Impulsive control strategies in biological control of pesticide.

By presenting and analyzing the 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 insecticide. For this scheme, two thresholds, E(1)() and E(1)() for E(1) are obtained. If E(1)>or=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, 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.