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

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.