Unveiling the dynamics of canard cycles and global behaviour in a singularly perturbed predator-prey system with Allee effect in predator

TL;DR

This paper provides a comprehensive mathematical analysis of a predator-prey model with Allee effect, revealing complex dynamics like canard cycles, bifurcations, and global stability, supported by numerical simulations.

Contribution

It introduces a detailed analysis of a modified Leslie-Gower predator-prey model with Allee effect, uncovering rich dynamical phenomena and bifurcation structures.

Findings

Existence of homoclinic and heteroclinic orbits.

Identification of canard limit cycles and relaxation oscillations.

Global stability analysis showing bistability and different population outcomes.

Abstract

In this article, we have considered a planar slow-fast modified Leslie-Gower predator-prey model with a weak Allee effect in the predator, based on the natural assumption that the prey reproduces far more quickly than the predator. We present a thorough mathematical analysis demonstrating the existence of homoclinic orbits, heteroclinic orbits, singular Hopf bifurcation, canard limit cycles, relaxation oscillations, the birth of canard explosion by combining the normal form theory of slow-fast systems, Fenichel's theorem and blow-up technique near non-hyperbolic point. We have obtained very rich dynamical phenomena of the model, including the saddle-node, Hopf, transcritical bifurcation, generalized Hopf, cusp point, homoclinic orbit, heteroclinic orbit, and Bogdanov-Takens bifurcations. Moreover, we have investigated the global stability of the unique positive equilibrium, as well as bistability, which shows that the system can display either 'prey extinction', 'stable coexistence', or 'oscillating coexistence' depending on the initial population size and values of the system parameters. The theoretical findings are verified by numerical simulations.