Almost periodic solutions of neutral-type differential system on time scales and applications to population models

Jing Ge; Xiaoliang Li; Bo Du; Famei Zheng; Jing Ge; Xiaoliang Li; Bo Du; Famei Zheng

doi:10.3934/math.2025180

AIMS Mathematics

2025, Volume 10, Issue 2: 3866-3883. doi: 10.3934/math.2025180

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Research article

Almost periodic solutions of neutral-type differential system on time scales and applications to population models

Jing Ge ¹,
Xiaoliang Li ^{2
,
,},
Bo Du ^{1
,
,},
Famei Zheng ¹

1.
School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, Jiangsu, China
2.
Jiyang College, Zhejiang Agriculture and Forestry University, Zhuji 311800, Zhejiang, China

Received: 09 January 2025 Revised: 10 February 2025 Accepted: 19 February 2025 Published: 26 February 2025
MSC : 34K14, 34K20

We first study almost periodic solutions of neutral-type differential system on time scales and establish some basic results for the considered system. Furthermore, based on these results, the dynamic behaviors of two classes of neutral-type biological population models including host-macroparasite model and Lasota–Wazewska model are obtained. It is worth mentioning that we study almost periodic solutions for neutral-type differential system on time scales. Furthermore, using the above study and exponential dichotomy method, we investigate two types of biological population models.
- almost periodic solution,
- time scales,
- global exponential stability,
- exponential dichotomy
Citation: Jing Ge, Xiaoliang Li, Bo Du, Famei Zheng. Almost periodic solutions of neutral-type differential system on time scales and applications to population models[J]. AIMS Mathematics, 2025, 10(2): 3866-3883. doi: 10.3934/math.2025180

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Abstract

We first study almost periodic solutions of neutral-type differential system on time scales and establish some basic results for the considered system. Furthermore, based on these results, the dynamic behaviors of two classes of neutral-type biological population models including host-macroparasite model and Lasota–Wazewska model are obtained. It is worth mentioning that we study almost periodic solutions for neutral-type differential system on time scales. Furthermore, using the above study and exponential dichotomy method, we investigate two types of biological population models.

References

[1]	G. Seifert, A condition for almost periodicity with some applications to functional-differential equations, J. Differ. Equations, 1 (1965), 393–408. https://doi.org/10.1016/0022-0396(65)90001-X doi: 10.1016/0022-0396(65)90001-X
[2]	A. M. Fink, Almost periodic differential equations, Berlin: Springer, 1974. https://doi.org/10.1007/BFb0070324
[3]	A. M. Fink, G. Seifert, Liapunov functions and almost periodic solutions for almost periodic systems, J. Differ. Equations, 5 (1969), 307–313. https://doi.org/10.1016/0022-0396(69)90045-x doi: 10.1016/0022-0396(69)90045-x
[4]	C. David, M. Cristina, Invariant manifolds, global attractors and almost periodic solutions of nonautonomous defference equations, Nonlinear Anal.-Theor., 56 (2004), 465–484. https://doi.org/10.1016/j.na.2003.09.009 doi: 10.1016/j.na.2003.09.009
[5]	Y. Y. Li, L. Yang, Existence and stability of almost periodic solutions for Nicholson's blowflies models with patch structure and linear harvesting terms on time scales, Asian-Eur. J. Math., 5 (2012), 1250038. https://doi.org/10.1142/S1793557112500386 doi: 10.1142/S1793557112500386
[6]	M. Almheidat, H. Yasmin, M. Al Huwayz, R. Shah, S. A. El-Tantawy, A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform, Open Phys., 22 (2024), 20240081. https://doi.org/10.1515/phys-2024-0081 doi: 10.1515/phys-2024-0081
[7]	A. S. Alshehry, H. Yasmin, R. Shah, A. Ali, I. Khan, Fractional-order view analysis of Fisher's and foam drainage equations within Aboodh transform, Eng. Computation., 41 (2024), 489–515. https://doi.org/10.1108/ec-08-2023-0475 doi: 10.1108/ec-08-2023-0475
[8]	H. Yasmin, A. Alrowaily, M. Areshi, R. Shah, S. A. El-Tantawy, On the analytical soliton-like solutions to (2+1)-dimensional fractional asymmetric Nizhnik–Novikov–Veselov system arising in incompressible fluids, Front. Phys., 12 (2024), 1443986. https://doi.org/10.3389/fphy.2024.1443986 doi: 10.3389/fphy.2024.1443986
[9]	X. Y. Gao, In an ocean or a river: Bilinear auto-B$\ddot{a}$cklund transformations and similarity reductions on an extended time-dependent (3+1)-dimensional shallow water wave equation, China Ocean Eng., 2 (2025), 1–10. https://doi.org/10.1007/s13344-025-0012-y doi: 10.1007/s13344-025-0012-y
[10]	X. Y. Gao, Hetero–B$\ddot{a}$cklund transformation, bilinear forms and multi-solitons for a (2+1)-dimensional generalized modified dispersive water-wave system for the shallow water, Chinese J. Phys., 92 (2024), 1233–1239. https://doi.org/10.1016/j.cjph.2024.10.004 doi: 10.1016/j.cjph.2024.10.004
[11]	X. Y. Gao, Symbolic computation on a (2+1)-dimensional generalized nonlinear evolution system in fluid dynamics, plasma physics, nonlinear optics and quantum mechanics, Qual. Theory Dyn. Syst., 23 (2024), 202. https://doi.org/10.1007/s12346-024-01045-5 doi: 10.1007/s12346-024-01045-5
[12]	S. Hilger, Analysis on measure chains–A unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56. https://doi.org/10.1007/BF03323153 doi: 10.1007/BF03323153
[13]	C. Lungan, V. Lupulescu, Random dynamical systems on time scales, Electronic Journal of Diﬀerential Equations, 2012 (2012), 86.
[14]	M. Adivar, Y. N. Raffoul, Stability, periodicity and boundedness in functional dynamical systems on time scales, Cham: Springer, 2020. https://doi.org/10.1007/978-3-030-42117-5
[15]	J. R. Graef, M. Hill, Nonoscillation of all solutions of a higher order nonlinear delay dynamic equation on time scales, J. Math. Anal. Appl., 423 (2015), 1693–1703. https://doi.org/10.1016/j.jmaa.2014.10.061 doi: 10.1016/j.jmaa.2014.10.061
[16]	J. M. Zhang, M. Fan, H. P. Zhu, Existence and roughness of exponential dichotomies of linear dynamic equations on time scales, Comput. Math. Appl., 59 (2010), 2658–2675. https://doi.org/10.1016/j.camwa.2010.01.035 doi: 10.1016/j.camwa.2010.01.035
[17]	R. P. Agarwala, C. H. Zhang, T. X. Li, Some remarks on oscillation of second order neutral differential equations, Appl. Math. Comput., 274 (2016), 178–181. https://doi.org/10.1016/j.amc.2015.10.089 doi: 10.1016/j.amc.2015.10.089
[18]	Y. K. Li, C. Wang, Almost periodic functions on time scales and applications, Discrete Dyn. Nat. Soc., 2011 (2011), 727068. https://doi.org/10.1155/2011/727068 doi: 10.1155/2011/727068
[19]	Y. K. Li, B. Li, Existence and exponential stability of positive almost periodic solution for Nicholson's blowflies models on time scales, SpringerPlus, 5 (2016), 1096. https://doi.org/10.1186/s40064-016-2700-9 doi: 10.1186/s40064-016-2700-9
[20]	F. Al-Dheleai, E. Elabbasy, New oscillation criteria for second order half-linear neutral type dynamic equations on time scales, Journal of Applied Mathematics and Physics, 9 (2021), 2376–2398. https://doi.org/10.4236/jamp.2021.99151 doi: 10.4236/jamp.2021.99151
[21]	W. G. Yang, W. W. Yu, J. D. Cao, F. E. Alsaadi, T. Hayat, Almost automorphic solution for neutral type high-order Hopfield BAM neural networks with time-varying leakage delays on time scales, Neurocomputing, 267 (2017), 241–260. https://doi.org/10.1016/j.neucom.2017.05.089 doi: 10.1016/j.neucom.2017.05.089
[22]	A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906–920. https://doi.org/10.1016/j.jmaa.2008.03.019 doi: 10.1016/j.jmaa.2008.03.019
[23]	L. P. Jiang, Generalized exponential dichotomy and global linearization, J. Math. Anal. Appl., 315 (2006), 474–490. https://doi.org/10.1016/j.jmaa.2005.05.042 doi: 10.1016/j.jmaa.2005.05.042
[24]	J. A. Ball, M. W. Raney, Discrete-time dichotomous well-linear systems and generalized Schur–Navanlinna–Pick interpolation, Complex Anal. Oper. Theory, 1 (2007), 1–54. https://doi.org/10.1007/s11785-006-0001-y doi: 10.1007/s11785-006-0001-y
[25]	S. N. Chow, H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach space, J. Differ. Equations, 120 (1995), 429–477. https://doi.org/10.1006/jdeq.1995.1117 doi: 10.1006/jdeq.1995.1117
[26]	A. L. Sasu, Integral equations on function spaces and dichotomy on the real line, Integr. Equ. Oper. Theory, 58 (2007), 133–152. https://doi.org/10.1007/s00020-006-1478-5 doi: 10.1007/s00020-006-1478-5
[27]	M. Bohner, A. Peterson, Dynamic equations on time scales: An introduction with applications, Boston: Birkh$\ddot{\mathrm{a}}$user, 2001. https://doi.org/10.1007/978-1-4612-0201-1
[28]	S. P. Lu, W. G. Ge, Z. X. Zheng, Periodic solutions to neutral differential equation with deviating arguments, Appl. Math. Comupt., 152 (2004), 17–27. https://doi.org/10.1016/S0096-3003(03)00530-7 doi: 10.1016/S0096-3003(03)00530-7
[29]	R. M. Anderson, R. M. May, Infectious diseases of humans: dynamics and control, Oxford: Oxford University Press, 1991. https://doi.org/10.1093/oso/9780198545996.001.0001
[30]	M. Elabbasy, S. H. Saker, K. Saif, Oscillation in host macroparasite model with delay time, Far East Journal of Applied Mathematics, 4 (2000), 119–142.
[31]	Z. J. Yao, Existence and global exponential stability of an almost periodic solution for a host-macroparasite equation on time scales, Adv. Differ. Equ., 2015 (2015), 41. https://doi.org/10.1186/s13662-015-0383-0 doi: 10.1186/s13662-015-0383-0
[32]	M. Wazewska-Czyzewska, A. Lasota, Mathematical problems of the dynamics of the red blood cells, Mathematica Applicanda, 4 (2016), 23–40.
[33]	K. Gopalsamy, S. I. Trofimchuk, Almost periodic solutions of Lasota–Wazewska-type delay differential equation, J. Math. Anal. Appl., 237 (1999), 106–127. https://doi.org/10.1006/jmaa.1999.6466 doi: 10.1006/jmaa.1999.6466
[34]	G. T. Stamov, On the existence of almost periodic solutions for the impulsive Lasota–Wazewska model, Appl. Math. Lett., 22 (2009), 516–520. https://doi.org/10.1016/j.aml.2008.07.002 doi: 10.1016/j.aml.2008.07.002
[35]	G. R. Liu, A. M. Zhao, J. R. Yan, Existence and global attractivity of unique positive periodic solution for a Lasota–Wazewska model, Nonlinear Anal.-Theor., 64 (2006), 1737–1746. https://doi.org/10.1016/j.na.2005.07.022 doi: 10.1016/j.na.2005.07.022

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