Asymptotic behaviour of the one-dimensional "rock-paper-scissors" cyclic cellular automaton

Abstract

The one-dimensional three-state cyclic cellular automaton is a simple spatial model with three states in a cyclic "rock-paper-scissors" prey-predator relationship. Starting from a random configuration, similar states gather in increasingly large clusters; asymptotically, any finite region is filled with a uniform state that is, after some time, driven out by its predator, each state taking its turn in dominating the region (heteroclinic cycles). We consider the situation where each site in the initial configuration is chosen independently at random with a different probability for each state. We prove that the asymptotic probability that a state dominates a finite region corresponds to the initial probability of its prey. The proof methods are based on discrete probability tools, mainly particle systems and random walks.

22 pages, figures in color

References in corpus (5)

• Evolutionary games on graphs
• Cyclic dominance in evolutionary games: A review
• Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model
• Self-organisation in cellular automata with coalescent particles: qualitative and quantitative approaches
• Clustering in the three and four color cyclic particle systems in one dimension