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Rick Durrett : Spatial evolutionary games with small selection coefficients

We use results of Cox, Durrett, and Perkins for voter model perturbations to study spatial evolutionary games on $\ZZ^d$, $d\ge 3$ when the interaction kernel is finite range, symmetric, and has covariance matrix $\sigma^2I$. The games we consider have matrices of the form ${\bf 1} + wG$ where ${\bf 1}$ is matrix of 1's and $w$ is small and positive. We prove that the effect of space is equivalent to (i) changing the entries of the game matrix and (ii) replacing the replicator ODE by a related PDE. The first idea is due to Ohtsuki and Nowak (for the pair approximation) while the second is well known in the theory of stochastic spatial processes. A remarkable aspect of our result is that the limiting PDE depends on the interaction kernel only through the values of two simple noncoalescence probabilities. Due to results of Aronson and Weinberger, and Fife and McLeod, we can analyze any 2x2 game. However, when there are three strategies the limiting object is a system of reaction diffusion equations, so we only have results for special cases.

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