## 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:

  Abdelkader, L.; Arino, O., Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn. continuous discrete impulsive systems, 7, 265-287, (2000) · Zbl 1011.34031  Angelova, J.; Dishliev, A., Optimization problems for one-impulsive models from population dynamics, Nonlinear anal., 39, 483-497, (2000) · Zbl 0942.34010  Bainov, D.D.; Simeonov, P.S., Systems with impulsive effects; stability, theory and applications, (1989), John Wiley & Sons New York · Zbl 0676.34035  Bainov, D.D., Simeonov, P.S., 1993. Impulsive Differential Equations. Longman Scientific & Technical, New York.  Erbe, L.H.; Freedman, H.I.; Liu, X.Z.; Wu, J.H., Comparison principles to models of single species growth, J. austral. math. soc. ser. B, 32, 382-400, (1991) · Zbl 0881.35006  Freedman, H.I., 1987. Deterministic Mathematical Models in Population Ecology. HIFR Consulting Ltd, Edmonton. · Zbl 0448.92023  Funasaki, E.; Kot, M., Invasion and chaos in a periodically pulsed mass-action chemostat, Theoret. popul. biol., 44, 203-224, (1993) · Zbl 0782.92020  Goh, B.S., Management and analysis of biological populations, (1980), Elsevier Scientific Publishing Company Amsterdam · Zbl 0453.92015  Kaul, S., On impulsive semidynamical systems, J. math. anal. appl., 150, 120-128, (1990) · Zbl 0711.34015  Kulev, G.K.; Bainov, D.D., On the asymptotic stability of systems with impulses by the direct method of lyapunove, J. math. anal. appl., 140, 324-340, (1989) · Zbl 0681.34042  Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002  Liu, X.Z., Practical stabilization of control systems with impulsive effects, J. math. anal. appl., 166, 563-576, (1992) · Zbl 0757.93073  Liu, X.Z.; Willms, A., Impulsive stabilizability of autonomous systems, J. math. anal. appl., 187, 17-39, (1994) · Zbl 0814.34010  Liu, X.Z.; Rohlf, K., Impulsive control of a lotka – volterra system, IMA J. math. control inform., 15, 269-284, (1998) · Zbl 0949.93069  Loos, G.; Joseph, D., Elementary stability and bifurcation theory, (1980), Springer New York  Panetta, J.C., A mathematical model of periodically pulsed chemotherapytumor recurrence and metastasis in a competition environment, Bull. math. biol., 58, 3, 425-447, (1996) · Zbl 0859.92014  Shulgin, B.; Stone, L.; Agur, Z., Pulse vaccination strategy in the SIR epidemic model, Bull. math. biol., 60, 1123-1148, (1998) · Zbl 0941.92026  Tiamos, E.C., Panavizas, G.C., Cook, R.J., 1992. Biological control of plant diseases, progress and challenges for the future. NATO ASI Series A: Life Sciences, Vol. 230. Plenum Press, New York.
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