Spatial heterogeneity promotes coexistence of rock-paper-scissor metacommunities

Abstract. The rock-paper-scissor game – which is characterized by three strategies R,P,S, satisfying the non-transitive relations S excludes P, P excludes R, and R excludes S – serves as a simple prototype for studying more complex non-transitive systems. For well-mixed systems where interactions result in fitness reductions of the losers exceeding fitness gains of the winners, classical theory predicts that two strategies go extinct. The effects of spatial heterogeneity and dispersal rates on this outcome are analyzed using a general framework for evolutionary games in patchy landscapes. The analysis reveals that coexistence is determined by the rates at which dominant strategies invade a landscape occupied by the subordinate strategy (e.g. rock invades a landscape occupied by scissors) and the rates at which subordinate strategies get excluded in a landscape occupied by the dominant strategy (e.g. scissor gets excluded in a landscape occupied by rock). These invasion and exclusion rates correspond to eigenvalues of the linearized dynamics near single strategy equilibria. Coexistence occurs when the product of the invasion rates exceeds the product of the exclusion rates. Provided there is sufficient spatial variation in payoffs, the analysis identifies a critical dispersal rate d∗ required for regional persistence. For dispersal rates below d∗, the product of the invasion rates exceed the product of the exclusion rates and the rock-paper-scissor metacommunities persist regionally despite being extinction prone locally. For dispersal rates above d∗, the product of the exclusion rates exceed the product of the invasion rates and the strategies are extinction prone. These results highlight the delicate interplay between spatial heterogeneity and dispersal in mediating long-term outcomes for evolutionary games. Author’s Summary: The rock-paper-scissor game, which might initially seem to be of purely theoretical interest, plays an important role in describing the behavior of various real-world systems including the evolution of alternative male mating strategies in the side-blotched lizard, the evolution of bacterial populations, and coexistence in plant communities. While the importance of dispersal in mediating coexistence for these intransitive communities has been documented in theoretical and empirical studies, these studies have, by in large, ignored the role of spatial heterogeneity in mediating coexistence. We introduce and provide a detailed analysis of models for evolutionary games in a patchy environment. Our analysis reveals that spatial heterogeneity coupled with low dispersal rates can mediate regional coexistence, despite species being extinction prone in all patches. The results suggests that diversity is maintained by a delicate interplay between dispersal rates and spatial heterogeneity.