Abstract
In this paper, we investigate a simple two-phenotype and two-patch model that incorporates both spatial dispersion and density effects in the evolutionary game dynamics. The migration rates from one patch to another are considered to be patch-dependent but independent of individual’s phenotype. Our main goal is to reveal the dynamical properties of the evolutionary game in a heterogeneous patchy environment. By analyzing the equilibria and their stabilities, we find that the dynamical behavior of the evolutionary game dynamics could be very complicated. Numerical analysis shows that the simple model can have twelve equilibria where four of them are stable. This implies that spatial dispersion can significantly complicate the evolutionary game, and the evolutionary outcome in a patchy environment should depend sensitively on the initial state of the patches.
Highlights
In order to explain the evolution of animal behavior, Maynard Smith and Price [1] developed the concept of evolutionarily stable strategy (ESS)
A vast amount of research has been devoted to analyze the influence of spatial diffusion on the evolutionary stability of ecology systems
They showed that in two-strategy coordination games, if the reaction term of the reaction-diffusion equation is taken as replicator dynamics, one strategy will drive out the other strategy in form of a traveling wave front, there is no simple rule to decide which strategy can survive
Summary
In order to explain the evolution of animal behavior, Maynard Smith and Price [1] developed the concept of evolutionarily stable strategy (ESS) (see [2,3,4,5]). For the evolutionary game dynamics in a patchy environment, Prior et al [6] mainly focused their analysis on the stability of the homogeneous states, where they assumed that all patches have the same payoff matrix and density-dependent background fitness. Their main results showed that a stable equilibrium (e.g. an evolutionarily stable strategy) of the non-dispersed frequency dynamics becomes a stable equilibrium of the large system if population density stabilizes at these fixed frequencies. Our main goal is to reveal the dynamical properties of the evolutionary game in a heterogeneous patchy environment
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