Dynamics of the Almost Periodic Discrete Mackey–Glass Model

Zhijian Yao,Debaldev Jana,Jehad Alzabut

doi:10.3390/math6120333

Abstract

This paper is concerned with a class of the discrete Mackey–Glass model that describes the process of the production of blood cells. Prior to proceeding to the main results, we prove the boundedness and extinction of its solutions. By means of the contraction mapping principle and under appropriate assumptions, we prove the existence of almost periodic positive solutions. Furthermore and by the implementation of the discrete Lyapunov functional, sufficient conditions are established for the exponential convergence of the almost periodic positive solution. Examples, as well as numerical simulations are illustrated to demonstrate the effectiveness of the theoretical findings of the paper. Our results are new and generalize some previously-reported results in the literature.

Highlights

Introduction and PreliminariesThe nonlinear delay differential equation:x 0 (t) = −αx (t) + β 1 + x n (t − τ ) (1)was proposed by Mackey and Glass in [1] as an appropriate model for the dynamics of hematopoiesis, which describes the process of the production of blood cells
Was proposed by Mackey and Glass in [1] as an appropriate model for the dynamics of hematopoiesis, which describes the process of the production of blood cells
X denotes the density of mature cells in blood circulation at time t and τ is the time delay between the production of immature cells in the bone marrow and their maturation for release in the circulating bloodstream

Summary

Introduction and Preliminaries

The authors employed the fixed point theorem in cones to study the existence, nonexistence, and uniqueness of positive almost periodic solutions. Their approach was based on using a new fixed point theorem without compactness restrictions. In [18], Guo used the Lyapunov functional method and differential inequality techniques to study the exponential stability of pseudo almost periodic solutions of Model (2).

Boundedness and Extinction of Solutions

Existence of the Almost Periodic Positive Solution

Exponential Convergence

Examples and Numerical Simulations

Conclusions