Fixation time of the rock-paper-scissors model: rigorous results in the well-mixed setting

Abstract

The rock-paper-scissors model simulates the effect of cyclic dominance in a finite population of size N and has received considerable attention in applied literature. In the well-mixed version of the model, population densities fluctuate around periodic orbits of a deterministic ODE approximation, and for large N the time to fixation (complete dominance by one species) has been observed by simulation to be approximately $\mathit{\tau }N$ where τ is a positive, finite random variable. We give a rigorous proof of this observation by establishing a slow diffusion limit for a conserved quantity of the deterministic approximation, together with a careful analysis of the behaviour at the boundary, both in the limit as $N\to \mathrm{\infty }$, and for large but finite N.