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Journal of Applied Nonlinear Dynamics 2(2) (2013) 193--206 | DOI:10.5890/JAND.2013.04.007

Elizabeth N. Wesson$^{1}$; Richard H. Rand$^{2}$

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

$^{2}$ Department of Mathematics, Department of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA

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Abstract

Models of evolutionary dynamics are often approached via the replicator equation, which in its standard form is given by ˙ xi = xi ( fi (x)−φ ) , i = 1, . . . ,n, where xi is the frequency of strategy i, fi is its fitness, and φ = Σn i=1 xi fi is the average fitness. A game-theoretic aspect is introduced to the model via the payoff matrix A by taking fi(x) = (A · x)i. This model is based on the exponential model of population growth, ˙ xi = xi fi, with φ introduced in order both to hold the total population constant and to model competition between strategies. We analyze the dynamics of analogous models for the replicator equation of the form ˙ xi = g(xi)( fi −φ ), for selected growth functions g.

Acknowledgments

Thanks to David Rand for his helpful ideas and intuition when we were beginning work on this project.

References

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