### A Comparison Method for Forced Oscillations.

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We propose and analyze a mathematical model of hematopoietic stem cell dynamics. This model takes into account a finite number of stages in blood production, characterized by cell maturity levels, which enhance the difference, in the hematopoiesis process, between dividing cells that differentiate (by going to the next stage) and dividing cells that keep the same maturity level (by staying in the same stage). It is described by a system of n nonlinear differential equations with n delays. We study...

Inequalities for some positive solutions of the linear differential equation with delay ẋ(t) = -c(t)x(t-τ) are obtained. A connection with an auxiliary functional nondifferential equation is used.

In the paper we study the existence of nonoscillatory solutions of the system ${x}_{i}^{\left(n\right)}\left(t\right)={\sum}_{j=1}^{2}{p}_{ij}\left(t\right){f}_{ij}\left({x}_{j}\left({h}_{ij}\left(t\right)\right)\right),n\ge 2,i=1,2$, with the property $li{m}_{t\to \infty}{x}_{i}\left(t\right)/{t}^{{k}_{i}}=const\ne 0$ for some ${k}_{i}\in \{1,2,...,n-1\},i=1,2$. Sufficient conditions for the oscillation of solutions of the system are also proved.