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

TL;DR

This paper develops a fully mechanistic stochastic diffusion model for predator-prey dynamics that captures extinction events and the coupling effects of predation, extending classical models with explicit covariance structure and boundary behavior.

Contribution

It introduces an absorbed, fully mechanistic diffusion approximation for predator-prey models with explicit extinction structure and negative cross-covariance, advancing beyond ad hoc noise assumptions.

Findings

Derived a diffusion approximation with explicit covariance structure.

Proved strong well-posedness and non-explosion of the model.

Demonstrated extinction probabilities and conditions for predator extinction.

Abstract

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 $\Sigma_{12}(N,P)=-mNP/(1+N)$ induced solely by the predation--conversion increment $(-1,1)$. We define the absorbed It\^o 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 $m\le c$.