zbMATH — the first resource for mathematics

Mathematics and dynamic analysis of an apparent competition community model with impulsive effect. (English) Zbl 1201.34018
Summary: By using the theories and methods of ecology and ordinary differential equation, an ecological model consisting of two preys and one predator with impulsive control strategy is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the prey-eradication periodic solution. It is proved that the system is permanent. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows rich dynamics, such as period-doubling bifurcation, period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises,etc. Moreover, the computation of the largest Lyapunov exponent demonstrates the chaotic dynamic behavior of the model. At the same time, we investigate the qualitative nature of the strange attractor by using Fourier spectra. All these results may be useful for the study of the dynamic complexity of ecosystems.

34A37 Ordinary differential equations with impulses
92D40 Ecology
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
Full Text: DOI
[1] Holt, R.D., Predation, apparent competition, and the structure of prey communities, Theor. popul. biol., 12, 197-229, (1977)
[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, Biol. control., 49, 605-626, (2004)
[8] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.C., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[9] Bainov, D.D.; Simeonov, P.C., Impulsive differential equations: asymptotic properties of the solutions, (1993), World Scientific Singapore · Zbl 0793.34011
[10] Wang, H.; Wang, W., The dynamical complexity of a ivler-type prey – predator system with impulsive effect, Chaos solitons fractals, 38, 1168-1176, (2008) · Zbl 1152.34310
[11] Wang, M.; Wang, X.; Lin, Y., Complicated dynamics of a predator – prey system with walt-type functional response and impulsive control strategy, Chaos solitons fractals, 37, 1427-1441, (2008) · Zbl 1142.34342
[12] Jiao, J.; Chen, L.; Li, L., Asymptotic behavior of solutions of second-order nonlinear impilsive differential equations, J. math. anal. appl., 331, 458-463, (2008) · Zbl 1135.34302
[13] Zhang, Y.; Xiu, Z.; Chen, L., Dynamic complexity of a two-prey one-predator system with impulsive effect, Chaos solitons fractals, 20, 131-139, (2005) · Zbl 1076.34055
[14] Sun, S.; Chen, L., Permanence and complexity of the eco-epidemiological model with impulsive perturbtion, Int. J. biomathematics, 1, 121-132, (2008) · Zbl 1166.92039
[15] Wang, F.; Hao, C.; Chen, L., Bifurcation and chaos in a monod – haldene type food chain chemostat with pulsed input and washout, Chaos solitons fractals., 32, 181-198, (2007) · Zbl 1130.92058
[16] 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
[17] Zhang, Y.; Chen, L., Chaos in three species food chain system with impulsive perturbations, Chaos solitons fractals, 20, 131-139, (2005)
[18] Wang, L.; Chen, L.; Nieto, J.J., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear anal. RWA, (2009)
[19] Yu, H.; Zhong, S.; Ye, M., Dynamic analysis of an ecological model with impulsive control strategy and distributed time delay, Math. comput. simulat., 80, 619-632, (2009) · Zbl 1178.92058
[20] Yu, H.; Zhong, S.; Ye, M.; Chen, W., Mathematical and dynamic analysis of an ecological model with an impulsive control strategyand distributed time delay, Math. comput. modell., 50, 1621635, (2009)
[21] Hwang, T.W., Uniqueness of limit cycles of the predator – prey system with the beddington – deangelis functional response, J. math. anal. appl., 290, 113-122, (2004) · Zbl 1086.34028
[22] Cantrell, R.S.; Consner, C., On the dynamics of predator – prey dodels with the beddington – deangelis functional response, J. math. anal. appl., 257, 206-222, (2001) · Zbl 0991.34046
[23] Grond, F.; Diebner, H.H.; Sahle, S.; Mathias, A., A robust, locally interpretable algorithm for Lyapunov exponents, Chaos solitions fractals, 16, 841-852, (2003)
[24] Sportt, J.G., Chaos and time-series analysis, (2003), Oxford University Press PP, pp. 116-117
[25] 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
[26] Lv, S.; Zhao, M., The dynamic complexity of a three species food chain model, Chaos solitons fractals, 37, 1469-1480, (2008) · Zbl 1142.92342
[27] 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)
[28] 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
[29] Masoller, C.; Sicaedi cshifino, A.C.; Romanelli, L., Charaterization of strange attractors of Lorenz model of general circulation of the atmospher, 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. 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.