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

TL;DR

This paper investigates how demographic stochasticity influences predator-prey dynamics near a Hopf bifurcation, revealing noise-induced mixed-mode oscillations and deriving a normal form to analyze stochastic effects on small amplitude oscillations.

Contribution

It derives a stochastic differential equation model from a discrete predator-prey system, analyzes noise-driven oscillations near a folded node, and links stochastic dynamics to a normal form for better understanding.

Findings

Noise induces mixed-mode oscillations near the Hopf bifurcation.

Distribution of small oscillations varies with noise intensity.

Normal form transformation links stochastic effects to deterministic bifurcations.

Abstract

The effect of demographic stochasticity, in the form of Gaussian white noise, in a predator-prey model with one fast and two slow variables is studied. We derive the stochastic differential equations (SDEs) from a discrete model. For suitable parameter values, the deterministic drift part of the model admits a folded node singularity and exhibits a singular Hopf bifurcation. We focus on the parameter regime near the Hopf bifurcation, where small amplitude oscillations exist as stable dynamics in the absence of noise. In this regime, the stochastic model admits noise-driven mixed-mode oscillations (MMOs), which capture the intermediate dynamics between two cycles of population outbreaks. We perform numerical simulations to calculate the distribution of the random number of small oscillations between successive spikes for varying noise intensities and distance to the Hopf bifurcation. We also study the effect of noise on a suitable Poincar\'e map. Finally, we prove that the stochastic model can be transformed into a normal form near the folded node, which can be linked to recent results on the interplay between deterministic and stochastic small amplitude oscillations. The normal form can also be used to study the parameter influence on the noise level near folded singularities.