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Truncation selection and diffusion on lattices

Evolutionary games on graphs have been extensively studied. A variety of graph structures, graph dynamics, and behaviours of replicators have been explored. These models have primarily been studied in the framework of facilitation of cooperation, and much previous research has shed light on this field of study. However, there has been little attention devoted to truncation selection as most models employ proportional selection (such as in the replicator equation) or imitate the best.' Here we systematically explore truncation selection on periodic square lattices, where replicators below a fitness threshold are culled and the reproduction probabilities are equal for all survivors. We employ two variations of this method: independent truncation, where the threshold is fixed; and dependent truncation, which is a generalization of imitate the best.' Further, we explore the effects of diffusion in our networks in the following orders of operation: contest-diffusion-offspring (CDO), and diffusion-contest-offspring (DCO). CDO and DCO frequently facilitate and diminish cooperation, respectively. For independent truncation, we find three regimes determined by the fitness threshold: cooperation decreases as we raise the threshold; polymorphisms and extinction can occur; and the entire population goes extinct. Further, we show how an intermediate sucker's payoff maximizes cooperation in the DCO independent truncation model. We find that dependent truncation affects games differently; lower levels reduce cooperation for the Hawk Dove game and increase it for the Stag Hunt, and higher levels produce the opposite effects. We compare these truncation methods to proportional selection, and show that they can facilitate cooperation. We conclude that truncation selection can impact the prevalence of cooperation in complex ways, and therefore merit further study.

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