Singular bifurcations in a modified Leslie-Gower model

TL;DR

This paper rigorously analyzes a predator-prey model with a weak Allee effect, revealing complex bifurcation structures including degenerate transcritical bifurcations and singular Hopf bifurcations using blow-up methods and numerical bifurcation analysis.

Contribution

It provides a detailed analysis of degenerate bifurcations in a slow-fast predator-prey system, employing blow-up techniques and intrinsic methods to determine bifurcation criticality.

Findings

Identification of a degenerate transcritical bifurcation organizing the dynamics.

Determination of the criticality of a singular Hopf bifurcation without normal form reduction.

Numerical confirmation of a nearby Takens-Bogdanov bifurcation point.

Abstract

We study a predator-prey system with a generalist Leslie-Gower predator, a functional Holling type II response, and a weak Allee effect on the prey. The prey's population often grows much faster than its predator, allowing us to introduce a small time scale parameter $\varepsilon$ that relates the growth rates of both species, giving rise to a slow-fast system. Zhu and Liu (2022) show that, in the case of the weak Allee effect, Hopf singular bifurcation, slow-fast canard cycles, relaxation oscillations, etc., exist. Our main contribution lies in the rigorous analysis of a degenerate scenario organized by a (degenerate) transcritical bifurcation. The key tool employed is the blow-up method that desingularizes the degenerate singularity. In addition, we determine the criticality of the singular Hopf bifurcation using recent intrinsic techniques that do not require a local normal form. The theoretical analysis is complemented by a numerical bifurcation analysis, in which we numerically identify and analytically confirm the existence of a nearby Takens-Bogdanov point.