Journal of Mathematical Biology

, Volume 34, Issue 5–6, pp 579–612 | Cite as

The dynamical theory of coevolution: a derivation from stochastic ecological processes

  • Ulf Dieckmann
  • Richard Law


In this paper we develop a dynamical theory of coevolution in ecological communities. The derivation explicitly accounts for the stochastic components of evolutionary change and is based on ecological processes at the level of the individual. We show that the coevolutionary dynamic can be envisaged as a directed random walk in the community's trait space. A quantitative description of this stochastic process in terms of a master equation is derived. By determining the first jump moment of this process we abstract the dynamic of the mean evolutionary path. To first order the resulting equation coincides with a dynamic that has frequently been assumed in evolutionary game theory. Apart from recovering this canonical equation we systematically establish the underlying assumptions. We provide higher order corrections and show that these can give rise to new, unexpected evolutionary effects including shifting evolutionary isoclines and evolutionary slowing down of mean paths as they approach evolutionary equilibria. Extensions of the derivation to more general ecological settings are discussed. In particular we allow for multi-trait coevolution and analyze coevolution under nonequilibrium population dynamics.

Key words

Coevolution Stochastic processes Mutation-selection systems Individual-based models Population dynamics Adaptive dynamics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abrams, P. A.: Adaptive responses of predators to prey and prey to predators: the failure of the arms-race analogy. Evolution40, 1229–1247 (1986)CrossRefGoogle Scholar
  2. Abrams, P. A., Matsuda, H., Harada, Y.: Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits. Evol. Ecol.7, 465–487 (1993)CrossRefGoogle Scholar
  3. Bailey, N. T. J.: The elements of stochastic processes. New York: John Wiley and Sons 1964zbMATHGoogle Scholar
  4. Baker, G. L., Gollub, J. P.: Chaotic dynamics: an introduction. Cambridge: Cambridge University Press 1990zbMATHGoogle Scholar
  5. Brown, J. S., Vincent, T. L.: Coevolution as an evolutionary game. Evolution41, 66–79 (1987a)CrossRefGoogle Scholar
  6. Brown, J. S., Vincent, T. L.: Predator-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 1987bGoogle Scholar
  7. Brown, J. S., Vincent, T. L.: Organization of predator-prey communities as an evolutionary game. Evolution46, 1269–1283 (1992)CrossRefGoogle Scholar
  8. Christiansen, F. B.: On conditions for evolutionary stability for a continuously varying character. Amer. Natur.138, 37–50 (1991)CrossRefGoogle Scholar
  9. Dawkins, R.: The selfish gene. Oxford: Oxford University Press 1976Google Scholar
  10. Dawkins, R., Krebs, J. R.: Arms races between and within species. Proc. Roy. Soc. Lond. B205, 489–511 (1979)CrossRefGoogle Scholar
  11. Dieckmann, U.: Coevolutionary dynamics of stochastic replicator systems. Jülich Germany: Berichte des Forschungszentrums Jülich (Jül-3018) 1994Google Scholar
  12. Dieckmann, U., Marrow, P., Law, R.: Evolutionary cycles in predator-prey interactions: population dynamics and the Red Queen. J. Theor. Biol.176, 91–102 (1995)CrossRefGoogle Scholar
  13. Ebeling, W., Feistel, R.: Physik der Selbstorganisation und Evolution. Berlin: Akademie-Verlag 1982Google Scholar
  14. Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys.57, 617–656 (1985)CrossRefMathSciNetGoogle Scholar
  15. Emlen, J. M.: Evolutionary ecology and the optimality assumption. In: Dupre, J. (ed.) The latest on the best, pp. 163–177. Cambridge: MIT Press 1987Google Scholar
  16. Eshel, I.: Evolutionary and continuous stability. J. Theor. Biol.103, 99–111 (1983)CrossRefMathSciNetGoogle Scholar
  17. Eshel, I., Motro, U.: Kin selection and strong stability of mutual help. Theor. Pop. Biol.19, 420–433 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  18. Falconer, R. A.: Introduction to quantitative genetics. 3rd Edition. Harlow: Longman 1989Google Scholar
  19. Fisher, R. A.: The genetical theory of natural selection. New York: Dover Publications 1958Google Scholar
  20. Futuyma, D. J., Slatkin, M.: Introduction. In: Futuyma, D. J., Slatkin, M. (eds.) Coevolution, pp. 1–13. Sanderland Massachusetts: Sinauer Associates 1983Google Scholar
  21. Gillespie, D. T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comp. Phys.22, 403–434 (1976)CrossRefMathSciNetGoogle Scholar
  22. Goel, N. S., Richter-Dyn, N., Stochastic models in biology. New York: Academic Press 1974Google Scholar
  23. Hofbauer, J., Sigmund, K.: Theory of evolution and dynamical systems. New York: Cambridge University Press 1988zbMATHGoogle Scholar
  24. Hofbauer, J., Sigmund, K.: Adaptive dynamics and evolutionary stability. Appl. Math. Lett.3, 75–79 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  25. Iwasa, Y., Pomiankowski, A., Nee, S.: The evolution of costly mate preferences. II. The “handicap” principle. Evolution45, 1431–1442 (1991)CrossRefGoogle Scholar
  26. Kimura, M.: The neutral theory of molecular evolution. Cambridge: Cambridge University Press 1983Google Scholar
  27. Kubo, R., Matsuo, K., Kitahara, K.: Fluctuation and relaxation of macrovariables. J. Stat. Phys.9, 51–96 (1973)CrossRefGoogle Scholar
  28. Lande, R.: Quantitative genetic analysis of multivariate evolution, applied to brain: body size allometry. Evolution33, 402–416 (1979)CrossRefGoogle Scholar
  29. Lawlor, L. R., Maynard Smith, J.: The coevolution and stability of competing species. Amer. Natur.110, 79–99 (1976)CrossRefGoogle Scholar
  30. Lewontin, R. C.: Pitness, 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 1979Google Scholar
  31. Lewontin, R. C.: Gene, organism and environment. In: Bendall, D. S. (ed.) Evolution from molecules to men, pp. 273–285. Cambridge: Cambridge University Press 1983Google Scholar
  32. Lewontin, R. C.: The shape of optimality. In: Dupre, J. (ed.) The latest on the best, pp. 151–159. Cambridge: MIT Press 1987Google Scholar
  33. Loeschke, V. (ed.): Genetic constraints on adaptive evolution. Berlin: Springer-Verlag 1987Google Scholar
  34. Mackay, T. F. C.: Distribution of effects of new mutations affecting quantitative traits. In: Hill, W. G., Thompson, R., Woolliams, J. A. (eds.) Proc. 4th world congress on genetics applied to livestock production, pp. 219–228. 1990Google Scholar
  35. Marrow, P., Cannings, C.: Evolutionary instability in predator-prey systems. J. Theor. Biol.160, 135–150 (1993)CrossRefGoogle Scholar
  36. Marrow, P., Dieckmann, U., Law, R.: Evolutionary dynamics of predator-prey systems: an ecological perspective. J. Math. Biol.34, 556–578 (1996)CrossRefzbMATHGoogle Scholar
  37. Marrow, P., Law, R., Cannings, C.: The coevolution of predator-prey interactions: ESSs and Red Queen dynamics. Proc. Roy. Soc. Lond. B250, 133–141 (1992)Google Scholar
  38. Maynard Smith, J.: Evolution and the theory of games. Cambridge: Cambridge University Press 1982zbMATHGoogle Scholar
  39. Maynard Smith, J., Burian, R., Kauffman, S. Alberch, P., Campbell, J., Goodwin, B., Lande, R., Raup, D., Wolpert, L.: Developmental constraints and evolution. Q. Rev. Biol.60, 265–287 (1985)CrossRefGoogle Scholar
  40. Maynard Smith, J., Price, G. R.: The logic of animal conflict. Nature Lond.246, 15–18 (1973)CrossRefGoogle Scholar
  41. 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)CrossRefGoogle Scholar
  42. Metz, J. A. J, Geritz, S. A. H., Iwasa, Y.: On the dynamical classification of evolutionarily singular strategies. University of Leiden Preprint (1994)Google Scholar
  43. Nicolis, J. S.: Dynamics of hierarchical systems. Berlin: Springer-Verlag 1986zbMATHGoogle Scholar
  44. Ott, E.: Chaos in dynamical systems. Cambridge: Cambridge University Press 1993zbMATHGoogle Scholar
  45. Rand, D. A., Wilson, H. B.: Evolutionary catastrophes, punctuated equilibria and gradualism in ecosystem evolution. Proc. Roy. Soc. Lond. B253, 239–244 (1993)Google Scholar
  46. Rand, D. A., Wilson, H. B., McGlade, J. M.: Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. Phil. Trans. Roy. Soc. Lond. B343, 261–283 (1994)Google Scholar
  47. Reed, J., Stenseth, N. C.: On evolutionarily stable strategies. J. Theor. Biol.108, 491–508 (1984)MathSciNetGoogle Scholar
  48. Rosenzweig, M. L., Brown, J. S., Vincent, T. L.: Red Queens and ESS: the coevolution of evolutionary rates. Evol. Ecol.1, 59–94 (1987)CrossRefGoogle Scholar
  49. Roughgarden, J.: The theory of coevolution. In: Futuyma, D. J., Slatkin, M. (eds.) Coevolution, pp. 33–64. Sunderland Massachusetts: Sinauer Associates 1983.Google Scholar
  50. Saloniemi, I.: A coevolutionary predator-prey model with quantitative characters. Amer. Natur.141, 880–896 (1993)CrossRefGoogle Scholar
  51. Schuster, H. G.: Deterministic chaos: an introduction. Weinheim: VCH Verlagsgesellschaft 1989zbMATHGoogle Scholar
  52. Serra, R., Andretta, M., Compiani, M., Zanarini, G.: Introduction to the physics of complex systems. Oxford: Pergamon Press 1986zbMATHGoogle Scholar
  53. Stearns, S. C.: The evolution of life histories. Oxford: Oxford University Press 1992Google Scholar
  54. Takada, T., Kigami, J.: The dynamical attainability of ESS in evolutionary games. J. Math. Biol.29, 513–529 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  55. Taper, M. L., Case, T. J.: Models of character displacement and the theoretical robustness of taxon cycles. Evolution46, 317–333 (1992)CrossRefGoogle Scholar
  56. Taylor, P. D.: Evolutionary stability in one-parameter models under weak selection. Theor. Pop. Biol.36, 125–143 (1989)CrossRefzbMATHGoogle Scholar
  57. van Kampen, N. G.: Fundamental problems in statistical mechanics of irreversible processes. In: Cohen, E. G. D. (ed.) Fundamental problems in statistical mechanics, pp. 173–202. Amsterdam: North Holland 1962Google Scholar
  58. van Kampen, N. G.: Stochastic processes in physics and chemistry. Amsterdam: North Holland 1981zbMATHGoogle Scholar
  59. 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 1990Google Scholar
  60. Vincent, T. L., Cohen, Y., Brown, J. S.: Evolution via strategy dynamics. Theor. Pop. Biol.44, 149–176 (1993)CrossRefzbMATHGoogle Scholar
  61. Wissel, C., Stöcker, S.: Extinction of populations by random influences. Theor. Pop. Biol.39, 315–328 (1991)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Ulf Dieckmann
    • 1
  • Richard Law
    • 2
  1. 1.Theoretical Biology Section, Institute of Evolutionary and Ecological SciencesUniversity of LeidenLeidenThe Netherlands
  2. 2.Department of BiologyUniversity of YorkYorkUK

Personalised recommendations