×

Dynamics of an ecological model with impulsive control strategy and distributed time delay. (English) Zbl 1251.92049

Summary: In this paper, using the theories and methods of ecology and ordinary differential equations, an ecological model with an impulsive control strategy and a distributed time delay is defined. Using the theory of the impulsive equations, small-amplitude perturbations, and comparative techniques, a condition is identified which guarantees the global asymptotic stability of the prey-\((x)\) and predator-\((y)\) eradication periodic solutions. It is proved that the system is permanent. Furthermore, the influences of impulsive perturbations on the inherent oscillation are studied numerically, an oscillation which exhibits rich dynamics including period-halving bifurcation, chaotic narrow or wide windows, and chaotic crises. Computation of the largest Lyapunov exponent confirms the chaotic dynamic behavior of the model. All these results may be useful for study of the dynamic complexity of ecosystems.

MSC:

92D40 Ecology
34K45 Functional-differential equations with impulses
34K35 Control problems for functional-differential equations
37N25 Dynamical systems in biology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baek, H., Dynamic complexities of a three-species Beddington-DeAngelis system with impulsive control strategy, Acta Appl. Math., 110, 1, 23-38 (2010) · Zbl 1194.34087
[2] Baek, H., Species extinction and permanence of an impulsively controlled two-prey one-predator system with seasonal effects, Biosystems, 98, 7-18 (2009)
[3] Bainov, D. D.; Simeonov, P. S., Impulsive Differential Equations: Asymptotic Properties of the Solutions (1993), World Scientific: World Scientific Singapore · Zbl 0793.34011
[4] Beddington, J. R., Mutual interference between parasites or predator and its effect on searching efficiency, J. Anim. Ecol., 44, 331-340 (1975)
[5] Chen, C. W., The stability of an oceanic structure with T-S fuzzy models, Math. Comput. Simul., 80, 402-426 (2009) · Zbl 1174.86002
[6] Chen, C. W., Modeling and control for nonlinear structural systems via a NN-based approach, Expert Syst. Appl., 36, 4765-4772 (2009)
[7] Chen, C. W.; Chiang, W. L.; Hsiao, F. H., Stability analysis of T-S fuzzy models for nonlinear multiple time-delay interconnected systems, Math. Comput. Simul., 66, 523-537 (2004) · Zbl 1049.93556
[8] DeAngelis, D. L.; Goldstein, R. A.; Neill, R. V., A model for trophic interaction, Ecology, 56, 881-892 (1975)
[9] Georgescu, P.; Zhang, H.; Chen, L. S., Bifurcation of nontrivial periodic solution for an impulsively controlled pest management model, Appl. Math. Comput., 202, 675-687 (2008) · Zbl 1151.34037
[10] 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
[11] Grond, F.; Diebner, H. H.; Sahle, S.; Mathias, A., A robust, locally interpretable algorithm for Lyapunov exponents, Chaos Solitons Fract., 16, 841-852 (2003)
[12] 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
[13] Holt, R. D., Predation, apparent competition, and the structure of prey communities, Theor. Popul. Biol., 12, 197-229 (1977)
[14] Holt, R. D., On the evolutionary stability of sink populations, Evol. Ecol., 11, 723-731 (1997)
[15] Ji, C. Y.; Jiang, D. Q.; Shi, N. Z., Analysis of a predator-prey model with modified Leslies-Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl., 359, 482-498 (2009) · Zbl 1190.34064
[16] Jiao, J. J.; Chen, L. S., A pest management SI model with biological and chemical control concern, Appl. Math. Comput., 196, 1018-1026 (2006) · Zbl 1104.92054
[17] Jiao, J.; Meng, X.; Chen, L., A new stage structured predator-prey Gompertz model with time delay and impulsive perturbations, Appl. Math. Comput., 196, 705-719 (2008) · Zbl 1131.92064
[18] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. C., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[19] Li, Z. Q.; Wang, W. M.; Wang, H. L., The dynamics of a Beddington-type system with impulsive control strategy, Chaos Solitons Fract., 29, 1229-1239 (2006) · Zbl 1142.34305
[20] Lin, M. L.; Chen, C. W., Application of fuzzy models for the monitoring of ecologically sensitive ecosystems in a dynamic semi-arid landscape from satellite imagery, Eng. Comput., 27, 5-19 (2010) · Zbl 1284.92115
[21] Lv, S. J.; Zhao, M., The dynamic complexity of a three-species food chain model, Chaos Solitons Fract., 37, 1469-1480 (2008) · Zbl 1142.92342
[22] Lv, S. J.; Zhao, M., The dynamic complexity of a host-parasitoid model with a lower bound for the host, Chaos Solitons Fract., 36, 911-919 (2008)
[23] Masoller, C.; Sicaedi-Schifino, A. C.; Romanelli, L., Characterization of strange attractors of the Lorenz model of the general circulation of the atmosphere, Chaos Solitons Fract., 6, 357-366 (1995) · Zbl 0905.58023
[24] Meng, X. Z.; Chen, L. S., The dynamics of a new SIR epidemic model concerning pulse vaccination strategy, Appl. Math. Comput., 197, 582-597 (2008) · Zbl 1131.92056
[25] 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
[26] Pei, Y.; Liu, S.; Liu, C.; Chen, L., The dynamics of an impulsive delay SI model with variable coefficients, Appl. Math. Model., 33, 2766-2776 (2009) · Zbl 1205.34094
[27] 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
[28] Shi, R. Q.; Chen, L. S., Staged-structured Lotka-Volterva predator-prey models for pest management, Appl. Math. Comput., 203, 258-265 (2008) · Zbl 1152.92029
[29] Song, X.; Guo, H., Extinction and permanence of a kind of pest-predator model with impulsive effect and infinite delay, J. Korean Math. Soc., 44, 327-342 (2007) · Zbl 1143.34052
[30] Song, X. Y.; Li, Y. Y., Dynamic complexities of a Holling II two-prey one-predator system with impulsive effect, Chaos Solitons Fract., 33, 463-478 (2007) · Zbl 1136.34046
[31] Sportt, J. G., Chaos and Time-Series Analysis (2003), Oxford University Press, pp. 116-117 · Zbl 1012.37001
[32] Yeh, K.; Chen, C. Y.; Chen, C. W., Robustness design of time-delay fuzzy systems using fuzzy Lyapunov method, Appl. Math. Comput., 205, 568-577 (2008) · Zbl 1152.93040
[33] Yu, H. G.; Zhao, M.; Lv, S. J.; Zhu, L. L., Dynamic complexity of a parasitoid-host-parasitoid ecological model, Chaos Solitons Fract., 39, 39-48 (2009) · Zbl 1197.37127
[34] Yu, H. G.; Zhong, S. M.; Agarwal, R. P., Mathematics and dynamic analysis of an apparent competition community model with impulsive effect, Math. Comput. Model., 52, 25-36 (2010) · Zbl 1201.34018
[35] Yu, H. G.; Zhong, S. M.; Agarwal, R. P., Mathematics analysis and chaos in an ecological model with an impulsive control strategy, Commun. Nonlinear Sci. Numer. Simul., 16, 776-786 (2011) · Zbl 1221.37207
[36] Yu, H. G.; Zhong, S. M.; Ye, M., Dynamic analysis of an ecological model with impulsive control strategy and distributed time delay, Math. Comput. Simul., 80, 619-632 (2009) · Zbl 1178.92058
[37] Zhang, Y. J.; Liu, B.; Chen, L. S., Extinction and permanence of a two-prey one-predator system with impulsive effect, Math. Med. Biol., 20, 309-325 (2003) · Zbl 1046.92051
[38] Zhao, M.; Zhang, L., Permanence and chaos in a host-parasitoid model with prolonged diapause for the host, Commun. Nonlinear Sci. Numer. Simul., 14, 4197-4203 (2009)
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.