing patch, we need the probability Pi that patch i at time t is occupied, given all other patch states, denoted by X_i' This probability can be calcu lated by Pi = P[X,(t) = l 1X-;] =
1
".
� f;'
(1 )
with f; the odds ratio (McCullagh and Neider, 1 989) fi =
P[Xi(t) = l IX-i] P[X;(t) = 1 , X_a = , P[Xi(t) = 0ILi] P[Xi(t) = 0, X-I]
(2)
The second equality in Eq. (2) follows from the rule for conditional probability, P[AIB] = P[A,B]/P[B], and by noting that P[X-;] drops out. By applying Eqs. (5.' 4) and (5 .1 5) to the numerator and denominator of Eq. (2) and observing that all terms cancel except those involving years t - 1 , t and t + 1 we obtain f r
P [Xr· (t) -= X(_ =_'_ t)_ -_ , )] P[_ I,X( t-_ i ( t)] ' X_ + ' )-,--, I Xi(_ t --_ --:l _ _ = -:-:�-:-: - : --: P[Xi(t) = 0 IX(t - 1 :-::)] -::)r P[X(t + 1 ) IXi(t) = 0, X- i(t
(3)
In this method, a 1 is filled in for the unknown state with probability Pi and a 0 with probability 1 - Pi' Simulating the missing values in this way is known as Gibbs sampling because we sample from the exact conditional distribution. The acceptance probability then equals 1 . Equation (3) is not cheap to calculate, as the second term in both the numerator and the denominator of fi involves the multiplication of N + , transition probabilities, as is evident from Eq. (5.1 5). If many patch states are missing for a partic ular year, it is computationally more efficient to simulate them jointly using the Metropolis-Hastings algorithm of Box 5.3. If a state is missing in' the last year, the corresponding term can simply be removed from the likelihood because the other states do not depend on it. Equivalently, the miss ing state is simulated as the other missing states but with Pi as defined by Eq. (1 ). O'Hara et al. (2002, personal communication) used Metropolis-Hastings to each missing patch in turn with proposals derived from Eq. (5.1 ), i.e., without conditioning on X(t + 1 ). This is less efficient than Gibbs sampling because Eq. (3) needs to be calculated for the acceptance probability.
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
0 1 220
first many missing missing values first term term dropped]. dropped]. There There are are very very many values in in the the extended extended data, data, but but apart apart from from that, that, there there is is nothing nothing to to prevent prevent aa standard standard Bayesian Bayesian analysis analysis with simulation. The with MCMC MCMC simulation. The validity validity of of choosing choosing arbitrary arbitrary fixed fixed states states for for year the Markov of aa SPOM the probabil year -L -L is is guaranteed guaranteed by by the Markov property property of SPOM that that the probability ity of of the the system system being being in in aa certain certain state state does does not not depend depend on on the the state state of of the the system system in in the the infinite infinite past. past. In In formula, formula, [X( - L)] P[X(1)] = lim l i m PP[X( [ X ( 1l) )] ] iP P[X(-L)] P[X( l )] = <x L---~oc L�
(5.22) (5.22)
X(-L). Thus, it is wise to take L large. As Eq. (5.1 (5.17), for any state X( -L). Thus, 7), Eq. (5.22) requires requires the the assumption assumption of of quasistationarity. quasistationarity. Second, each patch patch in each year Second, instead instead of of sampling sampling missing missing data data for for each in each year separ separately, ately, Ter Ter Braak Braak and and Etienne Etienne (2003) (2003) chose chose to to sample sample missing missing data data for for all all patches patches in (see Box in aa single single year year simultaneously, simultaneously, thus thus only only alternating alternating between between years years (see Box 5.3). 5.3). This This turned turned out out to to be be much much faster faster than than the the method method of of O'Hara O'Hara et et al. al. (2002). (2002).
BOX 5.3 Simulating Missing Patch Data by the Metropolis-Hastings Algorithm In the Metropolis-Hasting algorithm of Ter Braak and Etienne (2003), missing patch data for a particular year t are simulated jointly by proposing values for the missing states (proposals) and accepting the proposals with an acceptance probability r [Eq. (5.2 1 ) with e and X interchanged]. Years are updated in turn. If the metapopulation is believed to have low turnover probabilities (low E; and Cj), it is easy to generate sensible proposals for all missings in year t, given the current states in the year before and after. If the states of a patch i n the years before and after the miss ing value are the same, propose this state with high probability, say 0.99; if the states differ, choose "1 " with probability 1f2• This rule is applied to all missing patches in year t, yielding an N vector of proposed states denoted by X*(t) to distinguish it from the cur rent state. Of course, nonmissing data are not simulated, so that for these patches the proposed state and the current state are identical. The so-generated proposal X*(t) is accepted with acceptance probability r and X(t) is retained if the proposal is not accepted. To calculate the acceptance probability r, we need the ratio of the proposal distributions and the ratio of the likelihoods of x*(t) and X(t) [compare Eq. (5.21 )]. Because proposed patch states are generated independently, the proposal distribu tion Ju is the binomial probability .
Ju[X* (t) IX(t)]
=
N
rrp�i (t)(l ;= 1
•
Pi)(1 -Xi {t)) •
-
(1 )
J
with Pi the probability in the proposal scheme that a 1 is filled in, i.e., Pi is 0.99, 0.01 , or 0.5. The proposal distribution Ju[X(t)IX*(t)] is obtained as in Eq. (1 ) with X(t) and X*(t) interchanged. The ratio of the probability (likelihood) of X*(t) over that of X(t), given the states of all patch states in other years, is
P[X* (t)IX- tl
P[ X *(t),X-d
f = P[X(t)IX-t] = P[X(t),X- d =
P[X* (t)IX(t - 1 )] P[X(t + 1 JIX* (t)] P[X(t)lxCt
-
1 )] P[X(t + 1 ) IX(t)]
(2)
5. 5. STOCHASTIC STOCHASTICPATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
1121 21
and depends only on the states in the neighboring years t - 1 and t + 1 (see Box 5.2). In Eq. (2) X-t denotes the states of all patches in all years except year t. The acceptance ratio r is r =
f
lu_ < t)_ ] )IX [X_ ( t....;. _ _ . *_ 1u[X * (t)I X( t)]
(3)
In the proposal scheme of Ter Braak and Etienne (2003), the {Pi} are not as simple as below Eq. (1 ). Instead, Pi is calculated as in Eq. (1 ) of Box 5.2 but with the odds-ratio f; simplified to f..
tX....,. P[_ (_ )] (t_ _ _ _ '_ ) ] P[ t)_ = _'-,_X i (_ IX -,(,_ t_ +_ )I X i_ = ,_ , X_ __ '· ( t_ )_ '-,....,. ,= P[Xi ( t) = 0IX( t - ' )] P[X,(t + ' )IXi ( t) = O,X-i ( t)]'
(4)
which involves four instead of 2(N + 1 ) transition probabilities and is thus inexpensive to calculate. The proposal X*(t) is not generated with the correct conditional distribu tion, but that does not matter because the theoretical validity of the method hinges on the correct acceptance probability r, which is calculated via Eqs. (1 )-(3). Note that Eq. (1 ) depends on X(t) because Pi depends on X(t) via fi in Eq. (4). It is instructive to see which proposals are generated when Ei and Ci would be con stant over time (Le., when the connectivity of a patch remains the same over time). If X(t - ' ) 0 and X(t + ' ) 0, then fi C;f;/(l - Q2, which will be a small value if Ei and/or Ci is small so that Pi is also close to 0 so that with only a small probability a 1 is proposed in year t. Similar considerations show that if X(t - ' ) 1 and X(t + 1 ) 1 , Pi is close to 1 so that with a large probability a 1 is proposed in year t. If X(t - 1 ) *' X(t + 1 ) 0, then f; (1 - Ei)/(1 - Q so that if Ci and E; are equal or both small, Pi is approx imately one-half. The proposal mechanism is thus very similar to the one begun in this Box. It has the advantage of yielding good proposals for more variable metapopulations and does not require a tuning parameter such as the value 0.99 in our initial scheme. Software to implement the full Bayesian analysis is available in the archives of Ecology: http://www.esapubs.org/archive/ecol/E084/005. =
=
=
=
=
5.4 5.4
=
=
REMAINING REMAINING PROBLEMS PROBLEMS FOR FOR PREDICTIONS: PREDICTIONS: A A CASE CASE STUDY STUDY (2003)) uses the spatial information information The method of Ter Braak and Etienne (2003 of of the the first first year year of of aa dataset, dataset, reflecting reflecting the the history history of of the the metapopulation, metapopulation, as as well well as as the the turnover turnover information information in in the the following following years, years, it it can can handle handle missing missing data, data, and and it it provides provides aa joint joint probability probability distribution distribution of of the the model model parameters. parameters. Because Because missing missing data data are are treated treated as as parameters parameters that that need need to to be be estimated, estimated, aa probability points is probability distribution distribution of of the the occupancy occupancy at at the the missing missing data data points is also also provided. provided. This This is is very very convenient convenient if if one one wants wants to to make make predictions, predictions, as as shown shown later. Bayesian approach later. Furthermore, Furthermore, the the Bayesian approach allows allows for for many many extensions extensions of of the the dataset ((Moilanen, method, such as misclassifications of occupancies in the dataset Moilanen, 2002) and and temporal temporal stochasticity (regional stochasticity in the terminology of of 2002) 991). We Hanski, Hanski, 11991). We refer refer to to the the discussion discussion for for their their possible possible consequences consequences (see (see also also Ter Ter Braak Braak and and Etienne, Etienne, 2003 2003).). However, However, in in addition addition to to technical technical difficul difficulties ties of of the the MCMC MCMC approach approach [When [When has has the the MCMC MCMC converged? converged? How How many many
1122 22
RAM PAL S. S. ETIENNE RAMPAL ETIENNE ET ET AL. AL.
preyears added? Is perhaps an preyears (L) (L) should should be be added? Is there there perhaps an even even more more efficient efficient algor algorithm ithm than than the the one one presented presented by by Ter Ter Braak Braak and and Etienne Etienne (2003)?], (2003)?], there there are are (at (at least) least) three three fundamental fundamental difficulties difficulties that that need need to to be be considered considered before before cali calibrated SPOMs can brated SPOMs can be be used used for for prediction. prediction. First, First, Ter Ter Braak Braak et et al. al. (2003) (2003) invoked invoked the the assumption assumption of of quasistationarity quasistationarity to be able use occupancy The question to be able to to use occupancy datasets datasets to to their their fullest fullest extent. extent. The question is is whether this this assumption is warranted warranted and and what what impact impact it it has. even whether assumption is has. Second, Second, even if model parameters are precisely precisely known, known, predictions predictions of if the the model parameters are of occupancy, occupancy, turnover, will be turnover, or or the the time time to to extinction extinction will be uncertain uncertain due due to to the the inherent inherent sto stochasticity of of the model (termed (termed extinction-colonization chasticity the model extinction-colonization stochasticity stochasticity in in Hanski, 99 1 ) . If contribution of inherent stochasticity stochasticity to Hanski, 11991). If the the contribution of the the inherent to the the total total uncertainty predictions is more than uncertainty in in model model predictions is large, large, data data collection collection for for more than aa few few years be rather is possible years may may be rather fruitless. fruitless. Third, Third, in in making making predictions predictions it it is possible to to take into account take into account any any planned planned changes changes in in network network structure structure (habitat (habitat creation, creation, barrier barrier removal). removal). Indeed, Indeed, comparing comparing the the outcome outcome of of several several scenarios scenarios is is often often the main purpose al. 2002a). the main purpose (e.g., (e.g., Wahlberg Wahlberg et et al. 2002a). However, However, unpredictable unpredictable changes plans) may changes in in the the landscape landscape (habitat (habitat turnover, turnover, unknown unknown management management plans) may make reliable predictions make reliable predictions rather rather difficult. difficult. These These problems problems are are discussed discussed in in order order using using the the tree tree frog frog SPOM SPOM mentioned mentioned earlier earlier on on the the tree tree frog frog dataset dataset and and on on aa simulated simulated dataset dataset for for the the same same network network with with the the same same pattern pattern of of miss missing ing values values as as the the real real dataset. dataset.
The Quasistationarity Quasistationarity Assumption quasistationarity allowed allowed Hanski Hanski ((1994) to estimate estimate model model 1 994) to Assuming quasistationarity parameters parameters from from aa single single year year of of occupancy occupancy data data only, only, and and it it allowed allowed Ter Ter Braak Braak et 1998) and et al. al. ((1998) and Ter Ter Braak Braak and and Etienne Etienne (2003 (2003)) to to extract extract information information in in aa dataset about the dataset about the history history of of the the metapopulation, metapopulation, as as reflected reflected by by the the first first year year of of aa dataset. dataset. However, However, how how do do we we know know that that the the metapopulation metapopulation is is in in the the quasi quasistationary not? To stationary state state or or not? To answer answer this this question, question, we we need need to to know know more more about about the metapopulation (Moilanen, the history history of of the the metapopulation (Moilanen, 2000), 2000), or or we we can can attempt attempt to to find find the answer using dataset itself. itself. Assuming Assuming that information about past the answer using the the dataset that information about the the past history is history is usually usually unavailable, unavailable, it it seems seems worthwhile worthwhile to to explore explore the the dataset dataset for for information about the information about the presence presence or or absence absence of of quasistationarity. quasistationarity. If If we we can can find find such also avoid avoid the such information, information, we we also the objection objection that that assuming assuming quasistationarity quasistationarity makes predictions worthless because it makes predictions worthless because it is is aa self-fulfilling self-fulfilling prophecy. prophecy. We start by well as the We start by comparing comparing parameter parameter estimates estimates as as well as predictions predictions for for the complete dataset dataset assuming assuming quasistationarity quasistationarity (which (which is is denoted denoted by by QS6), QS6), the complete the first year 1 ) , and and the first year only only dataset dataset assuming assuming quasistationarity quasistationarity (QS (QS1), the turnover turnover dataset Figures 5.1 cumulative posterior posterior distributions distributions for dataset (TO). (TO). Figures 5.1 and and 5.2 5.2 show show cumulative for the tree frog Figure 5.1 shows that the tree frog SPOM SPOM and and dataset. dataset. Figure 5.1 shows that posterior posterior distributions distributions of the the model model parameters for QS1 QS1 are are wider wider than than those those for for QS6 and TO, TO, which which of parameters for QS6 and are but note )' This does not necessar are quite quite alike alike ((but note the the differences differences for for x x and and Q q2). This does not necessar2 ily ily mean mean that that the the predictions predictions must must also also be be different different because because the the model model param parameters eters may may be be highly highly correlated. correlated. Indeed, Indeed, Fig. Fig. 5.2A 5.2A shows shows that that the the variance variance in in distributions QS11 and distributions of of the the occupancy occupancy after after 100 100 years years is is similar similar for for QS and TO, TO, indi indicating about the cating that that there there is is nearly nearly as as much much information information about the occupancy occupancy in in QS1 QS1 as as in in TO. TO. Combining Combining them them in in QS6 QS6 gives gives aa slightly slightly narrower narrower distribution distribution (i.e., (i.e., steeper .2A. steeper cumulative cumulative distribution) distribution),, as as shown shown in in Fig. Fig. 55.2A.
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The The variances variances may may be be similar, similar, the the medians medians are are different, different, with with QS6 QS6 (0.129) (0.129) being 1 ) and (0.158). The being intermediate intermediate between between QS1 QS1 (0.094 (0.0941) and TO TO (0.158). The occupancies occupancies in in the 1 9 8 1-1983, 1986) 1 1, 0.1 1 1, 0.141, the 4 4 data data years years ((1981-1983, 1986) are are 0.1 0.111, 0.111, 0.141, and and 0.147, 0.147, so so TO TO seems seems to to reflect reflect this this upward upward trend. trend. This This is is not not accidental. accidental. Moilanen Moilanen (2000) (2000) ascertained ascertained that that parameter parameter estimation estimation based based on on aa few few years years of of simulated simulated A A
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turnover data data often often leads to the the prediction of aa substantially substantially higher higher or or lower lower turnover leads to prediction of occupancy, SPOM was occupancy, even even if if the the simulated simulated SPOM was in in the the quasistationary quasistationary state. state. It It remains to whether this change in remains to be be seen seen whether this change in occupancy occupancy is is still still noteworthy noteworthy when when the the full full posterior posterior distribution distribution is is known. known. To To illustrate illustrate this this change change in in occupancy, occupancy, we we simulated simulated the the tree tree frog frog system system using median values QS6 posteriors . 1 for 000 time using the the median values of of the the QS6 posteriors in in Fig. Fig. 55.1 for 11000 time steps steps to quasistationary state additional time to reach reach the the quasistationary state and and for for 66 additional time steps steps to to generate generate aa dataset similar to real dataset dataset similar to the the real dataset (see (see Fig. Fig. 5.3). 5.3). We We then then estimated estimated the the model model parameters TO. Figures parameters for for QS6, QS6, QS1, QS1, and and TO. Figures 5.4 5.4 and and 5.5 5.5 show show posterior posterior distri distributions corresponding predictions butions for for the the model model parameters parameters and and the the corresponding predictions of of patch patch occupancy similar to . 1 and occupancy and and turnover. turnover. These These figures figures are are similar to Figs. Figs. 55.1 and 5.2, 5.2, respec respectively, places in tively, except except that that TO TO and and QSl QS1 have have traded traded places in Fig. Fig. 5.5; 5.5; medians medians are are 0.158 1 ) , 0.139 QS6), and 0.158 (QS (QS1), 0.139 ((QS6), and 0.124 0.124 (TO). (TO). Again, Again, TO TO reflects reflects the the trend trend in in data, . 1 53, 0.1 33, 0.129). data, which which are are downward downward in in this this case case (0.158, (0.158, 00.153, 0.133, 0.129). Nevertheless, Nevertheless, because because the the posteriors posteriors of of the the occupancies occupancies for for QSl QS1 and and TO TO largely largely overlap, overlap, the the assumption assumption of of quasistationarity quasistationarity is is not not refuted refuted by by the the dataset. dataset. Therefore, TO, appears Therefore, the the use use of of QS6, QS6, which which contains contains more more data data than than TO, appears to to be be warranted. warranted. This This warrant warrant becomes becomes even even stronger stronger when when we we calculate calculate the the median median occupancy 00 iterations (Fig. 5.3), occupancy in in the the simulations simulations after after 1100 iterations (Fig. 5.3), which which is is 0.139, 0.139, precisely the the median median value value for for QS6. QS6. Obviously, Obviously, for for the the real real dataset dataset we we cannot cannot precisely perform perform this this check, check, but but there there too too we we find find aa large large overlap overlap of of the the posteriors posteriors for for QSl the assumption of quasi stationarity cannot QS1 and and TO, TO, so so the assumption of quasistationarity cannot be be refuted, refuted, and and using using QS6 QS6 seems seems the the best best choice. choice. If If the the posteriors posteriors have have little little overlap, overlap, we we can can interpret population may interpret this this as as aa sign sign that that the the meta metapopulation may not not be be in in the the quasistation quasistationary ary state, state, and and we we should should perhaps perhaps refrain refrain from from using using QS6 QS6 and and use use TO TO instead. instead. Until Until now, now, we we have have only only looked looked at at the the predictions predictions of of the the occupancy occupancy after after 100 100 years. years. Predictions Predictions of of the the turnover turnover in in 100 100 years, years, pictured pictured in in Figs. Figs. 5.2B 5.2B and and
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S.SB, QSl contains 5.5B, show show that that QS1 contains little little information information about about turnover. turnover. Common Common sense sense tells us that there there is is no turnover in snapshot data, tells us that no real real information information about about turnover in snapshot data, i.e., i.e., in QS1, QS1, and and that that any any found an artifact of the the model. In in found information information must must be be an artifact of model. In preliminary posteriors for for QSl preliminary MCMC M C M C simulations, simulations, we we sometimes sometimes found found posteriors QS1 con coninformation about turnover. It It turned out that that this this disappeared disappeared taining some some information about turnover. turned out taining when increased the the value value of when we we carried carried out out more more iterations iterations or or increased of L. L. Hence, Hence, if if contain information information about turnover, this this should be the posteriors posteriors for for QSl QS1 contain about turnover, should be the B B
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RAMPAL S. ETIENNE RAMPAL S. ETIENNE ET ET AL.
interpreted interpreted as as aa sign sign that that the the MCMC MCMC simulations simulations have have not not yet yet converged converged or or that that L L is is not not taken taken large large enough. enough. As As already already mentioned mentioned earlier, earlier, differences differences in in the the posteriors posteriors of of the the model model parameters parameters do do not not necessarily necessarily entail entail differences differences in in model model predictions predictions because because parameters parameters may may be be highly highly correlated. correlated. We We remarked remarked that that in in the the IFM IFM with with res rescue cue effect, effect, parameters parameters ee and and y y cannot cannot be be distinguished distinguished by by snapshot snapshot data; data; they they appear appear as as aa product product in in Eq. Eq. (5.6). (5.6). Although Although this this model model is is mathematically mathematically not not completely completely exact, exact, this this high high correlation correlation between between ee and and y y is is still still to to be be expected expected for Because the al. (2000) contains contains additional for QS1 QS1.. Because the extended extended IFM IFM of of Vos Vos et et al. additional , we need to correct for these parameters to observe this parameters parameters q q ll and and qq2, we need to correct for these parameters to observe this 2 correlation. correlation. The The appropriate appropriate transformation, transformation, relating relating Eq. Eq. (5.7) (5.7) to to Eq. Eq. (5.2) (5.2) and and Eq. Eq. (5.8) (5.8) to to Eq. Eq. (5.3), (5.3), is is ' (ql )log H log log ee' = = log log ee + + (ql - Ql ~l)log H1l ' q2 log log yy' = = log log yy + + ((q2 - Q2)log/4 )log H2
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Inherent Inherent Stochasticity Stochasticity Evidently, more data reflected in in Evidently, the the more data available, available, the the better better the the predictions, predictions, as as reflected steeper and 5.5). These cumulative steeper cumulative cumulative posteriors posteriors (Figs. (Figs. 5.2 5.2 and 5.5). These cumulative posteriors posteriors will, however, however, never never be be vertical vertical regardless regardless of of the the amount amount of of data data available available will, because because the the model model is is stochastic. stochastic. This This inherent inherent stochasticity stochasticity may may contribute contribute much much more more to to the the total total uncertainty uncertainty than than the the uncertainty uncertainty in in the the model model param parameters, useless. Hence, eters, which which would would make make further further data data collection collection rather rather useless. Hence, the the question arises how needed. Figure question arises how much much data data are are needed. Figure 5.7 5.7 shows shows predictions predictions for for occupancy occupancy and and turnover turnover for for the the simulated simulated tree tree frog frog system system when when the the model model
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parameters parameters are are known known exactly exactly compared compared to to the the predictions predictions based based on on QS6, QS6, shown shown earlier in Fig. 5.2 (real dataset) dataset) and and Fig. 5.5 (simulated (simulated dataset). As far as gained if more data as occupancy occupancy is is concerned, concerned, not not much much would would be be gained if more data became became available. However, available. However, more more data data would would convey convey more more information information about about turnover. population is turnover. This This is is important important if if the the meta metapopulation is at at risk risk of of extinction. extinction. A A metapopulation turnover will will go than aa metapopulation with with high high turnover go extinct extinct much much faster faster than meta population with low turnover and the same occupancy metapopulation occupancy (i.e., the same basic population capacity; basic reproduction reproduction number number Ro R0 or or meta metapopulation capacity; Ovaskainen Ovaskainen and and Hanski, 1 ) . Hence, 5.7B suggests that Hanski, 200 2001). Hence, Fig. 5.7B that data data overestimated overestimated the turnover turnover and population extinction and thus thus underestimated underestimated the the meta metapopulation extinction time, time, that that is, is, the the true true time parameterized with time to to metapopulation metapopulation extinction extinction is is larger larger than than the the model model parameterized with available available data data predicts. predicts.
Changes in Environment C h a n g e s in Environment The aforementioned aforementioned model model predictions predictions all assumed assumed that that the system e.g., using deci remained unaltered. To weigh several management management options options ((e.g., decision theory, Possingham, 11996, 996, 11997; 997; Possingham Possingham et aI., al., 2001 2001),), one needs to be be able able to to change change system system properties properties by, by, for for example, example, introducing introducing patches, patches, increasing increasing patch patch quality, or reducing reducing the effective interpatch interpatch distance distance by build building corridors. corridors. Because these changes do not not affect the model model parameters, parameters, their their effect effect can can be be studied studied easily easily with with model model simulations. simulations. Problems Problems enter, enter, however, however, when when unknown unknown changes changes need need to to be be dealt dealt with. with. In In our our tree frog frog example, example, a large proportion proportion of the ponds ponds have disappeared disappeared and and been been created created since since 1986 1986 because because of of changes changes in in land land use use and and succession. succession. Precise Precise data data are are unavailable. unavailable. We We only only know know that that in in the the year year 2002, 2002, 23 23 patches patches are are occupied 9 8 1 , 11982, 982, occupied (in (in comparison, comparison, the the number number of of occupied occupied patches patches in in 11981, 1983, 986 is 10 patches 1983, and and 11986 is 22, 22, 22, 22, 28, 28, and and 29, 29, respectively) respectively) of of which which just just 10 patches existed (and 88 occupied) 9 8 1-1986. So 13 new existed (and occupied) in in 11981-1986. So there there are are at at least least 13 new patches, patches, but but perhaps perhaps many many more more that that are are empty. empty. This This makes makes it it impossible impossible to to compare compare
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the occupancy found the predictions predictions of of the the model model and and the the occupancy found in in 2002 2002 quantitatively, quantitatively, let alone 98 1-1986 data to cali alone that data for 2002 2002 can be combined combined with 11981-1986 calibrate brate the the model model even even better. better. Qualitative Qualitative comparison comparison is, is, however, however, still still possible. possible. 5.8A shows shows the spatial occupancy occupancy pattern pattern in 11986 location of Figure 5.8A 986 and the location the the 23 23 occupied occupied patches patches in in 2002. 2002. In 11986, 986, eight pond population pond clusters were occupied. occupied. In 2002 2002 the meta metapopulation retreated to only three of these clusters (clusters I, II, and III). III). It is apparent that the weakest weakest clusters have become become extinct extinct in 2002. 2002. These are the relatively small and isolated VIII) isolated clusters (clusters V, VI VI,, VII, and the one-pond one-pond cluster VIII) and a cluster with occupied pond in 1986 with only one occupied 1986 (cluster IV) IV).. Let us exam examine ine whether whether the the model model is is able able to to predict predict this this spatial spatial pattern. pattern. Figure Figure 5.8B 5.8B shows shows the patches being the probability probability of of patches being occupied occupied in in 2002 2002 based based on on simulations simulations start starting with the pattern in 1986 1986 and using posteriors posteriors obtained obtained from the real dataset assuming QS6) . The eight clusters assuming quasistationarity quasistationarity ((QS6). clusters (except (except for clus cluster ter VIII) VIII) are are the the only only parts parts of of the the metapopulation metapopulation where where ponds ponds with with aa prob probability of being occupied occupied of more than 0.25 0.25 occur, which which is consistent consistent with with the 2002 data. 2002 data. Clusters Clusters II and and III, III, which which remained remained occupied occupied in in 2002, 2002, are are the the only only clusters clusters that that contain contain ponds ponds with with aa probability probability of of more more than than 0.50 0.50 or or even even 0.75 0.75 of of being occupied. occupied. Thus the model is able to locate the parts of of the metapopu metapopulation with the highest survival potential. The fact that clusters V, VI, VII, and
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986 and 2002: Fig. 5.8 5.8 Occupancy patterns for the tree frog system. (A) Comparison of 11986 2002: patches empty in 11986 986 (gray dots), patches occupied 986 (black dots), patches occupied occupied in 11986 occupied 2002 (black asterisks), asterisks), patches empty in 11986 occupied in 2002 2002 (gray aster asterin 11986 986 and 2002 986 but occupied isks), and patches occupied in 2002 (B) Probability PP isks), 2002 that did not exist in 11986 986 (open circles). (8) of occupancy in 2002 2002 as predicted 986 and using predicted from simulation simulation starting starting in 11986 using the model parameters of QS6: 0.5 (black dots), 0.5 Q S 6 : 0O :s -< P P< < 0.25 0.25 (gray dots), 0.25 0.25 :s --- P P< < 0.5 0.5 :s - P P< < 0.75 0.75 (open squares), 75 :s squares), and 0. 0.75 -< P P :s -< 11 (open circle).
5. 5. STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
11 29 29
VIII model predictions, VIII have have become become extinct extinct is is in in accordance accordance with with the the model predictions, as as in in these clusters clusters only ponds with than 0.25 these only one one or or two two ponds with aa probability probability of of more more than 0.25 were were present present (the (the probability probability of of occupancy occupancy of of the the pond pond in in cluster cluster VIII VIII is is even even smaller smaller than than 0.25) 0.25).. However, However, in in contrast contrast to to data, data, the the model model does does not not neces necessarily sarily predict predict the the retreat retreat of of the the tree tree frog frog from from cluster cluster IV, IV, as as four four ponds ponds in in this this cluster probabilities larger cluster have have occupancy occupancy probabilities larger than than 0.25. 0.25. At At least least one one occupied occupied pond pond would would have have been been expected. expected. It It is is apparent, apparent, however, however, that that cluster cluster IV IV shows shows the recolonization, as the best best potential potential for for recolonization, as the the cluster cluster is is relatively relatively large large and and has has many ponds with many ponds with aa high high probability probability of of occupancy. occupancy. To To increase increase colonization colonization probability, probability, the the spatial spatial cohesion cohesion with with cluster cluster II should should be be improved. improved. This This is is in in accordance Crombaghs and accordance with with the the actual actual protection protection policy policy in in the the region region ((Crombaghs and Lenders, 1 ). Thus, Lenders, 200 2001). Thus, the the model model is is aa useful useful tool tool to to determine determine the the optimal optimal spa spatial tial strategy strategy to to conserve conserve the the tree tree frog frog metapopulation. metapopulation.
5.5 5.5
DISCUSSION DISCUSSION SPOMs SPOMs were were introduced introduced as as metapopulation metapopulation models models that that combine combine simplicity simplicity with with sufficient sufficient realism realism (particularly (particularly pertaining pertaining to to spatial spatial structure), structure), such such that that they populations and they are are ideal ideal for for assessing assessing the the current current condition condition of of real real meta metapopulations and for for predicting predicting the the effect effect that that human human intervention intervention has has on on metapopulation metapopulation per persistence. sistence. Hence, Hence, it it is is interesting interesting to to see see what what the the literature literature teaches teaches us us about about the the four four parts, parts, mentioned mentioned earlier, earlier, of of applying applying SPOMs SPOMs to to real real metapopulations, metapopulations, that population data that is, is, to to see see what what meta metapopulation data have have been been used, used, what what SPOMs SPOMs have have been been developed, developed, what what parameterization parameterization method method has has been been used, used, and and whether whether the SPOMs were the SPOMs were useful useful and and successful successful in in their their predictions. predictions. Perhaps metapopulation is Perhaps the the best-documented best-documented metapopulation is the the Glanville Glanville fritillary fritillary (Melitaea land Islands Islands in (Melitaea cinxia) cinxia) metapopulation metapopulation on on the the A ~lland in southwest southwest Finland Finland (Hanski, 994; Hanski aI., 11996; 996; Moilanen 998). There (Hanski, 11994; Hanski et et al., Moilanen and and Hanski, Hanski, 11998). There are are some 4000 habitat 0some 4000 habitat patches, patches, forming forming several several subnetworks subnetworks of of which which one one 550patch network has been been studied has been been modeled the patch network has studied in in more more detail. detail. It It has modeled with with the IFM method of 1 994) . Because IFM and and parameterized parameterized with with the the incidence incidence method of Hanski Hanski ((1994). Because the the model model describes describes data data very very well, well, the the two two variables variables of of the the IFM, IFM, area area and and interpatch interpatch distance, distance, have have been been declared declared as as the the main main explanatory explanatory factors. factors. Other Other factors (Moilanen and 99 8 ) . factors do do not not appear appear to to play play aa significant significant role role (Moilanen and Hanski Hanski 11998). Eber 1996), studying Eber and and Brandl Brandl ((1996), studying the the tephritid tephritid fly fly Urophora Urophora cardui cardui iinn north northeastern eastern Bavaria, Bavaria, came came to to the the same same conclusion conclusion based based on on their their logistic logistic regression regression on turnover turnover data, data, although although distance distance did did not not matter matter much much because because distances distances on between between patches patches were were smaller smaller than than the the average average dispersal dispersal distance. distance. However, However, several authors authors have true. For several have suggested suggested that that it it is is not not generally generally true. For example, example, Sjogren-Gulve 1 996), applying Sj6gren-Gulve and and Ray Ray ((1996), applying their their logistic logistic regression regression method method to to aa dataset of pool frog dataset of aa pool frog (Rana (Rana lessonae) lessonae) metapopulation metapopulation on on the the Baltic Baltic coast coast of of east central central Sweden, Sweden, considered east considered additional additional variables variables (mean (mean water water temperature, temperature, presence presence of of forestry) forestry) that that were were necessary necessary to to obtain obtain aa good good fit. fit. Fleishman Fleishman et et al. al. (2002), (2002), applying applying logistic logistic regression regression to to an an Appache Appache silverspot silverspot butterfly butterfly (Speyeria nokomis nokomis apacheana) apacheana) metapopulation metapopulation in in the the Toiyable Toiyable Range, Range, (Speyeria Nevada, Nevada, even even found found that that patch patch area area and and isolation isolation are are not not important important at at all. all. Other Other factors factors that that are are better better indicators indicators of of habitat habitat quality quality than than area area had had much much more 1 999) used more explanatory explanatory power. power. Lindenmayer Lindenmayer et et al. al. ((1999) used the the IFM IFM to to model model
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meta populations of South metapopulations of four four marsupials marsupials in in Buccleuch Buccleuch State State Forest, Forest, New New South Wales, common brushtail moun Wales, Australia: Australia: common brushtail possum possum (Trichosurus (Trichosurus vulpecula), vulpecula), mountain Trichosurus caninus), tain brushtail brushtail possum possum ((Trichosurus caninus), common common ringtail ringtail possum possum (Pseudocheirus (Pseudocheirus peregrinus), peregrinus), and and greater greater glider glider (Petauroides (Petauroides volans). volans). They They 1 994) incidence method. For parameterized the parameterized the model model with with Hanski's Hanski's ((1994) incidence method. For two two of of these these four four species, species, the the parameterization parameterization produced produced unreliable unreliable results, results, which which 1 999) blamed blamed on absence of Lindenmayer et al. al. ((1999) on the the absence of patch patch quality quality in in the the Lindenmayer et model. These These case case studies demonstrate that model. studies demonstrate that the the important important variables variables to to be be put put in the the model model should should be be determined on aa case by case basis, using expert eco ecoin determined on case by case basis, using expert logical knowledge logical knowledge and and statistical statistical techniques. techniques. While While in in these these examples examples model model structure structure is is altered altered by by additional additional variables, variables, processes. Several structure can can also also be be affected affected by by different different processes. Several authors authors model structure report interesting structure in report interesting consequences of their data analysis for model structure this al. (200 1 ) applied maximization on this respect. respect. Crone Crone et et al. (2001) applied likelihood likelihood maximization on snapshot snapshot data metapopulation on 71 islands data of of aa field field vole vole (Microtus (Microtus agrestis) agrestis) metapopulation on 71 islands in in the the Tvarminne archipelago concluded that Tvfirminne archipelago in in the the Gulf Gulf of of Finland Finland and and concluded that colonization colonization and and extinction extinction are are highly highly correlated: correlated: when when aa population population is is on on the the brink brink of of extinction, extinction, the the voles voles disperse disperse en en masse masse to to other other islands. islands. Although Although the the source source population then population then goes goes extinct, extinct, other other populations populations can can benefit benefit from from immigration, immigration, the 977). Peltonen Hanski the rescue rescue effect effect (Brown (Brown and and Kodric-Brown, Kodric-Brown, 11977). Peltonen and and Hanski ((1991) 1991 ) compared species of compared three three species of shrews shrews (Sorex (Sorex araneus, araneus, Sorex Sorex caecutiens, caecutiens, and and Sorex Sorex minutus) minutus) on on islands islands in in three three Finish Finish lakes lakes and and on on the the mainland. mainland. They They found, 1 994) approach, found, using using aa predecessor predecessor of of the the Hanski Hanski ((1994) approach, that that the the coloniza colonization tion rates rates of of these these species species were were identical, identical, but but with with different different contributions contributions from from the of settlement. the probability probability of of arrival arrival and and the the probability probability of settlement. Hence, Hence, the the simi similarity larity of of the the colonization colonization rates rates is is perhaps perhaps coincidental, coincidental, in in which which case case these these dif different should really really be ferent contributions contributions should be modeled modeled explicitly. explicitly. Wahlberg Wahlberg et et al. al. (2002), who 1 999) Monte who employed employed Moilanen's Moilanen's ((1999) Monte Carlo Carlo simulation simulation method method on on snapshot 14 snapshot data data of of the the marsh marsh fritillary fritillary butterfly butterfly (Euphydryas (Euphydryas aurinia) aurinia) in in 1114 patches in Joutseno region region of southern Finland, made yet another import patches in the the Joutseno of southern Finland, made yet another important discovery: if landscape is ant discovery: if the the landscape is very very dynamic dynamic itself itself (patches (patches appear appear and and dis disappear) represent this. this. If dynamics are appear),, the the model model must must represent If patch patch dynamics are due due to to succession succession or or due due to to metapopulation metapopulation behavior behavior of of the the patches patches themselves themselves [e.g., [e.g., if if the the patches patches are are host host plants plants to to aa moth, moth, (Nieminen, (Nieminen, 1996) or or host host butterflies butterflies modeled in to (Lei and to aa parasitoid parasitoid (Lei and Hanski, Hanski, 1997)], this this can can be be modeled in aa fairly fairly straightforward case study, straightforward way, way, but but if, if, as as in in our our case study, patches patches disappear disappear because because of of human activities, any human activities, any model model may may be be as as good good as as any any other. other. 1 999) raised about model Moilanen Moilanen ((1999) raised perhaps perhaps the the most most important important point point about model structure. (regional stochasticity structure. Temporal Temporal stochasticity stochasticity (regional stochasticity in in Hanski, Hanski, 1991) in in model i.e., for model parameters, parameters, i.e., for each each transition transition between between states states there there are are different different values for is values for the the model model parameters, parameters, may may determine determine whether whether aa metapopulation metapopulation is expected to extinct or expected to go go extinct or not, not, so so it it is is crucial crucial whether whether it it should should be be incorporated incorporated in model. It 1 999) did in the the model. It is is relatively relatively straightforward straightforward to to do do so: so: Moilanen Moilanen ((1999) did this this by by multiplying multiplying patch patch area area by by aa normally normally distributed distributed factor factor with with variance variance (J cr22,, but putting aa certain variance (J� but it it may may also also be be done done by by putting certain white white noise noise e~ with with variance (r2 on on the the model model parameters. parameters. The The latter latter is is advocated advocated by by Carlsson Carlsson and and Kindvall Kindvall (2001 who used used Moilanen's 1 999) approach the grasshopper (2001),), who Moilanen's ((1999) approach on on the grasshopper Stauroderus Stauroderus scalaris scalaris in in 158 patches patches on on Oland C)land Island, Island, southeast southeast Sweden, Sweden, and and concluded concluded that that it it is is not not mechanistic mechanistic enough enough to to generate generate the the patterns patterns they they
5. 5.
STOCHASTIC STOCHASTIC PATCH PATCH OCCUPANCY OCCUPANCY MODELS MODELS
1131 31
found. It also straightforward straightforward to found. It is is also to estimate estimate the the corresponding corresponding parameters parameters (0" approaches. However, (or or or O"E (r~)) in in the the Monte Monte Carlo Carlo simulation simulation and and Bayesian Bayesian approaches. However, the given the Many years the estimates estimates may may not not be be very very accurate, accurate, given the available available data. data. Many years of of data data are are needed needed to to provide provide information information on on the the variability variability of of the the model model parameters. parameters. Because Because the the transitions transitions between between years years are are then then treated treated relatively relatively independently, independently, each each transition transition is is likely likely to to yield yield aa different different trend trend (Moilanen, (Moilanen, 2000), 2000), potentially potentially leading leading to to an an overestimate overestimate of of temporal temporal stochasticity. stochasticity. This This bias bias may may again again be be reduced reduced by by the the assumption assumption of of quasistationarity. quasistationarity. Now Now we we are are back apply? back at at the the problem problem mentioned mentioned in in the the case case study: study: does does this this assumption assumption apply? Because Because of of temporal temporal stochasticity, stochasticity, it it may may no no longer longer be be able able to to tell tell from from data data whether whether quasistationarity quasistationarity applies applies or or not, not, and and perhaps perhaps we we have have no no other other choice choice than than to to resort resort to to independent independent information information to to decide decide about about quasistationarity quasistationarity (Moilanen, (Moilanen, 2000) 2000).. Further Further study study is is evidently evidently needed. needed. Things Things may may even even get get worse worse when when data data themselves themselves contain contain errors. errors. Moilanen Moilanen (2002) (2002) mentioned mentioned three three errors errors that that can can be be made made in in data data collection. collection. First, First, patch patch areas areas can can be be estimated estimated wrongly. wrongly. Second, Second, unknown unknown habitat habitat patches patches may may be be located located within within or or around around the the study study area. area. Third, Third, occupancy occupancy observations observations may may be be false: false: truly truly occupied occupied patches patches may may have have been been observed observed as as being being empty empty (false (false zeros). zeros). The The other other possibility possibility is is that that truly truly empty empty patches patches are are observed observed as as occu occupied pied (false (false ones). ones). This This may may happen happen if if observation observation of of aa few few individuals individuals is is inter interpreted preted as as an an entire entire population. population. Moilanen Moilanen (2002) (2002) argued argued that that wrongly wrongly estimated seriously affect estimated patch patch areas areas can can seriously affect extinction extinction risk, risk, whereas whereas missing missing patches patches cause cause an an overestimation overestimation of of migration migration distances distances and and colonization colonization abil ability ity of of the the species. species. False False zeros zeros can can have have aa severe severe effect effect on on both both extinction extinction and and colonization colonization components, components, and and hence hence on on metapopulation metapopulation persistence. persistence. Like Like tem temporal poral stochasticity, stochasticity, such such data data errors errors can can be be dealt dealt with with in in the the Monte Monte Carlo Carlo simu simulation lation and and Bayesian Bayesian parameterization parameterization methods. methods. Moilanen Moilanen (2002) (2002) gives gives explicit explicit formulae formulae for for the the case case of of false false zeros, zeros, but but again, again, the the parameter parameter estimates estimates will will lose lose some some reliability. reliability. The The great great advantage advantage of of the the Bayesian Bayesian approach approach is is that that this this loss loss of of reliability reliability will will show show up up in in posterior posterior distributions distributions of of the the parameters parameters and and it it can can be be translated translated immediately immediately into into uncertainties uncertainties about about the the predictions. predictions. However, However, no no ready-made ready-made computer computer programs programs exist exist as as yet yet for for Bayesian Bayesian analy analysis sis of of these these extended extended models. models. Considering Considering all all these these allegations allegations at at the the information information richness richness of of data, data, we we have have to to face face the the important important question question of of how how much much data data (how (how many many patches patches and and how 1 99 8 ) how many many years) years) are are needed needed for for reliable reliable predictions. predictions. Ter Ter Braak Braak et et al. al. ((1998) showed showed bbyy simulation simulation that that the the curves curves for for extinction extinction and and colonization colonization [Eqs. [Eqs. (5.2) (5.2) and and (5.3)] (5.3)] are are discomfortingly discomfortingly variable variable when when estimated estimated from from 2 2 years years of of data 1 994) . data with with 50 50 patches, patches, thus thus mimicking mimicking data data and and analysis analysis of of Hanski Hanski ((1994). Tyre 0 1 ) conducted Tyre eett al. al. (20 (2001) conducted maximum maximum likelihood likelihood regression regression oonn snapshot snapshot data data of of the the hydrobiid hydrobiid snail snail Fonscochlea Fonscochlea zeidleri zeidleri in in only only 99 patches patches in in Bopeechee Bopeechee Springs, Springs, Australia, Australia, with with only only 33 patches patches occupied. occupied. Their Their objective objective was was to to com compare pare parameter parameter estimates estimates for for two two sets sets of of snapshot snapshot data: data: before before and and after after (trampling by impact (trampling by stock stock and and water water extraction) extraction).. They They con conhuman-driven impact cluded cluded that that the the method method has has too too little little power power to to detect detect this this impact. impact. The The reason reason they they mentioned mentioned is is that that the the number number of of patches patches is is too too small. small. However, However, the the rea reason son may may also also be be that that the the number number of of years years is is too too small. small. Thomas Thomas et et al. al. (2002a) (2002a) applied applied maximum maximum likelihood likelihood regression regression on on snapshot snapshot data data of of the the silver-studded silver-studded 33 patches in blue butterfly blue butterfly (Plebejus (Plebejus argus) argus) in in 33 patches on on the the Creuddyn Creuddyn Peninsula Peninsula in
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RAMPAL RAMPAL S. S. ETIENNE ETIENNE ET ET AL. AL.
north Wales. Wales. They They considered considered four four snapshots snapshots spread spread fairly fairly evenly evenly over over aa north period of 27 years and compared results obtained from one, two, three, or studfour snapshots. Their message was that predictions based on short-term stud ies may may considerably considerably underestimate underestimate turnover turnover and and caution caution must must therefore therefore be be ies taken when when extrapolating extrapolating to to long-term long-term dynamics. dynamics. They They found found themselves themselves sup suptaken ported 1 998), ported by by the the only only other other long-term long-term study study known, known, by by Moilanen Moilanen et et al. al. ((1998), population of the 66 patch patch meta metapopulation of the the American American on four snapshots in 20 years of the pika (Ochotona (Ochotona princeps) princeps) in in Bodie, Bodie, California. California. Although Although both both studies studies used used pika Hanski's 1 994) incidence Hanski's ((1994) incidence approach approach on on snapshot snapshot data data and and therefore therefore ignored ignored temporal temporal correlation correlation among among these these snapshots, snapshots, results results of of the the American American pika pika study were were recovered recovered by by Moilanen with his his Monte Monte Carlo Carlo simulation simulation study Moilanen ((1999) 1 999) with method method that that does does appropriately appropriately deal deal with with temporal temporal correlation. correlation. Hence, Hence, the the warning warning must must be be taken taken seriously. seriously. It It is is interesting, interesting, however, however, to to note note that that it it is is opposite to to Moilanen's Moilanen's (2000) (2000) warning warning that that short-term short-term data data may may indicate indicate aa opposite false trend. Still, both advocate the use of many years of data. These These results results justify justify the the question question of of whether whether there there is is some some rule rule of of thumb thumb about the the minimum minimum number number of of patches patches and and the the number number of of years years needed. needed. This This about question question is is not not easy easy to to answer answer for for (at (at least) least) five five reasons. reasons. First, First, there there is is obvi obviously aa trade-off trade-off between between the the number number of of patches patches and and the the number number of of years years ously needed. needed. Second, Second, consecutive consecutive years years may may contain contain aa different different amount amount of of infor information than than an equal number years with missing years years in between. Third, mation an equal number of of years with missing in between. Third, it matters matters whether whether the the assumption assumption of of quasistationarity quasistationarity is is made. Fourth, the the it made. Fourth, information information content content of of datasets datasets with with the the same same number number of of patches patches and and years years can can vary vary widely. widely. Fifth, Fifth, the the answer answer depends depends heavily heavily on on what what prediction prediction is is required to to be be reliable. reliable. For For example, example, our our results results in in the the case case study study (202 (202 patches, patches, required years, 11 additional year after missing years) years) show show that 3 consecutive consecutive years, additional year after 2 2 missing that both both patches, 11 year, and TO (202 QS1 (202 (202 patches, year, assumption assumption of of quasistationarity) quasistationarity) and TO (202 patches, 4 fairly reliable patches, 4 transitions) transitions) give give fairly reliable predictions predictions of of the the patch patch occupancy, occupancy, but turnover turnover is badly predicted predicted by by QSl only capable of statstat but is badly QS1.. Perhaps Perhaps we we are are only capable of ing a rule of thumb about what what datasets will not not give reliable predictions: thumb about predictions: netnet works with less than 25 patches patches and less than years of will most most works with less than 25 and less than 33 years of data data will probably be hopeless to parameterize. same time, we note note that the only probably be hopeless to parameterize. At At the the same time, we that the only way data you and to carry out way to be sure is to to simulate data you expect expect to to collect collect and to carry out the the parameter uncertainty analysis num parameter estimation estimation with with uncertainty analysis of of the the predictions predictions for for aa number of of simulated The Bayesian Bayesian approach is most most suited suited for for this ber simulated datasets. datasets. The approach is this purpur pose, demonstrated in O'Hara et pose, as as demonstrated in O'Hara et al. al. (2002). (2002). Once sufficiently sufficiently good good data data have have been been found found to parameterize the the model, model, we we Once to parameterize are ready for for one of the main purposes purposes of of developing developing the model: prediction, prediction, are ready one of the main the model: not not only only of of what what would would happen happen when when the the system system is is left left to to itself, itself, but but particuparticu larly larly to to assess assess the the impact impact of of conservation conservation strategies. strategies. For For example, example, O'Hara O'Hara et patches by et al. al. (2002) (2002) explored explored the the effects effects of of reducing reducing the the area area of of all all patches by some some factor. Cabeza Cabeza and and colleagues colleagues (Chapter (Chapter 22) 22) used used aa SPOM SPOM for selection factor. for optimal optimal selection of of sites sites to to conserve. conserve. In In our our case case study, study, we we studied studied the the consequences consequences of of insertinsert patches near near the the viable viable subnetworks. subnetworks. Whatever Whatever the the scenarios scenarios to to be be studied, studied, ing ing patches we which difwe urge urge the the use use of of uncertainty uncertainty analysis analysis because because it it is is the the only only way way in in which dif ferent ferent scenarios scenarios can can be be compared compared properly. properly.
FROM 6~ FROM META PO M ETAP O PULATIONS PU LATI O N S TO META COMMUNITIES TO M ETACO M M U N ITI ES Mathew and T homas E. Miller Mathew A. Leibold and Thomas
6. 6.11
INTRODUCTION INTRODUCTION Metapopulation thinking (Hanski and Gilpin, 11997, 997, Chapter 1) 1) has led to aa remarkable remarkable change change in in the the way way that that population population ecologists ecologists view view population population dynamics but is only now beginning to have similar effects on how ecologists view view community community dynamics. dynamics. In In population population biology, biology, the the shift shift in in scale scale from from con considering local population dynamics to population persistence in a network network of connected connected habitat habitat patches patches has has enabled enabled ecologists ecologists to to understand understand the the relative relative importance of different factors, such as habitat quality, species interactions, and migration in a scale-dependent context. One of the earliest and perhaps most obvious extensions of meta population theory was to pairs of interacting metapopulation 997; Hanski, 11999), 999), such as competitors (Levins and species (Nee et aI., al., 11997; Culver, 1971; Horn 972; Hastings, 1980) or predator Horn and MacArthur, 11972; predator and prey ((Caswell, Caswell, 11978). 978). The development from one- to two-species metapopula metapopulation tion models models is is important important in in that that it it adds adds aa significant significant level level of of complexity complexity and and realism: realism: the the colonization colonization and and extinction extinction rates rates of of the the focal focal species species become become functions of both habitat characteristics and the presence of the other species (Holt, 11993, 993, 11997). 997). Although (Holt, Although continued continued work work on on pairs pairs of of species species is is important, important, aa bigger bigger challenge challenge is addressing addressing patterns patterns in in community community structure structure involving involving more complex systems. To date, work work on metacommunities with this higher level of complexity is sparse and somewhat simplistic, but important important progress
Ecology, Ecology, Genetics, Genetics, and and Evolution Evolution Metapopulations of Metapopulations
33 1 33
Copyright Copyright 2004, Elsevier, Elsevier, Inc. Inc. 0-12-323448-4
MATHEW MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER MILLER
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has has been been made made toward toward understanding understanding patterns patterns of of biodiversity biodiversity at at different different scales associated phenomena population, community, scales and and associated phenomena at at the the population, community, and and ecosys ecosystem tem levels. levels. It It is is at at this this level level of of complexity complexity that that the the perspective perspective of of metacommunities metacommunities becomes just aa simple simple extension metapopula becomes substantially substantially different different from from just extension of of metapopulation 992; Holt, Holt, 11993). 993). We tion ecology ecology (Wilson, (Wilson, 11992; We use use the the term term metacommunity metacommunity to to refer communities (sets species that refer to to sets sets of of communities (sets of of species that interact interact with with each each other) other) occurring Gilpin and occurring at at discrete discrete sites, sites, linked linked by by migration migration ((Gilpin and Hanski, Hanski, 1991). 1991). The The crucial sites at crucial elements elements are are multiple multiple potentially potentially interacting interacting species, species, multiple multiple sites at which which such such interactions interactions might might occur, occur, and and migration migration by by individuals individuals of of at at least least some some of of the the species species to to link link species species interactions interactions among among sites. sites. In In its its simplest simplest form, form, aa metacommunity metacommunity may may consist consist of of networks networks of of discrete discrete habitats habitats such such as as islands that linked by all species species among islands that are are linked by migration migration of of some some or or all among the the habitats. habitats. This community will This kind kind of of meta metacommunity will have have aa structure structure that that emerges emerges from from the the dynamics metapopulations, dynamics of of the the interconnected interconnected habitats. habitats. However, However, as as with with metapopulations, our situations, including our definition definition can can fit fit aa broader broader array array of of situations, including mainland-island mainland-island metacommunities, metacommunities, where where migration migration from from the the mainland mainland may may largely largely determine determine community patterns on community patterns on islands. islands. Nevertheless, Nevertheless, this this chapter chapter focuses focuses on on meta metacommunities independent entities entities with communities as as independent with their their own own emergent emergent structure. structure. A between A community community perspective perspective creates creates an an important important distinction distinction between meta population and metapopulation and metacommunity metacommunity views. views. The The most most basic basic issue issue in in metapopu metapopulation theory lation theory is is to to address address what what determines determines the the persistence persistence of of aa metapopula metapopulation habitat patches Chapter 4), tion in in aa system system of of connected connected habitat patches ((Chapter 4), whereas whereas the the most most basic issue in basic issue in metacommunity metacommunity studies studies is is to to address address what what regulates regulates coexistence coexistence of of multiple multiple species species in in such such aa system. system. Excluding Excluding the the case case of of exact exact ecological ecological equivalence equivalence among among species species (Hubbell, (Hubbell, 2001 2001),), long-term long-term coexistence coexistence of of species species requires requires that that they they show show some some trade-off trade-off among among important important aspects aspects of of their their biology scale (see Chesson, 2000). biology at at some some scale (see Chesson, 2000). Theoretical Theoretical work work to to date date in in meta metacommunities shows communities shows that that dynamics dynamics involving involving trade-offs trade-offs are are surprisingly surprisingly constrained reasons. We constrained for for complex complex reasons. We also also examine examine how how our our current current empirical empirical knowledge knowledge is is related related to to the the theory, theory, particularly particularly in in relation relation to to the the role role of of different regulate metacommunities. different possible possible trade-offs trade-offs that that might might regulate metacommunities.
6.2 6.2
FOUR FOUR BASIC BASIC PERSPECTIVES PERSPECTIVES ON ON METACOMMUNITIES METACOMMUNITIES We We can can identify identify four four basic basic perspectives perspectives on on metacommunities metacommunities that that have have gen generally set community thinking thinking to erally set the the context context for for meta metacommunity to date. date. These These simplistic simplistic frameworks communities differ frameworks for for meta metacommunities differ from from one one another another in in the the degree degree of of spa spatial heterogeneity tial heterogeneity among among patches patches and and in in the the degree degree of of connectedness connectedness among among the relative to species (Table .1). the patches patches relative to the the migration migration rate rate of of the the component component species (Table 66.1). This This section section also also relates relates these these perspectives perspectives ttoo the the traits traits ooff the the component component species species and and how how these these traits traits covary covary to to produce produce trade-offs trade-offs at at different different scales. scales.
The The Patch-Dynamic Patch-Dynamic Perspective Perspective This species version This view view is is the the one one that that corresponds corresponds most most closely closely to to aa multi multispecies version of generally seen seen as homogeneous, with of "classic" "classic" metapopulations. metapopulations. Patches Patches are are generally as homogeneous, with local populations populations subject subject to local extinctions. local to local extinctions. Colonization Colonization of of patches patches by by new new
FROM METAPOPULATIONS METAPOPULATIONS TO METACOMMUNITIES METACOMMUNITIES 6. FROM
1 35 135
TABLE 6.1 6.1 TABLE
Four Simplified Simplified Perspectives Perspectives on on Metacommunities M etacommunities Arranged Arranged Four According to to whether whether Migration Migration Is Sufficient Sufficient to to Alter Alter Local Local Population Population Abundances Abundances According and whether whether Local or Patches Patches Are Are Heterogeneous Heterogeneous and Local Sites or
Migration Migration sufficient sufficient to to alter alter local abundances
Homogeneous patches Homogeneous
No Yes Yes
No
Yes
Species Species sorting Patch dynamics dynamics
Mass effect Neutral Neutral models models
species is is thought thought to to occur occur on on aa long long time time scale scale (through (through low low migration migration rates), rates), species so local local population population sizes sizes are are not not regulated regulated by by immigration immigration or emigration. so or emigration. Persistence the metacommunity thus depends depends on on how how colonization colonization Persistence of of species species in in the metacommunity thus of the the species species can can be be balanced against extinctions either to to interactions with of balanced against extinctions due due either interactions with other species or to to disturbance disturbance events events that that occur occur at at the the scale scale of of entire entire patches patches other species or ((Hanski, Hanski, 1999). 1 999). Local Local communities communities are are frequently "undersaturated" in in terms frequently "undersaturated" terms of number and/or to invasions invasions by by other other members members of the metameta of species species number and/or open open to of the community. If disturbance are sufficiently high, some species may may persist persist community. If disturbance rates rates are sufficiently high, some species in "fugitive species" being poor poor competitors. competitors. in such such metacommunities metacommunities as as "fugitive species" despite despite being To also have have sufficiently high migration migration rates rates to to allow allow them them To do do so, so, they they must must also sufficiently high to in the the metacommunity metacommunity with with superior superior competitors 1 980). to coexist coexist in competitors (Hastings, (Hastings, 1980). The perspective is is perhaps perhaps the the earliest view of com The patch-dynamic patch-dynamic perspective earliest view of aa meta metacommunity closely connected pioneer species species and munity and and is is closely connected to to ideas ideas about about fugitive fugitive or or pioneer and trade-offs abilities. Early trade-offs between between colonization colonization and and competitive competitive abilities. Early theoretical theoretical species coexistence work demonstrated demonstrated that that patch patch dynamics dynamics can can contribute contribute to to species coexistence work ((Caswell, Caswell, 11978; 978; Yodzis, 978). Recent Yodzis, 11978). Recent theoretical theoretical work work related related to to this this per perspective 1994), Kinzig al. ((1999), 1 999), Yu spective includes includes studies studies by by Tilman Tilman ((1994), Kinzig et etal. Yu and and Wilson Wilson (2001 (2001),) , Levine Levine and and Rees Rees (2002), (2002), and and Holt Holt (2002). (2002). The The primary primary result result of colon of these these studies studies is is that that although although aa trade-off trade-off between between competitive competitive and and colonabilities can (and potentially can allow allow for for coexistence coexistence of of species species (and potentially many many of of ization abilities them), them), the the range range of of parameters parameters that that allow allow coexistence coexistence by by this this mechanism mechanism is is surprisingly surprisingly limited. limited. The The patch-dynamic patch-dynamic perspective perspective is is also also associated associated with with certain certain types types of of ecosystems: ecosystems: the the concept concept of of habitat habitat patches patches connected connected by by migration migration underlies underlies our our understanding understanding of of hard-substrate hard-substrate intertidal intertidal systems systems (Paine 9 8 1 ) , the Connell, 11978; 978; (Paine and and Levin, Levin, 11981), the role role of of gaps gaps in in tropical tropical forests forests ((Connell, Denslow, 980), and Denslow, 11980), and effects effects of of small small mammals mammals on on certain certain plant plant communities communities (Platt 988; Huntley, 99 1 ) . (Platt and and Weis, Weis, 1977; 1977; Goldberg Goldberg and and Gross, Gross, 11988; Huntley, 11991).
The T h e Species-Sorting S p e c i e s - S o r t i n g Perspective Perspective If If connected connected habitat habitat patches patches are are heterogeneous heterogeneous in in some some important important charac character, ter, such such that that different different species species are are favored favored in in different different patches, patches, then then increased increased migration sorting" or migration can can lead lead to to aa ""sorting" or matching matching of of species species with with their their favored favored habitats. habitats. This This species-sorting species-sorting view view focuses focuses on on the the role role of of patch-type patch-type hetero heterogeneity geneity and and examines examines how how species' species' distributions distributions among among patches patches are are related related to to their their relative relative abilities abilities to to exist exist and and interact interact successfully successfully interact interact with with other other
11 336 6
MATHEW THOMAS E. MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS E. MILLER MILLER
species in the larger metacommunity Clements, 11916; 9 1 6; Gleason, 9 1 7; Tilman, metacommunity ((Clements, Gleason, 11917; Tilman, 1980). number of issues in 1980). A A number of issues in ecology ecology take take on on different different interpretations interpretations when when viewed viewed in in aa metacommunity metacommunity framework. framework. Some Some of of these these are are highlighted. highlighted. For For species species sorting sorting to to occur, occur, the the migration migration of of species species must must be be sufficient sufficient to to distrib distribute ute species species among among patches patches but but also also be be insufficient insufficient to to affect affect local local population population abundances. abundances. In In this this view, view, metacommunity metacommunity structure structure emerges emerges as as aa simple simple cumu cumulative outcome outcome of all local interactions, interactions, and and colonization-extinction colonization-extinction dynamics are only important important for determining determining the transient transient dynamics dynamics of the system until steady-state behavior metacommunity, it reaches a steady state. The steady-state behavior of the metacommunity, including including the distribution distribution of species among among patches, patches, is thus thus unrelated unrelated to rates of of colonization colonization and and extinction. extinction. Under Under species species sorting, sorting, immigration immigration may may have have within-patch diversity but but will be important important for for deter deterlittle effect on local, within-patch mining species identity identity within within patches. Work on this view includes that that by Tilman ala ((1993), 1 993), Holt 1 993, 11997), 997), and Leibold ((1996, 1 996, 1998). Tilman and and Pac Pacala Holt ((1993, and Leibold 1998).
The Mass Effects" The ""Mass Effects" Perspective Perspective In In this this view, view, patch-type patch-type heterogeneity heterogeneity is is also also important, important, but but migration migration is is suf sufficiently are ficiently high high that that local local population population abundances abundances of of component component species species are affected net emigration affected by by net emigration (producing (producing "source" "source" populations) populations) and and net net immigra immigrapopulations").) . Under Under these conditions, conditions, metacommunity metacommunity tion (yielding "sink populations" structure structure is is affected affected in in two two ways. ways. First, First, species species that that might might be be expected expected to to do do well closed community emigration or well in in aa closed community (with (with no no emigration or immigration) immigration) can can be be driven driven to extinction from superior by to extinction from local local patches patches where where they they are are competitively competitively superior by strongly strongly subsidized subsidized "sink "sink populations" populations" supported supported by by immigration immigration from from other other patches. Second, Second, such patches. such competitively competitively superior superior species species can can have have reduced reduced compet competitive itive abilities abilities in in metacommunities metacommunities because because of of the the emigration emigration of of individuals individuals (equivalent (equivalent to reduced local birth birth rates) as well. The overall result is that that mass effects can increase local diversity diversity (Mouquet and Loreau, 2002) 2002) by such "sub "subsidized" sidized" population population dynamics. dynamics. The The issue issue is is aa bit bit more more complicated, complicated, however, however, because require very emigration that itself because such such strong strong subsidy subsidy can can require very strong strong emigration that can can itself make local populations populations vulnerable to extinction. Thus Thus these mass effects effects can reduce reduce the the diversity diversity over over broader broader spatial spatial scales scales (the (the metacommunity metacommunity as as aa whole) whole) and reduce local local diversity mass effects and therefore therefore reduce diversity despite despite strong strong mass effects (Mouquet (Mouquet and and Loreau, 1 993), Loreau, 2002). 2002). Research Research work work on on this this perspective perspective includes includes that that by by Holt Holt ((1993), Amarasekare Amarasekare and Nisbet Nisbet (2002), and Mouquet Mouquet and Loreau (2002).
The The Neutral Neutral Perspective Perspective Of course, all presumes that Of course, all of of this this theory theory presumes that species species differ differ from from one one another another important ways, including their responses to local conditions conditions (patch-type in important specialization specialization and/or and/or competitive competitive superiority) superiority) and to the landscape landscape structure structure (migration (migration and background background extinction rates). One possible view is that that species not differ differ in any important important respect in any of these ways (Hubbell, 11997, do not 997, 200 1 ) . Although 2001). Although unlikely unlikely to to be be universally universally true true (see (see Zhang Zhang and and Lin, Lin, 1997; 1997; Yu Yu et aI., 998), this et al., 11998), this view view does does highlight highlight how how important important stochastic stochastic processes processes can can be be in in structuring structuring metacommunities metacommunities and and may may be be important important in in explaining explaining some some important important aspects aspects of of metacommunities, metacommunities, including including abundance abundance distributions distributions and and incidence aI., 2002). incidence functions functions (Hubbell, (Hubbell, 2001 2001;; see see Chave Chave et et al., 2002).
6. 6.
FROM M ETAPOPULATIONS TO TO METACOMMUNITIES METACOMMUNITIES FROM METAPOPULATIONS
11 337 7
A Missing Missing Synthesis Synthesis A Obviously, all all of of these these perspectives perspectives are are simplifications, simplifications, and and real real metacommetacom Obviously, munities are are likely likely to to be be structured structured by by some some mixture mixture of of the the processes processes that that are are munities by each each of of them. them. The The issue issue is is made made more more complex complex because because species species highlighted highlighted by in real real communities may interact interact with each other fundamen in communities may with each other despite despite having having fundamentally different migration rates. rates. For For example, example, aa predator predator may may have have aa migration migration tally different migration rate sufficiently high that that itit has has aa metapopulation metapopulation structure and significant significant rate sufficiently high structure and local population population regulation regulation due due to to migration, migration, while while itit interacts interacts with with prey prey that that local have much much lower lower migration migration rates rates and and in in which which local local population population regulation regulation is is have completely determined determined by by local local processes processes [see [see Holt Holt (1997) ( 1 997) and and van van Nouhuys Nouhuys completely and Hanski Hanski (2002) (2002) for for an an empirical empirical example]. example] . To To date, date, there there is is little little underunder and standing of of how how to to synthesize synthesize these these various various perspectives perspectives into into aa general general frameframe standing work, even even though though there there is is some some recognition recognition that that such such aa synthesis synthesis is is important important work, (Holt, 1997). 1 997). Surprisingly Surprisingly little little work work is is done done at at the the interface interface of of the the speciesspecies (Holt, sorting and patch-dynamics perspectives (but ( but see Holt, 1997; 1 997; Levine Levine and and Rees, Rees, sorting and patch-dynamics perspectives see Holt, 2002; Shurin Shurin et 2003 ). Work Work by by Amarasekare Amarasekare and and Nisbet Nisbet (200 1 ) and by 2002; et aI., al., 2003). (2001) and by Mouquet and and Loreau examines the interface between between species species sorting sorting Mouquet Loreau (2002) (2002) examines the interface and mass effects. It is is as yet unclear unclear how how to to integrate integrate these these various various perspecperspec and mass effects. It as yet tives with the stochastic processes described by Hubbell Hubbell (200 1 ), although although tives with the stochastic processes described by (2001), work by by Chave Chave et ai. (2002) (2002) indicates indicates that that the the effects can be be subtle. work et al. effects can subtle.
6.3 6.3
THEORETICAL METACOMMUNITIES THEORETICAL ISSUES ISSUES ABOUT ABOUT METACOMMUNITIES
The The Importance Importance of of Trade-Offs Trade-Offs about trade-offs trade-offs from a more conventional conventional It is helpful to start thinking about basis that basis that ignores ignores metacommunities. metacommunities. Traditional Traditional ecological ecological theory theory about about closed closed communities communities has has drawn drawn attention attention to to trade-offs trade-offs involving involving relative relative biotic biotic abilities. Within abilities. Within aa local local site, site, this this view view involves involves conventional conventional mechanisms mechanisms of of resource resource partitioning partitioning and and associated associated ecological ecological traits traits such such as as susceptibility susceptibility to to predators and and so forth. forth. At the local scale, the nature of these trade-offs trade-offs will predators determine species combinations stability of determine which which species combinations are are possible, possible, what what the the stability of the the system be, and responds to environmental change. system might might be, and how how the the community community responds to environmental change. Such Such differences differences in in responses responses and and impacts impacts on on resources resources and and other other fitness fitnessrelated related factors factors (this (this suite suite of of traits traits is is subsumed subsumed into into aa general general term term we we call call ""biotic biotic ability " ) can ability") can also also lead lead to to habitat habitat (patch-type) (patch-type) specialization specialization (involving (involving coexistence coexistence at at the the larger larger regional regional scale scale even even if if there there is is no no local local coexistence). coexistence). We species sorting" We can can think think of of this this process process as as one one of of ""species sorting" that that regulates regulates how how well well matched matched species species are are to to their their local local environments. environments. Although Although ecologists ecologists are are familiar familiar with with this this conventional conventional approach approach to to niche niche dynamics, dynamics, they they do do not not always always recognize recognize that that trade-offs trade-offs can can also also be be important important in in regulating regulating meta metacommunity community dynamics. dynamics. This This chapter chapter reviews reviews theory theory and and empirical empirical work work that that show show how how important important such such relative relative habitat habitat specialization specialization is is in in regulating regulating meta metacommunity community dynamics dynamics even even when when other other important important processes processes are are also also operating. operating. We We can can also also approach approach questions questions about about trade-offs trade-offs from from the the perspective perspective of of meta population theory metapopulation theory based based on on single single species. species. This This theory theory draws draws attention attention to to the the tension tension between between colonization colonization ability ability and and susceptibility susceptibility to to stochastic stochastic extinction extinction in in regulating regulating the the persistence persistence of of populations populations in in patchy patchy landscapes landscapes where where aa variety variety
MATHEW MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER MILLER
11 38 38
allow species to persist persist in a given given landscape ((Chapter of strategies can allow Chapter 4). Similarly, population theory Similarly, meta metapopulation theory oriented oriented toward toward small small numbers numbers of of interacting interacting species species has has shown shown that that trade-offs trade-offs involving involving colonization colonization ability ability and and biotic biotic ability ability within 980; Tilman, 994; Taneyhill, within patches patches can can also also be be important important (Hastings, (Hastings, 11980; Tilman, 11994; Taneyhill, 2000; 2000; but but see see Klausmeier, Klausmeier, 2001; 2001; Yu Yu and and Wilson, Wilson, 2001; 2001; Levine Levine and and Rees, Rees, 2002). 2002). These These dynamics dynamics affect affect species species sorting sorting in in two two opposite opposite ways: ways: increased increased competi competitive tive ability ability can can be be related related to to increased increased specialization, specialization, but but high high rates rates of of migration migration may decrease the degree to which local sites are occupied by species that are best matched matched to to local local conditions conditions (see (see Mouquet Mouquet et et aI., al., 2002). 2002). Thus Thus connectivity connectivity in in aa metacommunity metacommunity can can change change conditions conditions from from those those that that favor favor habitat habitat specializa specialization and coexistence to those that favor species with traits related to overall biotic ability ability in in the the metacommunity metacommunity as as aa whole whole (including (including dispersal dispersal as as well well as as average average extinction rates and biotic ability over all patches). The The aforementioned aforementioned considerations considerations suggest suggest that that three three broad broad categories categories of of species species differences differences are are all all potentially potentially important important in in regulating regulating the the distribution distribution of species in of species in metacommunities: metacommunities: patch patch specialization specialization by by means means of of biotic biotic ability, ability, migration migration among among patches, patches, and and likelihood likelihood of of local local extinctions. extinctions. These These admit admittedly broad categories and lead lead to tedly broad categories could could probably probably be be subdivided subdivided further further and to even even more process of could be more possible possible traits; traits; for for example, example, the the process of migration migration could be dissected dissected component (Chapter (Chapter 113). 3). into an emigration and an immigration component now consider consider multiple patch patch types where these species differences can If we now effects, then we can consider traits and and trade-offs trade-offs that that may be have various effects, related to these species related to these species differences. differences. For For example, example, expression expression of of each each of of the the two different types of patches produces three categories in two produces six traits or ways that species can differ that influence their persistence and coexistence: that 1. 1. 2. 2. 3. 4. 5. 5. 6.
patch-type 11 Biotic ability in patch-type Biotic ability in patch-type patch-type 2 Migration ability into into patch-type patch-type 1 Migration ability into patch-type 2 Susceptibility to stochastic extinction in patch-type patch-type 1 to stochastic stochastic extinction in patch-type Susceptibility to patch-type 2
It is is important important to to understand understand that biotic ability that can can lead lead to to extincextinc It that any any biotic ability that tions (traits (traits 11 and 2) in in many many metacommunity metacommunity models models is is substantially distinct tions and 2) substantially distinct from from susceptibility to to stochastic stochastic extinctions extinctions (traits (traits 55 and and 6). 6). The The latter latter is is viewed as as being being independent independent of of biotic biotic interactions, interactions, due due either either to to stochastic stochastic viewed effects effects acting acting on on the the demography demography of of small small populations populations or or (perhaps (perhaps just just as as commonly) commonly) to to environmental environmental change change within within local local sites, sites, including including disturbances disturbances that affect large that can can affect large populations populations as well. Taken in in pairwise pairwise combinations, combinations, this this list list of of six six traits traits can can lead lead to to 15 15 possible possible Taken trade-offs between between species, species, and and more more of of them them involve involve niche niche axes axes that that operate at trade-offs operate at the the larger larger regional scale. This This perspective perspective results in many many more more mechanisms mechanisms that that can can regulate regulate coexistence coexistence and and metacommunity dynamics than than are considconsid ered ered in in local-scale local-scale models. models. To To date, date, metapopulation metapopulation theory theory has has addressed addressed only only some of with any degree of of these possibilities possibilities with any degree of effort. effort. Current work work on on metacommunities metacommunities reveals reveals important important interdependent interdependent concon Current straints on on each each of of these these possibilities. possibilities. Two Two features features are are important. important. First, First, the the straints behavior of of the the system system in in relation relation to to these these traits traits can can be be complex complex because because the the behavior behavior behavior of of metacommunities metacommunities may may vary vary in in response response to to these these trade-offs trade-offs in in ways ways
6. 6.
FROM METAPOPULATIONS M ETAPOPULATIONS TO METACOMMUNITIES FROM TO METACOMMUNITIES
11 339 9
that are are not not always always consistent. consistent. For For example, example, the the colonization-rate-competitivecolonization-rate--competitive that trade-off can can be important important in allowing allowing species to to coexist in a metacommetacom ability trade-off munity when when local populations populations are are subject to to stochastic stochastic extinctions, extinctions, thus thus munity allowing competitively subordinate subordinate species to to coexist with with competitive competitive dominant dominant allowing species. Under Under this this scenario, scenario, the the constraint constraint for for coexistence coexistence in in the the metacommunity metacommunity species. is that that the the colonization colonization rate rate of of the the subordinate subordinate exceed exceed some some critical critical value value that that is allows it it to to exist exist as as aa fugitive fugitive species species in in the the metacommunity metacommunity (i.e., (i.e., that that the the migramigra allows tion rate rate time time scale scale be be much much shorter shorter than than that that at at which which species species interactions interactions lead lead tion to extinction). extinction). The The colonization-rate-competitive-ability colonization-rate--competitive-ability trade-off trade-off can can also also allow allow to species to to coexist coexist when when colonization rates are are high high enough enough to to alter local abunabun species colonization rates alter local dances (through mass mass effects). effects). However, However, under this scenario, colonization by by the the dances (through under this scenario, colonization subordinate species species must must be be low low enough enough to to prevent prevent it it from from subsidizing subsidizing large sink subordinate large sink populations in in patches patches occupied by competitive competitive dominants dominants (Amarasekare (Amarasekare and and populations occupied by Nisbet, 2001). 200 1 ) . Nisbet, Second, models models based based o one mechanism mechanism can can also bbee strongly strongly altered altered bbyy the the Second, onn one presence of mechanisms involving other trade-offs. trade-offs. For example, involving other example, Law and Leibold (in press) have found found that that a metacommunity metacommunity of species with with nontrannontran sitive sitive assembly rules rules (species (species A invades and and excludes excludes species species B, B, species B invades and and excludes species C, and species C invades and and excludes species A) shows cycles in in the the frequency frequency of of patch patch types types (a (a metacommunity metacommunity becomes becomes domdom shows cycles turn, by sites containing and then then inated, in turn, containing species A, species B, species C, and background stochastic extinctions, species A again). In the absence of background extinctions, the amplitude of these cycles in patch patch occupancy varies in a neutral neutral manner manner (such amplitude as neutral limit of the the Lotka-Volterra Lotka-Volterra predator-prey models) . as the the neutral limit cycles cycles of predator-prey models). However, in in the such extinctions, the cycles to aa stable stable However, the presence presence ooff such extinctions, the cycles converge converge to point point at at which which the the frequency frequency of of occupancy occupancy states states is is determined determined by by the the relative relative colonization and colonization and stochastic stochastic extinction extinction probabilities probabilities of of the the three three species. species. Thus Thus the (under nontransitive the outcome outcome of of aa model model of of species species interactions interactions (under nontransitive assembly) assembly) is background extinc is strongly strongly modified modified by by the the presence presence of of patch patch dynamics dynamics ((background extinctions tions unrelated unrelated to to species species interactions). interactions).
The The Role Role of of Migration: Migration: From From Patch Patch Dynamics Dynamics to to Species Species Sorting Sorting to to Source-Sink Source-Sink Relations Relations A broader broader review of the theory on metacommunities metacommunities is beyond the scope of of this this chapter chapter (such (such aa review review is is under under preparation preparation by by Holyoak, Holyoak, Leibold, Leibold, and and Holt), but two two important important conclusions can be drawn about the dynamics of metacommunities. Loreau and metacommunities. First, First, the the role role of of migration migration is is not not simple simple ((Loreau and Mouquet, 999; Mouquet Mouquet, 11999; Mouquet and Loreau, 2002; Leibold and Norberg, Norberg, 2003) 2003):: Fig. . 1 presents Fig. 66.1 presents aa hypothetical hypothetical relationship relationship between between migration migration rate rate and and diversity. diversity. Clearly, Clearly, when when there there is is no no migration, migration, then then the the metacommunity metacommunity per perspective spective is is not not needed, needed, the the dynamics dynamics of of local local communities communities are are entirely entirely inde independent pendent of of one one another, another, and and diversity diversity in in local local communities communities is is expected expected to to be be relatively relatively low low due due to to migration migration limitation. limitation. At At minimal minimal levels levels of of migration, migration, every every species species can can potentially potentially get get to to every every patch, patch, but but the the order order of of colonization colonization and and relative relative competitive competitive abilities abilities become become important important in in determining determining local local community community structure. structure. The The result result is is some some degree degree of of species species sorting sorting in in which which migration may not enhance diversity but can allow significant species replace replacement ment and and some some specialization. specialization. In In particular, particular, "fugitive "fugitive species" species" are are possible possible
MATHEW MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER MILLER
140 140 i
9
i en en w Z I o 0: en w (3 W 0... en
migration migration limitation limitation
f.
species species sorting sorting
i :
mass mass effects effects
i i
metacommunity metacommunity homegenization homegenization
--�--
i
9
9
i
MIGRATION MIGRATION RATE RATE 6.1 A hypothetical between migration Fig. hypothetical relationship relationship between migration rate and species richness illustrating illustrating Fig. 6.1 the major major mechanisms mechanisms by which migration affects community community structure. structure9 the which migration
under these conditions, conditions, species that that exist in the metacommunity metacommunity by virtue of under temporary temporary existence existence in patches that that have not yet been colonized colonized by patch-type specialists (Hastings, 11980). 980). In contrast, as migration increases until all species can disperse relatively quickly to all patches, the local assembly of communities colonization ((because because the communities is much much less affected by the the order order of of colonization the assembly process equilibrium behavior behavior relatively quickly, process can reach reach its final final equilibrium typically uninvasible community community but but also also possibly possibly aa pattern pattern of of repeated repeated typically an an uninvasible heteroclinic cycles), and heteroclinic and classic fugitive species are excluded by patch-type specialists. At yet higher rates, migration migration plays a different different role, being high enough to contribute growth rates. This allows species contribute significantly to local growth that inferior competitors that are locally inferior competitors to to persist through through mass effects (i.e., Mouquet and Loreau, 2002), 2002), increasing local diversity (Fig. (Fig. 6.1). 6 . 1 ) . At Mouquet and Loreau, At this migration may act to decrease diversity of the entire meta point, increased migration entire metacommunity, with consequential effects on local diversity as well. In the community, with consequential when migration migration is so high that that local populations populations are are completely completely extreme, when mixed, metacommunity metacommunity diversity (and thus thus local diversity) collapses into into that that mixed, expected expected in a single closed community. community.
The Importance Importance of of Habitat Habitat Heterogeneity Heterogeneity and Patch-Type Specialization The second second important conclusion is that habitat heterogeneity, heterogeneity, when The important conclusion that habitat when it is present, almost almost always always plays an an important important role role in maintaining maintaining diversity in present, metacommunities. Heterogeneity Heterogeneity allows allows habitat habitat specialization, specialization, which which can can metacommunities. provide a local local refuge refuge for for some some specialist specialist species and and increase increase metacommunity metacommunity provide (although not not local) local) diversity. At At higher higher rates rates of of migration, migration, such such refuges refuges can can (although act act as sources sources that that help help maintain maintain populations populations in less favorable favorable patches patches and and
ETAPOPULATIONS TO 6. 6. FROM FROM M METAPOPULATIONS TO METACOMMUNITIES METACOMMUNITIES
141
increase increase both both local local and and metacommunity metacommunity richness richness (Mouquet (Mouquet and and Loreau, Loreau, 2002). In addition, addition, metacommunity metacommunity dynamics dynamics involving stochastic stochastic extinctions extinctions at local scale at the the local scale prevent prevent the the occurrence occurrence of of alternative alternative stable stable states states in in local local communities Shurin et communities in in the the absence absence of of patch-type patch-type heterogeneity heterogeneity ((Shurin et aI., al., 2003 2003),), eliminating eliminating another another potentially potentially important important mechanism mechanism that that might might maintain maintain biodiversity in homogeneous homogeneous patch-type patch-type metacommunities. metacommunities. However, in high biodiversity the the presence presence of of patch-type patch-type heterogeneity, heterogeneity, which which can can prevent prevent exclusion exclusion at at the the metacommunity metacommunity scale, scale, such such alternative alternative stable stable states states can can act act to to increase increase compositional compositional diversity even among among patches patches of the same type. In both both cases, diversity diversity is increased only if patch-type patch-type heterogeneity heterogeneity is sufficient to allow the "source" populations sink" populations. "source" populations that that support support ""sink" populations. Further, even under under these these conditions, conditions, migration migration is is constrained constrained in in direct direct relation relation to to the the degree degree of of habitat habitat heterogeneity heterogeneity (Amaresekare (Amaresekare and and Nisbet, Nisbet, 2001 2001;; Mouquet Mouquet and and Loreau, Loreau, 2002; 2002; Levine Levine and and Rees, Rees, 2002 2002).) . Therefore, Therefore, even iiff we limit ourselves ttoo considering considering only the six types of metacommunity-related metacommunity-related traits listed earlier, we obtain obtain a complex complex set of results about about metacommunity metacommunity dynamics dynamics and and the the ways ways they they alter alter patterns patterns of of coexist coexistence ence and and biodiversity. biodiversity. This This theoretically theoretically rich rich area area of of understanding understanding the the effects effects of trade-offs trade-offs at different spatial scales is only now now being addressed, addressed, and a number number of of important important results results may may soon soon be be forthcoming. forthcoming.
AND EXPERIMENTAL 6.4 OBSERVATIONAL OBSERVATIONAL AND EXPERIMENTAL EVIDENCE EVIDENCE 6.4 ABOUT ETACOMMUNITIES ABOUT M METACOMMUNITIES Conventional Conventional Analyses Analyses of of Species Species Distributions Distributions in in Relation Relation to to Patch-Type Patch-Type Heterogeneity Heterogeneity Empirical work on metacommunities metacommunities predates predates the metacommunity metacommunity concept concept most basic types of data data that that describe metacommunities metacommunities can itself. One of the most be summarized incidence matrix," matrix," a matrix summarized as an ""incidence matrix describing which which species are Bray and 957; Leibold are found found at at each each of of aa variety variety of of sites sites ((Bray and Curtis, Curtis, 11957; Leibold and and Mikkelson, Mikkelson, 2002). 2002). There There are are sophisticated sophisticated ways ways of of analyzing analyzing such such data, data, often often in null models models ((Gotelli Gotelli and 996) or in relation relation to to null and Graves, Graves, 11996) or environmental environmental gradi gradients 998). Although ents (see, (see, e.g., e.g., Legendre Legendre and and Legendre, Legendre, 11998). Although these these data data have have strong strong empirical roots roots and have long been implicitly related related to mechanisms mechanisms important important in in metacommunity metacommunity dynamics, dynamics, the the link link between between incidence incidence matrices matrices and and metacommunities metacommunities has has generally generally not not been been made made explicit. explicit. A good summary summary of this large body of work work is beyond beyond the scope of this chapter, but but a few points points relevant relevant to the metacommunity metacommunity concept concept are highlighted. First, this body body of work work shows shows that that species associations associations in local highlighted. habitats aI., 2002 2002 and habitats are are typically typically not not random random (for (for summaries, summaries, see see Gotelli Gotelli et et al., and Leibold and Mikkelson Mikkelson 2002), 2002), suggesting that that species interactions interactions or similar similarity ity of of biotic biotic requirements requirements affects affects species species distributions distributions and and could could affect affect larger larger scale scale (metacommunity) (metacommunity) dynamics. dynamics. Second, Second, environmental environmental variability variability in in abiotic abiotic factors factors (heterogeneity (heterogeneity of of patch patch types) types) is is often often important important in in explaining explaining the the dis distribution 957; Whitaker, tribution of of species species among among sites sites (see, (see, e.g., e.g., Bray Bray and and Curtis, Curtis, 11957; Whitaker, abundance of individual individual species or suites of species can be shown shown 11975). 975) . The abundance to to covary covary significantly significantly with with factors factors such such as as nitrogen nitrogen availability availability or or soil soil
MATHEW MATHEW A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER
14Z 142
moisture. moisture. Nevertheless, Nevertheless, some some spatial spatial effects effects may may be due purely purely to effects of of migration aI., 11992; 992; Pinel aI., 11995). 995) . migration (Borcard (Borcard et et al., Pinel Alloul Alloul et et al., Important Important descriptive descriptive studies studies ooff biodiversity biodiversity aatt different different scales indicate indicate a between local and regional complex relationship relationship between regional patterns of of diversity. For example, Shurin (Shurin, 2000; 1) example, 2000; Shurin et aI., al., 2000; 2000; Shurin and Allen, 200 2001) has shown shown that commonly commonly observed observed linear relationships relationships between between local and regional uninvasible local com regional diversity diversity were not necessarily associated with with uninvasible communities, but rather could explained as resulting from could be explained from a complex complex assembly process process in in metacommunities metacommunities involving involving multitrophic-Ievel multitrophic-level food food webs webs (see (see also also Srivastava, 999; Mouquet aI., 2003 Srivastava, 11999; Mouquet et et al., 2003).). In In another another example, example, Chase Chase and and Leibold (2002) (2002) showed showed that local and regional regional diversity diversity had qualitatively qualitatively different different relationships relationships with with productivity; productivity; the local relationship relationship was was unimodal unimodal and regional relationship relationship increased increased monotonically (Fig. 6.2 and the the regional monotonically (Fig. 6.2).) . Steiner Steiner and and 3 ) showed Leibold (200 (2003) showed that this situation situation can result from from metacommunity metacommunity dynamics nontransitive invasion invasion cycles cycles (see 997) dynamics in in which which nontransitive (see Morton Morton and and Law, Law, 11997) are 1 997, 200 1 ) has are more more common common at at high high productivity. productivity. Hubbell Hubbell ((1997, 2001) has also also a a
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Fig. 66.2 Fig. . 2 Results Resultsfrom a survey of pond species diversity relative to in in situ situ primary productiv productivity at local local and regional scales. scales. (Top) Producers (vascular plants and macroalgae). (Bottom) (Bottom) Benthic animals (insects, crustaceans, amphibians, and so on). on). (a) Local, Local, within-pond, within-pond, species 30). Both relationships are significantly unimodal diversity (N (N = 30). unimodal (P (P < < 0.05). 0.05). The line represents the estimated quadratic function. (b) (b) Regional, within-watershed, within-watershed, species species diversity. For both producers (regression: N 0, RR22 = 74, PP < 0, N = = 110, = 0. 0.74, < 0.001 0.001)) and benthic animals (regression: N N = = 110, 2 RR2 = 0.75, ), regional species species diversity was related linearly to primary productivity. 0.75, P P< < 0.001 0.001), From Chase and Leibold (2002). (2002).
FROM M ETAPOPULATIONS TO ETACOMMUNITIES 6. FROM METAPOPULATIONS TO M METACOMMUNITIES 6.
143 143
reinterpreted reinterpreted patterns of diversity using a novel neutral metacommunity per perspective in which which species have equal effects on one another another and the habitat habitat is homogeneous 1 ). The ability of Hubbell's models homogeneous (see (see also Bell Bell 2000, 2000, 200 2001). to predict predict patterns of relative abundances abundances suggests that that many community patterns patterns are created by random random "sampling" of larger species pool and cannot cannot be used to imply mechanisms related to species interactions. interactions.
Observational Explicit Metacommunity Observational Studies Studies with with Explicit Metacommunity Approaches Approaches A few studies of unmanipulated unmanipulated systems have specifically used a metacom metacommunity framework for the interpretation interpretation of regional patterns. One of the more important aI. (2001 important of these is work by Cottenie et al. (2001)) on zooplankton zooplankton metacom metacommunity structure structure in a set of interconnected interconnected ponds. They found found that that individual ponds showed strong habitat habitat heterogeneity related to predator predator composition; some ponds ponds had dense planktivorous planktivorous fish populations populations that may have led to high through trophic cascades and other ponds had lower planktivo planktivoalgal densities through rous fish densities leading to higher densities of large grazers and consequent consequent lower algal densities. More importantly, they quantified quantified the amount amount of migra migraheterogeneous ponds to determine whether whether migration increases tion among heterogeneous diversity through through source-sink dynamics. Despite measuring high rates of migra migrapond to the other), they only rarely found found evi evition (via (via stream flow from one pond dence for such source-sink dynamics. This work work indicates that the strength of species sorting between these different patch types (pond types) is remarkably strong and cannot cannot be affected easily by the homogenizing effects of migration. Other Other patterns patterns in natural natural systems may take on novel significance when interpreted interpreted in the context of metacommunity metacommunity theory, including those related to abundance patterns patterns (Hubbell, 2001 2001;; Chave and Leigh, 2002), 2002), to similarity and coexistence (see 998), and to ecosystem attributes (see Leibold, 11998), attributes (see (see Mouquet Mouquet et al., aI., 2002; Leibold et al., aI., 11997). 997). Nevertheless, Nevertheless, using metacommunity metacommunity theory theory interpret natural natural patterns patterns of variation variation is speculative speculative until this body of to interpret theory and its principal mechanisms have been evaluated evaluated by means of experi experimental manipulations. A tremendously tremendously rich array of work work remains to be done, work in which which experiments experiments are specifically designed to test mecha mechaespecially work nisms of metacommunity dynamics.
Experimental Experimental Work Work with with Metacommunities Metacommunities A variety of experimental studies studies have have been been conducted on metacommuni metacommuni1 95 8 ) work ties, going back to Huffaker's ((1958) work with predator-prey predator-prey systems of relamites maintained on increasingly complex arrays of oranges. However, rela tively few such manipulative manipulative studies have been conducted conducted with with more than than two species in true metacommunities, metacommunities, where a closed system of local communities is linked by migration. Yet such studies are necessary for appropriate appropriate tests of factors identified identified by models as being important important for predicting species persist persistence and diversity in metacommunities. metacommunities. The theoretical theoretical work work described earlier indicates indicates that that metacommunity metacommunity dynamics dynamics are especially likely to result result from complex interactions among migration (or isolation), habitat habitat disturbance disturbance (or other mechanisms for local stochastic extinctions), patch-type heterogeneity, and the interrelation among species traits that might result in trade-offs.
MATHEW A A.. LEIBOLD LEIBOLD AND THOMAS EE.. MILLER MILLER
114141 44
Migration Migration among local communities communities is perhaps the hallmark of metacom metacommunities. Several theoretical studies explicitly predict that that increasing rates of migration will lead to increasing local diversity until the migration migration rate is so high that local variation among sites is swamped ((Caswell, Caswell, 11978; 978; Caswell and 991; Loreau and Mouquet, 1999; Mouquet Cohen, 11991; Mouquet and Loreau, 2002). Increased migration can increase diversity by at at least two mechanisms. If species are migration migration limited, then increased migration migration may allow species to gain access to communities where they were previously absent (Tilman, 11994; 994; Shurin, 2001; Miller et aI., al., 2002). Alternatively, migration may act to to augment or or replace local reproduction reproduction for some species, allowing them to persist in communities from which they would otherwise be excluded. At extremely high rates of migration, however, both local and regional (metacommunity) diversity will decline, decline, as regionally dominant dominant species exclude others. Several laboratory studies have varied migration rates among local communities, communities, with with mixed effects on local diversity. Warren ((1996a,b) 1996a,b) investigated the role of migration in two two laboratory studies of protist metacommunities, enforcing migration at various rates by moving fluid between separate local habitats. In both studies, diversity did increase with with increasing migration rate, but the effects were relatively small and depended on the magnitude magnitude of other factors, such as disturbance disturbance and habitat habitat size (Fig. (Fig. 6.3). Similarly, Forbes and Chase (2002) varied the degree of connectivity among zooplankton zooplankton communities in experimental microcosms and found that increased migration had no effect on local diversity but decreased regional diver diversity through a homogenization of local communities. Gonzalez and co-workers ((1998; 1 998; Gonzalez and Chaneton, 2002) experi experimentally communities of mentally varied varied migration migration rates rates in in fragmented fragmented communities of microarthro microarthropods that that occur in dense moss growth. They manipulated migration by arranging moss in isolated fragments fragments (very low migration), in fragments connected to larger areas by moss "corridors, "corridors,"" and in continuous continuous areas of moss (Fig. (Fig. 6.4). Lower migration resulted in the loss of rare species, and corridors corridors alleviated the effects of fragmentation greatly. This result indicates that that metacommunity metacommunity dynamics involving colonization-competition colonization-competition trade-offs
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6.3 Effects Fig. 6.3 Effects of disturbance and migration rate on local protist species species richness richness in meta metacommunities consisting of eight experimental 1100-ml 00-ml microcosms. Dispersal Dispersal took place in 2-ml aliquots of of fluid moved at 3-day intervals between either 6 (low dispersal, open bars) bars) or 24 (high dispersal, shaded bars) randomly chosen pairs of microcosms. Disturbance consisted of placing either 1I (low disturbance) or 2 (high disturbance) randomly chosen microcosms in hot water baths every 3 days to kill the protozoa (data adapted from 996a). from Warren, 11996a).
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Effect Effect of of fragmentation fragmentation on on species species richness richness in in aa moss moss invertebrate invertebrate community. community. Local Local communities continuous with communities were were either either continuous with surrounding surrounding moss, moss, isolated isolated but but connected connected with with moss moss corridors, corridors, or or isolated isolated with with corridor. corridor. Figure Figure adapted adapted from from Gonzalez Gonzalez and and Chaneton Chaneton (2002). (2002).
were probably not small were probably not important. important. Kneitel Kneitel and and Miller Miller (2003) (2003) studied studied the the small communities communities of of protists protists and and rotifers rotifers found found in in the the water-filled water-filled leaves leaves of of pitcher pitcher plants. plants. They They moved moved small small volumes volumes of of water water among among suites suites of of leaves leaves at at three three different different rates, rates, while while also also manipulating manipulating the the presence presence of of predators predators and and the the input input of resources. Local of resources. Local diversity diversity was was highest highest at at intermediate intermediate levels levels of of migration, migration, although predators reduced reduced the although the the presence presence of of predators the effects effects of of migration. migration. Metacommunity Metacommunity dynamics dynamics were were therefore therefore important important in in regulating regulating local local diver diversity, habitats must understood in sity, but but the the effects effects of of migration migration among among habitats must be be understood in light light of local biotic biotic factors. of local factors. Disturbance open" Disturbance can can immediately immediately reduce reduce densities, densities, but but it it can can also also create create ""open" habitats habitats that that are are available available to to all all species species for for colonization colonization and and may may allow allow fugitive Connell, 11978). 978). Disturbance fugitive species species to to persist persist in in metacommunities metacommunities ((Connell, Disturbance is is often often considered considered in in continuous continuous communities communities in in which which smaller smaller scale scale disturb disturbances ances create create open open patches patches in in aa larger larger community community matrix. matrix. Although Although not not usually usually described described as as doing doing so, so, such such disturbances disturbances create create aa metacommunity metacommunity scenario scenario for for the the community community of of species species that that depend depend on on disturbances disturbances for for persistence persistence in in the the community. propagules to community. Such Such species species must must persist persist by by moving moving or or dispersing dispersing propagules to new new locally locally disturbed disturbed sites sites before before they they are are excluded excluded from from their their current current sites. sites. disturbance in this type of metacommunity should should result in a The effects of disturbance maximum diversity of maximum diversity of species species at at some some intermediate intermediate rate rate of of local local habitat habitat disturbance 984). For 1 996a) labora disturbance (Sousa, (Sousa, 11984). For true true metacommunities, metacommunities, Warren's Warren's ((1996a) laboratory tory protist protist studies studies varied varied the the rate rate of of destruction destruction and and reestablishment reestablishment of of local local communities communities and and showed showed that that disturbance disturbance (local (local patch patch destruction) destruction) resulted resulted only species loss only in in species loss from from the the metacommunity, metacommunity, but but disturbance disturbance was was important important protist richness (Fig. 6.3). in determining the effects of migration on protist Heterogeneity Heterogeneity among among local local habitat habitat patches patches could could be be caused caused by by aa variation variation in in biotic biotic or or abiotic abiotic conditions conditions and and is is predicted predicted to to increase increase metacommunity metacommunity diversity local communities diversity by by increasing increasing the the variety variety of of local communities and and forming forming refuges refuges for habitat specialists Loreau, 2002). for habitat specialists (Mouquet (Mouquet and and Loreau, 2002). Only Only two two studies studies have have directly Forbes and and directly manipulated manipulated heterogeneity heterogeneity within within aa metacommunity. metacommunity. Forbes Chase Chase (2002) (2002) varied varied the the arrangement arrangement of of experimental experimental zooplankton zooplankton communi communities ties to to influence influence the the probability probability of of moving moving from from similar similar to to dissimilar dissimilar habitats: habitats: arrangement arrangement had had no no effect effect on on local local or or regional regional diversity. diversity. Miller Miller et et al. al. (2003 (2003))
46 11 46
MATHEW N D THOMAS MATHEW A A.. LEIBOLD LEIBOLD A AND THOMAS EE.. MILLER MILLER
used used suites suites of of small small laboratory laboratory microcosms microcosms with with protist protist communities communities linked linked by by weekly weekly migration. migration. Two Two types types of of heterogeneity heterogeneity were were maintained: maintained: either either all all local local communities community were communities composing composing aa given given meta metacommunity were maintained maintained at at the the same same medium medium resource resource level level or or equal equal numbers numbers of of local local communities communities provided provided low, low, medium, and high high resource and Chase medium, and resource levels. levels. Like Like Forbes Forbes and Chase (2002), (2002), they they found found that variation had local diversity: diversity: species that such such variation had no no effect effect on on local species were were found found to to spe specialize cialize on on different different resource resource levels levels and and migration migration among among different different resource resource lev levels els had had little little effect effect on on establishment. establishment. Heterogeneity Heterogeneity did did lead lead to to higher higher regional regional diversity, because aa greater diversity, however, however, because greater regional regional diversity diversity of of resources resources allowed allowed lowpersist in low- and and high-resource high-resource specialists specialists to to persist in appropriate appropriate communities. communities. If If we we consider consider all all these these experimental experimental studies studies to to date, date, two two somewhat somewhat sur surprising prising findings findings emerge. emerge. First, First, migration migration may may play play less less of of aa role role in in structuring structuring communities thought. Indeed, some experiments communities than than previously previously thought. Indeed, some experiments found found no no effects effects of of migration migration on on local local diversity diversity (e.g., (e.g., Forbes Forbes and and Chase, Chase, 2002) 2002) and and other other studies results do studies found found only only minimal minimal effects. effects. Such Such results do not not preclude preclude the the possibil possibility ity of of important important effects effects of of migration migration on on community community composition composition through through species Tilman and 993, Leibold, 9 9 8 ) or that species sorting sorting (e.g., (e.g., Tilman and Pacala, Pacala, 11993, Leibold, 11998) or it it may may be be that that migration rates been sufficiently that experimental experimental migration rates have have not not been sufficiently high high enough enough to to affect mass effects (Fig. 66.1). . 1 ) . Second, number of affect diversity diversity through through mass effects (Fig. Second, aa number of other other fac factors, such regional heterogeneity, tors, such as as trophic trophic interactions, interactions, disturbance, disturbance, and and regional heterogeneity, must be incorporated of metacommunities; must be incorporated into into our our understanding understanding of metacommunities; in in particu particular, interactions with lar, these these factors factors may may have have important important interactions with migration. migration.
Evidence of Evidence of the the Role Role of of Trade-Offs Trade-Offs In in Metacommunltles Metacommunities Of Of greatest greatest interest interest to to us us in in this this chapter chapter are are studies studies that that looked looked explicitly explicitly at at trade-offs trade-offs in in species species traits traits and and how how these these trade-offs trade-offs may may affect affect species species persist persistence ence and and coexistence coexistence at at local local and and metacommunity metacommunity scales. scales. To To date, date, remarkably remarkably few few such such studies studies have have been been conducted conducted and and most most deal deal with with aa very very small small number number of species. Yu 1 ) investigated of coexisting coexisting species. Yu et et al. al. (200 (2001) investigated factors factors contributing contributing to to the the coexistence coexistence of of two two ant ant genera genera that that specialize specialize on on the the same same plant plant host. host. Census Census and and experimental experimental data data demonstrated demonstrated that that Azteca Azteca sp. sp. queens queens were were better better dis dispersers, persers, whereas whereas Allomerus Allomerus cf. cf. demerarae demerarae colonies colonies were were more more fecund, fecund, presum presumably ably making making Allomerus Allomerus aa better better competitor competitor on on any any given given host host plant. plant. As As aa result, result, species species success success was was aa function function of of interplant interplant distance: distance: at at low low plant plant densities, densities, the the Azteca Azteca sp. sp. was was dominant, dominant, whereas whereas at at high high densities, densities, the the better better competitor, competitor, Allomerus, prevailed. Spatial heterogeneity host-plant densities densities allowed allowed Allomerus, prevailed. Spatial heterogeneity in in host-plant regional coexistence regional coexistence due due to to this this trade-off trade-off in in migration migration and and competitive competitive abilities. abilities. This possible "niche This result result is is intriguing intriguing because because it it highlights highlights aa novel novel possible "niche axis" axis" that that might communities, heterogeneity might be be important important in in meta metacommunities, heterogeneity in in the the local local density density of of patches community (Yu 1 ) . Lei patches within within aa meta metacommunity (Yu and and Wilson, Wilson, 200 2001). Lei and and Hanski Hanski ((1998) 1998) documented documented aa similar similar trade-off trade-off between between competition competition and and colonization colonization in parasitoids attacking in aa system system of of parasitoids attacking the the Glanville Glanville fritillary fritillary butterfly butterfly (Melitaea (Melitaea cinxia). local density large role community, as cinxia). Again, Again, local density may may play play aa large role in in this this meta metacommunity, as changes changes in in host host density density result result in in aa decline decline in in the the proportion proportion of of populations populations attacked attacked by by the the better better parasitoid parasitoid competitor competitor but but have have little little effect effect on on the the pro proportion (Fig. 6.5). portion of of populations populations attacked attacked by by the the better better colonizer colonizer (Fig. 6.5). Working Working on on the the community community of of herbivores herbivores on on ragwort ragwort (Senecio (Senecio jacobea), jacobea), Harrison Harrison et et al. al. ((1995) 1 995) found found weak weak evidence evidence for for aa colonization-competition colonization-competition trade-off trade-off and and
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Changes parasitoids, Cotesia Cotesia melitaearum melitaearum and Hyposoter Hyposoter Changes in the abundances abundances of two parasitoids, As the number of host host populations hortico/a, horticola, and their host, host, the butterfly butterfly Melitaea Mefitaea cinxia. cinxia. As declines, populations attacked attacked by the better competitor, competitor, C. declines, the proportion proportion of populations C. melitaearum, melitaearum, declines, proportion attacked attacked by the better colonizer, H. hortico/a, relatively declines, whereas the proportion colonizer, H. horticola, is relatively Fig. 6.5 Fig. 6.5
unaffected. From (1 998). From Lei Lei and Hanski Hanski (1998).
argued trade-off was was present, it was was not not important important in in regulating regulating argued that, that, even even if if aa trade-off present, it metacommunity They hypothesized hypothesized that that other mechanisms (possibly (possibly metacommunity dynamics. dynamics. They other mechanisms related related to to patch patch type type specialization specialization and and species species sorting) sorting) were were most most likely likely allow allowing ing for for coexistence coexistence in in this this system. system. Two Two studies studies have have more more directly directly tested tested for for the the importance importance of of trade-offs trade-offs that that operate spatial scales. scales. Amarasekare Amarasekare (2000) para operate at at different different spatial (2000) studied studied two two egg egg parasitoids insect. She sitoids that that both both use use the the same same host host insect. She hypothesized hypothesized that that aa trade-off trade-off at at either either of of two two different different scales scales might might explain explain the the coexistence coexistence of of the the two. two. At At the the inferior competitor competitor might might be better better able to find unparasitized unparasitized local scale, the inferior hosts predation. At scale, the hosts and and thus thus avoid avoid intraguild intraguild predation. At the the regional regional scale, the inferior inferior competitor allowing it persist in competitor may may have have higher higher migration migration abilities, abilities, allowing it to to persist in host host patches patches isolated isolated by by distance. distance. Manipulative Manipulative studies studies and and field field surveys surveys were were used used to to demonstrate demonstrate that that the the loss loss of of the the dominant dominant competitor competitor was was related related not not to to host host isolation, isolation, but but to to host host productivity. productivity. Coexistence Coexistence was was therefore therefore best best explained explained by by aa trade-off trade-off between between competitive competitive ability ability and and the the ability ability to to find find unparasitized unparasitized hosts hosts at at the the local local scale scale and and not not by by larger larger scale scale variation variation in in abundance. Again, Again, species species sorting sorting by by local local patch-type patch-type attributes attributes was was host abundance. surprisingly surprisingly important important in in this this system, system, which which might might at at first first have have appeared appeared to community structured to correspond correspond more more closely closely to to aa meta metacommunity structured by by aa coloniza colonization-biotic-ability tion-biotic-ability trade-off. trade-off. The The communities communities of of bacteria bacteria and and invertebrates invertebrates found found in in the the water-filled water-filled leaves of leaves of pitcher pitcher plants plants have have the the advantage advantage of of occurring occurring at at discrete discrete scales: scales: within individual local) and among individual leaves ((local) among leaves within within a population population (regional (regional or or metacommunity). metacommunity). Kneitel Kneitel (2003) (2003) proposed proposed to to quantify quantify the the strengths strengths of of trade-offs trade-offs that that may may operate operate at at these these two two different different scales. scales. He He quan quantified tified the the growth growth and and colonization colonization rate rate of of protists protists and and rotifers rotifers found found in in the the
1148 48
MATHEW MATHEW A A.. LEIBOLD LEIBOLD AND AND THOMAS THOMAS EE.. MILLER MILLER
midtrophic-Ievel midtrophic-level assemblage of these inquiline communities. To To quantify trade-offs, he also determined the competitive ability and tolerance of preda predation for each of five species using a combination combination of laboratory and field experi experiments. ments. Identical Identical experiments experiments were were conducted conducted at at each each of of two two resource resource levels, levels, representing different types of local patches. There was little evidence of a trade-off species interactions trade-off between colonization colonization ability and traits related to species (competitive (competitive ability ability or or predator predator tolerance). tolerance). Predator Predator tolerance tolerance and and competitive competitive ability generally showed showed a strong negative relationship relationship (Fig. 6.6). Further, this negative relationship strongest between relationship was strongest between competitive competitive ability at one resource level predator tolerance indicating that resource level and and predator tolerance at at the the other, other, indicating that this this trade-off trade-off specialization. According to Kneitel, these data sug sugcould lead to patch-type specialization. gest that species are specialized to do well under different biotic conditions: these species coexist coexist because of of a trade-off trade-off between competitive competitive ability and among-leaf scale. predator tolerance that operates at the among-leaf Clearly, Clearly, any any strong strong conclusions conclusions about about how how metacommunity metacommunity dynamics dynamics work work in natural systems are premature. Few studies have really tackled the difficult issues involved involved in evaluating evaluating complex complex interactions interactions among among migration, migration, patch heterogeneity, heterogeneity, and disturbance in naturally existing regional species assem assemblages. been done relate any blages. Further, Further, even even less less has has been done to to relate any of of these these results results to to the the pat pattern of of covariation covariation among among species traits (or to determine how how this covariation covariation possible trade-offs) trade-offs).. Nevertheless, Nevertheless, a few few generalizations warresults in possible generalizations may be war ranted. First, migration can have large effects effects on local and regional diversity, evibut its role may depend strongly on other factors. Second, some of the evi indicates that patch-type heterogeneity heterogeneity and its effect effect on species sorting dence indicates not always be the case (Hubbell, (Hubbell, 2001; 2001; Bell, are often very strong. This may not 2000), depending from some systems is interpreted 2000), depending on how how evidence evidence from some of of these systems (see Chave et aI., long history al., 2002; 2002; Chave Chave and Leigh, 2002) 2002).. Finally, despite a long
44 3.5 3.5
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c.. 13_
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Relationship between between the the average competitive competitive effect of inquiline Fig. 66.6 Fig. . 6 Relationship of pitcher pitcher plant inquiline species and predation. The The competitive and (a) (a) colonization rate and (b) (b) tolerance to to predation. competitive effect effect was the the species in suppressing suppressing the the growth growth of of competitors competitors in pairwise pairwise competicompeti average effect of each species tion. Predator Predator tolerance rate of each species in the tolerance is the the standardized standardized growth growth rate the presence of of larlar of the the mosquito mosquito Wyeomyia Wyeomyia smithii. smithii. The The colonization colonization rate rate was determined determined by by the the rate rate of of vae of migration migration of of each species species into into unoccupied unoccupied pitchers in the the field. field. Four Four of of the the species species are protoproto zoans (BO, Bodo; CO, CO, Colpoda; Co/poda; CI, Cyclidium; Cyclidium; and and CH, chrysomonad chrysomonad sp.); sp.); the the fifth fifth species is a (BO, Bodo; rotifer (Habrotochus rosea) rotifer (Habrotochus rosea) from from Kneitel (2003). (2003).
FROM METAPOPULATIONS METAPOPULATIONS TO TO METACOMMUNITIES METACOMMUNITIES 6. FROM
49 1149
coloin ecological theory, surprisingly little evidence supports the view that a colo nization-competition trade-off is important important in allowing species species coexistence or nization-competition otherwise influencing diversity. A lack of evidence should not necessarily be interpreted interpreted as as aa general general pattern; pattern; instead, instead, it it should should be be aa call call for for further further study study on the role of trade-offs operating at different spatial and temporal scales.
6.5 6.S
DISCUSSION DISCUSSION In an intriguing theoretical 1 992) theoretical paper paper on metacommumtles, metacommunities, Wilson ((1992) predicted predicted that that aa surprisingly surprisingly rich rich diversity diversity of of behaviors, behaviors, including including reciprocal reciprocal relations between local population population dynamics and metacommunity metacommunity dynamics, communities in metacommunities, metacommunities, and the persistence of alternative stable communities that might might affect the nature nature of ecosystems. Wilson ((1992) higher level selection that 1 992) showed that that these phenomena phenomena were possible under at least some metacommun metacommunframeworks, but but he did not not identify the conditions conditions that that would would influence ity frameworks, occurrence. Work done since then has explored only some of the possi possitheir occurrence. much remains remains to be done. bilities, and much Nonetheless, Nonetheless, research research in in this this area area is is of of great great potential potential importance importance because because it it that many aspects aspects of population population biology, community community is increasingly obvious that structure, structure, and and ecosystem ecosystem dynamics dynamics are are influenced influenced by by metacommunity metacommunity processes. processes. Furthermore, Furthermore, numerous numerous environmental environmental issues issues can can be be related related directly directly to these processes. Habitat Habitat fragmentation, fragmentation, for example, example, creates patches and isolation ((and therefore interpatch interpatch migration). Conversely, humans humans affects their isolation and therefore facilitate facilitate the the movements movements of of other other organisms, organisms, leading leading to to novel novel associations associations of of metacommunities. Furthermore, Furthermore, humans humans alter alter environmental environmental condi condispecies in metacommunities. tions in local patches patches in a number number of ways that that alter alter the local fitness fitness of organ organisms. local scales scales and isms. Finally, Finally, humans humans also also influence influence extinctions, extinctions, both both on on local and over over and global scales. These effects are just as likely to alter meta metalarger regional and community dynamics dynamics as they are to to affect local ecological processes. community This has tried metacommunity concepts This chapter chapter has tried to to emphasize emphasize the the growth growth of of metacommunity concepts from their their origins in meta metapopulation theory to current relevance in reevalupopulation theory to their current reevalu community theory. In particular, particular, we argue that that although although ating many many aspects of community metapopulation 4), metapopulation theory theory emphasizes emphasizes understanding understanding species species persistence persistence (Chapter (Chapter 4), metacommunity species coexist metacommunity approaches approaches are are directed directed toward toward explaining explaining species coexistence, diversity, and their their consequences. consequences. Understanding coexistence entails entails ence, diversity, and Understanding coexistence understanding trade-offs in in species' species' responses responses to to different different environmental environmental facfac understanding trade-offs tors, both both biotic and and abiotic. abiotic. Although Although trade-offs trade-offs have long served as the the corcor tors, community theory, theory, metacommunity metacommunity dynamics draw draw attention attention to to nerstone of community traits that that operate operate at at larger scales and and especially to to migration migration and and the the heterohetero traits geneity of of local local habitats. habitats. Among-habitat Among-habitat migration migration is is one one of of the the defining traits geneity defining traits of metacommunities and and may frequently to diversity (if of metacommunities may act act frequently to increase increase not not only only diversity migration rates rates are greater than than extinction extinction rates), rates) , but community composition composition migration are greater but community through species sorting. sorting. Habitat heterogeneity influences through Habitat heterogeneity influences the the scale at at which which species trade-offs trade-offs are important: important: homogeneity homogeneity of of habitats habitats may may require that that trade-offs allow coexistence within within local habitats, habitats, whereas whereas heterogeneity of of trade-offs habitats expands expands the the scale scale at at which which coexistence coexistence may may occur. habitats At this this stage, stage, we we are are still still learning learning much much about about the the roles roles of of altered altered migramigra At tion, tion, local local adaptation, adaptation, and and extinctions extinctions in metacommunities. metacommunities. Our Our knowledge knowledge to to
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MATHEW MATHEW A. A. LEIBOLD LEIBOLD AND AND THOMAS THOMAS E. E. MILLER MILLER
date of of these these effects effects shows shows that that they they can can be be complex complex and and multivaried, multivaried, but but studstud date ies have have also also produced produced aa surprising surprising richness richness of of results results that that we we are are only only beginbegin ies ning to to perceive. perceive. Nonetheless, Nonetheless, we we also also think think that that aa couple couple of of generalities generalities are are ning beginning to to emerge emerge from from both both aa theoretical theoretical and and an an empirical empirical perspective. perspective. beginning Experimental work work on on migration migration has has suggested suggested its its importance, importance, but but itit may may play play Experimental larger role role in in community community composition composition than than in in community community diversity. diversity. In In addiaddi aa larger tion, habitat habitat heterogeneity heterogeneity appears appears to to be be very very important important for for metacommunity metacommunity tion, dynamics, but but very very little little experimental experimental work work has has been been done done in in this this area. area. Finally, Finally, dynamics, very few few studies studies of of metacommunities metacommunities have have discussed discussed or or quantified quantified species species very trade-offs, aa clear clear call call for for further further experimental experimental work work in in this this area. area. trade-offs, A metacommunity metacommunity perspective, perspective, in in which which scale scale is is incorporated incorporated directly directly into into A our thinking thinking in in the the form form of of local local habitats habitats linked linked by by migration, migration, is is not not aa new new our view for for ecologists, ecologists, but but recent recent theoretical theoretical and and experimental experimental work work in in this this area area view demonstrates significant significant promise. promise. A A metacommunity metacommunity view view is is clearly important, demonstrates clearly important, not just just in in explaining explaining diversity diversity at at the the landscape landscape level level (something (something that that has has been been not obvious for for a long time), but but also also in constraining constraining the role that that disturbance, disturbance, obvious long time), the role migration, and and heterogeneity heterogeneity can can play play in regulating regulating biodiversity biodiversity at at local levels. migration, local levels. We that this may also also make make strong strong contributions to underunder We feel feel that this approach approach may contributions to standing community community composition composition through through understanding understanding traits traits that allow standing that allow coexistence at at different different spatial spatial scales. scales. A A rich rich array array of of issues issues awaits awaits both both theortheor coexistence etical and and empirical empirical investigations. etical investigations.
Part III Metapopulation Genetics
sdfsdf
O
SELECTION AND SELECTION AND DRIFT IN IN META PO PULATIONS M ETAPO PU LATIO N S Hanski and G aggiotti, by by Michael Michael C. Whitlock Whitlock Hanski and Gaggiotti,
7.1 7.1
INTRODUCTION INTRODUCTION The distribution distribution of of aa species species over over space space has has many many interesting interesting and and important important The evolutionary population genetic evolutionary consequences. consequences. All All of of the the basic basic population genetic forces forces m drift, drift, selection, recombination spatially selection, migration, migration, mutation, mutation, and and recombination m act act differently differently in in aa spatially structured enhanced or structured population. population. Genetic Genetic drift drift can can be be enhanced or diminished diminished relative relative to to aa panmictic population of panmictic population of the the same same total total size. size. Selection Selection can can be be more more or or less less effec effective. tive. Migration Migration is is impossible impossible without without aa spatial spatial context; context; the the consequences consequences of of muta mutations tions tend tend to to be be lowered, lowered, and and the the effective effective recombination recombination rate rate is is reduced. reduced. This This chapter chapter reviews reviews some some of of the the effects effects of of population population structure, structure, in in particular particular focus focusing ing on on how how selection selection and and drift drift are are changed changed by by the the fact fact that that species species exist exist in in space. space. This This chapter chapter takes takes aa heuristic heuristic and and largely largely nonmathematical nonmathematical look look at at these these issues, issues, trying trying to to express express intuitively intuitively some some recent recent results results in in spatial spatial population population genetics. genetics. This chapter chapter focuses focuses on on the the dynamics dynamics of of aa single single locus, locus, whereas whereas the the topics topics of of multi multiThis 1, and locus locus selection selection and and quantitative quantitative genetics genetics are are discussed discussed in in Chapters Chapters 9, 9, 111, and 12. One One very very important important summary summary statistic statistic about about the the effects effects of of population population structure structure turns turns out out to to be be one one of of the the oldest: oldest: Wright's Wright's FST• FST. There There are are several several ways ways to to define define FST, Fsv, but but they they all all are are standardized standardized measures measures of of the the genetic genetic differentiation differentiation among among populations. populations. Here Here let let us us define define FST Fsw as as the the variance variance in in allele Vamong ), standardized allele frequencies frequencies across across populations populations ((Vamong), standardized by by the the mean mean allele allele frequency frequency (p) (p):: FST Fsy == V Vamong/p(1 - pl. p). FST FST has has several several key key features features that that make make amongfp( 1 -
enetics, and Ecology, Ecology,G Genetics, and Evolution Evolution of of Metapopulations Metapopulations
53 1 53
evier, Inc. Copyright Copyright 2004, Els Elsevier, Inc. 0-12-323448-4 0-12-323448-4
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
11 54 54
it it useful useful and and interesting interesting for for the the study study of of evolution evolution in in structured structured populations. populations. FsT has has the the same same expectation expectation for for all all neutral neutral autosomal autosomal loci, loci, although although First, FST even neutral neutral loci loci can can vary vary substantially substantially around around this this expectation. expectation. Moreover, Moreover, even this this expectation expectation is is determined determined by by the the demographic demographic properties properties of of the the species, species, such such as as migration migration rates, rates, local local population population sizes, sizes, and and geography geography FST FsTcan can there therefore encapsulate encapsulate a lot of useful information information about about the demographic demographic history history of fore species. FST FsTtends tends to to be be larger larger if if local local populations populations are are not not connected connected by by high high aa species. rates rates of of migration migration and/or and/or if if local local population population sizes sizes are are small. small. Finally, Finally, FST FsT is is readily readily measurable measurable from from easily easily obtained obtained data data on on real real populations, populations, and and there there is is already already aa lot lot of of information information about about FST FsT in in nature. nature. There There are are other other useful useful ways ways to to view view the the information information conveyed conveyed by by FST FsT beyond beyond its its use use as as aa measure measure of of genetic genetic variance variance among among populations. populations. FST FsT is is indication of the amount amount of relatedness among individuals in the same also an indication demes. demes. If If FST FsT is is high, high, then then individuals individuals in in the the same same demes demes are are highly highly related related to to one one another; another; in in other other words, words, they they share share many many alleles. alleles. All All else else being being equal, equal, this this also pertains to alleles within aa diploid diploid individual: individual: if if FST FsT is is high, high, individuals individuals also pertains to alleles within to be homozygous homozygous than are more more likely to than would would be predicted predicted by by Hardy-Weinberg various interpretations Hardy-Weinberg frequencies. frequencies. These These various interpretations and and implications implications FSTare are useful useful in in interpreting interpreting the the results results that that follow. follow. It It turns turns out out that that because because of FST FsTrepresents represents both both the the relatedness relatedness of of individuals individuals within within aa de deme and the the excess excess FST me and homozygosity, homozygosity, it it is is often often the the only only extra extra parameter parameter needed needed to to describe describe how how population structure structure changes the pace of evolution. evolution. population This This chapter chapter reviews reviews the the effects effects of of spatial spatial population population structure structure on on the the amount amount response to selection. The greater greater part part of the chapter chapter then then of genetic drift and the response uses these these results results to to discuss discuss basic basic evolutionary evolutionary genetic genetic quantities quantities in in structured structured uses populations, such balance point point between between mutation mutation and and selection, selection, mutamuta populations, such as as the the balance inbreeding depression, depression, the probability of fixation tion load, load, inbreeding the probability fixation of of new new alleles, alleles, and and other basic quantities. out that these fundamental fundamental evolutionary other basic quantities. It It turns turns out that these evolutionary processes are are sometimes population structure. structure. processes sometimes strongly strongly affected affected by by even even aa weak weak population
7.2 7.2
GENE FREQUENCY CHANGE CHANGE IN IN METAPOPULATIONS METAPOPULATIONS GENE FREQUENCY Gene frequency can species by four mechanisms: mechanisms: selection, drift, Gene frequency can change change in in aa species by four selection, drift, introgression species, and introgression from from other other species, and mutation. mutation. This This section section reviews reviews mathmath ematical models structure in two more ematical models that that show show the the effect effect of of population population structure in the the two more important of of these these forces, forces, selection selection and and drift. drift. important
Genetic Genetic Drift Drift Genetic drift drift is is the the change change in in allele allele frequency frequency from from one one generation generation to to the the Genetic next caused caused by by random random sampling sampling of of alleles. alleles. Genetic Genetic drift is nondirectional, nondirection aI, next drift is meaning that that the the average average change change due due to to drift is zero, zero, but but as as the the population population size size meaning drift is gets small, frequency in generation can gets small, the the actual actual change change in in allele allele frequency in any any given given generation can be be relatively large. relatively large.
Effective Effective Population Population Size The smaller smaller the the effective effective population population size, size, the the more more random random effects effects can can The become become important. important. A A key key term term here here is is "effective" "effective" m - the the actual actual amount amount of of genetic drift in in aa population population is is determined determined not not only only by by the the actual actual number number genetic drift
7. 7. SELECTION SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS
11155 55
of individuals also by of individuals in in the the population, population, but but also by other other factors, factors, such such as as the the distri distribution species. The bution of of reproductive reproductive success success in in the the species. The effective effective size, size, Ne, of of aa pop population ulation is is defined defined as as the the size size of of an an ideal ideal population, population, which which would would be be expected expected to to have have the the same same amount amount of of genetic genetic drift drift as as the the population population in in question. question. An An ideal ideal population population is is one one in in which which each each of of the the alleles alleles in in the the offspring offspring generation generation have have an an equal equal and and independent independent chance chance of of having having come come from from each each of of the the parental alleles. ideal population population would parent parental alleles. An An ideal would function function as as though though each each parent allele number of pool, and allele contributes contributes an an equal equal and and large large number of copies copies to to aa gamete gamete pool, and then then offspring offspring would would be be formed formed by by random random draws draws from from this this gamete gamete pool. pool. Real Real populations populations are are not not ideal ideal though though for for several several reasons. reasons. First, First, and and most most importantly, importantly, in in real real populations, populations, each each individual individual is is not not expected expected to to contribute contribute equally equally to to the the next next generation: generation: some some are are very very fit fit and and have have aa high high reproductive reproductive suc success, cess, whereas whereas others others die die before before even even reproducing. reproducing. This This variance variance in in reproductive reproductive success increase the success tends tends to to reduce reduce the the effective effective population population size size and and therefore therefore increase the rate rate of of genetic genetic drift. drift. Second, Second, in in real real populations, populations, new new individuals individuals are are not not necessarily formed available alleles. alleles. For example, with necessarily formed at at random random from from the the available For example, with inbreeding, inbreeding, individuals individuals are are formed formed with with aa higher higher than than random random chance chance of of having having similar similar alleles alleles at at homologous homologous sites. sites. Such Such inbreeding inbreeding tends tends to to decrease decrease Ne because because each each individual individual effectively effectively carries carries fewer fewer copies copies of of alleles. alleles. Finally, Finally, both both variation variation in in reproductive success across generations, reproductive success and and nonrandom nonrandom mating mating can can be be inherited inherited across generations, and and the the correlations correlations in in reproductive reproductive success success which which result result can can also also affect affect Ne• Ne. In In structured structured populations, populations, these these three three factors factors are are even even more more important. important. When places, they When organisms organisms live live in in different different places, they are are likely likely to to experience experience different different conditions, variance in conditions, and and therefore therefore there there is is likely likely to to be be greater greater variance in reproductive reproductive success than than in in aa single single well-mixed population. Population structure causes causes success well-mixed population. Population structure kind of of inbreeding inbreeding because because locally locally mating mating individuals individuals are are likely likely to to be be related. related. aa kind Finally, Finally, if if local local conditions conditions are are correlated correlated positively positively from from one one generation generation to to the the next, next, variance variance in in reproductive reproductive success success will will also also be be correlated correlated among among parents parents and offspring, offspring, assuming assuming limited limited migration. migration. and The The effective effective size size of of structured structured populations populations has has been been well well reviewed reviewed by by Wang Wang and 1 999). and Caballero Caballero ((1999). The The Island Island Model Model
Describing Describing the the effective effective size size of of subdivided subdivided populations populations has has aa long long history, history, beginning in 11939. 939. In this paper, the effec beginning with with Sewall Sewall Wright Wright in In this paper, Wright Wright derives derives the effective species subdivided an island island model, be tive population population size size of of aa species subdivided by by an model, finding finding it it to to be
N e,Isiand Model Ne,Island Model = --
Nd Nd - ST
' 1 - FFST ' 1
-
(7. 1) (7.1)
where where N N is is the the number number ooff individuals individuals in in aa deme, deme, d d is is the the number number of of demes, demes, and and FST FST is is given, given, for for large large d d at at equilibrium, equilibrium, by by
FFsT, ST,Isiand Island Modei Model
== -~
11
44Nm Nm ++ l1"'
((7.2) 7.2)
Here, Here, m m is is the the migration migration rate rate among among demes. demes. In In the the island island model, model, each each deme deme contributes contributes aa proportion proportion m m of of its its individuals individuals to to aa migrant migrant pool pool and and then then receives receives the the same same number number of of migrants migrants chosen chosen randomly randomly from from that that migrant migrant
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
11 56 56
pool. It It is is important important to to note note that that these these are are not not random random proportions, proportions, but but that that pool. each deme deme gives gives and and receives receives exactly exactly Nm Nm individuals individuals to to and and from from the the migrant migrant each pool each each generation, generation, and and each each deme deme consists consists of of exactly exactly N N individuals. individuals. As As aa pool result, me contributes result, each each de deme contributes exactly exactly equally equally to to the the next next generation. generation. This This seemingly seemingly innocuous innocuous assumption assumption turns turns out out to to have have fairly fairly important important effects effects on on interpreting results results obtained obtained from from the the island island model. model. interpreting If each each deme deme contributes contributes exactly exactly equally equally to to the the next next generation, generation, then then there there If is no no variance variance in in reproductive reproductive success success among among demes. demes. We We know know from from classical classical is population population genetics genetics that that aa lower lower variance variance in in reproductive reproductive success success means means higher higher and in in fact fact this this is is the the case case with with structured structured populations as well. well. Look Look again again Ne, and populations as at 1 ) . Give at Eq. Eq. (7. (7.1). Give that that FST FsT is is aa quantity quantity that that ranges ranges between between 00 and and 11,, the the Ne for an an island island model model is is always always something something greater greater than than Nd, in in other other words words for greater than than the the total total number number of of individuals individuals in in the the metapopulation metapopulation as as aa whole. whole. greater This is is because because of of the the assumption assumption that that there there is is no no variance variance among among demes demes in in This reproductive success. success. reproductive
Model Assumptions Relaxing Island Model A more more general general model model of of the the effective effective size size of of structured structured populations populations has has A been derived derived (Whitlock (Whitlock and and Barton, Barton, 11997). The general general form form of of the the equation equation been 997). The for Ne in in aa species species that that has has reached reached demographic demographic equilibrium equilibrium is is given given by by for
Ne me = =
Ndd N
FsT i)·) + 22 2: 2: Nw4-( I I lNiW2(Ndl -FsT, ~.~ wwNN·p wiwI jNiINjPiIJNdI II·· + ~ 2:
--���--��------------���
il
Nd
,1
i1
j!
Nd
(7.3) (7.3)
shown to function of the local population sizes the relarela where where Ne m e is is shown to be be aa function of the local population sizes (Ni (mi), ) , the the FST predicted over over aa set tive contributions contributions of of each FsT predicted set of of demes demes tive each deme deme (wi), (Wi), the and the the correlation correlation among with properties such with demographic demographic properties such as as deme deme ii (FST), (FsT.i), and among which is is defined similar to to FsT, FST, but but instead using demes demes of of allelic allelic identity identity (Pi;, (Pii, which defined similar instead using covariance of of pairs demes).. This This equation few assumptions covariance pairs of of demes) equation makes makes few assumptions about about the of the the spatial spatial subdivision subdivision among among populations, allowing for for variable the nature nature of populations, allowing variable migration rates over different different population pairs, including including isolation migration rates over population pairs, isolation by by disdis tance, local changes changes in size, including local extinction, extinction, and tance, local in population population size, including local and new new population formation via via colonization or fission. population formation colonization or fission. While general, is aa bit for intuitive intuitive use. use. To aid in in explainexplain While general, Eq. Eq. (7.3 (7.3)) is bit unwieldy unwieldy for To aid ing few key ing aa few key features features of of this this result, result, let let us us use use aa simplified simplified version version of of this this equaequa tion few more tion that that makes makes aa few more assumptions. assumptions. If If all all demes demes have have the the same same size size as as each other, other, but but contribute contribute unequally unequally to to the the next next generation via differential differential each generation via migration, write V V as as the the variance variance among among demes demes in in the the expected expected migration, then then we we can can write reproductive success success of of individuals individuals from that deme deme (i.e., (i.e., V V == Var[wi]). Var [wil ). The The reproductive from that effective effective population population size size is is then then
_ Ne Ne = -
Nd
V ) ( 1l -( 1 ++ V)( (1
Nd
FST) + 2NFsTVdfi 2NFsTVd/( dd - 1) 1) FST)
(7.4) (7.4)
-
(Whitlock and Barton, (Whitlock and Barton, 1997). 1 997). Let us us examine examine two two extremes extremes using using Eq. Eq. (7.4). (7.4). If, If, as as in in the the traditional traditional island island Let model, model, the the variance variance among among demes demes in in reproductive reproductive success success is is zero, zero, then then Eq. Eq. (7.4) (7.4)
7. SELECTION SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS 7.
11557 7
reduces to to Eq. Eq. (7.1) ( 7. 1 ) [This [This is is true true not not only only for for the the island island model, model, but but for for any any reduces model for for which which each each deme deme is is equal equal in in size size and and contributes contributes exactly exactly equally equally to to model all other other demes, demes, provided provided that that all all demes demes are are ultimately ultimately reachable reachable by by each each deme deme all via migration migration (Nagylaki, (Nagylaki, 1982). 1 982). This This includes includes classic classic stepping stepping stone stone models.] models.] via At the the other other extreme extreme though, though, let let us us imagine imagine that that one one deme deme is is extremely extremely At successful and and produces produces all all of of the the offspring offspring that that fill fill all all dd of of the the demes. demes. In In this this successful case, we we would would intuitively intuitively predict predict that that the the effective effective population population size size of of the the case, whole system system should should be be the the same same as as the the size size of of the the single single successful successful deme, deme, and and whole Eq. (7.4), (7.4), with with appropriate appropriate modification, modification, shows shows us us that that this this is is in in fact fact the the case. case. Eq. (In (In this this extreme, extreme, the the FST would would be be zero zero and and the the variance variance among among demes demes of of 1 . ) This This extreme extreme example example tells tells us us allelic reproductive reproductive success success would would be be dd- 1.) allelic that Ne Ne can can be be much much smaller smaller than than the the census census size size with with population population structure. structure. that truth obviously obviously lies lies somewhere somewhere in in the the middle. middle. It It turns turns out out that that the the The The truth Ne isis boundary between between whether whether population population structure structure increases increases or or decreases decreases Ne boundary approximately whether whether or or not not demes demes have have greater greater or or less less variance variance in in reprorepro approximately ductive success success than than would would be be expected by aa Poisson Poisson distribution. In other other ductive expected by distribution. In words, if demic demic structure structure acts to increase increase variance variance in in reproductive reproductive success words, if acts to success relative to that that expected by chance, chance, N~ Ne would would be be reduced. reduced. Only Only if if the the effects effects relative to expected by of population population structure structure are are to to reduce reduce the the variance variance among among demes demes in in reproducreproduc of tive success success to to less less than than random random would would N~ Ne be be increased. increased. This This is is perhaps perhaps biobio tive logically yet this is the requirement for for the from the the simple logically unlikely, unlikely, yet this is the requirement the results results from simple island to hold qualitatively. In In real real species, is likely likely to to be be island model model to hold qualitatively. species, the the opposite opposite is true: demes are likely to to have have different resources, and true: different different demes are likely different amounts amounts of of resources, and different demes are to experience experience different different levels levels of of other other ecological fac different demes are likely likely to ecological factors that that might might affect success, such such as as levels levels of of parasitism, parasitism, disease, predation, tors affect success, disease, predation, weather weather fluctuations, fluctuations, and and other other catastrophes. catastrophes. Realistic Realistic ecology ecology implies implies higher higher than than random random variance variance among among demes demes in in reproductive reproductive success, success, and and therefore therefore the the effective effective size size of of aa subdivided subdivided species species is is likely likely to to be be reduced, reduced, perhaps perhaps substan substantially. tially. The The island island model model is is not not aa good good descriptor descriptor of of typical typical population population struc structure, Whitlock and 999). ture, for for this this and and many many other other reasons reasons (see (see Whitlock and McCauley, McCauley, 11999). -
Extinction n d Colonization Extinction aand Colonization
It It will will be be useful useful to to consider consider aa couple couple of of specific specific cases cases that that go go beyond beyond the the simple island model. One aspect of population structure that has attracted simple island model. One aspect of population structure that has attracted some some attention attention is is the the possibility possibility of of local local extinction extinction and and recolonization recolonization (Slatkin, (Slatkin, 11977; 977; Maruyama 980; Whitlock 997). The Maruyama and and Kimura, Kimura, 11980; Whitlock and and Barton, Barton, 11997). The mod models considered in these papers are similar: the basic structure is like an island els considered in these papers are similar: the basic structure is like an island model, model, except except that that each each deme deme has has some some chance chance per per generation generation of of going going extinct extinct independently of its genotype frequencies. An equal number of new independently of its genotype frequencies. An equal number of new demes demes are are colonized, colonized, either either in in the the same same places places recently recently vacated vacated by by the the extinction extinction events events or or in in other other vacant vacant sites, sites, by by aa small small number number of of individuals. individuals. As As aa major, major, unreal unrealistic istic simplification, simplification, each each new new deme deme then then immediately immediately grows grows back back to to N N individuals, individuals, like like all all other other demes. demes. With With local local extinction extinction and and recolonization, recolonization, population population structure structure contributes contributes in in an an obvious obvious way way to to the the variance variance in in reproductive reproductive success success among among demes. demes. Even Even though though this this model model is is based based on on the the island island model, model, even even aa small small rate rate of of extinc extinction tion is is enough enough to to cause cause the the effective effective population population size size of of the the species species to to be be reduced reduced rather rather than than increased. increased. The The main main reason reason is is perhaps perhaps obvious: obvious: with with extinction extinction and and recolonization, recolonization, some some demes demes have have zero zero reproductive reproductive success, success,
1158 58
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK 11.2 .2 1
0.8 NNe e 0.8 Nd 0.6 0.6 Nd ': -
0.4 0.4 0.2 0.2
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Fig. 7.1 7.1 The The effective effective size size (displayed (displayed as as aa proportion proportion of of the the census census size) size) of of aa metapopulation metapopulation with local local extinction extinction and and colonization. colonization. Here Here each each deme deme has has 1100 individuals, and and each each new new popu popuwith 00 individuals, lation lation is is founded founded by by four four individuals. individuals. These These colonists colonists have have aa probability probability ~ that that they they come come from from the /2 in the same same source source population, population, with with &= = 0 0 in in the the dotted dotted line, line, ~ = - 11/2 in the the dashed dashed line, line, and and = = 11 in in the the solid solid line. line. The The migration migration rate rate was was 0.01 0.01 in in all all examples. examples. As As the the local local extinction extinction rate rate increases, the the effective effective population population size size is is reduced reduced greatly. greatly. increases,
whereas manage to survive and whereas others others - - those those that that manage to survive and send send colonists colonists to to start start new new demes demes m have have aa reproductive reproductive jackpot. jackpot. Thus Thus there there is is aa great great deal deal of of vari variance among among demes demes in in reproductive reproductive success, success, which which causes causes the the effective effective size size to to ance This reduction reduction can be extreme extreme (Fig. (Fig. 7. 7.1). reduced. This can be 1 ). be reduced. Sources and Sinks Sources and Sinks
In In some some species, species, some some populations populations have have large large amounts amounts of of resources, resources, whereas whereas others that they replace themselves themselves without without migration migration others have have so so few few that they cannot cannot replace (Pulliam, 1988; Dias, 996; Holt and Gaines, 1 992). These so-called "sources" "sources" (Pulliam, 1988; Dias, 11996; Holt and Gaines, 1992). These so-called and population dynamics to be be different different from and "sinks," "sinks," respectively, respectively, cause cause the the population dynamics to from the demes do contribute equally migrant pool, pool, and and the island island model: model: demes do not not contribute equally to to the the migrant therefore there there is is variance in reproductive reproductive success. of patches patches of of therefore variance in success. If If the the quality quality of is even even more more resource is is correlated correlated positively positively over over time, time, then then the the effect on Ne is resource effect on extreme. extreme. The effects of source-sink source-sink structure and correlation correlation over over time time in in patch patch suitsuit The effects of structure and ability ability can can be be best best seen seen by by another another extreme extreme example. example. Imagine Imagine that that aa fraction fraction of other 80% of demes, demes, say say 20%, 20%, reside reside in in productive productive source source patches, patches, and and the the other 80% of of demes are are what what Bob Bob Holt Holt has has called called "black-hole" " black-hole" sinks sinks m - that that is, is, these these demes demes demes never never contribute contribute migrants migrants to to other other demes demes and and only only persist persist because because of of migramigra tion from from source source populations. populations. In In this this case, case, itit is is clear clear that that only only alleles alleles in in indiindi tion viduals in in source source populations populations can can contribute contribute to to future future generations generations and and so so the the viduals only only individuals individuals that that matter matter to to the the evolution evolution of of the the species species are are in in the the source source populations. Therefore Therefore the the Ne Ne of of the the species species should should reflect reflect only only the the effective effective populations. of this this species species should should be be only only size of of the the source source populations populations alone. alone. Thus Thus the the Ne of size 20% of of what what itit would would have have been been with with equal equal migration. migration. 20% To be be more more general, general, we we can can apply apply useful useful results results from from Nagylaki Nagylaki (1982), ( 1 982), who who To showed that that the the Ne Ne of of aa system system of of populations populations with with aa constant constant migration migration showed matrix could could be be described described with with the the left left eigenvector eigenvector of of that that matrix. matrix. (This (This matrix assumes aa few few technical technical details, details, such such that that all all demes demes are are ultimately ultimately reachable reachable assumes by migration migration from from all all other other demes, demes, even even ifif itit takes takes multiple multiple steps.) steps. ) Consider Consider aa by are case case where where migration migration is is via via aa migrant migrant pool pool so so all all emigrants emigrants from from all all demes demes are
7. 7.
SELECTION IN METAPOPULATIONS SELECTIONAND AND DRIFT DRIFT IN METAPOPULATIONS
1 59 59
1
0.8 0.8 N Nee N Ntot tot
0.6 0.6
0.4 0.4 0.2 0.2 0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.8
1
Relative Relative contribution contribution of of sink sink populations populations Fig. 7.2 7.2 The The effective effective size size of of aa species species in in which which 20% 20% of of the the demes demes are are sources sources and and the the rest rest axis varies varies the the contribution the sink are The x axis are sinks. sinks. The contribution of of the sink populations, populations, expressed expressed as as aa fraction fraction of of the Here each each 1100 00 demes 00 individuals, and each each receives the contribution contribution of of the the sources. sources. Here demes have have 1100 individuals, and receives five five immigrants immigrants per per generation generation sampled sampled from from the the migrant migrant pool. pool. As As the the contribution contribution of of sinks sinks reaches same as model with reaches zero, zero, the the effective effective size size of of the the system system is is the the same as an an island island model with only only the the 20 20 source source populations. populations.
mixed mixed together together and and then then moved moved on on to to recipient recipient demes demes at at random random with with respect respect to to where where they they originate. originate. Source Source demes demes contribute contribute aa large large number number to to the the migrant migrant pool, pool, whereas whereas "sink" "sink" demes demes contribute contribute aa fraction fraction of of that that number. number. For For simplicity, simplicity, each each deme deme receives receives aa constant constant number number of of immigrants immigrants from from the the migrant ensures that migrant pool. pool. This This ensures that the the FST Fsy among among sources sources and and among among sinks sinks are are approximately equal. Figure approximately equal. Figure 7.2 7.2 shows shows the the effective effective size size of of these these systems systems as as aa function function of of the the relative relative contribution contribution to to the the migrant migrant pool pool by by sinks. sinks. [To [To make make the 7.2, Nagylaki's 1 982) results the calculations calculations in in Fig. Fig. 7.2, Nagylaki's ((1982) results were were used, used, accounting accounting here. for for the the fact fact that that Nagylaki's Nagylaki's definition definition of of Ne Ne differs differs from from the the usage usage here. Nagylaki calculates the variance Nagylaki calculates the Ne that that would would give give the the same same amount amount of of variance within balance; in within aa deme deme at at mutation-migration-drift mutation-migration-drift balance; in other other work work including including in in this this chapter chapter Ne predicts predicts the the amount amount of of variance variance predicted predicted by by the the average average The second of these two two quantities allele frequency of the species as a whole. whole. The can can be be found found from from the the first first by by dividing dividing by by I-FsT• l-EsT. Details Details are are given given in in Whitlock Whitlock (2003).] (2003).1 Note Note that that with with this this form form of of source-sink source-sink structure, structure, the the effective effective size size of of the the populations when when the sinks species is just the effective size of just the source populations not contribute contribute to the future, future, and it reduces to the island model results when when do not "sinks" "sinks" contribute contribute equally equally to to sources. sources.
Selection Selection With With good good reason, reason, the the study study of of selection selection in in subdivided subdivided populations populations has, has, in in the the past, past, focused focused on on the the effects effects of of spatially spatially heterogeneous heterogeneous selection selection (e.g., (e.g., Felsenstein, 976) . A Felsenstein, 11976). A great great deal deal of of important important and and interesting interesting evolutionary evolutionary biol biology ogy results results from from variation variation in in selection selection over over space, space, but but population population structure, structure, perhaps surprisingly, perhaps surprisingly, has has aa lot lot of of interesting interesting effects effects even even on on uniform uniform selection. selection. Arguably, Arguably, most most loci loci have have approximately approximately similar similar selection selection in in different different demes, demes, even even though though the the more more obvious obvious and and more more polymorphic polymorphic cases cases may may reflect reflect spa spatially tially divergent divergent selection. selection. This This chapter chapter focuses focuses on on this this special special case case in in which which genotypes have same relative genotypes have the the same relative fitness fitness in in each each population population of of the the species. species.
1160 60
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
When When selection selection is is uniform uniform across across populations, populations, it it becomes becomes possible possible to to follow follow the population by the state state of of the the meta metapopulation by following following the the mean mean allele allele frequency frequency across across all all local local populations, populations, q. ~. Consider Consider simple simple selection selection between between two two alleles alleles at at the the same same locus, locus, with with the the fitnesses fitnesses of of the the three three genotypes genotypes given given by by 11 :: 11 + + h h ss :: 11 + + s. s. In this case, case, the the change change in in allele allele frequency frequency due due to to selection selection within within each each population population this is is aa third-order third-order function function of of q; q; therefore, therefore, to to understand understand how how the the mean mean allele allele fre frequency quency would would change change by by selection selection requires requires knowing knowing the the expected expected values values of of q, q, 2, and 3 . Fortunately, qq2, and qq3. Fortunately, under under most most circumstances circumstances the the dynamics dynamics of of the the expected expected value value of of qq33 can be be well well enough enough predicted predicted by by an an understanding understanding of of changes changes in in the the which reduces the problem to understanding q~ and and E[ E[q2]. The expected expected first two, which q2] . The value value of of qq22 may may seem seem like like an an exotic exotic quantity quantity to to keep keep track track of, of, but but remember remember that that 2] , and the the variance variance among among demes demes is is derived derived easily easily from from q ~ and and E[q E[q2], and FST Fsy is is derived derived easily from from the the variance variance in in allele allele frequency frequency and and q. ~. Thus, Thus, aa very very good good under undereasily standing of of the the change change in in allele allele frequency frequency across across aa meta metapopulation can be be standing population can obtained obtained by by knowing knowing q ~ and and FST. FST. Moreover, Moreover, as as long long as as the the selection selection coefficient coefficient is is not not much much greater greater than than the the rate rate of of migration migration into into aa deme, deme, the the FST FST predicted predicted from from neutral works extremely neutral theory theory works extremely well well to to predict predict allele allele frequency frequency change change in in struc structured tured populations. populations. These These conclusions conclusions are are derived derived and and discussed discussed in in greater greater detail detail (2002).) . in Whitlock (2002 [One note is these quantities, [One technical technical note is necessary: necessary: when when calculating calculating these quantities, it it is is essential weight each essential to to weight each individual individual equally. equally. The The usual usual calculations calculations of of FST FST weight each Most models weight each local local population population equally, equally, independent independent of of size. size. Most models of of population have assumed and therefore they population structure structure have assumed equal equal deme deme sizes, sizes, and therefore they predict right quantity. not measure predict the the right quantity. Most Most empirical empirical measures measures do do not measure the the appropriate appropriate FST FST exactly. exactly. This This may may be be an an important important issue issue in in some some cases; cases; for example, if smaller demes have higher extinction rates, rates, then then the the subsub for example, if smaller demes have higher extinction FST's would would properly properly be weighted set of the the population population with highest FsT's set of with the the highest be weighted least.] It help to look at at the the equation equation for for the the change mean allele allele frequency frequency It will will help to look change in in mean to selection. From Whitlock (2002),) , we get due to Whitlock (2002 2q ) + q ) ) s( 11 -- r)(FsT FST )(h( 11 -- 2g)+g)) r )(FsT ++ (( 11 -- FST)(h( As-q Asq = == p15 -q q s(
(7.5) (7.5)
where rr is the relatedness relatedness of of two two random random individuals individuals competing competing for for resources. resources. where is the Let us us consider consider the the various various parts parts of of this this equation equation in in turn. turn. First, First, we we see see that that Let the response response to to selection selection is is aa function function of of the the mean mean allele allele frequencies frequencies and and the the the strength of of selection selection io p~ q s. s. These These are are the the classic classic terms terms that that would would appear appear even even strength without population population structure: structure: the the response response to to selection selection is is proportional proportional to to the the without allelic and to to the the strength strength of of selection. selection. allelic variance variance pp qq and Next, we we find find that that the the response response to to selection selection is is proportional proportional to to one one minus minus Next, the the relatedness relatedness of of competing competing individuals. individuals. This This last last phrase phrase deserves deserves some some explanation. Consider Consider aa classic classic dichotomy dichotomy introduced introduced by by Dempster Dempster (1955; ( 1 955; explanation. see also also Christensen, Christensen, 1975) 1 975) between between local local and and global global competition competition for for see resources, i.e., i.e., soft soft versus versus hard hard selection. selection. With With soft soft selection, selection, each each deme deme concon resources, tributes aa number number of of individuals individuals to to the the next next generation generation (whether (whether via via resident resident tributes individuals or or migrants) migrants) independent independent of of the the genotypes genotypes of of the the deme. deme. With With hard hard individuals selection, each each deme deme contributes contributes to to the the next next generation generation in in proportion proportion to to selection, its its mean mean fitness fitness determined determined by by its its genotype genotype distribution. distribution. Under Under soft soft selection, selection, individuals are are competing competing locally locally for for resources, resources, and and therefore therefore there there is is individuals
7. 7.
SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS SELECTION
161 161
competition between between relatives. relatives. The The mean mean relatedness relatedness of of individuals individuals from from the the competition same deme deme (without (without inbreeding inbreeding within within demes) demes) is is given given by by rr === 2FsT/(1 2FsT/(1 ++ FsT). FST)' same At the the other other extreme, extreme, under under hard hard selection, selection, there there is is no no local local competition competition for for At resources, and and the the relatedness relatedness of of competing competing individuals individuals is is zero. zero. Putting Putting these these resources, equations into into Eq. Eq. (7.5), (7.5), we we find find that that hard hard selection selection is is always always more more effective effective equations than soft soft selection selection in in changing changing allele allele frequency. frequency. With With local local competition competition for for than resources, if if an an individual individual does does well well because because of of having having aa good good genotype, genotype, it it resources, will, through through competition, competition, reduce reduce the the resources resources available available to to other other individuals individuals will, in the the same same deme. deme. With population structure, these other other local local individuals individuals are are in With population structure, these likely to to share share alleles. alleles. Therefore Therefore the the event event that that would would have have boosted boosted the the numnum likely ber of of copies copies of of this this good good allele allele in in the the next next generation generation (the (the first individual ber first individual doing well) well) is is partially partially counterbalanced counterbalanced by by competition competition against against the the same same doing genotypes. genotypes. Note that that for the relatedness relatedness term, population structure structure tends tends to to Note for the term, increasing increasing population PST results weaken the the response response to to selection. selection. With With soft soft selection, selection, increasing increasing FsT results in in weaken greater relatedness relatedness and lower response response to to selection, selection, all else being greater and therefore therefore aa lower all else being eequal. qual. Finally, we we see the last last term term (FST + ((11 FST)(h ( l -- 2~) 2q) ++ q)) reflec Finally, see in in the (FsT + - FsT)(h(1 ~)) aa reflection the effects increasing homozygosity homozygosity on on the the response response to to selection selection in in tion of of the effects of of increasing structured populations. As As FST increases, so of individ structured populations. FsT increases, so does does the the proportion proportion of individuals that homozygous, even even for for the the same same mean mean allele allele frequency. Greater uals that are are homozygous, frequency. Greater increases the the magnitude magnitude of the response response to to homozygosity, the same homozygosity, for for the same q q,, increases of the selection. increase is is particularly particularly important important if if ~ q isis small small and and the the allele allele is selection. This This increase is at least partially recessive (h < 1/2). 112). In In these these cases, cases, with panmixia, most most at least partially recessive (h < with panmixia, alleles appear and selection therefore cannot alleles appear as as heterozygotes heterozygotes and selection therefore cannot discriminate discriminate increases, most selection is the the recessive recessive alleles. alleles. As As FST FsT increases, most of of the the selection is experienced experienced by by alleles alleles in in the the homozygous homozygous state, state, where where the the alleles alleles have have relatively relatively large large effects. effects. Thus, Thus, in in opposition opposition to to the the effect effect of of relatedness relatedness given given earlier earlier through through its its effects increasing homozygosity, effects on on increasing homozygosity, population population structure structure tends tends to to increase increase the recessive alleles, boost can can be be the response response to to selection. selection. For For nearly nearly recessive alleles, this this boost extremely large. extremely large. This This effect effect of of excess excess homozygosity homozygosity has has been been described described much much earlier earlier with with respect 974). In In respect to to inbreeding inbreeding within within populations populations (Ohta (Ohta and and Cockerham, Cockerham, 11974). fact, between the fact, with with hard hard selection, selection, there there is is no no distinction distinction between the effects effects of of inbreed inbreeding ing due due to to population population structure structure and and that that due due to to local local inbreeding; inbreeding; they they enter enter the the response response to to selection selection equations equations in in exactly exactly the the same same way. way. With With soft soft selection, selection, however, however, the the extra extra effects effects of of competition competition among among relatives relatives change change the the relation relationship ship between between F F and and response response to to selection. selection. The The balance balance between between these these two two effects effects (competition (competition among among relatives relatives and and With hard selection, there is no effect homozygosity) depends on the details. With of rate of of relatedness, relatedness, and and population population structure structure therefore therefore always always increases increases the the rate of response response to to uniform uniform selection. selection. With With soft soft selection, selection, response response to to selection selection can can be be either either increased increased or or decreased decreased depending depending on on the the dominance dominance coefficient coefficient of of the the locus locus under under selection selection and and FST' FsT. The The following following section section shows shows examples examples of of both. both. The The effects effects of of population population structure structure on on even even uniform uniform selection selection are are quite quite complicated. complicated. With With this this selection selection equation equation available, available, aa variety variety of of results results on on basic basic selection selection become become easy easy to to derive. derive. The The next next few few sections sections of of this this chapter chapter show show some some of of these these results. results.
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
1162 62
7.3 7.3
MAINTENANCE ENETIC VARIATION MAINTENANCE OF OF G GENETIC VARIATION IN IN SUBDIVIDED SUBDIVIDED POPULATIONS POPULATIONS One One of of the the oldest oldest questions questions in in population population genetics genetics is is "what "what forces forces are are most most important in in maintaining maintaining genetic genetic variation variation ??"" Population Population subdivision subdivision can can affect affect important the This section the maintenance maintenance of of genetic genetic variation variation in in aa variety variety of of ways. ways. This section reviews reviews aa few these briefly, case of few of of these briefly, focusing focusing on on the the case of spatially spatially uniform uniform selection. selection.
Mutation-Selection M u t a t i o n - S e l e c t i o n Balance Balance Estimates Estimates have have shown shown that that the the genomic genomic rate rate of of mutation mutation to to deleterious deleterious alle alleles les is is reasonably reasonably high, high, ranging ranging from from aa few few per per thousand thousand individuals individuals to to much much greater individual (Lynch greater than than one one per per each each new new individual (Lynch et et aI., al., 1999; 1999; Keightley Keightley and and Eyre-Walker, selection operates Eyre-Walker, 2000) 2000).. Although Although natural natural selection operates to to reduce reduce the the fre frequency of these deleterious quency of these deleterious alleles, alleles, they they are are not not immediately immediately eliminated eliminated com completely. result, some deleterious alleles pletely. As As aa result, some deleterious alleles are are always always segregating segregating in in populations at balance between populations at aa frequency frequency determined determined by by the the balance between mutation mutation and and selection. selection. Some Some have have argued argued that that levels levels of of standing standing genetic genetic variance variance observed observed in in natural natural populations populations could could be be explained explained largely largely by by this this mutation-selection mutation-selection balance. balance. Mutation Mutation is is likely likely not not much much affected affected by by population population structure, structure, but but the the prev previous section showed showed that ious section that the the efficacy efficacy of of selection selection can can be be affected affected greatly greatly by by subdivision. At allele is subdivision. At mutation-selection mutation-selection balance, balance, the the deleterious deleterious allele is likely likely to to be be rare, rare, which which simplifies simplifies Eq. Eq. (7.5) (7.5) to to /lsii As-q == -~ qs( -~s( 11 - r)(Fs r)(FsTT + + ((11 - F FsT)h) ST )h)
(7.6) (7.6)
The allele frequency frequency at at mutation-selection mutation-selection balance balance IS is then then The equilibrium equilibrium allele given by given by J.l. q == q = -s( - s ( 1 1 - r)(Fs r)(FsTT + + ((11 - F FsT)h) ST )h)
^ 2
-
(7.7) (7.7)
(Remember (Remember that that in in the the way way we we have have defined defined fitness fitness in in this this chapter, chapter, aa dele deleterious alleles in terious allele allele has has ss < < 0.) 0.) For For recessive recessive alleles in particular, particular, the the frequency frequency of of deleterious deleterious alleles alleles at at mutation-selection mutation-selection balance balance is is much much reduced reduced with with popu population lation structure structure due due to to the the more more effective effective selection selection against against homozygotes. homozygotes. See See Fig. Fig. 7.3, 7.3, for for some some examples. examples. As As aa result, result, the the amount amount of of variation variation maintained maintained by by mutation mutation selection selection balance balance can can be be reduced reduced greatly greatly in in large large metapopula metapopulations, tions, depending depending on on the the distribution distribution of of dominance dominance coefficients. coefficients. Most Most current current estimates mean dominance dominance coefficient deleterious alleles alleles give estimates of of the the mean coefficient of of mildly mildly deleterious give answers answers around around h h = - 0.1 0.1 (Houle (Houle et et aI., al., 1997; 1997; Garda-Dorado Garcia-Dorado and and Caballero, Caballero, 2000; 2000; Peters Peters et et aI., al., 2003), 2003), so so the the reduction reduction in in variance variance can can be be substantial substantial even even for for relatively relatively small small FS FsTT values. values. The model of genetic mechanism The predominant predominant model of the the genetic mechanism for for inbreeding inbreeding depression depression claims claims that that inbreeding inbreeding depression depression results results from from deleterious deleterious reces recessive mutation-selection balance. sive alleles alleles segregating segregating in in populations populations at at mutation-selection balance. With With population the reduction in mean population structure, structure, the reduction in mean deleterious deleterious allele allele frequency frequency
7. 7.
SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS SELECTION
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0.3
FST
Fig. 7 . 3 The equilibrium value of the frequency of a deleterious allele can be changed substantially by population structure. Here solid lines indicate pure soft selection and dashed lines indicate pure hard selection. With very recessive alleles, the equilibrium allele frequency is reduced greatly relative to the case in an undivided population (where C/---- - ~/hs). Parameter values used for these calculations were s -- - 0 . 1 , I~ -- 10 -6, and the three lines correspond to h = 0.4, 0.1, and 0.01 from top to bottom. From Whitlock (2002).
results in in aa potentially potentially large large reduction reduction in in the the amount amount of of inbreeding inbreeding depresdepres results PST values values (see ( see Whitlock, Whitlock, sion predicted predicted for relatively low low FsT sion for aa species, species, even even at at relatively 2002). 2002).
Balancing Selection Balancing Selection Balancing to increase the Balancing selection, selection, by by definition, definition, occurs occurs when when selection selection acts acts to increase the frequencies frequencies of of rare rare alleles. alleles. This This can can happen happen with with overdominance, overdominance, negative negative fre frequency-dependent are favored they are quency-dependent selection selection (where (where rare rare alleles alleles are favored because because they are rare), rare), or or by by spatially spatially heterogeneous heterogeneous selection. selection. Each Each of of these these are are affected affected by by the the spatial spatial population population structure. structure. Overdominance
With With overdominance, overdominance, the the heterozygote heterozygote is is the the most most fit fit genotype. genotype. For For this this sec section tion only, only, let let us us redefine redefine the the fitnesses fitnesses of of the the three three genotype genotype AA, Aa, and and aa as as 1-s 1-s :: 11 :: 1-t, l-t, such such that that the the fitness fitness of of the the two two homozygote homozygote genotypes genotypes is is reduced reduced by by aa factor factor ss or or t.t. With With overdominance overdominance in in aa large large randomly randomly mating mating population, population, there there is is an an intermediate intermediate equilibrium equilibrium allele allele frequency frequency that that stably stably maintains maintains vari variation ation in in the the population population as as aa result result of of the the heterozygote heterozygote being being selected selected for for when whenever ever one one or or the the other other of of the the two two alleles alleles becomes becomes too too rare. rare. In In structured structured populations, populations, the the extra extra homozygosity homozygosity caused caused by by population population structure structure can can change change the the dynamics dynamics of of the the maintenance maintenance of of variance. variance. Nonrandom Nonrandom mating mating causes causes the the marginal marginal fitnesses fitnesses of of the the two two alleles alleles to to be be determined determined more more by by their their homozygous homozygous effects effects and and less less by by their their effects effects in in heterozygotes. heterozygotes. As As aa result, result, if if the the two two homozygotes homozygotes fitnesses fitnesses are are not not equal equal (s (s =1= 4: t), t), then then the the allele allele associated associated with with the the fitter fitter homozygote homozygote will will have have aa higher higher frequency frequency than than expected expected under under random random mating. mating. Mathematically, Mathematically, that that frequency frequency is is given given by by �
ST s - tP tFsT
s
-
q= c)--~ ((ss + + t)( t)(11 - PFST)' ST ) ' -
(7.8) (7.8)
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
11 64 64
so so long long as as this this value value is is between between zero zero and and one, one, which which it it need need not not be be (Whitlock, (Whitlock, 2002). 2002). If If PST FSTis is large large enough, enough, the the expected expected equilibrium equilibrium leaves leaves the the population population fixed fixed for for the the allele allele with with the the most most fit fit homozygote. homozygote. Thus Thus population population structure structure tends tends to to reduce reduce the the amount amount of of variation variation maintained maintained by by overdominance. overdominance.
Frequency Dependence Dependence Frequency In In some some cases, cases, the the fitness fitness function function changes changes with with the the frequency frequency of of alleles alleles in in the the population population or or species; species; this this is is called called frequency-dependent frequency-dependent selection. selection. If If selec selection displays displays negative negative frequency frequency dependence, dependence, then then alleles alleles are are more more fit fit when when tion rare than when the same allele is common. In this case, selection can act to maintain maintain variation variation in in aa population population because because as as alleles alleles get get rare rare (as (as they they would would from the population) population),, their fitness increases and on the path to being lost from therefore their frequency climbs again. One One of of the the most most studied studied examples examples of of negative negative frequency frequency dependence dependence is is the the self-incompatibility self-incompatibility (51) (SI) alleles alleles common common to to many many species species of of plants. plants. With With 51, SI, pollen parent plant) that shares alleles with the maternal pollen (or, in some cases, cases, its parent plant) that alleles with plant ovules. These plant are are not not allowed allowed to to fertilize fertilize ovules. These processes processes presumably presumably evolved evolved as also prevent as aa mechanism mechanism to to prevent prevent self-fertilization, self-fertilization, but but they they also prevent unrelated unrelated individuals that alleles from alleles at individuals that share share alleles from mating. mating. As As aa result, result, rare rare alleles at the the 51 SI they are able to mate mate with more more other other indi indilocus have higher fitness because they viduals in the population. population. All else being equal, the system always favors new new viduals introduced into into the population, population, but but real species species have limited num numalleles being introduced bers alleles because because of bers of of 51 SI alleles of loss loss due due to to genetic genetic drift. drift. The The smaller smaller the the effective effective population size, the fewer 51 SI alleles maintained at equilibrium. population With population structure, alleles might With population structure, one one might might imagine imagine that that different different alleles might be be maintained increasing the the total in maintained in in different different populations, populations, thereby thereby increasing total diversity diversity in the species out that that this this is is true true for species with very low low the species as as aa whole. whole. It It turns turns out for species with very migration rates rates between but with realistic, intermediate intermediate levels levels of migra migration between demes, demes, but with realistic, of migration the total number number of of 51 SI alleles maintained than would would be tion the total maintained is is slightly lower than be expected 998; Schierup et ai., 2000; Muirhead, Muirhead, expected with with panmixia panmixia (Schierup, (Schierup, 11998; Schierup et al., 2000; 2001). 200 1).
Heterogeneous Selection Heterogeneous It the 1950s varying selection It has has been known known since at least least the 1 950s that that spatially spatially varying can maintain maintain genetic genetic variation, variation, especially especially if if there there is is soft selection (Levene, (Levene, can soft selection 1954; conditions for for this than was was com1 954; Dempster, Dempster, 1955). 1 955 ). The The conditions this are are narrower narrower than com monly Smith and strong, relamonly thought thought (Maynard (Maynard 5mith and Hoekstra, Hoekstra, 1980), 1980), requiring requiring strong, rela tively symmetric symmetric selection. selection. Felsenstein ( 1 976) and and Hedrick Hedrick (1986; ( 1 986; Hedrick Hedrick et et tively Felsenstein (1976) al., ai., 1976) 1 976) reviewed the the theory theory and and empirical evidence for for and and against the the maintenance of of genetic genetic variance variance by by heterogeneous heterogeneous selection. selection. maintenance A form of populations in A different different form of heterogeneous heterogeneous selection selection can can emerge emerge in in populations in which which there there is is already already aa lot lot of of genetic genetic differentiation differentiation among among populations. populations. In these these cases, cases, epistatic epistatic interactions loci can can cause cause different different alleles alleles to to In interactions between between loci be when the describing the be favored favored locally locally even even when the underlying underlying function function describing the relationrelation ship fitness and genotype is Chapters 99 and ship between between fitness and genotype is uniform uniform across across space space (see (see Chapters and 11). 1 1 ). This This sort sort of of heterogeneous heterogeneous selection selection depends depends on on there there being being selectively selectively and epistatically epistatic ally different different alleles alleles in in different local populations, populations, which which becomes becomes and different local important restricted gene important only only under under extremely extremely restricted gene flow flow or or extreme extreme drift. drift.
7. 7.
SELECTION IN METAPOPULATIONS SELECTIONAND AND DRIFT DRIFT IN METAPOPULATIONS
1165 65
One One special special case case of of epistasis epistasis that that may may be be quite quite common common is is that that generated generated on on approximately additively additively interacting interacting alleles that form form aa phenotype phenotype under under stabil stabilapproximately alleles that izing selection. Stabilizing izing selection. Stabilizing selection selection causes causes the the fitness fitness effects effects of of alleles alleles to to vary vary depending all other depending on on whether whether the the sum sum of of the the effects effects of of all other alleles alleles in in the the individual individual add add up up to to aa value value above above or or below below the the optimum optimum for for the the trait; trait; hence hence with with stabiliz stabilizing ing selection, selection, aa population population near near its its optimum optimum will will have have mainly mainly epistatic epistatic variance variance for 995). Barton for fitness fitness associated associated with with that that trait trait (Whitlock (Whitlock et et aI., al., 11995). Barton and and Whitlock Whitlock ((1997) 1 997) have stabilizing selection have shown shown that that with with uniform uniform stabilizing selection and and low low migration, migration, the the amount amount of of genetic genetic variance variance for for aa trait trait that that can can be be maintained maintained can can be be increased increased substantially epistasis. However, likely to substantially as as aa result result of of this this epistasis. However, this this is is only only likely to be be import important ant in in species species with with very very high high values values of of P FST, in the the range range of of P FST > -0.2. -0.2. ST > ST, in
7.4 7.4
ADAPTATION IN SUBDIVIDED ADAPTATION IN SUBDIVIDED POPULATIONS POPULATIONS Population Population structure structure can can affect affect the the pace pace of of adaptive adaptive evolution. evolution. We We have have already already discussed discussed the the conditions conditions under under which which the the response response to to selection selection is is increased increased or or decreased decreased with with population population structure. structure. The The subdivision subdivision also also allows allows novel 1 ), novel patterns patterns of of adaptation, adaptation, such such as as local local adaptation adaptation (see (see Barton, Barton, 200 2001), shifting 1 93 1 ), but Coyne et ai. ((1999) 1 999) and and shifting balance balance evolution evolution [Wright [Wright ((1931), but see see Coyne et al. Whitlock and Phillips Phillips (2000)], (2000)], and and more more rapid evolution with with epistatic epistatic inter interWhitlock and rapid evolution actions Bryant et 986; Goodnight, 9 8 8; see actions ((Bryant et aI., al., 11986; Goodnight, 11988; see Chapter Chapter 9). 9). More More funda fundamentally though, pace of mentally though, population population structure structure strongly strongly affects affects the the pace of evolution evolution even alleles that even for for those those alleles that are are uniformly uniformly selected selected without without any any complicating complicating interactions loci. This interactions with with other other loci. This section section reviews reviews the the effects effects of of population population structure structure on on the the probability probability of of fixation fixation of of new new mutations. mutations.
Probability Probability of of Fixation Fixation One One of of the the most most remarkable remarkable results results in in population population genetics genetics has has to to be be Haldane's 1927) result Haldane's ((1927) result that that aa new new beneficial beneficial allele allele with with heterozygous heterozygous benefit benefit of of hs hs has has only only about about 2hs 2hs chance chance of of ultimate ultimate fixation. fixation. Haldane Haldane assumed assumed that that the ideal (i.e., equaled its the species species in in question question was was ideal (i.e., its its census census size size equaled its effective effective size) size) and undivided. undivided. Even Even in in an an infinite infinite population, population, if if aa new new allele allele is is introduced introduced as as and only only aa single single copy, copy, the the fate fate of of that that allele allele is is partially partially determined determined by by stochastic stochastic changes changes in in the the numbers numbers of of copies copies of of the the allele allele left left in in each each generation. generation. It It turns turns out allele as (as aa rare out that that by by introducing introducing an an allele as aa single single copy copy (as rare mutation mutation would would likely be likely do), do), even even alleles alleles with with moderate moderate selective selective advantage advantage are are more more likely likely to to be lost population than 1 964; see also Crow lost stochastically stochastically from from the the population than fix. fix. Kimura Kimura ((1964; see also Crow and 970) modified and Kimura, Kimura, 11970) modified this this result result to to allow allow for for nonideal nonideal populations populations and and allowed allowed arbitrary arbitrary dominance dominance for for deleterious deleterious alleles alleles as as well. well. He He found found that that the allele is the probability probability of of fixation fixation of of aa beneficial beneficial allele is given given approximately approximately by by 2hsNiN, 2hsNe/N, where where N N is is the the census census size size of of the the population. population. In 11970, Maruyama achieved achieved the the first first results results on on the the probability probability of of fixation fixation In 970, Maruyama in populations. He in subdivided subdivided populations. He showed showed that that in in an an island island model, model, the the probabil probability ity of of fixation fixation for for an an additively additively acting acting allele allele was was simply simply s. s. (For (For additive additive alle alleles, les, h h = = 112, 1/2, so so this this result result is is equivalent equivalent to to the the 2hs 2hs of of Haldane.) Haldane.) Maruyama Maruyama ((1974) 1 974) and 9 8 1 ; Nagylaki, and others others (Slatkin, (Slatkin, 11981; Nagylaki, 1982) 1982) extended extended this this result result to to deal deal with with any any model model such such that that each each deme deme contributes contributes exactly exactly equally equally to to the the next next
MICHAEL MICHAEL C C.. WHITLOCK WHITLOCK
1166 66
generation; the generation; the probability probability of of fixation fixation with with population population structure structure with with this this restriction restriction remained remained s. s. This This was was viewed viewed by by some some as as an an invariant invariant result result of of popu population lation structure; structure; the the claim claim was was made made that that population population structure structure therefore therefore did did not beneficial alleles. not affect affect the the probability probability of of fixation fixation of of beneficial alleles. However, However, this this con conclusion was because other models of pos clusion was premature premature because other models of population population structure structure are are possible ((and and even model) and sible even more more reasonable reasonable than than the the island island model) and because because the the effects dominance were effects of of dominance were not not properly properly accounted accounted for. for. The The first first demonstra demonstration extinction and tion that that this this was was not not true true was was aa model model of of extinction and two two specific specific types types 1 993). In of Barton ((1993). of recolonization recolonization by by Barton In these these cases, cases, the the probability probability of of fixation fixation was reduced by relative to panmictic case. was much much reduced by population population structure structure relative to the the panmictic case. The The probability probability of of fixation fixation in in aa more more general general model model of of structured structured populations populations has has been been found found (Whitlock, (Whitlock, 2003). 2003). Based Based on on Kimura's Kimura's diffusion diffusion equations, equations, this this work work shows shows that that the the probability probability of of fixation fixation can can be be derived derived from from the the equations equations for response to earlier in for drift drift and and response to selection selection presented presented earlier in this this chapter. chapter. Moreover, Moreover, as as long long as as the the strength strength of of selection selection is is lower lower than than the the typical typical immigration immigration rate, rate, the the expected for for neutral neutral loci loci can can be be used used in in these these equations, equations, which which expands expands their their FFsT ST expected 112, the usefulness usefulness greatly. greatly. For For dominance dominance coefficients coefficients differing differing from from 1/2, the equations equations cannot obtained with cannot be be solved solved directly, directly, but but the the answers answers can can be be obtained with numerical numerical inte integration. space, this gration. In In the the interests interests of of space, this chapter chapter will will not not review review the the mathematics mathematics of of the focus on the general general equations, equations, but but will will focus on the the additive additive case, case, as as well well as as an an approxi approximation mation that that works works very very well well for for beneficial beneficial alleles alleles even even with with arbitrary arbitrary dominance. dominance. More details can More details can be be found found in in Whitlock Whitlock (2003). (2003). For For additive additive alleles, alleles, such such that that h h = = 112, 1/2, the the probability probability of of fixation fixation in in struc structured given by tured populations populations is is given by exp[ 2s( 1 11 - exp[-2s(1 - F FsT)Neq] ST )Neq] 2s( 1 -- FFsT)Ne] u[q]= 11 - exp[ exp[-2s(1 ST )Ne]
u [q ]
-
-
(7.9) (7.9)
-
for selection and for soft soft selection and
u[q]
11 --
exp[ -2s( 1 + exp[-2s(1 + F FsT)Neq] ST )Neq]
-2s( 1 ++ FFsT)Ne] u[q]= 11 - exp[ exp[-2s(1 ST )Ne]
(7.10) (7.10)
-
for for hard hard selection, selection, where where q is is the the initial initial allele allele frequency frequency of of the the allele allele in in the the metapopulation. metapopulation. If If the the population population starts starts with with aa single single copy copy of of the the new new allele, allele, the total of the then then q q = = 11/2Ntot, where Ntot mto t is is the total size size of the metapopulation. metapopulation. These These I2Ntot, where equations look fearsome, similar to equations for equations look fearsome, but but in in fact fact they they are are quite quite similar to the the equations for 1 964). There the the panmictic panmictic case case derived derived by by Kimura Kimura ((1964). There are are two two differences. differences. First, First, subdivided population, the the Ne Ne here here is is the the effective effective size size of of aa subdivided population, given given by by Eq. Eq. (7.3). (7.3). Second, Second, the the strength strength of of selection selection ss is is now now modified modified by by aa term term involving involving F FsT, ST, which of selection which reflects reflects the the change change in in the the efficacy efficacy of selection from from population population structure. structure. For can write For beneficial beneficial alleles, alleles, we we can write aa simple simple equation equation for for the the probability probability of of fixation fixation of of aa new new mutant, mutant, even even with with arbitrary arbitrary dominance: dominance:
1 - FST )h )NeINtot. )(FST ++ ((1-FsT)h)Ne/Ntot. -~ 2s( 2s(11 - rr)(FsT uu ==
(7.11) (7. 11)
Here Here it it iiss possible possible to to see see that that this this result result builds builds directly directly on on Kimura's. Kimura's. As As F FsT ST goes zero, this this approaches goes to to zero, approaches the the 2hsN)N 2hsNe/N given given earlier. earlier.
7. 7.
SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS SELECTION
1 67 161 A
0.0004 ,.c 0.0004
�
0 O .m ,.i-, cO x ;;;:;
9 0.0003 .---x 0.0003 '0 O
� 0.0002 0.0002
:c 43 co 03 .c
0.0001 se 0.0001 a..
13..
0.02 0.02
0.04 0.04
0.06 0.08 0.08 0.06
0.1 0.1
Extinction rate rate Extinction
B
0.0002 0.0002 0.0001 8 0.00018 0.0001 6 0.00016 0.00014 0.00014 0.000 1 2 0.00012 0.0001 0 0.00010 0.00008 0.00008 0.00006 0.00006 0.00004 0.00004 0.00002 0.00002 0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5
Migration rate rate Migration
Fig. 77.4 Examples of of the the fixation fixation probabilities probabilities of of nearly nearly recessive recessive beneficial beneficial alleles alleles (h (h = = 0.01) 0.01 ) Fig. . 4 Examples with soft selection. (A) Extinction and and recolonization. recolonization. In In this this example, example, the the migration migration rate rate between between with soft selection. (A) Extinction populations was 0.05, 0.05, colonization colonization occurred occurred by by four four individuals individuals with with aa probability probability of of common common oriori populations was gin /2, ss == 0.002, and there there were were 1100 00 demes demes with with 100 1 00 diploid diploid individuals each. (Each (Each point point gin of of 11/2, 0.002, and individuals each. represents results 1 07 simulations, so the standard error on the the left left represents results from from 107 simulations, so the standard error ranges ranges from from 6.9 6.9 x • 1100-6 -6 on to 0--66 on on the the right.) right.) As As the the extinction extinction rate the effective effective population population size size of of the to 3.9 3.9 xx 110 rate increases, increases, the the metapopulation decreases, decreases, and and therefore so does probability of (8) A A one-dimen metapopulation therefore so does the the probability of fixation. fixation. (B) one-dimenincreases as as the the sional model. With stepping-stone model, (and therefore sional stepping-stone stepping-stone model. With aa stepping-stone model, FST FST(and therefore Ne) increases migration rate drops so migration. This parmigration so the probability of fixation also also increases with with lower migration. This is par ticularly true true with with recessive recessive alleles, which are are expressed in the the homozygous homozygous state with the the ticularly alleles, which expressed often often in state with concomitant selection. (There are 100 1 00 demes demes with with 100 1 00 diploid individ concomitant increase increase in in the the efficacy efficacy of of selection. (There are diploid individand dots dots represent 1 06 simulations simulations each.) each.) uals each, uals each, ss = = 0.0002 0.0002 and represent 106
These have been been tested simulation in in aa wide wide variety variety of of models models of These results results have tested by by simulation of population population structure, structure, including including the the island island model, model, extinction-recolonization, extinction-recolonization, stepping-stone models, and source-sink models. They work remarkably well (see (see Figs Figs 7.4 7.4 and and 7.5). 7.5). The probability of fixation of beneficial alleles tends to to be much reduced with population structure. with population structure. This This is mainly a result of the fact that the effective
cc
"-= 03 � x 0 O
._x '0 0 ;;;:;
g ~ 9 :c
03 co .c
e a..
O a..
0.002 0.002 0.00175 0.00175 0.001 5 0.0015 0.00125 0.00125 0.001 0.001 0.00075 0.00075 0.0005 0.0005 0.00025 0.00025
•
0.2 0.2
0.6 0.8 0.4 0.4 0.6 0.8 Relative Relative contribution contribution of of sinks sinks
1
Fig. 00 demes, Fig. 7.5 7 . 5 The The probability probability of of fixation fixation in in aa source-sink source-sink model. model. Here Here there there are are 1100 demes, 20 20 of of which 00 individuals, which are are "sources" "sources" and and the the rest rest are are "sinks". "sinks". Each Each deme deme has has 1100 individuals, and and the the immigra immigration tion rate rate to to the the sources sources is is 0.2, 0.2, whereas whereas it it is is 0.25 0.25 in in sinks. sinks. Demes Demes exchange exchange migrants migrants by by aa modi modified fied island island model, model, where where each each sink's sink's contribution contribution to to the the migrant migrant pool pool is is aa fraction fraction of of that that of of each each source. source. As As this this asymmetry asymmetry increases, increases, the the effective effective population population size size is is reduced reduced and and the the probabil probability /2, and ity of of fixation fixation of of beneficial beneficial alleles alleles drops. drops. For For these these examples, examples, ss = = 0.002 0.002 and and hh = = 11/2, and dots dots represent 07 simulations. represent results results of of 1107 simulations.
11 68 68
MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
population size models of population size is is reduced reduced in in most most models of population population structure. structure. The The prob probability of loci, especially especially for ability of fixation fixation can can be be increased increased for for some some loci, for nearly nearly reces recessive alleles that sive alleles that can can be be expressed expressed more more strongly strongly in in structured structured populations populations because because of of increased increased homozygosity. homozygosity. Let us the island the island is an Let us return return to to the island model. model. As As mentioned mentioned earlier, earlier, the island model model is an extreme extreme description description of of population population structure structure because because it it allows allows no no variance variance among populations in among populations in reproductive reproductive success. success. For For additive additive alleles, alleles, Maruyama Maruyama and and successors successors found found the the probability probability of of fixation fixation to to be be simply simply ss in in an an island island model, model, the population. The the same same as as in in an an unstructured unstructured population. The more more general general model model predicts predicts that that the the probability probability of of fixation fixation should should be be ss ((11 - FST) FST) NefNtot Ne/Ntot (because (because the the island island model model in in its its basic basic form form as as used used by by Maruyama Maruyama is is also also aa soft soft selection selection model). model). Remember Remember that that the the island island model model has has the the unusual unusual property property of of having having aa larger larger Ne Ne than the probability of fix than census census size: size: Ne Ne= = Nto!(l Ntot/(1 - FST)' FST ). Putting Putting this this Ne N e into into the probability of fixation equation simply s. consistent; what ation equation simplifies simplifies it it to to simply s. The The results results are are consistent; what is is more more important island model important is is that that the the island model is is unrealistic unrealistic and and extreme. extreme. Most Most real real species species and so most will have lower will will have have Ne Ne < < Ntot, mtot, and so most will have lower probabilities probabilities of of fixation fixation of of bene beneficial ficial alleles alleles than than predicted predicted by by Maruyama's Maruyama's formula. formula. Probabilities Probabilities of of fixation fixation are are not population subdivision. not invariant invariant with with respect respect to to population subdivision. Relaxing Relaxing the the assumption assumption of of uniform uniform selection selection has has been been investigated investigated using using the island model (Barton, 11987; 9 8 7; Tachida the island model by by aa variety variety of of authors authors (Barton, Tachida and and lizuka, Iizuka, 11991; 99 1 ; Gavrilets Gavrilets and and Gibson, Gibson, 2002). 2002). Population Population structure structure tends tends to to increase increase the the probability probability of of fixation fixation relative relative to to that that expected expected by by the the mean mean fitness fitness of of the the alleles across demes. known what alleles across demes. It It is is not not yet yet known what effect effect heterogeneous heterogeneous selection selection would have have with with aa more more realistic realistic model model of of subdivision. subdivision. would Population Population structure structure also also substantially substantially affects affects the the time time taken taken for for fixation fixation of of new alleles (Whitlock, (Whitlock, 2003). new alleles 2003). 7.S 7.5
GENETIC GENETIC LOAD LOAD IN IN SUBDIVIDED SUBDIVIDED POPULATIONS POPULATIONS Genetic Genetic load load is is the the reduction reduction in in the the mean mean fitness fitness of of aa population population relative relative to to an an optimal optimal genotype genotype caused caused by by some some particular particular factor, factor, such such as as deleterious deleterious mutation, mutation, genetic genetic drift, drift, and and segregation segregation (Crow, (Crow, 1993). 1993). Load Load is is sometimes sometimes strongly strongly affected affected by by population population structure, structure, as as reviewed reviewed in in this this section. section.
Mutation M u t a t i o n Load Load Mutation Mutation load load is is the the reduction reduction in in mean mean fitness fitness caused caused by by recurrent recurrent delete deleterious mutations mutations in in aa population. Mutation load load is is usually usually calculated calculated at at muta mutarious population. Mutation tion-selection balance: is, it mean reduction reduction in tion-selection balance: that that is, it is is the the mean in fitness fitness associated associated with allele frequency with an an allele frequency predicted predicted by by the the equilibrium equilibrium between between mutation mutation and and selection. In panmictic populations, associated with allele that selection. In panmictic populations, the the load load associated with an an allele that is is not not completely completely recessive recessive is is L L = = 2J.L 2~ (where (where J.L ~ is is the the mutation mutation rate rate from from wild wild type deleterious allele; type to to deleterious allele; remarkably, remarkably, this this is is not not aa function function of of the the strength strength of of selection against selection against the the deleterious deleterious allele). allele). With With population population structure, structure, load load equations equations become become more more complicated complicated (Whitlock, 2002) 2002):: (Whitlock, L -FsT )+FsT)sq L= ~ --( 2(2h( h ( I1--FsT)+FST)S-q
(7. 12) (7.12)
11 69 69
7. AND DRIFT IN METAPOPULATIONS 7. SELECTION SELECTION AND DRIFT IN METAPOPULATIONS
"0 "0
.3
11.2 .2
ttl t~
0 .._1
9 (J) > ._>
�
1
(j) ~ 0.8 nII:
0.6 0.6
~
.
.
-"
~
. ----
--- _
-...- n
0.01 hh=O.01 =
- -
........
-
-
-
- - - - - - - -
0.05 0.05
0.1 0.1
Fig. F i g . 77.6 .6
0. 15 0.15 FFST ST
0.2 0.2
- - _ _ _
-
-
-- - -
0.25 0.25
0.3 0.3
The mutation in aa metapopulation the load in an The mutation load load in metapopulation relative relative to to the load at at aa similar similar locus locus in an undivided undivided population population (-2fL). (-21~). For For the the values values of of FST FSTlikely likely to to be be found found within within species species and and relatively relatively small values values of small of the the dominance dominance coefficient coefficient h, the the mutation mutation load load can can be be reduced reduced substantially substantially in in aa subdivided subdivided population. population. Solid Solid lines lines show show pure pure soft soft selection, selection, whereas whereas dashed dashed lines lines correspond correspond 0. 1 , fL 0--6, 6, and to example are to pure pure hard hard selection. selection. Parameters Parameters for for this this example are s = = -0.1, I~ = = 110 and the the three three pairs pairs 0.01 from of of curves curves correspond correspond to to h = = 0.4, 0.4, 0.1 0.1,, and and 0.01 from top top to to bottom. bottom.
where 7.7). Note where the the value value of of q ~ is is given given by by Eq. Eq. ((7.7). Note that that ss will will cancel cancel out out when when this substitution for q is made, but load remains a function of the dominance this substitution for ~ is made, but load remains a function of the dominance coefficient, case. Figure 7.6 shows coefficient, unlike unlike the the panmictic panmictic case. Figure 7.6 shows the the change change in in load load as as aa function function of of population population subdivision. subdivision. Load Load is is always always reduced reduced with with hard hard selection, selection, but but with with soft soft selection, selection, load load is is increased increased for for high high values values of of FST FST and and near additivity. With nearly recessive alleles, the reduction in load near additivity. With nearly recessive alleles, the reduction in load can can be be nearly %. nearly 5500 %.
Segregation Segregation Load Load Segregation Segregation load load is is the the reduction reduction in in fitness fitness caused caused by by the the inability inability of of aa pop population ulation to to be be composed composed entirely entirely of of heterozygotes heterozygotes even even when when these these genotypes genotypes are are the the most most fit. fit. As As such, such, segregation segregation load load requires requires overdominance. overdominance. With With population structure, there are even fewer heterozygotes population structure, there are even fewer heterozygotes in in aa species species than than under under Hardy-Weinberg Hardy-Weinberg conditions conditions so so the the segregation segregation load load would would be be more more pronounced. pronounced. Using Using the the same same notation notation as as in in the the overdominance overdominance section section given given earlier, earlier, the the segregation segregation load load is is expected expected to to be be L L = =
((11 + + FS FsT)st T )st , ss ++t t '
--
--
((7.13) 7. 1 3 )
which which reduces reduces to to the the segregation segregation load load in in aa panmictic panmictic population population when when FST FST = = 00 ((Crow, Crow, 11958). 958). Therefore, load is Therefore, the the segregation segregation load is ((11 + + FST) FST) times times as as great great in in aa subdivided population one, as expected by subdivided population as as in in an an undivided undivided one, as expected by the the increased increased number number of of homozygotes. homozygotes.
Drift Drift Load Load Drift Drift load load is is the the reduction reduction in in fitness fitness caused caused by by drift drift changing changing allele allele fre frequencies quencies away away from from those those favored favored by by selection. selection. An An extreme extreme form form of of drift drift load load results results from from fixation fixation of of deleterious deleterious alleles alleles by by drift. drift. Drift Drift load load has has received received aa lot lot
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MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
of attention attention in in the the last last several several years years because because of of the the possible possible mutational mutational meltmelt of down of of small small endangered endangered populations populations (Lande, ( Lande, 1994; 1 994; Lynch Lynch et et al., aI., 1995a,b). 1 995a,b). down rate that that deleterious deleterious alleles alleles accumulate accumulate in in aa species species is is aa function function of of the the The The rate efficacy of of selection selection and and of of the the effective effective population population size; size; the the smaller smaller these these two two efficacy values are are the the faster faster drift drift load load will will accumulate. accumulate. We We have have seen seen that that selection selection is is values often more more effective effective in in structured structured populations populations (although (although not not always), always), but but more more often importantly, the the effective effective population population size size tends tends to to be be reduced reduced by by structure. structure. importantly, Because the the latter latter of of these these two two effects effects turns turns out out numerically numerically to to be be more more imporimpor Because tant, in in most most cases, cases, population population structure structure increases increases the the rate rate of of accumulation accumulation of of tant, deleterious alleles alleles (Higgins (Higgins and and Lynch, Lynch, 2001; 200 1 ; Whitlock, Whitlock, 2003). 2003 ). This This is is most most deleterious pronounced in in cases cases with with large large variance variance in in reproductive reproductive success success among among demes, demes, pronounced such as as with with extinction extinction and and recolonization recolonization or or source-sink source-sink models. models. Figure Figure 7.7 7.7 such shows that that the change in in the probability of of fixation fixation of of deleterious deleterious alleles alleles can can shows the change the probability be reasonably reasonably large large (two(two- to to three fold), although although perhaps perhaps in in most most cases cases the the be three fold), change is less less than than aa doubling. doubling. change is
Migration Load Load Migration If the the local local population in aa deme deme is is well well adapted to local local conditions conditions and and if if If population in adapted to migrants to to this this population population come come from populations adapted to other condi migrants from populations adapted to other conditions, then alleles that into the the population population by by migration are likely likely to to tions, then the the alleles that come come into migration are be poorly adapted local conditions. mean fitness be poorly adapted to to local conditions. The The reduction reduction in in mean fitness that that Migration load load increases increases with with increasing increasing dif results is is called called migration results migration load. load. Migration differences the selection coefficients among populations and with migration migration ferences in in the selection coefficients among populations and with most important important type of rate. In some species, migration migration load is likely to to be the the most genetic load. Migration load in determining limits of genetic load. Migration load may may be be key key in determining the the range range limits of species species because because migration migration from from the the species species center center may may prohibit prohibit further further local local adaptation 963; Kirkpatrick 997). adaptation at at the the margins margins (Mayr, (Mayr, 11963; Kirkpatrick and and Barton, Barton, 11997).
B B
A 6• 10 -6 6x1 0-6 5x 10-6 5x1 0-6
tc: 0
'i 0.00003 "~ 0.00003 >< .O m
;._~0.00002 0.00002 o
:0
e
ct! .0 .13
J
4x 10-6 4x1 0-6 3x 10-6 3x1 0-6
._x ""
-6 2xl 00-6 2x1 -6 11xx10-6 10
0.00001
O
c.. 13_
0.02 0.02
0.04
0.06 0 . 0 6 0.08 0.08
Extinction Extinction rate rate
0.1 0.1
0.1 0.1
0.2
0.3 0.3
0.4
0.5
rate Migration rate
Fig. Fig. 7.7 7 . 7 The The probability probability of of fixation fixation of of deleterious deleterious alleles alleles with with (A) (A) extinction extinction and and colonization colonization or or (8) (B) aa one-dimensional one-dimensional stepping stepping stone stone model. model. (A) (A) The The three three lines lines plot, plot, from from bottom bottom to to top, top, the the predicted 1 , and predicted probability probability of of fixation fixation for for alleles alleles with with dominance dominance coefficients coefficients of of 0.5, 0.5, 0. 0.1, and 0.01 0.01,, respectively. 07 replicates respectively. The The symbols symbols mark mark simulation simulation results results over over aa minimum minimum of of 1107 replicates each, each, with with the the three three dominance dominance coefficients coefficients represented represented by by triangles, triangles, squares, squares, and and crosses, crosses, respectively. respectively. Other Other 1 , 1100 00 demes 00 diplOid parameters parameters used used for for these these examples were were 5s = = -0.0002, -0.0002, m m= = 0. 0.1, demes of of 1100 diploid indi individuals viduals each, each, and and colonization colonization by by four four individuals individuals with with aa probability probability of of common common origin origin equal equal to to 11/2. /2. The The probability probability of of fixation fixation isis increased increased substantially substantially by by the the reduction reduction in in Ne N e that that accompanies accompanies extinction 00 extinction dynamics. dynamics. (8) (B) The The parameters parameters in in these these examples examples were were hh = = 0.01 0.01,, 5s = = -0.0002 -0.0002 with with 1100 demes 00 diplOid 08 simulations. demes of of 1100 diploid individuals. individuals. Points Points represent represent the the results results of of 1108 simulations.
7. SELECTION SELECTION AND DRIFT DRIFT IN METAPOPULATIONS METAPOPULATIONS 7.
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Local Genetic Genetic Load Load and and the the Consequences Consequences of of Migration Migration Local In In subdivided subdivided populations, populations, weakly weakly deleterious deleterious alleles alleles can can rise rise by by drift drift to to keephigh frequencies within local populations, even if selection is effective at keep frequency low throughout the species. Crow (1948) (1948) proposed ing their overall frequency that this could be the mechanism for the commonly observed pattern of heter heterthat osis, osis, the the increase increase in in fitness fitness often often observed observed in in hybrids hybrids between between different different popu populations. lations. We We examined examined this this hypothesis hypothesis using using Wright's Wright's distribution distribution of of allele allele frequencies frequencies for for the the island island model model (Whitlock (Whitlock et et ai., al., 2000; 2000; Ives Ives and and Whitlock, Whitlock, 2002) 2002) and and found found that that Crow's Crow's hypothesis hypothesis was was extremely extremely credible. credible. We We referred referred to the reduction reduction in mean fitness caused by this local increase in the frequency local drift drift load load and showed that that reasonably large values of deleterious alleles local of of heterosis heterosis were were consistent consistent with with what what is is known known about about mutation mutation rates rates and and population population structure. structure. These These results results have have been been extended extended by by Morgan Morgan (2002) (2002) and and Gl~min (2003 (2003).). Morgan Morgan (2002) showed showed that that Glemin
(
)
Whybrid ( ((11- - hs hs)2)nVamong )2 nVamong Whybrid == 1 S Wlocal 1 -- S -
(7.14) (7.14)
where demes in where Vamo Vamong the variance variance among among demes in allele allele frequency frequency as as defined defined and and ng is the number of loci. With this we can write a prediction prediction for for the heterosis nn is the number in terms of FST Fsy and q: ~:
heterosis. =. Whybrid WhybYid heterOSIs - . 1l . ---
-Wlocal Wlocal
=
p ((1 I ( l -m hs hs )2 )21)nFSTQ nF~Tqp -1 l -s .
.
1- s
1
(7.15) (7.15)
If the the meta population itself large and and aatt equilibrium, then p == 1 1 If metapopulation itself iiss relatively relatively large equilibrium, then p ~and q ~ is approximately q ~ from and from Eq. (7.7). (7.7). Heterosis interesting biological consequence. If offspring formed formed by by Heterosis has has an an interesting biological consequence. If offspring crosses between between demes have selective advantage, then crosses demes have selective advantage, then the the offspring offspring of of migrants migrants increased fitness (Ingvarsson (Ingvarsson and and Whitlock, Whitlock, 2000; 2000; Morgan, Morgan, 2002). will have increased 2002). Thus of migration actual Thus the genetic effects of migration will will be increased increased relative relative to to the the actual migration rate for aa neutral neutral locus locus is is observed observed migration migration rate. rate. The The effective effective migration rate for approximately approximately erosislf, mee -eheter~ = mm ehet m
(7.16) (7.16)
where 7r is the harmonic harmonic mean mean recombination recombination rate rate between between the the neutral neutral locus locus where is the and and all all selected selected loci loci (Ingvarsson (Ingvarsson and and Whitlock, Whitlock, 2000). 2000). For For low low values values of of FST, FST, the magnification magnification of of the the effective rate of of migration migration can be severalfold. severalfold. This can can be counterbalanced counterbalanced or or reversed by sufficient sufficient local local adaptation adaptation or or strong differ be reversed by strong differences among among populations populations in in epistatic epistatic interactions. interactions. ences
Load Load in in Subdivided Subdivided Populations, Populations, a a Summary Summary Several types types of of load load are are affected by population population structure. structure. Mutation Mutation load load tends tends Several affected by to decline decline at at equilibrium equilibrium with with structure, structure, and and migration migration load load is is lowered lowered with with to lower migration migration rates, rates, whereas whereas drift drift load, load, segregation segregation load, load, and and local local drift drift load load lower tend cumulative, the tend to to increase. increase. Because Because these these different different genetic genetic loads loads are are cumulative, the mean mean
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MICHAEL MICHAEL C. C. WHITLOCK WHITLOCK
fitness of of the the population population with with three three different different types types of of genetic genetic load load isis approxiapproxi fitness mately (1 ( 1 -- L1) L1) (1 ( 1 - L2) L2 ) (1 ( 1 -- LL3 3) .) ' IfIf the the loads loads are are small small (they (they are are not not in in general general mately expected to to be) be) then then the the overall overall load load is is approximated approximated by by the the sum sum over over the the types types expected of load. load. Whether Whether population population structure structure increases increases or or decreases decreases mean mean fitness fitness on on of average depends depends on on aa large large number number of of circumstances. circumstances. If If habitat habitat conditions conditions vary vary average strongly, then then population population structure structure allows allows local local adaptation adaptation (in (in other other words, words, strongly, reducing migration migration load) load) and and this this effect effect can can be be paramount. paramount. However, However, if if migramigra reducing tion rates rates become become too too small small and and local local population population size size is is low, low, then then local local drift drift load load tion will become become very very important important and and essentially essentially the the population population will will suffer suffer from from will inbreeding depression. depression. Species-level Species-level drift drift load load could could become become important important ifif there there is is inbreeding lot of of variance variance among among demes demes in in reproductive reproductive success success and and if if the the total total census census size size aa lot of the the species species was was small small (so (so that that the the effective effective size size was was low), low), but but is is likely likely not not very very of important if if the the effective effective size size of of the the species species is is over over about about 10,000. 10,000. Mutation Mutation load load important may be be reduced reduced by by population population structure structure (at (at equilibrium), equilibrium), but but not not by by more more than than may half. In In some some species, species, for for example, example, those those in in which which the the genomic genomic deleterious deleterious aa half. mutation rate rate is is high, high, this this could could be be aa major major effect; effect; but but for for species species with with lower lower mutation mutation rates, rates, this this could could be be aa trivial trivial effect. effect. The The balance balance of of the the effects of these these mutation effects of processes will will depend depend on on the the specifics specifics of of the the species. species. processes -
7.6 7.6
CONCLUSIONS AND INCONCLUSIONS INCONCLUSIONS CONCLUSIONS AND The course of of evolution evolution is is changed changed quantitatively and qualitatively The course quantitatively and qualitatively by by the the subdivision of space. All the population population genetic genetic processes processes subdivision of populations populations over over space. All of of the that populations are substantially, and that act act in in unstructured unstructured populations are affected, affected, sometimes sometimes substantially, and some possible with populations. This some kinds kinds of of evolution evolution are are only only possible with structured structured populations. This chapter chapter focused focused on on the the former: former: quantitative quantitative changes changes in in evolutionary evolutionary rates rates from from population population subdivision. subdivision. Even Even with with uniform uniform selection, selection, the the rate rate of of genetic genetic drift drift and the are changed and the response response to to selection selection are changed substantially. substantially. For this chapter the probabil For some some of of the the quantities quantities described described in in this chapter (e.g., (e.g., Ne, the probability ity of of fixation fixation of of beneficial beneficial alleles), alleles), results results have have already already been been found found for for aa special special case case of of population population structure: structure: the the island island model. model. The The island island model model is is the the oldest oldest in in population population genetics, genetics, and and it it is is rightfully rightfully the the first first to to turn turn to to when when considering considering new new problems problems because because of of its its simplicity. simplicity. Unfortunately, Unfortunately, the the very very simplicity that that make make it it appealing appealing also also makes makes it it an an aberration. aberration. The The island island simplicity model model assumes assumes that that all all demes demes are are equal; equal; not not only only do do all all demes demes have have the the same same population population size size and and migration migration and and immigration immigration rates, rates, but but more more importantly, importantly, it it implicitly assumes assumes that that all all demes demes contribute contribute exactly exactly equally equally to to the the next next gener generimplicitly ation. ation. Clearly Clearly these these conditions conditions do do not not apply apply to to most most (or (or even even all) all) natural natural popu populations, lations, but but this this would would not not matter matter if if these these assumptions assumptions had had no no effect effect on on our our evolutionary evolutionary predictions. predictions. Unfortunately, Unfortunately, this this assumption assumption of of equal equal reproductive reproductive success success has has aa qualitative qualitative effect effect on on our our predictions, predictions, especially especially for for questions questions that that involve involve effective effective size. size. In In this this subtle subtle but but key key respect, respect, the the island island model model is is an an extreme extreme model, model, and and some some of of the the predictions predictions made made from from the the island island model model are are extreme extreme as as aa result. result. Fortunately, Fortunately, it it is is possible possible to to derive derive theory theory that that predicts predicts the the necessary necessary param parameters eters for for other other models models of of population population structure. structure. The The last last couple couple of of decades decades have have seen seen aa lot lot of of development development of of models, models, including including isolation isolation by by distance, distance, local local extinction, extinction, population population size size change, change, variable variable migration migration rates, rates, and and asymmetric asymmetric
7. 7.
SELECTION AND AND DRIFT DRIFT IN IN METAPOPULATIONS METAPOPULATIONS SELECTION
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migration. Even Even more more fortunately, fortunately, the the results results described described in in this this chapter chapter show show migration. that, at at least least for for weak weak selection, selection, most most of of the the effects effects of of population population structure structure can can that, be described described in in aa few few summary summary statistics, statistics, especially especially FsT PST and and N Ne• This is is be e. This extremely useful useful because because we we know know aa lot lot about about how how FsT PST isis changed changed by by various various extremely demographic processes processes and and we we have have the the theory theory to to predict predict the the effective effective size size for for demographic broad class class of of models. models. FsT PST in in particular particular has has been been very very well well studied, studied, with with aa broad many empirical empirical studies studies devoted devoted to to measuring measuring itit in in aa wide wide variety variety of of species species and and many large number number of of theoretical theoretical models. models. These These include include extinction extinction and and recolonrecolon aa large ization (Wade (Wade and and McCauley, McCauley, 1988; 1988; Whitlock Whitlock and and McCauley, McCauley, 1990), 1 990), populapopula ization tion fission fission and and fusion fusion (Whitlock, (Whitlock, 1994), 1 994), source-sink source-sink models models (Gaggiotti, (Gaggiotti, tion 1 996), and and stepping-stone stepping-stone models models (Kimura (Kimura and and Weiss, Weiss, 1964). 1 964). In In all all of of these these 1996), cases, FsT PST differs differs significantly significantly from from that that predicted predicted by by the the island island model, model, and and in in cases, most the the effective effective population population size size is is also also substantially substantially different different (and (and usually usually most much less less than than the the census census size). size). Moreover, Moreover, itit is is usually usually straightforward straightforward to to much calculate FsT PST even even for for aa novel novel system. system. calculate As an an aside, aside, the the reason reason that that FsT PST has has been been measured measured empirically empirically so so often often has has As little to do do with its importance to predict the effects effects of of population population structure structure on on little to with its importance to predict the selection or or drift. drift. FsT PST has has been been measured measured usually of the the false false hope hope that that selection usually because because of it could could be be used used to to estimate estimate the the number number of of migrants migrants coming coming into into aa population population it per generation (Whitlock and McCauley, 1999). 1 999). It It is is fortunate fortunate then that this this per generation (Whitlock and McCauley, then that effort has not been wasted, important not not to to throw effort has not been wasted, and and it it is is important throw the the evolutionary evolutionary baby the estimator bathwater. PST baby out out with with the estimator bathwater. FsT is is an an excellent excellent descriptor descriptor of of the the nature of population be calculated in genetic nature of population structure structure and and should should be calculated in genetic studies studies of of metapopulations. Unfortunately, Unfortunately, the be said properties as metapopulations. the same same cannot cannot be said for for its its properties as an an estimator estimator of of dispersal. dispersal. There There are are many many unresolved unresolved questions questions on on evolution evolution on on space. space. We We have have made made some some progress progress in in understanding understanding the the effects effects of of population population structure structure on on response response to to uniform uniform selection, selection, but but we we have have not not yet yet made made similar similar progress progress with with the the heterogeneous case. All heterogeneous selection selection case. All of of the the results results considered considered here here deal deal with with dis discrete crete populations populations in in which which organisms organisms are are grouped grouped into into demes demes with with the the space space between between them them empty. empty. Most Most of of the the questions questions presented presented here here have have not not solved solved for for the the spatial spatial case case in in which which individuals individuals are are spread spread continuously continuously over over space, space, aa much much more more challenging challenging topic. topic. These These results results all all assume assume weak weak selection, selection, yet yet some some of of the the most most interesting interesting cases cases involve involve selection selection coefficients coefficients stronger stronger than than migration migration rates. rates. We also also need need many many more more empirical empirical studies studies on on these these topics. topics. This This chapter chapter We has has not not reviewed reviewed the the empirical empirical literature literature at at all, all, but but most most of of the the theory theory pre presented sented here here remains remains untested untested experimentally. experimentally. Furthermore, Furthermore, we we need need better better measures measures of of some some key key parameters. parameters. The The dominance dominance coefficient coefficient has has aa ten tendency dency to to cancel cancel out out of of panmictic panmictic calculations, calculations, but but this this is is not not true true for for evolu evolution tion in in structured structured populations; populations; we we have have very very few few estimates estimates of of the the distribution distribution of of dominance dominance coefficients. coefficients. We We desperately desperately need need more more empirical empirical studies studies of of the the effective effective size size of of structured structured populations. populations. We We also also need need to to develop develop individual-weighted individual-weighted estimators estimators of of PST, FsT, as as has has been been shown shown to to be be required required by by this this theory. theory. The The subdivision subdivision of of aa species species over over space space can can affect affect its its evolution evolution strongly strongly and and in aa variety variety of of ways. ways. Because Because most most species species in in nature nature are are subdivided subdivided over over space, space, in itit behooves behooves us us to to understand understand this this nearly nearly ubiquitous ubiquitous feature feature of of the the natural natural world. world.
sdfsdf
META PO PU PULATIONS M ETAPO LATI O N S AND COALESCENT CO A LES CENT AND TH EORY EO RY John Wakeley Wakeley John
8.1 8.1
INTRODUCTION INTRODUCTION Coalescent Coalescent theory, theory, or or the the study study of of gene gene genealogies, genealogies, provides provides the the framework framework for rapidly moving for empirical empirical molecular molecular population population genetics. genetics. It It is is aa rapidly moving field field that that at at once long history once draws draws upon upon the the long history of of population population genetics genetics theory theory and and responds responds to to the the latest latest advances advances in in biotechnology. biotechnology. The The essence essence of of the the coalescent coalescent is is that that it it models models the the genealogical genealogical history history of of aa sample sample of of genetic genetic data data and, and, via via that that history, history, makes makes predictions predictions about about patterns patterns of of variation variation that that might might be be observed observed development of the coalescent among members of the sample. During the development approach 970s and 980s, there approach between between the the early early 11970s and the the early early 11980s, there was was aa switch switch in in viewpoint from the prospective view view taken by classical classical population population genetics to aa new one that begins with a sample and looks backward in time (Ewens, 11990). 990). The The immense immense practical practical benefit benefit of of this this was was that that it it was was no no longer longer neces necessary sary to to describe describe the the properties properties of of an an entire entire population population and and then then imagine imagine sam sampling piing from from it it in in order order to to make make predictions predictions about about aa sample sample of of genetic genetic data: data: only only the the direct direct ancestors ancestors of of the the sample sample mattered. mattered. The The aim aim of of this this chapter chapter is is to to describe describe the the basic basic features features of of coalescence coalescence in in unstructured unstructured populations, populations, to to dis discuss cuss how how this this forms forms aa basis basis for for inference inference about about population population history, history, and and then then to to discuss discuss the the ways ways in in which which metapopulation metapopulation structure structure changes changes these these basic basic fea features tures and and what, what, in in turn, turn, the the prospects prospects are are for for historical historical inference inference in in metapopu metapopulations. In taking taking the the coalescent coalescent approach, this this chapter chapter complements complements those of of
Ecology, Ecology,Genetics, Genetics,and and Evolution Evolution of of Metapopulations Metapopulations
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Copyright Copyright 2004, 2004, Elsevier, Elsevier,Inc. Inc. 0-12-323448-4 0-12-323448-4
JOHN JOHN WAKELEY WAKELEY
1176 76
Chapters 9, which consider classical, Chapters 77 and and 9, which consider classical, forward-time forward-time dynamics dynamics of of genetic genetic variation in variation in aa metapopulation metapopulation and and the the genetics genetics of of quantitative quantitative traits traits in in aa meta population, respectively. metapopulation, respectively.
8.2 8.2
COALESCENCE COALESCENCE IN IN PANMICTIC PANMICTIC POPULATIONS POPULATIONS Although Although the the seeds seeds of of genealogical genealogical thinking thinking and and coalescence coalescence date date back back at at least 1 972) and Karlin and 1 972) on least to to the the work work of of Ewens Ewens ((1972) and Karlin and McGregor McGregor ((1972) on the the sampling theory neutral alleles alleles under alleles model sampling theory of of selectively selectively neutral under the the infinite infinite alleles model of of mutation, obvious in 1 975) on mutation, and and are are obvious in the the work work of of Watterson Watterson ((1975) on the the number number of of segregating mutation without segregating sites sites in in aa sample sample under under the the infinite infinite sites sites model model of of mutation without recombination, 980s that familiar ancestral recombination, it it was was not not until until the the early early 11980s that the the familiar ancestral process known process known as as the the coalescent coalescent was was firmly firmly established. established. Almost Almost simultaneously, simultaneously, Kingman 1 982a,b,c) proved the convergence of the the ancestral Kingman ((1982a,b,c) proved the convergence of ancestral process process for for aa sample to process, which called the sample to this this simpler, simpler, pure pure death death process, which he he called the n-coalescent, n-coalescent, whereas 1 983a) and 1 983) explored whereas Hudson Hudson ((1983a) and Tajima Tajima ((1983) explored many many properties properties of of gene gene genealogies that biologists. The genealogies that are are of of direct direct interest interest and and use use to to biologists. The mathematically mathematically similar just before similar theory theory of of lines lines of of descent descent was was introduced introduced just before this this by by Griffiths Griffiths ((1980). 1 980). Tavare 1 984) reviewed reviewed these these early early mathematical Tavar~ ((1984) mathematical developments, developments, and and Hudson 1 990) and 1 ) reviewed Hudson ((1990) and Nordborg Nordborg (200 (2001) reviewed the the broader broader biological biological scope scope of of coalescent coalescent theory. theory. This section This section explores explores the the properties properties the the standard standard coalescent coalescent process process that that Kingman described and called called the n-coalescent (a (a bit of terminology that never quite biological practitioners). quite caught caught on, on, at at least least among among more more biological practitioners). Note Note that that ter terminology example, ancestral minology is is used used loosely loosely in in general general here: here: for for example, ancestral process, process, coales coalescent cent process, process, and and genealogical genealogical process process are are used used interchangeably, interchangeably, without without reference to standard coalescent reference to any any particular particular model model of of aa population. population. The The standard coalescent involves involves aa number number of of assumptions assumptions in in addition addition to to the the assumption assumption that that the the popu population panmictic, i.e., lation is is well well mixed, mixed, or or panmictic, i.e., mating mating randomly randomly across across the the entire entire range talking about about diploid also assumed range if if we we are are talking diploid species. species. It It is is also assumed that that variation variation is selectively neutral, neutral, that is selectively that the the effective effective size size of of the the population population has has not not changed changed that there is no recombination recombination within the locus under under study. over time, time, and that The choice model is is rather and typically The choice of of aa mutation mutation model rather flexible flexible and typically depends depends on on the the type of data that that is will be type of data is available available or or will be gathered. gathered. Deviations Deviations from from each each of of these these assumptions sections, although assumptions will will be be considered considered here here and and subsequent subsequent sections, although the the main focus chapter is main focus of of this this chapter is to to describe describe the the effects effects of of metapopulation metapopulation struc structure gene genealogies discuss the ture on on gene genealogies and and the the coalescent coalescent process process and and to to discuss the impli implications cations of of this this for for inference. inference.
The Structure of The Structure of Gene Gene Genealogies Genealogies For For aa wide wide variety variety of of population population models, models, which which differ differ in in terms terms of of important important biological distribution of of offspring number among biological properties, properties, such such as as the the distribution offspring number among members population and members of of the the population and whether whether generations generations are are overlapping overlapping or or dis discrete, 1 9 82a,b,c) proved crete, Kingman Kingman ((1982a,b,c) proved that that the the ancestral ancestral process process for for aa sample sample of of finite finite size size nn converges converges to to the the coalescent coalescent as as the the population population size size tends tends to to infin infinity. happen to ity. In In this this limit, limit, all all of of the the myriad myriad possible possible events events that that could could happen to the the sample looking looking back back in single generation reduce to either all all items sample in time time aa single generation reduce to two: two: either items
8. 8.
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have distinct parents parents or share aa common common ancestor. have distinct or two two members members of of the the sample sample share ancestor. The the popula The other other possibilities, possibilities, whose whose probabilities probabilities become become negligible negligible as as the population tion size size goes goes to to infinity, infinity, are are those those in in which which more more than than one one of of these these common common ancestor ancestor events events happens happens in in aa single single generation. generation. For For example, example, in in aa small small popu population, lation, two two pairs pairs of of samples samples may may have have common common ancestors ancestors or or more more than than two two members single common common ancestor members of of the the sample sample may may share share aa single ancestor in in the the immedi immediately ately previous previous generation, generation, and and the the probability probability of of this this cannot cannot be be neglected. neglected. In In aa large history of large population, population, the the genealogical genealogical history of aa sample sample is is simplified simplified greatly greatly because genealogies are because such such events events are are extremely extremely unlikely. unlikely. The The resulting resulting genealogies are easy easy to describe describe in in words words and and to to model model mathematically. mathematically. to This only for currently sampled items, but This simple simple process process holds holds not not only for the the currently sampled items, but also also for for the the lineages lineages ancestral ancestral to to them them that that existed existed at at some some time time in in the the past. past. Thus, Thus, the the genealogy genealogy of of aa sample sample under under the the standard standard coalescent coalescent is is simply simply aa series series of of com common mon ancestor ancestor events events between between pairs pairs of of lineages, lineages, by by which which the the sample sample of of n n items, items, or or lineages, lineages, can can be be traced traced back back to to aa single single common common ancestor. ancestor. An An example example geneal genealogy .l. ogy is is shown shown in in Fig. Fig. 88.1. Times Times ttoo Common Common Ancestry Ancestry
The The history history of of aa sample sample of of n n items items includes includes exactly exactly n n - 1i coalescent coalescent intervals. intervals. These These are are the the times times in in the the history history of of the the sample sample during during which which there there were were n, n, n - 11,,. .. .., . , 3, 3, 2 2 lineages lineages ancestral ancestral to to the the sample. sample. In In Fig. Fig. 8.1, 8.1, Tj Ti is is used used to to denote denote the lineages. For the time time during during which which there there were were ii ancestral ancestral lineages. For aa broad broad class class of of models models of of aa population population ~ the the "exchangeable" "exchangeable" models models of of Cannings Cannings ((1974) 1 9 7 4 )~ Kingman Kingman showed showed that that these these times times are are independent independent and and distributed distributed exponentially: exponentially: -
fTi(ti) = (~)e-(i~) ti
((8.1) 8.1)
when when time time is is measured measured iinn units units of of G/u G/or 22 generations, generations, where where G G iiss the the total total num number of of copies of each each genetic genetic locus locus in in the the population, population, and and u (r22 is is the the variance variance in in ber copies of offspring offspring number. number. Under Under the the commonly commonly used used Wright-Fisher Wright-Fisher model model (Fisher, (Fisher, 11930; 930; Wright, 93 1 ) of of aa diploid, Wright, 11931) diploid, monoecious monoecious organism, organism, G G is is equal equal to to 2N, 2N,
AA Fig. 8.1
BB
CG
DD
EE
An An example example genealogy genealogy of of five five items items under under the the standard standard coalescent. coalescent.
JOHN WAKELEY WAKELEY iOHN
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where N is equal equal to to 1. 1 . Strictly Strictly speaking, speaking, N isis the the number number of of individuals individuals and and 00".22 is where the coalescent coalescent is is aa model model for for aa haploid haploid population. population. However, However, itit holds holds exactly exactly the as stated stated for for the the diploid diploid Wright-Fisher Wright-Fisher model, model, due due to to the the assumptions assumptions of of ranran as dom mating dom mating and and monoecy, monoecy, and and itit holds holds when when there there are are two two sexes sexes if if N N is replaced with with the the appropriate appropriate effective effective population population size size (M6hle, (Mahle, 1998b). 1 998b). Thus, Thus, replaced on average, average, genealogies genealogies will will look look something something like like the the example example in in Fig. Fig. 8.1 8 . 1 in in on which T2 T2 >> T3 T3 >> T4 T4 > > Ts, Ts, although although the the expectation expectation is is even more skewed skewed than than which even more what is is shown shown in in the the Fig. Fig. 8.1, 8 . 1 , as as E[Ti] 2/[i(i -- 1)]. I}]. The The variances variances of of these these what E [ T;] == 2/[i(i times are are quite quite large large as as well. well. Nordborg Nordborg (2001) (200 1 ) displayed displayed several several realizations realizations times of the the coalescent coalescent process process to to illustrate illustrate this. this. of and TTji are are independent independent if if ii ~-:t j,j, only Eq. Eq. (8.1) ( 8 . 1 ) and and the the fact that T; Using fact that Ti and Using only many useful useful analytical analytical results results can can be be obtained, obtained, including the distribution distribution of of many including the ( TMRCA) and and the the the time time to to the the most most recent common ancestor of the the sample sample (TMRCA) the recent common ancestor of (TTotal), i.e., i.e., the the sum sum of of the the distribution of the the total total length of the the genealogy genealogy (Tvotat), distribution of length of lengths of of all all the the branches in the the genealogy. genealogy. Expressions these two prob lengths branches in Expressions for for these two probability not particularly particularly illuminating, illuminating, and are not not ability functions functions are are complicated, complicated, not and are reproduced here (see Tavare, 984). From From these distributions, or or directly from reproduced here (see Tavar6, 11984). these distributions, directly from Eq. ((8.1), 8 . 1 ), one can obtain the familiar the expected expected values values of of Eq. one can obtain the familiar expressions expressions for for the these quantities: these quantities:
E[TM E[TMRCA] RCA]
( - �),
= 2(11 _ 1 ) , = 2
-111 �1
E [ TTotal] = 22/~1= 2,; 7" --;-. E[rTotal] ;= 1 t
=
((8.2) 8.2) (8.3) (8.3)
As As seen seen later, later, the the second second of of these these determines determines the the expected expected number number of of poly polymorphic, morphic, or or segregating, segregating, sites sites at at aa locus locus when when mutations mutations occur occur according according to to the 975). the infinite infinite sites sites model model (Watterson, (Watterson, 11975). In addition, addition, it it is is not not too too difficult difficult to to derive derive the the expected expected total total length length of of In branches branches in in the the history history that that have have ii descendents descendents in in the the sample: sample:
E['ri]
2 =
_, l
(8.4) (8.4)
((Fu, Fu, 11995), 995), which which for for 11 ::5 -< ii ::5 --- nn - 1 are the individual individual terms in the expected expected sum 8 .3)]. In . 1 , the sum of of all all branch branch length length [Eq. [Eq. ((8.3)]. In Fig. Fig. 88.1, the branch branch above above the the asterisk, asterisk, up up to to the the root root of of the the tree, tree, has has three three descendents. descendents. It It is is the the only only branch branch in in that that particular particular genealogy genealogy that that can can contribute contribute to to 'T3, $3, whereas whereas all all the the other other branches branches contribute contribute to to either either 'T$1l or or 'T$2, and none none in in that that tree tree can can contribute contribute to to 'T4. $4. Under Under 2 , and the represents the opportunity for the infinite infinite sites sites model model of of mutation, mutation, 'T; Ti represents the opportunity for the the cre creation ation of of aa polymorphic polymorphic site site at at which which the the ancestral ancestral base base is is in in nn - ii copies copies and and the copies in 8.4) is the mutant mutant base base in in ii copies in the the sample. sample. Thus, Thus, Eq. Eq. ((8.4) is important important in in mak making ing predictions predictions about about base base frequencies frequencies at at polymorphic polymorphic sites. sites. Branching Pattern of Genealogies
Under Under the the standard standard coalescent coalescent model, model, every every pair pair of of ancestral ancestral lineages lineages has has an an equal equal chance chance of of being being the the pair pair that that coalesces coalesces at at each each common common ancestor ancestor event. event. In In fact, fact, the the simple simple ancestral ancestral process process and and the the rate rate factor factor (�) (~) in in specific specific follow follow
8. 8.
METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY METAPOPULATIONS
11 779 9
from the the fact fact that that each each pair pair of of lineages lineages in in the the coalescent coalescent limit limit coalesces coalesces with with from rate 11 independently independently of of all all other other pairs. pairs. Thus, Thus, every every possible possible random-joining random-joining rate tree is is equally equally likely likely under under the the coalescent. coalescent. In In addition, addition, the the structure structure of of the the tree genealogy and and the the coalescent coalescent times times are are independent independent of of one one another. another. The The many many genealogy useful results results of of the the coalescent, coalescent, some some of of which which are are discussed discussed in in Section Section 8.3, 8 .3 , folfol useful low from from these these facts facts together together with with the the specific specific distributions distributions of of coalescence coalescence low times described described earlier. earlier. In In fact, we have have already already seen seen one one such such result, result, Eq. Eq. (8.4), ( 8 .4), times fact, we in which which the the derivation derivation depends depends on on the the random-joining random-joining structure structure of of genealgeneal in ogies (see (see Fu, Fu, 1995). 1 995). ogies Genealogies and Recombination Genealogies and Recombination
If all all variation variation is is selectively selectively neutral, neutral, and and the other assumptions assumptions of of the stand If the other the standard model model are are true, true, then then the the marginal marginal distribution distribution of of the the genealogy genealogy at at any any ard E [ TMRCA] , nucleotide site site is is given given by by the the coalescent. coalescent. Thus, Thus, quantities such as as E[TMRcA], nucleotide quantities such that do do not not depend depend on on the the joint joint distribution distribution of of the the histories histories and E['ri] E[Tj] that E [ TTotatl, and E[TTotal], at multiple sites in turn turn depend depend on the rate rate of of recombination. recombination. The The joint at multiple sites do do not not in on the joint ancestral process at at two or more critically on recombination. ancestral process two or more sites sites depends depends critically on recombination. Therfore, that do the joint joint histories sites are are affected by Therfore, quantities quantities that do depend depend on on the histories of of sites affected by recombination. For example, the the variances of T MRCA, TTotal, TTotal, and at aa locus locus can can recombination. For example, variances of TMRCA, and TTji at be functions of of the the covariance in coalescence pairs of of sites be expressed expressed as as functions covariance in coalescence times times at at pairs sites in the the sequence, functions of the recombination recombination rate between in sequence, which which in in turn turn are are functions of the rate between the 983b; Hudson Kaplan, 1985). 1 985). McVean McVean (2002) (2002) provided provided the sites sites (Hudson, (Hudson, 11983b; Hudson and and Kaplan, aa simple simple genealogical genealogical derivation correlation in in coalescence coalescence time pair derivation of of the the correlation time for for aa pair of sites. of sites. Hudson ((1983b) 1 983b) and and others, others, including including Kaplan Kaplan and Hudson (1985), ( 1 985), Hudson and Hudson Griffiths 1 996), and 1 999), have Griffiths and and Marjoram Marjoram ((1996), and Wiuf Wiuf and and Hein Hein ((1999), have studied studied the the coalescent process locus. If coalescent process at at aa multisite multisite genetic genetic locus. If there there is is no no recombination, recombination, then then the the entire entire locus locus follows follows one one genealogy. genealogy. Recombination Recombination events, events, viewed viewed backward cause the backward in in time, time, cause the ancestral ancestral segments segments on on either either side side of of the the recombin recombination ation breakpoint breakpoint to to be be separated separated onto onto two two different different copies copies of of the the chromosome. chromosome. Genealogies Genealogies under under recombination recombination become become complicated complicated webs, webs, as as ancestral ancestral sites sites travel travel together together for for periods periods of of time time on on the the same same chromosome chromosome and and are are split split up up by by recombination, recombination, possibly possibly coming coming back back together together later later in in coalescent coalescent events. events. However, However, the the genealogy genealogy of of each each site site individually individually remains remains aa simple simple random randomjoining joining tree, tree, with with marginal marginal distribution distribution described described by by the the standard standard coalescent. coalescent. If If the the recombination recombination rate rate is is very very high, high, then then the the genealogy genealogy of of every every site site is is inde independent pendent of of the the genealogy genealogy of of every every other other site. site. The The effect effect on on the the covariances covariances in in coalescence coalescence times times at at pairs pairs of of sites sites in in the the sequence sequence is is predicted predicted from from these these con considerations. siderations. It It approaches approaches zero zero as as the the recombination recombination rate rate becomes becomes large large and and the the sites' sites' genealogies genealogies become become independent, independent, and and it it grows grows as as the the recombination recombination rate rate decreases. decreases. Extensions Extensions to to the the Coalescent Coalescent
The The basic basic coalescent coalescent technology technology of of modeling modeling the the genealogical genealogical process process for for aa sample been extended sample of of genetic genetic data data has has been extended in in many many different different directions. directions. Examples 1991 ), who Examples include include Slatkin Slatkin and and Hudson Hudson ((1991), who considered considered changes changes in in popu population 1 990), who lation size size over over time; time; Notohara Notohara ((1990), who gave gave aa general general mathematical mathematical model model of of coalescence coalescence in in aa geographically geographically structured structured population; population; Kaplan Kaplan et et al. al. ((1988), 1988), who 1 997), who who modeled modeled strong strong selection, selection, and and Krone Krone and and Neuhauser Neuhauser ((1997), who
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JOHN IOHN WAKELEY WAKELEY
described weak selection. described aa framework framework for for the the coalescent coalescent with with weak selection. Theoretical Theoretical works phenomena in isolation, but works tend tend to to treat treat these these phenomena in isolation, but it it will will often often be be neces necessary sary to to include include several several factors factors when when interpreting interpreting data. data. An An example example is is Kaplan Kaplan et al. ((1991), 19 9 1 ) , who balancing selection, et al. who used used aa model model that that includes includes balancing selection, recombin recombination, subdivision to ation, and and geographic geographic subdivision to explain explain the the decrease decrease in in levels levels of of poly polymorphism morphism with with distance distance from from aa selected selected site site in in samples samples of of the the Adh Adh gene gene in in Drosophila Drosophila melanogaster melanogaster (Kreitman, (Kreitman, 1983). 1983). In In addition, addition, the the standard standard coalescent coalescent has has been been obtained obtained under under aa variety variety of of circumstances, circumstances, such such as as the the case case of of aa two-sex two-sex diploid diploid population population mentioned mentioned earl earlier, in in which which it it is is not not obvious obvious at at first first that that such such aa simple simple model model should should hold hold ier, (M6hle, 998c). These coalescent derive (M6hle, 11998c). These "robustness" "robustness" results results for for the the coalescent derive from from aa lemma of M6hle ((1998) 1998) on chains with lemma of M6hle on the the convergence convergence of of discrete discrete Markov Markov chains with two two timescales timescales to to simpler, simpler, continuous continuous time time processes. processes. For For example, example, in in the the case case of of two two sexes, sexes, lineages lineages switch switch back back and and forth forth between between males males and and females females much much faster is obtained, only with faster than than they they coalesce, coalesce, and and the the standard standard coalescent coalescent is obtained, only with aa rescaled size that number of rescaled effective effective population population size that is is aa function function of of the the number of males males and and females population females in in the the population. population. The The result result for for genealogies genealogies in in aa meta metapopulation described in in Section Section 8.3 8.3 is is based based on on this this kind kind of of separation separation of of timescales. timescales. described
Mutation and Patterns Genetic Variation M u t a t i o n and Patterns of of Genetic Variation in in a a Sample Sample An An aligned aligned set set of of DNA DNA sequences sequences sampled sampled from from aa population population is is aa poten potentially about the tially rich rich source source of of information information about the history history and and current current demography demography of of the the population. population. The The coalescent, coalescent, together together with with aa model model of of mutation, mutation, can can be be used used to to make make predictions predictions about about levels levels and and patterns patterns of of variation variation in in aa sample. sample. The The most most frequently frequently used used mutation mutation model model for for DNA DNA sequences sequences is is the the infinite infinite sites model, which assumes assumes that that every every mutation mutation happens at aa previously previously unmu unmusites model, which happens at tated site. Thus, tared site. Thus, the the infinite infinite sites sites model model is is appropriate appropriate when when the the per-site per-site muta mutation rate is is low. low. Recombination Recombination can, of course, be an an important important factor factor in in tion rate can, of course, be determining patterns patterns of variously assumed assumed determining of genetic genetic variation, variation, and and workers workers have have variously no all (Watterson, 975), independent no recombination recombination at at all (Watterson, 11975), independent assortment assortment among among all all sites 969), or intermediate level sites (Kimura, (Kimura, 11969), or any any intermediate level of of recombination recombination (Hudson, (Hudson, 11983b). 983b). As As noted noted earlier, earlier, the the importance importance of of modeling modeling recombination recombination will will depend on on how how data data are are analyzed. analyzed. depend Undoubtedly about mutation Undoubtedly the the most most important important assumption assumption about mutation in in the the stand standard coalescent is neutral. Genetic ard coalescent is that that all all variation variation is is selectively selectively neutral. Genetic similarities similarities and to past past and present and differences differences among among sampled sampled sequences sequences are are aa view view to and present demography, demography, such such as as metapopulation metapopulation structure, structure, rather rather than than directly directly the the sub subject of natural selection, noted earlier earlier it it is is possible the ject of natural selection, although although as as noted possible to to extend extend the coalescent to to include include selection selection at at aa site under study. study. Due Due coalescent site linked linked to to the the locus locus under to measured under to the the way way in in which which time time is is measured under the the coalescent, coalescent, the the appropriate appropriate mutation mutation parameter, parameter, e, 0, is is similarly similarly scaled. scaled. Under Under the the Wright-Fisher Wright-Fisher model, model, e0 = = 4Nu, 4Nu, where where u u is is the the rate rate of of neutral neutral mutation mutation per per locus locus copy copy per per genera generation. Thus tion. Thus e0 is is equal equal to to twice twice the the average average number number of of mutations mutations introduced introduced into into the the population population each each generation. generation. The The extra extra factor factor of of two two is is due due to to the the histori historical cal importance importance of of the the notion notion of of heterozygosity, heterozygosity, the the expected expected value value of of which which in in randomly mating mating population population is is equal equal to to e. 0. aa randomly Because Because the the mutation mutation rate rate per per generation generation is is very very low, low, mutation mutation is is modeled modeled accurately accurately as as aa Poisson Poisson process process along along the the branches branches of of genealogy. genealogy. Specifically, Specifically,
METAPOPULATIONS AND COALESCENT COALESCENT THEORY THEORY S AND 8. METAPOPULATION
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the the number number of of mutations mutations on on aa lineage, lineage, lineages, lineages, or or entire entire genealogy genealogy of of given given Ot/2. Mutations under length tt follows the Poisson distribution with parameter etl2. neutrality, neutrality, by by definition, definition, do do not not affect affect the the reproductive reproductive rates rates of of individuals. individuals. Thus, Thus, the the genealogical genealogical process process and and the the mutation mutation process process can can be be treated treated separ separately. ately. This This allows allows predictions predictions to to be be made made easily easily under under the the coalescent coalescent about about many many measurable measurable aspects aspects of of DNA DNA sequence sequence polymorphism. polymorphism. The The reason reason for for generating generating such such predictions predictions is is of of course course twofold: twofold: it it builds builds our our understanding understanding how the forces that that maintain variation work, and the predictions can be of how about populations. populations. used for making inferences about Predictions about a b o u t Full Data Data Patterns Patterns Predictions
recomIt is possible under the coalescent with infinite sites mutation and no recom bination bination to to analytically analytically compute compute the the probability probability of of observing observing any any possible possible data data set, set, i.e., i.e., the the full full sample sample of of DNA DNA sequences, sequences, using using aa recursive recursive equation equation ((Griffiths Griffiths and 9 95 ) . The and Tavan\ Tavar~, 11995). The method method can can be be extended extended to to more more general general models Bahlo and models that, that, for for instance, instance, include include geographic geographic structure structure ((Bahlo and Griffiths, Griffiths, 2000). This analytic method method is infeasible except except for for small samples though though 2000). because the number of equations that must be because the number of equations that must be solved solved simultaneously simultaneously becomes becomes astronomically astronomically large large for for complex complex data data sets sets of of many many sequences. sequences. However, However, as as approach can be turned turned into a Monte Monte Carlo Section 8.3 describes, this general approach method of inference. method A A second second issue issue with with this this recursive recursive method method on on full full data data patterns patterns is is that that it it not lend itself to investigations of how how the forces that that produce produce and and main maindoes not act to shape patterns patterns of genetic variation. This This is a general con contain variation act cern cern rather rather than than aa problem problem with with this this particular particular analytic analytic method. method. While While there there obviously aa wealth wealth of of information information in in aa data of DNA DNA sequences, sequences, it it is is is obviously data set set of poorly known which aspects aspects of contain the the bulk bulk of of information information about about poorly known which of data data contain each factor of might have have been been important important in the history history of each factor of evolution evolution that that might in the of the the particular particular species species under under study. study. From From the the theoretical theoretical perspective, perspective, another another aspect aspect this problem is that of this that some of of the the parameters parameters in a complicated historical historical model might might be nonidentifiable nonidentifiable (Beaumont ( Beaumont et al., aI., 2003). 2003 ). There There is essentially just in population in which measure has has been been shown just one one result result in population genetics genetics in which aa measure shown to to contain of the information contain all of information about about a population population parameter. parameter. In the case of of allelic or or haplotypic haplotypic data data from an unstructured, allelic from an unstructured, constant-sized constant-sized population population in in which all variation variation is selectively neutral, numwhich neutral, Ewens Ewens (1972) ( 1972 ) showed showed that that the the num ber for 0, e, i.e., that that the the frequencies frequencies of of the the alleles ber of of alleles is a sufficient sufficient statistic for contain additional information. contain no no additional information. Identifying patterns in in data data that that correspond correspond to to particular particular phenomena phenomena and and Identifying patterns making statements, statements, even even approximate approximate ones, ones, about about the the sufficiency of statistics statistics making sufficiency of will likely be a major major focus focus of of research research in in the the future future given the the current current trends trends in inference inference discussed discussed in Section 8.3. 8 .3. Work Work under under the the coalescent has has focused focused on from the affect gross on how how various various deviations deviations from the standard standard model model affect gross summaries summaries of of the data data such such as as the the expected expected base base frequencies frequencies at at polymorphic polymorphic nucleotide nucleotide sites, sites, the which which form form the the basis basis of of the the "neutrality" "neutrality" tests tests of of Tajima Tajima (1989) ( 19 8 9 ) and and Fu Fu and and Li Li ( 1993). In In the the context context of of subdivided subdivided populations, populations, the the majority majority of of effort has (1993). effort has gone to to studies of Wright's Wright's (1951) ( 1 95 1 ) FF statistics, statistics, most most notably notably the the fixation fixation index index gone studies of FsT, FST, even even though though the the significance significance of of FsT FST in in most most situations situations is is unclear unclear (Whitlock (Whitlock and and McCauley, McCauley, 1999). 1 999). In In order order to to untangle untangle the the complex complex current current and and historical historical demography of of populations, populations, for for instance, instance, those those exhibiting exhibiting metapopulation metapopulation demography
JOHN JOHN WAKELEY WAKELEY
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dynamics DNA sequence be necessary dynamics using using summary summary statistics statistics of of DNA sequence data, data, it it will will be necessary at at aa minimum minimum to to expand expand the the battery battery of of such such measures measures to to include include at at least least as as many many measures measures as as the the number number of of parameters parameters affecting affecting the the population. population. Predictions Predictions about about Summary Summary Measures
The The measures measures of of DNA DNA sequence sequence polymorphism polymorphism that that have have received received the the most most attention attention in in theoretical theoretical studies studies and and the the most most use use in in empirical empirical work work are are the the number of total number sites S, the total number of of polymorphic polymorphic (or (or segregating) segregating) sites the average average number of pairwise pairwise differences differences 'IT, IT, and and the the number number of of polymorphic polymorphic nucleotide nucleotide sites, sites, 'TJi, ~qi, at at which frequent base base is size n. The which the the least least frequent is in in ii copies copies in in the the sample sample of of size The reason reason for for focusing focusing on on these these is is partly partly historical. historical. For For example, example, the the significance significance of of 'IT comes from can be used to comes from the the fact fact that that it it can be used to estimate estimate the the heterozygosity heterozygosity of of aa diploid 993). However, diploid population population (Tajima, (Tajima, 11993). However, concerns concerns about about the the efficiency efficiency of of inferences inferences made made from from sequence sequence data data have have also also been been important. important. For For example, example, 1994) studied possible estimators Fu Fu ((1994) studied the the properties properties of of various various possible estimators of of e0 using using lin linpopulations, the ear ear combinations combinations of of the the 'TJ ms. Again, for for subdivided subdivided populations, the focus focus has has i ' Again, been been on on FST, FsT, which which can can be be seen seen as as aa simple simple extension extension of of average average pairwise pairwise differences population ((Slatkin, Slatkin, 1991; differences to to aa structured structured population 1991; Wilkinson-Herbots, Wilkinson-Herbots, 11998). 998). Because Because mutational mutational and and genealogical genealogical processes processes can can be be treated treated separately separately under under neutrality, neutrality, predictions predictions about about these these and and other other summary summary measures measures can can be be made made by by conditioning conditioning on on the the genealogy genealogy or or on on some some relevant relevant aspect aspect of of the the geneal genealogy. ogy. Conditional Conditional on on the the genealogy, genealogy, the the number number of of mutations mutations in in the the history history of of the the sample sample is, is, again, again, Poisson Poisson distributed, distributed, and and under under the the infinite infinite sites sites model model each each mutation produces produces aa polymorphic polymorphic site. site. Thus, Thus, from from the the probability probability density density func funcmutation the tion total length tion for for the the total length of of the the genealogy, genealogy, Ttotal> Ttotat, the the probability probability function function for for the can be be obtained /(t)dt. Beyond number number of of segregating segregating sites, sites, S, can obtained as as foP{S .f~P{S = klt}hot klt}fTto,at(t)dt. Beyond a this, obtain analytical this, it it is is difficult difficult to to obtain analytical expressions expressions for for the the probability probability functions functions for measures of sequence polymorphism. for measures of DNA DNA sequence polymorphism. However, variances, and However, the the derivation derivation of of expected expected values, values, variances, and covariances covariances of of these these measures measures is is straightforward. straightforward. From From Eqs. Eqs. (8.3) (8.3) and and (8.4), (8.4), and and considering considering the Poisson nature process, we the Poisson nature of of the the mutation mutation process, we have have
(Watterson, (Watterson, 1975) 1975) and and
�1 1 E[S] S] ==0n~ e 2.J11. -;-. E[ i = 111 1 i=
(8.5) (8.5)
0
(8.6) (8.6)
E[Zi] = _, z
(Tajima, 989; Fu 993), where (Tajima, 11989; Fu and and Li, Li, 11993), where Zi Z i is is the the number number of of polymorphic polymorphic copies in sites base is sites at at which which the the mutant mutant base is in in ii copies in the the sample. sample. Typically, Typically, because is not the mutant is the the ancestral because it it is not known known which which is is the mutant and and which which is ancestral base, we base, we have have
E['TJ E[xliil] = =
e (lI + +
_1_. n - I
)
i) 11 + 8~)i,n-i i, n - i +
(8.7) (8.7)
8. 8. METAPOPULATIONS METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY
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for spectrum, that patterns Zi for the the "folded" "folded" site site frequency frequency spectrum, that is, is, when when the the patterns Z i and and denominator is one Z Z nn--i i are are indistinguishable. indistinguishable. The The Oi 8i,,n-i n_ i term t e r m in in the the denominator is equal equal to to one if if ii = = n n - ii and and zero zero otherwise. otherwise. It It is is needed needed in in order order to to avoid avoid counting counting Zi Zi twice twice variances and covariances of in case where in the the case where ii = = n n - ii = = nl2. n/2. The The variances and covariances of the the 'TJi Tli can can also (Fu, 1995). also be be obtained obtained (Fu, 1995). The The expected expected value value of of the the average average number number of of pairwise pairwise differences, differences, which which can can be be expressed expressed as as aa simple simple linear linear combination combination of of the the mutant mutant base base frequencies, frequencies, 7T ~r = 'i.7:/i En=-(i ((nn -- i) i) Zil\�), Zi/(~), is is equal equal to to e0 (Tajima, (Tajima, 11983), 983), and and this this is is of of course course identical identical to to the the expected expected value value of of S when when n n = = 2. 2. Tajima 1 983) also Tajima ((1983) also obtained obtained the the variance variance of of 7T. ~r. These These and and other other analytical analytical results results have have been been important important in in building building an an understanding understanding about about the the ancestral ancestral process for for aa sample sample and and in in making making inferences inferences about about populations, both when when process populations, both estimating population parameters estimating population parameters and and when when testing testing the the assumptions assumptions of of the the standard standard coalescent coalescent model. model. =
Making nferences Using Making IInferences Using the the Coalescent Coalescent Because Because of of its its close close connection connection to to samples samples of of genetic genetic data, data, the the coalescent coalescent approach approach provides provides aa natural natural framework framework for for inference inference about about the the structure structure and and history history of of populations populations (see (see Stephens, Stephens, 2001). 2001). Inferences Inferences can can of of course course be be made made using the the classical classical forward-time forward-time approach approach to to population population genetics, genetics, but but in in this this case case using it it becomes becomes aa two-step two-step procedure. procedure. First First the the properties properties of of the the entire entire population population are are considered and and then then the the process process of of sampling sampling from from the the population population is is modeled modeled and and considered the the properties properties of of such such samples samples determined. determined. In In some some cases, cases, the the classical classical approach approach may may be be preferable. preferable. For For instance, instance, much much of of the the ease ease and and computational computational efficiency efficiency of of the the coalescent coalescent evaporate evaporate when when weak weak selection selection acts acts on on variation variation (Krone (Krone and and Neuhauser, Neuhauser, 1997; 1997; Neuhauser Neuhauser and and Krone, Krone, 1997). 1997). However, However, the the convenience convenience and and efficiency efficiency of of the the coalescent coalescent approach approach under under neutrality, neutrality, which which stems stems from from the the fact fact that sample can can be reference to that the the genealogy genealogy of of aa sample be modeled modeled without without reference to the the rest rest of of the coalescent aa very the population, population, make make the the coalescent very powerful powerful inferential inferential tool. tool. Analytical Methods Methods
Where Where analytical analytical results results are are available, available, such such as as those those presented presented in in Section Section 8.3, 8.3, corresponding corresponding inferences inferences can can be be made. made. For For example, example, the the analytical analytical expression expression for for the the probability probability of of observing observing S segregating segregating sites sites in in aa sample sample of of size size n n can can be be used to to make make maximum maximum likelihood likelihood estimates estimates of of e0 under under the the assumption assumption of of no no used intralocus intralocus recombination recombination and and infinite infinite sites sites mutation. mutation. However, However, most most analytical analytical methods methods of of inference inference use use the the method method of of moments, moments, i.e., i.e., to to equate equate the the observed observed value of value of aa measure measure of of sequence sequence polymorphism polymorphism with with its its analytical analytical expectation expectation then then to to solve solve for for the the parameter parameter of of interest. interest. This This has has led led to to aa multitude multitude of of esti estie, of population based mators mators of of the the fundamental fundamental parameter, parameter, 0, of the the population based on on S (Watterson, 983), and 'TJl or (Watterson, 1975), 1975), 7T ~r (Tajima, (Tajima, 11983), and ~ql or other other combinations combinations of of the the 'TJi ~qi ((Fu Fu and and Li, Li, 1993; 1993; Fu, Fu, 1994). 1994). Among Among these, these, 7T ~r has has aa rather rather undesirable undesirable statisti statistical property: cal property: it it is is inconsistent. inconsistent. That That is, is, its its variance variance does does not not decrease decrease to to zero zero as as the 995). the sample sample size size tends tends to to infinity infinity (Tajima, (Tajima, 1983; 1983; Donnelly Donnelly and and Tavare, Tavar6, 11995). Therefore, Therefore, estimates estimates based based on on the the number number of of segregating segregating sites, sites, S, or or on on linear linear combinations combinations of of the the site site frequencies, frequencies, 'TJi, xli, are are preferable preferable to to those those made made using using pairwise pairwise differences. differences. These moment-based estimators unbiased and These moment-based estimators are are unbiased and easy easy to to implement. implement. They They also have applicable regardless also have the the advantage advantage of of being being applicable regardless of of the the recombination recombination
WAKELEY JOHN WAKELEY
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rate rate because because the the relevant relevant expected expected values values do do not not depend depend on on the the rate rate of of recomrecom bination. bination. In In fact, fact, their their accuracy accuracy will will increase increase with with the the rate rate of of recombination, recombination, as the the sites sites in in the the sequence sequence become become more more and and more more independent. independent. Of Of course, course, itit as is is not not satisfactory satisfactory to to make make only only point point estimates estimates of of parameters, parameters, and and recombinrecombin ation must must be be considered considered if if any any statement statement about about the the error error of of these these momentmoment ation based estimates estimates is is to to be be made. made. Obtaining Obtaining analytical analytical results results about about the the variances variances based of S, S, ~r, 1T, and and xli 'll i for for arbitrary arbitrary levels levels of of recombination recombination is is not not trivial. trivial. In In fact, fact, itit is is of only only for for ~r 1T that that such such results results are are available available (Hudson, (Hudson, 1987; 1987; Pluzhnikov Pluzhnikov and and Donnelly, 1996). 1 996). In In addition, addition, the the variance variance of of an an estimator estimator is is only only aa useful useful piece piece Donnelly, of information when its its errors are distributed normally or or at at least least when when its its disdis of information when errors are distributed normally tribution is is known known and and is is symmetric, which is is almost almost never never the the case case for meas tribution symmetric, which for measures of of sequence sequence polymorphism. polymorphism. ures Computational Methods Methods Computational
It is is straightforward straightforward and and extremely efficient to to simulate simulate genealogical histor It extremely efficient genealogical histories 1 9 83b, 11990) 990) because because it it is is not not necessary necessary to to simulate the entire entire ies (Hudson, (Hudson, 1983b, simulate the population, just just the sample history. ease of the population, the sample history. The The ease of simulations, simulations, together together with with the desire more about about how how the the errors errors of parameter estimates estimates are are distrib desire to to know know more of parameter distributed, has led led to explosion of computational coalescent coalescent methods methods of uted, has to aa recent recent explosion of computational of the field ment, it it does does appear focus inference. While While the field is is still still in in develope developement, appear that that the the focus inference. has somewhat over over time. time. The first of these methods methods used used Monte Monte Carlo Carlo has shifted shifted somewhat The first of these integration to to compute compute the the likelihood likelihood of full data data set set under under the the coalescent coalescent of aa full integration model. were used over genealogies genealogies by by simulating model. Simulations Simulations were used to to "integrate" "integrate" over simulating aa large number number of of them them and averaging the the results. results. The marks are are in in large and averaging The quotation quotation marks recognition of the of genealogies, continuously distributed recognition of the complexity complexity of genealogies, having having continuously distributed branch lengths lengths and and discrete discrete tree tree structures. This is is an an impossible task if if branch structures. This impossible task genealogies coalescent without genealogies are are simulated simulated using using the the standard standard coalescent without reference reference to to data because the genealogies make data because the overwhelming overwhelming majority majority of of genealogies make aa negligible negligible con contribution to the likelihood the sample sample size or the number number of poly polytribution likelihood unless the morphic morphic sites sites is is small. small. Two Two different different solutions solutions to to this this problem problem were were proposed. proposed. One One was was to to use use the the recursive equations recursive equations for for the the probability probability of of data data under under the the infinite infinite sites sites model, model, discussed .3, to discussed in in Section Section 88.3, to define define an an ancestral ancestral Markov Markov chain chain conditional conditional on on data data and and to to sample sample genealogies genealogies from from this this rather rather than than from from the the "uncondi "unconditional" coalescent process process (Griffiths 994a,b). The tional" coalescent (Griffiths and and Tavare, Tavar~, 11994a,b). The probability probability of of data data is is then then the the average average value value of of aa function function computed computed for for each each simulated simulated path, path, i.e., i.e., genealogy, genealogy, through through this this Markov Markov chain. chain. Because Because only only genealogies genealogies that that are are minimally minimally compatible compatible with with data data under under the the infinite infinite sites sites model model are are generated, generated, the likelihood can the likelihood can be be estimated estimated with with relative relative ease. ease. This This method method has has been been extended extended to to cover cover geographically geographically structured structured populations, populations, both both with with migration migration (Nath 996) and (Nath and and Griffiths, Griffiths, 11996) and without without (Nielsen, (Nielsen, 1998), 1998), and and loci loci that that undergo undergo recombination (Griffiths (Griffiths and and Marjoram, Marjoram, 11996). In the the case case of of recombination, recombination, recombination 996). In the the straightforward straightforward application application of of this this approach approach is is still still quite quite inefficient, inefficient, and and aa that makes better use of importance importance sampling has been more optimal scheme that proposed proposed (Fearnhead (Fearnhead and and Donnelly, Donnelly, 2001 2001).). The other other solution solution to to the the problem problem of of the the enormity enormity of of the the space space of of all all pos posThe sible sible genealogies genealogies was was to to use use aa Markov Markov chain chain Monte Monte Carlo Carlo (MCMC) (MCMC) method method to to focus focus on on genealogies genealogies that that do do contribute contribute substantially substantially to to the the likelihood likelihood (Kuhner (Kuhner et aI., al., 11995). The chain chain is is run run with with aa starting starting genealogy, genealogy, and and each each subsequent subsequent et 995). The
AND COALESCENT 8. 8. METAPOPULATIONS METAPOPULATIONS AND COALESCENT THEORY THEORY
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step involves the proposal of genealogy and acceptance accord step involves the proposal of aa new new genealogy and then then its its acceptance according proposed and ing to to aa probability probability that that depends depends on on how how new new genealogies genealogies are are proposed and on on the the relative relative contributions contributions of of the the current current and and the the new new genealogies genealogies to to the the likeli likelihood. hood. This This is is an an application application of of Metropolis-Hastings Metropolis-Hastings sampling. sampling. If If the the chain chain is is run run long long enough, enough, then then sampling sampling genealogies genealogies from from it it is is equivalent equivalent to to sampling sampling them likelihood of them with with respect respect to to their their relative relative contribution contribution to to the the likelihood of data. data. Mutation Mutation models models other other than than infinite infinite sites sites can can be be incorporated incorporated easily, easily, which which is is an an advantage advantage of of this this approach approach over over the the one one described described earlier. earlier. This This method method has has also also been been extended extended beyond beyond the the standard standard coalescent coalescent to to include include subdivison, subdivison, both both with 999) and 1) with (Beerli (Beerli and and Felsenstein, Felsenstein, 11999) and without without (Nielsen (Nielsen and and Wakeley, Wakeley, 200 2001) migration, migration, and and to to include include recombination recombination (Kuhner (Kuhner et et ai., al., 2000). 2000). By By definition, definition, these these full-data full-data likelihood likelihood methods methods extract extract the the greatest greatest pos possible sible information information from from data. data. However, However, despite despite being being made made feasible feasible by by focus focusing genealogies relevant ing on on genealogies relevant to to data, data, they they are are highly highly computationally computationally intensive, intensive, sometimes sometimes prohibitively prohibitively so. so. Further, Further, it it is is unclear unclear whether whether all all of of this this computa computation tion is is justified justified in in relation relation to to the the questions questions of of statistical statistical sufficiency sufficiency discussed discussed in in Section Section 8.3. 8.3. For For example, example, it it would would be be aa waste waste of of time time to to design design aa full-data full-data method method of of estimating estimating e0 if if the the data data were were allele allele counts counts under under the the infinite infinite alleles alleles model, model, as as all all the the information information about about e0 is is contained contained in in the the number number of of alleles, alleles, not not the the frequencies. frequencies. While While little little is is known known about about the the axes axes of of information information content content in in samples samples of of DNA DNA sequences, sequences, it it cannot cannot be be expected expected that that all all of of the the many many facets facets of contribute equally of polymorphism polymorphism will will contribute equally to to inferences inferences about about particular particular param parameters. eters. Work Work is is clearly clearly needed needed in in this this area, area, both both to to build build our our knowledge knowledge and and intuition intuition and and to to aid aid in in the the development development of of better better computational computational techniques techniques of of inference. inference. Partially Partially in in response response to to these these concerns concerns about about information information content, content, but but mostly mostly due due to to interest interest in in computational computational feasibility, feasibility, there there is is aa growing growing trend trend to to design measures of poly design computational computational methods methods of of inference inference using using summary summary measures of polymorphism 1997), morphism rather rather than than full full data. data. These These methods methods date date back back to to Fu Fu and and Li Li ((1997), Tavare 1 997), and 1 99 8 ) , and Tavar~ et et al. al. ((1997), and Weiss Weiss and and von von Haeseler Haeseler ((1998), and aa more more recent recent example is example is Beaumont Beaumont et et al. al. (2003) (2003).. They They use use simulated simulated genealogies genealogies to to compute compute the probability of the probability of observing observing aa set set of of summary summary measures measures that that are are identical identical to to or or sufficiently sufficiently close close to to the the values values observed observed in in data. data. The The advantage advantage of of this this approach is much larger randomly generated generated genealogies approach is that that aa much larger fraction fraction of of randomly genealogies have have aa chance chance of of producing producing the the observed observed data data summaries summaries than than the the fraction fraction that that contribute contribute significantly significantly to to the the likelihood likelihood of of the the full full data. data. Another Another recent recent trend trend in in inference inference is is the the growing growing popularity popularity of of the the Bayesian Bayesian approach aI., 2003 approach (see (see Beaumont Beaumont et et al., 2003).). The The difference difference between between the the likelihood likelihood and and Bayesian Bayesian approaches approaches is is less less in in the the mechanics mechanics of of the the computational computational methods methods than in than in the the interpretation interpretation of of the the output, output, i.e., i.e., as as aa likelihood likelihood surface surface or or as as aa pos posterior 5 . 1 , Chapter 5 ) . In terior probability probability distribution distribution (see (see also also Box Box 115.1, Chapter 115). In the the former former case, case, the the large large body body of of statistical statistical theory theory on on the the distribution distribution of of likelihood likelihood ratios, ratios, which which holds holds asymptotically asymptotically as as the the sample sample size size tends tends to to infinity, infinity, is is used used to to construct construct confidence confidence intervals intervals and and test test hypotheses. hypotheses. In In the the latter latter case, case, the the credible intervals credible intervals for for parameters parameters given given data data are are drawn drawn so so that that 95%, 95%, or or some some other other chosen chosen percentage, percentage, of of the the posterior posterior distribution distribution lies lies inside inside the the credible credible interval. interval. The The size size of of the the credible credible interval interval can can depend depend strongly strongly on on the the prior prior dis distribution tribution of of the the parameter, parameter, which which may may be be viewed viewed as as aa drawback drawback of of the the Bayesian approach. approach. In Bayesian methods, Bayesian In defense defense of of Bayesian methods, it it is is questionable questionable whether whether
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the available the asymptotic asymptotic theory theory of of likelihoods likelihoods is is valid valid except except when when data data are are available from loci. from aa large large number number of of independent independent loci. "Neutrality" "Neutrality" Tests Tests
The results for produced series series of The analytical analytical results for SS,, '11" ~r,, and and 11 TI1 (or Zl Z1),) , which which produced of 1 (or 8, have also spawned spawned aa series unbiased method of moments estimators unbiased method of moments estimators of of 0, have also series of of statistical coalescent model. model. Tajima's 1 989) D statistical tests tests of of the the standard standard coalescent Tajima's ((1989) D and and the the tests 1 993) are tests of of Fu Fu and and Li Li ((1993) are the the best best known, known, although although aa number number of of others others have proposed (Simonsen 995). All have been been proposed (Simonsen et et aI., al., 11995). All of of these these tests tests are are based based on on the the fact fact that that aa number number different different measures measures of of polymorphism, polymorphism, representing representing different different 8. Under aspects aspects of of data, data, can can all all be be used used to to estimate estimate the the single single parameter parameter 0. Under the the null null model, model, the the expected expected value value of of the the difference difference between between two two such such estimates estimates is is equal zero. Deviations Deviations from equal to to zero. from zero, zero, the the significance significance of of which which are are best best meas measured by by the the simulation scheme of of Simonsen et al. al. ((1995), lead to to rejection rejection of of ured simulation scheme Simonsen et 1 995), lead the the standard standard coalescent. coalescent. Unfortunately, Unfortunately, although although the the standard standard coalescent coalescent involves number of involves aa number of assumptions, assumptions, there there is is aa strong strong tendency tendency to to see see these these tests tests as as tests tests of of selective selective neutrality neutrality only. only. In measure deviations In fact, fact, all all of of these these tests tests simply simply measure deviations from from the the site-frequency site-frequency dis distribution 8.2 displays tribution predicted predicted by by the the coalescent. coalescent. Figure Figure 8.2 displays the the expectation expectation for for this this distribution for aa sample sample of of size size n = = 110, with the of the bars scaled, scaled, by by 0, with the heights heights of the bars distribution for dividing dividing E[Z;] E[Zi] by by E[S] E[S] so so that that they they sum sum to to one. one. The The various various test test statistics statistics detect detect deviations positive or deviations only only in in two two directions d i r e c t i o n s- positive or negative n e g a t i v e- and and if if the the standard standard model is is rejected rejected there there are are aa number number of of possible possible explanations, explanations, which which include include model selection 995). selection and and also also demographic demographic and/or and/or historical historical factors factors (Simonsen (Simonsen et et ai., al., 11995). Some Some statistics statistics assume assume that that information information about about the the ancestral ancestral state state at at each each site site is is known known and and are are thus thus functions functions of of the the mutant mutant base base counts, counts, Zi' Zi. Others Others do do not not make make this this assumption, assumption, instead instead assuming assuming that that the the patterns patterns Zi and and Zn-i Z,-i are are indistin indistinguishable 8.7). The 1989) D, guishable as as in in Eq. Eq. ((8.7). The latter, latter, which which include include Tajima's Tajima's ((1989) D, are are posi positive tive when when the the Zi around around ii = = nl2 n/2 are are inflated inflated relative relative to to Fig. Fig. 8.2 8.2 and and negative negative when when 1I are near either either ii = = 1 I or o r ii = = n n are inflated. inflated. The The statistics statistics that that assume assume the the ances ancesZi near tral tral states states are are known known have have the the potential potential to to detect detect differences differences between between inflated inflated Zi near near ii = = 11 and and inflated inflated Zi Zi near near ii = = n n - 11.. As As shown shown in in Section Section 8.3, 8.3, metapopula metapopulation tion dynamics dynamics can can produce produce aa wide wide variety variety of of site site frequency frequency distributions, distributions, putting putting the the status status of of these these "neutrality" "neutrality" tests tests in in further further jeopardy. jeopardy.
0.4 0.3 E[Z;J -E[SJ
0.2 0.1
2
Fig. 8.2 8.2 The expected expected site frequency distribution under the standard standard coalescent. coalescent.
AND COALESCENT 8. METAPOPULATIONS METAPOPULATIONS AND COALESCENT THEORY THEORY 8.
8.3 8.3
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COALESCENCE IN METAPOPULATIONS COALESCENCE IN METAPOPULATIONS The can be The word word metapopulation, metapopulation, implying implying aa "population "population of of populations," populations," can be applied applied very very broadly broadly to to any any geographically geographically structured structured species, species, particularly particularly ones 968a,b) . ones that that exhibit exhibit local local extinction extinction and and recolonization recolonization (Levins, (Levins, 11968a,b). Hanski Chapter 11)) discussed Hanski and and Gaggiotti Gaggiotti ((Chapter discussed the the current current metapopulation metapopulation con concept, cept, and and other other chapters chapters in in this this book book attest attest to to the the variety variety of of situations situations to to which which the concept has Because population been study the concept has been been applied. applied. Because population geneticists geneticists had had been studying metapopulations for decades before ing metapopulations for decades before the the word word metapopulation metapopulation was was intro introduced, terminology can confusing. In duced, the the terminology can be be confusing. In particular, particular, "population," "population," "subdivided "subdivided population," population, ...."structured structured population," population," and and "total "total population" population" are are often often used used interchangeably interchangeably to to refer refer to to aa metapopulation, metapopulation, and and any any of of "deme" ((Gilmour Gilmour and 939), "subpopulation," "deme" and Gregor, Gregor, 11939), "subpopulation," and and even even "popula "population" tion" are are used used to to refer refer to to geographically geographically local local populations populations within within aa metapopu metapopulation. population and lation. This This section section uses uses meta metapopulation and deme. deme. In In addition, addition, except except for for some work on some brief brief review review of of work on general general models models of of population population structure, structure, this this chapter common meta population notion notion that number of demes chapter adopts adopts the the common metapopulation that the the number of demes is making this assumption implicitly, is not not small. small. Instead Instead of of making this assumption implicitly, the the ancestral ancestral metapopulation number of metapopulation process process described described here here exists exists in in the the limit limit as as the the number of demes demes grows, grows, in in much much the the same same way way that that the the standard standard coalescent coalescent holds holds in in the the limit population size population approaches approaches infinity. limit as as the the population size of of an an unstructured unstructured population infinity. There genetics of There is is aa long long history history of of work work on on the the genetics of structured structured populations, populations, dat dating 1931 ). Much relevant to ing back back at at least least to to Wright Wright ((1931). Much of of this this work work is is relevant to the the discus discussion metapopulation structure, 1 940), who sion of of metapopulation structure, especially especially Wright Wright ((1940), who early early on on saw saw aa major potential recolonization in major potential role role for for extinction extinction and and recolonization in his his shifting shifting balance balance theory 1977) formulated theory of of evolution. evolution. Slatkin Slatkin ((1977) formulated the the basic basic population population genetic genetic model population dynamics commonly used model of of meta metapopulation dynamics that that is is still still commonly used and and identified identified the major possible the major possible effects effects of of extinction extinction and and recolonization recolonization on on genetic genetic variation: variation: ((1) 1 ) that that the the turnover turnover of of demes demes and and recolonization recolonization by by small small numbers numbers of of individ individuals uals can can decrease decrease overall overall levels levels of of variation variation and and (2) (2) that that the the movement movement of of founders population can among founders across across the the meta metapopulation can decrease decrease levels levels of of differentiation differentiation among demes demes in in aa manner manner similar similar to to migration. migration. Pannell Pannell and and Charlesworth Charlesworth (2000) (2000) pro provide (see also 7) which vide an an excellent excellent review review of of these these and and later later works works (see also Chapter Chapter 7) which have have metapopulation structure on well-known summaries of focused on the effects of meta population structure polymorphism 195 1 ) FST' polymorphism such such as as Wright's Wright's ((1951) FsT. approach to the case of population population The formal extension of the coalescent approach structure called the structure occurred occurred only only recently, recently, with with aa general general model model called the structured structured coalescent Nordborg, 11997; 997; Wilkinson-Herbots, 99 8 ) . The coalescent (Notohara, (Notohara, 1990; 1990; Nordborg, Wilkinson-Herbots, 11998). The structured include extinction structured coalescent coalescent does does not not include extinction and and recolonization recolonization of of demes, demes, only only migration migration between between them, them, but but it it could could be be reformulated reformulated to to do do so. so. The The back backme ii that ward ward migration migration rate, rate, mij, mij, is is defined defined to to be be the the fraction fraction of of de deme that is is replaced replaced by by migrants migrants from from deme deme jj each each generation. generation. The The structured structured coalescent coalescent exists exists in in the the limit limit as as the the sizes sizes of of demes demes go go to to infinity infinity but but the the scaled scaled backward backward migration migration rates, weak rates, Mjj Mij = 4Njmj 4Nimij,j, remain remain finite. finite. Thus, Thus, it it assumes assumes that that migration migration is is aa weak force, force, with with aa rate rate roughly roughly comparable comparable to to that that of of genetic genetic driftlcoalescence. drift/coalescence. This This is is not not aa weakness weakness of of the the model. model. If If Mij Mij = = 4Njmij 4Nimij does does not not remain remain finite finite as as the the Ni Ni goes goes to to infinity, infinity, then then migration migration is is aa much much faster faster process process than than drift/coales drift/coalescence, cence, and and the the dynamics dynamics of of the the metapopulation metapopulation converge converge on on those those of of an an unstructured 9 8 0 ) and unstructured population, population, both both forward forward (Nagylaki, (Nagylaki, 11980) and backward backward
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(Notohara, 993) in strong migration limit. In (Notohara, 11993) in time. time. This This is is known known as as the the strong migration limit. In prac practice, tice, the the effects effects of of structure structure are are very very difficult difficult to to detect detect once once the the rates rates Mij Mij are are greater Nordborg and greater than than about about 10. 10. Nordborg and Krone Krone (2002) (2002) studied studied aa structured structured coales coalescent cent in in which which some some of of the the Mij Mq remain remain finite finite whereas whereas others others increase increase without without bound process converges bound and and showed showed that that the the ancestral ancestral process converges to to aa structured structured coa coalescent population that lescent among among the the subunits subunits of of the the meta metapopulation that have have finite finite Mij• Mq. One the major of Wright 19 5 1 ) was was to focus attention One of of the major influences influences of Wright ((1951) to focus attention on on PST FsT as The connection as aa summary summary measure measure of of metapopulation metapopulation structure. structure. The connection to to coales coalescent 1 991 ), who cent theory theory was was made made by by Slatkin Slatkin ((1991), who showed showed that, that, in in the the limit limit of of small small mutation mutation rate, rate, inbreeding inbreeding coefficients coefficients such such as as PST FsTcan can be be expressed expressed in in terms terms of of expected expected pairwise pairwise coalescence coalescence times. times. Under Under the the infinite infinite sites sites mutation mutation model, model, the the expected expected values values of of pairwise pairwise differences differences are are linear linear functions functions of of e0 so so that that tak taking ing ratios ratios of of observed observed pairwise pairwise differences differences within within and and between between demes demes provides provides way of of estimating estimating migration migration parameters. For example, example, under under the the island island aa way parameters. For model metapopulation structure 93 1 ), in model of of metapopulation structure (Wright, (Wright, 11931), in which which every every deme deme exchanges me at exchanges migrants migrants with with every every other other de deme at the the same same rate rate and and all all demes demes are are of of the the same same size, size, expectations expectations of of the the average average number number of of pairwise pairwise differences differences within within and and between between demes, demes, 1Tw ~rw and and 1Tb, %, respectively, respectively, are are given given by by E[rrw]=0
and
E['n'b] = 0(1 + 1 ) ,
(8.8)
population. The where where e0 is is the the scaled scaled mutation mutation parameter parameter for for the the entire entire meta metapopulation. The sole sole migration migration parameter parameter of of the the island island model model can can be be estimated estimated as as = M=
--'"w=---_ __1T'rrw ,IT b
--
(8.9) (8.9)
,rr w
although although this this moment-based moment-based estimator estimator is is certainly certainly not not unbiased. unbiased. However, However, the the island model, many organisms, organisms, is island model, which which is is aa particularly particularly unrealistic unrealistic model model for for many is the the only only model model for for which which there there is is aa simple simple connection connection between between PST, Fsr, or or average average pairwise population (Whitlock pairwise differences, differences, and and the the parameters parameters of of the the meta metapopulation (Whitlock and and McCauley, 999). McCauley, 11999). Theoretical populations has Theoretical work work on on meta metapopulations has focused focused on on pairwise pairwise coalescence coalescence times times or or pairwise pairwise differences differences due due to to their their connection connection with with PST Fsr and and their their utility utility in estimating migration migration rates, rates, but small part in estimating but also also in in no no small part due due to to the the fact fact that that ana analytical lytical results results for for larger larger samples samples under under the the structured structured coalescent coalescent are are difficult difficult to to obtain. obtain. This This is is unfortunate unfortunate because, because, as as noted noted earlier, earlier, estimates estimates made made from from pair pairwise 983; wise differences differences have have relatively relatively poor poor statistical statistical properties properties (Tajima, (Tajima, 11983; Donnelly 995). In Donnelly and and Tavare, Tavar(}, 11995). In hindsight, hindsight, the the historical historical focus focus on on PST FST and, and, relatedly, relatedly, on on pairwise pairwise sequence sequence comparisons comparisons within within and and between between demes demes may may have have drawn drawn attention attention away away from from the the true true goal goal of of such such work, work, which which is is to to understand understand the the dynamics dynamics of of metapopulations metapopulations and and how how these these shape shape the the pat patterns terns of of genetic genetic polymorphism. polymorphism. To To some some degree, degree, this this is is an an unfair unfair statement. statement. The Eqs. ((8.8) 8 . 8 ) and example, can The profound profound importance importance and and utility utility of of Eqs. and (8.9), (8.9), for for example, cannot not be be questioned. questioned. At At the the same same time, time, it it is is clear clear that that these these simple simple measures measures of of polymorphism polymorphism are are not not sufficient sufficient to to untangle untangle the the complicated complicated demography demography of of metapopulations (Pannell (Pannell and 999; Pannell, Pannell, 2003 metapopulations and Charlesworth, Charlesworth, 11999; 2003).).
8. 8. METAPOPULATIONS METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY
89 11 89
The study study of of genealogies genealogies of of samples from aa metapopulation metapopulation will will help The samples from help ideniden tify likely to con tify the the patterns patterns in in DNA DNA sequences sequences or or other other genetic genetic data data that that are are likely to contain substantial about the tain substantial information information about the dynamics dynamics of of the the metapopulation. metapopulation. Again, Again, the the focus focus here here is is on on the the case case of of aa large large number number of of demes. demes. With With the the additional additional assumption assumption that that the the migrants migrants and/or and/or founders founders that that arrive arrive at at aa deme deme could could have have originated number of originated in in any any one one of of aa large large number of source source demes, demes, it it is is possible possible to to describe these histories, hold for describe the the general general features features of of these histories, which which are are expected expected to to hold for many metapopulation necesmany different different types of meta population structures. Beyond this it will will be neces sary sary to to make make more more detailed detailed assumptions assumptions about about the the structure structure of of the the metapopu metapopulation lation in in order order then then to to explore explore specific specific effects effects on on patterns patterns of of polymorphism. polymorphism. Sampling for for population genetic studies studies is is typically typically not not done done at at random random Sampling population genetic across metapopulation. Instead, number of across the the geographic geographic range range of of aa metapopulation. Instead, aa number of sam samples ples are are taken taken from from aa number number of of different different locations, locations, with with the the geographic geographic dis distances tances between between samples samples from from the the same same location location being being smaller smaller than than those those between locations. This between samples samples from from different different locations. This is is true true even even for for species species with with apparently and it apparently continuous continuous ranges ranges and it is is forced forced by by local local abundance abundance in in species species composed composed of of more more discrete discrete demes. demes. This This is is aa logical logical approach approach to to the the study study of of geographic geographic structure, structure, but but one one that that is is also also conditioned conditioned by by long-standing long-standing notions notions about 8 .9), which, about the the importance importance of of FST FsT and and results results such such as as Eq. Eq. ((8.9), which, again, again, come come directly directly from from the the island island model model of of aa metapopulation. metapopulation. A A consideration consideration of of sam sample genealogies populations may ple genealogies in in meta metapopulations may also also aid aid in in the the design design of of better better sam sampling piing strategies strategies for for studying studying geographic geographic structure. structure. Another Another noteworthy noteworthy aspect aspect of population composed of samples samples from from aa meta metapopulation composed of of aa large large number number of of demes demes or or distributed locations will distributed over over aa very very broad broad range range is is that that many many demes demes or or locations will not not be be sampled sampled at at all. all. Now Now consider consider the the locations locations of of the the lineages lineages ancestral ancestral to to the the sample sample at at some some time time in in the the past. past. Unlike Unlike the the locations locations of of the the present-day present-day samples, samples, which which are are under under experimental experimental control, control, these these will will be be determined determined by by the the history history and and dynamics population and, dynamics of of the the meta metapopulation and, of of course, course, the the depth depth of of time time considered. considered. Recently Recently in in the the past, past, samples samples from from the the same same place place will will tend tend still still to to be be close close together together and and will will be be relatively relatively likely likely to to coalesce. coalesce. Lineages Lineages from from different different loca locations tions are are less less likely likely to to share share aa common common ancestor ancestor recently. recently. In In the the more more distant distant past, lineages past, lineages originally originally sampled sampled from from the the same same location, location, if if they they have have not not coalesced, will coalesced, will have have instead instead moved, moved, by by migration migration and/or and/or extinction/recolo extinction/recolonization, to locations. In nization, to other other locations. In aa metapopulation metapopulation with with aa large large number number of of demes and and in in which which the the number number of of source source demes demes of of migrants migrants and and founders founders is is demes large, lineages are likely to me nor large, these these ancient ancient lineages are not not likely to be be in in the the same same de deme nor are are they they likely likely to to be be in in any any of of the the originally originally sampled sampled demes. demes. Their Their locations locations will will be be the the result of population according result of their their random random movement movement across across the the meta metapopulation according to to the the rates rates of of migration migration and and extinction/recolonization extinction/recolonization between between demes. demes. They They will will tend population that tend to to accumulate accumulate in in the the parts parts of of the the meta metapopulation that contribute contribute greatly greatly to to the the migrant migrant pool pool or or that that send send out out an an abundance abundance of of founders, founders, and and they they will will spend little little time time in in regions regions that that act act as as "sinks" "sinks" instead instead of of "sources" "sources" (Pulliam, (Pulliam, spend 11988). 9 8 8 ) . There There will will be be chances chances for for such such ancient ancient ancestral ancestral lineages lineages to to coalesce, coalesce, mediated mediated by by migration migration and and extinction/recolonization, extinction/recolonization, and and it it may may require require aa lot lot of of wandering wandering of of the the lineages lineages across across the the population population before before the the most most recent recent com common ancestor ancestor of of the the entire entire sample sample is is reached. reached. mon Thus, Thus, for for aa broad broad range range of of specific specific metapopulation metapopulation structures structures that that have have aa large number number of large of demes demes in in common, common, sample sample genealogies genealogies should should exhibit exhibit aa recent recent
JOHN WAKELEY WAKELEY JOHN
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burst of of coalescent coalescent events events among among samples samples taken taken from from the the same same locality locality folfol burst lowed by by aa more more ancient ancient historical historical process process for for the the remaining remaining ancestral ancestral linlin lowed eages. These These have have been been called called the the scattering scattering phase phase and and the the collecting collecting phase phase eages. (Wakeley, 1 999), and and details details of of them them depend depend on on the the details details of of the the dynamics dynamics of of (Wakeley, 1999), the metapopulation. metapopulation. The The next next section section describes an idealized idealized model model of of aa the describes an metapopulation in in which which this this behavior behavior emerges emerges in in the the limit limit as as the the number number of of metapopulation demes goes goes to to infinity infinity and and for which simple simple precise precise descriptions descriptions of of the the scat demes for which scattering and collecting phases are are possible. possible. A A simulation of Pannell Pannell (2003) (2003 ) tering and collecting phases simulation study study of showed, among among other other things, things, that that this this twofold twofold structure structure of of genealogies genealogies is is realreal showed, ized in in aa special special case case of of this model even even if if the the number number of of demes demes is is not not terribly terribly ized this model large. In In addition, addition, Ray Ray et et al. al. (2003) (2003 ) found found this this behavior behavior in in aa model model very very difdif large. ferent from from the the one one described described later. later. Ray Ray et et al. al. (2003) (2003) simulated simulated aa metapopulametapopula ferent tion that that expands expands from from aa single single deme deme over over aa two-dimensional two-dimensional grid and showed tion grid and showed that genetic genetic signatures signatures of of this expansion in in aa sample sample from single deme deme are are that this expansion from aa single strong only if if the the backward backward migration/colonization migration/colonization rate rate of of the the deme deme is is high. high. strong only If the escape the the recent If the migration migration rate rate is is low, low, then then few few lineages lineages will will escape recent burst burst of of scattering-phase be scattering-phase coalescent coalescent events, events, the the effective effective number number of of lineages lineages will will be small, and power to the expansion small, and the the power to detect detect the expansion will will be be low. low.
The Effect on Genealogies The Effect of of Metapopulation M e t a p o p u l a t i o n Structure Structure on Genealogies The same as as the model in The model model considered considered here here is is essentially essentially the the same the model in Wakeley Wakeley and although it it is is unnecessary unnecessary to to assume individual and Aliacar Aliacar (2001 (2001),), although assume that that the the individual demes large (Lessard (Lessard and Wakeley, 2003). 2003). The model assumes assumes that that there there demes are are large and Wakeley, The model are which resembles resembles the the metapopulation metapopulation described described by by are K K "regions," "regions," each each of of which Slatkin 1 977). Each Slatkin ((1977). Each region region may may have have different different values values of of all all parameters, parameters, and and among among regions regions there there is is some some explicit explicit geographic geographic structure. structure. For For ease ease of of discus discussion, case where sion, consider consider the the case where the the demes demes are are large large so so that that the the relevant relevant scaled scaled parameters parameters for for each each such such region, region, or or class class of of demes, demes, are are the the migration migration rate rate Mi Mi = = 4Nimi, 4 N i m i , the the extinction/recolonization extinction/recolonization rate rate Ei Ei = = 4Niei, 4miei, and and the the propagule propagule size size ki' ki, which which is is the the number number of of founders founders of of the the deme deme each each time time it it is is recolonized recolonized after after going going extinct. extinct. Note Note that that ki ki is is the the number number of of founding founding gametes gametes rather rather than than diploid 1 977), and diploid individuals, individuals, as as in in Slatkin Slatkin ((1977), and that that the the parameters parameters Mi M i and and Ei E i dif differ those in 1 ), in fer by by aa factor factor of of two two from from those in Wakeley Wakeley and and Aliacar Aliacar (200 (2001), in keeping keeping with with the the scaled scaled migration migration rate rate M M used used earlier earlier in in this this section. section. The The index index ii of of these 8.3 depicts these parameters parameters ranges ranges from from one one to to K. K. Figure Figure 8.3 depicts one one example example of of such such aa metapopulation. metapopulation. are D D demes demes total total in in the the metapopulation, metapopulation, and and aa fraction fraction [3i ~i of o f these these There are are class ii are in in class class or or region region i,i, where where k ]~iK=l[3i Thus there there are are D D ~[3ii demes demes of of class �l[3i == 11.. Thus so so that that in in the the limit limit D D� ~ 00 oo considered considered later, later, the the number number of of demes demes in in each each class class also also approaches approaches infinity. infinity. Looking Looking back back in in time, time, when when aa lineage lineage experiences experiences aa migration event event or or an an extinction/recolonization extinction/recolonization event, event, it it has has some some probability probability migration of come from me in of having having come from aa de deme in each each of of the the other other regions regions and and some some probabil probability ity of of coming coming from from aa deme deme in in its its own own region. region. That That is, is, mi mi= = kf=l ~K___1 mij mij and and eiei == kf= eij' and E~(=1leij, and these these probabilities probabilities of of movement, movement, given given that that aa migration migration event event or or an an extinction/recolonization extinction/recolonization event event has has occurred, occurred, are are given given by by m;/mi mij/mi and and ei/ej, eij/ei, respectively. respectively. Every Every deme deme in in aa region region has has an an equal equal chance chance of of being being the the source source deme deme of of aa migrant/colonist migrant/colonist from from that that region. region. The The only only constraints constraints on on the the structure structure of of movement movement are are that that lineages lineages can can get get from from any any of of these these K K
8. 8. METAPOPULATIONS METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY
1191 91
G t~
Fig. 8.3 One conforms to in the the text, One possible possible metapopulation metapopulation that that conforms to the the model model described described in text, in in this this case case with with K K= = 5 5 and and with with an an arbitrarily arbitrarily chosen chosen structure. structure.
regions regions to to any any other, other, given given enough enough time, time, and and that that there there is is aa nonzero nonzero probabil probabilthat a lineage will remain in the same region. Even very strongly con conity that strained strained patterns patterns of of movement movement among among regions, regions, such such as as the the one-dimensional one-dimensional stepping-stone 964), conform stepping-stone model model (Kimura (Kimura and and Weiss, Weiss, 11964), conform to to this this assumption. assumption. A A surprising surprising result result of of this this model model is is that that in in the the limit limit as as D D goes goes to to infinity, infinity, the the details population are details of of this this aspect aspect of of the the geographic geographic structure structure of of the the meta metapopulation are obliterated, obliterated, similarly similarly to to the the way way in in which which all all structure structure disappears disappears in in the the strong strong migration 980; Notohara, migration limit limit (Nagylaki, (Nagylaki, 11980; Notohara, 1993). 1993). Branching Pattern Pattern of of Genealogies
Under 1 998), Under this this model, model, with with the the aid aid of of the the convergence convergence result result of of Mahle M6hle ((1998), it it is is possible possible to to show show that that the the more more ancient ancient part part of of the the history, history, the the collecting collecting phase, 1 982a,b,c) coalescent phase, converges converges to to Kingman's Kingman's ((1982a,b,c) coalescent as as the the number number of of demes demes goes goes to to infinity, infinity, but but with with aa rate, rate, or or an an effective effective size, size, that that depends depends on on all all the the parameters 1 ). This parameters of of the the model model (Wakeley (Wakeley and and Aliacar, Aliacar, 200 2001). This had had been been found found previously include migration extinction and previously in in models models that that include migration but but not not extinction and recoloni recolonization 998, 11999, 999, 2001 zation (Wakeley, (Wakeley, 11998, 2001).). Simulations Simulations imply imply that that the the predictions predictions of of the the model model are are accurate accurate as as long long as as the the number number of of demes demes is is at at least least three three to to four four times 998; Pannell, deal times the the sample sample size size (Wakeley, (Wakeley, 11998; Pannell, 2003). 2003). Surprisingly, Surprisingly, aa great great deal of dynamics of of the the geographic geographic structure structure and and dynamics of the the metapopulation metapopulation ~ the the details details of unsampled of movement movement among among regions regions and and the the values values of of M, M, E, E, and and kk for for unsampled demes demes ~ is is manifest manifest only only through through the the single single effective effective size size of of the the metapopula metapopulation phase. This tion during during the the collecting collecting phase. This results results from from the the fact fact that that when when the the num number ber of of demes demes is is very very large, large, the the lineages lineages will will migrate migrate so so many many times times as as to to reach reach aa stationary stationary distribution distribution over over deme deme types, types, determined determined by by the the movement movement matri matrices ces for for migration migration and and extinction/recolonization, extinction/recolonization, before before two two of of them them end end up up in in the me and the same same de deme and have have the the chance chance to to coalesce. coalesce. Overall, Overall, then, then, if if time time is is scaled scaled by by the the effective effective size size and and e0 is is defined defined accordingly, accordingly, all all the the detailed detailed results results of of the the standard .2, hold standard coalescent coalescent model, model, including including those those discussed discussed in in Section Section 88.2, hold for for the these the these collecting-phase collecting-phase lineages. lineages.
JOHN JOHN WAKELEY WAKELEY
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In this this model, model, for for any any sample, sample, the the transition transition from from the the scattering scattering phase to In phase to this occurs as each ancestral this coalescent coalescent collecting collecting phase phase occurs as soon soon as as each ancestral lineage lineage is is in in aa separate Thus, the taken singly separate deme. deme. Thus, the history history of of aa sample sample taken singly from from different different demes demes in population of in aa meta metapopulation of this this sort sort is is also also described described by by the the standard standard coalescent. coalescent. The The only only evidence evidence of of the the structure structure in in this this "scattered" "scattered" sample sample will will be be in in the the magnitude magnitude of of e, 0, if, if, for for instance, instance, it it could could be be compared compared to to aa scattered scattered sample sample from population of from another another meta metapopulation of the the same same total total size size but but with with different different details details of of structure structure and and thus thus aa different different effective effective size. size. How How many many lineages lineages then then will will enter enter the the collecting collecting phase phase for for other other kinds kinds of of samples? samples? This This is is determined determined by by the the random random outcome outcome of of coalescent, coalescent, migration, migration, and and extinction/recolonization extinction/recolonization within within demes. demes. Assuming Assuming that that demes demes are are large large in in size, size, in in aa deme deme of of type type ii that that contains contains jj lineages lineages these these will will occur occur with with relative relative rates rates j(j j(j - 11),) , jM;, j M i , and and E;. E i. In In the the limit limit as as D D goes goes to to infinity, infinity, the the number number of of demes demes will will be number of lineages ancestral be much much greater greater than than the the number of lineages ancestral to to the the sample, sample, and and migration migration events events will will send send lineages lineages off off to to demes demes that that do do not not contain contain other other ancestral lineages. demes are size, these rates apply apply only ancestral lineages. If If the the demes are small small in in size, these rates only roughly roughly and multiple migration and it it will will be be possible possible for for multiple migration and and coalescent coalescent events events to to occur occur in in aa single single generation. generation. In In either either case, case, both both single single coalescent coalescent events events and and migration migration events sample closer transition to phase by events both both bring bring the the sample closer to to the the transition to the the collecting collecting phase by decreasing deme by decreasing the the number number of of lineages lineages in in the the deme by one. one. If If an an extinction/recolon extinction/recolonization me will ization event event occurs, occurs, whatever whatever lineages lineages remain remain in in the the de deme will be be related related me size possible that through through the the k; ki founders. founders. Even Even if if the the de deme size is is large, large, it it is is possible that several several common common ancestor ancestor events events will will occur occur in in this this step step because because k; ks may may not not be be large. large. one possible sample of Figure Figure 8.4 8.4 shows shows one possible scattering scattering phase phase for for aa sample of size size n n = = 88 from single deme series of events and from aa single deme in in which which aa series of three three coalescent coalescent events and two two migra migration events are tion events are followed followed by by an an extinction/recolonization extinction/recolonization event event in in which which the the remaining all coalesce. result is n' == 33 lineages lineages that remaining three three lineages lineages all coalesce. The The result is n' that will will enter collecting phase. lineages have enter the the collecting phase. These These three three lineages have different different numbers numbers of of descendents descendents in in the the sample, sample, or or different different "sizes." "sizes." Because Because whatever whatever labels labels we we might might assign assign to to these these collecting-phase collecting-phase lineages lineages are are arbitrary arbitrary ~ they they are are ) for the probability that exchangeable exchangeable ~ we we can can write write P(n'; P ( n ' ; a}, a l , a2, a 2 , . ... , , a an) for the probability that n n' lineages there there are are n' lineages are are the the end end of of the the scattering scattering phase phase and and among among these, these, al al have have one one descendent descendent in in the the sample, sample, a2 a2 have have two two descendents descendents in in the the sample, sample, and and so possible size so on. on. The The possible size configurations configurations are are all all those those that that satisfy satisfy !7= En=l1 iaj iai = = n n and 1 a; = and of of course course !7= Ei=lai = n'. n . -
.
n
.
!
a5 = 1 n' = 3
n =8
Fig. 8.4 phase for Fig. 8.4 A A realization realization of of the the scattering scattering phase for aa sample sample from from aa single single deme. deme. The The gray gray cylin cylinder represents the deme back through lines represent lineages ancestral to the sam through time time and lines sample. The ple. The two two attached attached boxes boxes represent represent the the kk = - 2 2 founders founders of of the the deme. deme.
8. 8.
METAPOPULATIONS AND AND COALESCENT COALESCENT THEORY THEORY METAPOPULATIONS
1 93 193
Different details details of of the the dynamics dynamics within within demes demes will will give give different different distribudistribu Different an ) . For For example, example, when when the the deme deme size size is is large large and and there there tions, PP(n'; tions, ( n ' ; aab l , aa22, ,. . 9. .9 , an). is no no extinction/recolonization, extinction/recolonization, the the distribution distribution is is identical identical to to Ewen's Ewen's (1972) ( 1972) is distribution, but but with with infinite infinite alleles alleles mutation mutation replaced replaced by by infinite infinite demes demes distribution, migration; the the number number of of alleles alleles becomes becomes the the number number of of collecting-phase collecting-phase lineline migration; ages (n') ( n ' ) and and the the counts of the the allele allele become become the the number number descendents descendents of of these these ages counts of lineages (ai) ( ai) in in the the sample sample (Wakeley, (Wakeley, 1998, 1 998, 1999). 1 999). However, However, if if the the rate rate of of lineages extinction/recolonization is is high high and and migration migration is is absent, absent, then an ) extinction/recolonization then PP(n'; ( n ' ; aab l , aab 2 , 9. 9. .9 , an) will be be the the result result of of tossing tossing nn balls balls into into kk boxes, boxes, with with ai being being the the number number of of will boxes that that contain contain ii balls balls (Wakeley (Wakeley and and Aliacar, Aliacar, 2001). 200 1 ) . Comparable Comparable levels levels of of boxes migration and extinction/recolonization extinction/recolonization combine combine both both these these effects, effects, and and there there migration and are of of course course many many other other possibilities, possibilities, depending depending on on the the local local dynamics dynamics within within are demes and and the the sizes sizes of of demes. demes. demes D goes goes to to infinity, infinity, the the scattering scattering phase phase occurs inde Finally, in in the the limit limit as as D Finally, occurs independently within each each sampled deme so so that probabilities PP(n'; an ) pendently within sampled deme that the the probabilities ( n ' ; aab s , aa22,, 9. .9 . , an) are simply simply multiplied over demes demes to the overall overall chance chance that that n' n' lineages lineages are multiplied over to obtain obtain the enter the collecting phase, with sample enter the collecting phase, with some some distribution distribution of of sizes, sizes, given given that that aa sample of size been sampled number of In sum, the topotopo of size n n has has been sampled among among some some number of demes. demes. In sum, the logical structure logical structure of of sample sample genealogies genealogies in in aa metapopulation metapopulation will will be be identical identical to to that the standard coalescent, except that now the number number of of (collecting (collecting that in in the standard coalescent, except that now the phase) is aa random random variable variable and and each each tip tip will will have have an an associassoci phase) tips tips of of the the tree tree is ated be greater than one one and and is equal to to the the number number of of ated stochastic stochastic size size that that can can be greater than is equal descendents branch in in the the sample. sample. descendents of of that that branch Times Times to to Common Common Ancestry Ancestry
The process. Its The collecting collecting phase phase is is aa metapopulation-wide metapopulation-wide process. Its effective effective size size is is roughly population, although roughly on on the the order order of of the the total total size size of of the the meta metapopulation, although low low rates can make than this this and some types types of of rates of of migration migration can make it it larger larger than and some extinction/recolonization make it situations it extinction/recolonization can can make it smaller. smaller. In In some some situations it is is import important consider these timescale of instance, in ant to to consider these effects effects on on the the timescale of the the coalescent, coalescent, for for instance, in the populations or species, where the context context of of divergence divergence between between two two meta metapopulations or species, where this this timescale reciprocal monophyly timescale determines determines the the probability probability of of reciprocal monophyly of of samples, samples, among among other other things things (Wakeley, (Wakeley, 2000). 2000). Here, Here, we we have have simply simply defined defined the the param parameter eter 60 for for the the metapopulation-collecting metapopulation-collecting phase, phase, and and the the importance importance of of its its effective effective size size is is mostly mostly in in comparison comparison to to that that of of the the scattering scattering phase. phase. This This more more recent phase, which which occurs within demes, depends on the effective sizes of demes. demes. Thus, Thus, the the effective effective size size of of the the collecting collecting phase phase is is about about D D times times larger larger than than that that of of the the scattering scattering phase. phase. In In the the limit limit as as D D goes goes to to infinity, infinity, the the duration duration of of the the scattering scattering phase phase becomes becomes negligible negligible in in comparison comparison to to that that of of the the col collecting phase. lecting phase. Clearly, Clearly, as as in in the the standard standard coalescent, coalescent, the the genealogy genealogy of of aa sample sample from from aa meta population contains exactly n 1 coalescent events. Under the limiting metapopulation contains exactly n - 1 coalescent events. Under the limiting process populations with process described described earlier, earlier, which which holds holds for for meta metapopulations with aa large large number number of of demes, demes, the the first first nn -- n' n' (scattering-phase) (scattering-phase) coalescent coalescent events events have have negligible negligible branch branch lengths, lengths, whereas whereas the the remaining remaining n' n' - 11 have have branch branch lengths lengths determined determined by the Kingman's coalescent process. The scattering phase becomes by the Kingman's coalescent process. The scattering phase becomes an an instant instantaneous aneous adjustment adjustment of of the the sample sample size size and and structure, structure, which which can can be be used used to to obtain obtain results results for for times times to to common common ancestry ancestry as as well well as as predictions predictions about about the the level level and and pattern pattern of of polymorphism polymorphism in in the the sample. sample. It It is is no no longer longer possible possible to to
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JOHN JOHN WAKELEY WAKELEY
write . 1 because write down down aa formula formula like like 88.1 because these these will will depend depend on on the the realization realization of of n' for n' for the the sample. sample. However, However, by by conditioning conditioning on on the the scattering scattering phase, phase, it it is is pos possible 8.2 through 8.4, for sible to to derive derive equations equations like like 8.2 through 8.4, for the the expected expected values values of of T TMRCA, MRCA , [n' ln] to n' lineages Ttotal, can write Ttota h and and 'Tj. a"i. We We can write P P[n'ln] to denote denote the the probability probability that that n' lineages enter enter the the collecting collecting phase, phase, i.e., i.e., without without regard regard to to their their sizes. sizes. Properties Properties of of T TMRCA and Ttotal Ttot~ldepend depend only only on on this this overall overall number number n', n', and and relatively relatively simple simple MRCA and analytic 998, 200 1). analytic expressions expressions can can be be obtained obtained in in some some cases cases (Wakeley, (Wakeley, 11998, 2001). Quantities Quantities such such aass 'Tj T i depend depend oonn the the sizes sizes ooff the the collecting-phase collecting-phase lineages. lineages. IInn the the context context of of aa metapopulation, metapopulation, these these frequency frequency measures measures should should be be redefined redefined to to represent joint frequencies 999). represent the the joint frequencies among among sampled sampled demes; demes; (e.g., (e.g., see see Wakeley, Wakeley, 11999).
The The Effect Effect of of Metapopulation M e t a p o p u l a t i o n Structure Structure on on Genetic Genetic Variation Variation in a a Sample Sample in The The level level and and pattern pattern of of genetic genetic variation variation in in aa sample sample from from aa metapopula metapopulation tion are are determined determined by by the the recent recent history history of of coalescent, coalescent, migration, migration, and and extinc extinction/recolonization tion/recolonization events events in in the the sampled sampled demes demes and and by by the the more more ancient ancient coalescent coalescent process process that that occurs occurs among among the the remaining remaining lineages. lineages. To To obtain obtain predictions predictions for for polymorphism polymorphism in in the the sample, sample, it it is is only only necessary necessary to to condition condition on phase. In on the the outcome outcome of of the the scattering scattering phase. In some some cases cases this this can can be be done done ana analytically, lytically, whereas whereas in in others others it it will will be be necessary necessary to to use use simulations. simulations. These These simulations are are nearly as straightforward straightforward as as simulations simulations of of the the standard standard coa coasimulations nearly as lescent, phase. First lescent, the the difference difference being being the the addition addition of of the the scattering scattering phase. First the the scattering phase is simulated, and his scattering phase is simulated, and all all branches branches during during this this period period of of the the history remaining lineages lineages are tory have have lengths lengths set set to to zero. zero. The The remaining are then then fed fed into into the the usual instance, as described in 1 990). The usual coalescent coalescent simulation, simulation, for for instance, as described in Hudson Hudson ((1990). The advantage advantage of of this this in in terms terms of of the the efficiency efficiency of of the the simulations simulations is is in in not not having having to to represent represent all all of of aa large large number number of of demes, demes, only only those those from from which which samples samples have taken. In have been been taken. In addition, addition, convergence convergence of of the the collecting collecting phase phase to to Kingman's rep Kingman's coalescent coalescent shows shows that that aa lot lot of of time time could could be be wasted wasted trying trying to to represent the myriad myriad details details of of movement movement of of lineage across the the metapopulation metapopulation resent the lineage across during during the the collecting collecting phase. phase. There There are are two two main main effects effects of of metapopulation metapopulation structure structure on on patterns patterns of of genetic genetic variation, variation, which which are are represented represented conveniently conveniently and and separately separately in in the the scattering phase. First, scattering phase phase and and the the collecting collecting phase. First, of of course, course, overall overall levels levels of of vari variation ation are are determined determined by by the the collecting-phase collecting-phase coalescent coalescent process, process, but but the the connection sample to connection of of the the sample to this this more more ancient ancient history history is is mediated mediated by by the the scat scattering overall levels tering phase. phase. In In particular, particular, the the overall levels of of polymorphism polymorphism in in aa sample sample are are greater n' is n' is n' is greater when when n' is larger larger and and smaller smaller when when n' is smaller. smaller. In In fact, fact, if if n' is equal equal to to one, one, which which is is possible possible only only if if all all samples samples come come from from the the same same deme, deme, there there will variation in surprising, but will be be no no variation in the the sample. sample. This This seems seems surprising, but it it is is an an under understandable consequence number of being very standable consequence of of the the number of demes demes being very large large and and e0 being being finite: finite: the the values values of of e0 for for individual individual demes demes must must be be infinitesimal. infinitesimal. This This is is prob probably ably appropriate appropriate for for low low mutation mutation rate rate data data such such as as DNA DNA sequence sequence data. data. The The alternative, alternative, that that demic demic e0 values values are are not not small, small, dictates dictates that that the the metapopulation metapopulationwide wide e0 approaches approaches infinity infinity as as the the number number of of demes demes grows, grows, predicting predicting an an infi infinite nite number number of of polymorphisms polymorphisms in in the the sample. sample. The The assumption assumption that that demic demic e0 values are small may values are not not small may be be appropriate appropriate for for loci loci with with higher higher mutation mutation rates, rates, such such as as microsatellites. microsatellites. In In this this case, case, the the model model would would predict predict an an infinite infinite number number
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of microsatellite microsatellite mutations mutations during during the the collecting collecting phase, phase, and and itit would would be be approappro of priate to to study sample probabilities probabilities of of identity identity as as in in Vitalis Vitalis and and Couvet Couvet priate study sample (2001 a,b). Under Under the the assumption assumption of of finite finite total-population total-population 00 made made here, here, all all (2001a,b). genetic variation variation is is ancestral ancestral in in the the sense sense that that itit is is not not due due to to mutations mutations that that genetic happened during during the the recent recent (scattering-phase) (scattering-phase) history history of of the the sampled sampled demes. demes. happened Scattered samples, taken taken singly some number number of of demes, demes, will will show show Scattered samples, singly from from some patterns of of polymorphism polymorphism identical identical to to those those in in aa completely completely unstructured unstructured patterns population, the the characteristics characteristics of of which which are are reviewed reviewed in in Section Section 8.2. 8.2. To the population, To the extent that that multiple multiple samples samples are are taken taken from from single single demes, demes, the the ancestral ancestral extent collecting-phase variation variation will will be be partitioned partitioned within within and and among among demes' demes' samsam collecting-phase ples. The The following following discussion discussion essentially essentially assumes assumes aa constant constant value value of of 00 for for the the pies. meta population; changes changes in in parameters parameters are are interpreted interpreted as as different different potential potential metapopulation; properties of of the the sampled sampled demes. demes. properties Full Data Data Patterns Patterns Full
the ancestral for samples samples from metapopula The connection connection of of the ancestral process process for from aa metapopulaThe tion samples from an unstructured unstructured population population means means that that the the tools tools tion to to that that of of samples from an of the standard be adapted use here. here. For For example, example, in of the standard coalescent coalescent can can be adapted for for use in prin principle, it should should be be straightforward straightforward to to use use the the recursive recursive approach approach of ciple, it of Griffiths Griffiths and 1 995), with with the the recognition recognition that that the the number number of migration and and and Tavare Tavar~ ((1995), of migration extinction/recolonization events in in the the history history is is not not fixed. fixed. In In other other words, words, it it extinction/recolonization events will be necessary necessary to to account account for for the the stochastic-scattering stochastic-scattering phase, phase, although although will be in case of scattered sample sample the standard coalescent coalescent methods methods can can be be in the the case of aa scattered the standard used directly. At present, present, this this remains remains one one of of several possible areas areas of of future used directly. At several possible future research. research. Summary Summary Measures Measures
Predictions summary measures, segregating sites Predictions for for summary measures, such such as as the the number number of of segregating sites S, average number pairwise differences site-frequency distri S, the the average number of of pairwise differences 1T, ~r, and and the the site-frequency distribution, can make bution, can be be made made by by modeling modeling the the scattering scattering phase. phase. It It is is possible possible to to make analytical predictions about them conditioning on analytical predictions about them by by conditioning on the the number number of of lineages lineages n phase. In n'' that that remain remain at at the the end end of of the the scattering scattering phase. In general, general, any any process process that that tends tends to to decrease decrease n n',' , such such as as restricted restricted migration migration and and extinction/recolonization extinction/recolonization with with aa small small number number of of founders, founders, will will tend tend to to decrease decrease the the number number of of (ances (ancestral) tral) polymorphisms polymorphisms found found to to be be segregating segregating in in the the sample. sample. Wakeley Wakeley and and Aliacar 1 ) showed Aliacar (200 (2001) showed that that larger larger values values of of M M produce produce larger larger average average values values of of S, S, as as do do larger larger values values of of E E if if the the number number of of founders, founders, k, k, is is large, large, whereas whereas increasing increasing E E when when kk is is equal equal to to one one decreases decreases the the average average value value of of S. S. In In add addition ition to to effects effects on on the the average average value value of of S, S, effects effects on on the the shape shape of of its its distribu distribution tion can can be be investigated investigated analytically analytically or or using using simulations. simulations. There There are are many many possible possible summary summary measures measures of of sequence sequence polymorphism polymorphism in in addition addition to to SS and and 1T, ~r, including including the the site site frequencies, frequencies, Z; Zi or or 11;, ~qi,and and it it is is hoped hoped that that the the study study of of gene gene genealogies genealogies in in aa metapopulation metapopulation will will aid aid in in the the development development of of new new statistics statistics that that capture capture the the essential essential features features of of the the dynamics dynamics of of the population. Figure the meta metapopulation. Figure 8.5 8.5 shows shows computer computer simulation simulation results results for for site site frequencies, frequencies, Z;, Zi, in in aa sample sample from from aa single single deme. deme. In In aa single-deme single-deme sample, sample, these these are are adequate adequate to to describe describe the the frequency frequency spectrum spectrum when when patterns patterns of of link linkage age among among sites sites are are not not aa concern. concern. Of Of course, course, there there is is likely likely to to be be some some extra extra
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M = 0. 1 2
0.2 E[Zi)
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8.5 Expected "unfolded" "unfolded" site frequencies, mutant base Fig. 8.5 frequencies, i.e., expected relative counts of the mutant sites, in a sample of size n = 115 from a single de deme with propagule propagule size size kk = = 2. at polymorphic sites, 5 from me with 2. information about about the the meta metapopulation in the the joint joint frequencies at information population contained contained in frequencies at two or or more more linked linked polymorphic sites. In In samples two polymorphic sites. samples from from multiple multiple demes, demes, there there may be contained in of alleles alleles among among demes. may be information information contained in the the joint joint frequencies frequencies of demes. However, the extent of of this this information among However, the extent information in in both both cases cases (among (among loci loci and and among demes) and the the potential potential to to construct construct sufficient sufficient statistics statistics demes) and sufficient or or nearly nearly sufficient for metapopulation not yet yet been explored. for metapopulation parameters parameters have have not been fully fully explored. demonstrates that that the the counts counts of of mutant mutant alleles in a sample sample from Figure 8.5 demonstrates from single deme contain substantial substantial information aa single deme contain information about about the the three three relevant relevant paramters of this of M, and kk for paramters of this model: model: the the values values of M, E, E, and for the the sampled sampled deme. deme. In In addition, the addition, the overall overall number number of of polymorphic polymorphic sites is is likely likely to to constitute constitute the the bulk of the O. In In Fig. Fig. 8.5, 8.5, site site frequencies frequencies are are bulk of the information information available available about about 0. shown shown as as fractions fractions for for the the total total number number of of polymorphic polymorphic sites. sites. In In the the case case of of low 0 . 1 ), when when extinction extinction and and recolonization recolonization are are weak weak forces, forces, low migration migration (M (M -= 0.1), i.e., i.e., when when E E is small, small, the the site-frequency site-frequency distribution distribution is U U shaped. shaped. This This is simisimi lar lar to to the the case case of of alleles alleles under under positive positive Darwinian Darwinian selection, selection, considered considered by by Fay Fay and Wu (2000) and Wu (2000) and and Kim Kim and and Stephan Stephan (2000), (2000), so so we we can can expect expect our our ability ability to to distinguish distinguish between between positive positive selection selection and and migration migration using using single-deme single-deme samsam ples ples to to be be low. low. At At the the other other extreme extreme for for small small M, M, when when extinction/recolonizaextinction/recoloniza tion stronger force force than migration, the tion is is aa much much stronger than migration, the site-frequency site-frequency distribution distribution has might be has aa mode mode at at the the middle middle frequencies. frequencies. This This is is similar similar to to what what might be expected in from two two demes expected in aa combined combined sample sample from demes in in aa metapopulation metapopulation or or if if balbal ancing operating between between two ancing selection selection were were operating two alleles alleles at at aa locus, locus, so so itit is is surprissurpris ing ing to to find find itit here here for for aa single-deme single-deme sample. sample. The The explanation explanation is is that that large large EE means means that that the the deme deme from from which which the the sample sample was was drawn drawn is is very very likely likely to to have have experienced lineages were were experienced aa recent recent extinction/recolonization extinction/recolonization event. event. The The nn lineages immediately immediately related related through through kk ancestors, ancestors, given given aa mode mode in in the the site site frequency frequency distribution distribution around around the the expected expected number number of of descendents descendents per per ancestor, ancestor, n/k, n/k, which which in in this this case case is is equal equal to to 7.5. 7.5. In 1 0 (left ( left side side of of Fig. Fig. 8.5), 8.5), if if EE is is small, small, the the site site frequenfrequen In the the case case of of M M == 10 cies cies are are close close to to those those predicted predicted for for aa sample sample from from aa panmictic panmictic population population
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shown in in Fig. Fig. 8.2. 8.2. As As the the extinction/recolonization extinction/recolonization rate rate increases, increases, aa mode mode shown again develops develops around around n/k. n/k. However, However, because because migration migration is is strong, strong, the the again panmictic pattern pattern continues continues to to hold hold and and the the interaction interaction of of these these two two patterns patterns panmictic produces an an average average site-frequency site-frequency distribution distribution that that has has three three modes. modes. On On the the produces one hand, hand, this this means means that that there there is is potentially potentially aa great great deal deal of of information information one about the the parameters parameters of of the the model model even even in in samples samples from from aa single single deme. deme. On On about the other other hand, hand, this this has has rather rather dire dire consequences consequences for for tests tests of of the the standard standard the neutral model model discussed discussed in in Section Section 8.2. 8.2. It It looks looks as as if if there there will will always always be be neutral neutral metapopulation metapopulation explanation explanation for any significant deviation in in these these aa neutral for any significant deviation statistics. statistics.
A Framework Framework for for Inference Inference in in Metapopulations Metapopulations A Clearly, there there is is great great potential potential to to adapt the many many useful useful methods methods that have Clearly, adapt the that have been developed developed for the standard standard coalescent coalescent model model to to the the case case of of aa metapopumetapopu been for the lation. means taking taking the the scattering phase into into account, lation. Again, Again, this this simply simply means scattering phase account, and and again this work work is is in in its its infancy. infancy. again this Analytical Analytical Methods Methods
As in the if predictions predictions about about summaries of polypoly As in the standard standard coalescent, coalescent, if summaries of morphism do the joint joint distribution of genealogical genealogical lengths morphism do not not depend depend on on the distribution of lengths at at pairs or or groups sites, then then they they will will be be accurate the rate rate of of pairs groups of of sites, accurate regardless regardless of of the recombination. Thus, unbiased method of moments moments estimators estimators of recombination. Thus, unbiased method of of metapopu metapopu(6, M, M, E, E, k) k) lation parameters could be be devised. devised. Because Because there there are at least least four lation parameters could are at four (0, parameters metapopulation model model used used here, here, so this will require the the parameters in in the the metapopulation so this will require proposal of proposal of some some new new summary summary statistics, statistics, tailored tailored to to samples samples from from metapopu metapopulations. lations. It It is is clear clear that that simple, simple, commonly commonly used used measures, measures, such such aa pairwise pairwise dif differences ferences within within and and between between demes, demes, will will not not suffice suffice (Pannell (Pannell and and Charlesworth, 999; Pannell, Charlesworth, 11999; Pannell, 2003). 2003). One One example example of of the the possibilities possibilities for for inference inference is is the the analytical analytical method method given given in 1 999), which bases inferences in Wakeley Wakeley ((1999), which bases inferences on on the the joint joint distribution distribution of of allele allele frequencies deme sample. frequencies among among demes demes in in aa multi multideme sample. It It was was assumed assumed that that the the demes demes were were not not subject subject to to extinction extinction and and recolonization, recolonization, only only migration, migration, and and predictions predictions like like those those shown shown in in Fig. Fig. 8.5 8.5 formed formed the the basis basis of of aa maximum maximum like likemethod of of inference inference using using data data from from unlinked unlinked loci loci such such as as RFLP RFLP or or lihood method SNP SNP data. data. The The model model also also incorporated incorporated aa change change in in the the effective effective population population size size at at some some time time in in the the past, past, illustrating illustrating the the ease ease with with which which such such complica complications tions can can be be treated treated when when part part of of the the history history of of the the sample sample is is given given by by the the stand standard ard coalescent coalescent process. process. Computational Computational Methods Methods
The The development development of of full full data data methods methods such such as as those those of of Griffiths Griffiths and and Tavare Tavar~ ((1994a,b) 1994a,b) and 1 995) appears and Kuhner Kuhner et et al. al. ((199.5) appears promising promising because because those those methods, methods, developed developed for for unstructured unstructured populations, populations, can can be be applied applied directly directly once once the the scat scattering populations with tering phase phase is is taken taken into into account. account. For For the the case case of of meta metapopulations with large large numbers numbers of of demes, demes, this this will will be be much much more more efficient efficient than than the the current current methods methods ((Beerli Beerli and 999; Bahlo and Felsenstein, Felsenstein, 11999; Bahlo and and Griffiths, Griffiths, 2000), 2000), which which require require the the estimation estimation of of migration migration rates rates between between every every possible possible pair pair of of demes demes and and which which
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so so far far have have assumed assumed that that the the sampled sampled demes demes are are the the only only demes demes in in the the metapopulation. Still, population would metapopulation. Still, aa full full data data method method for for aa meta metapopulation would have have to to deal with complicated data deal with more more complicated data and and aa greater greater number number of of potential potential histories histories of population requires, of the the sample sample than than aa panmictic panmictic population requires, so so aa method method of of this this sort sort is is expected expected to to have have aa greater greater number number of of potential potential drawbacks drawbacks than than panmictic panmictic methods. One One potential potential solution solution to to this this is is the the development development of of computational computational methods methods that that use use summary summary statistics, statistics, again again following following the the work work on on the the panmictic panmictic coa coalescent Fu and 997; Tavare aI., 11997; 997; Weiss 998; lescent ((Fu and Li, Li, 11997; Tavar~ et et al., Weiss and and von von Haeseler, Haeseler, 11998; Beaumont Beaumont et et aI., al., 2003 2003).) . These These should should be be much much more more efficient efficient computation computationally than being able able to ally than full full data data methods methods and and hold hold the the promise promise of of being to focus focus on on data data patterns patterns that that contain contain most most of of the the information information about about the the parameters parameters of of the the model, model, assuming assuming that that such such statistics statistics can can be be developed. developed. The The elucidation elucidation of of complicated .5 is goal. complicated patterns patterns like like those those displayed displayed in in Fig. Fig. 88.5 is aa step step toward toward this this goal. An An example example of of aa summary summary statistic statistic computational computational method method for for the the sort sort of of model model discussed discussed here here is is the the approximate approximate maximum maximum likelihood likelihood method method for for multilocus given in 1 ) in which numbers multilocus data data given in Wakeley Wakeley et et ai. al. (200 (2001) in which numbers of of poly polymorphisms morphisms were were used used to to make make inferences inferences about about 8, 0, the the distribution distribution of of those those polymorph isms among polymorphisms among demes demes was was used used to to make make inferences inferences about about migration migration parameters, parameters, and and the the overall overall frequency frequency of of polymorphisms polymorphisms in in the the sample sample was was used make inferences about aa possible used to to make inferences about possible change change in in effective effective size size at at some some time time in the past.
8.4 8.4
SUMMARY S U M M A R Y AND A N D CONCLUSIONS CONCLUSIONS It It is is important important to to remember remember that that the the results results presented presented in in this this chapter chapter hold hold only populations, within only for for large large meta metapopulations, within which which the the number number of of possible possible source source demes demes of of migrants migrants and and colonists colonists is is large. large. Whether Whether this this is is justified justified or or not not will will depend (see Chapters depend on on the the species species under under study study (see Chapters 1133 through through 23 23).) . If If the the num number small, the coalescent (Notohara, 990; ber of of demes demes is is small, the standard standard structured structured coalescent (Notohara, 11990; Nordborg, 11997; Wilkinson-Herbots, 11998) more appropriate appropriate frame frameNordborg, 997; Wilkinson-Herbots, 99 8 ) is a more work. The Kingman's coalescent work. The finding finding of of Kingman's coalescent as as part part of of the the history history of of any any sample sample from from aa large large metapopulation metapopulation immediately immediately makes makes applicable applicable aa plethora and inferential One interesting plethora of of theoretical theoretical and inferential results results and and methods. methods. One interesting consequence unfortunate, is consequence of of this, this, which which might might be be considered considered unfortunate, is that that many many of population are of the the details details of of the the dynamics dynamics of of the the meta metapopulation are folded folded into into aa sin single gle population population parameter: parameter: the the effective effective size size of of the the collecting collecting phase, phase, or or its its mutation-scaled mutation-scaled equivalent, equivalent, 8. 0. This This means means that that many many phenomena phenomena of of bio biological logical interest interest will will not not produce produce any any observable observable effect effect on on patterns patterns of of genetic genetic polymorphism. However, this is akin to the standard standard coalescent coalescent in which which polymorphism. However, the the distribution distribution of of offspring offspring numbers numbers among among individuals individuals in in the the metapopu metapopulation affects levels and only through lation affects levels and patterns patterns of of polymorphism polymorphism only through the the effect effective ive mutation mutation parameter, parameter, 80.. Outweighing Outweighing this this is is the the fact fact that that by by modeling modeling gene population, we about the gene genealogies genealogies in in aa meta metapopulation, we gain gain intuition intuition about the potential potential of of further further theoretical theoretical study study and and the the design design of of more more optimal optimal methods methods of of inference. inference. Even Even the the little little that that is is currently currently known known about about the the complex complex patterns patterns of variation in samples from pro of genetic genetic variation in samples from aa metapopulation, metapopulation, e.g., e.g., Fig. Fig. 8.5, 8.5, provides vides aa great great deal deal of of hope. hope.
POPULATION M ETAPO PU LATI0 N 9~ META
QUANTITATIVE GENETICS GENETICS:: THE QUANTITATIVE QUANTITATIVE GENETICS G EN ETICS OF POPULATION POPU kATION DIFFERENTIATION D IF FERENTIATI O N Charles Charles J. J. Goodnight Goodnight
9.1 9.1
INTRODUCTION INTRODUCTION The 1 930), although The field field of of quantitative quantitative genetics genetics can can be be traced traced to to Fisher Fisher ((1930), although the biometician" school the roots roots of of this this field field can can be be traced traced further further back back to to the the ""biometician" school of 1 ) . The of evolution, evolution, and and ultimately ultimately to to Darwin Darwin (Provine, (Provine, 200 2001). The original original goals goals of of quantitative genetics included explaining and describing the response to direc directional tional selection selection and and to to providing providing analytical analytical tools tools that that could could be be used used in in the the breeding of livestock. With respect to the goal of providing tools for breeders, quantitative quantitative genetics genetics has has been been stunningly stunningly successful. successful. In In developing developing quantitative quantitative genetics, genetics, Fisher Fisher made made the the assumption assumption that that popu populations lations were were large, large, unstructured, unstructured, and and mated mated randomly. randomly. These These assumptions assumptions are are inappropriate for meta populations, which are, by definition, structured metapopulations, structured and in which which the the individual individual demes demes are are frequently frequently small. small. Fisher's Fisher's methods methods remain remain
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valuable valuable in in that that they they predict predict the the response response to to selection selection within within demes; demes; however, however, this prediction not provide this prediction generally generally will will not provide aa useful useful description description of of the the evolution evolution of population as whole. Of many reasons reasons that of the the meta metapopulation as aa whole. Of the the many that this this may may be be true, true, such multilevel selection, such as as environmental environmental heterogeneity heterogeneity and and multilevel selection, one one of of the the most most interesting interesting is is the the effect effect of of population population structure structure on on the the underlying underlying effects effects of of genes genes on on the the phenotypes. phenotypes. In In particular, particular, when when there there is is epistasis, epistasis, defined defined as as inter interactions loci, the actions among among alleles alleles at at different different loci, the effect effect of of aa gene gene on on the the phenotype phenotype of of an an individual individual is is aa function function not not only only of of the the gene, gene, but but also also of of the the genetic genetic back background ground in in which which it it is is found. found. What What this this means means is is that that even even when when selection selection acts acts on phenotype in manner across all demes, demes, the on the the phenotype in the the same same manner across all the genetic genetic conse consequences quences of of that that selection selection may may be be different different in in different different demes. demes. Thus, Thus, an an allele allele favored favored by by selection selection in in one one deme deme may may be be eliminated eliminated by by the the same same selective selective regime me with regime in in aa second second de deme with aa different different genetic genetic background. background. The The goal goal of of metapopulation metapopulation quantitative quantitative genetics genetics is is to to describe describe the the vari variation ation among among demes demes in in the the phenotypic phenotypic effect effect of of alleles. alleles. Conceptually, Conceptually, it it is is aa question of question of what what is is the the variance variance in in the the phenotypic phenotypic effect effect of of aa particular particular allele allele when when it it is is "inserted" "inserted" into into the the different different demes demes in in aa metapopulation? metapopulation? This This vari variance me mean, ance must must be be corrected corrected for for the the effects effects of of the the overall overall de deme mean, which which will will necessarily allele across necessarily affect affect the the phenotypic phenotypic variance variance in in the the effects effects of of an an allele across aa metapopulation. more correct metapopulation. Thus, Thus, the the more correct quantity quantity is is the the phenotypic phenotypic variance variance in in the the effect effect of of an an allele allele relative relative to to the the effects effects of of other other alleles alleles at at the the same same locus locus measured allelic effects measured in in the the same same demes. demes. If If this this variance variance in in the the relative relative allelic effects is is zero, zero, or or in in an an experimental experimental situation situation small small and and not not significant, significant, then then allelic allelic effects effects measured measured in in one one deme deme are are indicative indicative of of allelic allelic effects effects (relative (relative to to other other alleles measured in deme. However, alleles at at the the same same locus) locus) measured in any any deme. However, if if the the variance variance in in the allelic effects nonzero, then allelic effects the allelic effects is is nonzero, then the the allelic effects measured measured in in one one deme deme are are not not predictive predictive of of the the relative relative allelic allelic effects effects in in other other demes. demes. When When there there is is variance variance in in the the effects effects of of alleles, alleles, phenotypic phenotypic selection selection acting acting uniformly all demes become aa diversifying uniformly in in all demes will will become diversifying selection selection at at the the genic genic level. level. That That is, is, selection selection favoring favoring an an allele allele in in one one deme deme may may lead lead to to aa decrease decrease in in the the frequency leads to frequency of of that that same same allele allele in in aa second second deme. deme. This This potentially potentially leads to aa selective restriction of migration between selective restriction of migration between demes, demes, as as the the offspring offspring of of migrants migrants will be will be of of low low fitness. fitness. This This could could interfere interfere with with ecological ecological and and demographic demographic processes, migrants. In processes, such such as as the the "rescue "rescue effect" effect" of of migrants. In the the extreme, extreme, the the fitness fitness of of the the offspring offspring of of migrants migrants may may be be so so low low that that interdemic interdemic gene gene flow flow is is elim eliminated, inated, effectively effectively turning turning the the different different demes demes into into separate separate species. species. Finally, Finally, it it is alleles need is important important to to note note that that variation variation in in the the effects effects of of alleles need not not be be corre correlated with with variation variation in in deme deme means. means. Thus, Thus, even even if if two two demes demes have have very very simi similated lar lar mean mean phenotypes, phenotypes, they they may may nevertheless nevertheless be be differentiated differentiated for for genic genic effects. effects. Conversely, mean phenotypic Conversely, two two demes demes with with very very different different mean phenotypic values values need need not not be be differentiated differentiated for for genic genic effects. effects. This describes traditional This chapter chapter briefly briefly describes traditional or or "Fisharian" "Fisharian" quantitative quantitative genet genetics some of ics and and uses uses this this as as aa framework framework to to discuss discuss some of the the modifications modifications of of this this theory when applying applying this theory that that are are necessary necessary when this theory theory to to aa metapopulation metapopulation rather than than aa single single panmictic panmictic population. It then then discusses discusses the the interpretation interpretation rather population. It of whereas traditional of metapopulation metapopulation quantitative quantitative genetics. genetics. In In particular, particular, whereas traditional or or Fisherian are naturally Fisherian quantitative quantitative genetics genetics are naturally related related to to measuring measuring the the response response to to selection, selection, metapopulation metapopulation quantitative quantitative genetics genetics is is more more naturally naturally related related to to the populations and, consequence, speciation. the differentiation differentiation of of populations and, as as aa consequence, speciation.
9. 9. METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVEGENETICS GENETICS
9.2 9.2
201 201
FISHERIAN QUANTITATIVE QUANTITATIVE GENETICS GENETICS FISHERIAN When When he he originally originally developed developed the the field field of of quantitative quantitative genetics, genetics, Fisher Fisher ((1930) 1 930) used used the the assumption assumption that that traits traits were were determined determined by by aa very very large large (in (in the the limit, infinite) number of loci each with a very small (in the limit infinitesimal) effect. effect. Under Under this this assumption, assumption, long-term long-term directional directional selection selection would would lead lead to to aa linear mean phenotype linear change change in in the the mean phenotype with with no no discernible discernible change change in in gene gene fre frequency limit of quency at at any any given given locus. locus. Indeed, Indeed, at at the the limit of an an infinite infinite number number of of loci, loci, with an infinitesimal effect on the phenotype, phenotype, changes in gene frequency each with would would be be infinitesimal infinitesimal as as well. well. More More importantly, importantly, he he assumed assumed that that there there was was no no population population structure structure and and that that populations populations were were very very large. large. Finally, Finally, in in order order work, there must be ran ranfor many of the relationships that Fisher described to work, dom mating (Falconer, 1985). Quantitative Quantitative genetics genetics is is built built on on the the idea idea of of partitioning partitioning the the phenotype phenotype into into components. components. If If the the ith ith individual individual has has aa phenotype phenotype Pi, Pi, then then this this can can be be divided divided into Gi) and into components components due due to to genetics genetics ((Gi) and the the environment environment (Ei): (Ei): Pi Pi = = f.L Ix + + G Gii + + Ei Ei
((9.1) 9.1)
where where f.L tx is is the the mean mean ooff the the population. population. IInn this this partitioning, partitioning, it it is is assumed assumed that that there there are are no no genotype genotype by by environment environment interactions interactions or or correlations. correlations. These These are are incorporated incorporated easily, easily, but but are are not not necessary necessary for for the the topics topics discussed discussed in in this this chap chapter. ter. The The genetic genetic component component can can be be further further divided divided into into components components including including the the breeding breeding value value (additive (additive effects, effects, A), A), aa component component that that can can be be attributed attributed to to interactions interactions between between alleles alleles at at the the same same loci loci (dominance, (dominance, D), D), and and interactions interactions between alleles alleles at loci (epistasis). Epistasis can can be be further further divided into between at different different loci (epistasis). Epistasis divided into components due to the the nature nature of the particular particular interaction. interaction. For For example, two components due to of the example, twolocus interactions interactions can can be be divided divided into into additive additive by by additive additive epistasis epistasis (AXA), (AXA), locus additive by by dominance dominance by by additive additive epistasis epistasis additive dominance epistasis epistasis (AXD), (AXD), dominance ( DXA) , and and dominance dominance by by dominance dominance epistasis epistasis ((DXD) DXD) (Table 9 . 1 ) . Similarly, Similarly, (DXA), (Table 9.1). three locus locus and and higher higher interactions interactions can, can, in in principle, principle, be be added. added. Thus, Thus, the the value value three of the phenotype of the ith individual individual becomes of the phenotype the ith Pi = Ai + Di + + AXAi AXDi + + DXAi DXAi + + DXDi +E Eii (9.2) (9.2) Pi = f.L IX + + Ai + Di AXA~ + + AXDi DXDi + + .. ... . + This partitioning partitioning of of the the phenotype phenotype into into components components is statistically using using This is done done statistically the the regression regression of of phenotype phenotype on on the the genetic variance variance components components (I-Iayman (Hayman and and Mather, 1955; 1 955; Goodnight, Goodnight, 2000a,b). 2000a,b). Understandably, Understandably, the the regression regression model model Mather, can potentially becoming complicated as genetic effects can potentially becoming quite quite complicated as more more of of the the genetic effects are are included. included. It is is important important to to emphasize emphasize that partitioning is is aa statistical statistical partitioning partitioning It that this this partitioning done done by by multiple multiple regression. regression. Further, Further, this this multiple multiple regression regression is is weighted weighted by by the the frequency of frequency of the the different different genotypes. genotypes. As a consequence, consequence, when when gene gene frequencies frequencies change, the the partitioning partitioning will will also also change. change. Thus, Thus, the the breeding breeding value value of of an an indiindi change, vidual is is not not only only aa property property of of the the genes genes that that make make up up that that individual, individual, but but also also vidual of population in which which it is measured measured (Falconer Mackay, 1996). of the the population (Falconer and and Mackay, 1 996). Variation is is necessary necessary if if there there is is to to be be evolution, evolution, and and as as aa result result it it is is the the partiparti Variation tioning of of the the phenotypic phenotypic variance variance that that is is of of interest. interest. Because Because the the phenotype phenotype has has tioning been divided divided into into genetic genetic and and environmental environmental components components using using aa least-squares least-squares been
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CHARLES CHARLESj.J. GOODNIGHT GOODNIGHT
TABLE 9.1 9.1 TABLE
The Eight Genetic Effects Effects Used Used in This ChapterQ Chapter a
Additive A A locus locus Additive
Additive B B locus locus Additive
Dominance A A locus locus Dominance
Dominance B B locus locus Dominance
Additive by by additive additive epistasis epistasis Additive
BjBj B1B1 BjB BIB2z B B2B2 zBz
BjB! BIB1 BIB B1B2 z B B2B2 zBz
BIB BIB I1 BIB BIB z2 B B2B2 zB z
BIBI BIB1 BjB BIB2z B B2B2 zBz
Additive by by ddominance o m i n a n c e epistasis epistasis Additive
BIBI BIB1 BIB BIB22 B2B22 B2B
D ominance b byy additive Dominance additive epistasis epistasis
BIB BIB I1 BI B22 B1B B2B22 B2B
D ominance b byy ddominance o m i n a n c e epistasis Dominance epistasis
a
BIBj1 BIB BIB22 BIB BzB2 B2B2
BjB ! BIB1 B ! B2z BIB B2Bz B2B2
A A2A2 zAz
A 1 Al A1A1
A 1 Az AIA2
11 11 11
00 00 0 0
A 1 Al AIA1
A 1 Az AIA2
11 0 0 -- 11
11 00 -- 11
11 00 -- 11
A 1 Al AIA1
A 1 Az AIA2
A AaAa zAz
- 11 - 11 - 11
A 1 Al AIA1
-- 11 -- 11 -- 11
A A2Aa zAz
1 1 11
- 11 - 11 -- 11
A 1 Az AIA2
A A2A2 zAz
- 11
-- 11 11 - 11
-- 11 11 -- 11
A 1 Al AIA1
A 1 Az AIA2
A A2A2 zAz
11 0 0 - 11
0 0 0 0 0 0
- 11 0 0 1
A 1 Al AIA1
A 1 Az AIA2
A A2A2 zAz
-- 11 11
0 0 0 0 0 0
1 --11
A 1 Al A1A1
A 1 Az AIA2
AzAz A2A2
00 -- 11
--1 1 00 11
00 --11
A1 Al A1A1
A1 Az AIA2
AzAz A2A2
1
11
--11 1 -- 11
11 -- 11 11
a These These effects effects fully fully describe describe any any two-locus two-locus two-allele two-allele genetic genetic effects. effects.
1
- 11
- 11
11
--11 1 -- 11
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METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS 9. METAPOPULATION
regression, regression, these these components components are are statistically statistically independent independent of of each each other. other. As As aa result, result, phenotypic phenotypic variance variance can can be be partitioned partitioned in in exactly exactly the the same same manner manner as as the the individual 930; Hayman 955; Falconer Falconer and individual phenotype phenotype (Fisher, (Fisher, 11930; Hayman and and Mather, Mather, 11955; and Mackay, 996; Goodnight, 998, 2000a,b): Mackay, 11996; Goodnight, 11998, 2000a,b): Vp Vp
VA A+ if- V Vo D + q- V VAX Jr- V VAX D + if- V VDX q- V VDX D + + . . .. . . + if- V VE OXAA + == V oxo AXO E AXAA +
(9.3) (9.3)
As As with with the the partitioning partitioning of of the the phenotype phenotype of of an an individual, individual, the the components components of phenotypic phenotypic variance variance are are statistical statistical properties properties of of aa population. As gene gene fre freof population. As quencies will also quencies change, change, the the partitioning partitioning among among variance variance components components will also change. change. Under Under Fisher's Fisher's assumption assumption of of large large population population size, size, and and many many loci loci each each with with small small effect effect gene gene frequencies frequencies will will not not change change appreciably, appreciably, and and the the variance variance components components will will remain remain approximately approximately constant. constant. The The primary primary utility utility of of quantitative quantitative genetics genetics is is that that it it can can be be used used to to predict predict the breeder's equation" equation" ((Falconer Falconer the response response to to selection selection using using the the standard standard ""breeder's and 996): and Mackay, Mackay, 11996): r-
h2s
(9.4)
where where rr is is the the response response to to selection selection measured measured as as the the difference difference in in mean mean pheno phenotype type between between parents parents and and offspring, offspring, ss is is the the selection selection differential differential measured measured as as the the difference difference in in mean mean phenotype phenotype between between the the selected selected parents parents and and all all parents, parents, and and hh 22 is is the the heritability. heritability. Heritability Heritability is is the the ratio ratio of of the the additive additive genetic genetic variance variance to to the the phenotypic phenotypic variance, variance, -VA. ~ . It It is is also also aa constant constant of of proportionalproportional-
Vp
/1
ity change due ity that that "converts" "converts" change due to to selection selection within within generations generations into into change change between Falconer and 996). between generations generations ((Falconer and Mackay, Mackay, 11996). For For predicting predicting the the response response to to selection, selection, only only two two variance variance components components and and the the selection selection differential differential are are needed. needed. The The two two variance variance components components are are the the additive additive genetic genetic variance variance and and the the phenotypic phenotypic variance. variance. The The phenotypic phenotypic vari variance ance is is simply simply the the variance variance in in the the trait trait observed observed in in the the population. population. Thus, Thus, the the additive additive genetic genetic variance, variance, the the genetic genetic variance variance that that can can contribute contribute to to the the resem resemblance blance between between parents parents and and offspring, offspring, is is important. important. In In addition, addition, it it is is often often use usedominance variance both both because it is often a large portion portion ful to measure the dominance of reasonable to of the the phenotypic phenotypic variance variance and and because because it it is is experimentally experimentally reasonable to measure. 1 930) felt measure. Fisher Fisher ((1930) felt that that the the epistatic epistatic variance variance components components could could be be rele relegated component. This emphasizes that gated to to the the environmental environmental variance variance component. This emphasizes that the the "environmental" "environmental" variance variance is is perhaps perhaps better better referred referred to to as as the the "residual "residual"" vari variance, ance, as as it it is is the the sum sum of of all all of of the the unmeasured unmeasured factors, factors, genetic genetic and and environ environmental, contributing variance. mental, contributing to to the the phenotypic phenotypic variance.
9.3 9.3
GENETIC GENETIC VARIANCE VARIANCE COMPONENTS COMPONENTS IN IN A A TWO-LOCUS TWO-LOCUS TWO-ALLELE TWO-ALLELE SYSTEM SYSTEM Consider Consider aa system system with with an an A A locus locus and and aa B B locus. locus. At At each each locus locus there there are are two two alleles, Al and and A and B1 Bl and and B In aa alleles, A1 A22 alleles alleles at at the the A A locus, locus, and B22 alleles alleles at at the the B B locus. locus. In system possible genotypes. purposes, it system such such as as this this there there are are nine nine possible genotypes. For For heuristic heuristic purposes, it is is
204 204
CHARLES CHARLES ). I. GOODNIGHT
convenient to array these as a three by three matrix (Table 99.1). . 1 ). Such a system equations (Goodnight, (Goodnight, 2000a,b). One can be fully described by nine orthogonal equations of these is the mean genotypic value, leaving eight equations to fully describe the genetic two-locus two-allele two-allele system always be genetic effects. effects. Thus, Thus, aa two-locus system can can always be divided divided into into independent genetic effects. Assuming a gene frequency of 0.5 for both alle alleeight independent les les at at both both loci loci and and aa population population in in two-locus two-locus Hardy-Weinberg-Castle Hardy-Weinberg-Castle equilib equilibrium, the eight genetic effects used in this chapter chapter are given in Table 9.1 9.1.. At a gene frequency orthogonal; however, frequency of of 0.5, 0.5, these these genetic genetic effects effects are are orthogonal; however, if if gene gene frequen frequencies deviate from from 0.5 for either locus, they will no longer be independent. independent. In gen general, eral, aa change change in in gene gene frequency frequency will will tend tend to to shift shift the the forms forms of of genetic genetic variation variation involving more interaction into forms forms involving less less interaction. The The statistical statistical shift shift of of genetic genetic variation variation into into forms forms involving involving less less interaction interaction means that periods of small small population will tend to cause cause dominance means that periods population size will dominance and epistasis to diminish and additive additive genetic variance to increase. Figure 9.1 is a pair of graphs of the additive genetic variance as a function function of the Wright's inbreeding coefficient, single locus effects (additive and and dominance coefficient, P, F, with single dominance effects, 1 ) shown Fig. 9.1a epistasis shown . 1 b. effects, Table Table 9. 9.1) shown in in Fig. 9.1a and and digenic digenic epistasis shown in in Fig. Fig. 99.1b.
Additive
11.00 .00
a a
OJ r () r
.lii 0.50 >� ._
>
0.00 .. -.,Ir----. ,I ----.,0.00 ,... -=--"11 I I 0.75 0.5 0 0.75 1 0.25 o Inbreeding Coefficient (F)
b 11.00 .00
AXA
OJ ()
lii 0.50 ._~
.� > >
0.00 +-"'--....::::'-r----;--:--" o 0
0.25
0.5
0.75
1
Inbreeding Coefficient (F)
Fig. 9.1 Additive genetic variance in a population as a function Fig. 9.1 function of of Wright's Wright's inbreeding coeff coefficient, F. F. Total genetic variance in the outbred population (F (F = 0) is standardized at 11.. (a) Single locus effects, additive effects (Additive) and dominance effects (Dominance), (b) Two locus interactions, additive by additive epistasis epistasis (AXA), additive by dominance and dominance by additive epistasis D), and dominance by dominance epistasis (DXD). epistasis (AX (AXD),
9. 9.
METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVE GENETICS GENETICS
205 205
When When there there are are only only additive additive effects, effects, the the additive additive genetic genetic variance variance declines declines lin linearly early as as aa function function of of F F (Fig. (Fig. 9.1a). 9.1a). For For dominance dominance (Fig. (Fig. 9.1a) 9.1a) and and all all forms forms of of epistasis b), the epistasis (Fig. (Fig. 9.1 9.1b), the additive additive genetic genetic variance variance increases increases as as aa function function of of F F until value at until it it reaches reaches aa maximum maximum value at an an intermediate intermediate level level of of inbreeding inbreeding before before declining. interaction, additive variance declines declining. In In the the absence absence of of gene gene interaction, additive variance declines in in direct direct proportion to to the the heterozygosity heterozygosity (equal (equal to to 1-F, l-F, where where F F is is Wright's Wright's inbreeding inbreeding proportion coefficient). In In the the presence presence of of gene gene interaction, interaction, the the change change in in additive additive genetic genetic coefficient). variance variance due due to to aa small small population population size size will will depend depend on on the the competing competing processes processes of of the the increase increase due due to to the the conversion conversion of of dominance dominance and and epistasis epistasis to to additive additive variance and overall loss variance and the the loss loss due due to to an an overall loss in in genetic genetic variation. variation. If If weighted weighted by by the the genotype genotype frequencies frequencies at at Hardy-Weinberg-Castle Hardy-Weinberg-Castle propor proportions, tions, the the genotypic genotypic values values of of each each genetic genetic effect effect in in Table Table 9.1 9.1 sums sums to to zero. zero. This This allows allows them them to to be be used used as as orthogonal orthogonal contrasts contrasts in in aa linear linear regression regression and, and, as as aa consequence, partitioning any consequence, provides provides aa simple simple method method for for partitioning any set set of of two-locus two-locus two-allele two-allele genotypic genotypic values values into into independent independent genetic genetic effects. effects. To To do do multiple multiple regression, the the nine nine observed observed genotypic genotypic values values are are used used as as dependent dependent variables variables regression, and and the the eight eight genetic genetic effects effects are are used used as as independent independent variables. variables. This This is is an an unusual unusual regression regression because because with with nine nine genotypic genotypic values values there there are are only only eight eight degrees each of degrees of of freedom. freedom. With With each of the the genetic genetic effects effects using using one one degree degree of of free freedom, there Thus, with only nine nine geno dom, there are are no no degrees degrees of of freedom freedom left left for for error. error. Thus, with only genotypic typic values, values, the the genetic genetic variance variance can can be be partitioned partitioned into into genetic genetic variance variance components, components, but but no no estimate estimate of of the the accuracy accuracy of of that that partitioning partitioning is is possible. possible. If, If, however, individuals are phenotypic value value of however, aa set set of of individuals are scored scored for for both both the the phenotypic of the the trait trait and and their their genotype genotype (possibly (possibly using using quantitative quantitative trait trait loci), loci), then then aa regression regression could could be be done done that that would would provide provide an an error error variance. variance. The The genotypic genotypic values values must must be be weighted weighted by by their their genotype genotype frequencies; frequencies; thus, thus, aa regression regression done done at at one one gene gene frequency frequency will will give give aa different different answer answer than than aa regression regression done done at at aa different different gene gene frequency. Again, this is the basis of conversion of a nonadditive nonadditive genetic vari varifrequency. ance (Fig. 9.1 ance into into an an additive additive genetic genetic variance variance (Fig. 9.1).). Finally, Finally, the the regression regression must must be be done done using using sequential sequential or or type type 11 sums sums of of squares. squares. Typical Typical regression regression packages packages use use iterative iterative or or type type 33 sums sums of of squares. squares. While While appropriate appropriate for for standard standard uses uses of of regression regression with with moderate moderate unbalance unbalance (type (type 11 and and type type 33 sums sums of of squares squares give give identical results squares give identical results for for balanced balanced data data sets), sets), type type 33 sums sums of of squares give incorrect incorrect results results for for the the genotypic genotypic regressions regressions that that are are unbalanced unbalanced due due to to changes changes in in gene gene frequency. frequency. An An issue issue with with using using type type 1I sums sums of of squares squares is is that that the the order order in in which which the regression model model will change the the independent independent variables variables are are entered entered into into the the regression will change the independent genetic vari variresults of the regression. In the case of regression on independent ance variables must ance components, components, the the order order in in which which the the variables must be be entered entered is is additive additive locus locus A A and and additive additive locus locus B B (order (order not not important); important); dominance dominance locus locus A A and and dominance dominance locus locus B B (order (order not not important); important); additive additive by by additive additive epistasis; epistasis; addi additive tive by by dominance dominance and and dominance dominance by by additive additive epistasis epistasis (order (order not not important); important); and and finally finally dominance dominance by by dominance dominance epistasis epistasis (Goodnight, (Goodnight, 2000a). 2000a).
9.4 9.4
METAPOPULATION METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS extend Fisherian Fisherian quantitative quantitative genetics genetics to to aa metapopulation metapopulation setting, setting, To extend assume that population structure consisting of assume that there there is is aa meta metapopulation structure consisting of aa large large set set of of demes demes linked linked by by migration. migration. Define Define the the local local breeding breeding value value of of the the ith ith individual individual
206 206
CHARLES J.I. GOODNIGHT CHARLES GOODNIGHT
Aij•
in in the the jth jth deme deme to to be be Aij. In In aa large large randomly randomly mating mating population, population, the the breeding breeding value mean phenotype individual. In value is is the the mean phenotype of of the the offspring offspring of of an an individual. In aa structured structured population, population, it it is is necessary necessary to to identify identify both both the the individual, individual, and and the the deme deme from from which which its its mates mates are are drawn. drawn. The The author author refers, refers, to to Aij as as the the local local breeding breeding value value to 1930). It to distinguish distinguish it it from from the the breeding breeding value value as as defined defined by by Fisher Fisher ((1930). It differs differs in that an individual individual has single Fisherian Fisherian breeding breeding value, value, which which is is an an in that an has only only aa single average population, whereas average across across the the meta metapopulation, whereas an an individual individual has has aa distinct distinct local local breeding breeding value value in in each each deme deme of of aa metapopulation. metapopulation. The The local local breeding breeding value value is is taken as population mean. mean. Using least-squares taken as aa deviation deviation from from the the meta metapopulation Using least-squares partitioning, partitioning, the the breeding breeding value value can can be be divided divided into into components components due due to to an an individual individual effect effect (Ai), aa deme deme effect effect (Ai), and and an an individual individual by by deme deme interaction interaction (Ai*j) (Wade (Wade and and Goodnight, Goodnight, 1998): 1998):
Aij
(Ai),
(Ai*j)
(Aj),
(9.5) (9.5)
Aij = Ai + Aj + Ai.j Within Within mean mean additive additive genetic genetic variance variance within within demes demes will will be be given given by by
1� (�P· ( ·(A
A*·)2 - A2.))
VA VA = = l~ ~j E, PiJ(Ai"· + t- Ai,j) ·f I f 2 - A2j J ..:,.; -4 If I I
f
Pij
(9.6) (9.6)
where me additive where V VAA is is the the mean mean within within de deme additive genetic genetic variance, variance, Pii is is the the fre frequency quency of of the the ith ith genotype genotype in in the the jth jth deme, deme, and and JJ is is the the number number of of demes. demes. The The effect effect of of population population size size on on this this equation equation is is worth worth discussing. discussing. First, First, when when the the population population is is unstructured, unstructured, Pij, the the frequency frequency of of the the ith ith genotype genotype in in the the jth jth population population is is simply simply the the frequency frequency of of the the ith ith genotype. genotype. In In aa structured structured popu population, will, by lation, genetic genetic drift drift will, by random random chance, chance, cause cause frequencies frequencies of of genotypes genotypes to to change change and and some some genotypes genotypes to to disappear. disappear. It rather than genotype fre It is is often often convenient convenient to to focus focus on on allele allele frequencies frequencies rather than genotype frequencies. quencies. Although Although selection selection acts acts on on entire entire phenotypes, phenotypes, much much of of the the recent recent genetical genetical data data has has taken taken the the form form of of mapping mapping the the chromosomal chromosomal regions regions that that affect affect quantitative quantitative traits, traits, i.e., i.e., quantitative quantitative trait trait loci loci (QTL). (QTL). These These studies, studies, which which are level focus whole phenotype phenotype level are naturally naturally related related to to aa gene gene level focus rather rather than than aa whole level focus, allow aa detailed focus, allow detailed dissection dissection of of two two locus locus interactions interactions that that would would be be diffi difficult or more traditional cult or impossible impossible using using more traditional quantitative quantitative genetic genetic methods. methods. In In sys systems tems without without gene gene interaction interaction and and random random mating, mating, the the frequencies frequencies of of individual individual alleles are adequate for alleles are adequate for describing describing the the genetic genetic variance. variance. Wright's Wright's inbreeding inbreeding coef coefmeasure of of the increase in correlation among among alleles randomly ficient, ficient, F, F, is is aa measure the increase in correlation alleles in in aa randomly mating and thus provides aa summary measure of mating population population and thus provides summary measure of the the expected expected change change in in allele allele frequencies. frequencies. In In an an additive additive system, system, the the additive additive genetic genetic variance variance is increases: is aa function function of of F F and and decreases decreases as as the the inbreeding inbreeding coefficient coefficient increases: n
Pij,
V� VA = = ((11 - ()VA f)VA
((9.7) 9.7)
where where VA V~i is is the the additive additive genetic genetic variance variance in in the the derived derived population. population. It It must must be be emphasized emphasized that that this this formula formula applies applies only only in in the the special special case case of of complete complete addi additivity tivity with with no no dominance dominance or or epistasis. epistasis. The The term term Aio should should remain remain approximately approximately constant constant with with inbreeding, inbreeding, as as it it is is an an average average of of breeding breeding value value across across all all demes. demes. This This will will not not be be true true if if there there
Ai.
9. 9.
207 207
METAPOPULATION METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS
is is inbreeding inbreeding depression depression or or other other factors factors that that cause cause shifts shifts in in breeding breeding values values associated associated solely solely with with increased increased F. F. Ai*/,j, however, however, is is aa function function of of the the inbreeding inbreeding in in the the population population when when there there is is Ai* equal zero. gene gene interaction. interaction. In In particular, particular, in in an an outbred outbred deme deme (F (F = -- 0), 0), Ai* Ai. j will will equal zero. As (F > As the the inbreeding inbreeding coefficient coefficient increases increases (F > 0), 0), demes demes will will become become progressively progressively more more differentiated differentiated and and Ai* Ai. j will will increase. increase. It It is is only only when when there there is is gene gene inter interaction inbreeding that nonzero. Figure action (dominance (dominance or or epistasis) epistasis) and and inbreeding that Ai* Ai. j is is nonzero. Figure 9.2 9.2 is is aa graph graph of of the the variance variance in in local local average average effects effects for for the the different different genetical genetical effects effects as as aa function function of of inbreeding inbreeding coefficient, coefficient, F. F. In In the the absence absence of of gene gene interaction, interaction, Ai* Ai. j is zero. For reason, additive is always always zero. For this this reason, additive effects effects are are not not shown. shown. For For the the additive additive by (and dominance dominance by interaction, the variance in by dominance dominance (and by additive) additive) gene gene interaction, the variance in local local average average effects effects is is shown shown separately separately for for the the "additive" "additive" locus locus (A (A locus locus in (B locus in the the additive additive by by dominance dominance interaction) interaction) and and the the "dominance" "dominance" locus locus (B locus in in the the additive additive by by dominance dominance interaction). interaction). The The "additive" "additive" locus locus refers refers to to the the locus locus with with additive additive genotypic genotypic values values within within the the genotype genotype of of its its interacting interacting pair. pair. For by dominance the BlBl For example, example, for for additive additive by dominance epistasis epistasis within within the B1B1 genotype, genotype, genotypic genotypic values values for for the the A A locus locus are are 11,, 0, 0, and a n d --1 1 for for the the AlAb A1A1, AlA AIA2, and A A2A2 2' and 2A2 genotypes, geno genotypes, respectively. respectively. In In contrast, contrast, the the "dominant" "dominant" locus locus has has dominant dominant genotypic values within typic values within genotypes genotypes of of the the interacting interacting pair. pair. For For example, example, within within the the AlAl locus has genotypic values A1A1 genotype, genotype, the the B B locus has genotypic values of of 11,, --1, 1 , and and 11 for for the the BlBb B1B1, BlB BIB2, and B B2B2 genotypes, respectively. respectively. 2, and 2B2 genotypes, Thus, there occur as demes differentiate. Thus, there are are two two competing competing processes processes that that occur as demes differentiate. The The loss loss of of genetic genetic variation variation is is reflected reflected in in the the Pi Pijj and and has has the the effect effect of of decreas decreasing ing the the additive additive genetic genetic variance, variance, whereas whereas when when there there is is gene gene interaction, interaction, the the
2.00 2.00 AXD AXD (dominance (dominance locus) locus) 11.50 .50 -
Q)u c .00 =~ 11.oo �> m o {:: til
0.50 0.50 Dominance AXA
AXD AXD (additive (additive locus) locus) 0.00 0.00
o
o-
0.25 0.'25
0.5 0'.5
0.75 0.'75
Inbreeding Coefficient Coefficient ((F) F)
i
Fig. 9.2 9 . 2 Variance due to to the allele by deme interaction, interaction, Var(a/kl), Var(oLffkl), as a function function of inbreed inbreeding coefficient, variance in in local ing coefficient, F, F, for for the the different different forms forms of of genetic genetic effects. effects. The The variance local average average effects effects for for additive additive effects effects is is zero zero for for all all values values of of F. F. Dominance Dominance effects effects (Dominance), (Dominance), additive additive by by addi additive D), tive epistasis epistasis (AXA), (AXA), additive additive by by dominance dominance and and dominance dominance by by additive additive epistasis epistasis (AX (AXD), and dominance dominance epistasis, dominance by dominance dominance epistasis (DXD) are �hown. shown. For additive by dominance additive additive and and dominance dominance loci loci are are listed listed separately. separately.
208 208
CHARLES CHARLES jJ.. GOODNIGHT GOODNIGHT
differentiation differentiation of of populations populations will will tend tend to to cause cause the the Ai*j Ai. j to to increase, increase, which which will will in in turn turn increase increase the the additive additive genetic genetic variance. variance. Thus, Thus, the the question question of of whether whether the genetic variance will increase following aa population the additive additive genetic variance will increase or or decrease decrease following population bottleneck bottleneck depends depends on on the the relative relative magnitude magnitude of of these these two two effects. effects. When When there there are (no dominance are only only additive additive effects effects (no dominance or or epistasis), epistasis), the the Ai*j Ai, j will will equal equal zero zero regardless inbreeding. With regardless of of the the level level of of inbreeding. With gene gene interaction, interaction, Ai*j Ai, j will will generally generally 1 9 8 8 ) showed be be nonzero nonzero and and will will increase increase with with increasing increasing F. F. Goodnight Goodnight ((1988) showed for 1/3, aa one for additive additive by by additive additive epistasis epistasis that that if if the the ratio ratio of of VAANA VAA/VA > > 1/3, onegeneration generation bottleneck bottleneck will will lead lead to to an an increase increase in in the the additive additive genetic genetic variance. variance. Similar Similar results results have have not not been been developed developed for for other other forms forms of of epistasis. epistasis.
9.5 9.5
METAPOPULATION METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS AND AND POPULATION POPULATION DIFFERENTIATION DIFFERENTIATION Some Some of of the the most most important important effects effects of of gene gene interaction interaction appear appear primarily primarily among demes. Typically, discussions of population population differentiation differentiation focus focus on on the the among demes. Typically, discussions of differentiation means of common formula differentiation of of the the means of the the populations. populations. That That is, is, aa common formula for for the demes (in (in an system) is the genetic genetic variance variance among among demes an additive additive system) is (e.g., (e.g., Hedrick, Hedrick, 2000) Vbet
--
8) (9. (9.8)
2f VA
variance between demes. Because Because the where where Vb Vbeett is is the the variance between demes. the additive additive genetic genetic variance variance deme mean is is aa portion portion of of the the phenotypic phenotypic variance, variance, Vb V b eett is is the the variance variance among among deme mean phenotypes. phenotypes. However, However, the the following following discussion discussion shows shows that that it it is is also also necessary necessary to to consider consider the the differentiation differentiation of of genetic genetic effects. effects. It It also also show show that that differenti differentiation me means. ation for for genetic genetic effects effects may may not not be be related related to to differentiation differentiation of of de deme means. The The increase increase in in additive additive genetic genetic variance variance following following bottlenecks bottlenecks in in systems systems with interaction must come from somewhere. The with gene gene interaction must come from somewhere. The additive additive genetic genetic vari variance earlier in ance was was given given earlier in terms terms of of breeding breeding value; value; however, however, in in aa randomly randomly mat mating population, population, the the breeding breeding value of an an individual is the the sum sum of of the the average average ing value of individual is effects underlying alleles. discussion it effects of of the the underlying alleles. In In the the following following discussion it will will be be conveni convenient allele is ent to to focus focus on on average average effects. effects. The The Fisherian Fisherian average average effect effect of of an an allele is defined population mean defined to to be be the the mean mean deviation deviation from from the the population mean of of individuals individuals that that received allele from parent, with received that that allele from one one parent, with the the allele allele received received from from the the other other parent parent having having come come at at random random from from the the population. population. Using Using this this relationship, relationship, the population can the additive additive genetic genetic variance variance in in aa randomly randomly mating mating population can be be shown shown to (Falconer and 996) to be be (Falconer and MacKay MacKay 11996)
VA- 2 ~
~
pkt~l
(9.9) (9.9)
k = loci l = alleles
where kk refers refers to to the the summation summation over over all all loci loci affecting affecting the the trait, trait, II refers refers to to the the where summation all alleles each of loci, Pkl is lth summation over over all alleles at at each of the the kk loci, is the the frequency frequency of of the the/th effect of of the the khh kith allele. allele kth locus, and fXkl allele at at the the kth locus, and ~kl is is the the average average effect allele. Examination Examination of equation reveals increase the of this this equation reveals that that there there are are only only two two ways ways to to increase the additive additive genetic One is number of genetic variance. variance. One is to to increase increase the the effective effective number of alleles. alleles. Although Although perhaps not increasing the perhaps not obvious obvious from from this this form form of of the the equation, equation, increasing the number number of of
9. 9.
209 209
M ETAPOPULATION QUANTITATIVE METAPOPULATION QUANTITATIVE GENETICS GENETICS
has the alleles mean Pkl has alleles and and decreasing decreasing the the mean the effect effect of of increasing increasing the the additive additive genetic genetic variance. However, genetic genetic drift drift leads to an an increase in F F and, and, on on aver aver variance. However, leads to increase in 1 age, will will decrease the effective effective number number of of alleles (defined as as ne ne =~) ((Crow and age, decrease the alleles (defined Crow and
=�)
Kimura, 970). As Kimura, 11970). As aa result, result, the the increase increase in in additive additive genetic genetic variance variance does does not not come possible way come from from this this source. source. The The other other possible way for for the the additive additive genetic genetic vari variance ance to to increase increase is is for for the the average average effects effects to to change. change. This This shift shift in in the the average average effects alleles is me interaction effects of of alleles is the the cause cause of of the the individual individual by by de deme interaction (Ai•j) (Ai,j) and and causes causes the the increase increase in in the the additive additive genetic genetic variance. variance. Thus, popu Thus, when when the the additive additive genetic genetic variance variance increases increases as as aa result result of of small small population lation size, size, this this is is evidence evidence that that there there is is an an individual individual by by deme deme interaction. interaction. What What this this means means is is that that genes genes have have different different effects effects in in different different demes demes in in aa genetically genetically differentiated differentiated metapopulation. metapopulation. The me interaction The gene gene by by de deme interaction requires requires aa bit bit more more discussion. discussion. In In moving moving to to aa metapopulation metapopulation setting, Fisherian setting, it it is is again again necessary necessary to to distinguish distinguish the the Fisherian average average effect effect from from local local average average effects. effects. Local Local average average effects, effects, like like local local breeding breeding values, values, are are similar similar to to average average effects, effects, but but defined defined separately separately for for each each deme. deme. Thus, Thus, the the local local average average effect effect of of an an allele allele measured measured in in aa deme deme is is the the mean metapopulation mean mean deviation deviation from from the the metapopulation mean of of individuals individuals that that have have the the allele allele, and allele in in question, question, with with the the other other allele, and all all alleles alleles at at other other loci loci having having come come at Goodnight, 2000a,b). at random random from from the the deme deme in in question question ((Goodnight, 2000a,b). In In theory, theory, meas measuring uring this this would would require require substituting substituting an an allele allele into into genotypes genotypes drawn drawn from from the the deme question without loci. While deme in in question without modifyi.ng modifying the the alleles alleles at at any any other other loci. While this this ideal can be ideal is is not not possible possible experimentally, experimentally, two two locus locus local local average average effects effects can be reconstructed data. Unfortunately, randomly mating reconstructed from from QTL QTL data. Unfortunately, whereas whereas in in aa randomly mating panmictic population the panmictic population the breeding breeding value value of of an an individual individual is is aa simple simple sum sum of of the the average average effects effects of of the the underlying underlying alleles, alleles, the the local local breeding breeding value value of of an an indi individual is is not not aa simple simple sum sum of of the the underlying underlying local local average average effects. effects. Factors Factors such such vidual as population com as linkage linkage disequilibrium disequilibrium generated generated by by drift drift in in aa structured structured population complicate this this summation. summation. Interestingly, Interestingly, this this is is an an example example of of how how reductionist reductionist plicate methods methods that that work work well well in in randomly randomly mating mating populations populations often often fail fail when when there there is population structure. is population structure. For purposes of For the the purposes of this this discussion, discussion, local local average average effects effects are are useful useful because because they interaction they help help in in describing describing the the behavior behavior of of the the different different forms forms of of gene gene interaction and data. Using and are are useful useful in in interpreting interpreting quantitative quantitative trait trait loci loci data. Using local local average average effects, mean additive metapopu effects, the the mean additive genetic genetic variance variance within within aa single single deme deme in in aa metapopulation lation becomes becomes
VA == 22 2: ~m VA
(
2: 2: jk/CiTk l 2 ~, PPik,~i2kljj = = demes es k k = = loci l o c i II = = alleles alleles -
( 2: ~ik-)2) k
Cijk .
((9.1o) 9.10)
where where jj iiss the the summation summation ooff demes, demes, kk iiss the the summation summation over over loci, loci, and and II iiss the the summation summation over over alleles alleles at at the the kth kth locus. locus. The The frequency frequency of of the the Ith lth allele allele at at the the kth /, Cijkl kth locus locus in in the the jth jth deme deme is is Pjk Pikl, ~ikl is is the the local local average average effect effect of of the the kith khh locus locus mean local in in the the jth jth deme, deme, and and Cij oqk. is the the deme deme mean local average average effect effect at at the the kth kth locus. locus. k . is As As with with the the local local breeding breeding value, value, the the local local average average effect effect can can be be divided divided into into components components due due to to aa deme deme effect, effect, Cij ogk., an allele allele effect, effect, Ci.kl o~.kl,, and and aa deme deme by by allele allele k ., an interaction, interaction, Cij*kl' cxi.kt,with with the the only only difference difference being being that that effects effects are are locus locus (k) (k) specific. specific. Using mean within Using this this formulation formulation the the mean within deme deme additive additive genetic genetic variance variance becomes becomes
2 2 1100
CHARLES ].J. GOODNIGHT GOODNIGHT
VA -- 2
j=
d~em E E Pjkl(~ es k = loci I=alleles
+ OL2k*l)-
OLjk,
(9.11)
It It is is particularly particularly interesting interesting to to examine examine the the among among demes demes variance variance in in local local average average effects. effects. Conceptually, Conceptually, this this is is the the equivalent equivalent of of measuring measuring the the local local average me and average effect effect of of aa single single allele allele in in each each de deme and measuring measuring the the variance variance in in these allele local local average local aver these within within allele average effects. effects. The The (mean) (mean) variance variance in in the the local average age effect effect of of an an allele allele at at the the kth kth locus locus is is
Var Var(( <Xkl) cxkl) == 22 ~, PPj(~jkl~-kI)) 22 j( <Xjkl - <X_kl
11 =- 22 E Pj( <xrk- + + <xr* kl - <X;kl) 11 Var(~jk. Var(oLj. kl) == Var ( <Xj* kl) (<Xjk_)) ++ Var
(9.12) (9.12)
The The variance variance due due to to the the deme deme effect effect on on the the kth kth locus, locus, Var( Var(ogk.), can be be dis dis<Xjk_), can j* ) tinguished allele by tinguished from from the the variance variance due due to to the the allele by deme deme interaction interaction Var( Var(o9.~1 ) by <X kl by examining examining the the among among demes demes variance variance in in the the mean mean local local average average effects: effects:
Var(o~k. V ar( <Xk_)) = 22 E p pj(/ <Xjj,. ,.,.)' 2 k_ -- <X_k_) ! --
= =
1
:t pj( �
_
Y \~ 2
E PJ( ~ PI<Xjkl - <X_kI ~ }
((9.13) 9.13)
The The allele allele bbyy deme deme interaction interaction can can then then bbee obtained obtained bbyy subtraction subtraction ((Goodnight, Goodnight, 2000a). 2000a). jk_ is Variance Variance in in the the <X 09k. is not not particularly particularly interesting. interesting. The The mean mean local local average average effect allele changes effect is is how how much much on on average average the the mean mean allele changes the the phenotype phenotype of of an an individual. If If aa de deme particularly high high frequency frequency of of alleles alleles conferring conferring aa individual. me has has aa particularly large phenotypic value, mean local local average average effect will be large phenotypic value, the the mean effect will be negative; negative; if if the the deme alleles, the local average deme has has aa low low frequency frequency of of these these alleles, the mean mean local average effect effect will will be be positive. positive. Indeed, Indeed, in in aa system system without without gene gene interaction, interaction, <X ogk. directly propor proporjk_ isis directly tional demes. More tional to to the the phenotypic phenotypic variance variance among among demes. More importantly, importantly, this this com component local ponent of of the the local local average average effects effects does does not not drive drive any any differentiation differentiation the the local average local average average effects effects of of alleles alleles because because the the local average effects effects of of all all alleles alleles at at aa locus will example, con locus will be be affected affected by by the the deme deme effect effect in in the the same same way. way. For For example, consider sider aa locus locus affecting affecting body body weight. weight. If, If, when when measured measured in in one one deme, deme, one one allele allele causes causes the the body body weight weight to to be be 33 g g heavier heavier than than aa second second allele, allele, this this 3-g 3-g differ difference which deme deme they they are are measured. The ence will will be be maintained maintained regardless regardless of of in in which measured. The mean mass of jk_; mean mass of individuals individuals will will vary vary as as aa result result of of the the variation variation in in the the <X o9~.; however, however, the the mean mean difference difference in in weight weight between between individuals individuals carrying carrying different different alleles alleles will will be be aa constant. constant. j*kl', that It It is is the the gene gene by by deme deme interaction, interaction, <X o9.kl that is is of of more more interest. interest. As As with with the the individual me interaction, individual by by de deme interaction, this this value value equals equals zero zero in in the the absence absence of of gene gene interactions. interactions. When When there there is is dominance dominance and and epistasis, epistasis, it it will will generally generally be be nonzero nonzero (Fig. (Fig. 9.2). 9.2). Unlike Unlike the the deme deme effect, effect, this this interaction interaction does does shift shift the the local local average average effects effects of of alleles alleles relative relative to to each each other. other. In In the the example example given given earlier, earlier, if if the the differ difference ence in in local local average average effect effect between between aa pair pair of of alleles alleles in in one one deme deme is is 33 g, g, this this will will
2 11 211
9. METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVEGENETICS GENETICS
not demes. In not be be predictive predictive of of the the difference difference in in other other demes. In aa second second deme deme the the differ difference may may be may be ence be 1I g g or or the the ordering ordering of of local local average average effects effects may be reversed. reversed. To local average To quantify quantify the the extent extent to to which which the the local average effects effects of of alleles alleles vary vary among among demes demes together together due due to to Ujke cxik. compared compared to to the the extent extent to to which which they they vary vary among independently due among demes demes independently due to to Uj*k og,kl,l' Goodnight Goodnight (2000a) (2000a) suggested suggested using using the the intraclass intraclass correlation correlation in in local local average average effects: effects: (9.14) (9.14)
cor(e kl ) = Var(cxk, )
This This intraclass intraclass correlation correlation will will vary vary between between zero zero and and one. one. If If it it is is one, one, this this indicates local average average effects locus are indicates that that the the local effects of of the the alleles alleles at at aa locus are maintaining maintaining their less than their relative relative ranking ranking in in all all demes. demes. If If the the correlation correlation is is less than one, one, it it indi indicates cates that that the the local local average average effects effects are are varying varying among among demes demes relative relative to to each each other. other. The The amount amount it it is is less less than than one one indicates indicates the the extent extent to to which which the the local local average average effects effects of of the the alleles alleles are are varying varying independently independently among among demes. demes. In In Fig. Fig. 9.3, 9.3, the the intraclass intraclass correlation correlation in in local local average average effects effects as as aa function function of of the the inbreeding inbreeding coefficient coefficient is is shown shown for for all all of of the the types types of of genetical genetical effects. effects. The The "additive" 1 ) and "additive" locus locus (A (A locus locus of of additive additive by by dominance dominance interaction, interaction, Table Table 9. 9.1) and the the "dominance" "dominance" locus locus (B (B locus locus of of additive additive by by dominance dominance interaction, interaction, Table Table 9.1 9.1)) o are separately for dominance and dominance by are shown shown separately for additive additive by by dominance and dominance by additive additive epistasis. epistasis. Several Several points points can can be be made made from from Fig. Fig. 9.3. 9.3. First, First, additive additive gene gene action action is is aa special special case case and and the the only only case case in in which which cor(ukl) cor(oLki) is is one. one. This This means means that that regard regardless inbreeding, if allele confers less of of the the level level of of inbreeding, if one one allele confers an an advantage advantage over over aa different different allele deme, it will maintain deme it allele in in one one deme, it will maintain that that advantage advantage regardless regardless of of in in what what deme it
/
1.00
0.75
AXD (Additive locus)
cO
-~ 0.50 L O
o
0.25
~~/./ /
0.00
0
, ~
Dominance D~,,D (Dominlant locus)
0.25 0.5 0.75 Inbreeding Coefficient (F)
1
Fig. Fig. 9.3 9.3 Intraclass Intraclasscorrelation in local local average average effects effects as as a function of inbreeding coefficient for the different forms forms of genetic effects. effects. Additive effects effects (Additive), dominance effects effects (Dominance), (Dominance), additive by additive epistasis epistasis (AXA), (AXA), additive by dominance and dominance by additive epistasis epistasis (AXD), (AXD), and dominance by dominance epistasis epistasis (DXD) are shown. For additive by dominance epistasis, epistasis, additive and dominance loci are listed listed separately. separately. Additive effects effects give a correlation of one, which indicates that the values values of alleles alleles relative to other alleles alleles at the same locus are a constant. All other effects have correlations less less than one (AXA, (AXA, AXD additive locus) or zero (all (all interactions involving dominance). dominance).
21 2122
CHARLES j. GOODNIGHT CHARLES J. GOODNIGHT
is is measured. measured. For For the the case case of of the the "additive" "additive" locus locus of of additive additive by by dominance dominance and and dominance epistasis, the approaches one dominance by by additive additive epistasis, the correlation correlation approaches one as as the the inbreed inbreeding one; however, ing coefficient coefficient (f) (f) approaches approaches one; however, this this is is very very different different from from the the addi additive case where one. Second, all interactions interactions involving tive case where the the correlation correlation is is fixed fixed at at one. Second, all involving dominance, locus of dominance and dominance, i.e., i.e., dominance, dominance, "dominance" "dominance" locus of additive additive by by dominance and dominance by by additive additive epistasis, and dominance dominance by by dominance dominance epistasis, have aa dominance epistasis, and epistasis, have correlation in all inbreeding correlation in local local average average effects effects of of zero zero for for all inbreeding coefficients. coefficients. This This means that local average means that to to the the extent extent that that the the local average effects effects of of alleles alleles vary vary among among demes, Thus, for these pure pure forms gene interaction interaction demes, they they vary vary independently. independently. Thus, for these forms of of gene involving dominance, rank involving dominance, regardless regardless of of the the inbreeding inbreeding coefficient, coefficient, the the relative relative ranking one deme ranking in ing of of alleles alleles in in one deme is is not not predictive predictive of of the the ranking in other other demes. demes. Actual Actual two two locus locus interactions interactions will will involve involve aa mixture mixture of of different different genetical genetical effects. effects. When When interacting interacting quantitative quantitative trait trait loci loci have have been been measured, measured, aa large large frac fraction 995; Goodnight, tion of of the the variation variation is is typically typically additive additive (Cheverud, (Cheverud, 11995; Goodnight, 2000b). 2000b). Thus, Thus, it it is is unlikely unlikely that that in in most most circumstances circumstances the the extreme extreme of of cor(ukl) cor(~t) = = 00 will will often often be be observed observed in in experimental experimental situations. situations. When there there is is dominance dominance or or epistasis, epistasis, the the intraclass correlation in in local When intraclass correlation local average less than average effects effects is is less than one. one. This This indicates indicates that that when when there there is is gene gene inter interaction, populations whereas when action, populations differentiate differentiate for for local local average average effects, effects, whereas when there there are differentiation occurs. are only only additive additive effects, effects, no no differentiation occurs. This This can can be be contrasted contrasted to to the differentiation different forms the differentiation of of population population means means (Fig. (Fig. 9.4) 9.4) for for the the different forms of of pure genetical pure genetical effects. effects. Additive Additive effects effects cannot cannot contribute contribute to to the the differentiation differentiation
4.00
AXA
3.50 3.50 3.00
Q)<.J
.�
Additive
2.50 2.00
>
11.50 .50
"~"'~AXD
11.00 .00
Dominance
0.50 oxo
0.00 0.00
-
0
0.25 0.25
0.5
0.75
Inbreeding Inbreeding Coefficient Coefficient ((F) F)
-
1
demes as Fig. 9.4 9 . 4 The The variance variance in in mean mean phenotype phenotype among among demes as aa function function of of inbreeding inbreeding coeffi coefficient for the different different forms of genetic genetic effects. effects. Additive Additive effects effects (Additive), dominance dominance effects (Dominance), (Dominance), additive by additive epistasis (AXA), additive additive by dominance dominance and dominance dominance by additive epistasis (AXD), and dominance dominance by dominance dominance epistasis (DXD) are shown. Additive Additive effects effects have have aa large large effect effect on on the the phenotypic phenotypic variance variance among among demes, demes, but but no no effect effect in in the the vari variance in local average effects. effects. Conversely, dominance dominance effects and DXD epistasis have little little effect on the the phenotypic the variance in in local on phenotypic variance among among demes, but but a much much larger effect on the average (Fig. 9.2). average effects effects (Fig. 9.2).
2 1133 2
9. METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVE GENETICS GENETICS
ooff average average effects, effects, but but can can contribute contribute to to the the differentiation differentiation ooff population population means, whereas whereas interactions interactions involving involving dominance dominance have have aa large large effect effect on on the the means, differentiation differentiation of of average average effects, effects, but but little little effect effect on on the the differentiation differentiation of of popu population means. means. lation
9.6 9.6
MIGRATION MIGRATION The The effects effects of of migration migration on on quantitative quantitative genetics genetics parameters parameters in in metapopu metapopulations have not been well studied. Whitlock et al. ( 1 993) used a descent lations have not been well studied. Whitlock et al. (1993) used a descent meas measure ure model model to to show show that that with with additive additive by by additive additive epistasis epistasis the the highest highest level level of of additive additive genetic genetic variance variance is is attained attained at at an an intermediate intermediate level level of of migration. migration. The The relationship relationship between between island island model model migration migration and and inbreeding inbreeding coefficient coefficient can be concerning the migration on on can be used used to to make make some some observations observations concerning the effects effects of of migration metapopulation quantitative genetic Following Hedrick Hedrick (2000) metapopulation quantitative genetic measures. measures. Following (2000) under an an infinite infinite alleles alleles model model in in aa meta metapopulation with an an infinite infinite number number of of under population with demes, migra demes, but but aa small small population population size size within within each each deme deme and and island island model model migra(F or, tion demes, the inbreeding coefficient tion among among demes, the change change in in inbreeding coefficient (F or, in in this this context, context, more more correctly correctly FST) Fsy) is is given given by by
1+(1 )Ft] l m,2 1
F(t+l ) = [~--~
((9.15) 9.15)
where em e population where N N iiss the the within within ddeme population size size and and m m iiss the the migration migration rate. rate. A A setting reasonable value of reasonable approximation approximation for for the the equilibrium equilibrium value of F F is is found found by by setting F(t F(t+l F(t ) and and ignoring ignoring terms terms on on the the order order of of m m 22.• Using Using this this approximation, approximation, + l)) == F(t) it it can can be be shown shown that that (Hedrick, (Hedrick, 2000) 2000) 11
/~
F= 1\
4Nm + + 11 4Nm
----
((9.16) 9. 1 6)
This result that migrants (Nm) not This emphasizes emphasizes the the classical classical result that it it is is the the number number of of migrants (Nm) not determines population the per-capita migration migration rate the per-capita rate (m) (m) that that determines population differentiation. differentiation. Solving Eq. Eq. ((9.16) for Nm N m gives gives Solving 9 . 1 6 ) for Nm = ~ Nm
11 - F F 4F 4F -
--
((9.17) 9 . 1 7)
Figure 9.5 additive genetic genetic variance Figure 9.5 is is aa graph graph of of the the additive variance (relative (relative to to aa variance variance of population) as of one one in in the the panmictic panmictic population) as aa function function of of the the number number of of migrants migrants per per deme .1. deme for for each each of of the the different different pure pure forms forms of of genetical genetical effects effects listed listed in in Table Table 99.1. IInn agreement agreement with with intuition, intuition, when when there there are are only only additive additive effects, effects, the the additive additive genetic number of genetic variance variance increases increases as as aa function function of of the the number of migrants migrants and and is is max maximum dominance to imum when when mixing mixing is is complete. complete. The The conversion conversion of of dominance to additive additive genetic maximal at genetic variance variance is is maximal at aa migration migration rate rate of of slightly slightly over over one one migrant migrant epistasis, the conversion of every every other other generation generation (Nm (Nm = = 0.575). 0.575). For For digenic digenic epistasis, the conversion of epistatic epistatic variance variance to to additive additive variance variance is is maximum maximum at at even even lower lower migration migration rates of of between between one one migrant migrant every every four four generations generations and and one one migrant migrant every every six six rates
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generations . 1 75). Nevertheless, generations (Nm (Nm = 0.25 0 . 2 5 -- 00.175). Nevertheless, for for several several forms forms of of gene gene interaction, interaction, notably notably dominance dominance and and additive additive by by additive additive epistasis, epistasis, the the conver conversion substantial, even sion of of nonadditive nonadditive variance variance to to additive additive variance variance can can be be substantial, even with with migration rates rates as as high high as as three three or or more more individuals individuals per per generation. generation. migration It also interesting It is is also interesting to to examine examine population population differentiation differentiation for for average average effects effects as as aa function function of of migration migration rate. rate. Figure Figure 9.6 9.6 shows shows the the variance variance in in local local average average effects effects as as aa function function of of number number of of migrants migrants per per deme deme for for dominance dominance and and each each of of the the different different forms forms of of epistasis epistasis (for (for additive additive effects, effects, this this value value is is zero zero for for all all migration migration rates) rates).. For For the the additive additive by by dominance dominance (and (and dominance dominance by by additive) additive) interaction, interaction, the the "additive" "additive" locus locus (A (A locus locus of of additive additive by by domin dominance 1 ) and B locus ance interaction, interaction, Table Table 9. 9.1) and the the "dominant" "dominant" locus locus ((B locus of of additive additive by 1 ) are by dominance dominance interaction, interaction, Table Table 9. 9.1) are plotted plotted separately. separately. From From Fig. Fig. 9.6 9.6 it it is is apparent apparent that that the the differentiation differentiation of of local local average average effects effects is is very very sensitive sensitive to to the the migration migration rate. rate. For For all all effects effects except except the the additive additive by by dominance dominance additive additive the differentiation differentiation of of local local average average effects effects declines declines with with migration migration rate, rate, locus, the and and with with the the exceptions exceptions of of dominance dominance and and additive additive by by additive additive epistasis, epistasis, the the differentiation differentiation for for average average effects effects is is very very low low with with as as little little as as one one migrant migrant per per generation. generation.
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Fig. Fig. 9.6 9 . 6 Variance Variancedue to the allele by deme interaction, Var(a!'k/), Var(otr, kl), as a function function of number number of migrants per per deme per generation. The variance in in local average for additive effects is average effects for zero for for all values values of Nm. Nm. Dominance Dominance effects (Dominance), (Dominance), additive by additive epistasis (AXA), additive additive by by dominance dominance and and dominance dominance by by additive additive epistasis epistasis (AXD), (AXD), and and dominance dominance by by domi dominance nance epistasis epistasis (DXD) (DXD) are are shown. shown. For For additive additive by by dominance dominance epistasis, epistasis, additive additive and and dominant dominant loci are listed separately. The variance in local average effects decreases decreases rapidly rapidly as a function function of the is very very small is less the number number of of migrants migrants and and is small unless unless the the number number of of migrants migrants per per deme deme is less than than one per generation.
It It is is important important to to note, note, however, however, that that these these calculations calculations assume assume that that there there is is no selection acting. potentially lower no selection acting. Selection Selection can can potentially lower the the effective effective migration migration rate. rate. Consider the Consider the situation situation of of aa metapopulation metapopulation with with aa migration migration rate rate of of three three indi individuals epistasis. In viduals per per generation generation and and considerable considerable additive additive by by additive additive epistasis. In this this case, the variance would would be elevated by conversion of epi case, the additive additive genetic genetic variance be elevated by the the conversion of epistasis variance (Fig. stasis to to additive additive genetic genetic variance (Fig. 9.5) 9.5) and and the the response response to to selection selection would would reflect reflect this. this. In In addition, addition, there there would would be be some some small small differentiation differentiation of of average average effects effects due due to to the the additive additive by by additive additive epistasis epistasis (Fig. (Fig. 9.6). 9.6). This This indi indicates cates that that to to some some degree, degree, directional directional selection selection would would be be driving driving aa differentia differentiation tion of of average average effects effects as as it it acted acted on on the the gene gene interaction. interaction. This This potentially potentially leads leads to Goodnight, 2003) to aa lowering lowering in in fitness fitness of of the the offspring offspring of of migrants migrants ((Goodnight, 2003),, which which in potential exists pos in turn turn would would lower lower the the effective effective migration migration rate. rate. The The potential exists for for aa posinteraction among among selection, migration, itive feedback to develop. Thus, the interaction drift may generate much more population population differentiation than predicted predicted and drift
216 216
CHARLES I.] . GOODNIGHT CHARLES
by the the apparent apparent migration migration rate rate alone. alone. This process has has not not been been explored explored and and by This process deserves more more attention. attention. deserves
9.7 9.7
EMPIRICAL EXAMPLE: EXAMPLE: THE THE METAPOPULATION M ETAPOPULATION QUANTITATIVE QUANTITATIVE EMPIRICAL GENETICS OF OF A A TEOSINTE-MAIZE TEOSINTE-MAIZE CROSS CROSS GENETICS Doebley et et al. a!. (1995) ( 1 995) identified identified quantitative quantitative trait loci for several traits traits in in aa Doebley trait loci for several wide cross cross between between teosinte teosinte (Zea (Zea mays mays ssp. ssp. parviglumis) parvig/umis) and and cultivated cultivated maize maize wide Zea mays mays ssp. ssp. mays). mays). The The details details of of the the mapping mapping procedure procedure are are ( "corn," Zea ("corn," described in in Doebley Doebley et et al. a!. (1995; ( 1 995; see see also also Doebley Doebley and and Stec, Stec, 1993). 1 993). Several Several described pairs of of QTL QTL regions regions were were shown shown to to interact interact epistatically. epistatically. One One pair pair of of epistaepista pairs tically interacting markers is is BV302 BV302 and and UMC107, UMC1 07, which which are are located located on on difdif tically interacting markers ferent chromosomes. chromosomes. One One trait trait these these loci loci affect affect is is "PEDS," "PEDS," the the percentage percentage of of ferent cupules with with the the pedicellate pedicellate (maize (maize like) like) spikelet spikelet (for (for aa more more complete complete descripdescrip cupules tion of trait, see see Doebley et a!., 1 995). tion of this this trait, Doebley et al., 1995). The PEDS for the different different two two locus locus genotypes genotypes are The genotypic genotypic values values of of PEDS for the are shown in Table 9.2. The values are are simply mean phenotypes of shown in Table 9.2. The genotypic genotypic values simply the the mean phenotypes of those individuals genotype of interest. Using Using the the regression regression procedure procedure those individuals with with the the genotype of interest. described into components components for for this this described earlier, earlier, the the partitioning partitioning of of genetic genetic variance variance into trait is three gene 9.3. This partitioning trait is shown shown for for three gene frequencies frequencies in in Table Table 9.3. This partitioning includes total genetic variance and percentage contributions contributions due includes total variance and due to to additive effects, epistasis. At three gene gene frequencies frequencies effects, dominance dominance effects, effects, and and digenic digenic epistasis. At all all three the mainly attributable attributable to to additive locus domdom the genetical genetical effects effects are are mainly additive and and single single locus inance effects. However, in in all epistasis accounts accounts for for aa substansubstan inance effects. However, all cases, cases, digenic digenic epistasis tial between 20.9 tial proportion proportion ((between 20.9 and and 39.4%) 39.4%) of of the the total total genetic genetic variance. variance. This This is variance due is reflected reflected in in Fig. Fig. 9.7, 9.7, which which is is aa graph graph of of additive additive genetic genetic variance due to to the the BV302 BV302 QTL QTL as as aa function function of of the the frequencies frequencies of of the the BV302 BV302 QTL QTL and and the the UMC1 07 QTL. 07 locus, UMC107 QTL. Within Within each each gene gene frequency frequency of of the the UMC1 UMC107 locus, the the addi additive variance of locus varies varies in manner consistent tive variance of the the BV302 BV302 locus in aa manner consistent with with aa standard standard locus 996); however, shape locus with with dominance dominance (e.g., (e.g., Falconer Falconer and and Mackay, Mackay, 11996); however, the the shape
TABLE TABLE 9.2 9 . 2 Genotypic Genotypic Values Values for PEDS, PEDS, the Percentage of Cupules with with the Pedicellate Pedicellate (Maize-like) Spikelet for aa Cross between Teosinte (T) (T) and Maize (M)Q (M) a UMCI07 UMC107 BV302
T/T TIT
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Note Note that that crosses crosses were were done done with with teosinte teosinte as as the the seed seed parent. parent. The The teosinte teosinte cytoplasm cytoplasm and and random random sampling sampling of of genes genes at at other other loci loci affecting affecting this this trait trait are are likely likely responsible responsible for for the the low low proportion proportion of of maize-like maize-like cupules cupules even even when when both both loci loci are are from from maize maize (data (data from from Doebley Doebley et et a!., al., 1995). 199.5).
9. 9.
221 1 17
METAPOPULATION QUANTITATIVE QUANTITATIVE GENETICS GENETICS METAPOPULATION
TABLE 9.3 9.3 Decomposition Decomposition into Genetic Variance Components Components for for a Pair of of Interacting Interacting TABLE Loci (UMC107 (UMCl 07 and BV302) Affecting Affecting the the Percentage Percentage of of Cupules Lacking the the Pedicellate (Maize-like) Spikelet.
Effect E ffect Additive Additive (UMC107) (UMC107) Additive Additive
(BV302) (BV302) Dominance Dominance (UMC107) (UMC107) Dominance Dominance
(BV302) (By302) AXA AXA AXD AXD DXA DXA DXD DXD Total genetic Total genetic vanance variance (percentage (percentage due to due to epistasis) epistasis)
aa
FFreq(BV302teo.) r e q ( B V 3 0 2 t e o . ) == 00.25 .25 = 00.25 Freq(UMC1 F r e q ( U M C 1 0 707teo.) teo.) = .25
FFreq(BV302teo.) r e q ( B V 3 0 2 t e o . ) == 00..55 FFreq(UMC107teo.) r e q ( U M C 1 0 7 t e o . ) -= 00.5 .5
FFreq(BV302teo.) r e q ( B V 3 0 2 t e o . ) == 00.75 .75 = 00.75 Freq(UMC1 F r e q ( U M C 1 0 707teo.) teo.) = .75
3.7209 (38.6%) (38.6%) 3.7209
0.6328 (21.6%) (21.6%) 0.6328
0.0258 (8.9%) (8.9%) 0.0258
2.7719 2.7719 (28.7%) (28.7%)
0.55 13 (18.8%) ( 1 8.8%) 0.5513
0.0697 (24.1%) (24.1 %) 0.0697
0.8016 (8.3%) (8.3%) 0.8016
0.4556 (15.5%) (15.5%) 0.4556
0.0665 (23.0%) (23.0% ) 0.0665
0.3433 (3.5%) (3.5%) 0.3433
0.1406 (4.8%) (4.8%) 0.1406
0.0137 (4.7%) (4.7%) 0.0137
1.4400 (14.9%) (14.9%) 1.4400 0.3 165 (3.3%) (3.3%) 0.3165 0.2109 0.2109 (2.2%) (2.2%) 0.0445 0.0445 (0.5%) (0.5%) 9.6501 9.6501 (20.9%) (20.9%)
0.4556 (15.5%) 0.4556 (15.5%) 0.3613 (12.3%) (12.3%) 0.3613 0.1953 (6.7%) 0.1953 (6.7%) 0.1406 (4.8%) 0.1406 (4.8%) 2.9331 %) 2.9331 (39.3 (39.3%)
0.0089 0.0089 0.0476 0.0476 0.0130 0.0130 0.0445 0.0445 0.2897 0.2897
(3.1 %) (3.1%) ( 1 6.4%) (16.4%) (4.5%) (4.5%) (15.4%) (15.4%) (39.4%) (39.4%)
Numbers iinn parentheses parentheses are percentage of of the the total total genetic genetic variance variance due the component. component. Total Total genetic genetic Numbers are the the percentage due ttoo the variance and percentage due to digenic digenic epistasis epistasis are also listed. listed. variance and percentage due to are also
of changes dramatically gene frequency frequency at 07 QTL of the the curve curve changes dramatically as as the the gene at the the UMCI UMC107 QTL changes. graph of total variance digenic epistasis changes. Figure Figure 9.8 9.8 is is aa graph of the the total variance in in digenic epistasis in in the the population. Note that greatest at gene fre population. Note that epistatic epistatic variance variance is is greatest at intermediate intermediate gene frequencies. and its quencies. Fixation Fixation of of either either locus locus leads leads to to aa loss loss of of epistasis epistasis and its conversion conversion to to additive additive and and dominance dominance variance variance at at the the locus locus that that is is still still segregating. segregating. When metapopulation undergoes When aa metapopulation undergoes genetic genetic drift, drift, gene gene frequencies frequencies at at both both loci loci in in the the different different demes demes will will change change randomly. randomly. The The expected expected distribution distribution of of gene gene frequencies frequencies can can be be described described using using aa Markov Markov chain chain and and Wright's Wright's inbreed inbreeding ing coefficient coefficient (F). (F). Figure Figure 9.9 9.9 is is aa graph graph of of the the effect effect of of drift drift (measured (measured by by F) F) on on mean mean additive additive genetic genetic variance variance and and the the variance variance in in local local average average effects effects (corrected Also shown (corrected for for demic demic effects) effects) for for the the BV302 BV302 QTL. QTL. Also shown for for comparison comparison is is the the variance variance in in deme deme means. means. Not Not that that due due to to epistatic epistatic interactions interactions with with UMCI07, maximum value UMC107, the the additive additive genetic genetic variance variance has has aa maximum value at at an an inter intermediate value of mediate value of F. F. The The early early increase increase in in additive additive genetic genetic variance variance is is due due to to aa conversion conversion of of epistatic epistatic variance variance to to additive additive variance. variance. At At higher higher values values of of F F the the effects effects of of fixation fixation within within demes demes at at both both loci loci overwhelm overwhelm the the conversion conversion of of epi epistasis stasis to to additive additive variance variance and and the the overall overall additive additive genetic genetic variance variance declines declines until until it it reaches reaches zero zero when when F F equals equals one. one. Along Along with with this this increase increase in in additive additive genetic local average genetic variance variance there there is is also also an an increase increase in in the the variance variance in in the the local average effects effects of of the the BV302 BV302 alleles. alleles. In In the the absence absence of of gene gene interaction, interaction, this this variance variance in local local average average effects effects would would be be zero. zero. The The variance variance in in local local average average effects effects in indicates population would indicates that that BV302 BV302 alleles alleles in in aa teosinte-maize teosinte-maize meta metapopulation would have have different different effects effects on on the the PEDS PEDS phenotype phenotype in in different different demes. demes.
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The locus (corrected deme mean) The local local average average effects effects for for the the BV302 BV302 locus (corrected for for deme mean) are are shown Fig. 9.10 shown in in Fig. 9.10 as as aa function function of of the the gene gene frequencies frequencies at at the the two two loci. loci. In In this this par particular case, maize ticular case, maize genes genes at at this this locus locus always always code code for for aa more more maize-like maize-like pheno phenotype; type; thus, thus, the the interaction interaction is is expressed expressed as as aa change change in in scale scale rather rather than than aa reversal reversal in in sign. sign. This This makes makes sense sense given given that that this this interaction interaction has has aa substantial substantial additive additive
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Inbreeding Inbreeding Coefficient Coefficient (F) (F) Fig. Fig. 9.9 9 . 9 Among Among deme deme phenotypic phenotypic variance variance (Deme (Deme Mean), Mean), variance variance in in local local average average effects effects for the BV302 for the BV302 locus locus corrected corrected for for demic demic effects effects (BV302), (BV302), and and mean mean within within deme deme additive additive genetic genetic variance variance (Va) (Va) as as aa function function of of an an inbreeding inbreeding coefficient. coefficient. The The additive additive genetic genetic variance variance shown shown is is the 07 locus. Maximum additive the sum sum of of the the effects effects for for the the BV302 BV302 locus locus and and the the UMC1 UMC107 locus. Maximum additive genetic genetic variance variance occurs occurs at at an an intermediate intermediate inbreeding inbreeding coefficient, coefficient, indicating indicating that that aa conversion conversion of of non nonadditive additive variance variance into into additive additive variance variance is is occurring. occurring. This This is is reflected reflected in in the the variance variance among among demes, demes, which which is is much much greater greater than than expected expected from from an an additive additive model. model. A A maximum maximum phenotypic phenotypic variance among demes under an model. It variance among demes of of 4.74 4.74 is is possible possible under an additive additive model. It is is also also reflected reflected in in the the vari variance which would ance in in local local average average effects effects for for the the BV302 BV302 locus, locus, which would be be zero zero in in an an additive additive model. model.
component component at at all all gene gene frequencies frequencies (Table (Table 9.3). 9.3). Nevertheless, Nevertheless, it it is is quite quite apparent apparent that 07 locus that the the UMC1 UMC107 locus has has aa very very large large effect effect on on the the average average effects effects of of BV302 BV302 alleles. alleles. The The consequences consequences of of this this can can be be seen seen by by considering considering the the difference difference between between the BV302 alleles alleles in the two two BV302 in demes demes with with different different frequencies frequencies of of the the UMCI07 UMC107 alleles. alleles. In In those those demes demes with with low low frequencies frequencies of of the the UMC107 UMC107 teosinte teosinte allele allele (and (and therefore therefore high high frequencies frequencies of of the the maize maize allele), allele), differences differences between between the the two two BV302 BV302 alleles alleles are are pronounced. pronounced. Selection Selection acting acting on on PEDS PEDS would would effectively effectively distinguish distinguish between between the the two two alleles. alleles. However, However, in in demes demes with with high high frequencies frequencies of of the the UMCI07, UMC107, tesinte tesinte allele allele selection selection would would be be much much less less effective, effective, as as the the difference difference between between the the alleles alleles is is much much smaller. smaller. As As an an aside, aside, the the teosinte teosinte ancestor ancestor of of corn corn may may have have had had genetical genetical effects effects at at the the BV302 BV302 locus locus similar similar to to the the back back corner corner in in Fig. Fig. 9.10. 9.10. For For this this set set of of genotype genotype frequencies, frequencies, there there is is almost almost no no difference difference between between the the two two alleles, alleles, and and the BV302 "maize" been nearly the BV302 "maize" gene gene would would have have been nearly neutral neutral with with respect respect to to the the PEDS PEDS phenotype. phenotype. As As the the domestication domestication of of teosinte teosinte progressed progressed and and it it became became more more maize maize like, like, the the frequency frequency of of the the UMC107 UMC107 maize maize allele allele would would have have presumably presumably increased. increased. This BV302 This in in turn turn would would have have acted acted to to magnify magnify the the differences differences between between the the BV302 alleles. alleles. Thus, Thus, this this is is an an interesting interesting case case where where selection selection converts converts formerly formerly neutral neutral variation variation into into large large differences differences that that can can respond respond to to selection. selection. In In this this case, case, selection, selection, rather rather than than using using up up additive additive genetic genetic variance, variance, generates generates new new additive additive genetic genetic vari variance. ance. These These results results also also suggest suggest why why in in the the past past there there was was considerable considerable debate debate over over the the origins origins of of maize maize (Beadle, (Beadle, 1980). 1980). Genes Genes coding coding for for aa "maize" "maize" phenotype phenotype are are nearly nearly neutral neutral in in aa teosinte teosinte genetic genetic background, background, and and the the pathway pathway for for selecting selecting maize maize from from teosinte teosinte is is not not clear. clear.
220 220
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9.8 9.8
GENE GENE INTERACTION I N T E R A C T I O N AND A N D SPECIATION SPECIATION Although Although there there is is aa substantial substantial literature literature on on specIatIOn speciation (reviewed (reviewed in in Templeton, 9 8 1 ; Turelli 1; Kondrashov, Templeton, 11981; Turelli et et ai., al., 2001; 2001; Schluter, Schluter, 200 2001; Kondrashov, 2001; 2001; Wu, 1 ), aa general Wu, 200 2001), general understanding understanding of of the the relationship relationship between between quantitative quantitative genetics speciation has elusive (Hedrick, genetics or or population population genetics genetics and and speciation has been been elusive (Hedrick, 2000) possible explanation explanation for 2000).. One One possible for this this is is that that although although the the importance importance of of gene interaction 963; interaction in speciation speciation is acknowledged acknowledged (Muller, 1939; Mayr, 11963; Templeton, 9 8 1 ; Futuyma, 986; Orr, 11995), 995), the majority Templeton, 11981; Futuyma, 11986; majority of models that that can be be applied applied to to the the study study of of speciation speciation assume assume only only additive additive gene gene action. action. For For example, in his review article, Wu Wu (20 (2001) "speciation genes," example, 0 1 ) focused on "speciation which which he he identified identified as as genes genes responsible responsible for for differential differential adaptation adaptation primarily primarily to ecological or environment. The to the the ecological or sexual sexual environment. The effect effect of of gene gene interaction interaction on on the the shift in the local average effects of alleles is a potential potential genetical mechanism mechanism for for speciation speciation that that does does not not depend depend on on shifts shifts in in the the ecological ecological or or sexual sexual envir environment. This is not 1 939; Orr, 11995) 995) model model onment. not a new idea; for for example, example, Muller's ((1939;
9. 9.
METAPOPULATION METAPOPULATIONQUANTITATIVE QUANTITATIVE GENETICS GENETICS
221 221
provided provided aa qualitative qualitative model model of of how how speciation speciation would would occur occur through through the the accu accumulation mulation of of mutations mutations that that caused caused reproductive reproductive incompatibility incompatibility between between two two populations. However, lack aa mechanism populations. However, these these models models lack mechanism other other than than random random mutation mutation for for the the accumulation accumulation of of incompatible incompatible mutations. mutations. The The model model dis discussed possible mechanism cussed in in this this chapter chapter suggests suggests that that aa possible mechanism for for speciation speciation is is that that genetic genetic drift drift generates generates small small differences differences between between demes demes and and directional directional selec selection tion works works to to magnify magnify these these differences differences to to the the point point that that the the two two populations populations become become reproductively reproductively isolated. isolated. Importantly, Importantly, directional directional selection selection may may be be uniform uniform at at the the phenotypic phenotypic level, level, but but nevertheless nevertheless diversifying diversifying at at the the genetic genetic level. That because of level. That is, is, although although selection selection may may favor favor the the same same phenotype phenotype because of the the different alleles would different genetic genetic backgrounds, backgrounds, different different alleles would be be favored favored in in the the two two populations. populations. To To see see how how genetic genetic drift drift coupled coupled with with directional directional selection selection can can lead lead to to spe speciation, .1; ciation, consider consider the the example example of of dominance dominance by by additive additive epistasis epistasis (Table (Table 99.1; Fig. . 1 1 a; Goodnight, Fig. 99.11a; Goodnight, 2000b). 2000b). With With dominance dominance bbyy additive additive epistasis, epistasis, the the A A locus is locus is overdominant, overdominant, neutral, neutral, or or underdominant, underdominant, depending depending on on the the genotype genotype at allele being at the the B B locus, locus, and and the the B B locus locus is is additive additive with with the the favored favored allele being depend dependent locus. ent on on the the genotype genotype at at the the A A locus. Consider population segregating locus, but Consider aa meta metapopulation segregating for for the the A A locus, but fixed fixed for for the the Bz B2 allele. . 1 1 b. In allele. This This situation situation corresponds corresponds to to the the bottom bottom row row in in Fig. Fig. 99.11b. In this this situ situation, ation, the the A A locus locus is is exhibiting exhibiting simple simple overdominance overdominance with with no no apparent apparent epis epistasis. tasis. Stabilizing Stabilizing selection selection will will tend tend to to drive drive the the gene gene frequency frequency at at the the A A locus locus to 0.5 for to approximately approximately 0.5 for both both alleles. alleles. Note, Note, however, however, that that because because demes demes within populations are within the the meta metapopulations are finite finite and and likely likely small, small, there there will will be be deviations deviations from this equilibrium gene frequency. from introduced into a deme mutation or by migration If a B1 Bl allele is introduced deme either by mutation it it will will be be neutral neutral provided provided that that the the gene gene frequency frequency at at the the A A locus locus is is exactly exactly 0.5 0.5 (Fig. 99.11c, middle column). column). Any Any deviations deviations from from aa gene gene frequency frequency of of 0.5 0.5 at at (Fig. . 1 1c, middle the locus will result in Bl allele . 1 1 c, the A A locus will result in directional directional selection selection favoring favoring the the B1 allele (Fig. (Fig. 99.11c, left resulting increase frequency of left and and right right columns). columns). The The resulting increase in in the the frequency of the the Bl B1 allele allele will will weaken weaken the the strength strength of of stabilizing stabilizing selection selection on on the the A A locus, locus, resulting resulting in in neutrality (Fig. 9. 1 1 b, mid neutrality and and ultimately ultimately disruptive disruptive selection selection at at the the A A locus locus (Fig. 9.11b, middle dle and and top top rows). rows). This This is is aa positive positive feedback feedback system, system, wherein wherein genetic genetic drift drift at at the the A A locus locus in weak weak directional directional selection selection at at the the B B locus. locus. This This directional directional selection selection results in on on the the B B locus locus has has the the effect effect of of weakening weakening the the strength strength of of stabilizing stabilizing selection selection at at the the A A locus, locus, which which in in turn, turn, when when coupled coupled with with genetic genetic drift, drift, will will increase increase the the strength locus. Once strength of of directional directional selection selection on on the the B B locus. Once the the frequency frequency of of the the Bl B1 allele allele exceeds exceeds 0.5, 0.5, the the A A locus locus will will experience experience disruptive disruptive selection selection and and be be fixed fixed quickly Al or quickly for for either either the the A1 or the the Az A2 allele. allele. If population some If this this process process occurs occurs in in several several demes demes within within the the meta metapopulation some of of the the demes demes will, will, by by random random chance, chance, become become fixed fixed for for the the Al A1 allele, allele, whereas whereas others others will will become become fixed fixed for for the the Az A2 allele. allele. If If the the strength strength of of destabilizing destabilizing selec selection tion is is strong strong enough, enough, this this could could be be sufficient sufficient to to cause cause reproductive reproductive isolation isolation and, and, as as aa consequence, consequence, speciation. speciation. Prior allele, there Prior to to the the introduction introduction of of aa Bl B1 allele, there would would be be no no reason reason to to con consider sider the the A A locus locus part part of of an an epistatic epistatic interaction, interaction, nor nor any any reason reason to to consider consider it it as Indeed, the as aa candidate candidate for for aa locus locus that that could could drive drive speciation. speciation. Indeed, the A A locus locus would would appear appear to to be be aa simple simple overdominant overdominant locus locus maintained maintained by by stabilizing stabilizing
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CHARLES J.J. GOODNIGHT CHARLES GOODNIGHT a
A, A, A, A2
A2 A2
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0
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Fig. 9.1 9.111 The The potential potential role role of of dominance dominance by by additive additive epistasis epistasis in in speciation. speciation. (a) (a) Genotypic Genotypic values for dominance epistasis. Gray arrows indicate the change values dominance by additive epistasis. change from from stabilizing selection selection to to disruptive disruptive selection selection at at the the A A locus locus that that occurs occurs as as the the frequency frequency of of the the B, B1 allele allele changes from from zero to to one. (b) Genotypic values for for the three three A locus genotypes when when the fre frequency of the B1 B, allele is 0, 0.5, and and 11.. When When the B, B1 allele is rare, there is stabilizing selection at common, there locus. at the the A A locus, locus, whereas whereas when when the the B, B1 allele allele is is common, there is is disruptive disruptive selection selection at at the the A A locus. (c) Genotypic values for the three BB locus genotypes when the frequency frequency of the A, A1 allele is 0, 0.5, and 0.5, the 0, 0.5, and 11.. At At aa frequency frequency at at the the A A locus locus of of 0.5, the BB locus locus is is neutral. neutral. However, However, if if frequency frequency at locus drifts locus will will be selection favoring at the the A A locus drifts from from 0.5, 0.5, then then the the BB locus be under under directional directional selection favoring the the B, allele. When B1 allele. When the B, B1 allele is rare, genetic drift at the A locus will interact interact with with directional selec selection at the BB locus, eventually leading to fixation and either the A, fixation of the B, B1 allele and A1 or the A2 allele. allele. Redrawn Redrawn from from Goodnight Goodnight (2000a). (2000a).
selection. population selection. Similarly, Similarly, at at the the end end of of the the divergence divergence process, process, the the meta metapopulation will Bl allele, allele, and will be be fixed fixed for for the the B1 and again again the the A A locus locus will will appear appear to to be be aa simple simple underdominant underdominant locus locus with with no no evidence evidence that that it it is is part part of of an an epistatic epistatic inter interaction. just by examining the action. Thus, Thus, just by examining the end end points points of of this this process, process, there there would would be be little little indication indication that that an an epistatically epistatically interacting interacting B B locus locus was was involved involved in in the the speciation speciation process. process. Finally, Finally, note note that that this this process process involves involves an an interaction interaction between between the the random random process process of of genetic genetic drift drift and and the the deterministic deterministic process process of of directional directional selection. selection. Random Bl allele allele is Random drift drift begins begins the the process process (the (the B1 is neutral neutral at at aa gene gene frequency frequency of of 0.5 0.5 at at the the A A locus) locus),, and and directional directional selection selection enhances enhances the the power power of of drift drift and greatly accelerates one of and greatly accelerates the the drive drive to to the the fixation fixation of of one of the the two two alleles alleles at at the the A A locus. locus.
METAPOPULATIONQUANTITATIVE QUANTITATIVE GENETICS GENETICS 9. METAPOPULATION
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SUMMARY S U M M A R Y AND A N D CONCLUSIONS CONCLUSIONS Fisher Fisher developed developed quantitative quantitative genetics genetics to to address address specific specific questions questions about about evolution within a single population. population. Indeed, Indeed, quantitative quantitative genetics in its original formulation solely with describing the process of adaptation formulation was concerned concerned solely adaptation to directional selection selection acting within populations. population populations. The goals of meta metapopulation quantitative quantitative genetics genetics are are different different from from those those of of Fisherian Fisherian quantitative quantitative genetics. genetics. Metapopulation Metapopulation quantitative quantitative genetics primarily primarily provides provides measures of popu population differentiation differentiation rather rather than than focusing on the response response to selection that that is central central to to Fisherian Fisherian quantitative quantitative genetics. genetics. One One of of the the central central observations observations of of meta population quantitative metapopulation quantitative genetics genetics is is that that populations populations can can be be differentiated differentiated both both for for population population means means and and for for local local average average effects. effects. The The first first is is well well known known from from studies studies of of genetic genetic drift drift in in additive additive systems. systems. The The second, second, while while it it has 963), had has been been observed observed qualitatively qualitatively (e.g., (e.g., Dempster, Dempster, 11963), had not not been been quanti quantified populations for local average fied in in the the past. past. The The differentiation differentiation of of populations for local average effects effects is is of of particular particular interest interest because because it it is is aa measure measure of of what what alleles alleles do do in in different different demes. This form form of population population differentiation, differentiation, unlike differentiation differentiation of popu population lation means, means, is is directly directly related related to to reproductive reproductive isolation isolation and and speciation. speciation. However, However, it it need need not not be be related related to to the the differentiation differentiation of of population population means. means. For For example, additive additive effects cause differentiation differentiation of population population means but no dif difno dif ferentiation average effects, fixation (E ferentiation of of local local average effects, whereas whereas at at fixation there is is no dif(F == 11)) there ferentiation ferentiation of population population means means due to dominance, dominance, but but the populations populations are strongly differentiated differentiated for local average effects. Population Population genetics and Fisherian genetics have been remarkably successful for Fisherian quantitative quantitative genetics have been remarkably successful for developing developing our populations, but our understanding understanding of of evolution evolution within within populations, but interestingly, interestingly, these these dis disciplines have also been remarkably remarkably unsuccessful unsuccessful at developing our our under understanding population changes lead to speciation standing of how how these within within population speciation and and evolution evolution above the species level. By providing providing new measures of population population differentiation, population quantitive differentiation, meta metapopulation quantitive genetics genetics may may shed shed light light on on this this important important subject.
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Part IV Evolutionary Dynamics in Metapopulations
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LIFE LI FE H ISTORY EVO LUTION IN EVOLUTION IN META PO PUL ATIONS M ETAPO PU LATI ONS Ophelie Oph~lie Ronce and Isabelle IsabeUe Olivieri
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INTRODUCTION INTRODUCTION Life history theory deals with with the the evolution of those those traits that that shape shape an organism's age schedules of birth and death Calow, 1998). Many death ((Calow, Many biological traits potentially potentially affect the patterns patterns of reproduction reproduction and mortality throughout throughout traits the life cycle. Life history traits therefore therefore constitute constitute a loosely defined set of morphological, morphological, developmental, developmental, or behavioral characteristics, characteristics, including, including, for instance, body size, growth patterns, patterns, size and age at maturity, reproductive reproductive effort, mating success, number, size, and sex of offspring, and rate of senes senes990s, life history evolution cence. Despite this diversity of traits, traits, by the early 11990s, had grown successfully into a very productive productive field organized around around a few central central questions questions with a very strong unifying theoretical theoretical background, background, grounded Stearns, grounded in both both optimization optimization principles and quantitative quantitative genetics genetics ((Stearns, 11992; 992; Roff, 11992). 992). Then, the realm of most studies of life history theory was that that of a single, large, undisturbed undisturbed and spatially homogeneous homogeneous population population 992; Kawecki, 1993). Through (see, however, Kawecki and Stearns, 11992; Through several 1 997) illustrated examples, Olivieri and Gouyon ((1997) illustrated how how disequilibrium and the populations might significantly affect spatial structure structure characteristic characteristic of meta metapopulations the evolution evolution of life history traits, a phenomenon phenomenon they called "the metapopu metapopulation effect." Five years later, despite increasing awareness awareness of the importance importance of meta population structure metapopulation structure and dynamics for the demography, genetics, and and
enetics, and Ecology, Ecology, G Genetics, and Evolution Evolution of of Metapopulations Metapopulations
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conservation of many species, little is understood understood about how much much these char characteristics acteristics have shaped basic life histories. The present chapter chapter reviews empir empirtheoretical studies studies published since since 11997 ical and theoretical 997 that have addressed the metapopulation context. We We first comment comment evolution of life history traits in a metapopulation generally about the development of this field of research since 11997. 997. First, studies of life history evolution in a meta population context remain metapopulation rare see rare as as compared compared to to the the blooming blooming of of metapopulation metapopulation demographic demographic studies studies ((see Chapters ) . Using Chapters 44 and and 55). Using aa literature literature search search engine engine (lSI (ISI Web Web of of Science Science v04), v04), combined key words words "metapopulation" "metapopulation" and ""life a search with the combined life history evolution" 997 compared evolution" gave gave only only 11 reference reference matching matching the the query query since since 11997 compared to, to, respectively, respectively, 962 962 and and 297 297 when when searching searching with with the the key key words words "metapopula "metapopula"life-history evolution" evolution" alone. This result reflects, not so much the tion" and "life-history scientific scientific production production on on the the subject, subject, but but rather rather the the fact fact that that life life history history evolu evolution in a metapopulation metapopulation context context is poorly poorly identified identified as a distinct distinct field of research. research. We We restrict restrict our our review review to to empirical empirical or or theoretical theoretical studies studies considering considering intraspecific intraspecific variation variation in life history traits traits in a landscape characterized characterized by spa spatial structure, structure, local extinction, extinction, and dispersal among among patches patches of habitat. habitat. Many Many studies in aa studies outside outside this this range range are are related related to to the the field field of of life life history history evolution evolution in metapopulation. instance, many models models of subdivided subdivided populations populations deal deal meta population. For instance, with with the evolutionary evolutionary consequences consequences of of spatial structure structure but do not not take into into account extinction-recolonization extinction-recolonization dynamics dynamics [see, e.g., Gandon Gandon ((1999) account 1 999) and Pen for models models of dispersal and and reproductive reproductive effort evolution] evolution].. (2000), respectively, for studies when we feel that that they point toward toward relevant and We refer to such studies unexplored unexplored aspects of life history evolution evolution in a metapopulation. metapopulation. Finally, life history evolution related to some evolution in a metapopulation metapopulation context context is closely related aspects of dynamics [see Tilman et al., of community community dynamics [see e.g., Tilman aI., (1997) ( 1 997) or or the discusdiscus sion about trade-offs in Chapter about colonization-competition colonization-competition trade-offs Chapter 6]. Testing predicpredic tions tions about about how how metapopulation meta population dynamics dynamics affect affect selection on life histories histories might actually actually be be achieved achieved more more easily easily by documenting documenting changes changes in in specific might specific composition within studying genetic differences composition within a community community rather rather than than by studying differences within a species. To To limit the the scope within scope of of the the present present chapter, chapter, we do do not not incorpoincorpo rate rate community-based community-based studies studies in our our review review but but we we invite invite the reader reader to to keep keep in mind when reading mind the the connection connection when reading Chapter Chapter 6. Second, the field field is largely dominated dominated by theory, with with very very little little empirical empirical Second, the is largely by theory, research due to to obvious obvious practical practical difficulties. difficulties. Most Most empirical empirical evidence of a research evidence of metapopulation effect effect on on life life history history evolution evolution in in natural natural systems systems is is indirect. indirect. metapopulation Metapopulation theory theory predictions predictions have have been been tested tested by comparing comparing mean mean phephe Metapopulation notypes among among populations populations that that have have been been founded founded for for different different times times (Cody (Cody notypes and and Overton, Overton, 1996; 1 996; Piquot Piquot et et al., aI., 1998; 1 998; Hill et et al., aI., 1999; 1 999; Hanski Hanski et et al., aI., 2002) 2002) or the mean or by comparing comparing the mean phenotypes phenotypes among among landscapes landscapes with with different different degrees of of fragmentation fragmentation (Thomas (Thomas et et al., aI., 1998; 1 998; Hill Hill et et al., aI., 1999; 1 999; Hanski Hanski et et al., aI., degrees 2002). Whether Whether those those phenotypic phenotypic differences differences are ultimately due evolution 2002). are ultimately due to to evolutionary investigated (but ary change change and and not not to to environmental environmental effects effects is still still too too rarely rarely investigated ( but et al., aI., 1999; 1 999; Hanski Hanski et et al., aI., 2002). 2002). Artificial Artificial see Thomas Thomas et et al., aI., 1998; 1998; Hill et metapopulations controlled conditions metapopulations of of short-lived short-lived organisms organisms in controlled conditions provide provide a fascinating opportunity opportunity to to witness witness evolutionary evolutionary change change and and test test metapopulametapopula fascinating tion theory predictions predictions more more accurately accurately (Warren, (Warren, 1996; 1 996; Buckling Buckling et et al., aI., 2000), 2000), tion theory but but such such projects, projects, though though growing growing in in numbers, numbers, are are still still in in the the process process of of develdevel opment opment (Lavigne (Lavigne et et al., aI., 2001). 200 1 ) . How How much much artificial artificial metapopulations metapopulations inform inform
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us about about the the relevance relevance of of metapopulation metapopulation theory theory for for life life history history evolution evolution in in us the real world world is is also also open open to to question. question. The The increasing increasing imbalance imbalance between between the real theoretical production production and and data data collection collection is somehow somehow worrying worrying for for the the develdevel theoretical opment of of the the discipline. discipline. Despite Despite our our efforts efforts to to integrate integrate relevant relevant empirical empirical opment examples, the the present present review, review, with with its its strong strong focus focus on on theory, theory, reflects reflects this this bias. bias. examples, The third point concerns the way way the part of the field The third point concerns the the theoretical theoretical part of the field has has develdevel history theory theory specific specific to to metapopulations metapopulations has has bloomed bloomed essentially oped. Life history around questions questions related related to to dispersal dispersal evolution evolution (we (we counted counted more more than than 40 40 around theoretical papers on dispersal dispersal evolution evolution published 1 997). Because theoretical papers on published since since 1997). Because aa whole whole chapter of of the the present present volume is devoted devoted to to dispersal dispersal (Chapter ( Chapter 13, 1 3, see see also also chapter volume is Chapter 16), 1 6), we we will not here review exhaustively exhaustively models of dispersal dispersal evolution. evolution. Chapter will not here review models of Instead, we we focus focus on on those those studies studies that that help help us us to to understand how the the evoevo Instead, understand better better how lution of dispersal contributes to to an an organism's organism's general history strategy. strategy. In In lution of dispersal contributes general life life history particular, this is illustrated illustrated with studies of of variation variation in in dispersal dispersal strategies strategies with with particular, this is with studies age and and the the interaction interaction of of dispersal dispersal with with other other life life history history characters. characters. age Comparatively, metapopulation context, more clas Comparatively, the the evolution, evolution, in in aa metapopulation context, of of other other more classical life Roy, 11999), 999), age sical life history history traits traits such such as as life life span span (Kirchner (Kirchner and and Roy, age at at matur maturity et al., aI., 2000), reproductive effort 1 997; ity (de (de Jong Jong et 2000), or or reproductive effort (Ronce (Ronce and and Olivieri, Olivieri, 1997; Ronce et aI., 2000c; 2000c; Crowley Crowley and 2002) has Ronce et al., and McLetchie, McLetchie, 2002) has received received little little attention attention to In particular, senescence patterns to date. date. In particular, the the evolution evolution of of senescence patterns or, or, more more generally, generally, of of age-specific reproductive major subjects history age-specific reproductive strategies, strategies, while while major subjects of of classical classical life life history theory, unexplored theoretical theoretical questions questions in the context theory, are are almost almost unexplored in the context of of aa metapop metapopulation (with of the ulation (with the the exception exception of the evolution evolution of of delayed delayed reproductive reproductive strategies strategies such 1 ). We such as as dormancy dormancy and and diapause; diapause; for for aa review, review, see see Olivieri, Olivieri, 200 2001). We suggest suggest reasons why why these these questions questions might promising investigation investigation areas. reasons might constitute constitute promising areas. Life Life than dispersal also deserve more attention because they may, history traits other than in some instances, than dispersal in some instances, be be easier easier to to measure measure empirically empirically than dispersal and and would would thus thus allow allow more more precise precise tests tests of of the the theory. theory. This This chapter chapter is is organized organized by by looking looking for for common common patterns patterns explaining explaining results obtained obtained in different different specific studies. Founding Founding events and small local results population population are population size size in in aa meta metapopulation are two two causes causes of of genetic genetic resemblance resemblance among among neighbors neighbors exploiting exploiting the the same same local local environment. environment. This This chapter chapter illus illustrates trates how how this this genetic genetic structure structure makes makes life life history history evolution evolution in in aa metapopula metapopulation tion deviate deviate from from that that expected expected in in aa single single large large panmictic panmictic population. population. Changes Changes in in population population age age structure structure and and density density following following disturbance disturbance and and recolonization population. Species recolonization are are major major features features of of life life in in aa meta metapopulation. Species whose whose biology is described described most adequately adequately using the metapopulation metapopulation framework framework also also often often occur occur in in habitats habitats subject subject to to successional successional changes. changes. Such Such variations variations in in selection selection pressures pressures associated associated with with colonization colonization and and succession succession have have deep deep implications implications for for life life history history evolution. evolution.
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RESEMBLANCE RESEMBLANCE BETWEEN BETWEEN NEIGHBORS NEIGHBORS population Fragmentation of the habitat is often associated with small population size size in in remnant remnant patches patches of of habitat. habitat. Patches Patches of of the the now now classic classic example example of of meta population, the Finnish populations of the butterfly Melitaea metapopulation, Melitaea cinxia cinxia in in the land Islands, the A Aland Islands, contain contain at at most most aa few few sib sib families. families. Both Both genetic genetic and and demo demographic graphic stochastic stochastic processes processes take take an an increasing increasing importance importance in in small small populations. populations.
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In In aa system system of of small small and and poorly poorly connected connected populations, populations, genetic genetic drift drift results results in in both both the the loss loss of of genetic genetic diversity diversity within within each each local local population population and and an an increasing increasing variance in allelic frequencies among populations. populations. Similarly, a transient transient reduc reducpopulation size size associated with disturbance disturbance and/or recolonization recolonization by a tion in population few founders founders can leave a significant signature in the genetic composition composition of popu populations long after population population regrowth 998; Ingvarsson and Giles, Giles, regrowth (Ingvarsson, 11998; Extinction-recolonization processes can either attenuate attenuate or accelerate the 11999). 999). Extinction-recolonization effect population differentiation, effect of of drift drift on on population differentiation, depending depending on on the the details details of of recolo recolonization, dispersal, and the length of the period of transient transient growth growth following recolonization 988; Whitlock recolonization (Slatkin, 1977; Wade and McCauley, 11988; Whitlock and McCauley, 990; Whitlock 997; Ingvarsson, 997; Pannell McCauley, 11990; Whitlock and and Barton, Barton, 11997; Ingvarsson, 11997; Pannell and and Charlesworth, 999). How metapopulation dynamics Charlesworth, 11999). How metapopulation dynamics affect affect the the structuring structuring of of genetic populations is genetic diversity diversity within within and and among among populations is reviewed reviewed in in Chapters Chapters 7, 7, 8, 8, and and 99 (see (see also also Pannell Pannell and and Charlesworth, Charlesworth, 2000). 2000). Both Both founder founder effects effects and and sub subsequent sequent genetic genetic drift drift within within local local populations populations have have the the result result that that two two individu individuals interacting in the same patch patch of habitat have a higher probability probability of sharing alleles belonging to alleles than than individuals individuals belonging to different different patches. patches. Such Such aa genetic genetic structure structure holds been holds major major implications implications for for the the evolution evolution of of life life history history traits, traits, which which have have been explored incompletely and are often neglected. This section illustrates those con consequences evolution of sequences through through three three examples examples concerned, concerned, respectively, respectively, with with the the evolution of dispersal, life span, and allocation strategies. dispersal, life span, and sex sex allocation strategies.
Dispersal Dispersal Multiple Multiple Causes for for the the Evolution Evolution of of Dispersal Dispersal
Dispersal metapopulation context Dispersal is is often often considered considered in in aa metapopulation context as as aa risky risky behav behavior, compensated by ior, compensated by the the potential potential benefit benefit of of founding founding aa new new population population in in an an empty empty patch patch of of habitat. habitat. Such Such aa view, view, adopted adopted by by early early students students of of dispersal dispersal evolution 971; Roff, 975) , appeared evolution (van Valen, 1971; Gadgil, 11971; Roll, 11975), appeared to be some somehow 1 977) discovered how too too simple simple after after Hamilton Hamilton and and May May ((1977) discovered that that selection selection should populations even should favor favor frequent frequent dispersal dispersal behavior behavior in in subdivided subdivided populations even in in the the absence 1 986) and absence of of empty empty patches. patches. Further Further theoretical theoretical work work by by Frank Frank ((1986) and Taylor ((1988), 1 98 8 ), in particular, particular, allowed a better understanding understanding and and quantification quantification of dispersal in demographically stable of the the forces forces selecting selecting for for dispersal in demographically stable but but genetically genetically structured structured populations. populations. Dispersal in such a theoretical theoretical context context can be seen as an altruistic act by which an individual individual risks its own fitness to alleviate alleviate kin competition competition within within the the natal natal patch. patch. As As with with any any altruistic altruistic act, act, such such aa behavior behavior is favored favored as long as the individual individual fitness fitness cost endured endured by the disperser is smaller smaller than than the the inclusive inclusive fitness fitness benefit benefit of of its its departure departure for for its its kin. kin. As As resources resources freed freed by by the the departure departure of of an an individual individual are are shared shared among among all all its its neighbors, neighbors, the the inclusive inclusive fitness fitness benefit benefit will will depend depend on on its its relatedness relatedness to to other other residents residents in in the the natal natal patch patch compared compared to to its its relatedness relatedness with with the the occupants occupants of of patch ((Gandon and Rousset, Rousset, 11999). its new patch Gandon and 999). Dispersal Dispersal is is aa complex complex character character with with multiple multiple consequences, consequences, whose whose evolution evolution is 1 ). Kin is affected affected by by multiple multiple causes causes (Clobert (Clobert et et ai., al., 200 2001). Kin competition competition avoidance avoidance and recolonization mutually exclusive selective forces recolonization of empty patches are not mutually acting on dispersal evolution. However, theoretical theoretical studies addressing the evolu evolution of dispersal often consider one force or the other major explanation other as the major explanation
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for dispersal dispersal by by either either neglecting neglecting genetic genetic drift drift (Olivieri (Olivieri et et al., aI., 1995; 1 995; Holt Holt and and for McPeek, 1996; 1 996; Doebeli Doebeli and and Ruxton, Ruxton, 1997; 1 997; Parvinen, Parvinen, 1999; 1 999; Ronce Ronce et et al., aI., 2000b; 2000b; McPeek, Mathias et et al., aI., 2001; 200 1 ; Parvinen, Parvinen, 2002; 2002; Kisdi, Kisdi, 2002) 2002) or or ignoring ignoring complex complex metapopmetapop Mathias ulation dynamics dynamics (Ezoe, 1998; 1 998; Gandon, Gandon, 1999; 1 999; Gandon Gandon and and Rousset, Rousset, 1999; 1 999; ulation Hovestadt et et al., aI., 2001; 2001 ; Leturque Leturque and Rousset, 2002). 2002). Such decisions are often Hovestadt and Rousset, are often linked to to trivial trivial technical technical choices, choices, such such as as modeling modeling population population numbers numbers as as aa linked continuous rather rather than than discrete variable or using deterministic deterministic rather rather than than stosto continuous chastic models models [see [see Ronce Ronce et et al. ai. (2001) (2001 ) for for further discussion on on this this topic]. topic]. chastic further discussion simulation models incorporate the the kin Individual-based evolutionary simulation models necessarily incorporate phenomena associated with drift and and genetic structure, structure, although selection phenomena with drift although this is not always clearly acknowledged acknowledged (see, e.g., Travis and and Dytham, Dytham, 1998). not A Kin Selection Model Model for for the the Evolution Evolution of of Dispersal Dispersal A Kin Selection in a a Metapopulation Metapopulation in
One how kin selection and One might might wonder wonder about about how and local extinctions extinctions interact interact in aa metapopulation importance in in explaining patterns metapopulation and and about about their their relative relative importance explaining patterns of early on, on, Comins Comins et et ai. of dispersal dispersal in in recurrently recurrently disturbed disturbed systems. systems. Quite Quite early al. ((1980) 1980) incorporated incorporated the two forces forces in in the the same model. An An analytical by the two same model. analytical model model by Gandon and Michalakis Michalakis ((1999), 1 999), building building on ai. ((1980), 1 980), Gandon and on the the work work of of Comins Comins et et al. helped clarify this question Table 10.1 10.1 for for the the main main assumptions their helped clarify this question (see (see Table assumptions of of their model) . The evolutionarily stable dispersal rate, the fraction fraction of of indiindi model). The evolutionarily stable (ES) (ES) dispersal rate, i.e., i.e., the their natal natal patch patch before viduals leaving their before reproduction, reproduction, can be expressed as a simple function function of of the the extinction extinction frequency, the extramortality extramortality or or "cost" "cost" assoasso simple frequency, the ciated the average relatedness among born in in the ciated with with dispersal, dispersal, the average relatedness among individuals individuals born the same patch, patch, and the probability of common common origin of immigrants. When immi immigrants grants in in the the same same patch patch have have aa null null probability probability of of common common origin origin (the (the migrant 977), the migrant pool pool model, model, see see Slatkin, Slatkin, 11977), the ES ES dispersal dispersal rate rate increases increases with with higher extinction extinction rates local population higher rates and and higher higher within within local population relatedness. relatedness. This This happens, patches are happens, respectively, respectively, because because more more empty empty patches are available available for for coloniza colonization tion and and because because kin kin competition competition is is more more intense intense for for philopatric philopatric individuals, individuals, as as was was predicted predicted by by previous previous models models that that have have considered considered kin kin competition competition phenomena phenomena or or the the extinction-recolonization extinction-recolonization dynamics dynamics separately. separately. However, However, more complex patterns emerge those two more complex patterns emerge due due to to the the interaction interaction of of those two forces. forces. In particular, for a very low probability of surviving migration, the ES dispersal dispersal rate rate can can increase increase with with increasing increasing dispersal dispersal cost cost (Fig. (Fig. 10.1.A), 10.1.A), whereas whereas previous previous models models predicted predicted that that dispersal dispersal should should always always decrease decrease with with increasing increasing cost but see 980). Gandon 1 999) cost of of dispersal dispersal ((but see Comins Comins et et aI., al., 11980). Gandon and and Michalakis Michalakis ((1999) explained this unexpected pattern by a simple kin selection argument. As the dis dispersal persal mortality mortality increases, increases, aa larger larger fraction fraction of of the the individuals individuals competing competing in in the the same patch are philopatric (because immigration is very low), which increases the the probability probability of of competing competing with with related related individuals individuals in in the the natal natal patch. patch. In In aa sys system 986), the tem with with no no empty empty patches patches (Frank, (Frank, 11986), the lower lower inclusive inclusive fitness fitness of of philopatric philopatric individuals individuals is is compensated compensated by by the the increasing increasing difficulty difficulty of of immigrat immigrating into extant populations. The empty patches created by local extinction, as in Gandon and Michalakis ((1999), 1 999), however, represent an extra benefit for dis dispersers, leading to increasing dispersal rates for very high dispersal costs. Interactions Interactions between kin competition and metapopulation dynamics are also complex because because the average level of relatedness among among individuals born more complex
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Effect on ES dispersal rate
Gandon and Michalakis ( 1 999) Spatially implicit Infinite island model of dispersal Catastrophic local extinction independent of population size All patches recolonized before disturbance Local carrying capacity reached at foundation
Dispersal mortality increases
Increases
None
Decreases, then increases
Patch carrying capacity decreases
Increases
None
Increases
Heino and Hanski (200 1 ) Spatially explicit Distance-dependent dispersal model based on focal butterfly species behavior Local extinction due to demographic stochasticity Not all patches recolonized Local carrying capacity reached at foundation
Dispersal mortality per unit distance increases or distance between patches increases
Increases
Increases
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Increases
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LIFEHISTORY HISTORY EVOLUTION 110. 0. LIFE
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Fig. Fig. 11 0. O.11 Effect of dispersal cost on the evolution evolution of dispersal. (A) ES ES dispersal rate as a func function of of dispersal cost as predicted predicted by a spatially implicit implicit kin selection model model for different different values 6, 32, 64, and 1128; 28; generated of patch carrying capacity: capacity: from from top top to bottom bottom N = = 11,, 2, 4, 8, 116, generated from model by and Michalakis 999), with values; here from the the model by Gandon Gandon and Michalakis (1 (1999), with different different parameter parameter values; here extinc extinc0.2 and fecundity dispersal propensity tion rate e = = 0.2 fecundity n = 5. 5. (S) (B) ES ES dispersal propensity as a function function of dispersal mor mortality per unit unit of distance as predicted by realistic spatially explicit explicit simulations based on the biology of Finnish Melitea of Finnish Melitea butterflies for two two hypothetical hypothetical patch patch networks characterized, respect respectivel y, by large interpatch (sparse) or short short interpatch ively, interpatch distances (sparse) interpatch distances distances (dense). Redrawn from from Heino and Hanski (2001 (2001);); for original original parameters, values see their Fig. Fig. 4A.
in same patch depends on in the the same patch depends on the the intensity intensity of of founder founder effects effects and and subsequent subsequent drift within within local populations. Local population population size, fecundity, extinction rate, dispersal cost, and the dispersal rate itself all affect indirectly the evolution of dispersal through their their effect on relatedness. Consistent with with what what has been observed in subdivided populations with with no extinction, the ES dispersal rate always decreases with increasing local population size (see (see Fig. 110.1A) 0.lA) because genetic genetic drift drift is is then then weaker weaker and and within-local within-local population population relatedness relatedness decreases. decreases. A Gandon and 1 ) showed A sensitivity sensitivity analysis analysis ((Gandon and Michalakis, Michalakis, 200 2001) showed that that population population extinctions may have a higher qualitative or quantitative impact on dispersal evolution than kin competition or other other forces, such as inbreeding avoidance. This populations made This is is especially especially true true of of meta metapopulations made of of local local populations populations of of large large size, with weak founder effects. In situations situations with strong strong genetic spatial struc structure, kin selection phenomena phenomena are likely to play a considerable role in the evolution of patterns of dispersal. Spatially Spatially Explicit Explicit Simulations Simulations for for a Butterfly Butterfly Metapopulation Metapopulation
Most Most of the qualitative conclusions of the simple theoretical study of Gandon and Michalakis ((1999) 1 999) have been reproduced by a much more realistic model of dispersal evolution (Heino and Hanski, 200 1 ) based on the biology of Finnish 2001) metapopulations of Melitaea (see Table 10.1 for a Melitaea cinxia cinxia and Melitaea Melitaea diamina diamina (see comparison of the assumptions and predictions of the two models). Each local population in these butterfly metapopulations is composed of few sib families, and mating occurs mostly within local populations before dispersal (for a detailed level of relatedness among description of the system, see Hanski, 1999). The level individuals born in the same patch is thus likely to be high and kin competition
234 234
ELIE RONCE OPH OPHELIE RONCEAND AND ISABELLE ISABELLEOLIVIERI OLIVIERI
avoidance avoidance may may be be aa potent potent force force in in dispersal dispersal evolution evolution in in those those species, species, together together with the the frequent frequent local local extinctions. extinctions. Both Both the the effect effect of of migration migration mortality (Fig. (Fig. with 110.1B) 0. lB) and and the the patch patch carrying carrying capacity capacity on on the the evolution evolution of of dispersal dispersal propensity (see Table Table 10.1) 10.1) can can be be interpreted interpreted with the the same kin kin selection arguments arguments as as in in the the (see analytical 1999). However, analytical model of of Gandon Gandon and and Michalakis Michalakis ((1999). However, an an alternative alternative explanation 1 ) is that, in their simulations, the explanation provided by by Heino and Hanski (200 (2001) number number of of empty empty patches patches increases increases when when recolonization recolonization becomes becomes more more difficult difficult because because of of higher higher dispersal dispersal cost cost or or when when demographic demographic stochasticity stochasticity increases increases because of aa small local local population size size (see (see Table 10.1). More numerous empty patches represent more opportunity opportunity for colonization colonization and favor the most dispers dispersing genotypes. Contrary to analytical studies, it is difficult to disentangle the respective role of kin selection phenomena and extinction-recolonization dynam dynam(see, however, the attempts to remove the genetic structure ics in simulation results (see, artificially in Heino and Hanski, 200 1 ). Spatially realistic models, designed to fit 2001). the biology of a focal species, species, offer nonetheless the major advantage that they can be compared to data more readily and may help build a more precise test of the theory. The predicted average dispersal propensity for the well-studied M. M. diamina diamina metapopulation was not significantly different from that estimated empirically, which is encouraging even though it does not provide a critical test of the theory. Predictions of the model concerning regional variation in dispersal propensity among different clusters of patches could, in the future, be compared to the observed variation in butterfly behavior among such clusters.
Evolutionary Consequences Consequences of Landscape Landscape Fragmentation Evolutionary Predictions of the previous models under Predictions of the previous models have have deep deep implications implications for for our our understanding of the effect fragmentation on the evolution of life life histories (see effect of fragmentation Table 10.1). 1 0. 1 ). For fragmentation were comcom For a long time the the evolutionary effects of fragmentation pared to syndromes of insularity. A rapid loss of dispersal ability of plants in pared Overton, 1996) happen in island populations populations (Cody and Overton, 1 996) was expected to also happen fragmented habitats habitats where where the increasing cost of of dispersal would would select against fragmented dispersal. Previous models (Comins et al., aI., 1980; 1980; Gandon Gandon and and Michalakis, Michalakis, 1999; 1 999; Heino and Hanski, 2001) 200 1 ) show show that that the expected relationship relationship between disperdisper Heino and Hanski, propensity and and dispersal mortality mortality or or distance between patches patches is more more comcom sal propensity than previously imagined (see (see also Leimar and and Norberg, Norberg, 1997). 1 997). Another Another plex than complicating factor factor is is that that fragmentation fragmentation is is also also associated associated with with habitat habitat loss, loss, complicating and thus thus with with a likely likely reduction reduction in local population population size: size: previous previous models models and showed that that a reduction reduction in local population population size selects for for increased increased dispersal. showed How How do do the the effects effects of of increasing dispersal cost cost and and decreasing decreasing local population population combine in a fragmented fragmented landscape landscape to to act act on on the the evolution evolution of of dispersal? dispersal? A size combine M. diamina diamina metapopulation metapopulation (Heino (Heino and and Hanski, Hanski, simulated habitat habitat change change in in the the M. simulated 200 1 ) led to to complex complex predictions predictions depending depending on on the the type type of of change change (removal (removal of of 2001) whole patches, patches, reduction reduction in patch patch area, area, or or quality). quality). In some some instances, instances, the the effect effect whole of of decreasing decreasing local local population population size size and and increasing increasing distance distance between between patches patches counteract each each other other completely completely so that that we we would would not not expect expect any any evolutionary evolutionary counteract change change in in dispersal dispersal in in the the new new landscape. landscape. Empirical evidence evidence for for aa correlation correlation between between habitat habitat fragmentation fragmentation and and disdis Empirical persal propensity propensity in in natural natural systems systems is ambiguous ambiguous (see (see Table Table 10.1 1 0 . 1 and and also also persal Plebejus argus argus butterflies, butterflies, total total Chapter 20). 20). Thomas Thomas et et al. al. (1998) ( 1 998) found found that that in in Plebejus Chapter
LIFEHISTORY HISTORY EVOLUTION EVOLUTION 110. 0. LIFE
235 235
mass, potentially associated with flight ability, increased in the most frag fragHesperia mented heathlands. Comparing two metapopulations of the butterfly Hesperia 1 999) found that the relative size of the thorax was larger comma, comma, Hill et al. ((1999) sugin the landscape characterized by larger distances between patches, which sug gests that higher fragmentation is associated with higher dispersal. The fact that famsuch patterns were observed in controlled conditions, together with strong fam ily effects, suggests that morphological differences are based genetically and that these characters have indeed evolved (or could evolve in the future) in response to fragmentation. Whether selection on dispersal is the major expla explanation for such patterns patterns is, however, difficult to establish. Hanski et al. (2002) found no obvious relationship between morphological measurements and and esti estimated dispersal propensity for Glanville fritillary butterflies, M. M. cinxia. cinxia. Contrary to Hill et al. ((1999), 1 999), they found no differences in migration propen propensity among meta populations characterized by different degrees of connectivity. metapopulations
Life Span Span Life Evolution of of Reproductive Effort Effort in Genetically Genetically Structured Structured Populations Populations
Does Does the the genetic genetic structure structure generated generated by by drift drift and and founder founder effects effects affect affect the the evolution of life history traits other than dispersal? A model of a subdivided population (Pen, 2000) suggests that that the same processes are likely to alter the evolution of reproductive effort effort in a metapopulation. metapopulation. Pen's (2000) model does not 0.2). Adults not incorporate extinction-recolonization extinction-recolonization dynamics (see (see Table 110.2). are sessile and competitively superior to juveniles. They allocate their resources functions, which generate between reproductive and and maintenance physiological functions, trade-off between fecundity and and survival. In a genetically structured popula a trade-off structured population, surviving surviving adults the recruitment recruitment of of related juveniles in in the the same same tion, adults prevent prevent the related juveniles patch, whereas dispersing juveniles compete with unrelated individuals. A patch, with unrelated allocation to offspring production instead of survival can then then be seen higher allocation offspring production another mechanism mechanism for for kin competition competition avoidance. avoidance. As a result, result, Pen (2000) as another found that that the ES reproductive found reproductive effort increases with the increasing level of of among competitors patch. We conjecture that that results relatedness among competitors in the same patch. obtained that the exisexis obtained by Pen (2000) could be generalized to other other life cycles in that tence of a strong strong genetic structure structure should favor the allocation allocation of resources resources to to tence favor the history stages with with the highest highest dispersal propensity. propensity. Because of of their their indiindi life history rect rect effect on on within-local within-local population population relatedness, a lower lower dispersal of of juveniles, higher dispersal dispersal cost, and and smaller local population population size all select for higher higher for higher reproductive effort effort in his model. model. We reached similar conclusions conclusions (see (see Table Table reproductive We reached 1 0 .2), although although for different reasons, reasons, in a metapopulation metapopulation model model with with 10.2), for entirely different but no no genetic structure structure (Ronce (Ronce and and Olivieri, 1997). 1997). Both Both local extinctions but that increasing fragmentation fragmentation of the the habitat habitat would would lead lead to to the the models suggest that evolution evolution of of a higher higher reproductive effort effort and and shorter shorter life span span for for species in which adults adults disperse less than than juveniles. Such a prediction should should be checked which by examining examining rigorously rigorously the the interactions interactions between between kin kin selection and and extincextinc tion-recolonization dynamics dynamics for for the the evolution evolution of of reproductive reproductive effort effort as tion-recolonization Gandon and and Michalakis Michalakis (1999) ( 1 999) did did for for dispersal. From From an an empirical empirical point point of of Gandon would be interesting interesting to to compare compare patterns patterns of of allocation allocation to to reproductive reproductive view, itit would
Yes
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No No Yes No transient population growth after colonization Yes Transient population growth after colonization Dynamic number of colonizers Yes Transient population growth after colonization Fixed number of colonizers Yes Transient population growth after colonization Dynamic number of colonizers
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110. 0. LIFE LIFEHISTORY HISTORY EVOLUTION EVOLUTION
237 237
and maintenance maintenance structures in landscapes of different fragmentation fragmentation levels, as was done for for traits related related to dispersal. Evolution Evolution of of Life Life Span In in a Metapopulatlon Metapopulation with with Sterilizing Sterilizing Parasites
With a different set of assumptions assumptions than in Pen (2000), Kirchner and Roy ((1999) 1 999) showed that resemblance between neighbors generated by founder effects could profoundly modify the evolution of life life span in a metapopulation. metapopulation. Contrary Contrary to Pen (2000), they assumed no direct trade-off between fecundity and adult survival (see (see Table 10.2). Their evolutionary scenario involves interactions with sterilizing pathogens. In a large panmictic population, a long life life span car carries an epidemiological cost, as increasing adult survival increases the prevalence of the pathogen in the population. This happens because infected infected hosts remain in the population longer. Despite this cost expressed at the scale of the population, longer. population, longer lived individuals still enjoy a higher fitness than short-lived hosts and alle alleles increasing longevity invade the population population readily. The situation is different in aa meta population with metapopulation with strong strong founder founder effects effects and and little little dispersal dispersal among among extant extant local populations. Then, the variance in gene frequencies among local popula populations. frequencies populations advantages the short-lived genotypes. When short-lived individuals have mainly short-lived neighbors, the local prevalence prevalence of pathogens is low and the average fecundity per individual is much higher than in local populations populations domin dominated by long-lived individuals where levels of infection infection and sterility are both high. Because of the higher offspring production production of local populations populations with with a high frequency of short-lived genotypes, new populations populations in empty patches are founded founded more often by such genotypes. Selection among local populations populations favors a short life span, whereas selection within local populations populations still advantages geno genotypes with a long life life span. The frequency of short-lived individuals in the metapopulation metapopulation depends depends critically critically on on the the rates rates of of dispersal dispersal and and local local extinction. extinction. that have explained the coexistence of Note that, differently from other models that different life history strategies in a metapopulation by a competition-colonization competition-colonization trade-off (Olivieri and Gouyon, 11997; 997; Lehman and Tilman, 11997; 997; Jansen and Mulder, 11999), 999), the trade-off here emerges from the genetic structure of the meta population, as short-lived individuals indeed enjoy a higher fecundity only metapopulation, when they are surrounded by other short-lived individuals. Predictions of the model could be tested by comparing survival rates of uninfected individuals belonging to Silene (see Chapter 119) 9 ) differing in the preva Silene alba alba metapopulations metapopulations (see prevaUstilago violacea violacea or in their connectivity. connectivity. lence of their sterilizing parasites Ustilago Life History between Life Span History Syndromes: Syndromes: Correlation Correlation between and Dispersal Dispersal
Starting from Grime ((1977), 1977), much effort in life history history theory theory and ecology has been devoted to identifying patterns patterns of covariation covariation among species life his history traits traits and relating relating these patterns patterns to characteristics characteristics of their habitat habitat ((Southwood, Southwood, 11988; 988; Taylor et aI., 11990; 990; Silvertown 993; etal., Silvertown and Franco, 11993; Westoby, 11998; 998; Charnov, 2002). This led to to the definition of syndromes, such as Hutchinson, 11951), 95 1 ), the as the the syndrome syndrome of of the the "fugitive "fugitive species" species" ((Hutchinson, the "ruderal "ruderal strategy" ((Grime, Grime, 11977), 977), or the "colonizer syndrome" (Baker and Stebbins, 11965), 965), which which associates associates high dispersal dispersal ability, high fecundity, and short short life
ELIE RONCE OPH RONCE AND AND ISABELLE OPHELIE ISABELLEOLIVIERI OLIVIERI
238 238
span as a set of coadapted coadapted traits traits symptomatic of unstable habitats. habitats. How How does metapopulation synthe genetic structure of meta population affect the evolution of such syn dromes? Based on previous theoretical theoretical studies, one would dromes? Based on the the two two previous studies, one would predict predict aa positive correlation correlation between lifespan and dispersal ability, contrary contrary to the col colonizer syndrome (see 0.2). This happens because higher (see Table 110.2). higher dispersal decreases genetic differentiation differentiation between local populations, which selects for a longer life span. Conversely, what what are the consequences of increased life span for for the evolution of dispersal? Despite the potential consequences for life his history tory evolution, our understanding understanding of how how complex life cycles with with several age classes affect the genetic structure structure in a metapopulation metapopulation is still very incomplete ((but but see Rousset, 11999). 999). Several models of subdivided populations, populations, assuming that that only juveniles disperse, have shown that that increasing adult survival rates led among juveniles born born in the the same patch, favor favorto a higher level of relatedness among 2000; ing the evolution of increasing juvenile dispersal (Taylor and Irwin, 2000; Irwin and and Taylor, 2000; Ronce et aI., al., 2000a) 2000a).. They therefore predict again a positive 0.2). positive association association between between life life span span and and dispersal dispersal (see (see Table Table 110.2). As such studies do do not incorporate incorporate extinction-recolonisation extinction-recolonisation dynamics, it is, however, difficult to conclude about the generality of the syndrome (see Section 110.3 0.3 for a further further discussion of syndromes). Better predictions could also be achieved by letting letting both life span and dispersal evolve jointly in the same metapopulation. metapopulation. However, theoretical studies addressing this question do not take into account the genetic structure structure of the metapopulation metapopulation (see (see Table 110.2 0.2 and 0.3 ). Artificial and Section Section 110.3). Artificial microcosms microcosms using using mutant mutant strains strains of of Caenorhabditis Caenorhabditis elegans elegans with with modified dispersal and life history history traits traits represent an exciting perspective for testing experimentally the role of kin structure and extinction-recolonization extinction-recolonization dynamics in the evolution of life life history syndromes ((Delattre Delattre and Felix, 200 1 ; Friedenberg, 2003). 2001; Variation Variation of of Allocation Allocation Strategies Strategies with with Age Age
Iteroparity (i.e., the existence of several reproductive episodes within an organism's life span) can also lead to the evolution of offspring dispersal strat strategy varying with maternal age (Ronce et aI., 998, 2000a) al., 11998, 2000a),, in part part because relatedness with other other juveniles born in the same patch increases with with the number number of times their parent parent reproduced reproduced in the patch. We conjecture that that vari variation in the intensity of kin competition competition could also alter the evolution of age agespecific reproductive reproductive effort in such subdivided populations, populations, and thus affect How those results generalize to metapopulations, metapopulations, with patterns of senescence. How large disequilibrium in age structure generated by local extinction, is an open question (see (see Section 110.3 0.3 for predictions, predictions, not not based on genetic resemblance, concerning concerning age-specific allocation strategies).
Sex Sex Allocation Allocation Spatial Spatial Structure Structure for for Genes Involved Involved In in Sex Determination Determination
Similarly, Similarly, several several theoretical theoretical studies studies have have investigated investigated how how genetic genetic structure structure and could affect and context-dependent context-dependent fitness fitness could affect the the evolution evolution of of resource resource allocation allocation between metapopulation. Such between sexes sexes in in aa metapopulation. Such studies studies focus focus particularly particularly on on the the case of hermaphrodites coexist case of gynodioecy gynodioecy or or of of androdioecy, androdioecy, in in which which hermaphrodites coexist with, with,
110. 0. LIFE LIFEHISTORY HISTORY EVOLUTION EVOLUTION
239 239
respectively, female or male individuals (Pannell, 1997a; McCauley McCauley and Taylor, 11997; 997; Couvet 998; Pannell, 2000; McCauley et al., 2000). Couvet et al., 11998; 978) and Gynodioecious species, such as Thymus Thymus vulgaris vulgaris (Dommee (Domm6e et al., 11978) Beta al., 1988), and androdioecious species, such as Beta maritima maritima (Boutin et etal., Datisca 990), Mercuria 997b), Datisca glomerata glomerata (Liston et al., 11990), Mercuria annua annua (Pannell, 11997b), Schizopepon 999), all Schizopepon bryoniafolius bryoniafolius (Akimoto (Akimoto et et al., al., 11999), all occur occur in in recurrently recurrently dis disturbed habitats. In gynodioecious species, species, gender turbed habitats. In gynodioecious gender is is often often determined determined by by epista epistatic interactions 98 8; Frank, interactions between nuclear and cytoplasmic factors (Kaul, 11988; 11989; 989; Charlesworth and Laporte, 1998). High levels of genetic differentiation among among patches for both neutral cytoplasmic markers in linkage disequilibrium with cytoplasmic male sterility alleles ((Cuguen with Cuguen et al., 11994; 994; Manicacci et al., 11996; 996; McCauley et al., 2000) and nuclear genes involved in sex determination determination (Manicacci et al., 11997) 997) suggest that that founding events and limited gene flow metapopulations, such as those of leave a strong signature in gynodioecious metapopulations, B. maritima, Sex ratio ratio is, is, moreover, highly vari B. maritima, T. T. vulgaris, vulgaris, or or Silene Silene vulgaris. vulgaris. Sex moreover, highly variable among populations populations in such species (Frank, 1989). Despite these common features features shared shared by by many many well-studied well-studied gynodioecious gynodioecious systems, systems, theoretical theoretical stud studies disagree about the consequences of genetic structure for the evolution of sex ratio population. Some ratio in in aa meta metapopulation. Some models models predict predict that that the the frequency frequency of of females females would be lower lower in a metapopulation metapopulation than than expected in a large panmictic popu population (Pannell, 11997a; 997a; McCauley and Taylor, 11997; 997; McCauley et al., 2000), 2000), whereas others reach the opposite conclusion ((Couvet Couvet et al., 11998). 998). We here suggest which specific assumptions may be responsible for those discrepancies and and show show how how different different models models shed shed light light on on different different mechanisms mechanisms acting acting on on sex ratio evolution in a genetically structured structured metapopulation. A list of the models' main assumptions predictions is 0.3. models' main assumptions and and predictions is given given in in Table Table 110.3. Local Local Sex Ratio Ratio Variation Variation and and Pollen Pollen Limitation Limitation
All All four four models models focus focus on on the the evolutionary evolutionary consequences consequences of of the the recolonization recolonization of disturbed patches by a small number of individuals. Such founder effects generate a large variation in the local sex ratio (see 0.3). How How does the (see Table 110.3). spatial clustering of female and hermaphrodite genotypes affect the evolution of the sex ratio at the meta population scale? In S. metapopulation S. vulgaris, vulgaris, both the pollination pollination rate rate of of females females and and the the number number of of viable viable seeds seeds per per fruit fruit born born by by hermaphro hermaphrodites were shown to increase with the frequency of hermaprodites hermaprodites in the neigh neighborhood 998; McCauley et al., 2000), suggesting that borhood (McCauley and Brock, 11998; that seed production production is indeed limited by the availability of outcross pollen in that that species. Because of increased pollen competition, the siring success of an her hermaphrodite plant is, however, correlated negatively with the local frequency of hermaphrodites 998). Using frequency-dependent hermaphrodites (McCauley and Brock, 11998). male and female fitness components as in S. S. vulgaris, vulgaris, McCauley and collabo collaborators 997) modeled rators (2000; McCauley and Taylor, 11997) modeled the combined combined effects of ratio variation variation in a gynodioecious metapopula metapopulapollen limitation and local sex ratio tion of an annual plant where where seed dispersal is global but pollen flow flow strictly localized. They concluded that that the expected frequency of females in the meta population was panmictic population metapopulation was lower lower than than in in aa large large panmictic population and and declined declined with with increasing increasing variance variance in in local local sex sex ratio. ratio. A A higher higher variance variance in in the the local local sex sex ratio at foundation foundation increases the spatial aggregation of females. Most Most of them are then found in patches with numerous other females and suffer from pollen
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limitation. Similar Similar arguments explain the limitation. arguments have have been been used used to to explain the predicted predicted lower lower fre frequency feminizing parasites quency of of sex sex ratio ratio distorters distorters or or feminizing parasites in in genetically genetically structured structured animal metapopulations animal metapopulations as as compared compared to to panmictic panmictic populations populations (Hatcher (Hatcher et et aI., al., 2000). 2000). Note Note that that this this prediction prediction goes goes in in aa direction direction opposite opposite to to the the classic classic local local mate mate competition competition hypothesis, hypothesis, which which states states that that in in the the presence presence of of aa strong strong spa spatial tial genetic genetic structure, structure, higher higher sperm sperm or or pollen pollen competition competition among among related related indi individuals but see viduals should should favor favor the the evolution evolution of of female-biased female-biased sex sex ratios ratios ((but see de de Jong Jong et McCauley and 1 997), polymorphism et aI., al., 2002). 2002). In In the the model model of of McCauley and Taylor Taylor ((1997), polymorphism could could not not be be maintained maintained at at equilibrium equilibrium at at both both nuclear nuclear and and cytoplasmic cytoplasmic sex sex loci. loci. The The prediction prediction of of lower lower female female frequency, frequency, however, however, held held whether whether the the determinism 997) or purely determinism of of sex sex was was purely purely nuclear nuclear (McCauley (McCauley and and Taylor, Taylor, 11997) or purely cytoplasmic 997; McCauley McCauley et cytoplasmic (McCauley (McCauley and and Taylor, Taylor, 11997; et aI., al., 2000). 2000). Reproductive Reproductive Assurance and and Recolonization Recolonization
The The same same conclusion conclusion of of aa lower lower frequency frequency of of females females in in aa gynodioecious gynodioecious metapopulation Pannell ((1997a) 1 997a) in pollen metapopulation was was reached reached by by Pannell in aa model model with with no no pollen limitation Pannell ((1997a), 1997a), however, limitation (see (see Table Table 10.3). 10.3). Pannell however, assumed assumed that that females females fail fail to to found found new new populations populations in in the the absence absence of of hermaphrodites. hermaphrodites. Successful Successful immigration immigration of of female female genotypes genotypes in in aa patch patch can can only only occur occur after after the the arrival arrival of of hermaphrodite hermaphrodite genotypes. genotypes. As As aa result, result, the the frequency frequency of of hermaphrodite hermaphrodite geno genotypes types in in recently recently founded founded populations populations is is higher higher than than expected expected on on the the simple simple basis pool. Hermaphrodites basis of of their their frequency frequency in in the the migrant migrant pool. Hermaphrodites benefit benefit more more than than females females from from the the relaxed relaxed competitive competitive conditions conditions and and higher higher recruitment recruitment rates local populations. variation in rates that that prevail prevail in in recently recently founded founded local populations. Stochastic Stochastic variation in the composition of population therefore the composition of founders founders in in such such aa meta metapopulation therefore tends tends to to favor favor cosexual hermaphrodites, hermaphrodites, at expense of cosexual at the the expense of unisexuals, unisexuals, such such as as females females in in gyn gynodioecious 997a). odioecious species, species, but but also also males males in in androdioecious androdioecious species species (Pannell, (Pannell, 11997a). Such Such an an argument argument bears bears close close connections connections to to Baker's Baker's law law and and the the reproductive reproductive assurance 99 8 ) . assurance concept concept (Pannell (Pannell and and Barrett, Barrett, 11998). Intragenomic Conflicts Conflicts and and Founder Founder Effects Effects Intragenomic
Assumptions 1 99 8 ) are Assumptions of of Couvet Couvet et et aI. al. ((1998) are very very similar similar to to those those used used by by Pannell ((1997a), 1997a), as 0.3. However, aI. ((1998) 1998) Pannell as can can be be seen seen in in Table Table 110.3. However, Couvet Couvet et et al. reached reached strikingly strikingly different different conclusions. conclusions. For For some some parameter parameter sets, sets, the the pre predicted population is dicted female female frequency frequency in in the the meta metapopulation is lower lower than than expected expected in in aa large most of range, the large panmictic panmictic population, population, but, but, for for most of the the explored explored parameter parameter range, the reverse holds. In can reverse prediction prediction holds. In particular, particular, relatively relatively high high frequency frequency of of females females can be values that allow the be maintained maintained in in aa metapopulation metapopulation for for parameter parameter values that do do not not allow the presence presence of of females females in in aa single single panmictic panmictic population. population. Such Such discrepancies discrepancies are are ultimately models. Pannell ultimately due due to to the the mode mode of of sex sex inheritance inheritance in in the the two two models. Pannell ((1997a) 1997a) assumed 1 99 8 ) assumed pure pure nuclear nuclear control control of of sex, sex, whereas whereas Couvet Couvet et et aI. al. ((1998) considered considered the the case case where where sex sex is is determined determined bbyy both both nuclear nuclear and and cytoplasmic cytoplasmic loci. 1 997), polymorphism loci. Contrary Contrary to to McCauley McCauley and and Taylor Taylor ((1997), polymorphism at at both both types types of parameters. of loci loci was was protected protected in in the the metapopulation metapopulation for for aa large large range range of of parameters. Why Why are are assumptions assumptions about about the the genetic genetic architecture architecture of of sex sex so so important important ?? In aI. ((1998), 1 998), founding In Couvet Couvet et et al. founding events events have have qualitatively qualitatively different different conse consequences models in quences than than envisioned envisioned in in all all previous previous models in this this section. section. Stochastic Stochastic varia variation tion in in the the identity identity of of founders founders not not only only generates generates phenotypic phenotypic correlations correlations among among
ELIE RONCE OPHIr:LIE RONCE AND ISABELLE ISABELLEOLIVIERI OLIVIERI OPH
242 242
neighbors, neighbors, but but also also results results in in local local variation variation in in the the mode mode of of sex sex inheritance. inheritance. Depending alleles borne Depending on on the the diversity diversity of of cytoplasmic cytoplasmic and and nuclear nuclear alleles borne by by founders founders of can show variation locally, of aa new new population, population, sex sex can show no no heritable heritable variation locally, have have aa strict strict nuclear or nuclear or cytoplasmic cytoplasmic inheritance, inheritance, or or be be determined determined by by variation variation at at both both types types of of loci. loci. In In local local populations populations with with aa strict strict cytoplasmic cytoplasmic inheritance inheritance of of sex, sex, female female frequency frequency is is predicted predicted to to reach reach aa frequency frequency close close to to 100% 100% ultimately, ultimately, as as soon soon as as females 94 1 ). In females produce produce more more seeds seeds than than hermaphrodites hermaphrodites (Lewis, (Lewis, 11941). In contrast, contrast, in in aa population expression, females population in in which which only only nuclear nuclear alleles alleles can can modify modify sex sex expression, females will will be be progressively progressively eliminated eliminated if if they they produce produce fewer fewer than than twice twice the the number number of of seeds seeds of of an an hermaphrodite hermaphrodite and, and, whatever whatever their their fecundity fecundity advantage, advantage, will will never never exceed exceed 50% 941 ). Such 50% of of the the local local population population at at equilibrium equilibrium (Lewis, (Lewis, 11941). Such aa discrepancy discrepancy is is explained explained by by the the fact fact that that cytoplasmic cytoplasmic genes genes are are usually usually transmitted transmitted through through seeds whereas nuclear transmitted through pollen and seeds only, only, whereas nuclear genes genes are are transmitted through both both pollen and seeds. seeds. In al. ((1998), 1 998), the generated by In Couvet Couvet et et al. the initial initial variation variation in in sex sex ratio ratio generated by found founding selection within ing events events is is thus thus exaggerated exaggerated by by further further selection within the the established established local local populations. production, local populations populations with with a populations. Because of their higher seed production, sex highly biased females grow local popu sex ratio ratio highly biased in in favor favor of of females grow faster faster than than other other local populations, lations, consistent consistent with with the the observed observed larger larger size size of of female-dominated female-dominated patches patches in 996) . The in natural natural T. T. vulgaris vulgaris populations populations (Manicacci (Manicacci et et aI., al., 11996). The combination combination of of within-local within-local population population selection selection favoring favoring aa large large frequency frequency of of females females in in local local populations populations with with aa cytoplasmic cytoplasmic inheritance inheritance of of sex sex and and the the more more dynamic dynamic growth populations results increase in growth of of such such local local populations results in in an an overall overall increase in female female abund abundance scale of ance at at the the scale of the the metapopulation. metapopulation. Note Note that that founding founding events events here here affect affect the ratio not the evolution evolution of of the the metapopulation metapopulation sex sex ratio not because because female female fitness fitness is is affected affected by by their their neighbors neighbors phenotype, phenotype, but but because because founding founding events events allow allow cytoplasmic cytoplasmic male male sterility sterility alleles alleles to to escape escape the the control control of of nuclear nuclear genes genes in in some some local local populations. populations. Conclusion
Because Because these these different different studies studies have have described described different different facets facets of of the the evolu evolutionary consequences of combine their tionary consequences of founding founding events, events, it it is is difficult difficult to to combine their mes messages to sages to appreciate appreciate their their relevance relevance for for sex sex ratio ratio evolution evolution in in real real metapopulations. metapopulations. In In gynodioecious gynodioecious metapopulations, metapopulations, observed observed female female fre frequency often much much higher than expected quency is is often higher than expected in in aa single single panmictic panmictic population population ((Couvet Couvet et 990) and et aI., al., 11990) and is is still still higher higher than than predicted predicted by by the the metapopulation metapopulation model 1 998) for model of of Couvet Couvet et et ai. al. ((1998) for the the same same female female fecundity fecundity advantage. advantage. The The observed observed decline decline in in female female abundance abundance with with time time since since foundation foundation observed observed for for T. 990; see see also 997) is T. vulgaris vulgaris (Belhassen (Belhassen et et aI., al., 11990; also Olivieri Olivieri and and Gouyon, Gouyon, 11997) is con consistent ai. ((1998), 1 998), but sistent with with the the predictions predictions of of Couvet Couvet et et al. but not not those those of of Pannell Pannell ((1997a). 1 997a). Further Further theoretical theoretical and and empirical empirical work work is is needed needed to to estimate estimate the the rel relative ative impact impact of of frequency frequency dependent dependent fitness fitness and and variation variation in in sex sex transmission transmission mode populations. mode for for the the evolution evolution of of sex sex ratio ratio within within meta metapopulations.
Genetic Genetic Resemblance: Resemblance: Conclusion Conclusion We We find find it it useful useful to to distinguish distinguish two two types types of of genetic genetic resemblance resemblance among among neighbors: relatedness relatedness for expression of neighbors: for genes genes directly directly affecting affecting the the expression of the the trait trait of of interest and relatedness relatedness for for genes affecting selection on the trait trait only indirectly. interest
1 0. 10.
LIFE HISTORY HISTORY EVOLUTION EVOLUTION LIFE
243 2 43
In the the first case, the the fact fact that that an an individual individual with with aa given given life life history history trait trait is is In first case, more likely likely to to be be surrounded surrounded by by individuals individuals with with the the same same phenotype phenotype will will more modify selection selection on on that that trait trait whenever whenever selection selection is is frequency frequency dependent. dependent. modify Examples reviewed reviewed here here have have shown shown that that frequency-dependent frequency-dependent selection selection can can Examples affect the the evolution evolution of of aa large large variety variety of of life life history history traits. traits. When When aa life life history history affect character has has aa complex complex inheritance inheritance mode, mode, as as sex sex in in some some plants plants and and animals, animals, character variation of the genetic composition of local local populations can result in differdiffer variation of the genetic composition of populations can result in ences in in the the transmission transmission of of such such character, character, with with potentially potentially important important conseconse ences quences for evolution. Whether Whether intragenomic intragenomic conflicts conflicts affect affect the the evolution evolution quences for its evolution. of life characters in in addition to sex sex allocation allocation is, is, however, however, open open to to of life history history characters addition to question (but ( but see see the the imprinting imprinting phenomena phenomena and and parental parental conflicts conflicts about about question maternal investment investment during pregnancy in in mammals; Hurst et et al., aI., 1996). 1 996). We We maternal during pregnancy mammals; Hurst have not not discussed discussed here here the the evolutionary evolutionary consequences consequences of of the the second second type type of of have genetic resemblance, namely namely that genes affecting affecting fitness fitness but but not not genetic resemblance, that concerning concerning genes directly the the expression of the the life trait of of interest. In particular, particular, the the loss loss directly expression of life history history trait interest. In of founding events and subsequent can of genetic genetic diversity diversity associated associated with with founding events and subsequent drift drift can result fixation of have result in in the the local local or or global global fixation of deleterious deleterious mutations, mutations, which which can can have dramatic effects on population viability Saccheri et 998; Nieminen Nieminen et aI., dramatic effects on population viability ((Saccheri et aI., al., 11998; et al., 2001 Higgins and Lynch, 200 1 ) . Several (Fowler and Whitlock, 2001;; Higgins and Lynch, 2001). Several theoretical theoretical (Fowler and Whitlock, 11999; 999; Whitlock Whitlock, 2002; 2002; Couvet, empirical Whitlock et et aI., al., 2000; 2000; Whitlock, Couvet, 2002) 2002) and and empirical aI., 2002; Groom uninger, 2000) 2000) studies have investigated investigated (Haag et al., Groom and and Pre Preuninger, how metapopulation functioning affects how meta population functioning affects mutational mutational load load and and inbreeding inbreeding depression. Very little inbreeding depression depression and and hetero depression. Very little is is known known about about how how inbreeding heterosis the evolution life history in aa metapopulation. metapopulation. sis indirectly indirectly affect affect the evolution of of life history traits traits in Gandon 1 999) studied kin competition competition avoidance avoidance and inbreeding avoid Gandon ((1999) studied how how kin and inbreeding avoidance ance interact interact to to influence influence the the evolution evolution of of dispersal dispersal in in aa subdivided subdivided population population with (see also with no no local local extinction extinction (see also Perrin Perrin and and Mazalov, Mazalov, 2000). 2000). Could Could similarly similarly the the evolution evolution of of higher higher reproductive reproductive effort effort or or age-specific age-specific reproductive reproductive strate strategies inbreeding avoidance gies be be understood understood as as inbreeding avoidance mechanisms? mechanisms?
110.3 0.3
CHANGING CHANGING LIVING LIVING CONDITIONS CONDITIONS The The recognition recognition of of the the changing changing and and ephemeral ephemeral nature nature of of life life is is deeply deeply rooted rooted in in the the metapopulation metapopulation concept, concept, which which acknowledges acknowledges that, that, just just as as indi individuals, populations populations do not persist forever. Evolution of many life history traits, 994), traits, such such as as dispersal, dispersal, dormancy dormancy (Venable (Venable and and Brown, Brown, 1988; 1988; Rees, Rees, 11994), iteroparity 994), clutch iteroparity (Rees, (Rees, 11994), clutch size size (Orzack (Orzack and and Tuljapurkar, Tuljapurkar, 2001 2001),) , or or age age at at maturity 1 ) can maturity (Lytle, (Lytle, 200 2001) can be be understood understood as as adaptations adaptations to to this this fundamental fundamental uncertainty. Bet-hedging strategies diminish the risks of genotype extinction extinction by spreading spreading reproduction reproduction over over several several years years or or several several sites. sites. This This section section focuses focuses on variability in on the the evolutionary evolutionary consequences consequences of of aa different different type type of of variability in aa metapopulation: metapopulation: we we are are interested interested in in changes changes associated associated with with return return to to the the equilibrium condition condition within within disturbed disturbed populations. populations. Founding Founding events events not not only only equilibrium leave leave aa signal signal in in the the genetic genetic composition, composition, but but also also in in the the demographic demographic struc structure ture of of aa population. population. It It may may be be useful useful to to consider consider the the fact fact that that local local popula populations tions have have aa history, history, characterized characterized by by aa transient transient period period following following recolonization, recolonization, where where density, density, age age structure, structure, and and genetic genetic diversity diversity change change with with time, time, and and aa quasistationary quasistationary period period where where those those population population variables variables are are
ELIE RONCE OPHELIE RONCE AND ISABELLE ISABELLEOLIVIERI OLIVIERI OPH
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Fig. 110.2 population size size for for hypothetical "fast" "fast" (A and B) 0.2 Temporal and spatial variability in population "slow" (C and D) species. Metapopulations Metapopulations of the two two species are characterized by the same and "slow" persistence time for local populations, equal to 11/e time probability probability of expected persistence / e if e is the per time extinction. extinction. Demes of the fast species go through through a short period of transient growth growth (A: hatched area) but spend most of their lifetime in a quasistationary state; as a result, most of the occupied metapopulation at equilibrium equilibrium are at this stationary density (B). Demes patches (in black) in the metapopulation species spend a large amount amount of time in the transient phase phase (C); such temporal of the slow species amount of spatial variability in density in the metapopulation metapopulation at dynamics translates into a large amount equilibrium (D). Density now now varies not only among among empty (white) and occupied patches (black equilibrium varies not but differs differs among among occupied occupied patches patches depending time since their colonization or hatched), but depending on time their colonization growing, black for for demes at their stationary density). (hatched for for demes still growing,
approximately constant state will be approximately constant (see Fig. 10.2). 1 0.2). Note Note that that this this stationary stationary state reached at different different times times for for the different different parameters parameters (in particular, reached particular, genetic equilibrium will be reached after demographic equilibrium). If local equilibrium reached long after demographic equilibrium). extinctions and and recolonization synchronized perfectly extinctions recolonization events are not not synchronized perfectly among among different patches, such temporal different temporal changes within within populations populations translate translate into into increased spatial heterogeneity increased heterogeneity at at the the scale of of the the metapopulation metapopulation (see (see Fig. 10.2). differ not 10.2). Populations Populations in different different patches patches of of habitat habitat then then differ not only on on the the basis differences in patch basis of of intrinsic intrinsic differences patch quality quality or or stochastic stochastic factors, factors, but but also also because different amounts of elapsed since because different amounts of time time have have elapsed since their their foundation. foundation. Theoretical starting with Theoretical studies, studies, starting with the the classical classical Levins (1969) ( 1 969) metapopulametapopula tion often assume tion model, model, often assume that that the the transient transient period period in population population dynamics dynamics is relatively short as compared compared to to the the mean mean local local population population life span span (see, (see, e.g., relatively short Gandon the biology Gandon and and Michalakis, Michalakis, 1999). 1 999). Such an an assumption assumption fits fits well well the biology of of some studied metapopulations metapopulations (Heino some empirically empirically studied (Heino and and Hanski, Hanski, 2001). 200 1 ) . In that that case, we we can can safely ignore ignore the the heterogeneity heterogeneity among among patches patches recolonized recolonized at at difdif ferent ferent times and and assume assume that that local local populations populations in the the landscape landscape have have all reached some state (Drechsler reached some demographic demographic stationary stationary state (Drechsler and and Wissel, Wissel, 1997). 1 997). One One may, may, however, however, wonder wonder about about the the generality generality of of this this assumption: assumption: do do the the majormajor ity ity of of species species living living in a metapopulation metapopulation spend spend most most of of their their time time in populapopula tions tions at at equilibrium equilibrium or or in in populations populations still still recovering recovering from from the the last last disturbance? disturbance? Do Do metapopulation metapopulation dynamics dynamics select select for for life history history traits traits that that the transient transient period period following following recolonization recolonization to to be be short? short? Or Or is the the very very allow allow the concept concept of of stationary stationary demographic demographic state state a mere mere abstraction, abstraction, a concept concept difficult difficult to to reconcile reconcile with with the the ever ever changing changing demographic demographic conditions conditions that that prevail, prevail, for for instance, instance, in successional successional systems? systems? Answers Answers t0 to those those questions questions are are still unclear. unclear.
110. 0. LIFE LIFEHISTORY HISTORY EVOLUTION EVOLUTION
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Through Through several several theoretical theoretical and and empirical empirical examples, examples, this this section section illustrates illustrates how periods of transient transient dynamics or, more more generally, varying ecological con conhow foundation can affect affect the evolution of life histories in ditions with time since foundation aa metapopulation. general, spatial spatial or temporal heterogeneity metapopulation. In In general, or temporal heterogeneity in in selection selection has three possible possible consequences: consequences: (i) it may may facilitate facilitate the coexistence coexistence of geno genotypes types with with different different life life history history strategies strategies in in the the same same metapopulation, metapopulation, (ii) (ii) it it may lead lead to strategies that may to the the evolution evolution of of strategies that represent represent aa compromise compromise between between conflicting iii) it conflicting selection selection pressures, pressures, or or ((iii) it may may promote promote the the evolution evolution of of plastic plastic genotypes, genotypes, which which can can express express different different life life history history strategies strategies depending depending on on their their location location or or at at different different times times in in their their life. life. Such Such consequences consequences are are illus illustrated trated through through the the example example of of the the evolution evolution of of sex sex allocation allocation (i), (i), age age at at matur maturity (i and dispersal (ii (ii and ity (i), (i), reproductive reproductive effort effort (i and ii), ii), and and dispersal and iii). iii).
Coexistence Coexistence Sex Allocation Allocation and and Inbreeding Inbreeding
Pannell Pannell (2000) (2000) suggested suggested that that aa change change in in inbreeding inbreeding levels levels within within local local populations populations following recolonization recolonization could could allow allow the the invasion invasion of males males in a hermaphrodite metapopulation. metapopulation. Androdioecy Androdioecy has always been a puzzle for hermaphrodite evolutionary also Section 0.2). Because evolutionary biologists biologists (see (see also Section 110.2). Because males males transmit transmit their their genes pollen, they genes only only through through pollen, they must must produce produce and and disperse disperse more more than than twice twice the the quantity quantity of of pollen pollen produced produced by by hermaphrodites hermaphrodites to to be be maintained maintained in in aa population 1 978 ), but see Vassiliadis et aI., population [Charlesworth [Charlesworth and and Charlesworth Charlesworth ((1978), al., (2000) for (2000) for an an exception]. exception]. This This condition condition is is less less difficult difficult to to achieve achieve if if her hermaphrodites do not allocate their resources male and maphrodites not allocate resources fairly between between male and female female functions functions and and invest invest more more in in ovule ovule production. production. In In large large populations, populations, her hermaphrodites are only if self maphrodites are expected expected to to have have aa female-biased female-biased investment investment only if they they self a large fraction hermaphro fraction of their their ovules. Invasion of male phenotypes phenotypes in a hermaphrodite population increasing the level level of population is, however, made made more more difficult difficult by increasing selfing Charlesworth, 11984). 984). These selling in in such such hermaphrodites hermaphrodites ((Charlesworth, These paradoxical paradoxical requirements requirements could could explain explain why why androdioecy androdioecy is is so so rare. rare. Pannell Pannell (2000) (2000) claimed claimed that that metapopulation metapopulation dynamics dynamics can can generate generate the the situation situation with with aa high high female investment investment and and low inva female low selfing selfing rate rate in in hermaphrodites hermaphrodites necessary necessary for for invasion by argument involves involves the selfing rate goes sion by males. males. His His argument the change change in in selling rate that that goes along change in density following along with with the the change in density following recolonization. recolonization. Delayed Delayed selfing selling consists in consists in various various mechanisms mechanisms favoring favoring outcrossing outcrossing in in dense dense populations populations but but allowing pollen. Such allowing self-pollination self-pollination in in the the absence absence of of outcross outcross pollen. Such aa strategy strategy is is thought thought to to be be particularly particularly adaptive adaptive in in metapopulations metapopulations with with recurrent recurrent bottle bottlenecks necks that that severely severely limit limit the the availability availability of of pollen pollen donors donors as as well well as as that that of of pollinators. pollinators. Strong Strong inbreeding inbreeding in in generations generations following following recolonization recolonization tends tends to to favor favor the evolution evolution of increased increased female allocation allocation in hermaphrodites, hermaphrodites, whereas to whereas the the low low selfing selfing rate rate in in denser denser and and older older populations populations allows allows males males to invade in those those populations. populations. Such a scenario is consistent with with the fact fact that known androdioecious both plants that known androdioecious species, species, both plants and and animals, animals, are are frequently frequently found and found in in recurrently recurrently disturbed disturbed habitats, habitats, have have highly highly variable variable sex sex ratios, ratios, and are are known known or or suspected suspected to to vary vary in in selfing selfing rates rates in in aa density-dependent density-dependent way (see the Pannell, 2000 verbal model model of Pannell (2000) way (see the review review in in Pannell, 2000).) . The The verbal of Pannell (2000) does does not not allow allow us us to to conclude conclude rigorously rigorously that that coexistence coexistence of of males males with with
ELIE RONCE OPH OPHI~LIE RONCE AND ISABELLE ISABELLEOLIVIERI OLIVIERI
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hermaphrodites hermaphrodites in in such such aa metapopulation metapopulation is is stable, stable, but but suggests suggests that that aa change change population dynamics dynamics may cre crein inbreeding levels during transient stages of population ate ate some some evolutionary evolutionary window window favorable favorable to to the the establishment establishment of of different different life life history history strategies. strategies. Age Age at at Reproduction Reproduction
Changes in population period of growth Changes population density during during the transient transient period growth fol following recolonization also affect history strategies lowing recolonization can can also affect the the evolution evolution of of life life history strategies because it recruitment of uveniles born because it affects affects the the probability probability of of successful successful recruitment of jjuveniles born in populations times. The populations established established for for different different times. The evolution evolution of age at repro reproduction duction in the the perennial perennial monocarpic monocarpic Carlina Carlina vulgaris vulgaris provides provides an illustra illustration tion (de (de Jong Jong et et ai., al., 2000). 2000). In In semelparous semelparous organisms, organisms, age age at at reproduction reproduction represents delaying reproduction represents aa compromise compromise between between the the benefits benefits of of delaying reproduction to to attain aa greater greater size size and higher fecundity attain and thus thus higher fecundity and and the the risk risk of of dying dying before before reproduction. reproduction. Single population population models models based on this basic trade-off, trade-off, how however, largely overpredict overpredict the age of first reproduction reproduction observed observed in natural natural populations of populations of C. C. vulgaris. vulgaris. In In the the system system studied studied by by de de Jong Jong et et ai. al. (2000), (2000), the the rare rare thistles thistles grow grow in in ephemeral ephemeral patches patches at at the the edge edge of of willow willow scrub. scrub. Local populations are Local populations are founded founded by by seeds, seeds, grow grow to to reach reach aa peak peak density, density, and and then decline to 99 6 ) . Recruitment then decline to extinction extinction (Klinkhamer (Klinkhamer et et ai., al., 11996). Recruitment of of seeds seeds is is highly highly variable variable among among patches, patches, being being high high in in the the period period of of population population growth growth but but very very low low in in the the period period of of decline. decline. Rosette Rosette survival survival is is not not affected affected by density, but varies by density, but varies widely widely between between patches patches within within aa year. year. Peaks Peaks of of den density are asynchronized asynchronized among among patches patches and and are probably probably related related to vegeta vegetation succession. tion succession. de de Jong Jong et et ai. al. (2000) (2000) incorporated incorporated these these features features in in aa metapopulation metapopulation model model with with local extinctions extinctions and limited dispersal between between patches. patches. The The predicted predicted age closer to age at at maturity maturity for for C. C. vulgaris vulgaris was was then then closer to that that observed observed in in natural natural popu populations than lations than the the age age predicted predicted by by the the single single population population model model for for the the same same rosette instances, they rosette mortality. mortality. In In some some instances, they found found that that two two genotypes genotypes with with dif different reproduction could could coexist coexist in same metapopulation. ferent ages ages at at reproduction in the the same metapopulation. In In the the period of following colonization, plants reproducing early are period of growth growth following colonization, plants reproducing early are favored favored because their progeny benefits from benign competitive because their progeny benefits from more more benign competitive conditions conditions than than progeny born later in Once safe become scarce, progeny born later in denser denser populations. populations. Once safe sites sites have have become scarce, it, however, delay reproduction reproduction to it, however, pays pays to to delay to reach reach aa higher higher fecundity fecundity and and the the late-reproducing genotypes are late-reproducing genotypes are advantaged. advantaged. Reproductive Reproductive Effort
The latter was obtained The latter results results bear bear close close resemblance resemblance to to what what was obtained about about the the evolution evolution of of reproductive reproductive effort effort in in aa metapopulation metapopulation (Ronce (Ronce and and Olivieri, Olivieri, 11997). 997). We We indeed indeed similarly similarly found found situations situations where where two two genotypes genotypes with with dif different population with ferent reproductive reproductive effort effort strategies could coexist coexist in a meta metapopulation with a transient but neither neither in an unstructured unstructured population transient period period of growth, growth, but population nor nor in in aa metapopulation metapopulation where where the the stationary stationary density density would would be be reached reached imme immediately diately upon upon recolonization. recolonization. Within Within each each population, population, genotypes genotypes with with aa higher reproductive are selected for during reproductive effort effort are during the period period of growth growth because they occupy more rapidly, they are are replaced because they occupy space space more rapidly, but but they replaced progressively progressively by by genotypes genotypes with with aa higher higher investment investment in in survival survival as as soon soon as as juvenile juvenile
241 247
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Time since foundation two genotypes with with different different reproductive reproductive efforts in populations Fig. 110.3 0. 3 Frequency of two populations established for for different different times. In white, the genotype genotype with with the larger reproductive reproductive effort (49% (49% reproduction); in black, the genotype genotype with with the lower lower reproductive reproductive of resources allocated to reproduction); (41% to reproduction). The arrow arrow indicates the the time at which the effort (41 % of resources allocated to stationary ceiling density is reached within within local populations. same model populations. Generated from from the same model (1997), with different different parameters as in their Fig. Fig. 9. as in Ronce and Olivieri (1 997), with
recruitment becomes adult survival (Fig. 10.3). 1 0. 3 ) . Such a recruitment becomes severely limited by adult prediction is consistent frequency in recently recently founded founded popupopu prediction consistent with the higher higher frequency lations lations of genotypes investing in sexual sexual reproduction reproduction rather rather than than asexual asexual growth observed in the erectum along along a succesSparganium erectum succes growth as observed the macrophyte macrophyte Sparganium sional gradient (Piquot (Piquot et aI., al., 11998). sional gradient 99 8 ) . competition-colonization trade-off trade-off (see Cases ooff coexistence coexistence mediated mediated bbyy a competition-colonization and Gouyon, Gouyon, 11997) at the Olivieri and 997) often involve conflicting selection pressures pressures at level of the meta metapopulation of each population population and and at at the level of population [see [see Olivieri et al. ai. (1995) ( 1 995) for polymorphism and and Jansen Jansen and and Mulder Mulder (1999) ( 1 999) for for dispersal dispersal polymorphism reproductive present example example (Ronce and Olivieri, 1997), 1 997), reproductive effort]. The present together with that Jong et ai. (2000) (200 0 ) about about age at maturity, suggests that that together with that of of De Jong et al. at maturity, the succession succession of of antagonistic antagonistic selection selection pressures pressures with with time time within within each each local the population may broaden broaden the the range of conditions conditions allowing allowing the the maintenance maintenance of of population may range of protected polymorphism polymorphism in a metapopulation. metapopulation. Empirical Empirical evidence evidence on on sideside protected Uta stansburiana, stansburiana, similarly suggest that that temporal temporal changes changes in blotched lizards, lizards, Uta blotched help maintain maintain polymorphism polymorphism for for clutch clutch size strategies strategies (Sinervo ( Sinervo et et al., aI., density help 2000). 2000) . Orange-throated Orange-throated females producing producing large clutches clutches of of small offspring offspring are favored during the the period period of of population population growth, whereas yellow-throated yellow-throated are favored during growth, whereas females producing fewer offspring producing fewer offspring of of higher higher quality quality are are favored favored at at high high The previous previous theoretical theoretical models models also also lead us to to think think differently densities. The about colonization: colonization: successful colonization is a complex complex process, process, which which about successful colonization not only only arriving arriving in empty empty patches patches of of habitat, habitat, but but also an an efficient efficient implies not strategy of of space space occupation occupation once once arrived arrived [see also also Bolker Bolker and and Pacala Pacala (1999) ( 1 999) strategy on on the the same same topic]. topic] .
ELIE RONCE OPH OPHF_LIE RONCE AND ISABELLE ISABELLEOLIVIERI OLIVIERI
248 248
Evolutionary Evolutionary Compromises Compromises Length Length of of the the Growth Growth Period: Effect of of Productivity Productivity
Classic life history theory has for long shown that that an increase in produc producthat fecundity increases, leaving survival unchanged, unchanged, should select tivity, such that for a higher reproductive reproductive effort in an exponentially exponentially growing growing population population (Charnov 973). In (Charnov and and Shaffer, Shaffer, 11973). In aa density-regulated density-regulated population population where where juve juveniles establish only in safe micro sites freed by the death of an adult, microsites adult, increasing productivity has, however, however, no effect on the evolution of reproductive reproductive effort (see Kisdi and Meszena, 11995) 995) because the number number of recruited offspring is then limited by adult mortality and not by productivity. In a meta population metapopulation where populations populations go through through a period of transient growth growth after recoloniza recolonization, the evolutionary pattern pattern is strikingly different from from that that predicted predicted in a population. The ES reproductive effort increases but then decreases with single population. 0AB, and similar pattern increasing productivity [Fig. [Fig. 110.4B, pattern in Ronce and Olivieri ((1997) 1 997) and Ronce et aI., al., (2000c), but see Crowley and McLetchie, (2002) (2002),, for a different conclusion]. conclusion]. Such a nonmonotonic nonmonotonic response reflects the conflict in selection pressures in the metapopulation. metapopulation. Within each growing local popula population, increasing productivity selects for increasing reproductive reproductive effort, but the period of growth growth during which higher reproductive reproductive efforts are favored favored also becomes shorter as local populations populations grow faster and and reach their stationary state sooner (Fig. 0 AA). At any point of time, the fraction of local popula (Fig. 110.4A). populations in the landscape landscape still in the transient transient phase of growth declines with increasing productivity. Lower reproductive reproductive efforts are selected for at the scale of the metapopulation. metapopulation. This example illustrates how changes in demographic parameters affect the evolution of life life history traits, not only by changing selection pressures within local populations, populations, but also by changing the very composition of the landscape. Longer periods periods of transient transient growth tend to select for lower dispersal dispersal rates in the metapopulation Olivieri et al., aI., 11995, 995, Ronce et aI., metapopulation ((Olivieri al., 2000b,c; see also Crowley and McLetchie, 2002). In particular, ES dispersal rates rates decrease decrease with 0AB, and also Fig. 6 in Crowley and decreasing productivity (see (see Fig. 110.4B, McLetchie, 2002). The availability of safe sites within recently founded, low lowdensity local populations populations indeed makes the venture of risking death to reach an an empty empty patch patch of of habitat less less worthwhile worthwhile compared compared to to aa situation situation where where almost all occupied patches are fully crowded 0.4A). Ellner and crowded (Fig. (Fig. 110.4A). Schmida ((1981) 1 98 1 ) used a similar argument argument to explain the rarity of long-range dispersal adaptations aI. ((1990) 1 990) adaptations in desert floras. Consistently, Wilson et al. showed that the relative frequency of plant plant species whose seeds are dispersed by by wind wind or or vertebrates vertebrates increased increased along along aa fertility fertility gradient gradient in in Australian Australian forests, forests, whereas the frequency of species with ant-dispersed seeds or no special dis dispersal device decreased. Coupling between between Local Local and Regional Dynamics: Dynamics: Dispersal and Extinctions Extinctions
Dispersal has been perceived as an adaptation adaptation to habitat habitat instability. The fre fre962; Denno, quency of of winged winged species species in in insect communities communities (Southwood, (Southwood, 11962; Denno, 11994) 994) or of winged morphs 994; morphs in populations populations of the same species (Roff, 11994; Denno aI., 11996) 996) decreases with increasing stability of the Denno et al., the habitat. habitat. Most Most
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models predict that ES dispersal rates should increase monotonically monotonically with with an increasing frequency of of local extinction extinction (see, e.g., Gandon Gandon and Michalakis, growth fol fol11999). 999). Such models generally assume that the period of transient growth lowing lowing recolonization recolonization is so short that it can be neglected neglected altogether. By relax relaxing this assumption, two two deterministic models (Ronce et aI., al., 2000b; 2000b; Parvinen et aI., al., 2003) 2003) found found that the ES dispersal rate may vary nonmonotonically nonmonotonically with with the frequency of of catastrophic catastrophic extinctions. extinctions. On the one hand, more frequent frequent dis disturbances turbances create more empty empty patches patches in the landscape and thus more coloniz colonizwhich selects for increasing increasing dispersal ability. On the other ing opportunities, which hand, disturbances also affect local dynamics indirectly by reducing the total metapopulation size and thus the average number of of immigrants arriving in any given patch. As a result, local populations populations start off off smaller and grow grow longer before reaching some equilibrium density. The longer growth growth period and higher frequency of of low-density low-density local populations populations tend to select against
250
ELIE RaNCE AND ISABELLE OPH OPHELIE RONCE AND ISABELLEOLIVIERI OLIVIERI
frequent dispersal. The conflict between the effects of disturbance at the two spatial scales results in a hump-shaped pattern of dispersal evolution (Ronce et aI., al., 2000b). Note that that it is not so much the inclusion of a transient phase of local population growth that that created this pattern, pattern, but the coupling between local and global dynamics in the metapopulation. metapopulation. For instance, Crowley and McLetchie (2002) assumed a fixed number of founders per new population and no subsequent immigration (see (see Table 110.2): 0.2): they found found that ES dispersal rates always increased with increasing frequency of local extinction because increasing the fraction of empty patches in the landscape then had no effect on the local dynamics. It has, however, been questioned whether whether the decreasing part of the hump humpshaped curve relating dispersal to local extinctions could be observed in nature, the most serious objection being that in the presence of demographic stochasticity, metapopulations may not persist sufficiently long in this high disturbance regime to disturbance regime to observe observe the the evolution evolution of of declining declining dispersal dispersal rates rates (Heino and Hanski, 2001 2001;; Poethke et aI., al., 2003 2003).). Our deterministic model (Ronce et al., aI., 2000b) assumes that that the global landscape composition composition affects the local dynamics, but does not incorporate the reverse feedback (but see Parvinen et al., aI., 200 3 ) . This might not describe accurately the more general 2003). situation where environmental fluctuations combine with demographic sto stochasticity (see Heino Heino and 1 ). Poethke chasticity to to lead lead to to extinction extinction (see and Hanski, Hanski, 200 2001). Poethke et et ai. al. (2003 (2003)) showed that, in the latter situation, the pattern pattern of covariation between dispersal rates and extinction rates is much more complex than previously envisioned. This happens because the evolving dispersal rate itself then affects the probability of extinction. In particular, increasing environmental variabil variability ity may may select select for for larger larger dispersal dispersal rates, rates, which which in in turn turn reduces reduces the the local local extinc extinction because of Kodric-Brown, 11977). 977). tion rate rate because of the the rescue rescue effect effect (Brown (Brown and and Kodric-Brown, Dispersal and local extinctions, both dynamic parameters, may then covary positively or negatively along gradients of environmental variability or dis disai. (2003) suggested that the two latter param persal mortality. Poethke et al. parameters would be better predictors predictors of the ES dispersal rates than the frequency of local extinction. Habitat Habitat Templates Templates and and Life History History Syndromes Syndromes
Taking into consideration transient population dynamics deeply affects our understanding of life life history syndromes. In particular, we showed that pre predicted patterns of association between reproductive effort and dispersal were strikingly different in ""fast fast species" where local populations reach their equi equilibrium density immediately upon recolonization and in "slow species" where the length of the growth period varies dynamically with landscape and life his history tory characteristics characteristics (Ronce (Ronce et et aI., al., 2000c). 2000c). First, First, both both changes changes in in dispersal dispersal and and reproductive effort effort can affect population dynamics and the length of the tran transient phase or, more generally, the distribution of population densities in the landscape. Evolution of one trait may thus modify the selection pressures on the other indirectly through changes in demographic dynamics, just as they do through changes in the genetic structure of the metapopulation (see (see Section 110.2). 0.2). Two models (Ronce et aI., al., 2000c; Crowley and McLetchie, 2002), how however, suggest that such evolutionary interactions may be of minor importance
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Fig. 10.5 1 0.5 Covariation Covariation among among dispersal and reproductive reproductive effort effort evolving evolving jointly jointly along along envienvi Fig. ronmental gradients. g radients. (A) Along a gradient of habitat fragmentation fragmentation in a fast fast species ronmental (A) Along of increasing habitat (in black) black) where where ceiling ceiling density density is is reached reached immediately immediately upon upon recolonization recolonization and and in in aa slow slow (in species (in (in white) white) with with variable variable periods periods of transient growth; the dispersal dispersal cost cost varies varies from from 00 to species of transient growth; the to 0.9. (B) Along a gradient of increasing disturbance in a fast species 0.9. species (in black) and in a slow white); the local extinction rate varies from respectively, the fast species (in white); from 0.1 to to 0.9 or 0.6 for, for, respectively, species or the slow species. Parameters and results results for the ES ES life life history history traits traits are are as as in in Fig. Fig. 33 species or the slow species. Parameters and for the and 44 in in Ronce Ronce et and et al. al. (2000c). (2000c).
(see Table 0.2). Second, because both both traits respond in in opposite opposite directions to (see Table 110.2). Second, because traits respond directions to change in the length of the transient transient period, we may often expect negative corcor relations between reproductive reproductive effort and dispersal along environmental environmental gra grarelations dients (see Fig. Fig. 10.5), 1 0.5), contrary to the the classical colonizer syndrome syndrome (for (for further further dients (see contrary to classical colonizer discussion see Ronce Ronce et aI., 2000c). 2000c). Again, Again, aa clearer understanding of of such life discussion see et al., clearer understanding such life history syndromes syndromes may history may be be achieved achieved by by acknowledging acknowledging the the fact fact that that aa change change in landscape structure in the the landscape structure or or quality quality will will affect affect the the distribution distribution of of population population numbers at two two different spatial scales. The fraction of occupied patches patches in the numbers landscape occupied space space within patches can can be landscape and and the the fraction fraction of of occupied within occupied occupied patches be considered considered two two measures of the degree of resource resource monopolization monopolization in the metapopulation, reflecting, metapopulation, reflecting, respectively, respectively, the the intensity intensity of of global global or or local local com competition. Unexpected life history syndromes can emerge because the evolution of responds differently of long-distance long-distance dispersal, dispersal, contrary contrary to to reproductive reproductive effort, effort, responds differently to change in intensity of local scales. to change in the the intensity of competition competition at at the the global global and and local scales. While While decreased competition competition at the global scale favors increased allocation to repro reproduction and dispersal, decreased competition locally selects for more repro reproduction duction but but less less dispersal. dispersal.
Plasticity Plasticity Density-Dependent Life Histories
If If the the benefits benefits of of dispersal dispersal vary vary with with local local density, density, as as suggested suggested by by previous previous theoretical work, it is quite natural to expect the evolution of dispersal strate strategies predictions and gies conditional conditional on on local local density. density. Theoretical Theoretical predictions and empirical empirical evi evidence dence for for density-dependent density-dependent dispersal dispersal strategies strategies are are reviewed reviewed in in detail detail in in Chapter 3 . While attention Chapter 113. attention to density-dependent dispersal in a metapopula metapopulation has certainly increased in recent years (see, e.g., theoretical work by J
252 252
ELIE RaNCE AND ISABELLE OPH OPHI~LIE RONCE AND ISABELLEOLIVIERI OLIVIERI
and Gyllenberg, 200 1 ; Poethke 2001; Poethke and and Hovestadt, Hovestadt, 2002), the same conclusion conclusion not hold hold about about the evolution of other other plastic life history characteristics. characteristics. does not Yet given the conflicting conflicting selection pressures acting on reproductive reproductive effort or maturity in growing and and crowded crowded populations, populations, one might might expect simi simiage at maturity larly decisions may larly that that such such life life history history decisions may depend depend on on the the density density of of neighbors. neighbors. Quintana-Ascencio al. ((1998) 1998) found Quintana-Ascencio et al. found that that the reproductive reproductive effort of trans transplanted planted Hypericum Hypericum cumilcola cumilcola changed changed with with time time since since fire fire and and the the density density of of conspecifics conspecifics in patches patches of Florida Florida rosemary rosemary scrub. Variation Variation in mean life his hiswith population population age observed in natural natural populations populations is often often consistent consistent tories with both succession through both with the succession through time of genotypes with different different allocation expression of plastic allocation strategies (see, (see, e.g., strategies and with the expression Houssard 995). Houssard and and Escarre, Escarr~, 11995). Time n d Individual Time since Foundation Foundation aand Individual Age Age
Many Many population population characteristics characteristics other other than than density might might change with with time since foundation, consequences for the evolution conditional disper foundation, which consequences evolution of conditional dispersal behaviors or generally conditional conditional life sal behaviors or more more generally life history history strategies strategies have have been been lit little explored. explored. Drastic changes in the age structure structure may occur during during the transient phase following recolonization, transient phase following recolonization, especially especially in in those those species species where where the the age structure structure of migrants migrants does not not reflect the stable age structure structure of local popu populations 996). If competitive lations (see, e.g., Joly and Grolet, 11996). competitive abilities or, more gen generally, demographic parameters, such as fecundities demographic parameters, fecundities or survival rates, vary among will generate among age age classes, classes, the the disequilibrium disequilibrium in in age age structure structure will generate aa large large variation in recruitment variation recruitment probabilities probabilities or expected reproductive reproductive success among among populations founded populations founded for for different different times, even if they have the same density. Patterns genetic diversity diversity may change with (Whitlock and Patterns of genetic with population population age (Whitlock McCauley, 11990), 990), affecting the level of within local population population relatedness compared compared to genetic resemblance resemblance to individuals individuals from from other other local populations. populations. One One might might then then expect expect the the evolution evolution of of variable variable kin kin competition competition avoidance avoidance depending population age. We are, however, aware of no study trying to depending on population relate patterns patterns of change through through time in the genetic structure structure of populations populations to expression of to the the expression of altruistic altruistic behaviors. behaviors. The The probability probability of of extinction extinction of of aa local also depend depend on since foundation, foundation, either either local population population may may also on the the time time elapsed elapsed since through change in population through a change population size and demographic demographic stochasticity stochasticity or because of negative or positive temporal temporal autocorrelation autocorrelation in disturbance disturbance events. In many either exploiting many species species either exploiting an an ephemeral ephemeral and and nonrenewable nonrenewable resource resource or or subject to successional replacement, replacement, the very quality of the habitat habitat deteriorates deteriorates with time (Valderde and Silvertown, 1998). This led Olivieri and Gouyon Gouyon ((1997) 1 997) to predict that that ES conditional conditional dispersal in a metapopulation metapopulation should should increase like that increase with with population population age. age. A A marginal marginal value value argument argument like that used used by by Metz and Gyllenberg (200 1 ) and Poethke (2001) Poethke and and Hovestadt Hovestadt (2002) when study studying density-dependent density-dependent dispersal enabled us to predict predict the ES reaction reaction norm norm for dispersal age for dispersal as as aa function function of of local local population population age for aa species species subject subject to to succes successional replacement, replacement, with no constraint sional constraint on the shape of the reaction reaction norm norm (0. Ronce, S. Brachet, 1. Gouyon and (O. Ronce, S. Brachet, I. Olivieri, Olivieri, P.-H. P.-H. Gouyon and J. J. Clobert, Clobert, unpublished unpublished result). Our Our analysis does not not incorporate incorporate kin selection effects. In most most 1 997) of an cases, we verified the general prediction prediction of Olivieri and Gouyon Gouyon ((1997) increasing increasing dispersal dispersal rate with with time since foundation. foundation. In details, the patterns patterns of
253 253
LIFEHISTORY HISTORY EVOLUTION 110. 0. LIFE
variation variation in in optimal optimal dispersal dispersal rates rates can can be be more more complex complex than than envisioned envisioned in in during the population their simple study, with several outbreaks of dispersal during density is lifetime (Fig. 110.6.A). 0.6.A). Such oscillations occur when equilibrium density reached reached before before the the stable stable age age structure: structure: the the peaks peaks match match the the oscillations oscillations in in age age structure 0.6A). A structure during during the the transient transient period period (Fig. (Fig. 110.6A). A consequence consequence of of such such con conditional ditional dispersal dispersal strategy strategy is is that that high high rates rates of of emigration emigration in in old old populations populations accelerate the rate of successional replacement and shorten shorten the life span of the accelerate population. population. This This population population level level pattern pattern interestingly interestingly parallels parallels what what happens happens in the evolution of senescence at the individual scale. These These highly highly theoretical theoretical considerations considerations do do not not help help us us understand understand how how indi individuals viduals may may acquire acquire information information about about the the age age of of their their population population and and adjust adjust candidate cue for population age their dispersal strategy accordingly. A candidate popucould be the age of the individual itself in a long-lived species in which popu lations lations take take aa long long time time before before reaching reaching their their stable stable age age structure. structure. We We indeed indeed showed that in the presence of strong variance in age structure among popula populashowed founded for for different times, juvenile dispersal strategies conditional on tions founded
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Individual age Individual age Fig. . 6 Evolution Fig. 11 00.6 Evolution of of conditional conditional dispersal dispersal strategies strategies conditional conditional on on individual individual age age or or deme deme (A) Variation Variation of of the the mean mean dispersal dispersal rate rate per per local local deme deme as as aa function function of of time time since since foundafounda age. age. (A) tion for plastic tion for plastic ES ES dispersal dispersal strategies. strategies. Large Large dots: dots: ES ES dispersal dispersal strategy strategy conditional conditional on on deme deme age age when information about when individuals individuals have have access to to complete complete information about time time since since foundation. foundation. Small Small dots: mean age of of individuals individuals per per deme deme as a function function of of time time since foundation. foundation. Continuous Continuous line: mean mean dispersal rate per local local deme deme when when dispersal dispersal varies varies with with individual individual age age (see B) and and individuals individuals have have dispersal rate per (see B) no of demographic information about about time time since since foundation. foundation. (B) (B) Variation Variation of demographic parameters parameters as as aa funcfunc no information tion age-specific fecundity. age-specific survival tion of of individual individual age. age. Long Long dashes: dashes: age-specific fecundity. Short Short dashes: dashes: age-specific survival rate. Continuous line: line: age-specific dispersal rate when individuals individuals do do not not have have access to to information information Continuous age-specific dispersal rate when about time time since since foundation foundation (generating (generating the the continuous continuous line line in in A). A). Demes Demes persist persist for for 80 80 yr yr with with aa about probability of of local local extinction extinction of of 0.1 0.1 per per year. After 80 yr, local local demes demes go go extinct extinct and and remain remain probability After 80 uncolonizable until until the the arrival of the the next next disturbance disturbance (which (which occurs occurs with with aa probability probability of of 0.01) 0.01 ) to to uncolonizable arrival of mimic mimic the the effect effect of of successional successional replacement. replacement. All All demes demes have have aa constant constant size size of of 200 200 adults. adults. Only Only juveniles patches or juveniles disperse disperse and and establish establish either either in in empty empty patches or within within occupied occupied patches patches in in microsites microsites freed freed by by the the death death of of adults. adults. Seventy Seventy percent percent of of the the dispersing dispersing juveniles juveniles die during during migration. migration.
OPH ELIE RONCE RONCE AND AND ISABELLE ISABELLE OLIVIERI OLIVIERI OPHF.LIE
254 254
maternal age age could could be be selected selected for for in in aa metapopulation metapopulation (Fig. (Fig. 10.6.B; 1 0.6.B; O. O. Ronce, Ronce, maternal S. Brachet, Brachet, I. 1. Olivieri, Olivieri, P.-H. P.-H. Gouyon Gouyon and and J. J. Clobert, Clobert, unpublished unpublished result). result). A A S. variation in mean mean dispersal dispersal rates rates among among populations populations due due to to the the combination combination of of variation age structure structure and and age-specific age-specific dispersal dispersal rates rates results results in in aa pattern pattern mimmim variable variable age icking the the optimal optimal reaction reaction norm norm that that would would evolve evolve if if individuals individuals had had complete complete icking information about about the the time time of of foundation foundation of of their their population population (Fig. (Fig. 10.6.A). 1 0.6.A). information This example example illustrates illustrates how how metapopulation metapopulation processes processes may alter the the evolution evolution This may alter of age-specific age-specific life history strategies. Similar processes processes might might affect affect the the evoluevolu of life history strategies. Similar tion of of age-specific age-specific reproductive reproductive effort effort or or rates rates of of senescence. senescence. How How much of the the tion much of classic life life history history theory theory would would be be modified modified in in aa metapopulation metapopulation with with aa varivari classic able age age structure? structure? able
Change in in Living Living Conditions: Conditions: Conclusion Conclusion Change Previous authors authors have have found found it it useful useful to to distinguish distinguish between between selection selection Previous pressures the local level to to understand understand the the evolution evolution of of pressures acting acting at at the local and and global global level life history population ((Olivieri Olivieri and 997). We have life history traits traits in in aa meta metapopulation and Gouyon, Gouyon, 11997). We have view by by showing of extended this view showing through through several several examples examples how how aa variation variation of extended this selection within local local populations, populations, ultimately selection pressures pressures within ultimately due due to to the the disequi disequilibrium generated librium generated by by local local extinction, extinction, could could affect affect the the evolution evolution of of life life his history that the tory traits. traits. The The same same theoretical theoretical examples examples suggest suggest that the evolutionary evolutionary transient dynamics may depend between importance of such such transient dynamics may depend on on the the coupling coupling between importance of landscape processes, extinction and recolonization, and local landscape processes, such such as as extinction and recolonization, and local growth. patches of the growth. Immigration Immigration and and recolonization recolonization of of empty empty patches of habitat habitat are are the two connect the dynamics within local population population to two events events that that connect the dynamics within each each local to the the regional state regional state of of the the metapopulation. metapopulation. Although Although the the demographic demographic conse consequences quences of of this this coupling coupling have have started started to to be be explored explored both both theoretically theoretically and and empirically see, e.g., e.g., the empirically ((see, the relationship relationship between between species species abundance abundance and and distri distribution 999), little about its bution as as discussed discussed in in Hanski, Hanski, 11999), little is is known known about its impact impact on on life life history evolution. We have presented a few examples where this coupling counterintuitive patterns patterns of evolution. However, much results in initially counterintuitive achievement population theory realized by achievement in in meta metapopulation theory has has been been realized by uncoupling uncoupling the the local local and and regional regional dynamics, dynamics, relying relying on on the the timescale timescale difference difference between between the the two 997; see two processes processes (Drechsler (Drechsler and and Wissel, Wissel, 11997; see also also Chapter Chapter 4). 4). Theory Theory based based on on such such assumptions assumptions has has often often led led to to remarkably remarkably robust robust conclusions conclusions 2002). Some Some authors authors have have also also argued argued that that this this very very uncoupling uncoupling (Keeling, 2002). might define define what what is is aa "real" "real" or or "classic" "classic" metapopulation metapopulation rather rather than than aa sim simple 997). We ple patchy patchy population population (Harrison (Harrison and and Taylor, Taylor, 11997). We do do not not share share this this point point of of view. view. Rather, Rather, we we think think that that details details of of the the focal focal species species biology biology deter determine mine the the degree degree of of coupling coupling between between processes processes at at different different spatial spatial scales scales and and that that any any feedback feedback between between local local and and global global dynamics dynamics should should be be neglected neglected only only with with great great caution. caution. This This is is especially especially true true of of modeling modeling exercises. exercises. For For instance, instance, Crowley Crowley and and McLetchie McLetchie (2002) (2002) consider consider that that the the average average number number of of founders founders per per new new population population is is independent independent of of the the global global demographic demographic state state of of the the metapopulation, metapopulation, while while we we assume assume that that the the number number of of empty empty patches patches in in the the landscape landscape is is independent independent of of the the distribution distribution of of local local popula population tion sizes sizes (Ronce (Ronce et et aI., al., 2000b,c). 2000b,c). More More realistically, realistically, both both parameters parameters vary vary dynamically dynamically with with the the number number of of migrants migrants in in the the metapopulation: metapopulation: Both Both mod models els therefore therefore miss miss part part of of the the real real picture. picture. Data Data are are sorely sorely needed needed to to assess, assess,
110. 0. LIFE LIFEHISTORY HISTORY EVOLUTION EVOLUTION
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in a diversity diversity of organisms, organisms, the extent extent and and importance importance of the transient transient phase phase of population population dynamics dynamics and the degree degree of coupling coupling between between regional and and local dynamics.
110.4 0.4
GENERAL GENERAL CONCLUSION CONCLUSION
Perspectives Perspectives for for Theoretical Theoretical Studies Studies of of Life Life History History Evolution Evolution in in a a Metapopulation Metapopulation The The two two parts parts of of this this chapter chapter unfortunately unfortunately reflect reflect aa major major division division in in the theoretical population context. Studies oretical studies of life history history evolution in a meta metapopulation that that have investigated the effect of genetic structure on life history evolution have generally ignored the complex population dynamics generated by extinc extinction tion and and recolonization recolonization and and vice vice versa. versa. Such Such aa theoretical theoretical gap gap is is essentially essentially explained by the difficulty to analyze kin selection models with with complex complex demography. Individual-based simulations allow the consideration consideration of both both demographic demographic and genetic stochasticity, but it remains difficult to disentangle processes and to identify the mechanisms leading to some evolutionary evolutionary pat pattern. Without Without a mechanistic understanding understanding of such patterns, patterns, it may be difficult to determine their generality. Two methods have been proposed proposed to integrate genetic drift drift and demographic demographic stochasticity in the same analytical framework framework (Metz and Gyllenberg, 2001; Rousset and Ronce, 2004). 2004). The latter bears more resemblance resemblance to to classic classic kin kin selection selection models, models, expressing expressing selection selection gradients gradients as as functions of probability probability of genetic identity between between pairs of individuals found found in the same or different local populations. populations. The effect of kin selection processes can probabilities of can then then be be assessed assessed readily readily by by setting setting all all probabilities of identity identity to to zero. zero. The The application particular class of models application of such methods methods is still restricted to a particular ((infinite infinite island models with no age structure) and relies on extensive numernumer ical ical computations. Still, they they offer offer the the exciting exciting possibility possibility to to better better identify identify and and disentangle the effects of demography demography and genetic structure structure acting acting on the evo evolution of life history traits in a meta population. Finally, with the exception of metapopulation. models for for dispersal evolution (Heino and Hanski, Hanski, 2001 2001;; Travis and Dytham, Dytham, 11999), 999), the evolution of life history history traits traits in a metapopulation metapopulation context context has essentially been modeled modeled in a spatially implicit way. We therefore therefore know know very little about about how how the details of spatial configuration configuration of patches may affect the previous models' predictions. A better better understanding understanding of this question could be achieved achieved by by aa careful careful comparison comparison of of spatially spatially explicit explicit simulations simulations and and analyt analytical models as we did here with the work of Gandon 1 999) Gandon and Michalakis ((1999) and Heino 1 ). Heino and Hanski Hanski (200 (2001). Most this chapter make predictions Most of of the the theoretical theoretical studies studies reviewed reviewed in in this chapter make predictions about population, about evolutionary evolutionary stable life history history strategies strategies in a meta metapopulation, evolutionary end points, points, without without worrying worrying about about realistic realistic amounts amounts of i.e., evolutionary genetic variation variation maintained maintained in such such metapopulations, metapopulations, which eventually fuels fuels evolutionary evolutionary change. change. Both Both genetic genetic drift drift and and demographic demographic asymmetries asymmetries not not only affect the direction direction of selection, but but also constrain constrain the evolutionary see Chapters 6) . Although response to selection ((see Chapters 9 and and 116). Although an increasing num number ber of of theoretical theoretical studies studies address address questions questions related related to to the the balance balance among among selection, dispersal, and and drift in a metapopulation metapopulation context, context, little is still
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ELIE RONCE OPH AND ISABELLE OPHI~LIE RONCE AND ISABELLEOLIVIERI OLIVIERI
known known about about how how constraints constraints on selection selection affect the evolution evolution of of particular particular Boughton ((1999) provided a nice empirical empirical example in the life history traits. Boughton 1999) provided checkerspot butterfly checkerspot butterfly Euphydryas Euphydryas editha, editha, where where extinction-recolonization extinction-recolonization dynamics dynamics and and patterns patterns of of gene gene flow flow severely severely constrain constrain the the evolution evolution of of host host preference. preference.
Perspectives Perspectives for for Empirical Empirical Studies Studies of of Life Life History History Evolution Evolution in a Metapopulation in a Metapopulation As As stated stated in in the the introduction, introduction, the the present present chapter chapter strongly strongly reflects reflects the the bias bias affecting affecting the the field, field, with with aa scarcity scarcity of of empirical empirical studies. studies. Data Data are are needed needed to to assess assess the the validity validity of of model model assumptions assumptions about about the the genetic genetic structure structure or or the the demography thought to have evolved demography of of species species thought to have evolved in in aa metapopulation metapopulation context. context. Obtaining Obtaining the the latter latter type type of of information information can can be be extremely extremely demanding demanding as as it it often often requires requires long-term long-term surveys surveys at at multiple, multiple, and and sometimes sometimes very very large, large, spatial spatial scales. scales. Such Such data data accumulated accumulated on on aa few few systems systems now now allow allow us us to to ask ask evolu evolutionary (as the evolution tionary questions questions about about such metapopulations metapopulations (as evolution of dispersal bserving evolutionary and and host host preference preference in in M. M. cinxia). cinxia). O Observing evolutionary changes changes at at the the scale scale of population is challenging task. of aa meta metapopulation is aa challenging task. Patterns Patterns of of variation variation in in life life history history traits traits along along environmental environmental or or successional successional gradients gradients offer offer an an alternative alternative for for testing evolutionary evolutionary model model predictions, predictions, but but it requires evaluating evaluating the relative role of role of genetic genetic differences differences and and plasticity plasticity in in the the production production of of such such patterns. patterns. Proving mechanism as in the explained the Proving that that the the same same mechanism as envisioned envisioned in the model model explained the observed represents another observed pattern pattern represents another difficulty difficulty of of this this approach, approach, as as for for instance instance seed size or flight ability ability may be selected for reasons not for reasons not related to dispersal. Manipulation Manipulation of of natural natural systems systems offers offers aa much much more more powerful powerful test test of of evo evolutionary scenarios. For instance, Sinervo et al. (2000) manipulated manipulated offspring lutionary size assess selection selection on along cycles size in in lizards lizards to to assess on this this trait trait along cycles of of low low and and high high den density in lizard populations. heritable variation variation for life populations. There is a large amount amount of heritable history strategies in their system due to the coexistence coexistence of very distinct female reproductive documented existing patterns reproductive tactics. Sinervo et al. (2000) documented patterns of genetic genetic covariation covariation among among different different life life history history characters, characters, but but also also artificially artificially modified phenotypes of selection scenarios. modified the the phenotypes of individuals individuals to to test test selection scenarios. In In the the same same study, measured the component of study, they they measured the frequency-dependent frequency-dependent component of selection selection acting acting Combining such approaches approaches with a spatial perspective on on offspring size. Combining such such selective selective processes processes would would provide provide aa fascinating fascinating opportunity opportunity for for empirical empirical studies population. Parasites studies of of life life history history evolution evolution in in aa meta metapopulation. Parasites exploiting exploiting indi individual hosts or groups groups of hosts have been pointed pointed out out as organisms organisms particu particularly amenable for metapopulation theory larly amenable for testing testing metapopulation theory (Grenfell (Grenfell and and Harwood, Harwood, 11997). 997). Many Many studies studies have have now now documented documented interspecific interspecific or or intraspecific intraspecific vari variation ation in in parasites parasites life life history history traits traits in in relation relation to to the the structure structure or or dynamics dynamics of of their ai., 11997; 997; Morand, 2002; Parker their hosts hosts populations populations (see, (see, e.g., e.g., Sorci Sorci et et al., Morand, 2002; Parker et et aI., al., 2003; 2003; Thomas Thomas et et aI., al., 2002). 2002). An An experimental experimental test test of of theoretical theoretical predictions predictions about populations using about life life history history evolution evolution in in meta metapopulations using host-parasite host-parasite systems systems and and manipulating manipulating the the characteristics characteristics of of the the host host population population (see, (see, e.g., e.g., Koella Koella and 999) should and Agnew, Agnew, 11999) should be be further further encouraged. encouraged. We potential directions We conclude conclude by by indicating indicating potential directions for for empirical empirical investigation investigation of of life life history history evolution evolution in in aa metapopulation, metapopulation, summarizing summarizing questions questions emerging emerging from particular models from our our review. review. First, First, in in order order to to judge judge the the generality generality of of particular models
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predictions, one one would would wish wish to to have have aa better better sense sense of of the the importance importance of of trantran predictions, sient population population dynamics, dynamics, of of the the degree degree of of coupling coupling between between local local and and global global sient dynamics, and and of of the the extent extent of of among-local among-local population population differentiation differentiation for for dynamics, quantitative characters characters in in diverse diverse species. species. As As seen seen in in this this review, review, kin kin selection selection quantitative and demographic demographic disequilibrium disequilibrium can can profoundly profoundly modify modify our our expectations expectations concon and cerning life evolution, but but the the quantitative quantitative importance of such such phephe cerning life history history evolution, importance of nomena in in real real metapopulations metapopulations is is somehow unclear. Second, if details details of of nomena somehow still still unclear. Second, if the population population dynamics and genetic structure matter so much, much, it it casts casts doubts doubts the dynamics and genetic structure matter so on the the potential potential of of broad broad scale scale comparative comparative studies studies to to identify identify aa robust robust pattern on pattern of variation variation in in life life history history syndromes syndromes associated associated with with metapopulation metapopulation of processes (for aa discussion discussion of of such syndromes, see see Poethke et al., aI., 2003; 2003; Ronce Ronce processes (for such syndromes, Poethke et et al., aI., 2000c). 2000c). Focusing Focusing on on well-studied well-studied systems, systems, for are et for which which such such details details are known and where quantitative predictions predictions can can be be achieved, achieved, sounds as aa more more known and where quantitative sounds as promising alternative. In In particular, particular, the the present review points points toward toward two two promising alternative. present review main questions questions deserving deserving more more empirical empirical exploration. exploration. The The first first concerns concerns the the main patterns of traits other than dispersal in relation relation to to patterns of evolution evolution for for life life history history traits other than dispersal in landscape fragmentation. reproductive landscape fragmentation. Candidate Candidate traits traits include, include, for for instance, instance, reproductive effort, maturity, and size. The question is is effort, longevity, longevity, age age at at maturity, and clutch clutch size. The second second question related evolution of plasticity or, life hishis related to to the the evolution of plasticity or, more more generally, generally, of of conditional conditional life tory in aa metapopulation. metapopulation. tory strategies strategies as as adaptations adaptations to to aa variable variable environment environment in Of particular evolution of senescence patterns or age-specific Of particular interest interest is is the the evolution of senescence patterns or age-specific reproductive for species species subject reproductive strategies strategies for subject to to extinction-recolonization extinction-recolonization dynamics dynamics or processes. or successional successional processes.
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11
SELECTION SELECTION IN IN M ETAPO PU LATI O N S:: META PO PULATIONS THE COEVOLUTION COEVOLUTION OF PH ENOTYPE ENOTYPE OF AND CONTEXT AND CONTEXT Michael JJ.. Wade
11.1 1 1 .1
INTRODUCTION INTRODUCTION Most live in metapopulations, interacting interacting locally locally with con Most organisms organisms live in metapopulations, with conspecifics in ways that that affect affect individual fitness and in genetic contexts that vary fitness and contexts that from to another another (Loveless (Loveless and Hamrick, Hamrick, 1984; 1 984; Whitlock, 1992; 1 992; from one deme to 1 996; Hanski Hanski and and Gilpin, 1997; 1 997; Ingvarsson, 1999; 1 999; Wade and Goodnight, Kelly, 1996; and Goodnight, 1998). As As aa result, result, natural natural selection selection in in metapopulations metapopulations always always differs differs from from 1998). that in large, large, randomly randomly mating mating populations, populations, whether whether or or not not there there is is an an added added that in component of higher level selection acting among demes. This difference difference component higher level between metapopulations and that popu between adaptive evolution in metapopulations that in nonsubdivided poputradition lations has been overlooked by evolutionary genetic theory with its tradition of partitioning partitioning phenotypic variation into into dichotomous dichotomous genetic and and environenviron (e.g., Falconer and MacKay, 1996). 1 996). This traditional traditional approach approach mental factors (e.g., founders whenever whenever some of the factors for variation variation in individual founders factors responsible for and environmental environmental at at the same time. time. It It is phenotype or or fitness fitness are both both genetic and phenotype factors, often called "indirect "indirect genetic genetic effects" effects" (IGEs) (IGEs) (Moore (Moore et et al., ai., 1997; 1 997; these factors,
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MICHAEL MICHAEL J.I. WADE WADE
Wolf 998, 11999), 999), that Wolf et et aI., al., 11998, that create create causal causal pathways pathways between between the the genes genes in in one one individual individual and and the the phenotypes phenotypes expressed expressed by by others, others, even even if if unrelated, unrelated, and and that that permit permit the the coevolution coevolution of of phenotype phenotype and and context context that that is is unique unique to to metapopu metapopulations. lations. Even Even individual individual traits traits with with no no additive additive genetic genetic variance variance can can evolve evolve in in metapopulations an IGE) metapopulations when when the the mean mean of of aa trait trait in in the the social social partners partners ((an IGE) evolves (Moore et aI., al., 11998; et aI., al., 200 2001). evolves 998; Agrawal et 1). An An IGE IGE can can bbee the the outcome outcome ooff aann overt overt social social interaction interaction between between individ individuals, uals, as as in in altruism altruism or or mutualism, mutualism, or or aa less less conspicuous conspicuous competitive competitive inter interaction, action, such such as as competition competition for for sunlight sunlight or or nutrients nutrients in in plants. plants. The The evolution evolution of interactions interactions among among individuals individuals typically typically has has been been investigated investigated in in the the context context of of 964a,b), aa concep of optimality optimality and and game-theoretic game-theoretic models models (e.g., (e.g., Hamilton, Hamilton, 11964a,b), conceptual tual context context that that does does not not deal deal very very effectively effectively with with genotype-by-environment genotype-by-environment interactions, interactions, epistasis, epistasis, or or indirect indirect genetic genetic effects effects (Wolf (Wolf and and Wade, Wade, 2001 2001;; Shuster Shuster and and Wade, Wade, 2003). 2003). An An IGE IGE iiss aann environmental environmental source source ooff phenotypic phenotypic variation variation that that evolves evolves itself itself in in response response to to selection selection and and random random genetic genetic drift. drift. Thus, Thus, IGEs IGEs are are fundamental fundamental to to the the coevolution coevolution of of phenotype phenotype and and context. context. Whenever Whenever there there are are IGEs, IGEs, the the phenotype phenotype of of an an individual individual depends depends both both on on its its own own genotype genotype and and on on the the genotypes genotypes of of those those with with which which it it interacts interacts (Griffing, (Griffing, 11967, 967, 11977, 977, 11989; 989; Moore Moore et 998), permitting et aI., al., 11998), permitting coevolution coevolution of of individual individual and 1 . 1 and 1 .2). The and social social context context (see (see Fig. Fig. 111.1 and 111.2). The magnitude magnitude of of an an IGE IGE depends depends on the the sign sign of of its its effect effect on on the the individual's individual's phenotype, the reciprocity reciprocity of of the the on phenotype, the interactions individuals, and interactions among among individuals, and the the genetic genetic correlations correlations with with other other traits traits in the the individual individual and and in in those those with with which which it it interacts. interacts. However, However, like like the the coevo coevoin lution Gomulkiewicz et lution of of traits traits in in different different species species ((Gomulkiewicz et aI., al., 2000; 2000; Wade, Wade, 2003) 2003),, genetic genetic correlations correlations are are not not necessary necessary for for IGEs IGEs ttoo affect affect the the rate rate oorr direction direction of evolutionary response to local selection. Whenever different differently to to the same context, Whenever different individuals individuals respond respond differently the same context, component to to the the evolution evolution of of interactions. interactions. then IGEs IGEs introduce then introduce aa nonlinear nonlinear component In large, randomly mating mixing populations, populations, these these effects IGEs are are of In large, randomly mating and and mixing effects of of IGEs of limited importance limited importance because because all all individuals individuals experience experience essentially essentially the the same same average context context (Wade (Wade and and Goodnight, Goodnight, 1998). 1 998). In In metapopulations, metapopulations, however, however, average nonlinear interactions interactions become become more more important important because because they they can can enhance enhance the the nonlinear among-deme variation variation in in the the contexts contexts experienced experienced by by individuals individuals and and are are among-deme Environment Environment
.~~Nurture �
�
Phenotype Phenotype
Genes Genes "
Standard Decomposition Decomposition of of the the IIndividu Phenotype: Standard n d i v i d u a l al Phenotype: zI= aI+ eI
caused by by variation variation in in aalI or or in in el: el: Variation in in zz,I isis caused Variation V(zl)= V(a,) ++ V(el) V(z,) = V(al) V(e,) Heritability: Heritability: h2 IJ2(zl) V(a,)I V(z,) (z,) = V(al)/V(zl) =
Standard decomposition decomposition of of an an individual individual phenotype phenotype into into genetic genetic (al) (al) and and envirenvir Fig. 111.11 1 1 .1 Standard Fig. onmental onmental (el) (el) components. components.
261 2.61
111. 1 . SELECTION SELECTION IN METAPOPULATIONS METAPOPULATIONS (a)
Genes Genes
\ \
/ /
Environment Environment
Jal Partners' Soc Social Mean Phenotype
Zse \. NN~
ZsP
Environment
/
Abiotic f Environment (b) (b)
./ / Genes j Genes
�
Social Partners ~ Abiotic Environment Genes
______ Individual Individual
Phenotype zJ zl
~
� ~ Phenotype Nurture Nurture -'~
Individual Individual
Phenotype
Genes Decomposition of the Indi Individual vidual Phenotype: z,= zl= a,+ al+ e,+ el+ ('If)(Asp+ (~)(Asp+ Esp) Esm) Variation in z, z / is caused by
V(z/) = V(a,) V(al) + V(e,) V(el) + 22~12 Cov(al, Asm) V[E/]) V(z,) '1f12 Cov(a" Asp) + (~) ('Iff2 ((V[Asm] V[Asp] + V[E,D h2(zl) = { V V(al) Cov(al, Asp) Asp) + ('Iff (~)2V(Asp)}/V(zl) If(z,) (a,) + 2~ 2'1f Cov(a" V(Asp)}!V(z,) Fig. f i g . 11 11 ..2 2 (a)The influence influence of environment environment on an individual's phenotype can be partitioned into abiotic and social social components (IGEs). (IGEs). Importantly, the social social component itself can be decomposed into genetic and environmental components. Because Because the environment environment contains genes, the individual phenotype and the social social context can coevolve. (b) The decomposition decomposition of an individual phenotype into genetic (al), IGE components (al), environmental (el), (el), and IGE components ('lt12ZSP)' (~12Zse). The mean phenotype of the social social partner, Zsp, Zsp, is partitioned partitioned into genetic (Asp) (Asp) and environ environmental mental (Esp) (Esv) components. components.
more more likely likely to to create create higher higher levels levels of of selection selection (e.g., (e.g., Aviles Aviles et et a!., al., 2002). 2002). In In short, short, the the evolutionary evolutionary role role of of IGEs IGEs in in metapopulations metapopulations is is always always greater greater than than it is in large panmictic panmictic populations, populations, which which are lacking in conspicuous conspicuous genetic structure see Fig. 111.4). 1 .4). structure ((see In considering considering the role of of IGEs in metapopulations, metapopulations, this chapter chapter shows that that it 1 ) how evolution is it is is important important to to determine determine ((1) how trait trait evolution is affected affected by by local local con context, 3 ) how text, (2) (2) which which traits traits contribute contribute to to local local context, context, ((3) how trait trait and and context context are are correlated variation in correlated ecologically ecologically and and genetically, genetically, and and (4) (4) whether whether variation in mean mean fit fitness that the ness leads leads to to selection selection among among contexts. contexts. This This chapter chapter emphasizes emphasizes that the traits traits affected 1 ) may affected by by context context ((1) may be be the the same same or or different different from from those those traits traits con contributing tributing to to context context (2) (2) and and that that these these may may or or may may not not be be genetically genetically correl correlated. absence of ated. Furthermore, Furthermore, even even in in the the absence of among-deme among-deme selection selection (4), (4), meta populations permit metapopulations permit the the coevolution coevolution of of phenotype phenotype and and context context where where the the more more traditional traditional evolutionary evolutionary genetic genetic models models do do not not (cf. (cf. discussion discussion in in Wade, Wade, 11996; 996; Wade 998; Goodnight Wade and Goodnight, Goodnight, 11998; Goodnight and and Wade, 2000). 2000). Finally, the chapter coevolution of chapter discusses discusses how how these these considerations considerations might might influence influence the the coevolution of species in metacommunities, metacommunities, where the presence or absence of interacting interacting species variation in species contributes contributes to to aa variation in local local context. context.
262 262
11 11..2 2
MICHAEL J. WADE MICHAEL J. WADE
KINDS K I N D S OF OF CONTEXTS CONTEXTS It It is is fundamental fundamental in in evolutionary evolutionary genetic genetic theory theory that that the the environment environment affects expressed as individual phenotype. affects how how aa nuclear nuclear genotype genotype is is expressed as an an individual phenotype. The The general concept concept of general of "environment" "environment" includes includes intraindividual intraindividual cytoplasmic cytoplasmic organelles, organelles, such such as as mitochondria mitochondria and and chloroplasts, chloroplasts, as as well well as as endosymbionts endosymbionts such epistatically with olbachia and Wolbachia and Buchnera, Buchnera, which which can can interact interact epistatically with the the such as as W host expression of phenotype. Classical Classical host nuclear nuclear genome genome and and affect affect expression of the the host host phenotype. physiological epistasis, epistasis, based physiological based on on interactions interactions between between genes genes in in the the same same genome (see Chapter nonclassical epistasis, genome (see Chapter 9), 9), or or nonclassical epistasis, based based on on interactions interactions between genes in maternal and genomes (Wolf, between genes in the the maternal and zygotic zygotic genomes (Wolf, 2000; 2000; Wade, Wade, 2000), 2000), are are ubiquitous ubiquitous intraindividual intraindividual genetic genetic contexts, contexts, wherein wherein the the genetic genetic back background ground at at one one locus locus modifies modifies the the phenotypic phenotypic expression expression of of another another locus. locus. Variation Variation in in these these heritable heritable contexts contexts can can cause cause aa focal focal gene gene to to change change its its Mendelian identity from Mendelian phenotypic phenotypic identity from dominant, dominant, to to recessive, recessive, to to additive, additive, to to neutral, 1 ; see see also neutral, or or to to overdominant overdominant (Wade, (Wade, 200 2001; also Chapter Chapter 99).) . The The variance variance in in context phe context caused caused by by additive-by-additive additive-by-additive epistasis epistasis alone alone can can change change aa gene's gene's phenotypic major to its effect notypic effect effect from from major to minor minor and and change change the the sign sign of of its effect on on fitness fitness so deme but but aa ""bad" bad" gene in another (Wade, so that that it it is is aa "good" "good" gene gene in in one one deme gene in another (Wade, 2002; 995, 2000) intraindividual genetic genetic 2002; Goodnight, Goodnight, 11995, 2000).. These These effects effects of of the the intraindividual environment always more important in populations, where environment are are always more important in meta metapopulations, where subdivi subdivision leads to large populations, populations, sion leads to among-deme among-deme variance variance in in context, context, than than in in large where recombination diminishes variation experi where panmixia panmixia with with recombination diminishes the the contextual contextual variation experienced single genes genes (Fig. 1 1 .3). Indeed, Indeed, the enduring controversy controversy at enced by by single (Fig. 11.3). the enduring at the the foun foundation of theory between between S. S. Wright A. Fisher Fisher (Coyne dation of evolutionary evolutionary theory Wright and and R. R. A. (Coyne et 997, 2000; Wade and and Goodnight, 998; Goodnight et aI., al., 11997, 2000; Wade Goodnight, 11998; Goodnight and and Wade, Wade, 2000) 2000) can varia can be be seen seen as as aa controversy controversy over over the the evolutionary evolutionary significance significance of of such such variations tions in in genetic genetic context. context. The The patterns patterns of of migration migration within within and and among among demes demes and local population variance in and the the local population size size affect affect the the variance in context. context.
Large, Panmictic Population
Metapopulation
Fig. 11 11 ..33 A large, randomly mating "Fisherian" population (left) and a genetically sub subdivided meta population (right). metapopulation (right). The different size circles in the right panel represent demes of different age (smaller is younger) and different abundance. Demes with dotted borders are local extinctions. Small arrows connecting different demes represent migration, whereas larger arrows indicate colonization events.
263 263
111. 1 . SELECTION IN M ETAPOPULATIONS SELECTION IN METAPOPULATIONS
The The abiotic abiotic environment, environment, external external to to the the individual, individual, is is the the most most commonly commonly considered considered context context in in relation relation to to phenotypic phenotypic variation variation among among individuals. individuals. It It is is the the reference reference context context for for most most discussions discussions of of adaptation adaptation and, and, in in most most evolu evolutionary tionary theory, theory, the the external external environment environment is is treated treated as as temporally temporally and and spatially spatially invariant, invariant, at at least least on on average. average. The The traditional traditional partitioning partitioning of of the the variation variation in in aa phenotype, i ) and phenotype, Zl, zl, into into additive additive genetic, genetic, al (with (with mean mean A A1) and environmental, environmental, eb (Fig. 111.1): 1 . 1 ): el, components components reflects reflects this this viewpoint viewpoint (Fig. zl = al + el.
((11.1) 11.1)
Genotype-by-environment Genotype-by-environment interactions interactions (G (G X x E E)) are are the the most most prevalent prevalent format format for introduction of for the the introduction of variation variation in in environmental environmental context context into into evolutionary evolutionary models 998). In models (Schlichting (Schlichting and and Pigliucci, Pigliucci, 11998). In spatially spatially heterogeneous heterogeneous habitats, habitats, geo geographic graphic variation variation in in parameters parameters of of the the niche niche can can lead lead to to variation variation in in local local select selectMigration among among habitats with G G X x E E ive pressures (Holt and Gaines, Gaines, 11992). 992). Migration constrains constrains evolution evolution because because genotypes genotypes adapted adapted to to one one habitat habitat are are moved moved to to other other habitats habitats where where they they are are less less fit fit by by migration. migration. Habitats Habitats where where abundance abundance is is greatest greatest will will tend tend to to dominate dominate evolution evolution in in those those where where abundance abundance is is lower lower if if migration 992). Under migration is is proportional proportional to to local local abundance abundance (Holt (Holt and and Gaines, Gaines, 11992). Under some some circumstances, circumstances, however, however, 'adaptive 'adaptive plasticity' plasticity' can can evolve evolve as as aa response response to to spatial plays aa central understanding of evo spatial variation. variation. Plasticity Plasticity plays central role role in in our our understanding of the the evolution of those involved involved in adaptive response lution of polyphenisms, polyphenisms, particularly particularly those in an an adaptive response to to temporal and par temporal and spatial spatial heterogeneity heterogeneity of of the the abiotic abiotic environment. environment. The The standard standard parE, CCaxe, titioning titioning of of phenotypic phenotypic variation variation is is expanded expanded to to include include aa term term for for G G X x E, axe, Zl -- a l + e l q- Claxe.
((11.2) 1 1 .2 )
This This approach approach iiss appropriate appropriate for for the the abiotic abiotic environment environment but but not not for for biotic biotic environments. environments. This This formulation formulation permits permits aa genetic genetic response response to to abiotic abiotic environ environmental mental variation, variation, but but does does not not allow allow the the reciprocal reciprocal because because the the environment, environment, although variable, is although variable, is not not evolutionarily evolutionarily dynamic. dynamic. Only Only the the genotype genotype changes changes evolutionarily 1 1 .2 ) cannot evolutionarily if if fitness fitness varies varies with with context context in in this this way. way. Thus, Thus, Eq. Eq. ((11.2) cannot adequately adequately account account for for the the context context provided provided by by conspecifics conspecifics or or by by other other species species in ecological contexts in an an ecological ecological community community because because these these ecological contexts (unlike (unlike el) el) also also have ecologically important have aa genetic genetic component. component. Thus, Thus, many many ecologically important contexts contexts differ differ from they are are evolutionarily and can can coevolve with Zl' from G G X x E E because because they evolutionarily dynamic dynamic and coevolve with zl. Interactions Interactions between between males males and and females, females, between between parents parents and and offspring, offspring, between between age age cohorts, cohorts, between between competitors, competitors, between between predators predators and and prey, prey, or or between between hosts hosts and and pathogens pathogens all all contribute contribute to to individual individual phenotypic phenotypic and and fitness fitness variation animals. Notably, variation in in plants plants and and animals. Notably, these these contexts contexts contain contain genes genes and, and, when me to population, genes when they they vary vary from from de deme to deme deme across across aa meta metapopulation, genes and and con context 994; Wolf text can can coevolve coevolve (Thompson, (Thompson, 11994; Wolf et et aI., al., 2003; 2003; Wade, Wade, 2003) 2003)..
111.3 1 .3
EVOLVING EVOLVING CONTEXTS CONTEXTS In In order order to to investigate investigate evolving evolving contexts, contexts, additional additional terms terms must must be be included included in in the the standard standard formulation formulation given given earlier. earlier. Imagine Imagine aa second second trait, trait, Zz2, which is is 2 , which sensitive sensitive to to the the local local mean mean value, value, Zli, Zli, of of the the first first trait. trait. The The first first trait, trait, Zb Zl, is is the the
264 264
M ICHAEL I.). WADE MICHAEL WADE
"context" "context" and and its its local local mean mean value value influences influences individual individual values values of of phenotype, phenotype, Zz2, in that that deme deme as as follows: follows: 2 , in Z2i = a2i + xP12Zli + e2i.
((11.3) 1 1 .3)
The The second second term, term, 'l'1 xIt12Z1i is the the IGE, IGE, where where the the coefficient, coefficient, '1'1 xlt122 ,, represents represents 2Z1i,, is the magnitude of linear effect the scale scale or or magnitude of the the linear effect of of Z1i Z l i relative relative to to the the direct direct effects effects whose have been set to both lin whose coefficients coefficients have been set to 11.. The The effects effects of of Z1i Z l i ccan a n include include both linear nonlinear components ear and and nonlinear components (d. (cf. Agrawal Agrawal et et ai., al., 2001). 2001). Whenever Whenever it it con contains possible to tains nonlinear nonlinear terms, terms, then then it it is is not not generally generally possible to scale scale the the global global means, means, Z1 Z1 and and Z Z2, to zero zero without without also also aa temporal temporal and and spatial spatial rescaling rescaling of of 2 , to the all other 1 1.3) each the coefficients coefficients of of all other terms terms in in Eq. Eq. ((11.3) each generation. generation. Thus, Thus, the the presence an evolvable context means additive presence of of an evolvable context means that that the the coefficients coefficients of of additive .0), must be reevaluated genetic assumed to genetic effects, effects, such such as as aa2i (here assumed to be be 11.0), must be reevaluated 2i (here from remaining constant from generation generation to to generation generation instead instead of of remaining constant as as is is done done in in standard standard theory. theory. It It is is the the nonzero nonzero values values of of '1'1 XItl2 that permit permit the the environment environment 2 that to evolve (Fig. 1 .2). to co coevolve (Fig. 111.2). Substituting 1 1. 1 )) into 1 1 .3 ), we Substituting Eq. Eq. ((11.1 into Eq. Eq. ((11.3), we find find that that 2~2i =
a2i + xP12(A1i + Eli) + e2i.
((11.4) 1 1.4)
This This is is the the effect effect of of IGEs: IGEs: an an individual's individual's phenotypic phenotypic value, value, ZZ2, depends on on the the 2, depends genetic and value of 1 .2). genetic and phenotypic phenotypic value of its its local, local, social social ccontext o n t e x t 'l'1 xI~12Zli (Fig. 111.2). 2Z1i (Fig. Furthermore, Furthermore, with with genetic genetic subdivision subdivision (FST (FsT > > 0), 0), the the average average social social context context itself will Thus, there itself will vary vary from from deme deme to to deme deme across across aa metapopulation. metapopulation. Thus, there can can be be heritable heritable variation variation in in the the additive additive genetic genetic values values of of social social context context from from deme to is deme to deme, deme, affecting affecting the the evolution evolution of of traits traits sensitive sensitive to to context. context. If If FST FST is Wright's population genetic (aj) isis the Wright's measure measure of of meta metapopulation genetic subdivision subdivision and and V V(ai) the vari variance ance in in aj aj in in aa population population without without subdivision subdivision (i.e., (i.e., FST FST-= 0), 0), then then in in the the absence of absence of epistasis, epistasis, the the average average variance variance within within demes demes is is ((11 - FST) FsT)V(aj) and V(aj) and the the variance variance among among demes demes is is 2FsTV(aj) 2FsTV(ai) (Hartl (HaM and and Clark, Clark, 1997). 1997). ia; X• ai When When there there is is additive-by-additive additive-by-additive epistasis, epistasis, V V(ai aj),l , the the average average addi additive tive genetic genetic variance variance within within demes demes can can be be greater greater than than (( 11 -- FST) FsT)V(ai) V(aj) (Goodnight, 990, 1995, (Goodnight, 1987, 1987, 11990, 1995, 1999) 1999) by by an an amount amount approximately approximately equal equal to to 993). With 4FsT(I1 -- FST) FsT)V(ai aj) (Whitlock (Whitlock et et ai., al., 11993). With nonlinear nonlinear interactions interactions 4FsT( Via; xx aj) among contributing to among genes, genes, the the identity identity of of the the genes genes contributing to the the local local additive additive vari variance me to ance will will vary vary from from de deme to deme deme as as will will the the coefficients coefficients of of aa1l and and aa22 in in Eqs. Eqs. ((11.1)-(11.3) 1 1. 1 )-( 1 1.3) ((Goodnight, Goodnight, 2000). 2000). If If the the IGE IGE term, term, 'l'1 xt~1221i is itself itself partitioned partitioned 2Z1i,, is into linear and into linear and nonlinear nonlinear components, components, then then not not only only will will the the coefficients coefficients change deme to change from from deme to deme deme and and from from generation generation to to generation, generation, but but the the strictly strictly additively , will now have an added epistatic additively determined determined phenotypes, phenotypes, Z1 zl and and Zz2, will now have an added epistatic 2 (i.e., (i.e., a1 al x x aa2) component. 2 ) component.
11 11. .4 4
SELECTION S E L E C T I O N IN IN M M EETAPOPULATIONS TAPOPULATIONS To populations is To see see how how selection selection in in meta metapopulations is different different with with IGEs, IGEs, consider consider selection selection only only within within demes. demes. We We need need to to first first def