Stochasticity induced mixed-mode oscillations and distribution of recurrent outbreaks in an ecosystem

Abstract: The effect of stochasticity, in the form of Gaussian white noise, in a predator-prey model with two distinct time-scales is presented. A supercritical singular Hopf bifurcation yields a Type II excitability in the deterministic model. We explore the effect of stochasticity in the excitable regime, leading to dynamics that are not anticipated by its deterministic counterpart. The stochastic model admits several kinds of noise-driven mixed-mode oscillations which capture the intermediate dynamics between two cycles of population outbreaks. Depending on the strength of noise, the prey population exhibits intermediate to high-amplitude fluctuations (related to moderate or severe outbreaks respectively). We classify these fluctuations as isolated or intermittent or as clusters depending on their recurrences. We study the distribution of the random variable $N$, representing the number of small oscillations between successive spikes, as a function of the noise intensity and the distance to the Hopf bifurcation. The distribution of $N$ is "asymptotically geometric" with the corresponding parameter related to the principal eigenvalue of a substochastic Markov chain. Finally, the stochastic model is transformed into its "normal form" which is used to obtain an estimate of the probability of repeated outbreaks.