Analysis of long-term transients and detection of early warning signs of major population changes in a two-timescale ecosystem

TL;DR

This paper investigates early warning signs of sudden population shifts in a predator-prey ecosystem using a two-timescale model, singular Hopf bifurcation analysis, and normal form reduction to identify conditions predicting regime changes.

Contribution

It introduces a novel normal form analysis near singular Hopf bifurcation to identify early warning indicators of regime shifts in a two-timescale ecosystem model.

Findings

Chaotic mixed-mode oscillations observed as long transient behaviors.

Normal form analysis reveals a separatrix surface predicting large amplitude oscillations.

Conditions derived for early warning signs of population regime shifts.

Abstract

Identifying early warning signs of sudden population changes and mechanisms leading to regime shifts are highly desirable in population biology. In this paper, a two-trophic ecosystem comprising of two species of predators, competing for their common prey, with explicit interference competition is considered. With proper rescaling, the model is portrayed as a singularly perturbed system with fast prey dynamics and slow dynamics of the predators. In a parameter regime near singular Hopf bifurcation, chaotic mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations are observed as long-lasting transients before the system approaches its asymptotic state. To analyze the dynamical cause that initiates a large amplitude oscillation in an MMO orbit, the model is reduced to a suitable normal form near the singular-Hopf point. The normal form possesses a separatrix surface that separates two different types of oscillations. A large amplitude oscillation is initiated if a trajectory moves from the "inner" to the "outer side" of this surface. A set of conditions on the normal form variables are obtained to determine whether a trajectory would exhibit another cycle of MMO dynamics before experiencing a regime shift (i.e. approaching its asymptotic state). These conditions can serve as early warning signs for a sudden population shift in an ecosystem.