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On the global attractivity controversy for a delay model of hematopoiesis. (English) Zbl 1131.34327

In this short paper it is shown, with the help of the general theory of monotone systems, and under some specific hypotheses for function $f$ that the trivial equilibrium of the delayed differential equation

${x}^{\text{'}}\left(t\right)=-\mu x\left(t\right)+f\left(x\left(t-r\right)\right)$

is stable and attracts a large subset of positive solutions, for all the values of the delay $r$. Such hypotheses are satisfied by some values of $m$ and $n$ in the delayed differential equation

${p}^{\text{'}}\left(t\right)=\frac{\beta {p}^{m}\left(t-r\right)}{1+{p}^{n}\left(t-r\right)}-\gamma p\left(t\right)$

which is proposed to describe the dynamics of hematopoiesis and reduces for $m=1$ to the classical Mackey-Glass equation. Therefore, this paper serves to clarify the controversy on the global attractiveness of positive soltions to this model, as affirmed in [S. H. Saker, Appl. Math. Comput. 136, 241-250 (2003; Zbl 1026.34082)]. This claim is not valid in general due to the structure of the solutions around the zero, and to put it in evidence it is not necessary to construct particular counterexamples as has been done in [S. J. Yang, B. Shi and M. J. Gai, Appl. Math. Comput., 168, 973–980 (2005; Zbl 1084.34543)].

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 92D25 Population dynamics (general)