Abstract
We study the phenomenon of cyclic dominance in the paradigmatic Rock–Paper–Scissors model, as occurring in both stochastic individualbased models of finite populations and in the deterministic replicator equations. The meanfield 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 individualbased 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 stochasticallydominated and meanfield behaviour.
Original language  English 

Pages (fromto)  157168 
Number of pages  12 
Journal  Journal of Theoretical Biology 
Volume  432 
Early online date  28 Jul 2017 
DOIs  
Publication status  Published  7 Nov 2017 
Keywords
 Cyclic dominance ecology
 Limit cycle
 Mean field model
 Replicator equation
 Stochastic differential equation
 Stochastic simulation
ASJC Scopus subject areas
 Statistics and Probability
 Medicine(all)
 Modelling and Simulation
 Immunology and Microbiology(all)
 Biochemistry, Genetics and Molecular Biology(all)
 Agricultural and Biological Sciences(all)
 Applied Mathematics
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Profiles

Jonathan Dawes
 Department of Mathematical Sciences  Professor
 Centre for Networks and Collective Behaviour
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Water Innovation and Research Centre (WIRC)
 Institute for Mathematical Innovation (IMI)
 Institute for Policy Research (IPR)
 Water Informatics: Science and Engineering Centre for Doctoral Training (WISE)
 Centre for Mathematical Biology
Person: Research & Teaching