Permanence in a discrete Lotka-Volterra competition model with deviating arguments. (English) Zbl 1156.39300

Chen, Fengde

Nonlinear Anal., Real World Appl. 9, No. 5, 2150-2155 (2008).

Summary: We propose a discrete Lotka-Volterra competition model with deviating arguments, sufficient conditions which ensure the permanence of the system are obtained.

MSC:

39A12	Discrete version of topics in analysis
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25	Periodic solutions to ordinary differential equations
92D25	Population dynamics (general)

Keywords:

nonautonomous; Lotka-Volterra competition model; discrete; permanence

References:

[1]	R.P. Agarwal, Difference Equations and Inequalities: Theory, Method and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker, New York, 2000.
[2]	Chen, F.D., On a periodic multi-species ecological model, Appl. math. comput., 171, 1, 492-510, (2005) · Zbl 1080.92059
[3]	Chen, F.D., Positive periodic solutions of neutral lotka – volterra system with feedback control, Appl. math. comput., 162, 3, 1279-1302, (2005) · Zbl 1125.93031
[4]	Chen, F.D., On a nonlinear non-autonomous predator – prey model with diffusion and distributed delay, J. comput. appl. math., 180, 1, 33-49, (2005) · Zbl 1061.92058
[5]	Chen, F.D., Permanence and global attractivity of a discrete multispecies lotka – volterra competition predator – prey systems, Appl. math. comput., 182, 1, 3-12, (2006) · Zbl 1113.92061
[6]	Chen, F.D., Permanence in nonautonomous multi-species predator – prey system with feedback controls, Appl. math. comput., 173, 2, 694-709, (2006) · Zbl 1087.92059
[7]	Chen, F.D., Permanence of a delayed nonautonomous gilpin – ayala competition model, Appl. math. comput., 179, 1, 55-66, (2006) · Zbl 1096.92041
[8]	Chen, F.D., Some new results on the permanence and extinction of nonautonomous gilpin – ayala type competition model with delays, Nonlinear anal. real world appl., 7, 5, 1205-1222, (2006) · Zbl 1120.34062
[9]	Chen, F.D., Average conditions for permanence and extinction in nonautonomous gilpin – ayala competition model, Nonlinear anal. real world appl., 7, 4, 895-915, (2006) · Zbl 1119.34038
[10]	Chen, F.D., Note on the permanence of a competitive system with infinite delay and feedback controls, Nonlinear anal. real world appl., 8, 2, 680-687, (2007) · Zbl 1152.34366
[11]	Chen, F.D.; Shi, C.L., Global attractivity in an almost periodic multi-species nonlinear ecological model, Appl. math. comput., 180, 1, 376-392, (2006) · Zbl 1099.92069
[12]	Chen, F.D.; Shi, J.L., Periodicity in a logistic type system with several delays, Comput. math. appl., 48, 1-2, 35-44, (2004) · Zbl 1061.34050
[13]	Chen, F.D.; Xie, X.D.; Shi, J.L., Existence uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. comput. appl. math., 194, 2, 368-387, (2006) · Zbl 1104.34050
[14]	Chen, X.X.; Chen, F.D., Stable periodic solution of a discrete periodic lotka – volterra competition system with a feedback control, Appl. math. comput., 181, 2, 1446-1454, (2006) · Zbl 1106.39003
[15]	Chen, Y.M.; Zhou, Z., Stable periodic solution of a discrete periodic lotka – volterra competition system, J. math. anal. appl., 277, 1, 358-366, (2003) · Zbl 1019.39004
[16]	Fan, M.; Wang, K., Periodic solutions of a discrete time nonautonomous ratio-dependent predator – prey system, Math. comput. modelling, 35, 9-10, 951-961, (2002) · Zbl 1050.39022
[17]	Fan, Y.H.; Li, W.T., Permanence for a delayed discrete ratio-dependent predator – prey system with Holling type functional response, J. math. anal. appl., 299, 2, 357-374, (2004) · Zbl 1063.39013
[18]	Huo, H.F.; Li, W.T., Permanence and global stability for nonautonomous discrete model of plankton allelopathy, Appl. math. lett., 17, 9, 1007-1013, (2004) · Zbl 1067.39009
[19]	Li, Y.K.; Lu, L.H., Positive periodic solutions of discrete \(n\)-species food-chain systems, Appl. math. comput., 167, 1, 324-344, (2005) · Zbl 1087.39012
[20]	Montes de Oca, F.; Vivas, M., Extinction in a two dimensional lotka – volterra system with infinite delay, Nonlinear anal. real world appl., 7, 5, 1042-1047, (2006) · Zbl 1122.34058
[21]	Muroya, Y., Persistence and global stability in lotka – volterra delay differential systems, Appl. math. lett., 17, 795-800, (2004) · Zbl 1061.34057
[22]	Muroya, Y., Partial survival and extinction of species in discrete nonautonomous lotka – volterra systems, Tokyo J. math., 28, 1, 189-200, (2005) · Zbl 1081.39014
[23]	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
[24]	Wang, L.; Wang, M.Q., Ordinary difference equation, (1991), Xinjiang University Press China, (in Chinese)
[25]	Wang, W.D.; Lu, Z.Y., Global stability of discrete models of lotka – volterra type, Nonlinear anal., 35, 1019-1030, (1999) · Zbl 0919.92030
[26]	Yang, X.T., Uniform persistence and periodic solutions for a discrete predator – prey system with delays, J. math. anal. appl., 316, 1, 161-177, (2006) · Zbl 1107.39017
[27]	Zhou, Z.; Zou, X., Stable periodic solutions in a discrete periodic logistic equation, Appl. math. lett., 16, 2, 165-171, (2003) · Zbl 1049.39017