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Impulsive control strategies in biological control of pesticide. (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 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 τ 0 for the pulse period τ is obtained. We can reach the same target as above by controlling the period impulsive effect τ<τ 0 , even if close to τ 0 . Our theoretical results are confirmed by numerical simulations.

MSC:
92D40Ecology
93C15Control systems governed by ODE
93C95Applications of control theory
34A37Differential equations with impulses