Invariant distribution and noise-induced regime shift in prey-predator model with Allee effect and cooperative hunting

In the real world, the predator–prey relationship is essentially a continuous strategic game driven by survival. Such interactions are often reflected in group defense behaviors of prey and cooperative hunting strategies of predators, as observed in nature — for example, large marine fish jointly hunting sardines, or wolf packs coordinating to hunt deer. These are typical cases where prey exhibit collective defense while predators display cooperative behavior. In this study, we construct a stochastic predator–prey model that incorporates both the Allee effect and predator cooperation, with multiplicative environmental noise acting on the prey’s growth rate and the predator’s mortality rate. Due to the presence of the Allee effect, the noise terms become nonlinear and state-dependent, making the system substantially more complex than those with simple linear perturbations. We rigorously prove the existence and uniqueness of global positive solutions, demonstrate their boundedness, and derive sufficient conditions for species extinction. In addition, by introducing a new energy function adapted to the nonlinear noise structure, we establish the existence of an ergodic invariant distribution. Importantly, we perform numerical experiments to clarify the extinction boundary for different parameters under both weak and strong Allee effects. Furthermore, we conduct detailed numerical exper- iments to investigate stochastic bifurcations, including both P-bifurcation and D-bifurcation. These results offer an ecologically reasonable interpretation of how environmental randomness influences the coexistence and stability of interacting species.