Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. P. Agarwal, P. J. Y. Wong: Advanced Topics in Difference Equations. Mathematics and its Applications 404, Kluwer Academic Publishers, Dordrecht, 1997.
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.
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.
Y. Chen: Periodic solutions of a delayed, periodic logistic equation. Appl. Math. Lett. 16 (2003), 1047–1051.
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.
F. Chen, X. Xie, X. Chen: Permanence and global attractivity of a delayed periodic logistic equation. Appl. Math. Comput. 177 (2006), 118–127.
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.
S. S. Cheng, W. T. Patula: An existence theorem for a nonlinear difference equation. Nonlinear Anal., Theory Methods Appl. 20 (1993), 193–203.
X. Ding, C. Lu: Existence of positive periodic solution for ratio-dependent N-species difference system. Appl. Math. Modelling 33 (2009), 2748–2756.
M. Fan, K. Wang: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Comput. Modelling 35 (2002), 951–961.
H. I. Freedman: Deterministic Mathematical Models in Population Ecology. Monographs and Textbooks in Pure and Applied Mathematics 57, Marcel Dekker, New York, 1980.
R. E. Gaines, J. L. Mawhin: Coincidence Degree, and Nonlinear Differential Equations. Lecture Notes in Mathematics 568, Springer, Berlin, 1977.
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.
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.
Y. Li, L. Lu: Positive periodic solutions of discrete n-species food-chain systems. Appl. Math. Comput. 167 (2005), 324–344.
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.
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.
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.
J. D. Murray: Mathematical Biology. Biomathematics 19, Springer, Berlin, 1989.
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.
Z. Teng: Persistence and stability in general nonautonomous single-species Kolmogorov systems with delays. Nonlinear Anal., Real World Appl. 8 (2007), 230–248.
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.
Y. Xie, X. Li: Almost periodic solutions of single population model with hereditary effects. Appl. Math. Comput. 203 (2008), 690–697.
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.
X. Yang: The persistence of a general nonautonomous single-species Kolmogorov system with delays. Nonlinear Anal., Theory Methods Appl. 70 (2009), 1422–1429.
S. Zhang, G. Zheng: Almost periodic solutions of delay difference systems. Appl. Math. Comput. 131 (2002), 497–516.
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.
L. Zhu, Y. Li: Positive periodic solutions of higher-dimensional functional difference equations with a parameter. J. Math. Anal. Appl. 290 (2004), 654–664.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work has been supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183 and 11361072, and it was also supported by IRTSTYN.
Rights and permissions
About this article
Cite this article
Li, Y., Yang, L. & Wu, W. Almost periodic solutions for a class of discrete systems with Allee-effect. Appl Math 59, 191–203 (2014). https://doi.org/10.1007/s10492-014-0049-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10492-014-0049-3