Relaxation oscillations of a piecewise-smooth slow-fast Bazykin's model with Holling type Ⅰ functional response.

Abstract

In this paper, we consider the dynamics of a slow-fast Bazykin's model with piecewise-smooth Holling type Ⅰ functional response. We show that the model has Saddle-node bifurcation and Boundary equilibrium bifurcation. Furthermore, it is also proven that the model has a homoclinic cycle, a heteroclinic cycle or two relaxation oscillation cycles for different parameters conditions. These results imply the dynamical behavior of the model is sensitive to the predator competition rate and the initial densities of prey and predators. In order to support the theoretical analysis, we present some phase portraits corresponding to different values of parameters by numerical simulation. These phase portraits include two relaxation oscillation cycles, an unstable relaxation oscillation cycle surrounded by a stable homoclinic cycle; the coexistence of a heteroclinic cycle and an unstable relaxation oscillation cycle. These results reveal far richer and much more complex dynamics compared to the model without different time scale or with smooth Holling type Ⅰ functional response.