Multi-scale dynamics of a singularly perturbed piecewise-smooth predator–prey model with weak predator interference

Abstract

In this paper, we focus on the dynamics of a piecewise-smooth predator–prey model with weak predator interference. By non-dimensional transformation, the model can be rewritten as a regular-singular system with a regularly perturbed system for $u<1$ and a singularly perturbed system for $u\ge 1$. Based on the analysis, the regular-singular system has two saddle boundary equilibriums and at most three positive equilibriums. When the positive equilibrium with $u<1$ is a stable focus, the system undergoes a saddle–node bifurcation and a boundary equilibrium bifurcation. Furthermore, as the parameters cross the bifurcation curves, the system has a small-amplitude hyperbolically unstable limit cycle, which is surrounded by a stable relaxation oscillation cycle, a homoclinic cycle and a heteroclinic cycle, respectively. Finally, we provide the phase portraits to illustrate our theoretical results.