Spatial evolutionary games
with small selection coefficients
Rick Durrett May 5, 2014
Abstract
Here we will use results of Cox, Durrett, and Perkins [56] for voter model perturba- tions to study spatial evolutionary games on Zd, d ≥ 3 when the interaction kernel is finite range, symmetric, and has covariance matrix σ2I. The games we consider have payoff matrices of the form 1 + wG where 1 is matrix of all 1’s and w is small and positive. Since our population size N = ∞, we call our selection small rather than weak which usually means w = O(1/N). We prove that the effect of space is equiv- alent to replacing the replicator ODE by a related PDE where the reaction term is the replicator equation for a game matrix with some of the entries changed. The first idea is well known in the theory of stochastic spatial processes [58, 16, 62, 63]. The second is inspired by work of Ohtsuki and Nowak [28] (for the pair approximation). A remarkable aspect of our result is that the modifications of the game matrix depend on the interaction kernel only through the values of two simple probabilities for an associated coalescing random walk.
link: http://www.math.duke.edu/~rtd/evog/spaceg.pdf
Rick Durrett May 5, 2014
Abstract
Here we will use results of Cox, Durrett, and Perkins [56] for voter model perturba- tions to study spatial evolutionary games on Zd, d ≥ 3 when the interaction kernel is finite range, symmetric, and has covariance matrix σ2I. The games we consider have payoff matrices of the form 1 + wG where 1 is matrix of all 1’s and w is small and positive. Since our population size N = ∞, we call our selection small rather than weak which usually means w = O(1/N). We prove that the effect of space is equiv- alent to replacing the replicator ODE by a related PDE where the reaction term is the replicator equation for a game matrix with some of the entries changed. The first idea is well known in the theory of stochastic spatial processes [58, 16, 62, 63]. The second is inspired by work of Ohtsuki and Nowak [28] (for the pair approximation). A remarkable aspect of our result is that the modifications of the game matrix depend on the interaction kernel only through the values of two simple probabilities for an associated coalescing random walk.
link: http://www.math.duke.edu/~rtd/evog/spaceg.pdf
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