A finite population destroys a traveling wave in spatial replicator dynamics.

• We derive a finite population spatial replicator dynamic for arbitrary games. • We analyze the effects of nite populations on generalized rock-paper- scissors. • We show the finite population spatial replicator dynamic destroys a trav- eling wave. We derive both the finite and infinite population spatial replicator dynamics as the fluid limit of a stochastic cellular automaton. The infinite population spatial replicator is identical to the model used by Vickers and our derivation justifies the addition of a diffusion to the replicator. The finite population form generalizes the results by Durett and Levin on finite spatial replicator games. We study the differences in the two equations as they pertain to a one-dimensional rock-paper-scissors game. [ABSTRACT FROM AUTHOR]