Asymptotic behaviour of positive solutions of the model which describes cell differentiation

S. Georgiev, University of Sofia, Sofia, Bulgaria

E. J. Qualitative Theory of Diff. Equ., No. 6. (2000), pp. 1-17.

Abstract: In this paper we will study the asymptotic behaviour of positive solutions to the system
$$\left|
\begin{array}{lcr}
x_1^{\prime}(t)={{A(t)}\over {1+x_2^n(t)}}-x_1(t)\\
x_2^{\prime}(t)={{B(t)}\over {1+x_1^n(t)}}-x_2(t),
\end{array}
\right.\leqno{(1)}$$
where $A$ and $B$ belong to ${\cal C}_+$ and ${\cal C}_+$ is the set of continuous functions $g:{\cal R}\longrightarrow {\cal R}$, which are bounded above and below by positive constants. $n$ is fixed natural number. The system (1) describes cell differentiation, more precisely - its passes from one regime of work to other without loss of genetic information. The variables $x_1$ and $x_2$ make sense of concentration of specific metabolits. The parameters $A$ and $B$ reflect degree of development of base metabolism. The parameter $n$ reflects the highest row of the repression's reactions.

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