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Mathematical modelling to control a pest population by infected pests. (English) Zbl 1205.34065

Summary: In this paper, we formulate and investigate the pest control models in accordance with the mathematical theory of epidemiology. We assume that the release of infected pests is continuous and impulsive, respectively. Therefore, our models are the ordinary differential equations and the impulsive differential equations. We study the global stability of the equilibria of the ordinary differential equations. The permanence of the impulsive differential equations is proved. By means of numerical simulation, we obtain the critical values of the control variable under different methods of release of infected pests. Thus, we provide a mathematical evidence in the management of an epidemic controlling a pest.

MSC:

34D20 Stability of solutions to ordinary differential equations
92D25 Population dynamics (general)
92D30 Epidemiology
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References:

[1] DeBach, P., Biological control of insect pests and weeds, (1964), Rheinhold New York
[2] DeBach, P.; Rosen, D., Biological control by natural enemies, (1991), Cambridge University Press Cambridge
[3] Freedman, H.J., Graphical stability, enrichment and pest control by a natural enemy, Math. biosci., 31, 3-4, 207-225, (1976) · Zbl 0373.92023
[4] Huffaker, C.B., New technology of pest control, (1980), Wiley New York
[5] Barclay, H.J., Models for pest control using predator release, habitat management and pesticide release in combination, J. appl. ecol., 19, 2, 337-348, (1982)
[6] Grasman, J.; Van Herwaarden, O.A.; Hemerik, L.; Van Lenteren, J.C., A two-component model of host cparasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control, Math. biosci., 169, 2, 207-216, (2001) · Zbl 0966.92026
[7] Tang, S.Y.; Xiao, Y.N.; Chen, L.S.; Cheke, R.A., Integrated pest management models and their dynamical behaviour, Bull. math. biol., 67, 1, 115-135, (2005) · Zbl 1334.91058
[8] Van Lenteren, J.C., Integrated pest management in protected crops, (), 311-320
[9] Anderson, R.M.; May, R.M., Population biology of infectious diseases: part I, Nature, 280, 361-367, (1979)
[10] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. biol., 25, 4, 359-380, (1987) · Zbl 0621.92014
[11] Zhang, X.A.; Chen, L.S., The periodic solution of a class of epidemic models, Comput. math. appl., 38, 3-4, 61-71, (1999) · Zbl 0939.92031
[12] Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: periodic solutions and applications, (1993), Longman Scientific and Technical Burnt Mill · Zbl 0793.34011
[13] Kulev, G.K.; Bainov, D.D., On the asymptotic stability of systems with impulses by the direct method of Lyapunov, J. math. anal. appl., 140, 2, 324-340, (1989) · Zbl 0681.34042
[14] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[15] Simeonov, P.S.; Bainov, D.D., Stability with respect to part of the variables in system with impulsive effect, J. math. anal. appl., 117, 1, 247-263, (1986) · Zbl 0588.34044
[16] Goh, B.S., Management and analysis of biological populations, (1980), Amsterdam Oxford New York · Zbl 0453.92015
[17] Wickwire, K., Mathematical models for the control of pests and infectious disease: a survey, Theor. pop. biol., 11, 2, 182-238, (1977) · Zbl 0356.92001
[18] Zhang, Z., Qualitative theory of differential equations, Translations of mathematical monographs, vol. 101, (1992), American Mathematics Society Providence, RI
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