Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-09T09:33:48.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Chaos in sociobiology

Published online by Cambridge University Press:  17 April 2009

J.R. Christie ,
K. Gopalsamy  and
Jibin Li
J.R. Christie
Affiliation:
School of Information Science and Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide SA 5001, Australia
K. Gopalsamy
Affiliation:
School of Information Science and Technology, The Flinders University of South Australia, GPO Box 2100, Adelaide SA 5001, Australia
Jibin Li
Affiliation:
Institute of Appplied Mathematics of Yunnan Province, Department of Mathematics, Kunming Institute of Technology, Yunnan 650093, People's Republic of China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that the dynamical game theoretic mating behaviour of males and females can be modelled by a planar system of autonomous ordinary differential equations. This system occurs in modelling “the battle of the sexes” in evolutionary biology. The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov's method, a sufficient condition is obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Finally, subharmonic Melnikov theory is used to obtain a criterion for the existence of subharmonic periodic orbits of the perturbed system.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 3 , June 1995 , pp. 439 - 451
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Bertozzi, A.L., An extension of the Smate-Birkhoff homoclinic theorem, Melnikov's method, and chaotic dynamics in incompressible fluids, A.B. thesis (Princeton University, Princeton, New Jersey, 1987).Google Scholar
[2]Bertozzi, A.L., ‘Heteroclinic orbits and chaotic dynamics in planar fluid flows’, SIAM J. Math. Anal. 19 (1988), 1271–1294.CrossRefGoogle Scholar
[3]Byrd, P.F. and Friedman, M.D., Handbook of elliptic integrals for scientists and engineers (Springer-Verlag, Berlin, Heidelberg, New York, 1971).CrossRefGoogle Scholar
[4]Dawkins, R., The selfish gene (Oxford University Press, Oxford; 1976).Google Scholar
[5]Foster, D. and Young, P., ‘Stochastic evolutionary game dynamics’, Theoret. Population Biol. 38 (1990), 219–232.CrossRefGoogle Scholar
[6]Gradshteyn, I.S. and Ryzhik, I.M., Table of integrals, series, and products (Academic Press, New York, 1980).Google Scholar
[7]Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer-Verlag, Berlin, Heidelberg, New York, 1983).CrossRefGoogle Scholar
[8]Hofbauer, J. and Sigmund, K., The theory of evolution and dynamical systems (Cambridge University Press, Cambridge, 1988).Google Scholar
[9]Schuster, P. and Sigmund, K., ‘Coyness, philandering and stable strategies’, Anim. Behav. 29 (1981), 186–192.CrossRefGoogle Scholar
[10]Taylor, P.D. and Jonker, L.B., ‘Evolutionary stable strategies and game dynamics’, Math. Biosci. 40 (1978), 145–156.CrossRefGoogle Scholar
[11]Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos (Springer-Verlag, Berlin, Heidelberg, New York, 1990).CrossRefGoogle Scholar
[12]Wiggins, S., Chaotic transport in dynamical systems (Springer-Verlag, Berlin, Heidelberg, New York, 1992).CrossRefGoogle Scholar