Canard Phenomena of a Singularly Perturbed Leslie-Gower Model of Generalized Holling Type II with Prey Harvesting and Weak Allee Effect of Predator

This paper investigates the dynamics of a slow-fast Leslie-Gower predator-prey model incorporating a generalized Holling type II functional response, prey harvesting, and a weak Allee effect in the predator population. Combining the normal form theory of slow-fast systems and the geometric singular perturbation theory, we address the rich canard phenomena including the existence of canard cycles, singular Hopf bifurcation, homoclinic and heteroclinic orbits, birth of canard explosion. We also employ the entry-exit function to demonstrate the presence of relaxation oscillations. Furthermore, we observe much richer new dynamical phenomena, specifically examining the transformation of bistability via Hopf bifurcation. Our results demonstrate that variations in parameters and initial population sizes can lead to different long-term outcomes, ranging from predator extinction to stable coexistence. The theoretical results are confirmed by numerical simulations.