The Evolution of Cooperation in Spatially Heterogeneous Populations

A challenging problem in sociobiology is to understand the emergence of cooperation in a nonsocial world. Recent models of the iterated Prisoner's Dilemma (IPD) game conclude that population mixing due to individual mobility limits cooperation; however, these models represent space only implicitly. Here we develop a dynamical IPD model where temporal and spatial variations in the population are explicitly considered. Our model accounts for the stochastic motion of individuals and the inherent nonrandomness of local interactions. By deriving a spatial version of Hamilton's rule, we find that a threshold level of mobility in selfish always-defect (AD) players is required to beget invasion by social "tit for tat" (TFT) players. Furthermore, the level of mobility of successful TFT newcomers must be approximately equal to or somewhat higher than that of resident defectors. Significant mobility promotes the assortment of TFT pioneers on the front of invasion and of AD intruders in the core of a cooperative cluster. It also maximizes the likelihood of TFT retaliation. Once this first step whereby TFT takes over AD is completed, more generous and perhaps more suspicious strategies may outperform and displace TFT. We derive the conditions under which this continued evolution of more robust cooperative strategies occurs.