Metapopulation model for rock-paper-scissors game: Mutation affects paradoxical impacts

The rock-paper-scissors (RPS) game is known as one of the simplest cyclic dominance models. This game is key to understanding biodiversity. Three species, rock (R), paper (P) and scissors (S), can coexist in nature. In the present paper, we first present a metapopulation model for RPS game with mutation. Only mutation from R to S is allowed. The total population consists of spatially separated patches, and the mutation occurs in particular patches. We present reaction-diffusion equations which have two terms: reaction and migration terms. The former represents the RPS game with mutation, while the latter corresponds to random walk. The basic equations are solved analytically and numerically. It is found that the mutation induces one of three phases: the stable coexistence of three species, the stable phase of two species, and a single-species phase. The phase transitions among three phases occur by varying the mutation rate. We find the conditions for coexistence are largely changed depending on metapopulation models. We also find that the mutation induces different paradoxes in different patches.