Bifurcations of Relaxation Oscillations in a Nonsmooth Slow–Fast Biological Model with Predator Mutual Interference: Canard Explosion and Homoclinic Scenarios

In this paper, we study the bifurcations (the birth and the annihilation) of relaxation oscillations in a nonsmooth, slow–fast biological model with predator mutual interference. We find that the number of relaxation oscillations will change via the bifurcation events including canard explosion and two homoclinic bifurcations under variations of the control parameter. During this dynamical process, the number of positive equilibria changes and their positions on the critical curve move. Based on geometric singular perturbation theory, canard theory and entry–exit function, we capture this dynamical process completely in a continuous way. Depending on the values of parameters, the number of relaxation oscillations in this model may be zero, one and at most two. All the theoretical predictions are verified by numerical simulations.