Persistence, extinction and noise-induced steady-state transitions in a stochastic generalist predator–prey model

Understanding the dynamics of the predator and prey populations is essential for maintaining ecosystem stability and biodiversity, both in theory and in practice. Because generalist predators consume multiple prey species, they can remain in the ecosystem even when certain prey are absent. In this work, we formulate a stochastic generalist predator–prey model incorporating a sigmoidal functional response. We first prove the global existence and uniqueness of solutions, as well as the boundedness of positive solutions. We then derive sufficient conditions for the persistence and extinction of both the predator and the prey populations. Using the stochastic sensitivity functions (SSF) method, we construct confidence ellipses for the random state and identify the critical noise intensity at which steady-state transitions occur. Finally, we investigate the mean first passage time (MFPT) and probability of the system’s extinction. Our results show that higher noise intensity accelerates the extinction process and increases the likelihood of extinction. We further find that increasing predator conversion efficiency makes it less probable for the predator population to drop to a low level and reduces the chance of simultaneous collapse of both prey and predator populations. In particular, noise has a stronger impact on prey than on generalist predators.