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...

We present several results dealing with the asymptotic behaviour of a real two-dimensional system $x^{\text{'}}\left(t\right)=𝖠\left(t\right)x\left(t\right)+{\sum }_{k=1}^m{𝖡}_k\left(t\right)x\left({\theta }_k\left(t\right)\right)+h(t,x\left(t\right),x\left({\theta }_1\left(t\right)\right),\cdots ,x\left({\theta }_m\left(t\right)\right))$ with bounded nonconstant delays $t-{\theta }_k\left(t\right)\ge 0$ satisfying ${lim}_{t\to \infty }{\theta }_k\left(t\right)=\infty$, under the assumption of instability. Here $𝖠$, ${𝖡}_k$ and $h$ are supposed to be matrix functions and a vector function, respectively. The conditions for the instable properties of solutions together with the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the real system considered to one equation with...

In this article, stability and asymptotic properties of solutions of a real two-dimensional system $x^{\text{'}}\left(t\right)=𝐀\left(t\right)x\left(t\right)+𝐁\left(t\right)x\left(\tau \left(t\right)\right)+𝐡(t,x\left(t\right),x\left(\tau \left(t\right)\right))$ are studied, where $𝐀$, $𝐁$ are matrix functions, $𝐡$ is a vector function and $\tau \left(t\right)\le t$ is a nonconstant delay which is absolutely continuous and satisfies $\underset{t\to \infty }{lim}\tau \left(t\right)=\infty$. Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented.

We study asymptotic properties of solutions of the system of differential equations of neutral type.

The main result of the present paper is obtaining new inequalities for solutions of scalar equation $\dot{x}\left(t\right)=b\left(t\right)x(t-\tau \left(t\right))$. Except this the interval of transient process is computed, i.e. the time is estimated, during which the given solution $x\left(t\right)$ reaches an $\epsilon$ - neighbourhood of origin and remains in it.