×

The dynamics of a Beddington-type system with impulsive control strategy. (English) Zbl 1142.34305

Summary: In this paper, by using the theories and methods of ecology and ordinary differential equation, a prey-predator system with Beddington-type functional response and impulsive control strategy is established. Conditions for the system to be extinct are given by using the theories of impulsive equation and small amplitude perturbation skills. It is proved that the system is permanent via the method of comparison involving multiple Liapunov functions. Furthermore, by using the method of numerical simulation, the influence of the impulsive control strategy on the inherent oscillation are investigated, which shows rich dynamics, such as period doubling bifurcation, crises, symmetry-breaking pitchfork bifurcations, chaotic bands, quasi-periodic oscillation, narrow periodic window, wide periodic window, period-halving bifurcation, etc. That will be useful for study of the dynamic complexity of ecosystems.

MSC:

34A37 Ordinary differential equations with impulses
92D40 Ecology
93C10 Nonlinear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. C., Theory of impulsive differential equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[2] Tang, S.; Xiao, Y.; Clancy, D., New modelling approach concerning integrated disease control and cost-effectivity, Nonlinear Anal, 63, 439-471 (2005) · Zbl 1078.92059
[3] Lu, Z.; Chi, X.; Chen, L., Impulsive control strategies in biological control of pesticide, J Theor Popul Biol, 64, 39-47 (2003) · Zbl 1100.92071
[4] Liu, B.; Zhang, Y.; Chen, L., Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control, Chaos, Solitons & Fractals, 22, 123-134 (2004) · Zbl 1058.92047
[5] Sun, J.; Qiao, F.; Wu, Q., Impulsive control of a financial model, J Phys Lett A, 335, 282-288 (2005) · Zbl 1123.91325
[6] Zhang, Y.; Xiu, Z.; Chen, L., Dynamics complexity of a two-prey one-predator system with impulsive effect, Chaos, Solitons & Fractals, 26, 131-139 (2005) · Zbl 1076.34055
[7] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math Comput Model, 26, 59-72 (1997) · Zbl 1185.34014
[8] Lenci, S.; Rega, G., Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation, Chaos, Solitons & Fractals, 11, 2453-2472 (2000) · Zbl 0964.70018
[9] Beddington, J. R., Mutual interference between parasites or predators and its effect on searching efficiency, J Animal Ecol, 44, 331-340 (1975)
[10] Cantrell, R. S.; Cosner, C., On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J Math Anal Appl, 257, 206-222 (2001) · Zbl 0991.34046
[11] Hwang, T. W., Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J Math Anal Appl, 281, 395-401 (2003) · Zbl 1033.34052
[12] Hwang, T. W., Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J Math Anal Appl, 290, 113-122 (2004) · Zbl 1086.34028
[13] Liu, Z.; Yuan, R., Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response, J Math Anal Appl, 296, 521-537 (2004) · Zbl 1051.34060
[14] Qui, Z.; Yu, J.; Zou, Y., The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response, Math Bios, 187, 175-187 (2004) · Zbl 1049.92039
[15] Dimitrov, D. T.; Kojouharov, V. H., Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, J Appl Math Comp, 162, 523-538 (2005) · Zbl 1057.92050
[16] Huisman, G.; Deboer, R. J., A formal derivation of the Beddington functional response, J Thero Biol, 185, 389-400 (1997)
[17] Bainov, D. D.; Simeonov, P. S., Impulsive differential equations: asymptotic properties of the solutions (1993), World Scientific: World Scientific Singapore · Zbl 0793.34011
[18] Wang, W., Computer algebra system and symbolic computation (2004), Gansu Sci-Tech Press: Gansu Sci-Tech Press Lanzhou, China, [in Chinese]
[19] Wang, W.; Lian, X., A new algorithm for symbolic integral with application, Appl Math Comput, 162, 949-968 (2005) · Zbl 1076.65022
[20] Wang, W.; Lin, C., A new algorithm for integral of trigonometric functions with mechanization, Appl Math Comput, 164, 71-82 (2005) · Zbl 1069.65149
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.