Stability Analysis of Anti-Periodic Solutions of the Time-Varying Delayed Hematopoiesis Model with Discontinuous Harvest
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Stability Analysis of Anti-Periodic Solutions of the Time-Varying Delayed Hematopoiesis Model with Discontinuous Harvesting Terms Fanchao Kong1 · Juan J. Nieto2 · Xiangying Fu3
Received: 17 October 2018 / Accepted: 26 March 2020 © Springer Nature B.V. 2020
Abstract This paper is concerned with a time-varying delayed hematopoiesis model with discontinuous harvesting terms. The harvesting terms considered in our hematopoiesis model are discontinuous which are totally different from the previous continuous, Lipschitz continuous or even smooth ones. By means of functional differential inclusions theory, inequality technique and the non-smooth analysis theory with Lyapunov-like approach, some new sufficient criteria are given to ascertain the existence and globally exponential stability of the anti-periodic solution for our proposed hematopoiesis model. Some previously known works are significantly extended and complemented. Moreover, simulation results of two topical numerical examples are also delineated to demonstrate the effectiveness of the theoretical results. Keywords Anti-periodic solution · Hematopoiesis model · Functional differential inclusions theory · Discontinuous harvesting terms · Global exponential stability
1 Introduction 1.1 Previous Works Hematopoiesis model which arisen in blood cell production, was firstly introduced and studied by Mackey and Glass [18]. They proposed some autonomous delay differential equations to describe the models, one of the equations is x (t) = −ax(t) +
b , t ≥0 1 + x n (t − τ )
B F. Kong
[email protected]; [email protected]
1
School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui 241000, P.R. China
2
Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain
3
College of Science, Jiujiang University, Jiujiang, Jiangxi 332005, P.R. China
F. Kong et al.
where x(t) denotes the density of mature cells in blood circulation, the cells are lost from the circulation at a rate a, and the flux of the cells into the circulation from the stem cell compartment depends on the density of mature cells at the previous time t − τ , τ is the time delay between the production of immature cells in the bone marrow and their maturation for release in circulating bloodstreams. Due to its applications in our daily lives, in last years, the qualitative properties for hematopoiesis model and its generalized models have been extensively investigated in literature, see for example [1], [21], [26], [27]. Meanwhile, In 1991, Gyori and Ladas [11] investigate the global attractivity of unique positive equilibrium of the following equation x (t) = −ax(t) +
b , t ≥ 0, 1 + x n (t − τ )
and Gyori and Ladas gave the open problem of extending the results to equations with several delays. In order to solve this problem, the following equation was come up with: x (t) = −ax(t) +
m i=1
bi (t) , n > 0, 1 + x n (t − τi (t))
(1.1)
and researchers have paid lots of attention on the qualitative properties of (1.1). For e
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