The breakdown of linear quasi-cycles: Demographic noise and absorbing boundaries in finite predator–prey systems

Louis Shuo Wang; Jiguang Yu; Ye Liang; Jilin Zhang; Louis Shuo Wang; Jiguang Yu; Ye Liang; Jilin Zhang

doi:10.3934/era.2026190

Electronic Research Archive

2026, Volume 34, Issue 6: 4248-4289. doi: 10.3934/era.2026190

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The breakdown of linear quasi-cycles: Demographic noise and absorbing boundaries in finite predator–prey systems

1.
Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
2.
College of Engineering, Boston University, Boston, MA 02215, USA
3.
Department of Industrial and Systems Engineering, The University of Iowa, Iowa City, IA 52242, USA
4.
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
† These authors contributed equally to this work

Received: 04 March 2026 Revised: 19 April 2026 Accepted: 06 May 2026 Published: 19 May 2026

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 their 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. Our exact nonlinear simulations demonstrate that the dominant near-Hopf phenomenon is the profound breakdown of the linear noise approximation itself. While the linear noise approximation predicts unbounded variance and spectral amplification, the true nonlinear quasi-cycles remain strictly bounded, rapidly driving the system into absorbing extinction boundaries. We conclude that accurate early warning inference in finite ecosystems depends not on resolving fine-scale stoichiometric covariance, but on properly accounting for nonlinear saturation and noise-induced boundary hitting.
- complex systems,
- stochastic dynamical systems,
- Hopf bifurcation,
- diffusion approximation,
- covariance structure,
- linear noise approximation,
- power spectral density,
- multiscale modeling
Citation: Louis Shuo Wang, Jiguang Yu, Ye Liang, Jilin Zhang. The breakdown of linear quasi-cycles: Demographic noise and absorbing boundaries in finite predator–prey systems[J]. Electronic Research Archive, 2026, 34(6): 4248-4289. doi: 10.3934/era.2026190

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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 their 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. Our exact nonlinear simulations demonstrate that the dominant near-Hopf phenomenon is the profound breakdown of the linear noise approximation itself. While the linear noise approximation predicts unbounded variance and spectral amplification, the true nonlinear quasi-cycles remain strictly bounded, rapidly driving the system into absorbing extinction boundaries. We conclude that accurate early warning inference in finite ecosystems depends not on resolving fine-scale stoichiometric covariance, but on properly accounting for nonlinear saturation and noise-induced boundary hitting.

References

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