Evolution of cooperation in prisoner's dilemma game under the coupling of aspiration and imitation rules

Considering the important roles played by aspiration and imitation rules, we explore how cooperation evolves in the spatial prisoner's dilemma game. In detail, the strategy updating rule couples the inner aspiration payoff and the external neighborâs payoff through the parameter α. Intriguingly, we find that there exists an optimal value of α leading to the highest density of cooperators for each aspiration level A. By analyzing the dynamical strategy changes of the system, we find that an optimal α helps individuals form cooperator clusters and curbs the further spread of high-payoff defectors, and thus improves the density of cooperators. Moreover, when $0.1\leq A\leq0.5$ , the optimal value of α, which leads to the highest density of cooperators, decreases with A, the optimal range of α gradually expands and the highest cooperation level is on the rise