Regime shifts in a modified Holling–Tanner predator–prey model: The roles of prey refuge and environmental stochasticity

In a modified Holling–Tanner predator–prey model, the introduction of a constant prey refuge can lead to complex dynamics, including multi-type bistability and complex bifurcations, which is highly susceptible to environmental perturbations. In this paper, we conduct a comprehensive exploration of the effects of environmental stochasticity on the critical transitions of the model when it exhibits bistability, particularly in conjunction with the capacity of the prey refuge, using the stochastic sensitivity function (SSF) technique. Specifically, we focus on two types of bistability: one between two positive equilibria and the other between an interior equilibrium and a limit cycle surrounding it. The critical values of noise intensity and prey refuge for the occurrence of regime shifts are identified. If we further consider stable coexistence as our target, the parameter ranges for the prey refuge and noise intensity are also determined. In particular, we conclude that a larger value of the prey refuge parameter is beneficial while a moderate value of noise intensity is more advantageous for this target.