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\AtBeginDocument{{\noindent\small
2004-Fez conference on Differential Equations and Mechanics \newline
{\em Electronic Journal of Differential Equations},
Conference 11, 2004, pp. 167--173.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{167}
\begin{document}
\title[\hfilneg EJDE/Conf/11 \hfil Stability and Hopf bifurcation]
{Stability and Hopf bifurcation in \\
a haematopoietic stem cells model}
\author[H. Talibi A. \& R. Yafia\hfil EJDE/Conf/11 \hfilneg]
{Hamad Talibi Alaoui, Radouane Yafia} % in alphabetical order
\address{Hamad Talibi Alaoui \hfill\break
Universit\'{e} Chouaib Doukkali Facult\'{e} des Sciences\\
D\'{e}partement de Math\'{e}matiques et Informatique \\
B.P. 20, El Jadida, Morocco}
\email{talibi@math.net}
\address{Radouane Yafia \hfill\break
Universit\'{e} Chouaib Doukkali Facult\'{e} des Sciences\\
D\'{e}partement de Math\'{e}matiques et Informatique \\
B. P. 20, El Jadida, Morocco}
\email{yafia\_radouane@hotmail.com}
\date{}
\thanks{Published October 15, 2004.}
\subjclass[2000]{34K18}
\keywords{Haematopoietic stem cells model; delayed differential equations;
\hfill\break\indent Hopf bifurcation; periodic solutions}
\begin{abstract}
We consider the Haematopoietic Stem Cells (HSC) Model
with one delay, studied by Mackey \cite{m78,m96}
and Andersen and Mackey \cite{M1}.
There are two possible stationary states in the model. One of them
is trivial, the second $E^{*}(\tau )$, depending on the delay,
may be non-trivial . This paper investigates the stability of
the non trivial state as well as the occurrence of the Hopf
bifurcation depending on time delay.
We prove the existence and uniqueness of a critical values
$\tau_{0}$ and $\overline{\tau}$ of the delay such that
$E^{*}(\tau )$ is asymptotically stable for $\tau <\tau _{0}$
and unstable for $\tau _{0}<\tau <\overline{\tau }$.
We show that $ E^{*}(\tau_{0})$ is a Hopf bifurcation critical
point for an approachable model.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}
\section{Introduction}
The population of haematopoietic stem cells (HSC) give
rise to all of the different elements of the blood: the white
blood cells, red blood cells, and platelets, which may be either
actively proliferating or in a resting phase. After entering the
proliferating phase, a cell is committed to undergo cell division
at a fixed time $\tau $ later. The generation time $\tau $ is
assumed to consist of four phases, $G_{1}$ the pre-synthesis
phase, $S$ the DNA synthesis phase, $G_{2}$ the post-synthesis
phase and $M$ the mitotic phase. Just after the division, both
daughter cells go into the resting phase called $G_{0}$-phase.
Once in this phase, they can either return to the proliferating
phase and complete the cycle or die before ending the cycle.
The dynamics of the (HSC) are governed by the coupled differential
delay equation (see \cite{M1,m78,m96,M2}):
\begin{equation}
\begin{gathered}
\frac{dN}{dt}=-\delta N-\beta (N)N+2e^{-\gamma \tau }\beta (N_{\tau })N_{\tau }\\
\frac{dP}{dt}=-\gamma P+\beta (N)N-e^{-\gamma \tau }\beta (N_{\tau })N_{\tau}
\end{gathered} \label{1}
\end{equation}
where $\beta $ is a monotone decreasing function of $N$ which has
the explicit form of a Hill function,
\begin{equation}
\beta (N)=\beta _{0}\frac{\theta ^{n}}{\theta ^{n}+N^{n}}\,. \label{2}
\end{equation}
The symbols in equation (\ref{1}) have the following
interpretation. $N$ is the number of cells in non-proliferative
phase, $N_{\tau }=N(t-\tau )$, $P$ the number of cycling
proliferating cells, $\gamma $ the rate of cells loss from
proliferative phase, $\delta $ the rate of cells loss from
non-proliferative phase, $\tau $ the time spent in the proliferative phase,
$\beta $ the feedback function, rate of recruitment from
non-proliferative phase, $\beta _{0}$ the maximum recruitment
rate, and $\theta $ and $n$ the control shape of the feedback
function.
\section{Stability without delay $\tau =0$}\label{s1}
For $\tau =0$ the equation (\ref{1}) reads to
\begin{equation}
\begin{gathered}
\frac{dN}{dt}=-\delta N+\beta (N)N \\
\frac{dP}{dt}=-\gamma P\qquad \qquad
\end{gathered} \label{5}
\end{equation}
\begin{theorem} \label{t1}
Assume $\delta \in (0,\beta _{0}]$. The system
\eqref{5} has a positive equilibrium $(N^{*},0)=(\beta^{-1}(\delta ),0)$
which is asymptotically stable. The trivial one $(0,0)$ is unstable.
\end{theorem}
\begin{proof}
The characteristic equation of the linearized equation of (\ref{5}) around
$E^{*}=(N^{*},0)$, has two roots given by
$ \lambda _{1}=-\delta+\alpha '(N^{*})$ and
$ \lambda _{2}=-\gamma $,
where
\begin{equation}\label{00}
\alpha (N)=\beta (N)N\end{equation}
and
$\alpha '(N)$ its derivative.
Since $\beta $ is a decreasing function, $E^{*}$ is asymptotically
stable.
For the trivial equilibrium, the roots of the
characteristic equation of the linearized equation of (\ref{5})
around $(0,0)$ are
$ \lambda _{1}=-\delta +\alpha '(0)$ and $ \lambda _{2}=-\gamma$.
Since $\alpha ' (0)=\beta _{0}>\delta $, $(0,0)$ is unstable.
\end{proof}
\section{Stability for positive delay} \label{s2}
Normalizing the delay $\tau $ by the time scaling
$t\to \frac{t}{\tau }$, effecting the change of variables
$u(t)=N(t\tau )$ and $v(t)=P(t\tau )$, the system (\ref{1}) is transformed into
\begin{equation}
\begin{gathered}
\dot{u}(t)=\tau [-\delta u(t)-\alpha (u(t))+2e^{-\gamma \tau }\alpha (u(t-1))] \\
\dot{v}(t)=\tau [-\gamma v(t)+\alpha (u(t))-e^{-\gamma \tau }\alpha (u(t-1))]
\end{gathered} \label{61}
\end{equation}
where $\alpha$ is given by equation $(\ref{00})$.
Let
\begin{itemize}
\item[(H0)] $\delta<\frac{\beta_{0}}{2}$
and denote by $\overline{\tau }=\frac{1}{\gamma }
\ln \big( \frac{2}{1+\frac{2\delta }{\beta _{0}}}\big)$.
\end{itemize}
Note that (H0) implies that for each $0<\tau <\overline{\tau}$,
$\alpha '(u^{*})<0$ and $\beta_{0}(2e^{-\gamma \tau}-1)>\delta$
and system (\ref{61}) has a unique positive equilibrium
$E^{*}(\tau )=(u^{*}(\tau),v^{*}(\tau ))$ with
$$
u^{*}(\tau )=\theta \big( \frac{\beta _{0}(2e^{-\gamma \tau }-1)-\delta }
{\delta }\big) ^{1/n},\quad v^{*}(\tau )=\frac{\delta u^{*}}{\gamma }
\big( \frac{1-e^{-\gamma \tau }}{2e^{-\gamma \tau }-1}\big)
$$
and the characteristic equation of the linearized equation associated with
\eqref{61} around $E^{*}(\tau )$ is
\begin{equation}
W(\lambda ,\tau )=(\lambda +\tau \gamma )(\lambda -\tau a(\tau )-\tau b(\tau
)e^{-\lambda })=0, \label{63}
\end{equation}
with
$ a(\tau )=-(\delta +\alpha '(u^{*})) \mbox{ and }
b(\tau )=2e^{-\gamma \tau }\alpha '(u^{*}) $ and
$$
\alpha '(u^{*})=\frac{\delta }{\beta _{0}(2e^{-\gamma
\tau }-1)^{2}}\left[ \beta _{0}(1-n)(2e^{-\gamma \tau }-1)+n\delta\right].
$$
Since $\tau \gamma >0$, the stability of the positive equilibrium
$E^{*}(\tau ) $ follows from the study of roots of the equation
\begin{equation}
\Delta (\lambda ,\tau )=\lambda -\tau a(\tau )-\tau b(\tau )e^{-\lambda }=0
\label{64}
\end{equation}
corresponding to the characteristic equation associated to the
first equation in (\ref{61}). To obtain the switch of stability of
$E^{*}(\tau )$, one needs to find the imaginary root of equation
(\ref{64}). Let $\lambda =i\zeta $, then
$\Delta (i\zeta ,\tau )=0\text{\qquad } $ if and only if
\begin{equation}
\begin{gathered}
\zeta =\arccos \big( -\frac{a(\tau )}{b(\tau )}\big) \in (0,\pi )\quad
\text{for }0\leq | \frac{a(\tau )}{b(\tau )}| \leq 1 \quad \text{and} \\
\tau \sqrt{b^{2}(\tau )-a^{2}(\tau )}=\arccos (-\frac{a(\tau
)}{b(\tau )}) \quad \text{for }0\leq | \frac{a(\tau )}{b(\tau)}| <1 .
\end{gathered} \label{c1}
\end{equation}
Let
\begin{itemize}
\item[(H1)] $a(\tau )<0$ and $| b(\tau)| <-a(\tau )$ for all
$\tau >0$.
\item[(H2)] $\tau a(\tau )<1$, and $|a(\tau )| < | b(\tau )| $ for all
$\tau>0$.
\end{itemize}
\begin{theorem} \label{p1}
Under assumption (H0), we have:\\
$(1)$ The trivial equilibrium $(0,0)$ is unstable for $0<\tau<\overline{\tau }$.
\\
$(2)$ \begin{itemize}
\item[(i)] If $a$ and $b$ satisfy (H1), then $E^{*}(\tau )$ is asymptotically
stable for $0<\tau<\overline{\tau}$. \\
\item[(ii)] If $a$ and $b$ satisfy (H2), $n$ is
sufficiently large and $\gamma $ is close enough to $0$, there exists a unique
$\tau _{0}$ in $]0,\overline{\tau }[$ such that $E^{*}(\tau )$ is
asymptotically stable for $\tau \in ]0,\tau _{0}[$ and unstable for
$\tau \in (\tau _{0},\overline{\tau })$.
\end{itemize}
\end{theorem}
\begin{proof}
(1) The characteristic equation of the linearized
equation associated to (\ref {61}) around $(0,0)$ is
\begin{equation}
\lambda +\tau (\delta +\beta _{0})-2\tau e^{-\gamma \tau }\beta
_{0}e^{-\lambda }=0 \label{e4}
\end{equation}
From (H0), we have $\beta _{0}(2e^{-\gamma \tau}-1)>\delta $, thus
(\ref{e4}) has a real root which is positive. Then $(0,0)$ is unstable.
(2) part (i): Let $\lambda =\mu +i\nu $ be a
root of equation $\Delta(\lambda,\tau)=0$ for $0<\tau <\overline{\tau }$.
We have
\begin{equation}
\begin{gathered}
\mu -\tau a(\tau )-\tau b(\tau )e^{-\mu }\cos (\nu )=0 \\
\nu +\tau b(\tau )e^{-\mu }\sin (\nu )=0
\end{gathered} \label{65}
\end{equation}
If there exists a root $\mu _{0}\geq 0$ of (\ref{65}), then
$-a(\tau )\leq b(\tau )e^{-\mu _{0}}\cos (\nu )$.
Since $-1\leq \cos (\nu )\leq 1$ and $0-1$, (ii) $c+d>0$, and (iii) $ \sqrt{d^{2}-c^{2}}<\zeta$,
where $\zeta $ is the root of $\zeta =-c\tan \zeta $, $0<\zeta <\pi $,
if $c\neq 0$ and $\zeta =\frac{\pi }{2}$ if $c=0$.
\end{lemma}
\begin{lemma} \label{l1}
Under hypotheses (H0) and (H2), for
$n$ sufficiently large and $\gamma $ close enough to $0$, there
exists a unique solution $\tau _{0}$ of the second equation of
(\ref{c1}) in $]0,\overline{\tau }[$, such that $i\zeta _{0}$ is a
purely imaginary root of equation (\ref{64}), with $\zeta
_{0}=\arccos (-\frac{a(\tau _{0})}{b(\tau _{0})})$. Furthermore,
the following inequalities hold
\begin{equation}
\begin{gathered}
\tau \sqrt{b^{2}(\tau )-a^{2}(\tau )}<\arccos (-\frac{a(\tau )}{b(\tau )})
\quad\text{for }\tau \in (0,\tau _{0}) \\
\tau \sqrt{b^{2}(\tau )-a^{2}(\tau )}>\arccos (-\frac{a(\tau )}{b(\tau )})
\quad\text{for }\tau \in (\tau _{0},\overline{\tau })
\end{gathered} \label{c2}
\end{equation}
\end{lemma}
\begin{lemma} \label{l2}
Let $f:(0,\pi )\to \mathbb{R}$ be defined by $f(x
)=\alpha \tan x $, $\alpha <1$ and $\alpha \neq 0$. Then, $f$ has
a unique fixed point $ \zeta \in (0,\pi )$, such that: \\
For $0<\alpha <1$, $f(x)< x $ if $x \in (0, \zeta )\cup (\frac{\pi }{2},\pi )$
and $f(x)> x $ if $x \in ( \zeta ,\frac{\pi}{2})$;\\
and for $\alpha <0$, $f(x )< x $ if
$ x \in (0,\frac{\pi }{2})\cup ( \zeta ,\pi )$
and $f(x )> x $ if $ x \in (\frac{\pi }{2}, \zeta )$.
\end{lemma}
\begin{proof}[Proof of (2) part (ii) of theorem \ref{p1}]
We only have to verify the three conditions
(i), (ii) and (iii) of lemma \ref{t11}. The assertions (i) and
(ii) follow from (H2) with $c=-\tau a(\tau )$ and $d=-\tau b(\tau )$.
For condition (iii), let $\tau \in (0,\tau _{0})$ and
$f(\zeta )=\tau a(\tau )\tan \zeta $. From the first equation of
(\ref{c2}) we have:
If $a(\tau )=0$, the first inequality of (\ref{c2})
becomes $-\tau b(\tau )< \frac{\pi }{2}$, and (iii) is satisfied.
If $0<\tau a(\tau )<1$ or $a(\tau )<0$, since
\[
f\big( \arccos (-\frac{\ a(\tau )}{b(\tau )})\big) =\tau\sqrt{\ b(\tau
)^{2}- a(\tau )^{2}},
\]
the first equation of (\ref{c2}) implies that
\[
f\big( \arccos (-\frac{\ a(\tau )}{b(\tau )})\big) <\arccos
(-\frac{\ a(\tau )}{b(\tau )}),
\]
with $\arccos (-\frac{a(\tau )}{b(\tau )})\in (0,\pi )$.
From lemma \ref{l2} and the
graph of $f$, if $ \zeta $ is the fixed point of $f$ in $(0,\pi)$, we have,
\begin{equation}
f\big( \arccos (-\frac{\ a(\tau )}{b(\tau )})\big) < \zeta ,
\label{c5}
\end{equation}
that is
$\sqrt{\ (\tau b(\tau ))^{2}-(\tau a(\tau ))^{2}}< \zeta$,
which leads to the desired assertion. This complete the stability
of $E^{*}(\tau )$ for $0<\tau <\tau _{0}$.
To prove the unstability of $E^{*}(\tau )$ in (2) part (ii), for
$ \tau_{0}<\tau <\overline{\tau }$, we will show that the characteristic
equation (\ref{64}) has at least one root with positive real part.
Let $\tau _{0}<\tau <\overline{\tau }$. If all the roots of the
characteristic equation (\ref{64}) have negative real parts, the
properties (i), (ii) and (iii) of lemma \ref{t11} are satisfied.
From the second equation of (\ref{c2}) and from (\ref{c5}) we
have
\begin{gather*}
f\big( \arccos (-\frac{\ a(\tau )}{b(\tau )})\big)
>\arccos (-\frac{a(\tau )}{b(\tau )}) \\
f\big( \arccos (-\frac{\ a(\tau )}{b(\tau )})\big) <\overline{\zeta }
\end{gather*}
Henceforth, from lemma \ref{l2} and the graph of $f$, we have
$$
\arccos (-\frac{\ a(\tau )}{b(\tau )})<\overline{\zeta }, \quad
\text{and} \quad
\arccos (-\frac{\ a(\tau )}{b(\tau )})>\overline{\zeta }
$$
which is impossible.
Now, suppose that there is one
root with zero real part with all the remaining roots having
negative real parts. From (\ref{c1}) and lemma \ref{l1} we deduce
that $\tau =\tau _{0}$, which contradicts the assumption
$\tau>\tau _{0}$. Then $E^{*}(\tau )$ is unstable for $\tau
_{0}<\tau <\overline{\tau }$
\end{proof}
\begin{proof}[Proof of Lemma \ref{l1}]
In view of (H0) and (H2), to find a root of second equation of (\ref{c1}) is
equivalent to find a root of the equation
\begin{equation}
\tau =-\frac{\arccos (-\frac{a(\tau )}{b(\tau )})}{b(\tau )
\sin (\arccos (-\frac{a(\tau )}{b(\tau )}))}. \label{c4}
\end{equation}
Let $y(\tau )=\arccos (-\frac{a(\tau )}{b(\tau )})$, and
$F(\tau)=-\frac{y(\tau )}{b(\tau )\sin (y(\tau ))}$.
Besides, in the hypotheses (H0) and (H2), $F$ is continously
differentiable on $\tau _{0} \in [0,\overline{\tau }]$. As
$F(0)>0$, for sufficiently large $n$ and
$F(\overline{\tau })<\overline{\tau }$ for $\gamma $ close enough to $0$,
then there exits at least one solution $\tau _{0}$ of equation (\ref{c4}) in
$]0,\overline{\tau }[$. Now, for the uniqueness of $\tau _{0}$,
let $ g(\tau )=\tau -F(\tau )$, then
\[
g'(\tau )=1-\frac{y'(\tau )b(\tau )\sin (y(\tau
))-y(\tau )b'(\tau )\sin (y(\tau ))}{(b(\tau )\sin (y(\tau
)))^{2}}
-\frac{y(\tau )b(\tau )\cos ((\tau ))y'(\tau )}{(b(\tau )\sin
(y(\tau )))^{2}}
\]
where
\[
y'(\tau )=-\sqrt{1-\left( \frac{a(\tau )}{b(\tau )}\right) ^{2}}
\frac{a'(\tau )b(\tau )-a(\tau )b'(\tau )}{%
b^{2}(\tau )}.
\]
Since $\lim_{\gamma \to0}\frac{d}{d\tau}\alpha'(u^{*})=0$,
from (\ref{63}), we have
$$
{\lim_{\gamma \to 0}}b'(\tau )=0\quad \text{and}\quad {\lim_{\gamma \to 0}}a'(\tau )=0 .
$$
Then
${\lim_{\gamma\to 0}}g'(\tau )=1>0$, for $0\leq \tau \leq \overline{\tau }$.
Since $g'>0$ and $g$ is an increasing function on the interval
$]0,\overline{\tau }[$ for $\gamma $ close enough to $0$,
$\tau_{0}$ is unique in $]0,\overline{\tau }[$. By the continuity
property of $F$, we have $F(\tau)>\tau$ for $\tau\in ]0,\tau_{0}[$
and $F(\tau)<\tau$ for $\tau\in ]\tau_{0},\overline{\tau }[$.
\end{proof}
\section{Hopf Bifurcation Occurrence}
Below, we will show that the following system has a Hopf
bifurcation at $\tau =\tau _{0}$,
\begin{equation}\label{h1}
\begin{gathered}
\frac{dN}{dt}=-\delta N-\beta (N)N+2e^{-\gamma \tau _{0}}\beta (N_{\tau
})N_{\tau } \\
\frac{dP}{dt}=-\gamma P+\beta (N)N-e^{-\gamma \tau _{0}}\beta
(N_{\tau })N_{\tau }
\end{gathered}
\end{equation}
This system is equivalent to
\begin{equation}
\begin{gathered}
\dot{u}(t)=\tau [-\delta u(t)-\alpha (u(t))+2e^{-\gamma \tau
_{0}}\alpha (u(t-1))] \\
\dot{v}(t)=\tau [-\gamma v(t)+\alpha (u(t))-e^{-\gamma \tau
_{0}}\alpha (u(t-1))]
\end{gathered} \label{c3}
\end{equation}
with $u(t)=N(t\tau )$ and $v(t)=P(t\tau )$. System (\ref{c3}) has
a unique positive equilibrium
$E^{*}=(u^{*},v^{*})=(u^{*}(\tau_{0}),v^{*}(\tau_{0}))$, for all $\tau>0$.
By the translation $z(t)=(u(t),v(t))-(u^{*},v^{*})$, system
(\ref{c3}) is written as a functional differential equation (FDE)
in $C:=C([-1,0],\mathbb{R}^{2})$:
\begin{equation}
\dot{z}(t)=L(\tau )z_{t}+f_{0}(z_{t},\tau ) \label{62}
\end{equation}
where $L(\tau):C\to \mathbb{R}^{2}$ is a linear
operator and
$f_{0}:C\times\mathbb{R}\to\mathbb{R}^{2}$ are given
respectively by
\[
L(\tau )\varphi =\tau \begin{pmatrix}
-(\delta +\alpha '(u^{*}))\varphi _{1}(0) +2e^{-\gamma \tau
_{0}}\alpha '(u^{*})\varphi _{1}(-1) \\
-\gamma \varphi _{2}(0)+\alpha '(u^{*})\varphi _{1}(0)
-e^{-\gamma\tau _{0}}\alpha '(u^{*})\varphi _{1}(-1)
\end{pmatrix}
\]
\[
f_{0}(\varphi ,\tau )=\tau \begin{pmatrix}
-\alpha (\varphi _{1}(0)+u^{*})+\alpha '(u^{*})\varphi
_{1}(0)-2e^{-\gamma \tau _{0}}\alpha '(u^{*})\varphi _{1}(-1) \\
+2e^{-\gamma \tau _{0}}\alpha (\varphi _{1}(-1)+u^{*})-\delta u^{*} \\
\alpha (\varphi _{1}(0)+u^{*})-\alpha '(u^{*})\varphi _{1}(0)
-e^{-\gamma \tau _{0}}\alpha (\varphi _{1}(-1)+u^{*}) \\ +e^{-\gamma \tau
_{0}}\alpha '(u^{*})\varphi _{1}(-1)-\gamma v^{*}.
\end{pmatrix}
\]
for $\varphi =(\varphi _{1},\varphi _{2})\in C$.
Now, we apply the Hopf bifurcation theorem, see \cite{JH}, to show the
existence of a non-trivial periodic solution to (\ref{c3})
bifurcating from the non trivial equilibrium $E^{*}$. We use the
delay as a parameter of bifurcation. Therefore, the periodicity is
a result of changing
the type of stability, from stationary solution to limit cycle.
Let \begin{itemize}
\item[(H3)] $a(\tau _{0})<\frac{1}{\overline{\tau }}$ and
$\left| a(\tau )\right| < \left| b(\tau )\right|, \mbox{ for }
0<\tau<\overline{\tau}$.
\end{itemize}
\begin{theorem} \label{p2}
Under hypotheses (H0) and (H3) if $n$ is sufficiently
large and $\gamma $ is close enough to $0$, then, for
$\tau \in ]0,\tau_{0}[$, $E^{*}$ is asymptotically stable;
it is unstable for $\tau \in ]\tau _{0},\overline{\tau }[$,
where $\tau _{0}$ is stated in lemma \ref{l1}.
\end{theorem}
The proof of the above theorem follows the same procedure as
that the proof of theorem \ref{p1} (2) (ii). Therefore, we omit it.
\begin{theorem} \label{t2}
Assume (H0) and (H3) hold, $n$ is sufficiently large and $\gamma $
is sufficiently small. There exists $\varepsilon _{0}>0$ such that,
for each $0\leq \varepsilon <\varepsilon _{0}$, equation (\ref{c3}) has a
family of periodic solutions $p(\varepsilon )$ with period
$T=T(\varepsilon )$, for the parameter values
$\tau =\tau(\varepsilon )$ such that $p(0)=E^{*} $,
$T(0)=\frac{2\pi }{\zeta_{0}}$ and $\tau (0)=\tau _{0}$, where $\tau _{0}$
stated in lemma \ref{l1} and $\zeta _{0}=\arccos \big( -\frac{a(\tau
_{0})}{b(\tau _{0})}\big)$.
\end{theorem}
\begin{proof}
We apply the Hopf bifurcation theorem introduced in \cite{JH}.
From the expression of $f_{0}$ in (\ref{62}), we have
\[
f_{0}(0,\tau )=0\qquad \text{and}\qquad \frac{\partial f_{0}(0,\tau )}{%
\partial \varphi }=0,\text{ for all }\tau >0
\]
The linearized equation associated
to (\ref {c3}) around $E^{*}$ has the following characteristic
equation:
\begin{equation}
\Delta _{0}(\lambda ,\tau )=\lambda -\tau a(\tau _{0})-\tau b(\tau_{0})
e^{-\lambda }=0, \label{71}
\end{equation}
Firstly, let $\lambda =i\zeta $. From (\ref{c1}) and lemma \ref{l1},
we have
\[
\Delta _{0}(i\zeta ,\tau )=0\;\Longleftrightarrow \;
\zeta _{0}=\arccos \big( -\frac{a(\tau _{0})}{b(\tau _{0})}\big)
\text{ and } \tau =\tau _{0}
\]
where $\tau _{0}$ is unique in $(0,\overline{\tau })$. Thus,
the characteristic equation (\ref{71}) has a pair of simple
imaginary roots $\lambda _{0}=i\zeta _{0}$ and $\overline{\lambda
}_{0}=-i\zeta _{0}$ at $\tau=\tau_{0}$.
Lastly, we need to verify the transversality condition.
From (\ref{71}), $\Delta _{0}(\lambda _{0},\tau _{0})=0$ and
$\frac{\partial }{\partial \lambda }\Delta _{0}(\lambda _{0},\tau
_{0})=1-\tau_{0}a(\tau _{0})+\lambda _{0}\neq 0$.
According to the implicit function theorem, there exists a complex
function $\lambda=\lambda(\tau)$ defined in a neighborhood of $\tau _{0} $,
such that $\lambda (\tau _{0})=\lambda _{0}$ and
$\Delta _{0}(\lambda (\tau ),\tau )=0$ and
\begin{equation}
\lambda '(\tau )=-\frac{\partial \Delta _{0}(\lambda ,\tau
)/\partial \tau }{\partial \Delta _{0}(\lambda ,\tau )/\partial \lambda },
\label{e2}
\end{equation}
for $\tau$ in a neighborhood of $\tau_{0}$.
Let $\lambda (\tau )=p(\tau )+iq(\tau )$. From (\ref{e2}) we have
\[
p'(\tau )_{/\tau =\tau _{0}}=\ \frac{\tau _{0}(b^{2}(\tau
_{0})-a^{2}(\tau _{0}))}{ \left( 1+\tau _{0}b(\tau _{0})\cos \zeta
_{0}\right)
^{2}+\left( \tau _{0}b(\tau _{0})\sin \zeta _{0}\right) ^{2} }
\]
From (H3), we conclude that $p'(\tau )_{/\tau =\tau _{0}}>0$.
\end{proof}
\section{Discussions}
It's known (Mackey (1997) \cite{m96}) that when taking $\gamma$ as
a bifurcation parameter and allowing $\gamma$ to increase, a
supercritical Hopf bifurcation of (\ref{1}) is followed by an
inverse Hopf bifurcation. Considering the delay $\tau$ as a
parameter of bifurcation makes the study of bifurcation more
complicated.\\ In \cite{M1} the following conditions of stability
of the non-trivial steady state of (\ref{1}) were proposed (Hayes
(1950) \cite{h50}) $ |\frac{a(\tau)}{b(\tau)}|>1$ or
$ |\frac{a(\tau)}{b(\tau)}|\leq 1$ and
$\tau <\frac{\arccos(-\frac{a(\tau)}{b(\tau)})}{\sqrt{b(\tau)^{2}
-a(\tau)^{2}}}$ where
$ 0< \tau <\frac{1}{\gamma}\ln(\frac{2}{1+\frac{\delta}{\beta_{0}}})$,
$\delta< \beta_{0}$.
In sections \ref{s1} and \ref{s2} of this paper
it's shown that if the loss rate $\gamma$ from proliferating cells
is smaller and the control shape $n$ is large, then the steady
state $E^{*}(\tau)$ may be stable for $\tau=0$ and hence it's
stable for $0<\tau<\tau_{0}$ and unstable for
$\tau_{0}<\tau<\overline{\tau}$, where
$\overline{\tau}=\frac{1}{\gamma}\ln (
\frac{2}{1+\frac{2\delta}{\beta_{0}}}), 2\delta < \beta_{0}$. But
at $\tau=\tau_{0}$ we cannot give any result of stability of
$E^{*}(\tau_{0})$, because the dependance of $E^{*}(\tau)$ on the
delay $\tau$, which makes the study of the Hopf bifurcation more
difficult.
In the rest of the paper to study the Hopf
bifurcation around the critical value $\tau=\tau_{0}$, we propose
the approchable model (\ref{h1}) of (\ref{1}). Then
$E^{*}(\tau_{0})$ is the unique non-trivial steady state of
(\ref{h1}) for all $ 0<\tau<\overline{\tau}$, which is stable for
$0 <\tau<\tau_{0}$ and unstable for $\tau_{0}<\tau<\overline{\tau}$
and the Hopf bifurcation occures at
$ \tau=\tau_{0}$.
The results proposed in this paper should
hopefully improve the understanding of the qualitative properties
of the description delivered by model (\ref{1}). So far we have
now a description of stability properties and Hopf bifurcation
with a detailed analysis of the influence of delays terms.
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\end{document}