  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.
