Hybrid control for the prey in a spatial prey-predator model with cooperative hunting and fear effect time lag

In the ecosystem, the chase of the predator with cooperation contributes to fear psychology in the prey, resulting in behavioral changes such as a decrease in the birth rate. We construct a spatially diffusive model with delay to investigate the combined perturbation of these factors. Initially, we establish the existence of positive solutions and examine the stability of steady-state solutions under varying conditions. The bifurcation dynamics of the positive solutions have been analyzed. Turing instability, arising from the random diffusion of the species, generates spatially irregular patterns characterized by patchy distribution of prey and predators in the spatial domain. Hopf bifurcation, resulting from the diffusive rate and delay, contributes to spatially periodic solutions where the number of species will spatially oscillate. The combined influence of diffusion and delay results in the emergence of Turing and Hopf bifurcation phenomena. In this case, the combined effect amplifies the spatially heterogeneous distribution of prey and predator. Our results reveal the heterogeneous behaviors of prey and predator under the coupled influence of cooperation hunting and fear effects. In this paper, we will also study a hybrid control scheme for controlling the generations of Turing patterns and Hopf bifurcation. Our theoretical results and numerical simulations demonstrate that the control scheme can mitigate the negative influence of combined factors and promote the species' stability.