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Oscillatory dynamics in rock-paper-scissors games with mutations.

Mauro Mobilia1

  • 1Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK. M.Mobilia@leeds.ac.uk

Journal of Theoretical Biology
|January 20, 2010
PubMed
Summary

This study explores rock-paper-scissors games with mutations, revealing that mutations can stabilize species coexistence or lead to limit cycles. Noise-induced resonance generates persistent, large-amplitude oscillations in stochastic models.

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Area of Science:

  • Ecology
  • Evolutionary Game Theory
  • Mathematical Biology

Background:

  • The rock-paper-scissors game is a fundamental model for studying species coexistence and cyclical dynamics.
  • Mutations introduce variations that can alter the stability and behavior of ecological systems.
  • Understanding oscillatory dynamics is crucial for predicting species persistence and ecosystem stability.

Purpose of the Study:

  • To investigate the oscillatory dynamics of a generic three-species rock-paper-scissors game with mutations.
  • To analyze the impact of mutation rates on species coexistence and system stability.
  • To explore the emergence of persistent oscillations in stochastic models due to demographic fluctuations.

Main Methods:

  • Mean-field analysis to identify different dynamic regimes based on mutation rates.
  • Hopf bifurcation and heteroclinic cycle analysis to characterize oscillations.
  • Individual-based stochastic modeling and diffusion theory to account for demographic noise.
  • Numerical simulations and analytical calculations to study quasi-cycles and escape times.

Main Results:

  • High mutation rates lead to a stable fixed point with all species coexisting.
  • Low mutation rates can induce limit cycles via Hopf bifurcations.
  • In the absence of mutations, heteroclinic cycles cause large-amplitude oscillations, though these are not robust to noise.
  • Stochastic models exhibit persistent, large-amplitude erratic oscillations (quasi-cycles) driven by noise-induced resonance.
  • Average escape times to reach these quasi-cycles were computed.

Conclusions:

  • Mutations play a critical role in stabilizing or destabilizing ecological dynamics in rock-paper-scissors games.
  • Stochastic effects, particularly noise-induced resonance, can generate robust, persistent oscillations not predicted by deterministic models.
  • The study provides insights into the complex interplay between deterministic forces, mutations, and demographic stochasticity in ecological systems.