Multi-scale dynamics of a piecewise-smooth Bazykin's prey-predator system

In this paper, we study a Bazykin's prey-predator model with piecewise-smooth Holling type I functional response and small predator's competition rate. By using non-dimensional transformation, the model can be rewritten as a multi-scale system which is a regularly perturbed system for x<1 and a singularly perturbed system for x>1. We are keen on the complex dynamics when the system has a focus in the region x<1. Using geometric singular perturbation theory, we show that the system has a relaxation oscillation cycle, a homoclinic cycle and a heteroclinic cycle under different parameter conditions, which separately enclose a small-amplitude hyperbolically unstable limit cycle near the focus. Meanwhile, we also prove that the system undergoes saddle-node bifurcation and boundary equilibrium bifurcation. Furthermore, we present some phase portraits with different parameter values by numerical simulation, which support our theoretical analysis. These results reveal far richer and much more complex dynamics compared to the model without different time scales or with smooth Holling type I functional response.