Quasi-Stationary Distribution of a Prey–Predator Model Driven by Demographic Stochasticity

In this paper, we develop a stochastic predator–prey model driven by demographic stochasticity, where prey are subject to predation by both generalist and specialist predators. We begin by analyzing the asymptotic dynamics of the system in a stable environment using a deterministic framework, focusing on boundary dynamics and coexistence equilibria. With the introduction of demographic noise, we demonstrate that population extinction occurs within finite time. To capture the transient dynamics prior to extinction, we employ quasi-stationary distributions. By studying the stability of the sub-Markov semi-group of the stochastic system, we establish key conditions for the existence, uniqueness, and convergence of the quasi-stationary distribution. The quasi-stationary distribution serves as a bridge between the eventual extinction and the transient, time-dependent behavior of the species.