Singular bifurcations in a slow-fast modified Leslie-Gower model

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 ɛ 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. 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. • We investigate a slow-fast Leslie-Gower predator-prey system • The model has a generalist predator, a type II response, and a weak Alle effect • The dynamics are mainly organized by singular transcritical and Hopf bifurcations • Our analysis describes degenerate canard cycles and relaxation oscillations • The criticality of the Hopf bifurcation is obtained without a normal form • We support our theoretical results with a detailed numerical bifurcation analysis