Stability and Bifurcation Analysis of a Type of Hematopoietic Stem Cell Model

The observed dynamical property illustrates that state feedback control may stabilize invariant attractor to stable state in a simple version of hematopoietic stem cell model. The stability character of the positive steady state is analyzed by the computation of the rightmost characteristic roots in complex plane. Hopf bifurcation points are tracked as the roots curve crossing imaginary axis from the left half plane to the right half plane continuously. The bifurcation direction and stability of the bifurcating periodical solution are discussed by norm form computation combined with the center manifold theory. Furthermore, the numerical simulation verifies that instead of chaos, system is stabilized to period-1, 2, 3, 4 and period-7 periodical solutions in some delay windows, and the continuous of periodical solutions is also numerical simulated with varying free parameters continuously.


Introduction
Delay differential equations have been broadly focused in every fields of scientific investigation work since time delay is a natural factor in the reality life [1] [2] [3]. For example, cell's maturation time in biology, the reflection time for driving in road traffic, etc. In engineering control fields, people try to design delay control apparatus to let the original system stabilized to stable periodical orbits or produce new bifurcation behavior [4] [5]. The more attention work is the state feedback control in engineering control studying, the periodical oscillation behavior is induced via either time delay state feedback or difference feedback control [6] [7].
As is well known, the new periodical oscillation phenomena occur at Hopf point since system lost its stability. As for the eigenvalue problem of DDEs, Hopf bifurcation occurs as the imaginary roots cross the imaginary axis from the left half plane to the right half plane. With single time delay effects, people try to analyze Hopf bifurcation of linear DDEs by computing imaginary roots by the algebra method [8] [9]. Wang and Hu have shown the high analyzing technique by applying Sturm criterion with Maple language computation [10] [11]. Kuang computes the eigenvalue problem of a type DDEs by locating imaginary roots of Hopf bifurcation with consideration of delay-dependant physical parameter [12]. Together with Cookie's work [13], Kuang's work invokes people's big interest in studying biological model with delay-dependent nonlinear coefficient. In fact, it is ubiquitous to introduce delay-dependent physical parameter in biological models since nonlinear birth rate to specify the loss rate in specie's growth stage [14] [15] [16]. The stability switching always brings forth the periodical oscillation phenomena and complex dynamical behavior. A standard and much studied work of DDE is the Mackey-Glass equation which is proposed to model the production of white blood cells and given by For which an invariant attractor is observed with parameters such as With state feedback control of time delay, system (1) is described as Herein, x denotes the concentration of white blood cells, a, c are Hill coefficients, and b represents death rate.
Since in one respect, with state feedback control to perturb system's dynamics to produce new bifurcation behavior; and in another respect, mathematically, to reflect the function of perturbation of state difference biologically since migration phenomena in model analysis.
The different dynamical character of hematopoietic stem cell model is displayed as shown in Figure 1(a) and Figure 1(b). The observed invariant attractor is becoming asymptotically stable steady state as exerting feedback control with strength 0.2 K = − . As varying the free parameter, the steady state may loss its stability into instability state to induce the oscillation phenomena arising in system. To further analyze the bifurcation mechanism of the stability of the equilibrium solution, the characteristic root with zero real part appearing in its characteristic equation is analyzed analytically. Hence, we endeavour to calculate the rightmost characteristic root wisdomly from geometrical point view which to decide the asymptotically stability of the equilibrium solution.
With the consideration of delay-dependent physical parameter in DDEs, we solve the eigenvalue problem of Hopf bifurcation by a new method which is illustrated as geometrical criterion in paper [17] [18]. Numerically, DDE-Biftool can compute the rightmost roots of the linear characteristic equation. We devote to compute the rightmost characteristic roots of the linear DDE of system (1) and (1) to give out the asymptotically stable condition, and make earnest endeavors to draw curve of the rightmost characteristic roots continuously as varying parameter K. Hopf bifurcation of Equation (1) and (1) are also analyzed to show the bifurcation of periodical oscillation solutions with small amplitude. Furthermore, based on the fundamental theory of functional differential equations [19] [20] [21], the bifurcating periodical solutions is computed via norm form analytical technique combined center manifold theory. The numerical simulation displays that the observed perioid-1, 2, 3, 4 and perioid-7 solution and chaos solution in different delay window underlying state feedback control. The whole paper is organized as the listed. In Section 2, the distribution of characteristic roots in a band is calculated via geometrical analyze technique and the rightmost characteristic roots determines the stability of the positive steady state, and Hopf bifurcation is analyzed in Section 3. Based on the fundamental theory of DDEs, the dimension reduction system of system (1) and (1) is computed and analyzed combined with the center manifold theory. The bifurcation direction and the stability of the bifurcating periodical solution are derived via formal norm analytical technique. The numerical simulation has shown the continuous of oscillation solutions as varying free parameters.

The Computation of the Rightmost Characteristic Root
The characteristic equation of the linear DDEs of Equation (1) and Equation (1) can be written as For the fixed value of Hence we derive the following lemma 2.1, Define the line  : then the intersection point * S determines the corresponding characteristic Setting ( ) ( ) x t x t τ − = , the positive steady state of Equation (1) and Equation Hence one has Equation (2) is rewritten as By the second equation in Equations (2) We also draw the following conclusion: The switching of stability of the only equilibrium solution is shown in Figure  2(a) and Figure 2 The solution is asymptotically stable for small time delay then change to be unstable as increasing strength K. We draw the stable regime with green as shown in Figure 3 and red regime represents the unstable regime. It can be seen Hopf bifurcation occurs as stability property of the equilibrium solution is changed with varying parameter a and b and the amplifying picture denotes the stability property of equilibrium solution with the specified value of 0.1 K = − as varying b continuously. Hopf bifurcation line is also drawn with blue line and the discussion of Hopf bifurcation is given in Section 3.

Hopf Bifurcation
As shown in Figure 2 and and further to obtain Hopf line in ( ) Hopf line is shown in Figure 4 with blue color. By Equation (16), one obtains that Following we will compute the transversal condition for Hopf bifurcation. The characteristic Equation (2) can be rewritten as That is, On another respect, differentiate Equation (3) with respect to S to get By Equation (3) and Equation (3) one can compute that

Bifurcation of Periodical Solutions
As we have discussed in Section 3, Hopf bifurcation occurs with one pair of imaginary roots iω with zero real part of the characteristic equation. Suppose with the phase space Equation (3) can be written as its opearator differential form The solution operator of Equation (4) is a strong continuous semigroup with infinitesimal generator The adjoint operator in the conjugate space  . Therefore, for any t x C ∈ , it is written as t t x zq zq y = + + , substitute it into operator differential Equation (4) to obtain By Equation (4) Figure 5(a). The bifurcating stable periodical solution at 0.13 b = is shown in Figure 5 . Comparing with the period of oscillating solution before bifurcation, a periodical solution with two times period is produced since floquent multiplier is arriving at −1 at the bifurcation  Figure 5(d). The state feedback control in system (1) produce stable periodical orbits within some time delay windows, which may function hematopoietic stem cells system to visualize stable oscillation solution instead of chaos.

Conclusions
The dynamics of a hematopoietic stem cell model with delay state feedback control is discussed. Underlying super-critical Hopf bifurcation, system lost its stability to experience periodical oscillation behavior. The curve of the rightmost characteristic roots is continuously simulated with varying free parameters, hence Hopf point is found as the roots curve cross the imaginary axis from the left half plane to the right half plane. Furthermore, the stable and unstable regime of the steady state is partitioned by Hopf bifurcation curve. The continuous bifurcating periodical solution is carried out with varying free parameters. It is discovered that period-1, 2, 3, 4, and period-7 solution arises underlying adding-period bifurcation and period-doubling bifurcation of periodical solutions. The results visualized the stable periodical orbits instead of chaos under the state feedback control with time delay.
In this paper, we discussed the hematological system model underlying delay state feedback control. However, for simplicity, the feedback delay is uniform with the mature delay as in stem cells growth stage. We will further discuss the hematological system which contains two different time delays in later paper.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.