×

Almost periodic solutions for a class of discrete systems with Allee-effect. (English) Zbl 1324.92066

In this paper, the authors use Mawhin’s continuation theorem to obtain a sufficient condition for the existence of positive almost periodic solutions for a delayed discrete time population model, which is characterized as being subject to Allee effects.
Although the treatment of the problem is adequate and the results are valid, neither this discrete model nor the continuous one from which it is derived, however, appear to be models with Allee effects. They do not exhibit the hallmark of Allee effects, namely a decline in individual fitness at low population density, which usually implies the existence of a critical threshold below which populations cannot escape extinction. Instead, these models appear to be single-species models with multiple intraspecies competition terms.

MSC:

92D40 Ecology
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] R. P. Agarwal, P. J. Y. Wong: Advanced Topics in Difference Equations. Mathematics and its Applications 404, Kluwer Academic Publishers, Dordrecht, 1997. · Zbl 0878.39001 · doi:10.1007/978-94-015-8899-7
[2] J. O. Alzabut, G. T. Stamov, E. Sermutlu: Positive almost periodic solutions for a delay logarithmic population model. Math. Comput. Modelling 53 (2011), 161-167. · Zbl 1211.34084 · doi:10.1016/j.mcm.2010.07.029
[3] D. Cheban, C. Mammana: Invariant manifolds, global attractors and almost periodic solutions of nonautonomous difference equations. Nonlinear Anal., Theory Methods Appl. 56 (2004), 465-484. · Zbl 1065.39026 · doi:10.1016/j.na.2003.09.009
[4] Y. Chen: Periodic solutions of a delayed, periodic logistic equation. Appl. Math. Lett. 16 (2003), 1047-1051. · Zbl 1118.34327 · doi:10.1016/S0893-9659(03)90093-0
[5] W. Chen, B. Liu: 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. · Zbl 1207.92042 · doi:10.1016/j.cam.2010.10.007
[6] F. Chen, X. Xie, X. Chen: Permanence and global attractivity of a delayed periodic logistic equation. Appl. Math. Comput. 177 (2006), 118-127. · Zbl 1101.34058 · doi:10.1016/j.amc.2005.10.040
[7] L. Chen, H. Zhao: Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients. Chaos Solitons Fractals 35 (2008), 351-357. · Zbl 1140.34425 · doi:10.1016/j.chaos.2006.05.057
[8] S. S. Cheng, W. T. Patula: An existence theorem for a nonlinear difference equation. Nonlinear Anal., Theory Methods Appl. 20 (1993), 193-203. · Zbl 0774.39001 · doi:10.1016/0362-546X(93)90157-N
[9] X. Ding, C. Lu: Existence of positive periodic solution for ratio-dependent N-species difference system. Appl. Math. Modelling 33 (2009), 2748-2756. · Zbl 1205.39001 · doi:10.1016/j.apm.2008.08.008
[10] M. Fan, K. Wang: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Comput. Modelling 35 (2002), 951-961. · Zbl 1050.39022 · doi:10.1016/S0895-7177(02)00062-6
[11] H. I. Freedman: Deterministic Mathematical Models in Population Ecology. Monographs and Textbooks in Pure and Applied Mathematics 57, Marcel Dekker, New York, 1980. · Zbl 0448.92023
[12] R. E. Gaines, J. L. Mawhin: Coincidence Degree, and Nonlinear Differential Equations. Lecture Notes in Mathematics 568, Springer, Berlin, 1977. · Zbl 0339.47031
[13] H. Huo, W. Li: Existence and global stability of periodic solutions of a discrete predator-prey system with delays. Appl. Math. Comput. 153 (2004), 337-351. · Zbl 1043.92038 · doi:10.1016/S0096-3003(03)00635-0
[14] Y. Li, X. Fan: 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. · Zbl 1205.34086 · doi:10.1016/j.apm.2008.05.013
[15] Y. Li, L. Lu: Positive periodic solutions of discrete n-species food-chain systems. Appl. Math. Comput. 167 (2005), 324-344. · Zbl 1087.39012 · doi:10.1016/j.amc.2004.06.082
[16] Y. Li, T. Zhang, Y. Ye: 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. · Zbl 1228.39012 · doi:10.1016/j.apm.2011.04.034
[17] X. Meng, L. Chen: Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays. J. Theoret. Biol. 243 (2006), 562-574. · Zbl 1447.92355 · doi:10.1016/j.jtbi.2006.07.010
[18] X. Meng, J. Jiao, L. Chen: Global dynamics behaviors for a nonautonomous Lotka-Volterra almost periodic dispersal system with delays. Nonlinear Anal., Theory Methods Appl. 68 (2008), 3633-3645. · Zbl 1155.34042 · doi:10.1016/j.na.2007.04.006
[19] J. D. Murray: Mathematical Biology. Biomathematics 19, Springer, Berlin, 1989. · Zbl 0682.92001 · doi:10.1007/978-3-662-08539-4
[20] Y. G. Sun, S. H. Saker: Existence of positive periodic solutions of nonlinear discrete model exhibiting the Allee effect. Appl. Math. Comput. 168 (2005), 1086-1097. · Zbl 1087.39015 · doi:10.1016/j.amc.2004.10.005
[21] Z. Teng: Persistence and stability in general nonautonomous single-species Kolmogorov systems with delays. Nonlinear Anal., Real World Appl. 8 (2007), 230-248. · Zbl 1119.34056 · doi:10.1016/j.nonrwa.2005.08.003
[22] W. Wu, Y. Ye: 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. · Zbl 1221.34185 · doi:10.1016/j.cnsns.2009.02.022
[23] Y. Xie, X. Li: Almost periodic solutions of single population model with hereditary effects. Appl. Math. Comput. 203 (2008), 690-697. · Zbl 1166.34327 · doi:10.1016/j.amc.2008.05.085
[24] J. Yan, A. Zhao, W. Yan: 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. · Zbl 1086.34066 · doi:10.1016/j.jmaa.2004.09.038
[25] X. Yang: The persistence of a general nonautonomous single-species Kolmogorov system with delays. Nonlinear Anal., Theory Methods Appl. 70 (2009), 1422-1429. · Zbl 1161.34355 · doi:10.1016/j.na.2008.02.023
[26] S. Zhang, G. Zheng: Almost periodic solutions of delay difference systems. Appl. Math. Comput. 131 (2002), 497-516. · Zbl 1029.39011 · doi:10.1016/S0096-3003(01)00165-5
[27] W. Zhang, D. Zhu, P. Bi: Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses. Appl. Math. Lett. 20 (2007), 1031-1038. · Zbl 1142.39015 · doi:10.1016/j.aml.2006.11.005
[28] L. Zhu, Y. Li: Positive periodic solutions of higher-dimensional functional difference equations with a parameter. J. Math. Anal. Appl. 290 (2004), 654-664. · Zbl 1042.39005 · doi:10.1016/j.jmaa.2003.10.014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.