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.

Original languageEnglish
Pages (from-to)157-168
Number of pages12
JournalJournal of Theoretical Biology
Early online date28 Jul 2017
Publication statusPublished - 7 Nov 2017


  • Cyclic dominance ecology
  • Limit cycle
  • Mean field model
  • Replicator equation
  • Stochastic differential equation
  • Stochastic simulation

ASJC Scopus subject areas

  • Statistics and Probability
  • General Medicine
  • Modelling and Simulation
  • General Immunology and Microbiology
  • General Biochemistry,Genetics and Molecular Biology
  • General Agricultural and Biological Sciences
  • Applied Mathematics


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