×

Mathematical and dynamic analysis of an ecological model with an impulsive control strategy and distributed time delay. (English) Zbl 1185.37203

Summary: On the basis of the theories and methods of ecology and ordinary differential equations, an ecological model with an impulsive control strategy and distributed time delay is established. By using impulsive equation theories, small amplitude perturbation skills and the comparison technique, we get the condition which guarantees the global asymptotical stability of the prey \((x)\) and predator \((y)\) eradication periodic solutions. Further, the influences of impulsive perturbations on the inherent oscillations are studied numerically; these show rich dynamics features, such as period-halving bifurcation, a chaotic band, a periodic window, chaotic crises, etc. Moreover, the computation of the largest Lyapunov exponent shows the chaotic dynamic behavior of the model. Meanwhile, we investigate the qualitative nature of the strange attractor by using Fourier spectra. All of these results may be useful in the study of the dynamic complexity of ecosystems.

MSC:

37N25 Dynamical systems in biology
34A37 Ordinary differential equations with impulses
92D25 Population dynamics (general)
34D05 Asymptotic properties of solutions to ordinary differential equations
92D40 Ecology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Holt, R. D., Predation, apparent competition, and the structure of prey communities, Theor. Popul. Biol., 12, 197-229 (1997)
[2] Holt, R. D., On the evolutionary stability of sink populations, Evol. Ecol., 11, 723-731 (1997)
[3] Holt, R. D., Spatial heterogeneity, indirect interactions, and the coexistence of prey species, Am. Nat., 124, 377-406 (1984)
[4] Holt, R. D.; Lawton, J. H., The ecological consequences of shared nature enemies, Ann. Rev. Ecol. Syst., 25, 495-520 (1994)
[5] Rand, T. A.; Louda, S. M., Exotic weed invasion increases the susceptibility of native plants attack by a biocontrol herbivore, Ecology, 85, 1548-1554 (2004)
[6] Koss, A. M.; Snyder, W. E., Alternative prey disrupt biocontrol by a guild of generalist predators, Biol. Control., 32, 243-251 (2005)
[7] Harmon, J. P.; Andow, D. A., Indirect effects between shared prey, predictions for biological control, Biocontrol., 49, 605-626 (2004)
[8] Liu, Z.; Chi, X.; Chen, L., The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Math. Comput. Modelling, 36, 1039-1057 (2002) · Zbl 1023.92026
[9] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math. Comput. Modelling, 26, 59-72 (1997) · Zbl 1185.34014
[10] Georgescu, P.; Morosanu, G., Impulsive perturbations of three-trophic prey-dependent food chain system, Math. Comput. Modelling, 48, 975-997 (2008) · Zbl 1187.34071
[11] Zhou, Y.; Liu, H., Stability of periodic solutions of an SIS model with pulse vaccination, Math. Comput. Modelling, 38, 299-308 (2003) · Zbl 1045.92042
[12] Georgescu, P.; Zhang, H.; Chen, L., Bifurcation of nontrivial periodic solution for an impulsively controlled pest management model, Appl. Math. Comput., 202, 675-687 (2008) · Zbl 1151.34037
[13] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[14] Bainov, D. D.; Simeonov, P. S., Impulsive Differential Equations: Periodic Solution and Application (1993), World Scientific: World Scientific Singapore · Zbl 0793.34011
[15] Pei, Y.; Liu, S.; Li, C.; Chen, L., The dynamics of an impulsive delay SI model with variable coefficients, Appl. Math. Modelling, 33, 2766-2776 (2009) · Zbl 1205.34094
[16] Guo, H.; Chen, L., The effects of impulsive harvest on a predator-prey system with distributed time delay, Commun. Nonlinear Sci. Numer. Simul., 14, 2301-2309 (2009) · Zbl 1221.34218
[17] Song, X.; Guo, H., Extinction and permanence of a kind of pest-predator models impulsive effect and infinite delay, J. Korean Math. Soc., 44, 327-342 (2007) · Zbl 1143.34052
[18] Meng, X.; Jiao, J.; Chen, L., The dynamics of an age structured predator-prey model with disturbing pulse and time delays, Nonlinear Anal., 9, 547-561 (2008) · Zbl 1142.34054
[19] Jiao, J.; Meng, X.; Chen, L., A new stage structured predator-prey Gomportz model with time delay and impulsive perturbations, Appl. Math. Comput., 196, 705-719 (2008) · Zbl 1131.92064
[20] Li, Z.; Wang, W.; Wang, H., The dynamics of a Beddington-type system with impulsive control strategy, Chaos Solitons Fractals, 29, 1229-1239 (2006) · Zbl 1142.34305
[21] Grond, F.; Diebner, H. H.; Sahle, S.; Mathias, A., A robust, locally interpretable algorithm for Lyapunov exponents, Chaos Solitons Fractals, 16, 841-852 (2003)
[22] Sportt, J. G., Chaos and Time-series Analysis (2003), Oxford University Press, pp. 116-117
[23] Rosenstein, M. T.; Collins, J. J.; De Luca, C. J., A practical method for calculating largest Lyapunov exponents from small data sets, Physica D, 65, 117-134 (1993) · Zbl 0779.58030
[24] Lv, S.; Zhao, M., The dynamic complexity of a three species food chain model, Chaos Solitons Fractals., 37, 1469-1480 (2008) · Zbl 1142.92342
[25] Lv, S.; Zhao, M., The dynamic complexity of a host-parasitoid model with a lower bound for the host, Chaos Solitons Fractals, 36, 911-919 (2008)
[26] Yu, H.; Zhao, M.; Lv, S.; Zhu, L., Dynamic complexity of a parasitoid-host-parasitoid ecological model, Chaos Solitons Fractals, 39, 39-48 (2009) · Zbl 1197.37127
[27] Masoller, C.; Sicaedi Schifino, A. C.; Romanelli, L., Characterization of strange attractors of Lorenz model of general circulation of the atmosphere, Chaos Solitons Fractals, 6, 357-366 (1995) · Zbl 0905.58023
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.