2015/02/11 by Danielle F. P. Toupo, Steven H. Strogatz · 82 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · Physics and Astronomy · Social Sciences · #Bifurcation #Biology #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #Game theory #Genetics #Hopf bifurcation #Limit (mathematics) #Limit cycle #Mathematical analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematical economics #Mathematics #Mutation #Mutation rate #Nonlinear system #Physics #Pitchfork bifurcation #Quantum mechanics #Replicator equation #Statistical physics #Zero (linguistics) #Zero-sum game #math.DS #nlin.AO
paper · pdf · doi:10.1103/physreve.91.052907
published in Physical Review E 91(5), 052907 (American Physical Society) · 6 pages, 5 figures
arxiv created 2015/02/11 · openalex publication_date 2015/05/11 · arxiv updated 2015/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.