*(English)*Zbl 1100.92071

Summary: By presenting and analyzing a pest-predator model under insecticides used impulsively, two impulsive strategies in biological control are put forward. The first strategy: the pulse period is fixed, but the proportional constant ${E}_{1}$ changes, which represents the fraction of pests killed by applying insecticides. For this scheme, two thresholds, ${E}_{1}^{**}$ and ${E}_{1}^{*}$ for ${E}_{1}$ are obtained. If ${E}_{1}\ge {E}_{1}^{*}$, both the pest and predator (natural enemies) populations go to extinction. If ${E}_{1}^{**}<{E}_{1}<{E}_{1}^{*}$, the pest population converges to the semi-trivial periodic solution while the predator population tends to zero. If ${E}_{1}$ is less than ${E}_{1}^{**}$ but even if close to ${E}_{1}^{**}$, there exists a unique positive periodic solution via bifurcation, which implies both the pest and the predator populations oscillate with a positive amplitude. In this case, the pest population is killed to the maximum extent while the natural enemies are preserved to avoid extinction.

The second strategy: the proportional constant ${E}_{1}$ is fixed $({E}_{1}<{E}_{1}^{*}$ firstly), but the pulse period changes. For this scheme, one threshold ${\tau}_{0}$ for the pulse period $\tau $ is obtained. We can reach the same target as above by controlling the period impulsive effect $\tau <{\tau}_{0}$, even if close to ${\tau}_{0}$. Our theoretical results are confirmed by numerical simulations.