Full-Covariance Chemical Langevin Predator--Prey Diffusion with Absorbing Boundaries

Many stochastic Rosenzweig–MacArthur predator–prey models inject ad hoc independent (diagonal) noise and therefore cannot 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 on $\mathbb N_0^2$ 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-numbers limit recovers the classical RM ODE, while central-limit scaling yields a chemical-Langevin diffusion with explicit drift and full state-dependent covariance. A distinctive signature is the strictly negative cross-covariance $Σ_{12}(N,P)=-mNP/(1+N)$ induced solely by the predation–conversion increment $(-1,1)$. We define the absorbed Itô SDE by freezing trajectories at the first boundary hit and prove strong well-posedness, non-explosion, and moment bounds up to absorption. Extinction has positive probability from every interior state, and predator extinction is almost sure when $m\le c$.

💡 Research Summary

This paper addresses a fundamental shortcoming in many stochastic predator‑prey models based on the Rosenzweig‑MacArthur (RM) framework: the common practice of injecting ad‑hoc independent (diagonal) noise fails to capture the intrinsic coupling that arises from predation events, where a single interaction simultaneously reduces prey abundance and increases predator abundance. Starting from a fully mechanistic continuous‑time Markov chain (CTMC) defined on the non‑negative integer lattice ℕ₀², the authors specify four elementary reaction channels: (B) prey birth, (C) prey competition death, (D) predator death, and (E) coupled predation‑conversion. The stoichiometric vectors are ΔB=(1,0), ΔC=(−1,0), ΔD=(0,−1), ΔE=(−1,1). The corresponding propensity functions are λB(N)=N, λC(N)=N²/k, λD(P)=cP, and λE(N,P)=mNP/(1+N), where k>0 is the prey carrying capacity, m>0 the predation efficiency, and c>0 the predator mortality rate.

Applying Kurtz’s density‑dependent scaling with system size Ω, the scaled state ZΩ=Ω⁻¹XΩ satisfies a law‑of‑large‑numbers (LLN) limit that recovers the classical RM ODE: \