Spatial Pattern Dynamics due to the Fitness Gradient Flux in Evolutionary Games

Abstract

We introduce a non-diffusive spatial coupling term into the replicator equation of evolutionary game theory. The spatial flux is based on motion due to local gradients in the relative fitness of each strategy, providing a game-dependent alternative to diffusive coupling. We study numerically the development of patterns in 1D for two-strategy games including the coordination game and the prisoner's dilemma, and in 2D for the rock-paper-scissors game. In 1D we observe modified travelling wave solutions in the presence of diffusion, and asymptotic attracting states under a frozen strategy assumption without diffusion. In 2D we observe spiral formation and breakup in the frozen strategy rock-paper-scissors game without diffusion. A change of variables appropriate to replicator dynamics is shown to correctly capture the 1D asymptotic steady state via a nonlinear diffusion equation.