Diffusively-Coupled Rock-Paper-Scissors Game with Mutation in Scale-Free Hierarchical Networks

Takashi Nagatani,Genki Ichinose

doi:10.1155/2020/6976328

Abstract

We present a metapopulation dynamic model for the diffusively-coupled rock-paper-scissors (RPS) game with mutation in scale-free hierarchical networks. We investigate how the RPS game changes by mutation in scale-free networks. Only the mutation from rock to scissors (R-to-S) occurs with rate μ. In the network, a node represents a patch where the RPS game is performed. RPS individuals migrate among nodes by diffusion. The dynamics are represented by the reaction-diffusion equations with the recursion formula. We study where and how species coexist or go extinct in the scale-free network. We numerically obtained the solutions for the metapopulation dynamics and derived the transition points. The results show that, with increasing mutation rate μ, the extinction of P species occurs and then the extinction of R species occurs, and finally only S species survives. Thus, the first and second dynamical phase transitions occur in the scale-free hierarchical network. We also show that the scaling law holds for the population dynamics which suggests that the transition points approach zero in the limit of infinite size.

Highlights

Rock-paper-scissors (RPS) games have been extensively investigated in various fields, such as physics and ecology
We study the metapopulation dynamics and dynamical phase transitions of the diffusively-coupled RPS game with mutation in a hierarchical network with scale-free and self-similar properties where the dynamics are represented by reaction-diffusion equations. e hierarchical structure of a network is described by the simple recursion formula for the reaction-diffusion equations
Summary e RPS games without mutation have been investigated by reaction-diffusion equations on one-dimensional and twodimensional lattices

Summary

Introduction

Rock-paper-scissors (RPS) games have been extensively investigated in various fields, such as physics and ecology. The dynamical stability for the coexistence of three species has been studied in networks. The dynamical phase transitions between coexistence and extinction have not been investigated for the RPS games with mutation in large-scale networks. It is poorly understood how the metapopulation dynamics and phase transitions change in a large-scale network. How does the network’s structure affect the dynamical phase transition in the diffusively-coupled RPS game with mutation? We investigated the scaling properties since the properties are unknown in the dynamical phase transitions of RPS games with mutation. We study the metapopulation dynamics and dynamical phase transitions of the diffusively-coupled RPS game with mutation in a hierarchical network with scale-free and self-similar properties where the dynamics are represented by reaction-diffusion equations. We clarify the dependence of the transition points (thresholds) on the highest degree

Mean-Field and Metapopulation Models

Metapopulation in Hierarchical Network

Numerical Result