Canards and Homoclinic Bifurcations for a Singularly Perturbed Rosenzweig–MacArthur Model with the Generalist Predator

In this paper, we investigate the multi-scale dynamics of a singularly perturbed Rosenzweig–MacArthur model with a generalist predator and identify dynamical phenomena, including equilibrium bifurcations, supercritical or subcritical singular Hopf bifurcations, canard explosion bifurcations and homoclinic bifurcations. Specifically, the system exhibits a globally stable node, a headless canard cycle evolving into a homoclinic cycle, a headed canard cycle encompassing either a headless canard cycle or a homoclinic cycle, and so on. Notably, near the boundary equilibrium, these cycles exhibit a diminutive beard-shaped structure whenever it aligns with the transcritical non-normally hyperbolic point. The numerical simulations confirm the occurrence of a canard explosion, relaxation oscillation, and an inverse canard explosion phenomena not previously reported in singularly perturbed systems with both a transcritical point and a canard point. In brief, our results demonstrate that the generalist predation can cause richer bifurcations and dynamics.