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

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.

Figures (5)

• Figure 1: Mechanism overview. Environmental enrichment drives the deterministic backbone toward Hopf. Stoichiometric coupling determines the diffusion covariance. Two drift-matched closures differing only in off-diagonal covariance propagate distinct fluctuation geometry through LNA diagnostics and extinction-relevant metrics.
• Figure 2: Microscopic Mechanism of Predation Noise. (a) In the coupled closures, a single predation event simultaneously reduces prey and increases predator populations, generating a structurally negative cross-covariance. (b) In the split-channel comparator, these are treated as independent events, yielding zero cross-covariance despite matching the deterministic drift.
• Figure 3: Covariance Matrix Structure. Heatmap representation of the predation contribution to the diffusion covariance. The mechanistic full-covariance model (a) exhibits negative cross-covariance (blue), whereas the split-channel comparator (b) explicitly zeroes out this off-diagonal coupling (white), preserving only the diagonal variances (red).
• Figure 4: Stochastic Sensitivity Ellipses Near the Hopf Threshold. Fluctuation geometry driven by different diffusion closures. The fully coupled mechanistic model (solid blue) yields a tilted ellipse due to negative cross-covariance, indicating that unexpected prey decreases often coincide with predator increases. The split-channel comparator (dashed gray) incorrectly predicts orthogonal, independent fluctuations.
• Figure 5: Time-Series Near the Hopf Bifurcation. Simulation of population fluctuations using the linear noise approximation. (Top) The fully coupled mechanistic model (with negative cross-covariance) generates clear, amplified quasi-cycles, serving as a reliable noisy-precursor indicator of the impending Hopf bifurcation. (Bottom) The split-channel comparator (diagonal noise only) fails to effectively excite these resonant cycles, resulting in a muted, random appearance that obscures the early warning signals of system destabilization.

Theorems & Definitions (11)

• Proposition 4.1: Structural predation-induced cross-covariance
• proof
• Remark D.1: Equivalent notation used in the $2\times2$ Lyapunov appendix.
• Proposition E.1: Drift equivalence does not imply covariance equivalence
• proof
• Remark E.2: Relevance to the predation closures in this paper
• Proposition E.3: Structural negativity of the predation-induced prey--predator cross-covariance
• proof
• Corollary E.4: Cross-covariance agreement of Bernoulli-coupled and effective coupled closures
• proof