Crossing the Threshold: the role of density dependence and demographic stochasticity in the evolution of cooperation

We make two simplifications to a joint population model developed by M. Doebeli et al. of two populations whose growth rates depend on total population density and pay-offs governed by the Iterated Prisoner’s Dilemma. One population uses the ‘Always Defect’ strategy and the second uses the ‘Tit for Tat’ (TFT) strategy. In the deterministic model, there are two simple basins of attraction that lead to the extinction of one or the other population. In particular, a small TFT population cannot spread from rarity. We compute the boundary between these two regions. If, on the other hand, the growth rate of the TFT population is stochastic, then it is possible for the TFT population to become established if the growth rate at any given time is sufficiently large to allow the TFT population to cross the threshold computed in the deterministic model. We describe the factors that increase the likelihood of TFT establishment and explain why density dependence is an essential feature of the model. In particular, we show that if the relative advantage of defecting is small compared to the benefits of cooperating, then there is an increased likelihood that cooperation will evolve.