Dissatisfaction-driven replicator dynamics of the evolutionary snowdrift game in structured populations

Abstract

We study the evolutionary snowdrift game with an update rule of dissatisfaction-driven replicator on regular lattices, in which there exists a threshold value K of dissatisfaction dividing the update rule into unconditional imitation and probability replication. It is found that under the random initial state with fc0=0.5, the equilibrium cooperation frequency fc against the cost-to-benefit ratio r has a step-like piecewise structure when K is small and then neighboring plateaus gradually connect together as K increases, and finally fc will take the typical continuous form observed in the game system with the conventional replicator. By analyzing the stability and microscopic evolution of local strategy configurations, we interpret qualitatively both the emergence and disappearance of discontinuous transitions at several critical values of r. And we also enumerate some snapshots of spatial patterns of strategy distribution to exhibit how the competitive system evolves from one dynamical behavior to another. Numerical simulations show that the system can evolve to a mixed state with cooperators and defectors coexisting in the case of r<2/3, while it can certainly end up in a pure phase only with D agents in the case of r>3/4. We further investigate the dynamics of our model by the master equation approach and then calculate the numerical computation results of fc, in perfect accord with those obtained by numerical simulations.