### 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.

Original language | English |
---|---|

Pages (from-to) | 157-168 |

Number of pages | 12 |

Journal | Journal of Theoretical Biology |

Volume | 432 |

Early online date | 28 Jul 2017 |

DOIs | |

Publication status | Published - 7 Nov 2017 |

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### 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

### Cite this

**Demographic noise slows down cycles of dominance.** / Yang, Qian; Rogers, Tim; Dawes, Jonathan H.P.

Research output: Contribution to journal › Article

*Journal of Theoretical Biology*, vol. 432, pp. 157-168. https://doi.org/10.1016/j.jtbi.2017.07.025

}

TY - JOUR

T1 - Demographic noise slows down cycles of dominance

AU - Yang, Qian

AU - Rogers, Tim

AU - Dawes, Jonathan H.P.

PY - 2017/11/7

Y1 - 2017/11/7

N2 - 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.

AB - 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.

KW - Cyclic dominance ecology

KW - Limit cycle

KW - Mean field model

KW - Replicator equation

KW - Stochastic differential equation

KW - Stochastic simulation

UR - http://www.scopus.com/inward/record.url?scp=85028043954&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1016/j.jtbi.2017.07.025

U2 - 10.1016/j.jtbi.2017.07.025

DO - 10.1016/j.jtbi.2017.07.025

M3 - Article

VL - 432

SP - 157

EP - 168

JO - Journal of Theoretical Biology

JF - Journal of Theoretical Biology

SN - 0022-5193

ER -