Finite and Infinite Population Spatial Rock-Paper-Scissors in One Dimension

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 the one-dimensional rock-paper-scissors game. In particular, we show that a constant amplitude traveling wave solution exists for the infinite population case and show how population collapse prevents its formation in the finite population case. This is closely related to recent work by Postlethwaite and Rucklidge on traveling waves in a similar but distinct rock-paper-scissor setting. Additional solution classes in variations of rock-paper-scissors are also studied.