Beyond diagonal noise: A better predator-prey modeling framework with cross-covariance

Area of Science:

• Ecology
• Mathematical Biology
• Theoretical Ecology

Background:

• Continuous ecological models often use phenomenological diffusion terms lacking microscopic justification.
• Standard diagonal diffusion approximations misrepresent the geometry of demographic fluctuations in population dynamics.

Purpose of the Study:

• To derive a stochastic Rosenzweig-MacArthur model from a microscopic, integer-valued Markov chain.
• To reveal the exact diffusion covariance structure dictated by event stoichiometry in ecological interactions.
• To formalize the distinction between open-domain and absorbed formulations in stochastic population modeling.

Main Methods:

• Derivation of a stochastic Rosenzweig-MacArthur model from a Bernoulli-coupled continuous-time Markov chain.
• Mathematical proof of negative predator-prey cross-covariance from coupled predation-conversion events.
• Development of a two-stage Lyapunov well-posedness architecture to analyze model dynamics.

Main Results:

• Standard diagonal-noise approximations in ecological models are shown to be mathematically and biologically limited.
• Coupled ecological events inherently generate a negative predator-prey cross-covariance.
• A rigorous framework is established to distinguish between survival-conditioned and extinction-permitting population dynamics.

Conclusions:

• This work provides a mathematically exact template for covariance-consistent and boundary-aware ecological modeling.
• The study bridges microscopic event stoichiometry with macroscopic diffusion processes.
• Ad hoc noise constructs are replaced with a definitive approach for stochastic population modeling.