Decay of invincible clusters of cooperators in the evolutionary prisoner's dilemma game

Two types of invincible clusters of cooperators are defined in the one-dimensional evolutionary Prisoner's Dilemma game. These invincible clusters can either be peaceful or aggressive. The survival of these invincible clusters is discussed in the context of the repeated Prisoner's Dilemma game with imitation and asynchronous updating procedure. The decay rates for these two types of clusters are analyzed numerically, for all enumeration of the configuration for small chain size. We find characteristic difference in the decay patterns of these two types of invincible clusters. The peaceful invincible clusters experience monotonic exponential decay, while the aggressive ones shows an interesting minimum in the density of cooperators before going through a slow exponential decay at long time. A heuristic argument for the existence of the minima is provided.