Full-covariance chemical Langevin predator–prey diffusion with absorbing boundaries

Abstract

Abstract Many stochastic predator–prey models introduce independent diagonal noise at the stochastic level. Such formulations are useful phenomenological models, but they do not explicitly encode the event-level coupling created by predation and biomass conversion. We derive an absorbed, fully mechanistic diffusion approximation and its extinction structure from a continuous-time Markov chain (CTMC) on N02 with four reaction channels: prey birth, prey competition death, predator death and a coupled predation–conversion event. Absorbing coordinate axes are imposed to represent the irreversibility of demographic extinction. Under Kurtz density-dependent scaling, the law-of-large-number (LLN) limit recovers the classical Rosenzweig–MacArthur (RM) ordinary differential equation, while central-limit scaling yields a chemical-Langevin diffusion with explicit drift and full state-dependent covariance. A direct consequence of the coupled predation–conversion event is the negative cross-covariance Σ12(N,P)=−mNP/(1+N) induced solely by the predation–conversion increment (−1,1). We define a stopped-and-frozen Itô diffusion up to the first boundary hit and establish strong well-posedness, non-explosion and moment bounds under this convention. Extinction has positive probability from every interior state, and the first boundary-hitting time is almost surely finite when m≤c.