The inappropriate use of insecticides may lead to disastrous pest
outbreaks. Aiming at avoiding the outbreak of pests by optimizing the
control strategies, we propose a novel mathematical model base on the
idea of pulse in this study, namely a discrete-time model of
pest-natural enemies, which considers the implementation of spraying
insecticides within the time interval between two generations of pests.
We first investigates the existence and stability of fixed points, and
then using the central manifold theory, we proved that the system can
exist period-doubling bifurcation. The main theoretical results are
verified by numerical simulations. The experiments show that if the
insecticidal time is within a certain range, the lower insecticidal rate
stimulates the growth of pests and natural enemies, while the larger
insecticidal rate can inhibit the growth of pests and natural enemies.
In addition, the effects of pest growth rate, timing of pesticide
spraying, and pesticide intensity on the system are comprehensively
discussed. The main research provides good reference significance to
choose an appropriate time to prevent the pest outbreaks.