Spatial heterogeneity promotes coexistence of rock-paper-scissor metacommunities

TL;DR

This paper investigates how spatial heterogeneity and dispersal influence the coexistence of strategies in rock-paper-scissor metacommunities, revealing conditions under which coexistence or extinction occurs based on invasion and exclusion rates.

Contribution

It introduces a framework linking invasion and exclusion eigenvalues to coexistence, identifying a critical dispersal rate for regional persistence in spatially heterogeneous environments.

Findings

Coexistence occurs when invasion rates' product exceeds exclusion rates' product.

A critical dispersal rate $d^*$ determines whether strategies persist or go extinct.

Spatial variation in payoffs is essential for coexistence despite local extinction risks.

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.