Differential migration in a finite-island model and the evolution of cooperation

In this paper, we investigate the evolution of cooperation in a haploid population subdivided into d demes, each composed of N≥2 individuals interacting according to a Prisoner's Dilemma. Using a discrete-time Moran process with migration of offspring, we analyze the fixation probability ρ_{C} of a single cooperative mutant. Our model incorporates a baseline migration probability m, with cooperators exhibiting an additional migration tendency proportional to the local frequency of defectors. Using a two-time-scale approach, we derive a diffusion approximation valid for a large number of demes d to analytically determine ρ_{C}. We show that, in the absence of differential migration and for a single-round Prisoner's Dilemma, the fixation of cooperation remains generally disfavored by selection in the sense that ρ_{C}<1/(Nd) even as the baseline migration probability increases. In contrast, in a repeated Prisoner's Dilemma, selection favors the fixation of cooperation when the number of repetitions n exceeds a critical threshold n^{*}, which is inversely related to m and N. The probability of fixation ρ_{C} increases as the level of differential migration increases, setting a threshold beyond which the fixation of cooperation is favored even in a single-round game. This threshold is modulated by the size of the deme, the probability of baseline migration and the cost of cooperation, with larger demes and a higher probability of baseline migration requiring a higher differential migration level. Numerical simulations validate the accuracy of our diffusion approximation, showing strong agreement with the simulated fixation probabilities for large d. These findings highlight the pivotal role of repeated interactions, migration strategies, and population structure in promoting the evolution of cooperation.