## 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.

### MSC:

 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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### References:

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