Almost periodic solutions of a discrete Lotka-Volterra model via exponential dichotomy theory

Lini Fang; N'gbo N'gbo; Yonghui Xia; Lini Fang; N'gbo N'gbo; Yonghui Xia

doi:10.3934/math.2022210

AIMS Mathematics

2022, Volume 7, Issue 3: 3788-3801. doi: 10.3934/math.2022210

Previous Article Next Article

Research article

Almost periodic solutions of a discrete Lotka-Volterra model via exponential dichotomy theory

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received: 03 October 2021 Accepted: 01 December 2021 Published: 09 December 2021
MSC : 34D10, 34K14, 39A10, 39A11

In this paper, we consider a discrete non-autonomous Lotka-Volterra model. Under some assumptions, we prove the existence of positive almost periodic solutions. Our analysis relies on the exponential dichotomy for the difference equations and the Banach fixed point theorem. Furthermore, by constructing a Lyapunov function, the exponential convergence is proved. Finally, a numerical example illustrates the effectiveness of the results.
- almost periodic solution,
- difference exponential dichotomy,
- Banach fixed point,
- discrete Lotka-Volterra model,
- exponential convergence
Citation: Lini Fang, N'gbo N'gbo, Yonghui Xia. Almost periodic solutions of a discrete Lotka-Volterra model via exponential dichotomy theory[J]. AIMS Mathematics, 2022, 7(3): 3788-3801. doi: 10.3934/math.2022210

Related Papers:

Abstract

In this paper, we consider a discrete non-autonomous Lotka-Volterra model. Under some assumptions, we prove the existence of positive almost periodic solutions. Our analysis relies on the exponential dichotomy for the difference equations and the Banach fixed point theorem. Furthermore, by constructing a Lyapunov function, the exponential convergence is proved. Finally, a numerical example illustrates the effectiveness of the results.

References

[1]	S. Ahmad, On almost periodic solutions of the competing species problems, Proc. Amer. Math. Soc., 102 (1988), 855–861. doi: 10.1090/S0002-9939-1988-0934856-5. doi: 10.1090/S0002-9939-1988-0934856-5
[2]	J. K. Hale, Periodic and almost periodic solution of functional differential equations, Arch. Ration. Mech. An., 15 (1964), 289–304. doi: 10.1007/BF00249199. doi: 10.1007/BF00249199
[3]	T. Yoshizawa, Stability properties in almost periodic system of functional differential equations, In: Functional differential equations and bifurcation, 799 (1980), 385–409. doi: 10.1007/BFb0089326.
[4]	T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, New York: Springer, 1975. doi: 10.1007/978-1-4612-6376-0.
[5]	Y. Hamaya, Existence of an almost periodic solution in a difference equation by Liapunov functions, Nonlinear Stud., 8 (2001), 373–380.
[6]	S. Gao, K. Peng, C. Zhang, Existence and global exponential stability of periodic solutions for feedback control complex dynamical networks with time-varying delays, Chaos Solitons Fractals, 152 (2021), 111483. doi: 10.1016/j.chaos.2021.111483. doi: 10.1016/j.chaos.2021.111483
[7]	S. Zhang, Existence of almost periodic solution for difference systems, Differ. Equ., 16 (2000), 184–206.
[8]	S. Zhang, G. Zheng, Almost periodic solutions of delay difference systems, Appl. Math. Comput., 131 (2002), 497–516. doi: 10.1016/S0096-3003(01)00165-5. doi: 10.1016/S0096-3003(01)00165-5
[9]	T. Q. Zhang, W. B. Ma, X. Z. Meng, T. H. Zhang, Periodic solution of a prey-predator model with nonlinear state feedback control, Appl. Math. Comput., 266 (2015), 95–107. doi: 10.1016/j.amc.2015.05.016. doi: 10.1016/j.amc.2015.05.016
[10]	T. H. Zhang, T. Q. Zhang, X. Z. Meng, Stability analysis of a chemostat model with maintenance energy, Appl. Math. Lett., 68 (2017), 1–7. doi: 10.1016/j.aml.2016.12.007. doi: 10.1016/j.aml.2016.12.007
[11]	Y. H. Xia, Almost periodic solution of a population model: Via spectral radius of matrix, Bull. Malays. Math. Sci. Soc., 37 (2014), 249–259.
[12]	F. Chen, Periodic solutions and almost periodic solutions for a delay multispecies Logarithmic population model, Appl. Math. Comput., 171 (2005), 760–770. doi: 10.1016/j.amc.2005.01.085. doi: 10.1016/j.amc.2005.01.085
[13]	Q. Wang, Y. Wang, B. Dai, Existence and uniqueness of positive periodic solutions for a neutral Logarithmic population model, Appl. Math. Comput., 213 (2009), 137–147. doi: 10.1016/j.amc.2009.03.028. doi: 10.1016/j.amc.2009.03.028
[14]	Y. Li, C. Wang, Almost periodic functions on time scales and applications, Discrete Dyn. Nat. Soc., 2011 (2011), 1–20. doi: 10.1155/2011/727068. doi: 10.1155/2011/727068
[15]	S. Gao, R. Shen, T. R. Chen, Periodic solutions for discrete-time Cohen-Grossberg neural networks with delays, Phys. Lett. A, 383 (2019), 414–420. doi: 10.1016/j.physleta.2018.11.016. doi: 10.1016/j.physleta.2018.11.016
[16]	Y. Hamaya, Existence of an almost periodic solution in a difference equation with infinite delay, J. Differ. Equ. Appl., 9 (2003), 227–237. doi: 10.1080/1023619021000035836. doi: 10.1080/1023619021000035836
[17]	Y. Hamaya, Bifurcation of almost periodic solutions in difference equation, J. Differ. Equ. Appl., 10 (2004), 257–279. doi: 10.1080/10236190310001634794. doi: 10.1080/10236190310001634794
[18]	Y. Xia, Z. Huang, M. Han, Existence of almost periodic solutions for forced perturbed systems with piecewise constant argument, J. Math. Anal. Appl., 333 (2007), 798–816. doi: 10.1016/j.jmaa.2006.11.039. doi: 10.1016/j.jmaa.2006.11.039
[19]	W. Liu, T. Chen, Positive periodic solutions of delayed periodic Lotka-Volterra systems, Phys. Lett. A, 334 (2005), 273–287. doi: 10.1016/j.physleta.2004.10.083. doi: 10.1016/j.physleta.2004.10.083
[20]	C. Corduneanu, Almost periodic discrete processes, Libertas Math., 2 (1982), 159–170.
[21]	K. Gopalsamy, S. Mohamad, Canonical solutions and almost periodicity in a discrete logistic equation, Appl. Math. Comput., 113 (2000), 305–323. doi: 10.1016/S0096-3003(99)00093-4. doi: 10.1016/S0096-3003(99)00093-4
[22]	Y. H. Xia, S. S. Chen, Quasi-uniformly asymptotic stability and existence of almost periodic solutions of difference equations with applications in population dynamic systems, J. Differ. Equ. Appl., 14 (2008), 59–81. doi: 10.1080/10236190701470407. doi: 10.1080/10236190701470407
[23]	X. Meng, J. Jiao, L. Chen, Global dynamics behaviors for a nonautonomous Lotka-Volterra almost periodic dispersal system with delays, Nonlinear Anal.-Theor. Methods Appl., 68 (2008), 3633–3645. doi: 10.1016/j.na.2007.04.006. doi: 10.1016/j.na.2007.04.006
[24]	C. Niu, X. Chen, Almost periodic sequence solutions of a discrete Lotka-Volterra competitive system with feedback control, Nonlinear Anal.: Real World Appl., 10 (2009), 3152–3161. doi: 10.1016/j.nonrwa.2008.10.027. doi: 10.1016/j.nonrwa.2008.10.027
[25]	Y. Xue, X. Xie, F. Chen, R. Han, Almost periodic solution of a discrete commensalism system, Discrete Dyn. Nat. Soc., 2015 (2015), 295483. doi: 10.1155/2015/295483. doi: 10.1155/2015/295483
[26]	J. Alzabut, Y. Bolat, T. Abdeljawad, Almost periodic dynamics of a discrete Nicholson's blowflies model involving a linear harvesting term, Adv. Differ. Equ., 2012 (2012), 158. doi: 10.1186/1687-1847-2012-158. doi: 10.1186/1687-1847-2012-158
[27]	D. Henry, Geometric theory of semilinear parabolic equations, New York: Springer, 1981. doi: 10.1007/BFb0089647.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)