Almost periodic oscillations in a generalized Mackey–Glass model of respiratory dynamics with several delays

Abstract

By using some analytical techniques, modified inequalities and Mawhin's continuous theorem of coincidence degree theory, some simple sufficient conditions for the existence of at least one positive almost periodic solution of a generalized Mackey–Glass model of respiratory dynamics are obtained. Further, the global attractivity of positive almost periodic solution of the above model is also studied. To the best of the author's knowledge, so far, the result of this paper is completely new. Finally, three examples are given to illustrate the main results in this paper.

Existence and global attractivity of positive almost periodic solutions for a kind of fishing model with pure-delay

“…Lemma 2.5. (Zhang [27]) Assume that x ∈ AP (R) andx > 0, then for ∀t 0 ∈ R, there exists a positive constant T 0 independent of t 0 such that…”

“…Hence, if we consider the effects of the environmental factors, almost periodicity is sometimes more realistic and more general than periodicity. In recent years, the almost periodic solution of the continuous models in biological populations has been studied extensively (see [16,21,26,27,28,29,30,31,32] and the references cited therein). Example 1.1.…”

Organization and identification of solutions in the time-delayed Mackey-Glass model

“…As the delay further increases, a sequence of bifurcations generates oscillations with higher periods and eventually aperiodic behavior. 19 Over the years, the dynamics of the MG model has been investigated numerically, [20][21][22][23][24][25] and it has been used to generate high-dimensional chaotic signals. 26 The MG model has also been used as a toy model to study chaotic synchronization in delayed systems.…”

Dynamics of the Almost Periodic Discrete Mackey–Glass Model

“…Indeed, it has been realized that due to various seasonal effects and certain environmental factors in real-life situations, the study of biological models under periodic or almost periodic perturbations has become obligatory. Many authors have incorporated this idea into their investigations and employed several techniques such as the degree theory, the Lyapunov functional approach, and some fixed point theorems to study the dynamic behavior of this model and its modifications in the last few years [5][6][7][8][9][10][11][12][13][14][15][16]. In [17], Wang and Zhang studied the following Mackey-Glass model:…”