A delayed stage-structured Holling II predator-prey model with mutual interference and impulsive perturbations on predator. (English) Zbl 1198.34133

Summary: We investigate a delayed stage-structured Holling II predator-prey model with mutual interference and impulsive perturbations on predator. Sufficient conditions of the global attractivity of prey-extinction periodic solution and the permanence of the system are obtained. We also prove that all solutions of the system are uniformly ultimately bounded. Our results provide reliable tactical basis for the practical pest management.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.


34K12 Growth, boundedness, comparison of solutions to functional-differential equations
92D25 Population dynamics (general)
34K45 Functional-differential equations with impulses
34K60 Qualitative investigation and simulation of models involving functional-differential equations
Full Text: DOI


[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, vol. 66 (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] Hui, Jing; Zhu, Deming, Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos, Solitons & Fractals, 29, 1, 233-251 (2006) · Zbl 1095.92067
[8] Xiao, Y. N.; Chen, L. S., An SIS epidemic model with stage structure and a delay, Acta Math Appl Engl Ser, 16, 607-618 (2002) · Zbl 1035.34054
[9] Gao, Shujing; Chen, Lansun, Dynamic complexities in a single-species discrete population model with stage structure and birth pulses, Chaos, Solitons & Fractals, 24, 4, 1013-1023 (2005) · Zbl 1061.92059
[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] Liu, Zhijun; Tan, Ronghua, Impulsive harvesting and stocking in a MonodCHaldane functional response predatorCprey system, Chaos, Solitons & Fractals, 34, 2, 454-464 (2007) · Zbl 1127.92045
[12] Hethcote, H., The mathematics of infectious disease, SIAM Rev, 42, 599-653 (2002) · Zbl 0993.92033
[13] 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
[14] 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
[15] Murray, J. D., Mathematical biology (1989), Springer-Verlag: Springer-Verlag Berlin Heidelberg (NY) · Zbl 0682.92001
[16] 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
[17] 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
[18] Freedman, H. I.; Gopalsamy, K., Global stability in time-delayed single species dynamics, Bull Math Biol, 48, 485 (1986) · Zbl 0606.92020
[19] Rosen, G., Time delays produced by essential nonlinearity in population growth models, Bull Math Biol, 49, 253 (1987) · Zbl 0614.92015
[20] Wangersky, P. J.; Cunningham, W. J., On time large equations of growth, Proc Nat Acad Sci USA, 42, 699 (1956) · Zbl 0072.37005
[21] Fisher, M. E.; Goh, B. S., Stability results for delay-recruitment models in population dynamics, J Math Biol, 19, 117 (1984) · Zbl 0533.92017
[22] Yang, Kuang, Delay differential equation with application in population dynamics (1987), Academic Press: Academic Press NY, p. 67-70
[23] Cull, P., Global stability for population models, Bull Math Biol, 43, 47-58 (1981) · Zbl 0451.92011
[24] Jiao, Jianjun; Meng, Xinzhu; Chen, Lansun, Global attractivity and permanence of a stage-structured pest management SI model with time delay and diseased pest impulsive transmission, Chaos, Solitons & Fractals, 38, 3, 658-668 (2008) · Zbl 1146.34322
[25] DeBach, P., Biological control of insect pests and weeds (1964), Rheinhold: Rheinhold New York
[26] DeBach, P.; Rosen, D., Biological control by natural enemies (1991), Cambridge University Press: Cambridge University Press Cambrige
[27] Freedman, H. J., Graphical stability, enrichment, and pest control by a natural enemy, Math Biosci, 31, 207-225 (1976) · Zbl 0373.92023
[28] 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
[29] 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)
[30] Liu, Xianning; Chen, Lansun, Compex 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
[31] Liu, B.; Zhang, Y. J.; Chen, L. S., Dynamical behavior of a Lotka-Volterra predator-prey model concerning integrated pest management, Chaos, Solitons & Fractals, 6, 123-134 (2004) · Zbl 1058.92047
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.