×

Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays. (English) Zbl 1154.34372

Summary: We study the existence and global attractivity of positive periodic solutions for impulsive predator-prey systems with dispersion and time delays. By using the method of coincidence degree theorem, a set of easily verifiable sufficient conditions are obtained for the existence of at least one strictly positive periodic solution, and by means of a suitable Lyapunov functional, the uniqueness and global attractivity of the positive periodic solution are presented. Some known results subject to the underlying systems without impulses are improved and generalized.

MSC:

34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
34K20 Stability theory of functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bainov, D. D.; Simeonov, P. S., Impulsive Differential Equations: Periodic Solutions and Applications (1993), Longman Scientific, Technical: Longman Scientific, Technical New York · Zbl 0815.34001
[2] Wang, L. L.; Li, W. T., Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response, J. Comput. Appl. Math., 162, 341-357 (2004) · Zbl 1076.34085
[3] Chen, S.; Zhang, J.; Yong, T., Existence of positive periodic solution for nonautonomous predator-prey system with diffusion and time delay, J. Comput. Appl. Math., 159, 375-386 (2003) · Zbl 1039.34061
[4] Xu, R.; Chaplain, M. A.J.; Davidson, F. A., Periodic solution of a Lotka-Volterra predator-prey model with dispersion and time delays, Appl. Math. Comput., 148, 537-560 (2004) · Zbl 1048.34119
[5] Zhen, J.; Ma, Zh.; Han, M., The existence of periodic solutions of the \(n\)-species Lotka-Volterra competition systems with impulsive, Chaos Solitons Fractals, 22, 181-188 (2004) · Zbl 1058.92046
[6] Zhang, J.; Chen, L.; Chen, X., Persistence and global stability for two-species nonautonomous competition Lotka-Volterra patch-system with time delay, Nonlinear Anal. TMA, 37, 1019-1028 (1999) · Zbl 0949.34060
[7] Xu, R.; Chen, L., Persistence and stability for a two-species ratio-dependent predator-prey system with time delay in a two-patch environment, Comput. Math. Appl., 40, 577-588 (2000) · Zbl 0949.92028
[8] Song, X. Y.; Chen, L. S., Persistence and global stability for nonautonomous predator-prey systems with diffusion and time delay, Comput. Math. Appl., 35, 33-40 (1998) · Zbl 0903.92029
[9] Li, J.; Shen, J., Periodic solutions of the Duffing equations with delay and impulses, Acta Math. Appl. Sinica, 28, 1, 124-133 (2005), (in Chinese)
[10] 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
[11] Liu, X.; Chen, L., Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos Solitons Fractals, 16, 311-320 (2003) · Zbl 1085.34529
[12] Li, Y.; Zhou, Q., Periodic solutions to ordinary differential equations with impulses, Sci. China, 36, 7, 778-790 (1993) · Zbl 0787.34015
[13] Dong, Y.; Erxin, Z., An application of coincidence degree continuation theorem in existence of solutions of impulsive differential equations, J. Math. Anal. Appl, 197, 875-889 (1996) · Zbl 0853.34010
[14] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berli · Zbl 0339.47031
[15] Yan, J.; Zhao, A., Oscillation and stability of linear impulsive delay differential equation, J. Math. Anal. Appl., 227, 187-194 (1998) · Zbl 0917.34060
[16] Wang, W.; Ma, Z., Harmless delays for uniform persistence, J. Math. Anal. Appl., 158, 256-268 (1991) · Zbl 0731.34085
[17] Barbalat, L., Systemes d’equations differentielles d’oscillations nonlineaires, Rev. Roumaine Math. Pures Appl., 4, 267-270 (1959) · Zbl 0090.06601
[18] Yoshizawa, T., Stability Theory by Lyapunov’s Second Method (1966), Math. Soc.: Math. Soc. Japan, Tokyo · Zbl 0144.10802
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.