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Permanence and positive periodic solutions of a discrete delay competitive system. (English) Zbl 1190.92043

Summary: A discrete time non-autonomous two-species competitive system with delays is proposed, which involves the influence of many generations on the density of the species population. Sufficient conditions for permanence of the system are given. When the system is periodic, by using the continuity theorem of coincidence degree theory and constructing a suitable Lyapunov discrete function, sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions are obtained. As an application, examples and their numerical simulations are presented to illustrate the feasibility of our main results.

MSC:

92D40 Ecology
34K13 Periodic solutions to functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
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References:

[1] R. M. May, “Time delay versus stability in population models with two or three tropic levels,” Ecology, vol. 54, no. 2, pp. 315-325, 1973.
[2] S. Ahmad and I. M. Stamova, “Asymptotic stability of competitive systems with delays and impulsive perturbations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 686-700, 2007. · Zbl 1153.34044 · doi:10.1016/j.jmaa.2006.12.068
[3] F. Chen, “Global asymptotic stability in n-species non-autonomous Lotka-Volterra competitive systems with infinite delays and feedback control,” Applied Mathematics and Computation, vol. 170, no. 2, pp. 1452-1468, 2005. · Zbl 1081.92038 · doi:10.1016/j.amc.2005.01.028
[4] M. Fan, K. Wang, and D. Jiang, “Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments,” Mathematical Biosciences, vol. 160, no. 1, pp. 47-61, 1999. · Zbl 0964.34059 · doi:10.1016/S0025-5564(99)00022-X
[5] F. Montes de Oca and M. Vivas, “Extinction in two dimensional Lotka-Volterra system with infinite delay,” Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp. 1042-1047, 2006. · Zbl 1122.34058 · doi:10.1016/j.nonrwa.2005.09.005
[6] W. Zhang and M. Fan, “Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays,” Mathematical and Computer Modelling, vol. 39, no. 4-5, pp. 479-493, 2004. · Zbl 1065.92066 · doi:10.1016/S0895-7177(04)90519-5
[7] Z. Zhang and Z. Wang, “Periodic solution for a two-species nonautonomous competition Lotka-Volterra patch system with time delay,” Journal of Mathematical Analysis and Applications, vol. 265, no. 1, pp. 38-48, 2002. · Zbl 1003.34060 · doi:10.1006/jmaa.2001.7682
[8] Z. Liu, R. Tan, and Y. Chen, “Modeling and analysis of a delayed competitive system with impulsive perturbations,” The Rocky Mountain Journal of Mathematics, vol. 38, no. 5, pp. 1505-1523, 2008. · Zbl 1194.34093 · doi:10.1216/RMJ-2008-38-5-1505
[9] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. · Zbl 0952.39001
[10] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. · Zbl 0448.92023
[11] J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, Berlin, Germany, 1989. · Zbl 0682.92001
[12] H.-F. Huo and W.-T. Li, “Existence and global stability of periodic solutions of a discrete predator-prey system with delays,” Applied Mathematics and Computation, vol. 153, no. 2, pp. 337-351, 2004. · Zbl 1043.92038 · doi:10.1016/S0096-3003(03)00635-0
[13] F. Chen, “Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition predator-prey systems,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 3-12, 2006. · Zbl 1113.92061 · doi:10.1016/j.amc.2006.03.026
[14] X. Chen and F. Chen, “Stable periodic solution of a discrete periodic Lotka-Volterra competition system with a feedback control,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1446-1454, 2006. · Zbl 1106.39003 · doi:10.1016/j.amc.2006.02.039
[15] X. Liao, S. Zhou, and Y. N. Raffoul, “On the discrete-time multi-species competition-predation system with several delays,” Applied Mathematics Letters, vol. 21, no. 1, pp. 15-22, 2008. · Zbl 1129.92067 · doi:10.1016/j.aml.2007.03.003
[16] X. Liao, S. Zhou, and Y. Chen, “Permanence and global stability in a discrete n-species competition system with feedback controls,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1661-1671, 2008. · Zbl 1154.34352 · doi:10.1016/j.nonrwa.2007.05.001
[17] X. Liao, Z. Ouyang, and S. Zhou, “Permanence of species in nonautonomous discrete Lotka-Volterra competitive system with delays and feedback controls,” Journal of Computational and Applied Mathematics, vol. 211, no. 1, pp. 1-10, 2008. · Zbl 1143.39005 · doi:10.1016/j.cam.2006.10.084
[18] B. Dai and L. Huang, “Asymptotic behavior of solutions for a class of nonlinear difference equations,” Computers & Mathematics with Applications, vol. 36, no. 2, pp. 23-30, 1998. · Zbl 0933.39018 · doi:10.1016/S0898-1221(98)00113-8
[19] E. E. Crone, “Delayed density dependence and the stability of interacting populations and subpopulations,” Theoretical Population Biology, vol. 51, no. 1, pp. 67-76, 1997. · Zbl 0882.92025 · doi:10.1006/tpbi.1997.1309
[20] W. D. Wang, G. Mulone, F. Salemi, and V. Salone, “Global stability of discrete population models with time delays and fluctuating environment,” Journal of Mathematical Analysis and Applications, vol. 264, no. 1, pp. 147-167, 2001. · Zbl 1006.92025 · doi:10.1006/jmaa.2001.7666
[21] J. Zhang and H. Fang, “Multiple periodic solutions for a discrete time model of plankton allelopathy,” Advances in Difference Equations, vol. 2006, Article ID 90479, 14 pages, 2006. · Zbl 1134.39008 · doi:10.1155/ADE/2006/90479
[22] Z. Liu and L. Chen, “Periodic solutions of a discrete time nonautonomous two-species mutualistic system with delays,” Advances in Complex Systems, vol. 9, no. 1-2, pp. 87-98, 2006. · Zbl 1107.92053 · doi:10.1142/S0219525906000690
[23] Z. Liu and L. Chen, “Positive periodic solution of a general discrete non-autonomous difference system of plankton allelopathy with delays,” Journal of Computational and Applied Mathematics, vol. 197, no. 2, pp. 446-456, 2006. · Zbl 1098.92066 · doi:10.1016/j.cam.2005.09.023
[24] R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays,” Discrete and Continuous Dynamical Systems. Series B, vol. 4, no. 3, pp. 823-831, 2004. · Zbl 1116.92072 · doi:10.3934/dcdsb.2004.4.823
[25] Y. Chen, R. Tan, and Z. Liu, “Global attractivity in a periodic delayed competitive system,” Applied Mathematical Sciences, vol. 1, no. 33-36, pp. 1675-1684, 2007. · Zbl 1187.34111
[26] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. · Zbl 0752.34039
[27] X. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with delays,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 161-177, 2006. · Zbl 1107.39017 · doi:10.1016/j.jmaa.2005.04.036
[28] R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977. · Zbl 0339.47031
[29] M. Fan and K. Wang, “Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system,” Mathematical and Computer Modelling, vol. 35, no. 9-10, pp. 951-961, 2002. · Zbl 1050.39022 · doi:10.1016/S0895-7177(02)00062-6
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