Singular bifurcations in a modified Leslie-Gower model
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 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.
Keywords
Cite
@article{arxiv.2411.18059,
title = {Singular bifurcations in a modified Leslie-Gower model},
author = {Roberto Albarran García and Martha Alvarez-Ramírez and Hildeberto Jardón-Kojakhmetov},
journal= {arXiv preprint arXiv:2411.18059},
year = {2025}
}