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

TL;DR

This paper derives a stochastic predator-prey model from microscopic event dynamics, revealing the importance of cross-covariance structures and formalizing boundary behaviors, thus improving ecological modeling accuracy.

Contribution

It introduces a mathematically rigorous framework linking microscopic event stoichiometry to macroscopic stochastic models with explicit cross-covariance structures.

Findings

Coupled predation events generate negative predator-prey cross-covariance.

Standard diagonal-noise approximations are fundamentally limited.

A new boundary-aware modeling architecture is developed.

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.