Stochastic birth-death dynamics of insect outbreaks: finite-time extinction and quasi-stationary distribution

Abstract This paper constructs a stochastic extension of the classic deterministic Ludwig–Jones–Holling (LJH) model for spruce budworm population dynamics. We propose a continuous-time Markov chain model whose mean-field limit coincides with the LJH model, leading to a stochastic differential equation (SDE) with an absorbing boundary at extinction. We establish the well-posedness of the SDE and, using Feller’s boundary classification, prove that extinction occurs almost surely in finite time—a fundamental departure from deterministic predictions. Conditioned on non-extinction, the process converges exponentially to a unique quasi-stationary distribution (QSD), which characterizes the metastable outbreak state observed in nature. The associated Q -process is shown to be exponentially ergodic, providing a formal framework for analyzing recurrent outbreak statistics. We further obtain an asymptotic extinction rate and a scaling function for local extinction resistance, offering practical metrics for resilience assessment. Extensive numerical simulations validate the theoretical results.