Oscillatory Dynamics in Rock-Paper-Scissors Games with Mutations

TL;DR

This paper investigates the oscillatory behaviors in rock-paper-scissors games with mutations, revealing how mutation rates influence stability, oscillations, and noise-induced phenomena through mean-field and stochastic models.

Contribution

It provides a comprehensive analysis of oscillatory dynamics in mutated rock-paper-scissors games, combining mean-field theory with stochastic simulations to uncover noise-induced oscillations.

Findings

High mutation rates stabilize coexistence of all species.

Low mutation rates lead to limit cycles via Hopf bifurcation.

Stochastic effects induce large amplitude quasi-cycles through noise resonance.

Abstract

We study the oscillatory dynamics in the generic three-species rock-paper-scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude.