Dynamics analysis of a Filippov pest control model with time delay

The most critical factor for increasing crop production is the successful resistance of pests and pathogens which has massive impacts on global food security. Therefore, Filippov systems have been used to model and grasp control strategies for limited resources in Integrated Pest Management (IPM). Extensive studies have been done on these systems where the evolution is governed by a smooth set of ordinary differential equations (ODEs). As far as we know the time delay has not been considered in these systems, which we mean that a set of delay differential equations (DDEs). With this motivation, a Filippov prey–predator (pest–natural enemy) model with time delay is introduced in this paper, where the time delay represents the change of growth rate of the natural enemy before releasing it to feed on pests. The threshold conditions for the stability of the equilibria are derived by using time delay as a bifurcation parameter. It is shown that when the time delay parameter passes through some critical values, a periodic oscillation phenomenon appears through Hopf bifurcation. Further, by employing Filippov convex method we obtain the equation of sliding motion and address the sliding mode dynamics. Numerically, we demonstrate that the time delay plays a substantial role in discontinuity-induced bifurcation. More precisely, one can get boundary focus bifurcation from boundary node bifurcation through variation of the value of the time delay. Moreover, the time delay is used as a bifurcation parameter to obtain sliding–switching and sliding–grazing bifurcations. In conclusion, a Filippov system with time delay can give new insights into pest control models.