Investigating the Evolution of Cooperative Behaviour in a Minimally Spatial Model

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It is well known that the evolution of cooperative behaviour is dependant upon certain environmental conditions. One such condition that has been extensively studied is the use of a spatially structured population, whereby cooperation is favoured by a reduced number of interactions between cooperators and selfish cheaters. However, models that address the role of spatial structure typically use an individual-based approach, which can make analysis unnecessarily complicated. By contrast, non-spatial population genetics models usually consist entirely of a set of replicator equations, thereby simplifying analysis. Unfortunately, these models cannot traditionally be used to take account of spatial structure, since they assume that interaction between any pair of individuals in a population is equally likely. In this paper, we construct as model that is still based on replicator equations, but where spatial localisation with respect to the number of interactions between individuals is incorporated. Using this model, we are able to successfully reproduce the dynamics seen in more complex individual-based models.