A note on global attractivity of the periodic solution for a model of hematopoiesis (Q1739466)

This paper considers a generalized periodic hematopoiesis model of the form \[ x'(t)=-a(t)x(t) + \sum_{i=1}^m \frac{b_i(t)}{1+ x(t-\tau_i(t))^n},\qquad t\geq 0 \tag{1} \] with $m\in \mathbb{N}$, $n\in (0,1]$, and $a,b_i : \mathbb{R} \to (0,\infty)$ and $\tau_i : \mathbb{R} \to [0,\infty)$ $\omega$-periodic continuous functions for all $i \in \{ 1,\dots,m\}$. \par Existence of a unique positive $\omega$-periodic solution of (1) was proved in Theorem~2.1 of [\textit{G. Liu} et al., J. Math. Anal. Appl. 334, No. 1, 157--171 (2007; Zbl 1155.34041)]. Moreover, some sufficient conditions for the global attractivity (among positive solutions) of this solution were also obtained [Theorem~3.1, loc. cit.]. \par The result of the present paper is the following theorem, which improves the global attractivity results obtained in [loc. cit.]. \par Let \begin{align*} \bar{a}=\max_{t\in[0,\omega]} a(t),\qquad \underline{b}_i & =\min_{t\in[0,\omega]} b_i(t), \qquad \tau(t)=\max_{1\leq i\leq m}\tau_i(t),\qquad \tau=\sup_{t\geq 0} \tau(t),\quad \text {and}\\ X_1 & =x_1 \exp\left(-\sup_{t\in[0,\omega]} \int_{t-\tau}^t a(u) \, du\right), \end{align*} where $x_1$ is the unique positive solution of the equation $\bar{a}x=\sum_{i=1}^m \frac{\underline{b}_i}{1+x^n}$. \par Then provided that there exist $T > 0$ such that \[ \frac{n X_1^{n-1}}{(1+X_1^n)^2} \sup_{t\geq T} \int_{t-\tau(t)}^t \sum_{i=1}^m b_i(s) e^{-\int_s^t a(u)\,du} \,ds < 1, \] the positive $\omega$-periodic solution $\tilde{x}(t)$ of (1) is globally attractive (in the set of all positive solutions). \par The applicability of the theorem is also illustrated by an example.