The Geometry of Quasi-Cycles: How Stoichiometric Covariance Alters Pre-Bifurcation Signatures
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arXiv
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| Format: | Preprint |
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2026
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| author | Wang, Louis Shuo Yu, Jiguang Liang, Ye Zhang, Jilin |
| author_facet | Wang, Louis Shuo Yu, Jiguang Liang, Ye Zhang, Jilin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15662 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Geometry of Quasi-Cycles: How Stoichiometric Covariance Alters Pre-Bifurcation Signatures Wang, Louis Shuo Yu, Jiguang Liang, Ye Zhang, Jilin Dynamical Systems Numerical Analysis Probability 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. |
| title | The Geometry of Quasi-Cycles: How Stoichiometric Covariance Alters Pre-Bifurcation Signatures |
| topic | Dynamical Systems Numerical Analysis Probability |
| url | https://arxiv.org/abs/2603.15662 |