Evolution of cooperation in a multidimensional phenotype space

The emergence of cooperation in populations of selfish individuals is a fascinating topic that has inspired much theoretical work. An important model to study cooperation is the phenotypic model, where individuals are characterized by phenotypic properties that are visible to others. The phenotype of an individual can be represented for instance by a vector x=(x1,…,xn), where x1,…,xn are integers. The population can be well mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies of the individuals can depend on their distance in the phenotype space. A cooperator can choose to help other individuals exhibiting the same phenotype and defects otherwise. Cooperation is said to be favored by selection if it is more abundant than defection in the stationary state. This means that the average frequency of cooperators in the stationary state strictly exceeds 1/2. Antal et al. (2009c) found conditions that ensure that cooperation is more abundant than defection in a one-dimensional (i.e. n=1) and an infinite-dimensional (i.e. n=∞) phenotype space in the case of the Prisoner’s Dilemma under weak selection. However, reality lies between these two limit cases. In this paper, we derive the corresponding condition in the case of a phenotype space of any finite dimension. This is done by applying a perturbation method to study a mutation-selection equilibrium under weak selection. This condition is obtained in the limit of a large population size by using the ancestral process. The best scenario for cooperation to be more likely to evolve is found to be a high population-scaled phenotype mutation rate, a low population-scaled strategy mutation rate and a high phenotype space dimension. The biological intuition is that a high population-scaled phenotype mutation rate reduces the quantity of interactions between cooperators and defectors, while a high population-scaled strategy mutation rate introduces newly mutated defectors that invade groups of cooperators. Finally it is easier for cooperation to evolve in a phenotype space of higher dimension because it becomes more difficult for a defector to migrate to a group of cooperators. The difference is significant from n=1 to n=2 and from n=2 to n=3, but becomes small as soon as n≥3.