Nonlinear Dynamics of the Rock-Paper-Scissors Game with Mutations

Abstract

We analyze the replicator-mutator equations for the Rock-Paper-Scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game.

arXiv comment6 pages, 5 figures

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• Topologically protected dynamics in three-dimensional nonlinear antisymmetric Lotka-Volterra systems
• Chimera states in a network-organized public goods game with destructive agents
• Landscape and flux for quantifying global stability and dynamics of game theory
• Memory and limit cycles in rock-scissors-paper
• Amplitude death in coupled replicator map lattice: averting migration dilemma
• Element-wise and Recursive Solutions for the Power Spectral Density of Biological Stochastic Dynamical Systems at Fixed Points
• Dynamics of a linearly-perturbed May-Leonard competition model
• Non-Markovian Rock-Paper-Scissors games
• Community Formation in Wealth-Mediated Thermodynamic Strategy Evolution