zbMATH — the first resource for mathematics

Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations. (English) Zbl 1097.34038
The paper deals with a predator-prey model incorporating Holling-type-IV functional response. The basic aim of the authors is to analyze the effects of periodic variation in the intrinsic growth rate of the prey and impulsive perturbations on the predator.
The behaviour of the system is discussed. A bifurcation diagram is obtained for selected range of impulsive perturbations and periodic forcing as biological parameters. A number of simulations describing the complex dynamics and quasiperiodic oscillatory states and chaotic behaviour are given.

34C60 Qualitative investigation and simulation of ordinary differential equation models
34A37 Ordinary differential equations with impulses
92D25 Population dynamics (general)
34C28 Complex behavior and chaotic systems of ordinary differential equations
Full Text: DOI
[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, J.; Howell, J.A., Kineties of phenol oxidation by washed cell, Biotechnol bioeng, 23, 2039-2049, (1980)
[4] Tang, S.Y.; Chen, L.S., Quasi-periodic solutions and chaos in a periodically forced predator-prey model with age structure for predator, Int. J. bifur. chaos, 13, 4, 973-980, (2003) · Zbl 1063.37586
[5] Sabin, G.C.W.; Summer, D., Chaos in a periodic forced predator-prey ecosystem model, Math biosci, 113, 91-113, (1993) · Zbl 0767.92028
[6] Rinaldi, S.; Muratori, S.; Kuznetsov, Yu.A., Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities, Bull math biol, 55, 15-35, (1993) · Zbl 0756.92026
[7] Gakkhar, S.; Naji, R.K., Seasonally perturbed prey-predator system with predator-dependent functional response, Chaos, solitons & fractals, 18, 1075-1083, (2003) · Zbl 1068.92045
[8] Cushing, J.M., Periodic time-dependent predator-prey systems, SIAM J appl math, 10, 384-400, (1977) · Zbl 0348.34031
[9] Cushing, J.M., Periodic Kolmogorov systems, SIAM J math anal, 13, 811-827, (1987) · Zbl 0506.34039
[10] Bardi, M., Predator-prey models in periodically fluctuating environments, J math biol, 12, 127-140, (1981) · Zbl 0466.92019
[11] Van Lentern, J.C., Environmental manipulation advantageous to natural enemies of pests, (), 123-166
[12] Van lenteren, J.C., Integrate pest management in protected crops, (), 311-320
[13] 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
[14] 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
[15] Shulgin, B.; Stone, L.; Agur, Z., Pulse vaccination strategy in the SIR epidemic model, Bull math biol, 60, 1-26, (1998) · Zbl 0941.92026
[16] D’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic model, Math comput modell, 26, 59-72, (1997) · Zbl 0991.92025
[17] Panetta, J.C., A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment, Bull math biol, 58, 425-447, (1996) · Zbl 0859.92014
[18] Lakmeche, A.; Arino, O., Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn continuous discrete and impulsive syst, 7, 165-187, (2000) · Zbl 1011.34031
[19] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math comput modell, 26, 59-72, (1997) · Zbl 1185.34014
[20] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.C., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[21] Bainov, D.D.; Simeonov, D.D., Impulsive differential equations: periodic solutions and applications, (1993), Longman England · Zbl 0793.34011
[22] Ruan, S.; Xiao, D., Global analysis in a predator-prey system with non-monotonic functional response, SIAM J appl math, 61, 1445-1472, (2001) · Zbl 0986.34045
[23] Zhang, S.W.; Chen, L.S., The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, solitons & fractals, 23, 311-320, (2005)
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.