Skip to main content
Log in

Travelling waves and dominance of ESS's

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

We consider a simple two strategy game in which each pure strategy is an evolutionarily stable strategy (ESS). Under the usual dynamical equations, the large-time behaviour of the system will depend upon the initial conditions and the pay-off matrix. If spatial effects are included to give a reaction-diffusion system, we prove that travelling wavefronts can occur which in effect replace one ESS by another. The ‘strength’ or ‘dominance’ of each ESS which decides the ‘winner’ in a precisely defined sense is determined by its pay-off and by its diffusion rate. Good strategies have large pay-offs and small diffusion rates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Conley, C., Gardner, R.: An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model. Indiana J. Math. 33, 319–365 (1984)

    Google Scholar 

  • Dunbar, S.: Travelling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in R4. Trans. Am. Math. Soc. 286, 557–594 (1984)

    Google Scholar 

  • Fife, P.: Mathematical Aspects of Reacting and Diffusing Systems. (Lect. Notes Biomath., vol. 28) Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  • Fisher, R. A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 353–369 (1937)

    Google Scholar 

  • Gardner, R.: Existence of travelling wave solutions of predator-prey systems via the connection index. SIAM J. Appl. Math. 44, 56–79 (1984)

    Google Scholar 

  • Hadeler, K. P.: Diffusion in Fisher's population model. Rocky Mt. J. Math. 11, 39–45 (1981)

    Google Scholar 

  • Hastings, A.: Global stability in Lotka-Volterra systems with diffusion. J. Math. Biol. 6, 163–168 (1978)

    Google Scholar 

  • Hutson, V.: Stability in a reaction diffusion model of mutualism. SIAM J. Math. Anal. 17, 58–66 (1986)

    Google Scholar 

  • Kishimoto, K.: The diffusive Lotka-Volterra system with three species can have a stable non-constant equilibrium solution. J. Math. Biol. 16, 103–112 (1982)

    Google Scholar 

  • Maynard-Smith, J.: The theory of games and the evolution of animal conflicts. J. Theor. Biol. 47, 209–221 (1974)

    Google Scholar 

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

    Google Scholar 

  • Mischaikow, K., Hutson, V.: Travelling waves for mutualist species. SIAM J. Math. Anal. (Submitted)

  • Taylor, P. D., Jonker, L. B.: Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978)

    Google Scholar 

  • Vickers, G. T.: Spatial patterns and ESS's. J. Theor. Biol. 140, 129–135 (1989)

    Google Scholar 

  • Zeeman, E. C.: Population dynamics from game theory. In: Nitecki, A., Robinson, C. (eds.) Global Theory of Dynamical Systems. (Lect. Notes Math., vol. 819, pp. 471–497) Berlin Heidelberg New York: Springer 1980

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hutson, V.C.L., Vickers, G.T. Travelling waves and dominance of ESS's. J. Math. Biol. 30, 457–471 (1992). https://doi.org/10.1007/BF00160531

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00160531

Key words

Navigation