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 \(E_1\geq E_1^*\), 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.


92D40 Ecology
93C15 Control/observation systems governed by ordinary differential equations
93C95 Application models in control theory
34A37 Ordinary differential equations with impulses
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