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

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 f(c0) = 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 f(c) 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 f(c), in perfect accord with those obtained by numerical simulations. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

The study focuses on the evolutionary snowdrift game with an update rule of dissatisfaction-driven replicator on regular lattices. The equilibrium cooperation frequency fc against the cost-to-benefit ratio r has a step-like piecewise structure when dissatisfaction threshold K is small and transitions to a typical continuous form with increasing K. Analysis of stability and microscopic evolution helps interpret the emergence and disappearance of discontinuous transitions at critical values of r and snapshots of spatial strategy distribution patterns display the system's evolution. Numerical simulations show coexistence of cooperators and defectors for r < 2/3 and pure phase with only defectors for r > 3/4, with results consistent with those obtained by the master equation approach.