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

∣ x1(t) = A(t) 1+x 2 (t) − x1(t) x2(t) = B(t) 1+x 1 (t) − x2(t), where A and B belong to C+ and C+ is the set of continuous functions g : R −→ 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 x1 and x2 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. For more details on the interpretation of (1) one may see [1]. With C◦ we denote the space of continuous and bounded functions g : R −→ R. For g ∈ C◦ we define gL(∞) = lim inft−→∞g(t), gM(∞) = lim supt−→∞g(t), gL = inf{g(t) : t ∈ R}, gM = sup{g(t) : t ∈ R}.


Introduction
In this paper we will study the asymptotic behaviour of positive solutions to the system (1) x 1 (t) = A(t) 1+x n 1 (t) − x 2 (t), where A and B belong to C + and C + is the set of continuous functions g : R −→ 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.For more details on the interpretation of (1) one may see [1].With C • we denote the space of continuous and bounded functions g : R −→ R. For g ∈ C • we define g L (∞) = lim inf t−→∞ g(t), g M (∞) = lim sup t−→∞ g(t), g L = inf{g(t) : t ∈ R}, g M = sup{g(t) : t ∈ R}.

Preliminary results
Here and further next lemmas will pay important role.EJQTDE, 2000 No. 6, p. 1 Lemma 1. [2]Let g : (α, ∞) −→ R be a bounded and differentiable function.Then there exists a sequence {t n } ∞ n=1 such that t Lemma 2. [2]Let g ∈ C • be a differentiable function.Then there exists a sequence Proposition 1.Let (x 1 , x 2 ) be a positive solution of (1) and Proof.From lemma 1 there exists a sequence .
This completes the proof.
Remark.Proposition 1 shows that (1) is permanent, i. e. there exist positive constants α and β such that where (x 1 (t), x 2 (t)) is a positive solution of (1).In [3] was proved that permanence implies existence of positive periodic solutions of (1), when A(t) and B(t) are continuous positive periodic functions.
Let X 1 be a positive solution of the equation and X 2 be a positive solution of the equation Proposition 2.Let X 1 , X 2 be as above and In the same way we may prove other pair of inequalities.

3.Asymptotic behaviour of positive solutions
The results which are formulated and proved below are connected to (1) and to We notice that every solution to (1 Let also If (x 1 (t), x 2 (t)) and (x 1 * (t), x 2 * (t)) are positive solutions respectively of ( 1) and . Let .
We notice that β(t) −→ t−→∞ 0. For h 1 (t) we get the equation On the other hand . Let EJQTDE, 2000 No. 6, p. 5 For h 1 (t) and h 2 (t) we find the system (2) Since From last inequality and (2) we find that Therefore we get the contradiction The proof is complete. Let ) and (x 1 * (t), x 2 * (t)) are positive solutions respectively to (1) and (1 * ) and , where X 1 and X 2 as in proposition 2. Then .