A Filippov system describing the effect of prey refuge use on a ratio-dependent predator–prey model

Abstract

In this paper, a Filippov ratio-dependent predator–prey model is proposed to describe the effect on behavioral refuges caused by prey instinct anti-predator behavior. The proposed model extends the classical ratio-dependent predator–prey model by combining a prey–predator ratio that describes the behavioral refuges make sense once it is less than a certain threshold. One of the prominent mathematical features of our model, distinguishing from the classical one, is that there exist singular points on discontinuous surface whose characteristics determine the main dynamical behaviors of this model. The complete analysis of topological structures of orbits near all the singular points is presented. We show 14 types of system behaviors realized for various parameter values. In particular, globally stable pseudo-equilibrium and globally finite time stable canard cycle are shown to exist in some ranges of parameter values, which cannot be achieved in classical model.