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Zbl 1100.92071
Lu, Zhonghua; Chi, Xuebin; Chen, Lansun
Impulsive control strategies in biological control of pesticide.
(English)
[J] Theor. Popul. Biol. 64, No. 1, 39-47 (2003). ISSN 0040-5809

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 2000:
*92D40 Ecology
93C15 Control systems governed by ODE
93C95 Appl. of control theory
34A37 Differential equations with impulses

Keywords: biological control; periodic predator-free solution; threshold; asymptotically stable

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