A Filippov type predator prey system with selective harvesting of predator

Abstract

In this paper, piece-wise smooth predator prey Filippov system to describe the role of refuge to prey and additional food to predators is explored. Logistic growth of prey and Holling type II predation is considered. Selective proportional harvesting of predators is included. Refuge is provided to a fraction of preys when their density is below a threshold value. This reduces the food availability for predators. To compensate this, additional food is provided to predators. The dynamics on the boundary is computed using the Filippov regularization method. The stability and existence of different equilibrium states of the independent smooth subsystems are analyzed. The role of quality of additional food with respect to prey in the characterization of the equilibrium points as regular or virtual equilibrium points for the Filippov system is investigated. It is verified that if only refuge is provided to prey with no additional food to predators then both the interior equilibrium state will never be regular simultaneously for the Filippov system. It is observed from the sliding mode dynamics that there is no escaping region. It is proved that both refuge to prey and additional food to predators enhances the sliding region on the boundary. The analysis of boundary equilibrium points, tangent points and pseudoequilibrium points on the boundary is carried out. The role of effort on harvesting on the expansion or shrinking of the limit cycle is elaborated numerically. Boundary bifurcation for the interior equilibrium state of the subsystems is demonstrated. Numerical simulation is used to elaborate the dynamics of Filippov system.