Analytical models for well-mixed populations of cooperators and defectors under limiting resources

In the study of the evolution of cooperation, resource limitations are usually assumed just to provide a finite population size. Recently, however, agent-based models have pointed out that resource limitation may modify the original structure of the interactions and allow for the survival of unconditional cooperators in well-mixed populations. Here, we present analytical simplified versions of two types of agent-based models recently published: one in which the limiting resource constrains the ability of reproduction of individuals but not their survival, and a second one where the limiting resource is necessary for both reproduction and survival. One finds that the analytical models display, with a few differences, the same qualitative behavior of the more complex agent-based models. In addition, the analytical models allow us to expand the study and identify the dimensionless parameters governing the final fate of the system, such as coexistence of cooperators and defectors, or dominance of defectors or of cooperators. We provide a detailed analysis of the occurring phase transitions as these parameters are varied.