Promotion of cooperation mechanism on the stability of delay-induced host-generalist parasitoid model

Hunting cooperation widely exists in biological systems, which increases the possibility of the encounter between prey and predator populations. It is worth noting that predators are often divided into specialists and generalists. The research on considering cooperation mechanism in the specialist predator-prey model has received extensive attention. However, there is few attentions on the generalist predator-prey model. In this paper, we consider the classical delay-induced host-generalist parasitoid model to determine the impact of such a cooperation mechanism. Specifically, the existence and stability of the equilibria are studied without considering the delay factor. It is clearly proved that the system without delay is a stable system, where the cooperation mechanism will not cause instability. Further, to explore the potential role of the cooperation mechanism in the delay-induced system, we emphasize the dynamical variation induced by delay. We, therefore, investigate the underlying bifurcation properties under different cases, where the direction of Hopf bifurcation and stability of the periodic solutions are determined by the normal form theory and the center manifold theorem. Interestingly, a series of dynamical changes occur in the population, including periodic oscillation, irregular oscillation, and chaotic attractor when the double delays are introduced in both populations. Then through the numerical experiments, we examine the impacts of cooperation levels on the dynamic properties. Hence, it is conclusive that the cooperation mechanism will inhibit the generation of complex phenomena in the host-generalist parasitoid system, thereby promoting the stability of the biological system. The present study puts insights into the mechanism of predator cooperation in a more straightforward and practical manner.