Stochastic mixed-mode oscillations in a three-species predator-prey model

Area of Science:

• Mathematical Biology
• Ecology
• Dynamical Systems

Background:

• Predator-prey models are fundamental in ecology.
• Demographic stochasticity can significantly alter population dynamics.
• Singular Hopf bifurcations in deterministic models lead to complex behaviors.

Purpose of the Study:

• To investigate the impact of demographic stochasticity (Gaussian white noise) on a predator-prey model.
• To analyze the emergence of noise-driven mixed-mode oscillations (MMOs) near a singular Hopf bifurcation.
• To understand the interplay between deterministic dynamics and noise in ecological systems.

Main Methods:

• Derivation of stochastic differential equations (SDEs) from a discrete model.
• Analysis of a predator-prey model with one fast and two slow variables.
• Numerical simulations to study MMOs and oscillation distributions.
• Transformation of the stochastic model to a normal form near a folded node singularity.

Main Results:

• The stochastic model exhibits noise-driven MMOs near the Hopf bifurcation.
• Numerical simulations quantify the distribution of small oscillations between population spikes.
• The effect of noise intensity and proximity to the bifurcation on MMOs is analyzed.
• A normal form is derived, linking deterministic and stochastic small amplitude oscillations.

Conclusions:

• Demographic stochasticity can induce complex, noise-driven dynamics (MMOs) in ecological models.
• The study provides insights into how random fluctuations affect population cycles near bifurcations.
• The developed normal form offers a framework for analyzing noise effects in systems with folded singularities.