Study of the Existence of Global Attractors for the Wezewska, Czyewska and Lasota Models

: In this research we present a study of global attractors in mathematical models of differential equations, which are an important tool in mathematics; furthermore, taking advantage of the stability of the solutions, it was possible to determine the control of biomedical phenomena, among other aspects, present in various population groups. Likewise, differential equation models are used to simulate biological, epidemiological and medical phenomena, among others. The reference population groups used in this research are the family of population models given by the differential equations N'(t)=p(t, N (t)) - d(t, N (t)). A particular case of this family of differential equations is the mathematical model called the Wezewska, Czyewska and Lasota (WCL) model, whose form is given by: N'(t)=pe (-q) - µN (t). This model describes the survival of red blood cells (erythrocytes) in humans. The WCL model, in discrete variable, has a non-trivial global attractor. In this research we demonstrate, using the Schwarz derivative technique, the existence of at least one model global attractor. On the other hand, the results of the present investigation showed the existence of a single fixed point, as the only global attractor characterized by the equation N=pe (-qN) - µN.


Introduction
The study of global attractors in models of differential equations, as well as their existence, definition, stability and application in various spaces and models, has been treated in recent years by several researchers, as expressed in [1][2][3][4][5][6][7][8].
Similarly, fractional Lasota-Wazewska models have been studied in [9,10]. This article is motivated by works [1][2][3], where the authors study the existence of global attractors for equations in differences of the Wezewska, Czyewska and Lasota (WCL) model, on the existence of the fixed point. The authors, in the previous works, up to that date, have not found a rigorous test, they only present a graph that gives an idea of the existence, and the uniqueness is not proven, which is an open problem. It should be noted that models such as those treated in the works mentioned above were described, for the first time, using differential equations, in [11,12]. Similarly, epidemiological studies have been carried out using differential equation models, such as those discussed in [13], as well as in [14].
In this research, we consider the family of population models given by: Different biological and medical phenomena were described by differential equations between the 1950s and 1980s, such as the production of blood cells and the general approach, proposed by equation (1), to describe the dynamics of cell production [15]. In equation (1), p (t, N (t)) is the Wezewska, Czyewska and Lasota Models production rate of the cells at time t, provided that the population size is N and d(t, N(t)) is the rate of mortality. It was a common belief for researchers that mortality was proportional to the number of circulating blood cells; that is: d(t, N (t))=µN(t). However, in this research we consider p(t, N(t))=pe (-qN (t)) and the model = − (2) to prove the existence of a single fixed point, as a single global attractor, characterized by equation (2).

Methods
A documentary research was carried out with a qualitative approach; Since a study of global attractors in models of differential equations is presented, these tools of great importance in mathematics. Behavior and responses are observed, by collecting all possible bibliographic material on the subject, making a detailed study of it. The hypotheticaldeductive method, characteristic of mathematical research, was used, so that the conjectures could be answered during the development of the work, which led to the achievement of the objectives. The present work shows an applied form that confronts theory with reality [17,18].

Preliminaries
Consider the system of differential equations ´ = , con t > 0, where f is a map of C ∞ (I), with I=[a, b]. Let's consider the linearized system: Where x* is an equilibrium point of equation (4); that is, Definition 1. An equilibrium point or stationary solution of a differential equation is a solution y(x)=a (constant) for all x ∈ R. That is, the stationary solutions or equilibrium points are those whose graphs are straight horizontal. Definition 2. The equilibrium point x * is locally stable for Definition 3. We will say that equation (3) has a fixed point at c, if f (c)=c.
Definition 4. Let f: Ω ⊆ R n → R n be a C 1 field and let x 0 be an equilibrium point of x ′=f (x).
1. It is said that x 0 is stable, if for all ε > 0 there exists δ > 0, such that for all ∈ # , Φ is defined for all 2. It is said that x 0 is asymptotically stable, if x 0 is stable and there is also for all x ∈ B (x 0 , r).
Definition 5. An equilibrium point x* is globally asymptotically stable, if it is locally stable and alsolim →( = * . Theorem 1. The real part of the eigenvalue (∂f/∂x(x^*)) is negative if and only if x* is local asymptotically stable. Definition 6. The critical points ) or the fixed points care locally stable or globally asymptotically stable, if they verify the previous definitions.
Remember that the critical points ) are those where f '( ) )=0.
Definition 7. A function f: R → R is contractive, if there is a constant k <1 such that for any x 1 , x 2 ∈ R holds: That is, a contractive application is one that contracts the distances with a contraction ratio strictly less than unity. Where, in the set of real numbers, the distance between two of them is defined as follows: So the definition of contractive application is as follows: Theorem 2 (Contraction). We will say that the map T: E → E, E ⊂ R is a contraction or a contractive operator, if there exists a constant L ∈ [0, 1), such that |* − * | ≤ ,| − |, for all x, y ∈ E.

Theorem 3. (Contractive fixed point). If T: E → E is a contraction, then T has an unique fixed point on E.
Definition 8 (Derived from Schwarz). The Schwarz derivative is defined for a real function f of class C 3 , as: At all points where 0 ≠ 0.
This derivative has been used in differential delay equations with unimodal feedback, verifying that they do not satisfy the Schwarz derivative condition [16].  Proof. If µ > 0, then by Theorem 5 equation (3) has an unique critical point given by the equation − = 0, let this be x*, and by Theorem 6 x* is local asymptotically stable. Then T has an unique fixed point. So, the equation ´ = − , -1 < µ < 0 has an unique fixed point.

Global Analysis of the Asymptotic
Stability of the Fixed Point C Theorem 9. For p, q ∈ (0, 1) and -1 < µ < 0, the fixed point Therefore, C is globally asymptotically stable. We note that equation (2) is a scalar differential equation.

Conclusions
It is natural to establish a control in the parameters that are exhibited in the equation in order to guarantee existence of: fixed points, equilibrium points and critical points. The first two are those that, in general, are of extreme importance for the global study of stability. As well as, the technique used in these systems is what we know as the famous Schwarz derivative technique.
For p, q ∈ (0, 1) and -1 < µ < 0, the fixed point C for the An interesting open problem would be to pose an analogous model in R 2 as follows: Here it is notable that the system involves more parameters and it is obvious that a control can be established in the parameters to guarantee the fixed points and the equilibrium points. Known techniques to be used will be Dulac, Poincare-Bendixon.
For three-dimensional differential equations the best known technique is the Liapunov functions.
For dimensions in complicated spaces (normed, Banach, among others), more hypotheses must be added in the problem that support the stock conditions fixed points and critical points, and therefore their stability in the global sense.
Another important aspect for differential equations of the abstract type is to be able to establish a definition of generalized dichotomy, which deduces generalized global stability for spaces of the abstract type. These are problems of interest raised in [14].