Canard limit cycles and global dynamics in a singularly perturbed predator-prey system with non-monotonic functional response

In this paper, we define singular saddles, singular nodes and singular homoclinic cycles for general singular perturbation system by combining the associated reduced system and layer system. We perturb these objects singularly to get the existence and the types of singular points as well as the birth of canard limit cycles. Then global dynamics, especially, the existence and the nonexistence of canard limit cycles in a predator-prey system with non-monotonic functional response are analyzed. We show that, depending on the topologies of the critical curve and the relative positions of the two positive equilibria, the flows of the system exhibit quite different limiting behaviors and different slow-fast processes. We point out that the system can have one or two stable nodes, or canard limit cycles generated through the homoclinic mechanism or the Hopf breaking mechanism as its ω-limit sets. Under different parameter conditions and/or initial densities, the evolutions of the predator and the prey are different, in which, the separatrices on the parametric and phase planes are determined explicitly. Numerical simulations are also carried out to verify the theoretical results.