Dynamics Analysis for a Prey–Predator Evolutionary Game System with Delays

In this paper, we couple population dynamics and evolutionary game theory to establish a prey–predator system in which individuals in the predator population need to choose between group hunting strategies and isolated hunting strategies. This system includes two types of delay: fitness delay and hunting delay. In the absence of delays, we discuss the stability of boundary and interior equilibria. In addition, the condition that the non-delayed system undergoes transcritical bifurcation is obtained. For the delayed system, we explore the stability of the interior equilibrium and obtain the conditions for the existence of Hopf bifurcation. The conditions for determining the direction and stability of the Hopf bifurcation and the periodic variation in the periodic solution are introduced by using the normal form theory and center manifold theory. Finally, we simulate non-delayed and delayed systems. The results indicate that when the availability of prey is high, the isolated hunting strategy is the dominant strategy. When the availability of prey is low, mixed strategies appear and the proportion of the group hunting strategy increases as the availability of prey decreases. Furthermore, large delays lead to the disappearance of the mixed hunting strategy and its replacement by pure hunting strategies.