Beyond Diagonal Noise: A Better Predator-Prey Modeling Framework with Cross-Covariance

Abstract

The introduction of stochasticity into continuous ecological models frequently relies on phenomenological, diagonal diffusion terms that lack a rigorous microscopic basis. We demonstrate that this standard practice fundamentally misrepresents the geometry of demographic fluctuations. By deriving a stochastic Rosenzweig--MacArthur model directly from an integer-valued, Bernoulli-coupled continuous-time Markov chain, we isolate the exact diffusion covariance structure dictated by event stoichiometry. We mathematically prove that coupled predation--conversion events inherently generate a structurally negative predator--prey cross-covariance, exposing the severe mathematical and biological limitations of standard diagonal-noise approximations. Furthermore, we resolve a persistent ambiguity in stochastic population modeling by explicitly formalizing the bifurcation between open-domain formulations (for survival-conditioned interior dynamics) and absorbed formulations (for extinction-permitting dynamics). To rigorously support this distinction, we develop a tailored two-stage Lyapunov well-posedness architecture that separates non-explosion criteria from boundary-barrier positivity invariance. By bridging microscopic event stoichiometry with macroscopic boundary-degenerate diffusions, this work replaces ad hoc noise constructs with a definitive, mathematically exact template for covariance-consistent and boundary-aware ecological modeling.

Figures (9)

• Figure 1: Mechanistic roadmap and modeling bifurcation for the stochastic R--M framework. The diagram summarizes the paper's logic from the deterministic R--M backbone and Hopf organization, through a mechanistic CTMC/CME derivation of a full-covariance CLE, to the structural negative predator--prey cross-covariance induced by coupled predation--conversion. It then highlights the explicit modeling bifurcation between open-domain and absorbed diffusion formulations built from the same local coefficients, and the corresponding two-stage well-posedness architecture for the open-domain SDE.
• Figure 2: Deterministic R--M backbone and the Hopf bifurcation. (a) Bifurcation diagram over the carrying capacity $k$. The coexistence equilibrium $K_3$ loses stability at the Hopf threshold $k_H$, partitioning the parameter space into the stable regime $\Lambda_2$ and the oscillatory regime $\Lambda_1$. Solid curves denote the extrema of the stable limit cycle, and dashed lines denote the unstable equilibrium. (b) Phase portrait in the stable regime $\Lambda_2$ ($k < k_H$). The predator nullcline (vertical) intersects the prey nullcline (humped) to the right of its peak, resulting in trajectories spiraling inward to a stable coexistence state $K_3$. (c) Phase portrait in the oscillatory regime $\Lambda_1$ ($k > k_H$). Enrichment shifts the intersection to the left of the peak, destabilizing $K_3$ and generating a stable limit cycle that drives large-amplitude, low-density excursions.
• Figure 3: Mechanistic origin of the demographic diffusion covariance. (a) Microscopic state transitions on the count lattice. The exact Bernoulli-coupled mechanism strictly requires a diagonal jump vector $\nu_4^{(B)}$ (crimson) to represent a single predation--conversion event. The split-channel comparator falsely separates this into two independent, orthogonal jumps. (b) Macroscopic covariance geometry at the density scale. The diagonal jump in the coupled mechanism inherently tilts the local fluctuation geometry, generating a strictly negative predator--prey cross-covariance ($a_{12} < 0$). In contrast, the phenomenological split-channel closure produces an axis-aligned covariance ellipse ($a_{12} = 0$), failing to capture the true geometric structure of the demographic noise despite maintaining mean-field drift equivalence.
• Figure 4: Visualizing the modeling bifurcation: Absorbed versus Open-Domain formulations. Both panels simulate the exact same local chemical Langevin drift and covariance parameters ($\Omega=150, k=5.5$, a fluctuation-amplified regime) originating from the identical initial state $x_0$. (a) The absorbed formulation, where trajectories are permanently frozen upon first boundary contact ($\tau_{\partial D}$), marked by red stars. This is the mathematically appropriate framework for evaluating extinction probabilities and mean persistence times. (b) A numerical proxy for the open-domain formulation, where trajectories are conditioned to survive and continuously explore the interior domain $D=(0,\infty)^2$. This framework captures the geometry of quasi-stationary interior fluctuations but must not be conflated with the extinction-permitting dynamics strictly captured in (a).
• Figure 5: Geometric visualization of the two-stage Lyapunov well-posedness architecture. (a) The anti-explosion Lyapunov function $V_\infty(x)$ (Theorem \ref{['thm:maximal_no_interior_explosion_paperA']}) acts as a geometric bowl that grows radially to infinity, strictly confining the stochastic trajectory from escaping to infinite states without restricting its boundary approach. (b) The boundary-barrier Lyapunov function $V_b(x)$ (Proposition \ref{['prop:barrier_invariance_paperA']}) acts as a steep geometric cliff. The surface is extremely flat within the interior domain $D=(0,\infty)^2$, permitting natural quasi-stationary fluctuations, but spikes asymptotically to infinity precisely at the boundary faces $N=0$ and $P=0$, theoretically repelling the trajectory and ensuring positivity invariance almost surely.

Theorems & Definitions (15)

• Proposition 5.1: Structural predation cross-covariance under coupled versus split closures
• Remark 5.2: Bernoulli-coupled exact CTMC vs. effective $({-1},e)^\top$ closure
• Remark 5.4: Sufficient regimes for small cross-covariance
• Remark 6.1: Why these are different questions (and why the distinction matters)
• Theorem 7.1: Maximal strong solution and no interior explosion before boundary hitting
• Proposition 7.2: Positivity invariance under a boundary-barrier Lyapunov condition
• Lemma B.1: Leaving every compact subset of $D$ while norm-bounded implies boundary approach
• proof
• proof
• proof