Strategy evolution of a novel cooperative game model induced by reward feedback and a time delay

Rewarding cooperators and punishing defectors are effective measures for promoting cooperation in evolutionary game theory. Given that previous models treated rewards as constants, this does not reflect real-world dynamics changes. Therefore, this paper focused on the classical payoff matrix and examined the dynamic variable rewards affected by cooperation and defection strategies, as well as the impact of time delays. First, for the system without a time delay, we analyzed the existence and stability of numerous equilibrium points and explored transcritical bifurcations under various conditions. Second, for the time-delay system, we discussed a series of delayed dynamical behaviors including Hopf bifurcation, period, and the stability and direction of bifurcation. Finally, the changes of cooperation strategy were observed by numerical simulation, and some interesting results were obtained: (ⅰ) Under certain circumstances, even if the reward given to the cooperators reaches the maximum, the proportion of cooperators is still zero, which means that increasing rewards does not always promote cooperation. (ⅱ) The initial state can affect the choice of cooperation strategy and defection strategy. (ⅲ) The increase of the time delay makes the stable equilibrium point disappear and forms a stable limit cycle.