Advertisement

Monatshefte für Mathematik

, Volume 110, Issue 3–4, pp 189–206 | Cite as

Dynamical aspects of evolutionary stability

  • Immanuel M. Bomze
Article

Abstract

Selection is often viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to relate these stability notions with asymptotic stability in the so-called “replicator dynamics”, by generalizing results, which are well-known for elementary situations, to a fairly general setting applicable, e.g. to complex populations. Moreover, a purely dynamical characterization of evolutionary stability and uninvadability is presented.

Keywords

Equilibrium State General Setting Game Theory Asymptotic Stability Dynamical Aspect 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Akin, E.: Exponential families and game dynamics. Can. J. Math.34, 374–405 (1982).Google Scholar
  2. [2]
    Bomze, I. M.: A Functional Analytic Approach to Statistical Experiments. London: Longman 1990.Google Scholar
  3. [3]
    Bomze, I. M.: Cross entropy minimization in uninvadable states of complex populations. Technical Report80, Inst. f. Stat., Univ. Wien (1990).Google Scholar
  4. [4]
    Bomze, I. M., Pötscher, B. M.: Game Theoretical Foundations of Evolutionary Stability. Berlin-Heidelberg-New York: Springer. 1989.Google Scholar
  5. [5]
    Choquet, G.: Lectures on analysis. Reading, Mass.: Benjamin. 1969.Google Scholar
  6. [6]
    Hines, W. G. S.: Strategy stability in complex populations. J. Appl. Prob.17, 600–610 (1980).Google Scholar
  7. [7]
    Hofbauer, J., Schuster, P., Sigmund, K.: A note on evolutionary stable strategies and game dynamics. J. theor. Biol.81, 609–612 (1979).Google Scholar
  8. [8]
    Hofbauer, J., Sigmund, K.: The Theory of Evolution and Dynamical Systems. Cambridge: Univ. Press. 1988.Google Scholar
  9. [9]
    Maynard Smith, J.: Evolution and the Theory of Games. Cambridge: Univ. Press. 1982.Google Scholar
  10. [10]
    Maynard Smith, J., Price, G. R.: The logic of animal conflict. Nature, London246, 15–18 (1973).Google Scholar
  11. [11]
    Schuster, P., Sigmund, K.: Replicator dynamics. J. theor. Biol.100, 533–538 (1987).Google Scholar
  12. [12]
    Sigmund, K.: Game dynamics, mixed strategies, and gradient systems. Theor. Pop. Biol.32, 114–126 (1987).Google Scholar
  13. [13]
    Taylor, P. D., Jonker, L. B.: Evolutionarily stable strategies and game dynamics. Math. Biosci.40, 145–156 (1978).Google Scholar
  14. [14]
    Thomas, B.: Evolutionary stability: states and strategies. Theor. Pop. Biol.26, 49–67 (1984).Google Scholar
  15. [15]
    Thomas, B.: Evolutionarily stable sets in mixed strategists models. Theor. Pop. Biol.28, 332–341 (1985).Google Scholar
  16. [16]
    van Damme, E. E. C., Bomze, I. M.: A dynamical characterization of evolutionarily stable states. Technical Report61, Inst. f. Stat., Univ. Wien (1988).Google Scholar
  17. [17]
    Vickers, G. T., Cannings, C.: On the definition of an evolutionarily stable strategy. Preprint (1987).Google Scholar
  18. [18]
    Zeeman, E. C.: Population dynamics from game, theory. In: Nitecki, Z., Robinson, C. (eds.), Global Theory of Dynamical Systems. Berlin: Springer. 1980.Google Scholar
  19. [19]
    Zeeman, E. C.: Dynamics of the evolution of animal conflicts. J. theor. Biol.89, 249–270 (1981).Google Scholar
  20. [20]
    Price, G. R.: Selection and covariance. Nature, London227, 520–521 (1970).Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Immanuel M. Bomze
    • 1
  1. 1.Institut für Statistik und InformatikUniversität WienWienAustria

Personalised recommendations