On the impulsive controllability and bifurcation of a predator–pest model of IPM

From a practical point of view, the most efficient strategy for pest control is to combine an array of techniques to control the wide variety of potential pests that may threaten crops in an approach known as integrated pest management (IPM). In this paper, we propose a predator–prey (pest) model of IPM in which pests are impulsively controlled by means of spraying pesticides (the chemical control) and releasing natural predators (the biological control). It is assumed that the biological and chemical control are used with the same periodicity, but not simultaneously. The functional response of the predator is allowed to be predator-dependent, in the form of a Beddington–DeAngelis functional response, rather than to have a perhaps more classical prey-only dependence. The local and global stability of the pest-eradication periodic solution, as well as the permanence of the system, are obtained under integral conditions which are shown to have biological significance. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.