×

zbMATH — the first resource for mathematics

A food chain model with impulsive perturbations and Holling IV functional response. (English) Zbl 1066.92061
Summary: We investigate a three trophic level food chain system with Holling IV functional responses and periodic constant impulsive perturbations of the top predator. Conditions for extinction of the predator are given. By using the Floquet theory of impulsive equations and small amplitude perturbation skills, we consider the local stability of predator eradication periodic solutions. Further, influences of the impulsive perturbations on the inherent oscillations are studied numerically, which show the rich dynamics in the positive octant.

MSC:
92D40 Ecology
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Holling, C.S., The functional response of predator to prey density and its role in mimicry and population regulation, Mem ent sec can, 45, 1-60, (1965)
[2] Andrews, J.F., A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol bioeng, 10, 707-723, (1968)
[3] Sugie, W.; Howell, J.A., Kinetics of phenol by washed cell, Biotechnol bioeng, 23, 2039-2049, (1980)
[4] Klebanoff, A.; Hastings, A., Chaos in three species food chains, J math biol, 32, 427-451, (1994) · Zbl 0823.92030
[5] Xianning, L.; Lansun, C., Complex dynamics of Holling type II lokta-voltrra predator-prey system with impulsive perturbations on the predator, Chaos, solitons & fractals, 16, 311-320, (2003) · Zbl 1085.34529
[6] Bing, L.; Yujina, Z.; Lansun, C., Dynamic complexities of a Holling I predator-prey model concerning biological and chemical control, Chaos, solitons & fractals, 22, 123-134, (2004) · Zbl 1058.92047
[7] Shuwen, Z.; Lingzhen, D.; Lansun, C., The study of predator-prey with defensive ablity of prey and impulsive perturbations on the predator, Chaos, solitons & fractals, 23, 631-643, (2005) · Zbl 1081.34041
[8] Funasaki, E.; Kot, M., Invasion and chaos in a periodically pulsed mass-action chemostat, Theoret populat biol, 44, 203-224, (1993) · Zbl 0782.92020
[9] 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
[10] Bainov, D.; Simeonor, P., Impulsive differential equations; periodic solutions and applications, Pitman monogr surr pure appl math, 66, (1993)
[11] Shulgin, B.; Stone, L.; Agur, I., Pulse vaccination strategy in the SIR epidemic model, Bull math biol, 60, 1-26, (1998) · Zbl 0941.92026
[12] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math comput modell, 26, 59-72, (1997) · Zbl 1185.34014
[13] Lakmeche, A.; Arino, O., Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn contin discr impuls syst, 7, 265-287, (2000) · Zbl 1011.34031
[14] 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
[15] 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
[16] Sanyi, T.; Lansun, C., Density-dependent birth rate,birth pulse and their population dynamic consequences, J math biol, 44, 185-199, (2002) · Zbl 0990.92033
[17] Laksmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore
[18] Sanyi, T.; Lansun, C., Multiple attractors in stage-structured population models with birth pulses, Bull math biol, 65, 479-495, (2003) · Zbl 1334.92371
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.