A Finite Population Destroys a Traveling Wave in Spatial Replicator Dynamics

TL;DR

This paper derives finite and infinite population spatial replicator dynamics from a stochastic cellular automaton, highlighting how finite populations disrupt traveling wave solutions in spatial evolutionary games.

Contribution

It provides a new derivation of spatial replicator dynamics from cellular automata and extends finite population results to spatial settings, clarifying the impact of finite size.

Findings

Finite populations destroy traveling wave solutions

Infinite population model aligns with previous models by Vickers

Finite model generalizes Durett and Levin's finite game results

Abstract

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.