Dynamical response of a reaction–diffusion predator–prey system with cooperative hunting and prey refuge

Abstract

The present research is concerned with the combined outcome of the cooperative hunting and prey refuge in a spatiotemporal predator–prey model. Firstly, the problem is confirmed to be well-posed and some basic preliminaries are provided within the context of the temporal environment. Subsequently, both the local and the global stability of the temporal system including permanence are thoroughly investigated so as to emerge the fact that the competition between the hunting cooperation factor a and the refuge coefficient r can resolve the dynamics of the system. More precisely, global stability for all of the feasible non-negative equilibria corresponding to the temporal environment and the coexistence equilibrium in the spatiotemporal domain are explored in the event of the hunting cooperation factor a not exceeding the prey refuge coefficient r. However, the moment a exceeds r, where both the Hopf bifurcation and the Turing bifurcation are induced by hunting cooperation. Nevertheless, a distinct Turing instability mechanism is emerged when the prey diffusivity exceeds that of predator but interestingly, the opposite is customarily a reasonable constraint in many predator–prey models. Later on, the diffusion coefficient is chosen as a bifurcation parameter interpreting pattern transition and the amplitude equations close to the onset are thereby derived. The stability analysis is made use of to explain the selection of patterns among hot spot patterns, the mixture of hot spots and stripes patterns and the stripe patterns themselves. Finally, numerical simulations are performed to explore pattern selection influenced by the hunting cooperation factor, the prey refuge coefficient and the diffusivity as well. Some interesting dynamical complexities including the variation of the number of equilibria, the bifurcation scenario, etc, also emerge out from such quantitative simulations.