A delayed epidemic model with stage-structure and pulses for pest management strategy. (English) Zbl 1154.34394

Summary: From a biological pest management standpoint, epidemic diseases models have become important tools in control of pest populations. This paper deals with an impulsive delay epidemic disease model with stage-structure and a general form of the incidence rate concerning pest control strategy, in which the pest population is subdivided into three subgroups: pest eggs, susceptible pests, infectious pests that do not attack crops. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact periodic susceptible pest-eradication solution of the system and observe that the susceptible pest-eradication periodic solution is globally attractive, provided that the amount of infective pests released periodically is larger than some critical value. When the amount of infective pests released is less than another critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its attractivity. Our results indicate that besides the release amount of infective pests, the incidence rate, time delay and impulsive period can have great effects on the dynamics of our system.


34K45 Functional-differential equations with impulses
92D30 Epidemiology
Full Text: DOI


[1] Aiello, W. G.; Freedman, H. I., A time delay model of single species growth with stage structure, Math. Biosci., 101, 139-156 (1990) · Zbl 0719.92017
[2] Anderson, R. M.; May, R. M., Regulation and stability of host-parasite population interactions, I: regulatory processes, J. Anim. Ecol., 47, 219-247 (1978)
[3] Bainov, D. D.; Simeonov, P. S., Impulsive Differential Equations: Periodic Solutions and Applications (1993), Longman: Longman New York · Zbl 0815.34001
[4] Bence, J. B.; Nisbet, R. M., Space limited recruitment in open systems: the importance of time delays, Ecology, 70, 1434-1441 (1989)
[5] Bica, A. M.; Muresan, S., Smooth dependence by lag of the solution of a delay integro-differential equation from biomathematics, Commun. Math. Anal., 1, 64-74 (2006) · Zbl 1126.45004
[6] Bodnar, M.; Forys, U.; Urszula, Behaviour of solutions to Marchuk’s model depending on a time delay, Int. J. Appl. Math. Comput. Sci., 10, 97-112 (2000) · Zbl 0947.92015
[7] Cherry, A. J.; Lomer, C. J.; Djegui, D.; Schulthess, F., Pathogen incidence and their potential as microbial control agents in IPM of maize stemborers in West Africa, Biocontrol, 44, 301-327 (1999)
[8] Cui, J. A.; Chen, L. S.; Wang, W. D., The effect of dispersal on population growth with stage-structure, Comput. Math. Appl., 39, 91-102 (2000) · Zbl 0968.92018
[9] Cui, J. A.; Song, X. Y., Permanence of a predator-prey system with stage structure, Discrete Continuous Dynamical Syst. Ser. B, 4, 547-554 (2004) · Zbl 1100.92062
[10] Debach, P., Biological Control of Insect Pests and Weeds (1964), Chapman & Hall: Chapman & Hall London
[11] Debach, P.; Rosen, D., Biological Control by Natural Enemies (1991), Cambridge University Press: Cambridge University Press Cambridge
[12] d’Onofrio, A., On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Math. Lett., 18, 729-732 (2005) · Zbl 1064.92041
[13] Ferron, P., Pest control using the fungi Beauveria and Metarhizinm, (Burges, H. D., Microbial Control in Pests and Plant Diseases (1981), Academic Press: Academic Press London)
[14] Franco, D.; Liz, E.; Nieto, J. J.; Rogovchenko, Y. V., A contribution to the study of functional differential equations with impulses, Math Nachr., 218, 49-60 (2000) · Zbl 0966.34073
[15] Freedman, H. I., Graphical stability, enrichment, and pest control by a natural enemy, Math. Biosci., 31, 207-225 (1976) · Zbl 0373.92023
[16] Funasaki, E.; Kot, M., Invasion and chaos in a periodically pulsed mass-action chemostat, Theor. Pop. Biol., 44, 203-224 (1994) · Zbl 0782.92020
[17] Gao, S. J.; Chen, L. S.; Nieto, J. J.; Torres, A., Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24, 6037-6045 (2006)
[18] Goh, B. S., Management and Analysis of Biological Populations (1980), Elsevier: Elsevier Amsterdam-Oxford-New York
[19] Grasman, J.; Van Herwarrden, O. A.; Hemerik, L., A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control, Math. Biosci., 196, 207-216 (2001) · Zbl 0966.92026
[20] Halany, A., Differential Equations: Stability, Oscillations, Time Lags (1966), Academic Press: Academic Press New York · Zbl 0144.08701
[21] Hui, J.; Chen, L. S., Impulsive vaccination of SIR epidemic models with nonlinear incidence rates, Discrete Continuous Dynamical Syst. Ser. B, 3, 595-606 (2004) · Zbl 1100.92040
[22] Jiang, G. R.; Lu, Q. S.; Peng, L. P., Impulsive ecological control of staged-structured pest management system, Math. Biosci. Eng., 2, 329-344 (2005) · Zbl 1082.34005
[23] Lakmeche, A.; Arino, O., Bifurcation of nontrivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynamics of Continuous, Discrete Impulsive Syst., 7, 265-287 (2000) · Zbl 1011.34031
[24] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore, New Jersey, London, Hong Kong · Zbl 0719.34002
[25] Li, X.; Zhang, X.; Jiang, D., A new existence theory for positive periodic solutions to functional differential equations with impulse effects, Comput. Math. Appl., 51, 1761-1772 (2006) · Zbl 1156.34053
[26] Liu, B.; Chen, L. S.; Zhang, Y. J., The dynamics of a prey-dependent consumption model concerning impulsive control strategy, Appl. Math. Comput., 169, 305-320 (2005) · Zbl 1074.92042
[27] Liu, G.; Yan, J.; Zhang, F., Existence of positive periodic solutions for neutral functional differential equations, Nonlinear Anal., 66, 253-267 (2007) · Zbl 1109.34052
[28] Liu, S. Q.; Chen, L. S.; Agarwal, R., Recent progress on stage-structured population dynamics, Math. Comput. Model, 36, 1319-1360 (2002) · Zbl 1077.92516
[29] Liu, X. N.; Chen, L. S., Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos Solitons Fractals, 16, 311-320 (2003) · Zbl 1085.34529
[30] Liu, X. Z., Impulsive stabilization and applications to population growth models, J. Math., 25, 1, 381-395 (1995) · Zbl 0832.34039
[31] Liu, X. Z.; Rohof, K., Impulsive control of a Lotka-Volterra system, IMA J. Math. Control Inf., 15, 269-284 (1998) · Zbl 0949.93069
[32] Liu, Y., Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations, J. Math. Anal. Appl., 327, 435-452 (2007) · Zbl 1119.34062
[33] Luff, M. L., The potential of predators for pest control, Agr. Ecosyst. Environ., 10, 159-181 (1983)
[34] Magnusson, K. G., Destabilizing effect of cannibalism on a structured predator-prey system, Math. Biosci., 155, 61-75 (1999) · Zbl 0943.92030
[35] Nieto, J. J.; Rodriguez-Lopez, R., Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. Math. Anal. Appl., 318, 593-610 (2006) · Zbl 1101.34051
[36] Shulgin, B.; Stone, L.; Agur, Z., Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol., 60, 1123-1148 (1998) · Zbl 0941.92026
[37] Smith, H. L., Systems of ordinary differential equations which generate an order preserving flow, SIAM Rev., 30, 87-98 (1998)
[38] Tang, S. Y.; Chen, L. S., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol., 44, 185-199 (2002) · Zbl 0990.92033
[39] X.H. Tang, Z. Jiang, Periodic solutions of first-order nonlinear functional differential equations, Nonlinear Anal, in press, doi: 10.1016/j.na.2006.11.041.; X.H. Tang, Z. Jiang, Periodic solutions of first-order nonlinear functional differential equations, Nonlinear Anal, in press, doi: 10.1016/j.na.2006.11.041.
[40] Van Lenteren, J. C., Measures of success in biological control of anthropoids by augmentation of natural enemies, (Wratten, S.; Gurr, G., Measures of Success in Biological Control (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrdcht)
[41] Wang, W. D.; Chen, L. S., A predator-prey system with stage-structure for predator, Comp. Math. Appl., 33, 83-91 (1997)
[42] Xiao, Y. N.; Chen, L. S., Global stability of a predator-prey system with stage structure for the predator, Acta Math. Sin. Engl. Ser., 19, 1-11 (2003)
[43] Yan, J. R.; Zhao, A.; Nieto, J. J., Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems, Math. Comput. Model, 40, 509-518 (2004) · Zbl 1112.34052
[44] H. Zhang, L.S. Chen, Pest Management Through Continuous and Impulsive Control Strategies, Biosystems, in press, doi: 10.1016/j.biosystems.2006.09.038.; H. Zhang, L.S. Chen, Pest Management Through Continuous and Impulsive Control Strategies, Biosystems, in press, doi: 10.1016/j.biosystems.2006.09.038.
[45] Zhang, S. W.; Tan, D. J.; Chen, L. S., Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations, Chaos Solitons Fractals, 28, 367-376 (2006) · Zbl 1083.37537
[46] Zhang, W.; Fan, M., Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Math. Comput. Model, 39, 479-493 (2004) · Zbl 1065.92066
[47] Zhang, X.; Chen, L. S.; Neuman, A. U., The stage-structure predator-prey model and optimal harvesting policy, Math. Biosci., 101, 139-153 (2000)
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