Evolutionary dynamics of rock-paper-scissors game in the patchy network with mutations

Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.