Exponential stability of periodic solutions for a hematopoiesis model with two time delays

Authors

  • Abdelhafid Younsi

DOI:

https://doi.org/10.54021/seesv5n1-159

Keywords:

hematopoiesis models, periodic solutions, exponential convergence, time delays, concave operators

Abstract

 

In applied mathematics, many practical problems involving heat flow, species interaction, microbiology, neural networks, and more are associated with delay differential equations. To elucidate regulation and control mechanisms in physiological systems, Mackey and Glass introduced new mathematical models as a representation of hematopoiesis (blood-cell formation) in their influential and highly referenced 1977 paper. The presence of multiple delays, rather than a single delay, can lead to a new range of dynamics: an equation that was stable with one delay may become unstable when the delays are not equal, resulting in sustained oscillations. The dynamics of the non-autonomous Mackey-Glass model with two variable delays have not been thoroughly explored yet, which is presented as an open problem by Berezansky and Braverman. This paper focuses on the existence, uniqueness, and exponential stability of positive periodic solutions of a generalized class of the Hematopoiesis Model with two variable delays. Initially, using properties of cones and a fixed point theorem for a concave and increasing operator, we establish conditions ensuring the existence of a unique positive periodic solution for the model with almost periodic coefficients and delays. Subsequently, we employ a suitable Lyapunov functional method and differential inequality techniques to derive sufficient conditions guaranteeing the exponential stability of the unique positive periodic solution of the system. This approach covers a broad range of models that were previously studied. Additionally, an illustrative example with a numerical simulation is provided to validate the efficiency and reliability of the theoretical results. Our approach can be applied to the case of positive almost periodic solutions and positive pseudo almost periodic solutions of the model under consideration.

References

BARTHA, F. A.; KRISZTIN, T.; VIGH, A. Stable periodic orbits for the Mackey--Glass equation. Journal of Differential Equations, v. 296, p. 15-49, 2021.

BEREZANSKY, L.; BRAVERMAN, E. A note on stability of Mackey-Glass equations with two delays. Journal of Mathematical Analysis and Applications, v. 450, n. 2, p. 1208-1228, 2017.

BEREZANSKY, L.; BRAVERMAN, E.; IDELS, L. Mackey-Glass model of hematopoiesis with non-monotone feedback: Stability, oscillation and control. Applied Mathematics and Computation, v. 2196, p. 268-6283, 2013.

BEREZANSKY, L.; BRAVERMAN, E.; IDELS, L. Mackey-Glass model of hematopoiesis with monotone feedback revisited. Applied Mathematics and Computation, v. 219, n. 9, p. 4892-4907, 2013.

EDELSTEIN-KESHET, L. Mathematical models in biology. Philadelphia: SIAM, 2004.

FARIA, T.; OLIVEIRA, J. J. Global asymptotic stability for a periodic delay hematopoiesis model with impulses. Applied Mathematical Modelling, v. 79, p. 843-864, 2020.

FARIA, T.; OLIVEIRA, J. J. A note on global attractivity of the periodic solution for a model of hematopoiesis. Applied Mathematics Letters, v. 94, p. 1-7, 2019.

GUO, D. J.; LAKSHMIKANTHAM, V. Nonlinear Problems in Abstract Cones, v. 5 of Notes and Reports in Mathematics in Science and Engineering. Boston, Mass, USA: Academic Press, 1988.

GUO, D. Nonlinear Functional Analysis. Shandong, China: Science and Technology Press, 1985.

HALE, J. K.; VERDUYN LUNEL, S. M. Introduction to functional differential equations. New York: Springer, 1993.

JIANG, D.; WEI, J. Existence of positive periodic solutions for nonautonomous delay differential equations. Chinese Ann. Math. Ser. A, v. 20, p. 715-720, 1999. In Chinese.

KRISZTIN, T. Periodic solutions with long period for the Mackey-Glass equation. Electronic Journal of Qualitative Theory of Differential Equations, v. 2020, Terjedelem-12, 2020.

KRASNOSEL'SKII, M. A.; ZABREKO, P. P. Geometrical Methods of Nonlinear Analysis, vol. 263 of Fundamental Principles of Mathematical Sciences. Berlin, Germany: Springer, 1984.

KRASNOSEL'SKII, M. A. Positive Solution of Operators Equations. The Netherlands: Noordoff, Groningen, 1964.

LIU, G.; YAN, J.; ZHANG, F. Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis. Journal of Mathematical Analysis and Applications, v. 334, n. 1, p. 157-171, 2007.

LIANG, Z. D.; WANG, W. X.; LI, S. J. On concave operators. Acta Mathematica Sinica (English Series), v. 22, n. 2, p. 577-582, 2006.

MACKEY, M. C.; GLASS, L. Oscillation and chaos in physiological control system. Science, v. 197, p. 287-289, 1977.

PREOBRAZHENSKAYA, M. M. A relay Mackey-Glass model with two delays. Theor Math Phys, v. 203, p. 524-534, 2020.

SMITH, H. An introduction to delay differential equations with applications to the life sciences. Springer-Verlag, New York, 2011.

SAKER, S. H.; ALZABUT, J. O. On the impulsive delay hematopoiesis model with periodic coefficients. The Rocky Mountain Journal of Mathematics, p. 1657-1688, 2009).

SUN, J. Nonlinear Functional Analysis and Applications. Beijing, China: Science Press, 2007.

SUN, J. X. Some new fixed point theorems of increasing operators and applications. Applicable Analysis, v. 42, n. 3-4, p. 263-273, 1991.

TAN, Y.; ZHANG, M. Global exponential stability of periodic solutions in a nonsmooth model of hematopoiesis with time-varying delays. Mathematical Methods in the Applied Sciences, v. 40, n. 16, p. 5986-5995, 2017.

TAN, Y. Dynamics analysis of Mackey-Glass model with two variable delays. Math. Biosci. Eng, v. 17, n. 5, p. 4513-4526, 2020.

WANG, X.; LI, Z. Dynamics for a class of general hematopoiesis model with periodic coefficients. Appl. Math. Comput., v. 186, p. 460-468, 2007.

WAN, A.; JIANG, D.; XU, X. A new existence theory for positive periodic solutions to functional differential equations. Comput. Math. Appl., v. 47, p. 1257-1262, 2004.

WU, J.; LIU, Y. Fixed point theorems and uniqueness of the periodic solution for the hematopoiesis models. In: Abstract and Applied Analysis. 2012. v. 2012.

WU, X. M.; LI, J. W.; ZHOU, H. Q. A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis. Computers & Mathematics with Applications, v. 54, n. 6, p. 840-849, 2007.

WAN, A.; JIANG, D.; XU, X. A new existence theory for positive periodic solutions to functional differential equations. Comput. Math. Appl., v. 47, p. 1257-1262, 2004.

WANG, W. X.; LIANG, Z. D. Fixed point theorems for a class of nonlinear operators and their applications, Acta Mathematica Sinica. Chinese Series, v. 48, n. 4, p. 789-800, 2005.

WU, X.; LI, J.; ZHOU, H. A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis. Comput. Math. Appl., v. 54, p. 840-849, 2007.

WAN, A.; JIANG, D. Existence of positive periodic solutions for functional differential equations. Kyushu J. Math., v. 56, p. 193-202, 2002.

WU, X.; LI, J.; ZHOU, H. A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis. Comput. Math. Appl., v. 54, p. 840-849, 2007.

WAN, A.; JIANG, D. Existence of positive periodic solutions for functional differential equations. Kyushu J. Math., v. 56, p. 193-202, 2002.

YANG, X. T. Existence and global attractivity of unique positive almost periodic solution for a model of hematopoiesis. Applied Mathematics B, v. 25, n. 1, p. 25--34, 2010.

YAO, Z. New results on existence and exponential stability of the unique positive almost periodic solution for hematopoiesis model. Applied Mathematical Modelling, v. 39, n. 23-24, p. 7113-7123, 2015.

YAO, Z.; ALZABUT, J.; OBAIDAT, S. On periodic solutions of Mackey-Glass hematopoiesis model via concave and increasing operator (2022).

ZHAI, C. B.; LI, Y. J. Fixed point theorems for u0-concave operators and their applications. Acta Mathematica Scientia. Series A, v. 28, n. 6, p. 1023-1028, 2008.

Downloads

Published

2024-06-20

How to Cite

Younsi, A. (2024). Exponential stability of periodic solutions for a hematopoiesis model with two time delays. STUDIES IN ENGINEERING AND EXACT SCIENCES, 5(1), 3199–3226. https://doi.org/10.54021/seesv5n1-159