Global stability and sliding bifurcations of a non-smooth Gause predator–prey system

A non-smooth Gause predator–prey model with a constant refuge is proposed and analyzed. Firstly, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed. Then the relations between the existence of a regular equilibrium and a pseudo-equilibrium are studied, and the results indicate that the two types of equilibria cannot coexist. The sufficient and necessary conditions for the global stability of limit cycle, sliding touching cycle, canard cycle, focus point and pseudo-equilibrium are provided by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Furthermore, sliding bifurcations related to boundary node (focus) and touching bifurcations were investigated by employing theoretical and numerical techniques. Finally, we compare our results with previous studies on a non-smooth Gause predator–prey model without involving a carrying capacity for the prey population, and some biological implications are discussed.