A Holling II functional response food chain model with impulsive perturbations. (English) Zbl 1086.34043

A food chain system with Holling-type II functional response and periodic constant impulsive perturbations is considered. Conditions for the extinction of the predator as a pest are given. Local stability of predator eradication periodic solution is studied. Influences of the impulsive perturbation on the inherent oscillation is studied numerically in the positive octant. Rich dynamic behavior appeared. The dynamic behavior is found to be sensitive to the parameter values and to the initial values. The reviewer’s opinion is that despite its rich mathematical content, the language of this paper is poor.


34C60 Qualitative investigation and simulation of ordinary differential equation models
34A37 Ordinary differential equations with impulses
37N25 Dynamical systems in biology
34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
92D25 Population dynamics (general)
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[1] Van Lenteren, J.C.; Woets, J., Biological and integrated pest control in greenhouses, Ann ann ent, 33, 239-250, (1988)
[2] Van Lenteren, J.C., Measures of success in biological of anthropoids by augmentation of natural enemies, (), 77-89
[3] Brauer, F.; Soudack, A.C., Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, J math biol, 12, 101-114, (1981) · Zbl 0482.92015
[4] Brauer, F.; Soudack, A.C., Constant-rate stocking of predator-prey systems, J math biol, 11, 1-14, (1981) · Zbl 0448.92020
[5] Laksmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore
[6] Bainov D, Simeonor P. Impulsive differential equations: periodic solutions and applications. Pitman Monographs and Surreys in Pure and Applied Mathematics. 1993;66
[7] Shulgin, B.; Stone, L.; Agur, I., Pulse vaccination strategy in the SIR epidemic model, Bull math biol, 60, 1-26, (1998) · Zbl 0941.92026
[8] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math compu modell, 26, 59-72, (1997) · Zbl 1185.34014
[9] Funasaki, E.; Kot, M., Invasion and chaos in a periodically pulsed mass-action chemostat, Theor popul biol, 44, 203-224, (1993) · Zbl 0782.92020
[10] Lakmeche, A.; Arino, O., Bifurcation of non-trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. dynamics of continuous, Discrete impulsive syst, 7, 265-287, (2000) · Zbl 1011.34031
[11] Panetta, J.C., A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment, Bull math biol, 58, 425-447, (1996) · Zbl 0859.92014
[12] Roberts, M.G.; Kao, R.R., The dynamics of an infectious disease in a population with birth pulses, Math biosci, 149, 23-36, (1998) · Zbl 0928.92027
[13] Tang, S.Y.; Chen, L.S., Density-dependent birth rate,birth pulse and their population dynamic consequences, J math biol, 44, 185-199, (2002) · Zbl 0990.92033
[14] Klebanoff, A.; Hastings, A., Chaos in three species food chains, J math biol, 32, 427-451, (1994) · Zbl 0823.92030
[15] Hsu, S.B.; Hwang, T.W.; Kuang, Y., A ratio-dependent food chain model and its applications to biological control, Math biosci, 181, 55-83, (2003) · Zbl 1036.92033
[16] Gakkhar, S.; Naji, M.A., Order and chaos in predator to prey ratio-dependent food chain, Chaos, solitons & fractal., 18, 229-239, (2003) · Zbl 1068.92044
[17] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.C., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[18] May, R.M., Biological population with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-657, (1974)
[19] May, R.M.; Oster, G.F., Bifurcations and dynamic complexity in simple ecological models, Am nature, 110, 573-599, (1976)
[20] Collet, P.; Eeckmann, J.P., Iterated maps of the inter val as dynamical systems, (1980), Birkhauser Boston
[21] Venkatesan, A.; Parthasarathy, S.; Lakshmanan, M., Occurrence of multiple period-doubling bifurcation route to chaos in periodically pulsed chaotic dynamical systems, Chaos, solitons & fractals, 18, 891-898, (2003) · Zbl 1073.37038
[22] Vandermeer, J.; Stone, L.; Blasius, B., Categories of chaos and fractal basin boundaries in forced predator-prey models, Chaos, solitons & fractals, 12, 265-276, (2001) · Zbl 0976.92033
[23] Neubert, M.G.; Caswell, H., Density-dependent vital rates and their population dynamic consequences, J math biol, 41, 103-121, (2000) · Zbl 0956.92029
[24] Wikan, A., From chaos to chaos. an analysis of a discrete age-structured prey-predator model, J math biol, 43, 471-500, (2001) · Zbl 0996.92031
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