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Interim Report IR-06-006
Coevolution of Slow-fast Populations: Evolutionary Sliding, Evolutionary Pseudo-equilibria, and Complex Red Queen Dynamics
Fabio Dercole (fabio.dercole@polimi.it) Régis Ferrière (ferriere@biologie.ens.fr) Alessandra Gragnani (gragnani@polimi.it) Sergio Rinaldi (rinaldi@polimi.it)
Approved by Ulf Dieckmann
Program Leader, Evolution & Ecology January 2006
IIASA S TUDIES IN A DAPTIVE D YNAMICS N O. 113
ADN
The Adaptive Dynamics Network at IIASA fosters the develop- ment of new mathematical and conceptual techniques for under- standing the evolution of complex adaptive systems.
Focusing on these long-term implications of adaptive processes in systems of limited growth, the Adaptive Dynamics Network brings together scientists and institutions from around the world with IIASA acting as the central node.
Scientific progress within the network is collected in the IIASA Studies in Adaptive Dynamics series.
No. 1 Metz JAJ, Geritz SAH, Meszéna G, Jacobs FJA, van Heerwaarden JS: Adaptive Dynamics: A Geometrical Study of the Consequences of Nearly Faithful Reproduction. IIASA Working Paper WP-95-099 (1995). van Strien SJ, Verduyn Lunel SM (eds): Stochastic and Spatial Structures of Dynami- cal Systems, Proceedings of the Royal Dutch Academy of Sci- ence (KNAW Verhandelingen), North Holland, Amsterdam, pp. 183-231 (1996).
No. 2 Dieckmann U, Law R: The Dynamical Theory of Co- evolution: A Derivation from Stochastic Ecological Processes.
IIASA Working Paper WP-96-001 (1996). Journal of Mathe- matical Biology 34:579-612 (1996).
No. 3 Dieckmann U, Marrow P, Law R: Evolutionary Cy- cling of Predator-Prey Interactions: Population Dynamics and the Red Queen. IIASA Preprint (1995). Journal of Theoreti- cal Biology 176:91-102 (1995).
No. 4 Marrow P, Dieckmann U, Law R: Evolutionary Dy- namics of Predator-Prey Systems: An Ecological Perspective.
IIASA Working Paper WP-96-002 (1996). Journal of Mathe- matical Biology 34:556-578 (1996).
No. 5 Law R, Marrow P, Dieckmann U: On Evolution under Asymmetric Competition. IIASA Working Paper WP-96-003 (1996). Evolutionary Ecology 11:485-501 (1997).
No. 6 Metz JAJ, Mylius SD, Diekmann O: When Does Evo- lution Optimize? On the Relation Between Types of Density Dependence and Evolutionarily Stable Life History Parame- ters. IIASA Working Paper WP-96-004 (1996).
No. 7 Ferrière R, Gatto M: Lyapunov Exponents and the Mathematics of Invasion in Oscillatory or Chaotic Popula- tions. Theoretical Population Biology 48:126-171 (1995).
No. 8 Ferrière R, Fox GA: Chaos and Evolution. IIASA Preprint (1996). Trends in Ecology and Evolution 10:480- 485 (1995).
No. 9 Ferrière R, Michod RE: The Evolution of Cooperation in Spatially Heterogeneous Populations. IIASA Working Pa- per WP-96-029 (1996). The American Naturalist 147:692- 717 (1996).
No. 10 van Dooren TJM, Metz JAJ: Delayed Maturation in Temporally Structured Populations with Non-Equilibrium Dy-
No. 11 Geritz SAH, Metz JAJ, Kisdi É, Meszéna G: The Dy- namics of Adaptation and Evolutionary Branching. IIASA Working Paper WP-96-077 (1996). Physical Review Letters 78:2024-2027 (1997).
No. 12 Geritz SAH, Kisdi É, Meszéna G, Metz JAJ: Evo- lutionary Singular Strategies and the Adaptive Growth and Branching of the Evolutionary Tree. IIASA Working Paper WP-96-114 (1996). Evolutionary Ecology 12:35-57 (1998).
No. 13 Heino M, Metz JAJ, Kaitala V: Evolution of Mixed Maturation Strategies in Semelparous Life-Histories: The Crucial Role of Dimensionality of Feedback Environment.
IIASA Working Paper WP-96-126 (1996). Philosophi- cal Transactions of the Royal Society of London Series B 352:1647-1655 (1997).
No. 14 Dieckmann U: Can Adaptive Dynamics Invade?
IIASA Working Paper WP-96-152 (1996). Trends in Ecol- ogy and Evolution 12:128-131 (1997).
No. 15 Meszéna G, Czibula I, Geritz SAH: Adaptive Dynam- ics in a 2-Patch Environment: A Simple Model for Allopatric and Parapatric Speciation. IIASA Interim Report IR-97-001 (1997). Journal of Biological Systems 5:265-284 (1997).
No. 16 Heino M, Metz JAJ, Kaitala V: The Enigma of Frequency-Dependent Selection. IIASA Interim Report IR- 97-061 (1997). Trends in Ecology and Evolution 13:367-370 (1998).
No. 17 Heino M: Management of Evolving Fish Stocks.
IIASA Interim Report IR-97-062 (1997). Canadian Journal of Fisheries and Aquatic Sciences 55:1971-1982 (1998).
No. 18 Heino M: Evolution of Mixed Reproductive Strategies in Simple Life-History Models. IIASA Interim Report IR-97- 063 (1997).
No. 19 Geritz SAH, van der Meijden E, Metz JAJ: Evolution- ary Dynamics of Seed Size and Seedling Competitive Ability.
IIASA Interim Report IR-97-071 (1997). Theoretical Popu- lation Biology 55:324-343 (1999).
No. 20 Galis F, Metz JAJ: Why Are There So Many Cichlid Species? On the Interplay of Speciation and Adaptive Radi- ation. IIASA Interim Report IR-97-072 (1997). Trends in Ecology and Evolution 13:1-2 (1998).
No. 21 Boerlijst MC, Nowak MA, Sigmund K: Equal Pay for all Prisoners/ The Logic of Contrition. IIASA Interim
No. 22 Law R, Dieckmann U: Symbiosis Without Mutualism and the Merger of Lineages in Evolution. IIASA Interim Re- port IR-97-074 (1997). Proceedings of the Royal Society of London Series B 265:1245-1253 (1998).
No. 23 Klinkhamer PGL, de Jong TJ, Metz JAJ: Sex and Size in Cosexual Plants. IIASA Interim Report IR-97-078 (1997).
Trends in Ecology and Evolution 12:260-265 (1997).
No. 24 Fontana W, Schuster P: Shaping Space: The Possi- ble and the Attainable in RNA Genotype-Phenotype Mapping.
IIASA Interim Report IR-98-004 (1998). Journal of Theoret- ical Biology 194:491-515 (1998).
No. 25 Kisdi É, Geritz SAH: Adaptive Dynamics in Allele Space: Evolution of Genetic Polymorphism by Small Muta- tions in a Heterogeneous Environment. IIASA Interim Report IR-98-038 (1998). Evolution 53:993-1008 (1999).
No. 26 Fontana W, Schuster P: Continuity in Evolution: On the Nature of Transitions. IIASA Interim Report IR-98-039 (1998). Science 280:1451-1455 (1998).
No. 27 Nowak MA, Sigmund K: Evolution of Indirect Reci- procity by Image Scoring/ The Dynamics of Indirect Reci- procity. IIASA Interim Report IR-98-040 (1998). Nature 393:573-577 (1998). Journal of Theoretical Biology 194:561- 574 (1998).
No. 28 Kisdi É: Evolutionary Branching Under Asymmetric Competition. IIASA Interim Report IR-98-045 (1998). Jour- nal of Theoretical Biology 197:149-162 (1999).
No. 29 Berger U: Best Response Adaptation for Role Games.
IIASA Interim Report IR-98-086 (1998).
No. 30 van Dooren TJM: The Evolutionary Ecology of Dominance-Recessivity. IIASA Interim Report IR-98-096 (1998). Journal of Theoretical Biology 198:519-532 (1999).
No. 31 Dieckmann U, O’Hara B, Weisser W: The Evolution- ary Ecology of Dispersal. IIASA Interim Report IR-98-108 (1998). Trends in Ecology and Evolution 14:88-90 (1999).
No. 32 Sigmund K: Complex Adaptive Systems and the Evo- lution of Reciprocation. IIASA Interim Report IR-98-100 (1998). Ecosystems 1:444-448 (1998).
No. 33 Posch M, Pichler A, Sigmund K: The Efficiency of Adapting Aspiration Levels. IIASA Interim Report IR-98- 103 (1998). Proceedings of the Royal Society London Series B 266:1427-1435 (1999).
No. 34 Mathias A, Kisdi É: Evolutionary Branching and Co- existence of Germination Strategies. IIASA Interim Report IR-99-014 (1999).
No. 35 Dieckmann U, Doebeli M: On the Origin of Species by Sympatric Speciation. IIASA Interim Report IR-99-013 (1999). Nature 400:354-357 (1999).
No. 36 Metz JAJ, Gyllenberg M: How Should We Define Fit- ness in Structured Metapopulation Models? Including an Ap- plication to the Calculation of Evolutionarily Stable Dispersal Strategies. IIASA Interim Report IR-99-019 (1999). Pro- ceedings of the Royal Society of London Series B 268:499- 508 (2001).
No. 37 Gyllenberg M, Metz JAJ: On Fitness in Structured Metapopulations. IIASA Interim Report IR-99-037 (1999).
Journal of Mathematical Biology 43:545-560 (2001).
No. 39 Kisdi É, Geritz SAH: Evolutionary Branching and Sympatric Speciation in Diploid Populations. IIASA Interim Report IR-99-048 (1999).
No. 40 Ylikarjula J, Heino M, Dieckmann U: Ecology and Adaptation of Stunted Growth in Fish. IIASA Interim Report IR-99-050 (1999). Evolutionary Ecology 13:433-453 (1999).
No. 41 Nowak MA, Sigmund K: Games on Grids. IIASA Interim Report IR-99-038 (1999). Dieckmann U, Law R, Metz JAJ (eds): The Geometry of Ecological Interactions:
Simplifying Spatial Complexity, Cambridge University Press, Cambridge, UK, pp. 135-150 (2000).
No. 42 Ferrière R, Michod RE: Wave Patterns in Spatial Games and the Evolution of Cooperation. IIASA Interim Report IR-99-041 (1999). Dieckmann U, Law R, Metz JAJ (eds): The Geometry of Ecological Interactions: Simplifying Spatial Complexity, Cambridge University Press, Cambridge, UK, pp. 318-332 (2000).
No. 43 Kisdi É, Jacobs FJA, Geritz SAH: Red Queen Evo- lution by Cycles of Evolutionary Branching and Extinction.
IIASA Interim Report IR-00-030 (2000). Selection 2:161- 176 (2001).
No. 44 Meszéna G, Kisdi É, Dieckmann U, Geritz SAH, Metz JAJ: Evolutionary Optimisation Models and Matrix Games in the Unified Perspective of Adaptive Dynamics. IIASA Interim Report IR-00-039 (2000). Selection 2:193-210 (2001).
No. 45 Parvinen K, Dieckmann U, Gyllenberg M, Metz JAJ:
Evolution of Dispersal in Metapopulations with Local Density Dependence and Demographic Stochasticity. IIASA Interim Report IR-00-035 (2000). Journal of Evolutionary Biology 16:143-153 (2003).
No. 46 Doebeli M, Dieckmann U: Evolutionary Branch- ing and Sympatric Speciation Caused by Different Types of Ecological Interactions. IIASA Interim Report IR-00-040 (2000). The American Naturalist 156:S77-S101 (2000).
No. 47 Heino M, Hanski I: Evolution of Migration Rate in a Spatially Realistic Metapopulation Model. IIASA Interim Report IR-00-044 (2000). The American Naturalist 157:495- 511 (2001).
No. 48 Gyllenberg M, Parvinen K, Dieckmann U: Evolution- ary Suicide and Evolution of Dispersal in Structured Metapop- ulations. IIASA Interim Report IR-00-056 (2000). Journal of Mathematical Biology 45:79-105 (2002).
No. 49 van Dooren TJM: The Evolutionary Dynamics of Di- rect Phenotypic Overdominance: Emergence Possible, Loss Probable. IIASA Interim Report IR-00-048 (2000). Evolu- tion 54:1899-1914 (2000).
No. 50 Nowak MA, Page KM, Sigmund K: Fairness Versus Reason in the Ultimatum Game. IIASA Interim Report IR- 00-57 (2000). Science 289:1773-1775 (2000).
No. 51 de Feo O, Ferrière R: Bifurcation Analysis of Pop- ulation Invasion: On-Off Intermittency and Basin Riddling.
IIASA Interim Report IR-00-074 (2000). International Jour- nal of Bifurcation and Chaos 10:443-452 (2000).
No. 52 Heino M, Laaka-Lindberg S: Clonal Dynamics and Evolution of Dormancy in the Leafy Hepatic Lophozia Sil- vicola. IIASA Interim Report IR-01-018 (2001). Oikos 94:525-532 (2001).
No. 53 Sigmund K, Hauert C, Nowak MA: Reward and Pun-
No. 54 Hauert C, De Monte S, Sigmund K, Hofbauer J: Os- cillations in Optional Public Good Games. IIASA Interim Report IR-01-036 (2001).
No. 55 Ferrière R, Le Galliard J: Invasion Fitness and Adap- tive Dynamics in Spatial Population Models. IIASA Interim Report IR-01-043 (2001). Clobert J, Dhondt A, Danchin E, Nichols J (eds): Dispersal, Oxford University Press, pp. 57-79 (2001).
No. 56 de Mazancourt C, Loreau M, Dieckmann U: Can the Evolution of Plant Defense Lead to Plant-Herbivore Mutual- ism. IIASA Interim Report IR-01-053 (2001). The American Naturalist 158:109-123 (2001).
No. 57 Claessen D, Dieckmann U: Ontogenetic Niche Shifts and Evolutionary Branching in Size-Structured Populations.
IIASA Interim Report IR-01-056 (2001). Evolutionary Ecol- ogy Research 4:189-217 (2002).
No. 58 Brandt H: Correlation Analysis of Fitness Land- scapes. IIASA Interim Report IR-01-058 (2001).
No. 59 Dieckmann U: Adaptive Dynamics of Pathogen-Host Interacations. IIASA Interim Report IR-02-007 (2002).
Dieckmann U, Metz JAJ, Sabelis MW, Sigmund K (eds):
Adaptive Dynamics of Infectious Diseases: In Pursuit of Viru- lence Management, Cambridge University Press, Cambridge, UK, pp. 39-59 (2002).
No. 60 Nowak MA, Sigmund K: Super- and Coinfection:
The Two Extremes. IIASA Interim Report IR-02-008 (2002).
Dieckmann U, Metz JAJ, Sabelis MW, Sigmund K (eds):
Adaptive Dynamics of Infectious Diseases: In Pursuit of Viru- lence Management, Cambridge University Press, Cambridge, UK, pp. 124-137 (2002).
No. 61 Sabelis MW, Metz JAJ: Taking Stock: Relating The- ory to Experiment. IIASA Interim Report IR-02-009 (2002).
Dieckmann U, Metz JAJ, Sabelis MW, Sigmund K (eds):
Adaptive Dynamics of Infectious Diseases: In Pursuit of Viru- lence Management, Cambridge University Press, Cambridge, UK, pp. 379-398 (2002).
No. 62 Cheptou P, Dieckmann U: The Evolution of Self- Fertilization in Density-Regulated Populations . IIASA In- terim Report IR-02-024 (2002). Proceedings of the Royal Society of London Series B 269:1177-1186 (2002).
No. 63 Bürger R: Additive Genetic Variation Under Intraspe- cific Competition and Stabilizing Selection: A Two-Locus Study. IIASA Interim Report IR-02-013 (2002). Theoret- ical Population Biology 61:197-213 (2002).
No. 64 Hauert C, De Monte S, Hofbauer J, Sigmund K: Vol- unteering as Red Queen Mechanism for Co-operation in Pub- lic Goods Games. IIASA Interim Report IR-02-041 (2002).
Science 296:1129-1132 (2002).
No. 65 Dercole F, Ferrière R, Rinaldi S: Ecological Bistabil- ity and Evolutionary Reversals under Asymmetrical Competi- tion. IIASA Interim Report IR-02-053 (2002). Evolution 56:1081-1090 (2002).
No. 66 Dercole F, Rinaldi S: Evolution of Cannibalistic Traits: Scenarios Derived from Adaptive Dynamics. IIASA Interim Report IR-02-054 (2002). Theoretical Population Bi- ology 62:365-374 (2002).
No. 67 Bürger R, Gimelfarb A: Fluctuating Environments and the Role of Mutation in Maintaining Quantitative Genetic
No. 68 Bürger R: On a Genetic Model of Intraspecific Com- petition and Stabilizing Selection. IIASA Interim Report IR- 02-062 (2002). Amer. Natur. 160:661-682 (2002).
No. 69 Doebeli M, Dieckmann U: Speciation Along Environ- mental Gradients. IIASA Interim Report IR-02-079 (2002).
Nature 421:259-264 (2003).
No. 70 Dercole F, Irisson J, Rinaldi S: Bifurcation Analysis of a Prey-Predator Coevolution Model. IIASA Interim Report IR-02-078 (2002). SIAM Journal on Applied Mathematics 63:1378-1391 (2003).
No. 71 Le Galliard J, Ferrière R, Dieckmann U: The Adaptive Dynamics of Altruism in Spatially Heterogeneous Populations.
IIASA Interim Report IR-03-006 (2003). Evolution 57:1-17 (2003).
No. 72 Taborsky B, Dieckmann U, Heino M: Unex- pected Discontinuities in Life-History Evolution under Size- Dependent Mortality. IIASA Interim Report IR-03-004 (2003). Proceedings of the Royal Society of London Series B 270:713-721 (2003).
No. 73 Gardmark A, Dieckmann U, Lundberg P: Life- History Evolution in Harvested Populations: The Role of Nat- ural Predation. IIASA Interim Report IR-03-008 (2003).
Evolutionary Ecology Research 5:239-257 (2003).
No. 74 Mizera F, Meszéna G: Spatial Niche Packing, Char- acter Displacement and Adaptive Speciation Along an En- vironmental Gradient. IIASA Interim Report IR-03-062 (2003). Evolutionary Ecology Research 5:363-382 (2003).
No. 75 Dercole F: Remarks on Branching-Extinction Evolu- tionary Cycles. IIASA Interim Report IR-03-077 (2003).
Journal of Mathematical Biology 47:569-580 (2003).
No. 76 Hofbauer J, Sigmund K: Evolutionary Game Dynam- ics. IIASA Interim Report IR-03-078 (2003). Bulletin of the American Mathematical Society 40:479-519 (2003).
No. 77 Ernande B, Dieckmann U, Heino M: Adaptive Changes in Harvested Populations: Plasticity and Evolution of Age and Size at Maturation. IIASA Interim Report IR- 03-058 (2003). Proceedings of the Royal Society of London Series B-Biological Sciences 271:415-423 (2004).
No. 78 Hanski I, Heino M: Metapopulation-Level Adaptation of Insect Host Plant Preference and Extinction-Colonization Dynamics in Heterogeneous Landscapes. IIASA Interim Report IR-03-028 (2003). Theoretical Population Biology 63:309-338 (2003).
No. 79 van Doorn G, Dieckmann U, Weissing FJ: Sympatric Speciation by Sexual Selection: A Critical Re-Evaluation.
IIASA Interim Report IR-04-003 (2004). American Natu- ralist 163:709-725 (2004).
No. 80 Egas M, Dieckmann U, Sabelis MW: Evolution Re- stricts the Coexistence of Specialists and Generalists - the Role of Trade-off Structure. IIASA Interim Report IR-04-004 (2004). American Naturalist 163:518-531 (2004).
No. 81 Ernande B, Dieckmann U: The Evolution of Pheno- typic Plasticity in Spatially Structured Environments: Implica- tions of Intraspecific Competition, Plasticity Costs, and Envi- ronmental Characteristics. IIASA Interim Report IR-04-006 (2004). Journal of Evolutionary Biology 17:613-628 (2004).
No. 82 Cressman R, Hofbauer J: Measure Dynamics on a
No. 83 Cressman R: Dynamic Stability of the Replicator Equation with Continuous Strategy Space. IIASA Interim Report IR-04-017 (2004).
No. 84 Ravigné V, Olivieri I, Dieckmann U: Implications of Habitat Choice for Protected Polymorphisms. IIASA Interim Report IR-04-005 (2004). Evolutionary Ecology Research 6:125-145 (2004).
No. 85 Nowak MA, Sigmund K: Evolutionary Dynamics of Biological Games. IIASA Interim Report IR-04-013 (2004).
Science 303:793-799 (2004).
No. 86 Vukics A, Asbóth J, Meszéna G: Speciation in Mul- tidimensional Evolutionary Space. IIASA Interim Report IR-04-028 (2004). Physical Review 68:041-903 (2003).
No. 87 de Mazancourt C, Dieckmann U: Trade-off Geome- tries and Frequency-dependent Selection. IIASA Interim Re- port IR-04-039 (2004). American Naturalist 164:765-778 (2004).
No. 88 Cadet CR, Metz JAJ, Klinkhamer PGL: Size and the Not-So-Single Sex: disentangling the effects of size on sex al- location. IIASA Interim Report IR-04-084 (2004). Ameri- can Naturalist 164:779-792 (2004).
No. 89 Rueffler C, van Dooren TJM, Metz JAJ: Adaptive Walks on Changing Landscapes: Levins’ Approach Extended.
IIASA Interim Report IR-04-083 (2004). Theoretical Popu- lation Biology 65:165-178 (2004).
No. 90 de Mazancourt C, Loreau M, Dieckmann U: Under- standing Mutualism When There is Adaptation to the Partner.
IIASA Interim Report IR-05-016 (2005). Journal of Ecology 93:305-314 (2005).
No. 91 Dieckmann U, Doebeli M: Pluralism in Evolutionary Theory. IIASA Interim Report IR-05-017 (2005). Journal of Evolutionary Biology 18:1209-1213 (2005).
No. 92 Doebeli M, Dieckmann U, Metz JAJ, Tautz D: What We Have Also Learned: Adaptive Speciation is Theoretically Plausible. IIASA Interim Report IR-05-018 (2005). Evolu- tion 59:691-695 (2005).
No. 93 Egas M, Sabelis MW, Dieckmann U: Evolution of Specialization and Ecological Character Displacement of Herbivores Along a Gradient of Plant Quality. IIASA Interim Report IR-05-019 (2005). Evolution 59:507-520 (2005).
No. 94 Le Galliard J, Ferrière R, Dieckmann U: Adaptive Evolution of Social Traits: Origin, Trajectories, and Corre- lations of Altruism and Mobility. IIASA Interim Report IR- 05-020 (2005). American Naturalist 165:206-224 (2005).
No. 95 Doebeli M, Dieckmann U: Adaptive Dynamics as a Mathematical Tool for Studying the Ecology of Speciation Processes. IIASA Interim Report IR-05-022 (2005). Journal of Evolutionary Biology 18:1194-1200 (2005).
No. 96 Brandt H, Sigmund K: The Logic of Reprobation: As- sessment and Action Rules for Indirect Reciprocity. IIASA Interim Report IR-04-085 (2004). Journal of Theoretical Bi- ology 231:475-486 (2004).
No. 97 Hauert C, Haiden N, Sigmund K: The Dynamics of Public Goods. IIASA Interim Report IR-04-086 (2004). Dis- crete and Continuous Dynamical Systems - Series B 4:575- 587 (2004).
No. 98 Meszéna G, Gyllenberg M, Jacobs FJA, Metz JAJ:
Link Between Population Dynamics and Dynamics of Dar- winian Evolution. IIASA Interim Report IR-05-026 (2005).
Physical Review Letters 95:Article 078105 (2005).
No. 99 Meszéna G: Adaptive Dynamics: The Continuity Ar- gument. IIASA Interim Report IR-05-032 (2005). Journal of Evolutionary Biology 18:1182-1185 (2005).
No. 100 Brännström NA, Dieckmann U: Evolutionary Dy- namics of Altruism and Cheating Among Social Amoebas.
IIASA Interim Report IR-05-039 (2005). Proceedings of the Royal Society London Series B 272:1609-1616 (2005).
No. 101 Meszéna G, Gyllenberg M, Pasztor L, Metz JAJ:
Competitive Exclusion and Limiting Similarity: A Unified Theory. IIASA Interim Report IR-05-040 (2005).
No. 102 Szabo P, Meszéna G: Limiting Similarity Revisited.
IIASA Interim Report IR-05-050 (2005).
No. 103 Krakauer DC, Sasaki A: The Greater than Two-Fold Cost of Integration for Retroviruses. IIASA Interim Report IR-05-069 (2005).
No. 104 Metz JAJ: Eight Personal Rules for Doing Science.
IIASA Interim Report IR-05-073 (2005). Journal of Evolu- tionary Biology 18:1178-1181 (2005).
No. 105 Beltman JB, Metz JAJ: Speciation: More Likely Through a Genetic or Through a Learned Habitat Preference?
IIASA Interim Report IR-05-072 (2005). Proceedings of the Royal Society of London Series B 272:1455-1463 (2005).
No. 106 Durinx M, Metz JAJ: Multi-type Branching Pro- cesses and Adaptive Dynamics of Structured Populations.
IIASA Interim Report IR-05-074 (2005). Haccou P, Jager P, Vatutin V (eds): Branching Processes: Variation, Growth and Extinction of Populations, Cambridge University Press, Cambridge, UK, pp. 266-278 (2005).
No. 107 Brandt H, Sigmund K: The Good, the Bad and the Discriminator - Errors in Direct and Indirect Reciprocity.
IIASA Interim Report IR-05-070 (2005).
No. 108 Brandt H, Hauert C, Sigmund K: Punishing and Ab- staining for Public Goods. IIASA Interim Report IR-05-071 (2005). Proceedings of the National Academy of Sciences of the United States of America 103:495-497 (2006).
No. 109 Ohtsuki A, Sasaki A: Epidemiology and Disease- Control Under Gene-for-Gene Plant-Pathogen Interaction.
IIASA Interim Report IR-05-068 (2005).
No. 110 Brandt H, Sigmund K: Indirect Reciprocity, Image- Scoring, and Moral Hazard. IIASA Interim Report IR-05- 078 (2005). Proceedings of the National Academy of Sci- ences of the United States of America 102:2666-2670 (2005).
No. 111 Nowak MA, Sigmund K: Evolution of Indirect Reci- procity. IIASA Interim Report IR-05-079 (2005). Nature 437:1292-1298 (2005).
No. 112 Kamo M, Sasaki A: Evolution Towards Multi-Year Periodicity in Epidemics. IIASA Interim Report IR-05-080 (2005). Ecology Letters 8:378-385 (2005).
No. 113 Dercole F, Ferrière R, Gragnani A, Rinaldi S: Co- evolution of Slow-fast Populations: Evolutionary Sliding, Evo- lutionoary Pseudo-equilibria, and Complex Red Queen Dy- namics. IIASA Interim Report IR-06-006 (2006).
Contents
Abstract... 2
Introduction ... 3
Coevolutionary dynamics of slow-fast populations: general results ... 4
A model of predator-prey eco-evolutionary dynamics... 6
Results and discussion ... 8
Concluding comments ... 12
References ... 14
Figure captions ... 18
Figures ... 19
Electronic Appendix ... 1
COEVOLUTION OF SLOW-FAST POPULATIONS:
EVOLUTIONARY SLIDING, EVOLUTIONARY PSEUDO-EQUILIBRIA, AND COMPLEX RED QUEEN
DYNAMICS
F. Dercole1,6, R. Ferri`ere2,3, A. Gragnani1, S. Rinaldi1,4,5
1DEI, Politecnico di Milano Via Ponzio 34/5, 20133 Milano, Italy Ph: +39 02 2399 3484; Fax: +39 02 2399 3412
fabio.dercole@polimi.it
2Mathematical Eco-Evolutionary Theory (MEET) Group Laboratoire d’Ecologie, Ecole Normale Sup´erieure, Paris, France
3Department of Ecology and Evolutionary Biology University of Arizona, Tucson AZ, USA
4Adaptive Dynamics Network IIASA, Laxenburg, Austria
5IEIIT, CNR, Politecnico di Milano, Milano, Italy
6To whom correspondence should be addressed
Running title: Coevolution of slow-fast populations
Abstract
We study the interplay of ecological and evolutionary dynamics in communities composed of popu- lations with contrasting timescales. In such communities, genetic variations of individual traits can cause population transitions between stationary and cyclic ecological regimes, hence abrupt varia- tions in fitness. Such abrupt variations raise ridges in the adaptive landscape where the populations are poised between equilibrium and cyclic coexistence and along which evolutionary trajectories can remain sliding for long times or halt at special points called evolutionary pseudo-equilibria.
These novel phenomena should be generic to all systems in which ecological interactions cause fit- ness to vary discontinuously. They are demonstrated by the analysis of a predator-prey community, with one adaptive trait for each population. The eco-evolutionary dynamics of the system show a number of other distinctive features, including evolutionary extinction and two forms of Red Queen dynamics. One of them is characterised by intermittent bouts of cyclic oscillations of the two populations.
Key words: slow-fast population dynamics, eco-evolutionary dynamics, Red Queen dynamics, evolutionary sliding and pseudo-equilibria, adaptive ridge, predator-prey coevolution.
1 Introduction
Understanding the determinants of population dynamics is an important theme throughout biology, from human health to conservation. In studying population dynamics, much research has addressed how ecological interactions affect population stability yet ignoring the genetic diversity and ensuing evolvability of populations. Ford (1949) was perhaps the first to document that evolutionary change and population dynamics can occur interdependently; Pimentel (1968), Stenseth and Maynard Smith (1984) and Metz et al. (1992) subsequently conceptualised the notion of the ecological and evolutionary dynamics of a population being entangled in a feedback loop. The dynamical interplay of ecology and evolution prompts three general questions (May and Anderson, 1983; Ferri`ere and Gatto, 1993, 1995; Abrams, 2000): (1) How does evolution of adaptive traits affect the ecological stability of a community? (2) Under which conditions are ecological interactions expected to beget fluctuations in a population’s genetic state? (3) How do eco-evolutionary dynamics respond to environmental change?
Although a significant number of studies have dealt with some aspects of these three questions, there has been so far no attempt to address them simultaneously in a unified framework. Moreover, most models of the adaptive evolution of traits related to inter- and/or intra-specific interactions have assumed stable ecological equilibria for all trait values in the relevant trait space. This study aims at developing a unified analysis of eco-evolutionary dynamics in communities containing “slow”
and “fast” populations, which allows us to relax the ecological equilibrium assumption.
Slow-fast systems are composed of populations whose ecological fluctuations develop on con- trasting timescales. Predator-prey communities offer many instances of contrasting ecological timescales. Prey are often smaller than predator, hence faster in growing and reproducing. In the plankton food chain, the turnover of algae is faster than that of most zooplankton species which, in turn, grow faster than fish (Scheffer, 1998). The Boreal forest is also rich of examples:
plants (forbs and grasses) have fast dynamics in comparison with most herbivores (hares, squirrel, and small rodents) which reproduce faster than their predators (lynx, coyote, and red fox) (Stenseth et al., 1997). The opposite case, namely that of slow prey and fast predator, is also frequently ob- served in nature—spruce budworm (Ludwig et al., 1978) and larch budmoth (Baltensweiler, 1971) provide typical examples among plant-insect interactions. Hereafter we investigate theoretically the coevolution of a slow predator and fast prey. This analysis will serve to demonstrate the general features of eco-evolutionary dynamics of slow-fast populations that we first outline qualitatively in
the following section.
2 Coevolutionary dynamics of slow-fast populations: general re- sults
Eco-evolutionary processes generally assume two main ingredients: genetically-based variations of individual traits generated through reproduction, and selection on this variation resulting from ecological interactions. This is a complex process because individual traits under consideration may affect both the birth process and the ecological interactions. The assumption of rare mutations of small effects allows one to approximate the dynamics of population densities and trait distributions with deterministic models.
In the limit of rare mutations of small effects, the rate of change of an adaptive trait over evolutionary time is proportional to the resident population birth output per unit time (proportional to the probability that a mutation occurs within a small time interval), and to the derivative, with respect to mutant’s trait, of the per-capita rate of increase of a mutant population per generation (when positive, proportional to the probability of mutant non-extinction) (Metz et al., 1996). In the general situation of a traitz, resident population densitynat equilibrium, per-capita birth rate β (i.e. 1/β is the expected time between birth), mutant’s trait z0 and instantaneous per-capita rate of increaseS, this translates into the so-called canonical equation of adaptive dynamics (Dieckmann and Law, 1996):
˙
z=k(βn) s
β
(1) where s = ∂S/∂z0|z0=z is called selection derivative, and k is a parameter proportional to the probability that an offspring is a mutant and to the variance of mutation. Since the resident population density is at equilibrium, this equation simplifies as bothβ cancel out.
Here we need an extension of eq. (1) to the case of more general resident population attractors.
The rigorous derivation of such an extension is a hard mathematical exercise that lies beyond the scope of this paper (see Dieckmann and Law, 1996, for a heuristic discussion of the problem).
However, as explained below, averaging the mutation term, (βn), and selection term, (s/β), over the attractor is appropriate in the case considered in this paper. In formulas, this results in
˙
z=khβni s
β
(2)
where brackets indicate temporal averaging over the resident population attractor.
At this point, the analysis of eco-evolutionary dynamics through eq. (2) would remain prob- lematic because, in general, the resident population attractor is not known analytically in closed form. Slow-fast systems represent a significant exception to this predicament. Indeed, any slow-fast population attractor can be approximated with the so-called singular attractor corresponding to completely separated timescales (Rinaldi and Scheffer, 2000), and this permits explicit calculation of the averages in eq. (2). The case of slow-predator-fast-prey limit cycles is particularly favourable because the cycle can be easily identified. Moreover, such cycles are characterised by long phases of slow motion of both populations alternating with fast phases of significant prey variation. Thus, slow-fast cycles are very long, so that mutant populations experience little variations in the resident state during their initial phase of growth or decline (with the only exception of particular mutations occurring during the short episodes of fast variation of the prey). This supports the use of eq. (2) which indeed takes the expectation of (βn) and (s/β) over all possible resident states at the time of mutant arising.
A key evolutionary consequence of slow-fast ecological interactions is that the selection pressure becomes discontinuous across the trait space when stationary and cyclic coexistence are possible for different combinations of the traits. This is so because the transition from an equilibrium to a singular cycle is discontinuous (Rinaldi and Scheffer, 2000). General implications for eco- evolutionary dynamics can be outlined by focusing on two populations, e.g. a prey and a predator, coevolving in a two-dimensional trait space. Predator-prey interactions have long been known for their potential to generate a whole spectrum of ecological dynamics in response to variation in individual trait values, from extinction to stable equilibria and cycles. Thus, the trait space splits in three regions: extinction of at least one population in E, stationary coexistence inS, and cyclic coexistence in C. The selection pressure driving the dynamics of the two traits is continuous inside region S and region C but it is discontinuous at the boundary separating S and C. Two different evolutionary gradients are associated to each point of the boundary: one is the vector tangent to the evolutionary trajectory driven by the selection pressure operating in region S and the other is the vector tangent to the evolutionary trajectory driven by the selection pressure operating in region C (see Fig. 1 for an example). If the transversal components of these two vectors with respect to the discontinuity boundary have the same sign, as in Fig. 1A (dotted part of the discontinuity boundary), the trajectory crosses the boundary and the populations switch
from cyclic to stationary (or viceversa). On the contrary, if the transversal components of the two vectors are of opposite sign, i.e. if the two evolutionary gradients are “pushing” in opposite directions, (solid part of the discontinuity boundary in Fig. 1), the traits are forced to remain on the boundary and “slide” on it. In other words, the boundary separating the two possible ecological regimes can raise an attractiveridgein the adaptive landscape, along which evolutionary trajectories from various ancestral conditions are canalised. The evolution on the ridge can be temporary, as in Fig. 1A where the sliding motion terminates at pointT, or permanent, when the sliding motion halts at a so-called pseudo-equilibrium, namely at a point P on the boundary (see Fig. 1B) where the two evolutionary gradients align. A pseudo-equilibrium has all the properties of an equilibrium (in particular it can be an attractor, a saddle or a repellor) even if the selection pressures do not vanish at that point.
The rest of the paper focuses on a specific predator-prey model to demonstrate patterns of evolutionary sliding and pseudo-equilibria, and to recast them among other distinctive features of predator-prey coevolution. Such features include enhancement of Red Queen dynamics through the increase of genetic variation of the prey, generic occurrence of evolutionary extinction in the predator, and coevolution acting against ecological destabilisation resulting from environmental enrichment.
3 A model of predator-prey eco-evolutionary dynamics
Our presentation of a specific model of slow-fast populations and their eco-evolutionary dynamics focuses on the main features of the model, while details on mathematical derivations and approxi- mations are relegated into the (electronic) Appendix.
The predator-prey model we consider is the so-called Rosenzweig-MacArthur model (Rosenzweig and MacArthur, 1963) composed of a logistic prey and a Holling-type II predator:
˙
x(t) =x(t)
r
1−x(t) K
− a y(t) h+x(t)
(3)
˙
y(t) =y(t)
b+e a x(t) h+x(t) −d
(4) wherex(t) andy(t) are prey and predator population densities at timet. In the absence of predator the prey population grows logistically (with intrinsic growth rater and carrying capacityK), while
in the absence of prey the predator population decays exponentially (the intrinsic birth rate b is smaller than the death ratedbut the maximum birth rate (b+ea) is greater thand). The maximum predation rate is a, the functional response half-saturation constant is h, and the extra natality resulting from predation is proportional to the predation rate through an efficiency coefficient e.
The presence of a saturating functional response makes stationary and cyclic coexistence possible for different parameter settings (Section A1 in Appendix). The limit cycle is not known analytically, yet if prey grow at a much faster rate than predators, it can be approximated by the singular limit cycle (derived algebraically in Section A2 in Appendix).
Letuandv denote the adaptive traits for the prey and predator, respectively. Assume that the prey has density- and trait-independent birth rate, while its death rate has a density-dependent component controlled byu. Thus, in eq. (3)ris constant whileK depends onu. We further assume thatK peaks atu0, for which the prey is most effective. Similarly, the predator intrinsic birth rate bis constant, while its death rate ddepends upon vand is minimum at v0, at which predators are best adapted to their environment. The predation rate is a function of both traits, and predator [prey] benefit [lose] most from the interaction when traits are balanced, i.e. when u and v are in a suitable relationship, which defines a “bidirectional axis of prey vulnerability” (Abrams, 2000).
This mechanism is present if, for example, the searching effectiveness of the predator depends upon both traits but with a certain degree of plasticity, so that the same effectiveness can be achieved for a continuum of pairs (u, v). Since the half saturation constanthis inversely related to searching effectiveness, h(u, v) must be minimum when u and v are balanced, i.e. u=v provided both traits are measured on an appropriate scale. These are standard assumptions for predator- prey community modelling (Abrams, 2000), which have the advantage of involving the minimum possible number of demographic parameters. In our analyses we use (Section A3 in Appendix):
K(u) =K0
2 u
u0
2 +u0
u
2 d(v) =d0
v v0
2 +v0
v 2
2 h(u, v) =h0+h1(u−v)2 (5)
The trait space (u, v) can be partitioned into the three regionsE,S, andCpreviously described (Section A4 in Appendix). At the boundary between S and C all quantities associated with the asymptotic regime of the slow-fast system are discontinuous.
In conclusion, taking into account that the prey birth rate is density-independent, the canonical
equation (2) can be specified for the two populations as follows
˙
u=kuhxi hsui (6)
˙ v=kv
b+ eax h+x
y
* sv b+ eax
h+x +
(7)
where superscriptsu andv are used to indicate the corresponding species. Of course, the selection derivativessu andsv must be explicitly computed (Section A5 in Appendix) and the time averages must be performed in order to transform eqs. (6, 7) into standard ordinary differential equations.
Moreover, since the time averages are different in the coexistence regions S and C of trait space, this operation must be performed twice (Sections A6, A7 in Appendix).
4 Results and discussion
State portraits for the eco-evolutionary model (6, 7) can be constructed by numerical simulations.
A typical example is shown in Fig. 2. It contains a small region (dark) where the predator under- goes evolutionary extinction (Matsuda and Abrams, 1994a; Ferri`ere, 2000; Dieckmann and Ferri`ere, 2004), and two evolutionary attractors: an equilibrium with low trait values and a cycle charac- terised by high trait values. The two basins of attraction are separated by the stable manifold of the saddle lying in region C. If the ancestral conditions are on the left of this manifold the traits converge to the equilibrium so that, after evolutionary transients, the populations coexist at ecological equilibrium, since the evolutionary equilibrium is in region S. However, for some ancestral conditions, one piece of the evolutionary orbit lies in regionC: this means that during the corresponding period of time the populations oscillate along an ecological cycle that slowly drifts on the evolutionary timescale. At evolutionary equilibrium prey evolutionary branching (Geritz et al., 1997, 1998; Dercole et al., 2003) may occur, but will not be investigated here.
Population dynamics associated with evolutionary trajectories in the other basin of attraction are radically different. Indeed, long periods of time characterised by slowly varying populations recurrently alternate with long periods of time during which populations fluctuate on an ecological cycle, as a consequence of the attracting evolutionary cycle being partly in region S and partly in region C.
Fig. 2 highlights the possibility of coevolution along adaptive ridges, which are segments of the
boundary separating cyclic from stationary coexistence. Sliding is a novel type of evolutionary dynamics with important ecological implications: when traits are sliding along an adaptive ridge, prey and predator are poised between stationary and cyclic coexistence, i.e. coevolution drives the populations toward and maintain them at the onset of their most complex dynamic behaviour. How sensitive these phenomena are to parameters is investigated by means of a thorough bifurcation analysis (Section A8 in Appendix). Synthesising the results yields a series of eight statements (the first three are general, while the others are specific of predator-prey systems) that we list hereafter and illustrate with selected state portraits (Fig. 3).
(i) Evolutionary sliding and pseudo-equilibria. Evolutionary sliding along the boundary separat- ing stationary from cyclic coexistence occurs for many parameter settings. The evolutionary sliding can be temporary (sliding segment) or halt at an evolutionary pseudo-equilibrium.
When the adaptive traits are sliding, or resting at a pseudo-equilibrium, the populations are in critically stable ecological states and their mean characteristics (densities, density- dependent parameters) can vary abruptly for small changes of individual traits.
(ii) Evolutionary extinction. There is always a subregion (dark in all state portraits) where the orbits tend toward the boundary of region E. This causes the predator to go extinct in the long run, a phenomenon that is not predictable on the basis of purely ecological arguments.
Evolution to extinction had been noted in predator-prey and competition models by Matsuda and Abrams (1994a,b); Dieckmann et al. (1995), and is also known to occur in models of mutualistic (Ferri`ere et al., 2002) and cannibalistic (Dercole and Rinaldi, 2002; Dercole, 2003) interactions. These examples highlight a common mechanism. Adaptive evolution is driven by the “marginal” benefit of performing better in interactions (predation, mutualism) than other conspecifics; yet the “direct”, physiological cost to the individual can become so great that eventually the population growth rate becomes negative, causing extinction.
(iii) Multiple evolutionary equilibria. The eco-evolutionary system most often possesses several equilibria: attractors, repellors and saddles, in addition to predator evolutionary extinction.
Two general implications can be drawn. First, in the long run, the same populations can reach different evolutionary states and develop different ecological dynamics due to to an- cient differences in their genotypic state. Experimental evolution inEscherichia coliprovides strong empirical support to this prediction (Travisano et al., 1995). Second, the co-occurrence
of predator evolutionary extinction and other viable evolutionary attractors provides a firm mathematical basis for the notion of evolutionary trapping suggested from empirical obser- vations (Colas et al., 1997; Schlaepfler et al., 2002): under given environmental conditions, the predator population can be trapped on an evolutionary trajectory heading to extinc- tion whereas alternative, ecologically safe evolutionary attractors could have been reached.
Schlaepfler et al. (2002) and Ferri`ere et al. (2004) have discussed the implications of evolu- tionary trapping in a conservation perspective.
(iv) Two forms of Red Queen dynamics. The first one (evolutionary cycle in regionS, see portraits B and A0) corresponds to slow periodic variations of the traits entraining slow population cycles. This form is well-known from Lotka-Volterra models that did not allow for other forms of Red Queen dynamics (Abrams, 2000). The second form (evolutionary cycle in region S and C, see portraits C and B0) corresponds to slow periodic variations of the traits accompanied by recurrent and long bouts of ecological oscillations. This complex pattern was predicted by Khibnik and Kondrashov (1997) who have named it “eco-genetically driven Red Queen dynamics”.
(v) Factors enhancing Red Queen dynamics. Our study confirms that a bidirectional axis of prey vulnerability is a potent mechanism for generating evolutionary cycles (Abrams, 2000). By increasing the impact of the traits on vulnerability the evolutionary attractor changes in a typical sequence (see portraits A, B, C): first an equilibrium associated with steady pop- ulations, then an evolutionary cycle with entrained population oscillations, and finally an evolutionary cycle associated with recurrent bouts of ecological oscillations. For a further in- crease of the vulnerability mechanism Red Queen dynamics suddenly disappear (see portraits D), a phenomenon that has gone unnoticed in previous studies. Increasing the probability of prey mutation, the variance of the distribution of prey mutational effects, and the maximum predation rate can also trigger and enhance Red Queen dynamics (see portraits (A, A0) and (B, B0)).
(vi) The predator chases the prey. All evolutionary cycles we have detected are counterclockwise.
Thus, the predator trait increases when the prey trait is large and decreases in the opposite case. This results from the bidirectional axis of prey vulnerability, and is, indeed, present in all studies where the prey has a most vulnerable phenotype depending upon predator’s trait
(see Marrow et al., 1992; Dieckmann et al., 1995; Dieckmann and Law, 1996; Abrams and Matsuda, 1997; Khibnik and Kondrashov, 1997).
(vii) Evolution toward ecological stability: The paradox of enrichment. Ecological theory predicts that predator-prey interactions should cause large amplitude cycles in rich environments (Rosenzweig, 1971). The “paradox of enrichment” emphasises that this does not occur in nature (e.g. Murdoch et al., 1998). Abrams and Walters (1996) found an ecological solution to the paradox for certain types of predator-prey communities, later confirmed by experimental findings (McCauley et al., 1999). Rosenzweig and Schaffer (1978) took a general, evolutionary approach to the problem, arguing that evolution should tend to restore ecological stability lost through enrichment. Evolution may actually play such a significant role in light of e.g.
Yoshida et al.’s (2003) findings on rapid evolutionary change in predator-prey systems. Our work substantiates, refines, and broadens Rosenzweig and Schaffer’s view in the case of slow predator and fast prey. In fact, if the system is at its evolutionary equilibrium in region S (portrait D), would evolutionary processes be absent (i.e. u and v being kept frozen), significant enrichment would destabilise the populations. This is clearly recognisable from portrait D0 where the point∗in regionCis the copy of the evolutionary equilibrium of portrait D. Interestingly, after enrichment the evolutionary processes act in the opposite direction and the final result (portrait D0) is that the traits tend to an evolutionary pseudo-equilibrium. In other words, the full destabilisation of the populations triggered by enrichment is opposed by the counteracting forces of evolution.
(viii) Evolution opposes permanent ecological oscillations. There seems to be no realistic environ- mental conditions under which an evolutionary attractor is entirely in region C, although evolutionary trajectories are often trapped on the boundary betweenS andC (see statement (i)). Only if the predation rate is almost independent of prey and predator traits (i.e. ifh1 is of the order of 10−3), there is an evolutionary equilibrium in C (notice that for h1 = 0, point (u0, v0) is a stable evolutionary equilibrium in C). Thus, evolution seems to oppose permanent ecological oscillations.
5 Concluding comments
Our analysis unravels novel evolutionary phenomena whose scope extends beyond predator-prey coevolution. This includes the possibility that coevolution guides the traits along adaptive ridges formed by segments of the boundary between the regions of stationary and cyclic coexistence (evolutionary sliding), or comes to a halt at special points of that boundary (evolutionary pseudo- equilibria).
Our conclusions for predator-prey coevolution are likely to be influenced by the specific model chosen to describe the interaction. The analysis, however, should in principle be repeatable for any ecological model involving slow-fast dynamics. Considering the dual case of slow prey (e.g. plants) and fast predator (e.g. insects), and how common recurrent insect-pest outbreaks are in natural or exploited forests, coevolution might well in this case have just the opposite effect on ecological dynamics, namely that of favouring cyclic coexistence.
Finally, the conjecture formulated by Ellner and Turchin on the basis of their analyses of pop- ulation time series (Ellner and Turchin, 1995), namely that “ecosystems might evolve toward the edge of chaos”, finds some support in the present study. Indeed, our findings suggest that ecosys- tems might evolve toward the edge of their most complex dynamic regime, which in the case of predator-prey models, is indeed cyclic coexistence. But the support could become even stronger once the present analysis is extended to tritrophic food chains with potentially chaotic ecological dynamics. This should be feasible since singular cycles and singular bifurcations responsible for chaotic dynamics have already been found in such communities (De Feo and Rinaldi, 1998). Bifur- cations leading to chaos are a likely cause of discontinuity in fitness across trait space. Complex Red Queen dynamics involving intermittent bouts of chaotic fluctuations of the populations would then rank among the expected outcomes, as predicted by Ellner and Turchin (1995).
Acknowledgements
The authors are grateful to Peter Abrams and to an anonymous reviewer for their comments and constructive criticisms on earlier drafts of this manuscript. F. D., A. G. and S. R. acknowledge funding support from MIUR (under project FIRB-RBNE01CW3M). R.F. acknowledges funding support from the French National Programmes “New Science Interfaces with Mathematics” and
“Biological Invasions”, and from the European Research and Training Network “ModLife” (HPRN-
CT-2000-00051).
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Figure captions
Figure 1
Evolution of the traits in the neighbourhood of the boundary separating stationary coexistence (region S) from cyclic coexistence (region C). A, evolutionary sliding toward T (solid boundary) and crossing (dotted boundary). B, evolutionary sliding toward the pseudo-equilibriumP.
Figure 2
A state portrait of the eco-evolutionary model (6, 7). There are three equilibria (a stable node (filled dot) and an unstable focus (empty dot) in region S and a saddle in region C) and one limit cycle (partly in region S and partly in region C). There are two attractors, the node and the cycle, and their basins of attraction are separated by the stable manifold of the saddle. There are three sliding segments, one stable (T1T2, stretched and magnified in the lower right panel) and two unstable (T3T4 and T5T6). Predator evolutionary extinction occurs in the dark region. Parameter values: ku = 0.1,kv = 1, r= 1,K0 = 1,h0 = 0.02, h1 = 0.02,d0 = 0.01, b= 0.001, e= 0.1,a= 5, u0 = 1,v0 = 3.
Figure 3
Eight state portraits of the eco-evolutionary model (6, 7). In the first row the parameters are ku = 0.1, kv = 1, r = 1,K0 = 1, h0 = 0.02, d0 = 0.01, b= 0.001, e= 0.1, a= 1.5,u0 = 1, v0 = 3, and h1 = 0.01 in A,h1 = 0.017 in B,h1 = 0.03 in C, h1 = 0.05 in D. State portrait A0 is obtained from A by increasingkufrom 0.1 to 0.145. State portrait B0is obtained from B by increasingafrom 1.5 to 3. State portrait C0 is obtained from C by increasingv0 from 3 to 3.3. State portrait D0 is obtained from D by increasingK0from 1 to 5. All dark regions correspond to predator evolutionary extinction. Evolutionary sliding is present in all panels, while pseudo-equilibria (squared points) are present in panels A, A0, B, D0.
C
S T
C
S P
A
B
Figure 1
0 10 0
5
T4
T3
T1
T5
T2
T1
T2
predatortrait,v
prey trait,u C
S E
T6
Figure 2
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A
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B
C
D
B
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0Figure 3
