# Complex oscillatory patterns near singular Hopf bifurcation in a two time-scale ecosystem

## TL;DR

This paper analyzes a two-time-scale ecological predator-prey model exhibiting complex oscillatory behaviors, including mixed mode oscillations and chaos, near singular Hopf bifurcations, with implications for understanding ecosystem regime shifts.

## Contribution

It introduces a detailed bifurcation analysis of a predator-prey model with explicit predator interference, revealing novel oscillatory phenomena near singular Hopf bifurcations.

## Key findings

- Presence of mixed mode oscillations and relaxation oscillations.
- Long transient chaotic dynamics near bifurcation points.
- Oscillations driven by variations in predator intraspecific competition.

## Abstract

We consider an ecological model consisting of two species of predators competing for their common prey with explicit interference competition. With a proper rescaling, the model is portrayed as a singularly perturbed system with one-fast (prey dynamics) and two-slow variables (dynamics of the predators). The model exhibits variety of rich and interesting dynamics, including, but not limited to mixed mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations, relaxation oscillations and bistability between a semi-trivial equilibrium state and a coexistence oscillatory state. Existence of co-dimenison two bifurcations such as fold-Hopf and generalized Hopf bifurcations make the system further intriguing. More interestingly, in a neighborhood of {\emph{singular Hopf}} bifurcation, long lasting transient dynamics in form of chaotic MMOs or relaxation oscillations are observed as the system approaches the periodic attractor born out of supercritical Hopf bifurcation or a semi-trivial equilibrium state respectively. The transient dynamics could persist for hundreds or thousands of generations before the ecosystem experiences a regime shift. The time series of population cycles with different types of irregular oscillations arising in this model stem from a biological realistic feature, namely, by the variation in the intraspecific competition amongst the predators. To explain these oscillations, we use bifurcation analysis and methods from {\emph{geometric singular perturbation theory}}.