Spatial evolutionary games with weak selection

Abstract

Recently, a rigorous mathematical theory has been developed for spatial games with weak selection, i.e., when the payoff differences between strategies are small. The key to the analysis is that when space and time are suitably rescaled, the spatial model converges to the solution of a partial differential equation (PDE). This approach can be used to analyze all 2 × 2 games, but there are a number of 3 × 3 games for which the behavior of the limiting PDE is not known. In this paper, we give rules for determining the behavior of a large class of 3 × 3 games and check their validity using simulation. In words, the effect of space is equivalent to making changes in the payoff matrix, and once this is done, the behavior of the spatial game can be predicted from the behavior of the replicator equation for the modified game. We say predicted here because in some cases the behavior of the spatial game is different from that of the replicator equation for the modified game. For example, if a rock-paper-scissors game has a replicator equation that spirals out to the boundary, space stabilizes the system and produces an equilibrium.cancer modeling | public goods game | bone cancer | rock-paper-scissors E volutionary games are often studied by assuming that the population is homogeneously mixing, i.e., each individual interacts equally with all the others. In this case, the frequencies of strategies evolve according to the replicator equation. See, e.g., Hofbauer and Sigmund's book (1). If ui is the frequency of players using strategy i, thenwhere Fi = j Gi,j uj is the fitness of strategy i, Gi,j is the payoff for playing strategy i against an opponent who plays strategy j , andF = i ui Fi is the average fitness. The homogeneous mixing assumption is not satisfied for the evolutionary games that arise in ecology or modeling solid cancer tumors, so it is important to understand how spatial structure changes the outcome of games. The goal of this paper is to facilitate applications of spatial evolutionary games by giving rules to determine the limiting behavior of a large class of 3 × 3 games. Our spatial games will take place on the 3D integer lattice Z 3 . The theory (2, 3) has been developed under the assumption that the interactions between an individual and its neighbors are given by an irreducible probability kernel p(x ) on Z 3 with p(0) = 0, that is finite range, symmetric p(x ) = p(−x ), and has covariance matrix σ 2 I . Here we will restrict our attention to the nearest neighbor case, in which p(x ) = 1/6 for x = (1, 0, 0), (−1, 0, 0), . . . (0, 0, −1).To describe the dynamics we let ξt (x ) be the strategy used by the individual at x at time t and letbe the fitness of x at time t. In birth-death dynamics, site x gives birth at rate ψt (x ) and sends its offspring to replace the individual at y with probability p(y − x ). In death-birth dynamics, the individual at x dies at rate 1 and is replaced by a copy of the one at y with probability proportional to p(y − x )ψt (y). The theory developed in ref. 3 can be applied...

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