Canard Cycles and Their Cyclicity of a Fast–Slow Leslie–Gower Predator–Prey Model with Allee Effect

Abstract

We study canard cycles and their cyclicity of a fast–slow Leslie–Gower predator–prey system with Allee effect. More specifically, we find necessary and sufficient conditions of the exact number (zero, one or two) of positive equilibria of the slow–fast system and its location (or their locations), and then we further completely determine its (or their) dynamics under explicit conditions. Besides, by geometric singular perturbation theory and the slow–fast normal form, we find explicit sufficient conditions to characterize singular Hopf bifurcation and canard explosion of the system. Additionally, the cyclicity of canard cycles is completely solved, and of particular interest is that we show the existence and uniqueness of a canard cycle, whose cyclicity is at most two, under corresponding precise explicit conditions.