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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 $E_1\ge E_1^*$, 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.

##### MSC:
 92D40 Ecology 93C15 Control systems governed by ODE 93C95 Applications of control theory 34A37 Differential equations with impulses
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##### References:
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