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We introduce a non-diffusive spatial coupling term into the replicatorequation of evolutionary game theory. The spatial flux is based on motion dueto local gradients in the relative fitness of each strategy, providing agame-dependent alternative to diffusive coupling. We study numerically thedevelopment of patterns in 1D for two-strategy games including the coordinationgame and the prisoner's dilemma, and in 2D for the rock-paper-scissors game. In1D we observe modified travelling wave solutions in the presence of diffusion,and asymptotic attracting states under a frozen strategy assumption withoutdiffusion. In 2D we observe spiral formation and breakup in the frozen strategyrock-paper-scissors game without diffusion. A change of variables appropriateto replicator dynamics is shown to correctly capture the 1D asymptotic steadystate via a nonlinear diffusion equation.

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