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A stage-structured Holling mass defence predator-prey model with impulsive perturbations on predators. (English) Zbl 1117.92053

Summary: We consider a stage-structured Holling mass defence predator-prey model with time delay and impulsive transmitting on predators. Sufficient conditions which guarantee the global attractivity of pest-extinction periodic solutions and permanence of the system are obtained. We also prove that all solutions of the system are uniformly ultimately bounded. Our results provide a reliable tactic basis for practical pest management.

MSC:

92D40 Ecology
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations
37N25 Dynamical systems in biology
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[1] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[2] Bainov, D.; Simeonov, P., Impulsive Differential Equations: Periodic Solutions and Applications (1993), Longman · Zbl 0815.34001
[3] Barclay, H. J., Models for pest control using predator release, habitat management and pesticide release in combination, J. Appl. Ecol., 19, 337-348 (1982)
[4] Paneyya, J. C., A mathematical model of periodically pulse chemotherapy: tumor recurrence and metastasis in a competition environment, Bull. Math. Biol., 58, 425-447 (1996) · Zbl 0859.92014
[5] d’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. Biol., 179, 57-72 (2002) · Zbl 0991.92025
[6] Roberts, M. G.; Kao, R. R., The dynamics of an infectious disease in a population with birth pulse, Math. Biol., 149, 23-36 (2002) · Zbl 0928.92027
[7] Xiao, Y. N.; Chen, L. S., A ratio-dependent predator-prey model with disease in the prey, Appl. Math. Comput., 131, 397-414 (2002) · Zbl 1024.92017
[8] Xiao, Y. N.; Chen, L. S., An SIS epidemic model with stage structure and a delay, Acta Math. Appl. English Series, 16, 607-618 (2002) · Zbl 1035.34054
[9] Xiao, Y. N.; Chen, L. S.; Bosh, F. V.D., Dynamical behavior for stage-structured SIR infectious disease model, Nonlinear Anal.: RWA, 3, 175-190 (2002) · Zbl 1007.92032
[10] Xiao, Y. N.; Chen, L. S., On an SIS epidemic model with stage-structure, J. Syst. Sci. Complex., 16, 275-288 (2003) · Zbl 1138.92369
[11] Lu, Z. H.; Gang, S. J.; Chen, L. S., Analysis of an SI epidemic with nonlinear transmission and stage structure, Acta Math. Sci., 4, 440-446 (2003) · Zbl 1032.92030
[12] Zaghrout, A. A.S.; Attalah, S. H., Analysis of a model of stage-structured population dynamics growth with time state-dependent time delay, Appl. Math. Comput., 77 (1996) · Zbl 0848.92017
[13] Aiello, W. G.; Freedman, H. I.; Wu, J., Analysis of a model representing stage-structured population growth with state dependent time delay, SIAM J. Appl. Math., 52, 3 (1992) · Zbl 0760.92018
[14] Murray, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag Berlin Heidelberg, New York · Zbl 0682.92001
[15] Aiello, W. G.; Freedman, H. I., A time-delay model of single-species growth with stage-structured, Math. Biosci., 101, 139 (1990) · Zbl 0719.92017
[16] Aiello, W. G., The existence of nonoscillatory solutions to a generalized, nonautonomous,delay logistic equation, J. Math. Anal. Appl., 149, 114 (1990) · Zbl 0711.34091
[17] Freedman, H. I.; Gopalsamy, K., Global stability in time-delayed single species dynamics, Bull. Math. Biol., 48, 485 (1986) · Zbl 0606.92020
[18] Rosen, G., Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 49, 253 (1987) · Zbl 0614.92015
[19] Wangersky, P. J.; Cunningham, W. J., On time large equations of growth, Proc. Nat. Acad. Sci. USA, 42, 699 (1956) · Zbl 0072.37005
[20] Fisher, M. E.; Goh, B. S., Stability results for delay-recruitment models in population dynamics, J. Math. Biol., 19, 117 (1984) · Zbl 0533.92017
[21] Yang, Kuang, Delay Differential Equation with Application in Population Dynamics (1987), Academic Press: Academic Press NY
[22] Cull, P., Global stability for population models, Bull. Math. Biol., 43, 47-58 (1981) · Zbl 0451.92011
[23] Wang, W., Global behavior of an SEIRS epidemic model with delays, Appl. Math. Lett., 15, 423-428 (2002) · Zbl 1015.92033
[24] DeBach, P., Biological Control of Insect Pests and Weeds (1964), Rheinhold: Rheinhold New York
[25] DeBach, P.; Rosen, D., Biological Control by Natural Enemies (1991), Cambridge University Press: Cambridge University Press Cambrige
[26] Freedman, H. J., Graphical stability, enrichment, and pest control by a natural enemy, Math. Biosci., 31, 207-225 (1976) · Zbl 0373.92023
[27] Grasman, J.; Van Herwaarden, O. A., A two-component model of host-parasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control, Math. Biosci., 169, 207-216 (2001) · Zbl 0966.92026
[28] Caltagirone, L. E.; Doutt, R. L., Global behavior of an SEIRS epidemic model with delays, The history of the vedalia beetle importation to California and its impact on the development of biological control, Ann. Rev. Entomol., 34, 1-16 (1989)
[29] Liu, Xianning; Chen, Lansun, Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator Chaos, Soliton Fract., 16, 311-320 (2003) · Zbl 1085.34529
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