Persistence, extinction, and chaotic dynamics in a predator-prey system with cooperative hunting and prey refuge

Building on the discrete host-parasitoid model of Wu and Zhao (2018), we have proposed a discrete-time predator-prey system with Ricker-type prey growth, prey refuge, and predator hunting cooperation. Analytical results showed that when the prey growth rate is moderate and the predator reproduction number exceeds one, the system is uniformly persistent: both predator and prey populations are guaranteed to remain above a strictly positive lower bound in the long term, for all positive initial conditions, regardless of the strength of hunting cooperation. We further established global asymptotic stability of the predator-free equilibrium under a sufficient condition requiring the predator nullcline to lie entirely beyond the normalized prey carrying capacity, and we identified a related condition under which a predator-extinct $2$-cycle is globally stable. Numerical simulations revealed that the influence of cooperation depends strongly on prey growth: low growth leads to sensitive, possibly chaotic oscillations; moderate growth stabilizes coexistence; and high growth allows cooperation to act as a "rescue" mechanism for predators while inducing complex, non-equilibrium behavior. These results considerably extend previous discrete predator–prey models with prey refuge by demonstrating that hunting cooperation does not alter the persistence threshold but significantly enriches global dynamics by permitting bistability, oscillatory coexistence, and chaotic attractors.