×

Chaos in three species food chain system with impulsive perturbations. (English) Zbl 1066.92060

Summary: We investigate a three species food chain system with periodic constant impulsive perturbations of mid-level predator. Conditions for extinction of lowest-level prey and top predator are given. By using the Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability of lowest-level prey and top predator eradication periodic solutions. Further, influences of the impulsive perturbations on the inherent oscillations are studied numerically, which show the rich dynamics (for example: period doubling, period halfing, non-unique dynamics) in the positive octant. The dynamical behavior is found to be very sensitive to the parameter values and initial value.

MSC:

92D40 Ecology
34A37 Ordinary differential equations with impulses
37N25 Dynamical systems in biology
34D05 Asymptotic properties of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Van Lenteren, J. C.; Woets, J., Biological and integrated pest control in greenhouses, Am. Ann. Ent., 33, 239-250 (1988)
[2] Van Lenteren, J. C., Measures of success in biological of anthropoids by augmentation of natural enemies, (Wratten, S.; Gurr, G., Measures of success in biological control (2000), Kluwer Academic publishers: Kluwer Academic publishers Dordrecht), 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: World Scientific Singapore
[6] Bainov, D.; Simeonor, P., Impulsive differential equations: periodic solutions and applications, Pitman Monogr. Surv. Pure Appl. Math., 66 (1993) · Zbl 0815.34001
[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. Comput. Model., 26, 59-72 (1997) · Zbl 1185.34014
[9] Funasaki, E.; Kot, M., Invasion and chaos in a periodically pulsed mass-action chemostat, Theoret. Populat. 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, Dyn. Contin., 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 & Fractals, 18, 229-239 (2003) · Zbl 1068.92044
[17] Ma, Z. E., Mathematical modelling and study of species ecology (Chinese) (1994), Anhui Education Publishing Company
[18] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. C., Theory of impulsive differential equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[19] 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
[20] May, R. M., Biological populations with non-overlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647 (1974)
[21] May, R. M.; Oster, G. F., Bifurcations and dynamic complexity in simple ecological models, Am. Natur., 110, 573-599 (1976)
[22] Collet, P.; Eckmann, J. A., Iterated maps of the interval as dynamical system (1980), Birkhauser: Birkhauser Boston
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.