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

TL;DR

This work replaces ad hoc diagonal noise in stochastic predator–prey models with a fully mechanistic diffusion derived from a four-channel CTMC that includes a coupled predation–conversion event. By applying Kurtz density-dependent scaling, it yields a diffusion with drift $\mu$ and a full covariance matrix $\Sigma$, whose off-diagonal entry $\Sigma_{12}=-\frac{mNP}{1+N}$ captures event-level coupling and a strictly negative cross-covariance as a structural fingerprint. The authors prove strong well-posedness up to extinction for the absorbed Itô SDE, establish positive extinction probability from any interior state (with predator extinction almost surely in the subcritical regime $m\le c$), and provide reproducible numerical diagnostics, including two Brownian factorizations and absorbed Euler–Maruyama simulations. The results highlight fundamental limitations of diagonal-noise surrogates and offer a principled pipeline from demographic events to extinction analysis in consumer–resource systems, with potential extensions to richer dynamics and data-informed inference.

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 $Σ_{12}(N,P)=-mNP/(1+N)$ 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 $m\le c$.

Figures (12)

• Figure 1: Mechanistic roadmap. The pipeline proceeds from event-level definition to macroscopic scaling, yielding a full-covariance diffusion with structural signature $\Sigma_{12}<0$.
• Figure 2: Mechanistic vs. Ad Hoc Noise. Path A derives the diffusion matrix from demographic events, revealing the intrinsic negative correlation caused by predation. Path B assumes independent noise, missing this structural coupling.
• Figure 3: Visual proof of dissipativity and the absorbing set. Illustration of the compact invariant region $\mathcal{A}_{\beta,\varepsilon,\delta}$ defined in Theorem \ref{['thm:absorbing_set_global_A']}. The region (light green) is bounded by the prey logistic constraint $N=k+\delta$ (green dashed line) and the linear Lyapunov constraint $N+\beta P=C$ (blue dashed line). The red arrows along the boundaries explicitly show the inward direction of the vector field. They demonstrate that any trajectory starting outside will eventually enter and remain within this set, thereby ensuring global existence and non-explosion for the deterministic backbone.
• Figure 4: The three dynamic regimes of the deterministic backbone. Numerical illustration of the stability thresholds detailed in Theorem \ref{['thm:det_predator_extinction']}. (Left) Regime I: Low carrying capacity ($k \le N^*$) leads to predator extinction; trajectories converge to $(k,0)$. (Center) Regime II: Intermediate carrying capacity ($N^* < k \le 1+2N^*$) ensures stable coexistence; trajectories spiral into the positive equilibrium $K_3$. (Right) Regime III: High carrying capacity ($k > 1+2N^*$) destabilizes $K_3$ via a Hopf bifurcation, resulting in a stable limit cycle. This transition illustrates the paradox of Enrichment, where increasing resource availability (larger $k$) induces large-amplitude oscillations.
• Figure 5: Absorption rule. The model treats extinction as irreversible. Once a trajectory hits either axis ($N=0$ or $P=0$), the diffusion is frozen (marked by 'X'), representing the demographic end of a species.

Theorems & Definitions (53)

• Theorem 1.1: Mechanistic diffusion and covariance structure
• Theorem 1.2: Strong well-posedness up to absorption and non-explosion
• Theorem 1.3: Extinction properties
• Remark 1.4: Roadmap to proofs
• Lemma 2.1: Local well-posedness
• proof
• Lemma 2.2: Positive invariance of $\overline U$
• proof
• Lemma 2.3: Logistic comparison for the prey
• proof