×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] 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
[2] Angelova, J.; Dishliev, A., Optimization problems for one-impulsive models from population dynamics, Nonlinear anal., 39, 483-497, (2000) · Zbl 0942.34010
[3] Bainov, D.D.; Simeonov, P.S., Systems with impulsive effects; stability, theory and applications, (1989), John Wiley & Sons New York · Zbl 0676.34035
[4] Bainov, D.D., Simeonov, P.S., 1993. Impulsive Differential Equations. Longman Scientific & Technical, New York.
[5] 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
[6] Freedman, H.I., 1987. Deterministic Mathematical Models in Population Ecology. HIFR Consulting Ltd, Edmonton. · Zbl 0448.92023
[7] Funasaki, E.; Kot, M., Invasion and chaos in a periodically pulsed mass-action chemostat, Theoret. popul. biol., 44, 203-224, (1993) · Zbl 0782.92020
[8] Goh, B.S., Management and analysis of biological populations, (1980), Elsevier Scientific Publishing Company Amsterdam · Zbl 0453.92015
[9] Kaul, S., On impulsive semidynamical systems, J. math. anal. appl., 150, 120-128, (1990) · Zbl 0711.34015
[10] 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
[11] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[12] Liu, X.Z., Practical stabilization of control systems with impulsive effects, J. math. anal. appl., 166, 563-576, (1992) · Zbl 0757.93073
[13] Liu, X.Z.; Willms, A., Impulsive stabilizability of autonomous systems, J. math. anal. appl., 187, 17-39, (1994) · Zbl 0814.34010
[14] Liu, X.Z.; Rohlf, K., Impulsive control of a lotka – volterra system, IMA J. math. control inform., 15, 269-284, (1998) · Zbl 0949.93069
[15] Loos, G.; Joseph, D., Elementary stability and bifurcation theory, (1980), Springer New York
[16] 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
[17] Shulgin, B.; Stone, L.; Agur, Z., Pulse vaccination strategy in the SIR epidemic model, Bull. math. biol., 60, 1123-1148, (1998) · Zbl 0941.92026
[18] 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.