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Evolutionary dynamics of predator-prey systems: an ecological perspective

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Abstract

Evolution takes place in an ecological setting that typically involves interactions with other organisms. To describe such evolution, a structure is needed which incorporates the simultaneous evolution of interacting species. Here a formal framework for this purpose is suggested, extending from the microscopic interactions between individuals — the immediate cause of natural selection, through the mesoscopic population dynamics responsible for driving the replacement of one mutant phenotype by another, to the macroscopic process of phenotypic evolution arising from many such substitutions. The process of coevolution that results from this is illustrated in the context of predator-prey systems. With no more than qualitative information about the evolutionary dynamics, some basic properties of predator-prey coevolution become evident. More detailed understanding requires specification of an evolutionary dynamics; two models for this purpose are outlined, one from our own research on a stochastic process of mutation and selection and the other from quantitative genetics. Much of the interest in coevolution has been to characterize the properties of fixed points at which there is no further phenotypic evolution. Stability analysis of the fixed points of evolutionary dynamical systems is reviewed and leads to conclusions about the asymptotic states of evolution rather different from those of game-theoretic methods. These differences become especially important when evolution involves more than one species.

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References

  • Abrams, P. A.: Is predator-prey coevolution an arms race? Trends Ecol. Evol.1, 108–110 (1986)

    Article  Google Scholar 

  • Abrams, P. A.: The importance of intraspecific frequency-dependent selection in modelling competitive coevolution. Evol. Ecol.3, 215–220 (1989)

    Article  Google Scholar 

  • Abrams, P. A., Matsuda, H., Harada, Y.: Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits. Evol. Ecol.7, 465–487 (1993)

    Article  Google Scholar 

  • Anderson, P. A., May, R. M.: Coevolution of hosts and parasites. Parasitology85, 411–426 (1982)

    Article  Google Scholar 

  • Bakker, R. T.: The deer flees, the wolf pursues: incongruencies in predator-prey coevolution. In: Futuyma, D. J., Slatkin, M. (eds.) Coevolution, pp. 350–382. Sunderland Massachusetts: Sinauer Associates 1983

    Google Scholar 

  • Brown, J. S., Vincent, T. L.: Coevolution as an evolutionary game. Evolution41, 66–79 (1987a)

    Article  Google Scholar 

  • Brown, J. S., Vincent, T. L.: Fredator-prey coevolution as an evolutionary game. In: Cohen, Y. (ed.) Applications of Control Theory in Ecology, pp. 83–101. Lecture Notes in Biomathematics 73. Berlin: Springer Verlag 1987b

    Google Scholar 

  • Brown, J. S., Vincent, T. L.: Organization of predatorprey communities as an evolutionary game. Evolution46, 1269–1283 (1992)

    Article  Google Scholar 

  • Carroll, L.: Through the Looking Glass and what Alice found there. London: Macmillan 1871 (Harmondsworth Middlesex: Penguin Books 1970)

    Google Scholar 

  • Christiansen, F. B.: On conditions for evolutionary stability for a continuously varying character. Amer. Natur.138, 37–50 (1991)

    Article  Google Scholar 

  • Darwin, C.: The Origin of Species by Means of Natural Selection. John Murray 1859 (Harmondsworth Middlesex: Penguin Books 1968)

    Google Scholar 

  • Dawkins, R., Krebs, J. R.: Arms races between and within species. Proc. R. Soc. Lond. B205, 489–511 (1979)

    Article  Google Scholar 

  • Dieckmann, U.: Coevolutionary dynamics of stochastic replicator systems. Berichte des Forschungszentrums Jülich, 3018, Jülich Germany 1994

  • Dieckmann, U., Law, R.: The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol.34, 579–612 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Dieckmann, U., Marrow, P., Law, R.: Evolutionary cycling in predator-prey interactions: population dynamics and the Red Queen. J. Theor. Biol.176, 91–102 (1995)

    Article  Google Scholar 

  • Dwyer, G., Levin, S. A., Buttel, L.: A simulation model of the poulation dynamics and evolution of myxomatosis. Ecol. Monogr.60, 423–447 (1990)

    Article  Google Scholar 

  • Edley, M. E., Law, R.: Evolution of life histories and yields in experimental populations ofDaphnia magna. Biol. J. Linn. Soc.34, 309–326 (1988)

    Google Scholar 

  • Ehrlich, P. R., Raven, P. H.: Butterflies and plants: a study in coevolution. Evolution18, 586–608 (1964)

    Article  Google Scholar 

  • Eshel, I.: Evolutionary and continuous stability. J. theor. Biol.103, 99–111 (1983)

    Article  MathSciNet  Google Scholar 

  • Eshel, I., Motro, U.: Kin selection and strong evolutionary stability of mutual help. Theor. Pop. Biol.19, 420–433. (1981)

    Article  MathSciNet  MATH  Google Scholar 

  • Falconer, D. S.: Introduction to Quantitative Genetics. 3rd Edition. Harlow: Longman 1989

    Google Scholar 

  • Fenner, F., Ratcliffe, F. N.: Myxomatosis. Cambridge University Press 1965

  • Fisher, R. A.: The Genetical Theory of Natural Selection. 2nd Edition. New York: Dover Publications 1958

    Google Scholar 

  • Futuyma, D. J., Slatkin, M.: Introduction. In: Futuyma, D. J., Slatkin, M. (eds.) Coevolution, pp. 1–13. Sunderland Massachusetts: Sinauer Associates 1983

    Google Scholar 

  • Gatto, M.: The evolutionary optimality of oscillatory and chaotic dynamics in simple population models. Theor. Pop. Biol.43, 310–336 (1993)

    Article  MATH  Google Scholar 

  • Godfray, H. C. J., Cook, L. M., Hassell, M. P.: Population dynamics, natural selection and chaos. In: Berry, R. J., Crawford, T. J., Hewitt, G. M. (eds) Genes in ecology, pp 55–86. Oxford: Blackwell Scientific Publications (1993)

    Google Scholar 

  • Hansen, T. F.: Evolution of stability parameters in single-species population models: stability or chaos? Theor. Pop. Biol.42, 199–217 (1992)

    Article  MATH  Google Scholar 

  • Harrison, G. W.: Global stability of food chains. Amer. Natur.114, 455–457 (1979)

    Article  MathSciNet  Google Scholar 

  • Hassell, M. P., Lawton, J. H., May, R. M.: Patterns of dynamical behaviour in single-species populations. J. Anim. Ecol.45, 471–486 (1976)

    Google Scholar 

  • Hofbauer, J., Sigmund, K.: Adaptive dynamics and evolutionary stability. Appl. Math. Lett.3, 75–79 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Hutchinson, G. E.: The Ecological Theater and the Evolutionary Play. New Haven Connecticutt: Yale University Press 1967

    Google Scholar 

  • Iwasa, Y., Pomiankowski, A., Nee, S.: The evolution of costly mate preferences. II. The “handicap” principle. Evolution45, 1431–1442 (1991)

    Article  Google Scholar 

  • Janzen, D. H.: When is it coevolution? Evolution34, 611–612 (1980)

    Article  Google Scholar 

  • Jayakar, S. D., Zonta L. A.: Coevolution at two trophic levels. In: Wöhrmann, K., Jain, S. K. (eds) Population Biology, pp. 349–366. Berlin: Springer Verlag 1990

    Google Scholar 

  • Kauffman, S. A.: The Origins of Order: Self-Organization and Selection in Evolution. New York: Oxford University Press 1993

    Google Scholar 

  • Kauffman, S. A., Johnsen, S.: Coevolution to the edge of chaos: coupled fitness landscapes, poised states, and coevolutionary avalanches. J. theor Biol.149, 467–505 (1991)

    Google Scholar 

  • Kitchell, J. A., Boggs, C. H., Kitchell, J. F., Rice, J. A.: Prey selection by natacid gastropods: experimental tests and application to the fossil record. Palaeobiology7, 533–552 (1981)

    Google Scholar 

  • Lande, R.: Quantitative genetic analysis of multivariate evolution, applied to brain:body size allometry. Evolution33, 402–416 (1979)

    Article  Google Scholar 

  • Levin, S. A., Segel, L. A., Adler, F. R.: Diffuse coevolution in plant-herbivore communities. Theor. Pop. Biol.37, 171–191 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Lewontin, R. C.: Fitness, survival, and optimality. In: Horn, D. J., Stairs, G. R., Mitchell, R. D. (eds.) Analysis of Ecological Systems, pp. 3–21. Ohio State University Press 1979

  • Marrow, P., Law, R., Cannings, C.: The coevolution of predator-prey interactions: ESSs and Red Queen dynamics. Proc. R. Soc. Lond. B250, 133–141 (1992)

    Google Scholar 

  • Marrow, P., Cannings, C.: Evolutionary instability in predator-prey systems. J. theor. Biol.160, 135–150 (1993)

    Article  Google Scholar 

  • Mather, K.: Genetical Structure of Populations. London: Chapman and Hall 1973

    Google Scholar 

  • Maynard Smith, J.: A comment on the Red Queen. Amer. Natur.110, 325–330 (1976)

    Article  Google Scholar 

  • Maynard Smith, J.: Evolution and the Theory of Games. Cambridge University Press 1982

  • Maynard Smith, J., Price, G. R.: The logic of animal conflict. Nature Lond.246, 15–18 (1973)

    Article  Google Scholar 

  • Metz, J. A. J., Nisbet, R. M., Geritz, S. A. H.: How should we define ‘fitness’ for general ecological scenarios? Trends Ecol. Evol.7, 198–202 (1992)

    Article  Google Scholar 

  • Metz, J. A. J, Geritz, S. A. H., Iwasa, Y: On the dynamical classification of evolutionarily singular strategies. University of Leiden Preprint, Leiden, The Netherlands (1994)

  • Michod, R. E.: Evolution of life histories in response to age-specific mortality factors. Amer. Natur.113, 531–550 (1979)

    Article  MathSciNet  Google Scholar 

  • Parker, G. A.: Arms races in evolution — an ESS to the opponent-independent costs game. J. theor. Biol.101, 619–648 (1983)

    Article  Google Scholar 

  • Parker, G. A.: Population consequences of evolutionarily stable strategies. In: Sibly, R. M., Smith, R. H. (eds) Behavioural Ecology: Ecological Consequences of Adaptive Behaviour, pp 33–57. Oxford: Blackwell Scientific 1985

    Google Scholar 

  • Parker, G. A., Maynard Smith, J.: Optimality theory in evolutionary biology. Nature Lond. 348, 27–33 (1990)

    Article  Google Scholar 

  • Pimentel, D.: Population regulation and genetic feedback. Science 159, 1432–1437 (1968)

    Google Scholar 

  • Pimm, S. L., Lawton, J. H., Cohen, J. E.: Food web patterns and their consquences. Nature, Lond.350, 669–674 (1991)

    Article  Google Scholar 

  • Rand, D. A., Wilson, H. B.: Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proc. Roy. Soc. Lond. B246, 179–184 (1991)

    Google Scholar 

  • Reed, J., Stenseth, N. C.: On evolutionarily stable strategies. J. theor. Biol.108, 491–508 (1984)

    MathSciNet  Google Scholar 

  • Rosenzweig, M. L.: Evolution of the predator isocline. Evolution27, 84–94 (1973)

    Article  Google Scholar 

  • Rosenzweig, M. L., Brown, J. S., Vincent, T. L.: Red Queens and ESS: the coevolution of evolutionary rates. Evol. Ecol.1, 59–94 (1987)

    Article  Google Scholar 

  • Roughgarden, J.: Theory of population genetics and evolutionary ecology: an introduction. New York: Macmillan 1979.

    Google Scholar 

  • Roughgarden, J.: The theory of coevolution. In: Futuyma, D. J., Slatkin, M. (eds) Coevolution, pp. 33–64. Sunderland Massachusetts: Sinauer Associates 1983.

    Google Scholar 

  • Roughgarden, J.: Community coevolution: a comment. Evolution41, 1130–1134 (1987)

    Article  Google Scholar 

  • Saloniemi, I.: A coevolutionary predator-prey model with quantitative characters. Amer. Natur.141, 880–896 (1993)

    Article  Google Scholar 

  • Slatkin, M., Maynard Smith, J.: Models of coevolution. Q. Rev. Biol.54, 233–263 (1979)

    Article  Google Scholar 

  • Slobodkin, L. B.: How to be a predator. Amer. Zool.8, 43–51 (1968)

    Google Scholar 

  • Slobodkin, L. B.: Prudent predation does not require group selection. Amer. Natur.108, 665–678 (1974)

    Article  Google Scholar 

  • Stanley, S. M., Van Valkenburg, B., Steneck, R. S.: Coevolution and the fossil record. In: Futuyma, D. J., Slatkin, M. (eds.) Coevolution, pp. 328–349. Sunderland Massachusetts: Sinauer Associates 1983

    Google Scholar 

  • Stenseth, N. C.: Darwinian evolution in ecosystems: a survey of some ideas and difficulties together with some possible solutions. In: Castri, J. L., Karlquist, A. (eds) Complexity, language and life: mathematical approaches, pp 105–145. Berlin: Springer-Verlag 1986

    Google Scholar 

  • Stenseth, N. C., Maynard Smith, J.: Coevolution in ecosystems: Red Queen evolution or stasis? Evolution38, 870–880 (1984)

    Article  Google Scholar 

  • Stokes, T. K., Gurney, W. S. C., Nisbet, R. M., Blythe, S. P.: Parameter evolution in a laboratory insect population. Theor. Pop. Biol.34, 248–265 (1988)

    Article  MATH  Google Scholar 

  • Takada, T., Kigami, J.: The dynamical attainability of ESS in evolutionary games. J. Math. Biol.29, 513–529 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Taper, M. L., Case, T. J.: Models of character displacement and the theoretical robustness of taxon cycles. Evolution46, 317–333 (1992)

    Article  Google Scholar 

  • Taylor, P. D.: Evolutionary stability in one-parameter models under weak selection. Theor. Pop. Biol.36, 125–143 (1989)

    Article  MATH  Google Scholar 

  • Turchin, P., Taylor, A. D.: Complex dynamics in ecological time series. Ecology73, 289–305 (1992)

    Article  Google Scholar 

  • van Kampen, N. G.: Stochastic processes in physics and chemistry. Amsterdam: North Holland 1981

    MATH  Google Scholar 

  • Van Valen, L.: A new evolutionary law. Evol. Theory1, 1–30 (1973)

    Google Scholar 

  • Vermeij, G. J.: Unsuccessful predation and evolution. Amer. Natur.120, 701–720 (1982)

    Article  Google Scholar 

  • Vermeij, G. J.: Intimate associations and coevolution in the sea. In: Futuyma, D. J., Slatkin, M. (eds.) Coevolution, pp. 311–327. Sunderland Massachusetts: Sinauer Associates 1983

    Google Scholar 

  • Vermeij, G. J.: Evolution and escalation: an ecological history of life. Princeton New Jersey: Princeton University Press 1987

    Google Scholar 

  • Vermeij, G. J., Covitch, A. P.: Coevolution of freshwater gastropods and their predators. Amer. Natur.112, 833–843 (1978)

    Article  Google Scholar 

  • Vincent, T. L.: Strategy dynamics and the ESS. In: Vincent, T. L., Mees, A. I., Jennings, L. S. (eds.) Dynamics of Complex Interconnected Biological Systems, pp. 236–262. Basel: Birkhäuser 1991

    Google Scholar 

  • West, K., Cohen, A., Baron, M.: Morphology and behaviour of crabs and gastropods from Lake Tanganyika, Africa: impllications for lacustrine predator-prey coevolution. Evolution45, 589–607 (1991)

    Article  Google Scholar 

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Author notes

  1. Paul Marrow

    Present address: Large Animal Research Group, Department of Zoology, University of Cambridge, Downing Street, CB2 3EJ, Cambridge, UK

  2. Ulf Dieckmann

    Present address: Wissenschaftskolleg zu Berlin, Wallotstrasse 19, D-14193, Berlin, Germany

Authors and Affiliations

  1. Theoretical Biology Section, Institute of Evolutionary and Ecological Sciences, University of Leiden, Kaiserstraat 63, 2311 GP, Leiden, The Netherlands

    Paul Marrow

  2. Theoretical Biology Section, Institute of Evolutionary and Ecological Sciences, University of Leiden, Kaiserstraat 63, 2311 GP, Leiden, The Netherlands

    Ulf Dieckmann

  3. Department of Biology, University of York, YO1 5DD, York, UK

    Richard Law

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  3. Richard Law
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Marrow, P., Dieckmann, U. & Law, R. Evolutionary dynamics of predator-prey systems: an ecological perspective. J. Math. Biology 34, 556–578 (1996). https://doi.org/10.1007/BF02409750

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