---

Journal of Applied Nonlinear Dynamics 4(2) (2015) 131--140 | DOI:10.5890/JAND.2015.06.003

Elizabeth Wesson$^{1}$; Richard Rand$^{2}$

$^{1}$ Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA

$^{2}$ Professor of Mathematics and Mech. & Aero. Eng., Cornell University, Ithaca, NY 14853, USA

Download Full Text PDF

 

Abstract

Evolutionary dynamics combines game theory and nonlinear dynamics to model competition in biological and social situations. The replicator equation is a standard paradigm in evolutionary dynamics. The growth rate of each strategy is its excess fitness: the deviation of its fitness from the average. The game-theoretic aspect of the model lies in the choice of fitness function, which is determined by a payoff matrix. Previous work by Ruelas and Rand investigated the Rock- Paper-Scissors replicator dynamics problem with periodic forcing of the payoff coefficients. This work extends the previous to consider the case of quasiperiodic forcing. This model may find applications in biological or social systems where competition is affected by cyclical processes on different scales, such as days/years or weeks/years. We study the quasiperiodically forced Rock-Paper-Scissors problem using numerical simulation, and Floquet theory and harmonic balance. We investigate the linear stability of the interior equilibrium point; we find that the region of stability in frequency space has fractal boundary.

References

1. [1]	Hofbauer, J. and Sigmund, K. (1998), Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK.

2. [2]	Sigmund, K. (2011), Introduction to Evolutionary Game Theory, In Evolutionary Game Dynamics, K. Sigmund, ed., Vol. 69 of Proceedings of Symposia in Applied Mathematics, American Mathematical Society, Chap. 1, 1-26.

3. [3]	Taylor, P. and Jonker, L. (1978), Evolutionarily Stable Strategies and Game Dynamics, Mathematical Biosciences, 40(1-2), 145-156.

4. [4]	Ruelas, R., Rand, D., and Rand, R. (2012), Nonlinear parametric excitation of an evolutionary dynamical system, J. Mechanical Engineering science (Proceedings of the Institution of Mechanical Engineers Part C), 226 1912-1920.

5. [5]	Ruelas, R., Rand, D., and Rand, R. (2013), Parametric Excitation and Evolutionary Dynamics, Journal of Applied Mechanics, 80, 051013.

6. [6]	Nowak, M.(2006), Evolutionary Dynamics, Belknap Press of Harvard Univ. Press, Cambridge, MA.

7. [7]	Rand, R., Yazhbin, M., and Rand, D. (2011), Evolutionary dynamics of a system with periodic coefficients, Communications in Nonlinear Science and Numerical Simulation, 16, 3887-3895.

8. [8]	Rand, R. (2012), Lecture Notes on Nonlinear Vibrations, Published online by The Internet-First University Press, http://ecommons.library.cornell.edu/handle/1813/28989.

9. [9]	Herstein, I. (1975), Topics in Algebra, 2nd ed., John Wiley & Sons, New York.

10. [10]	Zounes, R. and Rand, R. (1998), Transition Curves for the Quasi-periodic Mathieu Equation, SIAM Journal on Applied Mathematics, 40(4), 1094-1115.

11. [11]	Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.