Evolution of cooperation in a particular case of the infinitely repeated Prisoner's Dilemma with three strategies

We will study a population of individuals playing the infinitely repeated Prisoner's Dilemma under replicator dynamics. The population consists of three kinds of individuals using the following reactive strategies: ALLD (individuals which always defect), ATFT (almost tit-for-tat: individuals which almost always repeat the opponent's last move) and G (generous individuals, which always cooperate when the opponent cooperated in the last move and have a positive probability $q$ of cooperating when they are defected). Our aim is studying in a mathematically rigorous fashion the dynamics of a simplified version for the computer experiment in [Nowak, Sigmund, Nature, 355, pp. 250--53, 1992 ] involving 100 reactive strategies. We will see that as the generosity degree of the G individuals varies, equilibria (rest points) of the dynamics appear or disappear, and the dynamics changes accordingly. Not only we will prove that the results of the experiment are true in our simplified version, but we will have complete control on the existence or non-existence of the equilbria for the dynamics for all possible values of the parameters, given that ATFT individuals are close enough to TFT. For most values of the parameters the dynamics will be completely determined.