Permanence of a discrete multispecies Lotka-Volterra competition predator-prey system with delays. (English) Zbl 1156.39302

Summary: We propose a discrete multispecies Lotka-Volterra competition predator-prey system with delays. For general nonautonomous case, sufficient conditions are established for the permanence of the system.


39A12 Discrete version of topics in analysis
92D25 Population dynamics (general)
34D40 Ultimate boundedness (MSC2000)
Full Text: DOI


[1] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 75, 1530-1535 (1992)
[2] Chen, F. D., Permanence and global stability of nonautonomous Lotka-Volterra system with predator-prey and deviating arguments, Appl. Math. Comput., 173, 1082-1100 (2006) · Zbl 1121.34080
[3] Chen, F. D., Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems, Appl. Math. Comput., 182, 3-12 (2006) · Zbl 1113.92061
[4] X Chen, X.; D Chen, F., Stable periodic solution of discrete periodic Lotka-Volterra competition system with a feedback control, Appl. Math. Comput., 181, 1446-1454 (2006) · Zbl 1106.39003
[5] Chen, Y. M.; Zhou, Z., Stable periodic solution of a discrete periodic Lotka-Volterra competition system, J. Math. Anal. Appl., 277, 358-366 (2003) · Zbl 1019.39004
[6] Crone, E. E., Delayed density dependence and the stability of interacting populations and subpopulations, Theor. Popul. Biol., 51, 67-76 (1997) · Zbl 0882.92025
[7] Freedman, H. I.; Ruan, S., Uniform persistence in functional differential equations, J. Differential Equations, 115, 173-192 (1995) · Zbl 0814.34064
[8] Hofbauer, J.; Hutson, V.; Jansen, W., Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol., 25, 553-570 (1987) · Zbl 0638.92019
[9] Hofbauer, J.; Sigmund, K., Dynamical Systems and the Theory of Evolution (1988), Cambridge University Press: Cambridge University Press Cambridge, UK
[10] Hutson, V.; Moran, W., Persistence of species obeying difference equations, J. Math. Biol., 15, 203-213 (1982) · Zbl 0495.92015
[11] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[12] Liao, X. Y.; Zhou, S. F.; Chen, Y. M., Permanence for a discrete time Lotka-Volterra type food-chain model with delays, Appl. Math. Comput., 186, 279-285 (2007) · Zbl 1120.92046
[13] Lu, Z.; Takeuchi, Y., Permanence and global attractivity for competitive Lotka-Volterra systems with delay, Nonlinear Anal., 22, 847-856 (1994) · Zbl 0809.92025
[14] Lu, Z.; Wang, W., Permanence and global attractivity for Lotka-Volterra difference systems, J. Math. Biol., 39, 269-282 (1999) · Zbl 0945.92022
[15] Muroya, Y., Persistence and global stability for discrete models of nonautonomous Lotka-Volterra type, J. Math. Anal. Appl., 273, 492-511 (2002) · Zbl 1033.39013
[16] Saito, Y.; Hara, T.; Ma, W., Harmless delays for permanence and impersistence of a Lotka-Volterra discrete predator-prey system, Nonlinear Anal., 50, 703-715 (2002) · Zbl 1005.39013
[17] Saito, Y.; Ma, W.; Hara, T., A necessary and sufficient condition for permanence of a Lotka-Volterra discrete system with delays, J. Math. Anal. Appl., 256, 162-174 (2001) · Zbl 0976.92031
[18] Takeuchi, Y., Global Dynamical Properties of Lotka-Volterra Systems (1996), World Scientific: World Scientific Singapore · Zbl 0844.34006
[19] Turchin, P., Chaos and stability in rodent population dynamics: evidence from nonlinear time series analysis, Oikos, 68, 167-182 (1993)
[20] Turchin, P.; Taylor, A. D., Complex dynamics in ecological time series, Ecology, 73, 289-305 (1992)
[21] Wang, W.; Ma, Z., Harmless delays for uniform persistence, J. Math. Anal. Appl., 158, 256-268 (1991) · Zbl 0731.34085
[22] Wang, W. D.; Mulone, G.; Salemi, F.; Salone, V., Global stability of discrete population models with time delays and fluctuating environment, J. Math. Anal. Appl., 264, 147-167 (2001) · Zbl 1006.92025
[23] Yang, X., Uniform persistence and periodic solutions for a discrete predator-prey system with delays, J. Math. Anal. Appl., 316, 161-177 (2006) · Zbl 1107.39017
[24] Zeng, X. Y.; Shi, B.; Gai, M. J., A discrete periodic Lotka-Volterra system with delays, Comput. Math. Appl., 47, 491-500 (2004) · Zbl 1067.39024
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