Robust Model for Evolution in Structured Populations

The evolution of cooperation has been, and continues to be, the inspiration for evolutionary game theory. Why should cooperation persist when it is inherently exploitable? It has long been recognized that cooperation could be affected by population viscocity, or spatial structure. Networks provide a tractable and general model for spatial structure, where individuals occupy nodes and edges define interactions and replacement dynamics. In this con- text the two popular biologically-motivated mechanisms to model evolution with overlapping generations are Birth-Death (BD) and Death-Birth (DB). In both rules reproduction is proportional to fitness and death is random; only the order of the two events changes. Although superficially similar, the two rules may produce qualitatively different dynamics; one rule (DB) may favour cooperation while the other (BD) does not. Whether structure can promote the evolution of cooperation should not hinge on some technical detail of the modeling architecture. Here, we propose an intermediate rule where in each time step DB is used with probability δ, otherwise. This homotopy allows us to explore how both rules affect the evolutionary dynamics of cooperation. We make use of a conventional game that contains as subcases all popular game-theoretic models of cooperation: The Prisoner’s Dilemma, The Snowdrift Game, The Stag-Hunt Game, and Byproduct Mutualism. We find that of any mixed update between BD and DB the only qualitatively different update is using only BD (δ = 0). The critical value δ = 0 has a natural interpretation in terms of kin selection and kin competition. Furthermore, under any mixed BD-DB update (and weak selection) we show that cooperation is never inhibited for any payoff structure including the Snowdrift Game.