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In the study of the evolution of cooperation, resource limitations areusually assumed just to provide a finite population size. Recently, however,agent-based models have pointed out that resource limitation may modify theoriginal structure of the interactions and allow for the survival ofunconditional cooperators in well-mixed populations. Here, we presentanalytical simplified versions of two types of agent-based models recentlypublished: one in which the limiting resource constrains the ability ofreproduction of individuals but not their survival, and a second one where thelimiting resource is necessary for both reproduction and survival. One findsthat the analytical models display, with a few differences, the samequalitative behavior of the more complex agent-based models. In addition, theanalytical models allow us to expand the study and identify the dimensionlessparameters governing the final fate of the system, such as coexistence ofcooperators and defectors, or dominance of defectors or of cooperators. Weprovide a detailed analysis of the occurring phase transitions as theseparameters are varied.

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