  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.
