Effects of prey refuge and predator cooperation on a predator-prey system.

We develop and investigate a discrete-time predator-prey model with cooperative hunting among predators and a spatial prey refuge. The system can exhibit two positive equilibria if the magnitude of cooperation is large and the maximal reproduction number of predators is less than one. In such a scenario, the predator population may exhibit a strong Allee effect, and therefore the predator may survive if its density is above the threshold. When the positive equilibrium is unique, we prove that hunting cooperation can destabilize the equilibrium through a Neimark-Sacker bifurcation. Numerical findings indicate that a high degree of predator hunting cooperation can help rescue the predator population if the proportion of prey refuge is large, while hunting cooperation becomes destabilizing when the proportion of refuge is small. Despite hunting cooperation's destabilizing effect, it can facilitate predator persistence, particularly with a large proportion of prey refuge. Furthermore, there exists a wide parameter space where the predator-prey interaction may exhibit chaotic behaviour.