Demographic noise slows down cycles of dominance

Authored by Tim Rogers, Qian Yang, Jonathan H P Dawes

Date Published: 2017

DOI: 10.1016/j.jtbi.2017.07.025

Sponsors: Royal Society

Platforms: No platforms listed

Model Documentation: Other Narrative Mathematical description

Model Code URLs: Model code not found

Abstract

We study the phenomenon of cyclic dominance in the paradigmatic Rock-Paper-Scissors model, as occurring in both stochastic individual-based models of finite populations and in the deterministic replicator equations. The mean-field replicator equations are valid in the limit of large populations and, in the presence of mutation and unbalanced payoffs, they exhibit an attracting limit cycle. The period of this cycle depends on the rate of mutation; specifically, the period grows logarithmically as the mutation rate tends to zero. We find that this behaviour is not reproduced in stochastic simulations with a fixed finite population size. Instead, demographic noise present in the individual-based model dramatically slows down the progress of the limit cycle, with the typical period growing as the reciprocal of the mutation rate. Here we develop a theory that explains these scaling regimes and delineates them in terms of population size and mutation rate. We identify a further intermediate regime in which we construct a stochastic differential equation model describing the transition between stochastically-dominated and mean-field behaviour. (C) 2017 Elsevier Ltd. All rights reserved.

Tags

Competition Dynamics stochastic simulation Strategies Game Rock-paper-scissors Mean field model Cyclic dominance ecology Limit cycle Replicator equation Stochastic differential equation Heteroclinic cycles