Dynamics and bifurcations of a Filippov Leslie-Gower predator-prey model with group defense and time delay

In this paper, the Leslie-Gower model with nonmonotonic functional response is extended to a nonsmooth Filippov control system to reflect the integrated pest management. Different from traditional Filippov models, here, we incorporate time delay as to account for predator maturity time. The stability of the equilibria and the existence of Hopf bifurcation of the subsystems are investigated. Moreover, sliding mode dynamics and regular/virtual/pseudoequilibria are analyzed. Numerical simulations indicate that all solutions finally converge to either the regular equilibrium, the pseudoequilibrium or a stable periodic solution according to different values of time delays and threshold levels. A boundary bifurcation that switches a stable regular equilibrium or a stable limit cycle to a stable pseudoequilibrium can occur. Meanwhile, global bifurcations from the standard periodic solution to the sliding switching bifurcation and then to the crossing bifurcation are obtained as time delay is increased. The results show that Filippov control strategies could effectively control the number of pests under the prescribed threshold, however, time delay may challenge pest control by the occurring of the sliding switching and crossing bifurcations.