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Keywords:
discrete system; coincidence degree; almost periodic solution; Allee-effect
Summary:
In this paper, using Mawhin's continuation theorem of the coincidence degree theory, we obtain some sufficient conditions for the existence of positive almost periodic solutions for a class of delay discrete models with Allee-effect.
References:
[1] Agarwal, R. P., Wong, P. J. Y.: Advanced Topics in Difference Equations. Mathematics and its Applications 404 Kluwer Academic Publishers, Dordrecht (1997). MR 1447437 | Zbl 0878.39001
[2] Alzabut, J. O., Stamov, G. T., Sermutlu, E.: Positive almost periodic solutions for a delay logarithmic population model. Math. Comput. Modelling 53 (2011), 161-167. DOI 10.1016/j.mcm.2010.07.029 | MR 2739253 | Zbl 1211.34084
[3] Cheban, D., Mammana, C.: Invariant manifolds, global attractors and almost periodic solutions of nonautonomous difference equations. Nonlinear Anal., Theory Methods Appl. 56 (2004), 465-484. DOI 10.1016/j.na.2003.09.009 | MR 2035322 | Zbl 1065.39026
[4] Chen, Y.: Periodic solutions of a delayed, periodic logistic equation. Appl. Math. Lett. 16 (2003), 1047-1051. DOI 10.1016/S0893-9659(03)90093-0 | MR 2013071 | Zbl 1118.34327
[5] Chen, W., Liu, B.: Positive almost periodic solution for a class of Nicholson's blowflies model with multiple time-varying delays. J. Comput. Appl. Math. 235 (2011), 2090-2097. DOI 10.1016/j.cam.2010.10.007 | MR 2763127 | Zbl 1207.92042
[6] Chen, F., Xie, X., Chen, X.: Permanence and global attractivity of a delayed periodic logistic equation. Appl. Math. Comput. 177 (2006), 118-127. DOI 10.1016/j.amc.2005.10.040 | MR 2234504 | Zbl 1101.34058
[7] Chen, L., Zhao, H.: Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients. Chaos Solitons Fractals 35 (2008), 351-357. DOI 10.1016/j.chaos.2006.05.057 | MR 2357008 | Zbl 1140.34425
[8] Cheng, S. S., Patula, W. T.: An existence theorem for a nonlinear difference equation. Nonlinear Anal., Theory Methods Appl. 20 (1993), 193-203. DOI 10.1016/0362-546X(93)90157-N | MR 1202198 | Zbl 0774.39001
[9] Ding, X., Lu, C.: Existence of positive periodic solution for ratio-dependent $N$-species difference system. Appl. Math. Modelling 33 (2009), 2748-2756. DOI 10.1016/j.apm.2008.08.008 | MR 2502144 | Zbl 1205.39001
[10] Fan, M., Wang, K.: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Comput. Modelling 35 (2002), 951-961. DOI 10.1016/S0895-7177(02)00062-6 | MR 1910673 | Zbl 1050.39022
[11] Freedman, H. I.: Deterministic Mathematical Models in Population Ecology. Monographs and Textbooks in Pure and Applied Mathematics 57 Marcel Dekker, New York (1980). MR 0586941 | Zbl 0448.92023
[12] Gaines, R. E., Mawhin, J. L.: Coincidence Degree, and Nonlinear Differential Equations. Lecture Notes in Mathematics 568 Springer, Berlin (1977). MR 0637067 | Zbl 0339.47031
[13] Huo, H., Li, W.: Existence and global stability of periodic solutions of a discrete predator-prey system with delays. Appl. Math. Comput. 153 (2004), 337-351. DOI 10.1016/S0096-3003(03)00635-0 | MR 2064661 | Zbl 1043.92038
[14] Li, Y., Fan, X.: Existence and globally exponential stability of almost periodic solution for Cohen-Grossberg BAM neural networks with variable coefficients. Appl. Math. Modelling 33 (2009), 2114-2120. DOI 10.1016/j.apm.2008.05.013 | MR 2488268 | Zbl 1205.34086
[15] Li, Y., Lu, L.: Positive periodic solutions of discrete $n$-species food-chain systems. Appl. Math. Comput. 167 (2005), 324-344. DOI 10.1016/j.amc.2004.06.082 | MR 2170919 | Zbl 1087.39012
[16] Li, Y., Zhang, T., Ye, Y.: On the existence and stability of a unique almost periodic sequence solution in discrete predator-prey models with time delays. Appl. Math. Modelling 35 (2011), 5448-5459. DOI 10.1016/j.apm.2011.04.034 | MR 2806184 | Zbl 1228.39012
[17] Meng, X., Chen, L.: Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays. J. Theoret. Biol. 243 (2006), 562-574. DOI 10.1016/j.jtbi.2006.07.010 | MR 2306349
[18] Meng, X., Jiao, J., Chen, L.: Global dynamics behaviors for a nonautonomous LotkaVolterra almost periodic dispersal system with delays. Nonlinear Anal., Theory Methods Appl. 68 (2008), 3633-3645. DOI 10.1016/j.na.2007.04.006 | MR 2416071
[19] Murray, J. D.: Mathematical Biology. Biomathematics 19 Springer, Berlin (1989). MR 1007836 | Zbl 0682.92001
[20] Sun, Y. G., Saker, S. H.: Existence of positive periodic solutions of nonlinear discrete model exhibiting the Allee effect. Appl. Math. Comput. 168 (2005), 1086-1097. DOI 10.1016/j.amc.2004.10.005 | MR 2171764 | Zbl 1087.39015
[21] Teng, Z.: Persistence and stability in general nonautonomous single-species Kolmogorov systems with delays. Nonlinear Anal., Real World Appl. 8 (2007), 230-248. MR 2268081 | Zbl 1119.34056
[22] Wu, W., Ye, Y.: Existence and stability of almost periodic solutions of nonautonomous competitive systems with weak Allee effect and delays. Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 3993-4002. DOI 10.1016/j.cnsns.2009.02.022 | MR 2522901 | Zbl 1221.34185
[23] Xie, Y., Li, X.: Almost periodic solutions of single population model with hereditary effects. Appl. Math. Comput. 203 (2008), 690-697. DOI 10.1016/j.amc.2008.05.085 | MR 2458985 | Zbl 1166.34327
[24] Yan, J., Zhao, A., Yan, W.: Existence and global attractivity of a periodic solution for an impulsive delay differential equation with Allee effect. J. Math. Anal. Appl. 309 (2005), 489-504. DOI 10.1016/j.jmaa.2004.09.038 | MR 2154131 | Zbl 1086.34066
[25] Yang, X.: The persistence of a general nonautonomous single-species Kolmogorov system with delays. Nonlinear Anal., Theory Methods Appl. 70 (2009), 1422-1429. DOI 10.1016/j.na.2008.02.023 | MR 2474929 | Zbl 1161.34355
[26] Zhang, S., Zheng, G.: Almost periodic solutions of delay difference systems. Appl. Math. Comput. 131 (2002), 497-516. DOI 10.1016/S0096-3003(01)00165-5 | MR 1920242 | Zbl 1029.39011
[27] Zhang, W., Zhu, D., Bi, P.: Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses. Appl. Math. Lett. 20 (2007), 1031-1038. DOI 10.1016/j.aml.2006.11.005 | MR 2344777 | Zbl 1142.39015
[28] Zhu, L., Li, Y.: Positive periodic solutions of higher-dimensional functional difference equations with a parameter. J. Math. Anal. Appl. 290 (2004), 654-664. DOI 10.1016/j.jmaa.2003.10.014 | MR 2033049 | Zbl 1042.39005
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