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Article

Xu, Jiabo ; Teng, Zhidong ; Gao, Shujing
Almost sufficient and necessary conditions for permanence and extinction of nonautonomous discrete logistic systems with time-varying delays and feedback control. (English). Applications of Mathematics, vol. 56 (2011), issue 2, pp. 207-225
MSC: 39A22, 39A30, 92D25, 93B52 | MR 2810244 | Zbl 1223.92051 | DOI: 10.1007/s10492-011-0003-6
Keywords:
discrete system; permanence; extinction; feedback control; time-varying delay
Summary:
A class of nonautonomous discrete logistic single-species systems with time-varying pure-delays and feedback control is studied. By introducing a new research method, almost sufficient and necessary conditions for the permanence and extinction of species are obtained. Particularly, when the system degenerates into a periodic system, sufficient and necessary conditions on the permanence and extinction of species are obtained. Moreover, a very important fact is found in our results, that is, the feedback control and delays are harmless for the permanence and extinction of species for discrete single-species systems. This shows that in a discrete single-species system introducing the feedback control to factitiously control the permanence and extinction of species is useless.
References:
[1] Agarwal, R. P.: Difference Equations and Inequalities: Theory, Methods, and Applications. Marcel Dekker New York (1992). MR 1155840 | Zbl 0925.39001
[2] Agarwal, R. P., Li, W.-T., Pang, P. Y. H.: Asymptotic behavior of a class of nonlinear delay difference equations. J. Difference Equ. Appl. 8 (2002), 719-728. DOI 10.1080/1023619021000000735 | MR 1914599 | Zbl 1010.39003
[3] Bohner, M., Fan, M., Zhang, J.: Existence of periodic solutions in predator-prey and competition dynamic systems. Nonlinear Anal., Real World Appl. 7 (2006), 1193-1204. DOI 10.1016/j.nonrwa.2005.11.002 | MR 2260908 | Zbl 1104.92057
[4] Braverman, E.: On a discrete model of population dynamics with impulsive harvesting or recruitment. Nonlinear Anal., Theory Methods Appl., Ser. A 63 (2005), 751-759 (Electronic only). DOI 10.1016/j.na.2004.12.015 | Zbl 1222.92060
[5] Braverman, E., Saker, S. H.: Permanence, oscillation and attractivity of the discrete hematopoiesis model with variable coefficients. Nonlinear Anal., Theory Methods Appl. 67 (2007), 2955-2965. DOI 10.1016/j.na.2006.09.056 | MR 2348007 | Zbl 1125.39002
[6] Chen, F.: Permanence in a discrete Lotka-Volterra competition model with deviating arguments. Nonlinear Anal., Real World Appl. 9 (2008), 2150-2155. MR 2441771 | Zbl 1156.39300
[7] Chen, F.: Permanence of a single species discrete model with feedback control and delay. Appl. Math. Lett. 20 (2007), 729-733. DOI 10.1016/j.aml.2006.08.023 | MR 2314699 | Zbl 1128.92029
[8] Chen, F., Wu, L., Li, Z.: Permanence and global attractivity of the discrete Gilpin-Ayala type population model. Comput. Math. Appl. 53 (2007), 1214-1227. DOI 10.1016/j.camwa.2006.12.015 | MR 2327675 | Zbl 1127.92038
[9] Chen, X., Chen, F.: Stable periodic solution of a discrete periodic Lotka-Volterra competition system with a feedback control. Appl. Math. Comput. 181 (2006), 1446-1454. DOI 10.1016/j.amc.2006.02.039 | MR 2270776 | Zbl 1106.39003
[10] Chen, Y., Zhou, Z.: Stable periodic solution of a discrete periodic Lotka-Volterra competition system. J. Math. Anal. Appl. 277 (2003), 358-366. DOI 10.1016/S0022-247X(02)00611-X | MR 1954481 | Zbl 1019.39004
[11] Douraki, M. J., Mashreghi, J.: On the population model of the non-autonomous logistic equation of second order with period-two parameters. J. Difference Equ. Appl. 14 (2008), 231-257. DOI 10.1080/10236190701466504 | MR 2397323 | Zbl 1135.92026
[12] Emmert, K. E., Allen, L. J. S.: Population persistence and extinction in a discrete-time, stage-structured epidemic model. J. Difference Equ. Appl. 10 (2004), 1177-1199. DOI 10.1080/10236190410001654151 | MR 2100721 | Zbl 1067.92052
[13] Fan, M., Wang, Q.: Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator-prey systems. Disc. Cont. Dyn. Syst., Ser. B 4 (2004), 563-574. DOI 10.3934/dcdsb.2004.4.563 | MR 2073960 | Zbl 1100.92064
[14] 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 (2004), 357-374. DOI 10.1016/j.jmaa.2004.02.061 | MR 2098248 | Zbl 1063.39013
[15] Giang, D. V., Huong, D. C.: Nontrivial periodicity in discrete delay models of population growth. J. Math. Anal. Appl. 305 (2005), 291-295. DOI 10.1016/j.jmaa.2004.11.035 | MR 2128128 | Zbl 1096.92034
[16] Győri, I., Trofimchuk, S. I.: Global attractivity and persistence in a discrete population model. J. Difference Equ. Appl. 6 (2000), 647-665. DOI 10.1080/10236190008808250 | MR 1813210
[17] Huo, H.-F., Li, W.-T.: Permanence and global stability for nonautonomous discrete model of plankton allelopathy. Appl. Math. Letters 17 (2004), 1007-1013. DOI 10.1016/j.aml.2004.07.002 | MR 2087748 | Zbl 1067.39009
[18] Kocic, V. L., Ladas, G.: Global Behavior of Nonlinear Difference Equations of Higher Order with Application. Kluwer Academic Publishers Dordrecht (1993). MR 1247956
[19] Kon, R.: Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points. J. Math. Biol. 48 (2004), 57-81. DOI 10.1007/s00285-003-0224-8 | MR 2035520 | Zbl 1050.92045
[20] Li, Y. K., Zhu, L. F.: Existence of positive periodic solutions for difference equations with feedback control. Appl. Math. Lett. 18 (2005), 61-67. DOI 10.1016/j.aml.2004.09.002 | MR 2121555 | Zbl 1085.39009
[21] Liao, X., Ouyang, Z., Zhou, S.: Permanence of species in nonautonomous discrete Lotka-Volterra competitive system with delays and feedback controls. J. Comput. Appl. Math. 211 (2008), 1-10. DOI 10.1016/j.cam.2006.10.084 | MR 2386823 | Zbl 1143.39005
[22] Liao, X., Zhou, S., Chen, Y.: Permanence and global stability in a discrete $n$-species competition system with feedback controls. Nonlinear Anal., Real World Appl. 9 (2008), 1661-1671. MR 2422571 | Zbl 1154.34352
[23] Liao, L., Yu, J., Wang, L.: Global attractivity in a logistic difference model with a feedback control. Comput. Math. Appl. 44 (2002), 1403-1411. DOI 10.1016/S0898-1221(02)00265-1 | MR 1938776 | Zbl 1043.93053
[24] Liu, Z., Chen, L.: Positive periodic solution of a general discrete non-autonomous difference system of plankton allelopathy with delays. J. Comput. Appl. Math. 197 (2006), 446-456. DOI 10.1016/j.cam.2005.09.023 | MR 2260418 | Zbl 1098.92066
[25] Liz, E.: A sharp global stability result for a discrete population model. J. Math. Anal. Appl. 330 (2007), 740-743. DOI 10.1016/j.jmaa.2006.06.030 | MR 2302956 | Zbl 1108.92033
[26] Liz, E., Tkachenko, V., Trofimchuk, S.: Global stability in discrete population models with delayed-density dependence. Math. Biosci. 199 (2006), 26-37. DOI 10.1016/j.mbs.2005.03.016 | MR 2205557 | Zbl 1086.92045
[27] Lu, Z., Wang, W.: Permanence and global attractivity for Lotka-Volterra difference systems. J. Math. Biol. 39 (1999), 269-282. DOI 10.1007/s002850050171 | MR 1716315 | Zbl 0945.92022
[28] Merdan, H., Duman, O.: On the stability analysis of a general discrete-time population model involving predation and Allee effects. Chaos Solitons Fractals 40 (2009), 1169-1175. DOI 10.1016/j.chaos.2007.08.081 | MR 2526104 | Zbl 1197.39009
[29] Muroya, Y.: Persistence and global stability in discrete models of Lotka-Volterra type. J. Math. Anal. Appl. 330 (2007), 24-33. DOI 10.1016/j.jmaa.2006.07.070 | MR 2298155 | Zbl 1124.39011
[30] Muroya, Y.: Persistence and global stability in discrete models of pure-delay nonautonomous Lotka-Volterra type. J. Math. Anal. Appl. 293 (2004), 446-461. DOI 10.1016/j.jmaa.2003.12.033 | MR 2053890 | Zbl 1059.39007
[31] Papaschinopoulos, G., Schinas, C. J.: Persistence, oscillatory behavior, and periodicity of the solutions of a system of two nonlinear difference equations. J. Difference Equ. Appl. 4 (1998), 315-323. DOI 10.1080/10236199808808146 | MR 1657224 | Zbl 0913.39009
[32] Sadhukhan, D., Mondal, B., Maiti, M.: Discrete age-structured population model with age dependent harvesting and its stability analysis. Appl. Math. Comput. 201 (2008), 631-639. DOI 10.1016/j.amc.2007.12.063 | MR 2431960 | Zbl 1143.92041
[33] 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 (2001), 162-174. DOI 10.1006/jmaa.2000.7303 | MR 1820074 | Zbl 0976.92031
[34] Saker, S. H.: Periodic solutions, oscillation and attractivity of discrete nonlinear delay population model. Math. Comput. Modelling 47 (2008), 278-297. DOI 10.1016/j.mcm.2007.04.007 | MR 2378835 | Zbl 1130.92044
[35] Teng, Z.: Permanence and stability in non-autonomous logistic systems with infinite delay. Dyn. Syst. 17 (2002), 187-202. DOI 10.1080/14689360110102312 | MR 1927807 | Zbl 1035.34086
[36] Wang, W., Mulone, G., Salemi, F., Salone, V.: Global stability of discrete population models with time delays and fluctuating environment. J. Math. Anal. Appl. 264 (2001), 147-167. DOI 10.1006/jmaa.2001.7666 | MR 1868334 | Zbl 1006.92025
[37] Xia, Y., Cao, J., Lin, M.: Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses. Nonlinear Anal., Real World Appl. 8 (2007), 1079-1095. MR 2331428 | Zbl 1127.39038
[38] Xiong, X., Zhang, Z.: Periodic solutions of a discrete two-species competitive model with stage structure. Math. Comput. Modelling 48 (2008), 333-343. DOI 10.1016/j.mcm.2007.10.004 | MR 2431471 | Zbl 1145.34334
[39] Yang, X.: Uniform persistence and periodic solutions for a discrete predator-prey system with delays. J. Math. Anal. Appl. 316 (2006), 161-177. DOI 10.1016/j.jmaa.2005.04.036 | MR 2201755 | Zbl 1107.39017
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