Global asymptotical stability and sliding bifurcation analysis of a general Filippov-type predator-prey model with a refuge

Abstract

In this paper, a general Filippov-type predator-prey model with a refuge is presented. We propose a discontinuous predator-prey model incorporating a threshold policy by extending a general continuous predator-prey model. By employing the qualitative analysis theory related to Filippov systems, the necessary and sufficient conditions for the global asymptotical stability of a standard cycle, a touching cycle and a sliding cycle are obtained respectively. Furthermore, the sliding cycle is globally finite-time stable. Especially, several kinds of sliding bifurcations including boundary node bifurcation, boundary focus bifurcation and grazing bifurcation are studied. Moreover, two specific models are provided to verify the main results obtained from the general model.