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On almost periodic solutions for an impulsive delay logarithmic population model. (English) Zbl 1190.34087

Summary: By employing the contraction mapping principle and applying the Gronwall-Bellman inequality, sufficient conditions are established to prove the existence and exponential stability of positive almost periodic solutions for an impulsive delay logarithmic population model. An example with its numerical simulations has been provided to demonstrate the feasibility of our results.

MSC:

34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
34K45 Functional-differential equations with impulses
92D25 Population dynamics (general)
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[1] Samoilenko, A. M.; Perestyuk, N. A., Impulsive Differential Equations (1995), World Scientific: World Scientific Singapore · Zbl 0837.34003
[2] Bainov, D. D.; Covachev, V., Impulsive Differential Equations with a Small Parameter (1994), World Scientific · Zbl 0828.34001
[3] Benchohra, M.; Henderson, J.; Ntouyas, S. K., Impulsive Differential Equations and Inclusions, vol. 2 (2006), Hindawi Publishing Corporation: Hindawi Publishing Corporation New York · Zbl 1146.34055
[4] Nieto, J. J.; O’Regan, Donal, Variational approach to impulsive differential equation, Nonlinear Anal. RWA, 10, 680-690 (2009) · Zbl 1167.34318
[5] Zavalishchin, S. T.; Sesekin, A. N., (Dynamic Impulse Systems. Theory and Applications. Dynamic Impulse Systems. Theory and Applications, Mathematics and its Applications, vol. 394 (1997), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht) · Zbl 0880.46031
[6] Nieto, J. J.; Rodriguez-Lopez, R., Comparison results and approximation of solutions for impulsive functional differential equations, Dyn. Contin. Discrete Impuls. Syst. A, 15, 169-215 (2008) · Zbl 1153.34048
[7] Zeng, G.; Wang, F.; Nieto, J. J., Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response, Adv. Complex Syst., 11, 77-97 (2008) · Zbl 1168.34052
[8] Wei, H.; Jiang, Y.; Song, X.; Su, G. H.; Qiu, S. Z., Global attractivity and permanence of a SVEIR epidemic model with pulse vaccination and time delay, J. Comput. Appl. Math., 229, 1, 302-312 (2009) · Zbl 1162.92038
[9] Zhang, H.; Chen, L.; Nieto, J. J., A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear Anal. RWA, 9, 1714-1726 (2008) · Zbl 1154.34394
[10] Zhang, J.; Gui, Z., Periodic solutions of nonautonomous cellular neural networks with impulses and delays, Nonlinear Anal. RWA, 10, 3, 1891-1903 (2009) · Zbl 1160.92005
[11] Li, W.; Chang, Y.; Nieto, J. J., Solvability of impulsive neutral evolution differential inclusions with state-dependent delay, Math. Comput. Modelling, 49, 9-10, 1920-1927 (2009) · Zbl 1171.34304
[12] Akhmet, M. U.; Alzabut, J. O.; Zafer A, A., On periodic solutions of linear impulsive differential systems, Dyn. Contin. Discrete Impuls. Syst. A, 15, 5, 621-631 (2008) · Zbl 1171.34340
[13] Hernández M, E.; Henrìquez, H. R.; McKibben, M. A., Existence results for abstract impulsive second-order neutral functional differential equations, Nonlinear Anal., 70, 7, 2736-2751 (2009) · Zbl 1173.34049
[14] Zhang, L.; Li, H., Periodicity on a class of neutral impulsive delay system, Appl. Math. Comput., 203, 1, 178-185 (2008) · Zbl 1166.34332
[15] Saker, S. H.; Alzabut, J. O., On impulsive delay hematopoiesis model with periodic coefficients, Rocky Mountain J. Math., 39, 5, 1657-1688 (2009) · Zbl 1179.34092
[16] Ahmad, S.; Stamov, G. Tr., Almost periodic solutions of \(n\)-dimensional impulsive competitive systems, Nonlinear Anal. RWA, 10, 3, 1846-1853 (2009) · Zbl 1162.34349
[17] Ahmad, S.; Stamov, G. Tr., On almost periodic processes in impulsive competitive systems with delay and impulsive perturbations, Nonlinear Anal. RWA, 10, 2857-2863 (2009) · Zbl 1170.45004
[18] Li, Z.; Chen, F., Almost periodic solutions of a discrete almost periodic logistic equation, Math. Comput. Modelling, 50, 254-259 (2009) · Zbl 1185.39011
[19] Luo, B., Travelling waves of a curvature flow in almost periodic media, J. Differential Equations, 247, 2189-2208 (2009) · Zbl 1182.35073
[20] Alzabut, J. O.; Nieto, J. J.; Stamov, G. Tr., Existence and exponential stability of positive almost periodic solutions for a model of hematpoiesis, Bound. Value Probl., 2009 (2009), Article ID 127510 · Zbl 1186.34116
[21] Yuan, R., On almost periodic solutions of logistic delay differential equations with almost periodic time dependence, J. Math. Anal. Appl., 330, 780-798 (2007) · Zbl 1125.34055
[22] Stamov, G. T.; Petrov, N., Lyapunov-Razumikhin method for existence of almost periodic solutions of impulsive differential-difference equations, Nonlinear Stud., 15, 2, 151-163 (2008) · Zbl 1156.34057
[23] Stamov, G. T.; Stamova, I. M., Almost periodic solutions for impulsive neutral networks with delay, Appl. Math. Model., 31, 1263-1270 (2007) · Zbl 1136.34332
[24] Stamov, G. T., Almost periodic solutions of impulsive differential equations with time-varying delay on the PC-space, Nonlinear Stud., 14, 3, 269-270 (2007) · Zbl 1140.34440
[25] Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic Publisher: Kluwer Academic Publisher Boston · Zbl 0752.34039
[26] Chen, F. D., Positive periodic solutions of state-dependent delay logarithm population model, J. Fuzhou Univ. Nat. Sci. Ed., 31, 3, 1-4 (2003), (in Chinese)
[27] Chen, F., Periodic solutions and almost periodic solutions for a delay multispecies logarithmic population model, Appl. Math. Comput., 171, 760-770 (2005) · Zbl 1089.92038
[28] Li, J. W.; Cheng, S. S., Globally attractive periodic solution of a perturbed functional differential equation, J. Comput. Appl. Math., 193, 2, 652-657 (2006) · Zbl 1109.34051
[29] Wang, C.; Shi, J., Periodic solution for a delay multispecies Logarithmic population model with feedback control, Appl. Math. Comput., 193, 257-265 (2007) · Zbl 1193.34144
[30] Zhao, W., New results of existence and stability of periodic solution for a delay multispecies logarithmic population model, Nonlinear Anal. RWA, 10, 544-553 (2009) · Zbl 1154.34366
[31] Wang, Q.; Wang, Y.; Dai, B., Existence and uniqueness of positive periodic solutions for a neutral lograithmic population model, Appl. Math. Comput. (2009)
[32] Alzabut, J. O.; Abdeljawad, T., Existence and global attractivity of impulsive delay logarithmic model of population dynamics, Appl. Math. Comput., 198, 1, 463-469 (2008) · Zbl 1163.92033
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