Bifurcation of nontrivial periodic solutions for an impulsively controlled pest management model

In this paper, we investigate the existence of nontrivial periodic solutions for an integrated pest management model which is impulsively controlled by means of biological and chemical controls used in a periodic fashion. For this model, a nonlinear incidence rate is employed to describe the transmission of the disease which is induced through the use of the biological control, while the chemical control is used with the same periodicity as the biological control, although not at the same time. Our problem is treated by means of an operator theoretic approach, being reduced first to a fixed point problem. It is then shown that once a threshold condition is reached, the trivial periodic solution loses its stability and a nontrivial periodic solution appears via a supercritical bifurcation.