In this paper, we investigate the evolution of cooperation in a haploid population subdivided into $d$ demes, each composed of $N\ge 2$ individuals interacting according to a Prisoner's Dilemma. Using a discrete-time Moran process with migration of offspring, we analyze the fixation probability ${\rho }_C$ 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 ${\rho }_C$. 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 ${\rho }_C<1/\left(Nd\right)$ 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 ${\rho }_C$ 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.

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• Ecological population dynamics
• Evolutionary & population dynamics
• Evolutionary games
• Social dynamics