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arXiv · 2603.15662

The Geometry of Quasi-Cycles: How Stoichiometric Covariance Alters Pre-Bifurcation Signatures

Louis Shuo Wang·Jiguang Yu·Ye Liang·Jilin Zhang
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Abstract

Environmental enrichment can destabilize predator--prey coexistence through a Hopf bifurcation, yet real ecosystems are finite and intrinsically stochastic. We investigate how mechanistically derived demographic noise shapes near-Hopf dynamics in the Rosenzweig--MacArthur model by systematically comparing two diffusion closures that share identical deterministic drift but differ solely in predation-induced covariance structure. Starting from a continuous-time Markov chain description, we derive a full-covariance stochastic differential equation whose diffusion tensor inherits stoichiometric coupling, generating a negative prey--predator cross-covariance. This model is contrasted with a drift-matched diagonal-noise comparator. Using linear noise approximation, Lyapunov analysis, and matrix-valued power spectral density formulations, we propagate local covariance structure through the entire diagnostic chain, including stochastic sensitivity ellipses and a dimensionless noisy-precursor indicator. The results highlight that drift equivalence does not imply covariance equivalence and show how event-level noise geometry influences macroscopic behavior in nonlinear ecological systems. This work integrates bifurcation theory and stochastic analysis to advance multi-scale modeling of complex interacting systems.

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Louis Shuo Wang, Jiguang Yu, Ye Liang, Jilin Zhang. 2026-03-10. The Geometry of Quasi-Cycles: How Stoichiometric Covariance Alters Pre-Bifurcation Signatures. https://arxiv.org/abs/2603.15662

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