\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
2005-Oujda International Conference on Nonlinear Analysis.
\newline {\em Electronic Journal of Differential Equations},
Conference 14, 2006, pp. 241--248.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\setcounter{page}{241}
\begin{document}
\title[\hfilneg EJDE/Conf/14 \hfil Periodic solutions for small and large delays]
{Periodic solutions for small and large delays in a tumor-immune
system model}
\author[R. Yafia \hfil EJDE/Conf/14 \hfilneg]
{Radouane Yafia}
\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@math.net}
\date{}
\thanks{Published September 20, 2006.}
\subjclass[2000]{34K18}
\keywords{Tumor-Immune system competition;
delayed differential equations; \hfill\break\indent
stability; Hopf bifurcation; periodic solutions}
\begin{abstract}
In this paper we study the Hopf bifurcation for
the tumor-immune system model with one delay. This model is
governed by a system of two differential equations with one delay.
We show that the system may have periodic solutions for small
and large delays for some critical value of the delay parameter
via Hopf bifurcation theorem bifurcating from the non trivial
steady state.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{definition}{Definition}[section]
\section{Introduction}
In this paper, we consider a model that provides a description of
tumor cells in competition with the immune system. This
description is described by many authors, using ordinary and
delayed differential equations to model the competition between
immune system and tumor. In particular \cite{g03,kp98,kt94} other
similar models can be found in the literature, see,
\cite{f02,mzd95,wz02} provide a description of the modelling,
analysis and control of tumor immune system interaction.
Other authors use kinetic equations to model the competition
between immune system and tumor. Although they give a complex
description in comparison with other simplest models, they are,
for example, needed to model the differences of virulence between
viruses, see, \cite{ab96,abd02,bpr00,bpl00,bpl02,c99}. Several
other fields of biology use kinetic equations, for instance
\cite{dp} and \cite{dp03} give a kinetic approach to describe
population dynamics, \cite{abd02} deals with the development of
suitable general mathematical structures including a large variety
of Boltzmann type models.
The reader interested in a more complete bibliography about the
evolution of a cell, and the pertinent role that have cellular
phenomena to direct the body towards the recovery or towards the
illness, is addressed to \cite{ff94,gtp96}. A detailed description
of virus, antivirus, body dynamics can be found in the following
references \cite{b00,dh00,mn00,pw97}. The mathematical model
with which we are dealing, was proposed in a recent paper by M.
Galach \cite{g03}. In this paper the author developed a new simple
model with one delay of tumor immune system competition, this idea
is inspired from the paper of Kuznetsov and Taylor (1994)
\cite{kt94} and he recall some numerical results of Kuznetsov and
Taylor in order to compare them with those obtained in his paper,
see, \cite{g03}.
\section{Mathematical model} \label{s2}
The mathematical model describing the tumor immune system
competition is given by a system of two differential equations
with one delay, see Galach \cite{g03},
\begin{equation}\label{21}
\begin{gathered}
\frac{dx}{dt}=\sigma+\omega x(t-\tau)y(t-\tau)-\delta x \\
\frac{dy}{dt}=\alpha y(1-\beta y)-xy
\end{gathered}
\end{equation}
where the parameter $\omega$ describes the immune response to the
appearance of the tumor cells and the constant $\tau$ is the time
delay which the immune system needs to develop a suitable response
after the recognition of non-self cells. Time delays in connection
with the tumor growth also appear in Bodnar and Fory\'{s}
\cite{bf001} and \cite{bf002}, Byrne \cite{by97},
Fory\'{s} and Kolev \cite{fk02} and Fory\'{s} and
Maciniak-Czochra \cite{fm02}. For the meaning of the
parameters $\alpha$, $\beta$, $\delta$ and $\sigma$, see Kuznetsov
and Taylor \cite{kt94} and Kirschner and
Panetta \cite{kp98}.
For $\tau=0$ system \eqref{21} becomes a system of ordinary
differential equations:
\begin{equation}\label{22}
\begin{gathered}
\frac{dx}{dt}=\sigma+\omega xy-\delta x \\
\frac{dy}{dt}=\alpha y(1-\beta y)-xy
\end{gathered}
\end{equation}
In \cite{g03}, the author study the existence, uniqueness and
nonnegativity of solutions and he show the nonexistence of
nonnegative periodic solution of system \eqref{22}, using the
Dulac-Bendixon criteria, see \cite{p91}. The possible
nonnegative steady states of system \eqref{22} and their stability
are
summarized in the Table 1; see also \cite{g03},
\begin{table}[ht]
\caption{Nonnegative steady states of system
\eqref{22} and their stability}
\begin{center}
\begin{tabular}{|c|p{5cm}|c|c|c|}
\hline
Region & Conditions & $P_{0}$ & P$_{1}$ & $P_{2}$ \\
\hline
1 & $\omega>0$, \quad $\alpha\delta<\sigma$ & stable & & \\
\hline
2 & $\omega>0$, \quad $\alpha\delta>\sigma$ & unstable & & stable \\
\hline
3 & $\omega<0$,\quad$ \alpha\delta>\sigma$, \newline
$ \alpha(\beta\delta-\omega)^{2}+4\beta\omega\sigma>0$ & unstable & & stable \\
\hline
4 & $\omega<0$, \quad $\alpha\delta<\sigma$, \quad $ \omega+\beta\delta<0$,
$\alpha(\beta\delta-\omega)^{2}+4\beta\omega\sigma>0$ & stable & unstable & stable \\
\hline
5 & $\omega<0$, \quad $\alpha(\beta\delta-\omega)^{2}+4\beta\omega\sigma>0$ &
stable & & \\
\hline
\end{tabular}
\end{center}
\end{table}
For $\tau>0$, the existence and uniqueness of
solutions of system \eqref{21} for every $t>0$ are established in
\cite{g03}, using the results presented in Hale \cite{h97}.
Based on the results of
Bodnar \cite{bd00}, in \cite{g03} the author showed that:
(1) If $\omega\geq 0$, these solutions are nonnegative for any
nonnegative initial conditions (biologically realistic case).
(2) If $\omega<0$, there exist nonnegative initial condition such
that the solution becomes negative in a finite time interval.
Our goal in this paper is to consider the case (1) when
$\omega>0$, which is the most biologically meaningful one. We
study the asymptotic behavior of the possible steady states
$P_{0}$ and $P_{2}$ with respect to the delay $\tau$. We establish
that, the Hopf bifurcation may occur by using the delay as a
parameter of bifurcation. We prove this result for small and large
delays.
This paper is organized as follows. In section \ref{s3}, we recall
some results about the absolute and conditional stability of delay
equations and the zeros of second order transcendental
polynomials. In section \ref{s4}, we investigate the results
presented in section \ref{s3} to prove the stability of the
possible steady states (trivial and non-trivial) of the delayed
system \eqref{21}. The main result of this paper is given in
section \ref{s5} and section \ref{s6}. Based on the Hopf
bifurcation theorem, we show the occurrence of Hopf bifurcation
for small and large delays.
\section{Stability of delay equations and zeros of second order
transcendental polynomials}\label{s3}
In this section we recall some results on
the stability of delay equations and on the zero of second order
transcendental polynomials.
\subsection{Absolute and conditional stability}\label{s31}
Consider the following general nonlinear delay differential system
\begin{equation}\label{31}
\frac{dx}{dt}=f(x(t),\quad x(t-\tau)),
\end{equation}
where $x\in \mathbb{R}^{n}$, $\tau$ is constant,
$f:\mathbb{R}^{n}\times \mathbb{C}^{n}\rightarrow \mathbb{R}^{n}$
is smooth enough to guarantee the existence and uniqueness of
solutions of \eqref{31} under the initial condition
\begin{equation}\label{32}
x(\theta)=\varphi(\theta), \quad \theta\in [-\tau,0],
\end{equation}
where $C=C([-\tau,0],\mathbb{R}^{n})$. Suppose $f(x^{*},x^{*})=0$,
that is $x=x^{*}$ is a steady state of system \eqref{31}.
\begin{definition} \label{def3.1} \rm
The steady state $x=x^{*}$ of system \eqref{31} is called
absolutely stable (i.e., asymptotically stable independent of the
delay $\tau$) if it is asymptotically stable for all delays
$\tau>0$. $x=x^{*}$ is called conditionally stable (i.e.,
asymptotically stable depending of the delay $\tau$) if it is
asymptotically stable for $\tau$ in some interval, but not
necessarily for all delays $\tau>0$.
\end{definition}
The linearized system of \eqref{31} at $x=x^{*}$ has the
form
\begin{equation}\label{33}
\frac{dX}{dt}=A_{0}X+A_{1}X(t-\tau),
\end{equation}
where $X\in \mathbb{R}^{n}$, $A_{i}$ ($i=0,1$) is an $n\times n$
constant matrix. Then the characteristic equation associated with
system \eqref{33} takes the form
\begin{equation}\label{34}
\det[\lambda I-A_{0}-A_{1}e^{-\lambda \tau}].
\end{equation}
The location of the roots of some transcendental equation
\eqref{34} in it is general form has been studied by many authors,
see Baptistini and T\'{a}boas \cite{bt97}, Bellman and Cooke
\cite{bc63}, Boese \cite{b95}, Brauer \cite{b87}, Cooke and van
den Driessche \cite{cd86}, Cooke and Grossman \cite{cg82}, Huang
\cite{h85}, Mahaffy \cite{m82}, Ruan and Wei \cite{rw} an dthe
references therein. The following result, which was proved by Chin
\cite{c60}, gives necessary and sufficient conditions for the
absolute stability of system \eqref{33}.
\begin{lemma}\label{l1}
The system \eqref{33} is absolutely stable if and only if
\begin{itemize}
\item[(i)] $\mathop{\rm Re}\lambda I-A_{0}-A_{1})<0$
\item[(ii)] $\det [i\zeta-A_{0}-A_{1}e^{-i \zeta \tau}]\neq 0 $
for all $\zeta >0$.
\end{itemize}
\end{lemma}
\subsection{A second degree transcendental polynomial}\label{s32}
In this section, we state some results on the second degree
transcendental polynomial (see Ruan and Wei \cite{rw}). For most
system with discrete delay, the characteristic equation of the
linearized system at a steady state is a second degree
transcendental polynomial equation of the following form:
\begin{equation}\label{35}
\lambda^{2}+p\lambda+r+(s\lambda+q)e^{-\lambda \tau}=0
\end{equation}
where $p,r,q$ and $s$ are real numbers. It is known that the
steady state is asymptotically stable if all roots of the
characteristic equation \eqref{35} have negative real parts.
Let define the following hypotheses:
\begin{itemize}
\item[(H1)] $p+s>0$.
\item[(H2)] $q+r>0$.
\item[(H3)] ($s^{2}-p^{2}+2r<0$ and $r^{2}-q^{2}>0$) or
$(s^{2}-p^{2}+2r)^{2}<4(r^{2}-q^{2})$.
\item[(H4)] $r^{2}-q^{2}<0$ or ($s^{2}-p^{2}+2r>0$ and
$(s^{2}-p^{2}+2r)^{2}=4(r^{2}-q^{2})$).
\item[(H5)] $r^{2}-q^{2}>0 $, $s^{2}-p^{2}+2r>0$ and
$(s^{2}-p^{2}+2r)^{2}>4(r^{2}-q^{2})$.
\end{itemize}
\begin{theorem}[\cite{rw}] \label{t1}
Let $\tau_{j}^{\pm}$ ($j=0,1,2,\dots $) defined by
$$
\tau_{j}^{\pm}=\frac{1}{\zeta_{\pm}}\arccos \big\{
\frac{q(\zeta_{\pm}^{2}-r)-ps\zeta_{\pm}^{2}}{s^{2}\zeta_{\pm}^{2}+q^{2}}
+\frac{2j\pi}{\zeta_{\pm}}\big\}, \quad
j=0,1,2,\dots
$$
where $\zeta_{\pm}$ is given by
$$
\zeta_{\pm}=\frac{1}{2}(s^{2}-p^{2}+2r)\pm
\frac{1}{2}[(s^{2}-p^{2}+2r)^{2}-4(r^{2}-q^{2})]^{\frac{1}{2}}.
$$
(i) If (H1)-(H3) hold, then all roots of equation
\eqref{35} have negative real parts for all $\tau\geq 0$.
\noindent (ii) If (H1), (H2) and (H4) hold, then when
$\tau\in [0,\tau_{0}^{+})$ all roots of equation \eqref{35} have
negative real parts, when $\tau=\tau_{0}^{+}$ equation \eqref{35}
has a pair of purely imaginary roots $\pm i \zeta_{\pm}$, and when
$\tau>\tau_{0}^{+}$ equation \eqref{35} has at least one root with
positive real part.
\noindent (iii) If (H1), (H2) and (H5) hold, then there is a
positive integer $k$ such that there are $k$ switches from
stability to instability to stability; that is, when $\tau\in
[0,\tau_{0}^{+})$, $ (\tau_{0}^{-},\tau_{0}^{+})$, \dots , $
(\tau_{k-1}^{-},\tau_{k}^{+})$,
all roots of equation \eqref{35} have negative real parts, and
when $\tau\in (\tau_{0}^{+},\tau_{0}^{-})$,
$(\tau_{1}^{+},\tau_{1}^{-})$, \dots,
$(\tau_{k-1}^{+},\tau_{k-1}^{-})$, and $\tau>\tau_{k}^{+}$,
equation \eqref{35} has at least one root with positive real part.
\end{theorem}
\begin{remark} \label{rmk3.1} \rm
Theorem \ref{t1} was obtained by Cooke and Grossman \cite{cg82} in
analyzing a general second order equation with delayed friction
and delayed restoring force. for other related work, see,
Baptistini and T\'{a}boas \cite{bt97}, Bellman and Cooke
\cite{bc63}, Boese \cite{b95}, Brauer \cite{b87}, Cooke and van
den Driessche \cite{cd86}, Cooke and Grossman \cite{cg82}, Huang
\cite{h85}, Mahaffy \cite{m82}, Ruan and Wei \cite{rw}, etc.
\end{remark}
\section{Steady states and stability for positive delays}\label{s4}
Consider the system \eqref{21}, and suppose that $\omega>0$. From
the table 1 (see, section \ref{s2}), we distingue between two
cases: $\alpha\delta<\sigma$
and $\alpha\delta>\sigma$.
\noindent\textbf{Case 1: $\omega>0$ and $\alpha\delta<\sigma$.}
The system \eqref{21} has a unique positive equilibrium $P_{0}$
given by $P_{0}=(\frac{\sigma}{\delta},0)$ and the linearized
system around $P_{0}$ takes the form
\begin{equation}\label{41}
\begin{gathered}
\frac{dx}{dt}=\omega \frac{\sigma}{\delta}y(t-\tau)-\delta x \\
\frac{dy}{dt}=(\alpha-\frac{\sigma}{\delta})y
\end{gathered}
\end{equation}
which leads to the characteristic equation
\begin{equation}\label{42}
W(\lambda)=(\lambda+\frac{\sigma}{\delta}-\alpha)(\lambda+\delta).
\end{equation}
Then we have the following result.
\begin{proposition}
Under the hypotheses $\omega > 0$ and $\alpha \delta < \sigma$,
the equilibrium point $P_{0}$ is absolutely stable.
\end{proposition}
\begin{proof} From the characteristic equation \eqref{42} and lemma
\ref{l1}, it is easy to obtain the result (see,
\cite{h93,h97}).
\end{proof}
\textbf{Case 2: $\omega>0$ and $\alpha\delta>\sigma$.}
In this case, system \eqref{21} has two equilibrium points (see,
Table 1) $P_{0}=(\frac{\sigma}{\delta},0)$ and
$P_{2}=(x_{2},y_{2})$ where
$$
x_{2}=\frac{-\alpha(\beta\delta-\omega)+\sqrt{\Delta}}{2\omega},\quad
y_{2}=\frac{\alpha(\beta\delta+\omega)-\sqrt{\Delta}}{2\alpha\beta\omega}
$$
with
$\Delta=\alpha^{2}(\beta\delta-\omega)^{2}+4\alpha\beta\sigma\omega$.
From the characteristic equation \eqref{42}, we deduce the
following result.
\begin{proposition} \label{prop4.2}
Under the hypotheses $\omega>0$ and $\alpha\delta>\sigma$, the
equilibrium point $P_{0}$ is unstable for all positive time delay.
\end{proposition}
\begin{proof} For the proof of this proposition, from the characteristic
equation \eqref{42}. It is obvious to check the result.
\end{proof}
In the next, we shall study the stability of the non-trivial
equilibrium point $P_{2}$.
Let $u=x-x_{2}$ and $v=y-y_{2}$, by linearizing system \eqref{21}
around the non-trivial equilibrium point $P_{2}$, we obtain the
linear system
\begin{equation}\label{43}
\begin{gathered}
\frac{du}{dt}=\omega x_{2}v(t-\tau)-\omega y_{2} u(t-\tau)
-\delta u \\
\frac{dv}{dt}=-y_{2}u+(\alpha-2\alpha\beta y_{2}-x_{2})v
\end{gathered}
\end{equation}
The characteristic equation of equation \eqref{43} has the form
\begin{equation}\label{44}
W(\lambda,\tau)=\lambda^{2}+p\lambda+r+(s\lambda+q)e^{-\lambda
\tau}=0,
\end{equation}
which is the same equation presented in section \ref{s32}.
Where $p=\delta+\alpha\beta y_{2}>0$, $r=\delta\alpha\beta y_{2}>0$,
$s=-\omega y_{2}<0$ and $q=\alpha \omega y_{2}(1-2\beta y_{2})>0$.
The stability of the equilibrium point $P_{2}$ is a result of the
localization of the roots of the equation
$$
W(\lambda,\tau)=0.
$$
Then we have the following theorem.
\begin{theorem} \label{t2}
Assume $0<\frac{\omega}{\beta}<\alpha$, $\alpha\delta>\sigma$,
$\alpha>0$ and $\beta>0$. Then
$P_{2}$ is conditionally stable.
\end{theorem}
\begin{proof} From the expressions of $p$, $q$, $s$ and $r$ and the
paper \cite{g03}, we have $p+s>0$ and $q+r>0$. Therefore, the
hypotheses (H1), (H2) are satisfied and the steady state
$P_{2}$ is asymptotically stable (see section \ref{s32}) for
$\tau=0$.
Since $r^{2}-q^{2}=\alpha^{2}
y_{2}^{2}(\delta^{2}\beta^{2}-\omega^{2}(1-2\beta y_{2})^{2})$,
the sign of $r^{2}-q^{2}$ is deduced from the sign of
$(\delta\beta-\omega^{2}(1-2\beta
y_{2}))=-2\alpha\omega-\sqrt{\Delta}$ which is negative.\\
Therefore, $r^{2}-q^{2}<0$
Now, we compute $s^{2}-p^{2}+2r$.
From the expressions of $p$, $s$ and $r$, we have
$$
s^{2}-p^{2}+2r=(\omega-\alpha\beta)(\omega+\alpha\beta)y^{2}-\delta^{2}.
$$
As $\frac{\omega}{\beta}<\alpha$, we have $$s^{2}-p^{2}+2r<0$$ and
the hypothesis (H4) of section \ref{s32} is satisfied. From
theorem \ref{t1} (ii), the equilibrium point $P_{2}$ is
conditionally stable and there exist $\tau_{l}$ such that: $P_{2}$
is asymptotically stable for $\tau\in [0,\tau_{l}) $ and unstable
for $\tau>\tau_{l}$. For $\tau=\tau_{l}$ the characteristic
equation \eqref{44} has a pair of purely imaginary roots $\pm
i\zeta$, where
\begin{gather} \label{45}
\tau_{l}=\frac{1}{\zeta_{l}}\arccos \big\{
\frac{q(\zeta_{l}^{2}-r)-ps\zeta_{l}^{2}}{s^{2}\zeta_{l}^{2}+q^{2}}\big\},
\\ \label{46}
\zeta_{l}=\frac{1}{2}(s^{2}-p^{2}+2r)+
\frac{1}{2}[(s^{2}-p^{2}+2r)^{2}-4(r^{2}-q^{2})]^{1/2}.
\end{gather}
\end{proof}
In the next sections, we will study the occurrence of Hopf
bifurcation for smaller and larger delays.
\textbf{Notation:} The index $s$ is designed for small time
delays and the index $l$ is designed for large time delays.
Let $z(t)=(u(t),v(t))=(x(t),y(t))-(x_{2},y_{2})$, then the system
\eqref{21} is written as a functional differential equation (FDE)
in $C:=C([-\tau,0], \mathbb{R}^{2})$:
\begin{equation}\label{53}
\frac{dz(t)}{dt}=L(\tau)z_{t}+f(z_{t},\tau)
\end{equation}
where $L(\tau):C\to \mathbb{R}^{2}$ is a linear
operator and $f:C\times \mathbb{R}\to \mathbb{R}^{2}$
are given respectively by
$$L(\tau)\varphi=\begin{pmatrix}
\omega y_{2} \varphi_{1}(-\tau)+\omega x_{2} \varphi_{2}(-\tau)
-\delta \varphi_{1}(0) \\
-y_{2} \varphi_{1}(0)+(\alpha-2\alpha\beta y_{2}-x_{2})\varphi_{2}(0)
\end{pmatrix}
$$
and
$$
f(\varphi,\tau)=\begin{pmatrix}
\sigma+\omega \varphi_{1}(-\tau)\varphi_{2}(-\tau)
+\omega x_{2}y_{2}-\delta x_{2}
\\
-\alpha\beta\varphi_{2}^{2}(0)+\alpha y_{2}-\alpha\beta y_{2}^{2}
-\varphi_{1}(0)\varphi_{2}(0)-x_{2}y_{2}
\end{pmatrix}
$$
for $\varphi=(\varphi_{1},\varphi_{2})\in C$.
\section{Hopf bifurcation occurrence for small delays}\label{s5}
For small delays, let
$e^{-\lambda\tau}\simeq 1-\lambda\tau$, then the characteristic
equation \eqref{44} becomes
\begin{equation}\label{51}
W_{0}(\lambda,\tau)=(1-s\tau)\lambda^{2}+(p+s-q\tau)\lambda+r+q=0.
\end{equation}
Since the equilibrium point $P_{2}$ is asymptotically stable for
$\tau=0$, by Rouche's theorem, there exist $\tau_{s}$ such that
$P_{2}$ asymptotically stable for $\tau<\tau_{s}$ and unstable for
$\tau>\tau_{s}$, where $\tau_{s}$ is the value for which the
characteristic equation \eqref{51} has a pair of purely imaginary
roots.
Let $\lambda=i\zeta$, then $W_{0}(i\zeta,\tau)=0$ if and only if
\begin{equation}\label{52}
\begin{gathered}
(1-s\tau)\zeta^{2}-q-r=0, \\
p+s-q \tau=0\,.
\end{gathered}
\end{equation}
Then, from the second equation of \eqref{52}, we have
$\tau_{s}=\frac{p+s}{q}$ and
$\zeta_{s}=\sqrt{\frac{q(q+r)}{q-s(p+s)}}$.
We deduce the
following result of stability of the non-trivial equilibrium point
$P_{2}$ for small delays.
\begin{theorem}\label{t3}
Assume $0<\omega/\beta<\alpha$, $\alpha\delta>\sigma$,
$\alpha>0$ and $\beta>0$. Then, there exists $\tau_{s}$ such that:
$P_{2}$ is asymptotically stable for $\tau\in [0,\tau_{s}) $ and
unstable for $\tau>\tau_{s}$. For $\tau=\tau_{s}$ the
characteristic equation \eqref{51} has a pair of purely imaginary
roots $\pm i\zeta_{s}$, where $\tau_{s}=\frac{p+s}{q}$ and
$\zeta_{s}=\sqrt{\frac{q(q+r)}{q-s(p+s)}}$.
\end{theorem}
\begin{proof} Since $q-s(p+s)=q+\omega\delta y_{2}+\omega
y_{2}^{2}(\alpha\beta-\omega)$ and $q>0$ and from the hypothesis
$\alpha\beta>\omega$, we have $p+s>0$ and $q-s(p+s)>0$. Then the
quantities of $\tau_{s}=\frac{p+s}{q}$ and
$\zeta_{s}=\sqrt{\frac{q(q+r)}{q-s(p+s)}}$ are well defined.
\end{proof}
Now, we apply the Hopf bifurcation theorem, see \cite{h93}, to
show the existence of a non-trivial periodic solutions of system
\eqref{53} bifurcating from the non trivial steady state $P_{2}$.
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.
Next we state the main result of this paper for small delays.
\begin{theorem}\label{t52}
Assume $0<\omega/\beta<\alpha$, $\alpha\delta>\sigma$,
$\alpha>0$ and $\beta>0$. There exists $\varepsilon _{s}>0$ such that,
for each $0\leq \varepsilon <\varepsilon _{s}$, equation \eqref{53} has a
family of periodic solutions $p_{s}(\varepsilon )$ with period
$T_{s}=T_{s}(\varepsilon )$, for the parameter values
$\tau =\tau(\varepsilon )$ such that $p_{s}(0)=P_{2} $,
$T_{s}(0)=\frac{2\pi }{\zeta_{s} }$ and $\tau (0)=\tau _{s}$, where
$\tau_{s}=\frac{p+s}{q}$ and
$\zeta_{s}=\sqrt{\frac{q(q+r)}{q-s(p+s)}}$ are given in equation
\eqref{52}.
\end{theorem}
\begin{proof} We apply the Hopf bifurcation theorem introduced in
\cite{h93}. From the expression of $f$ in \eqref{53}, we have
\[
f(0,\tau )=0\quad \text{and}\quad
\frac{\partial f(0,\tau )}{\partial \varphi }=0,\quad \text{for all }\tau >0
\]
From equation \eqref{52} and theorem \ref{t3}, the characteristic
equation \eqref{51} has a pair
of simple imaginary roots $\lambda _{s}=i\zeta_{s} $ and
$\overline{\lambda }_{s}=-i\zeta_{s}$ at $\tau=\tau_{s}$.
Next, we need to verify the transversality condition.
From equation \eqref{51}, $W_{0}(\lambda _{s},\tau _{s})=0$ and
$\frac{\partial }{\partial \lambda }W _{0}(\lambda _{s},\tau
_{s})=2 \lambda _{s}(1-s\tau_{s})\neq 0$. According to the
implicit function theorem, there exists a complex function
$\lambda=\lambda(\tau)$ defined in a neighborhood of $\tau _{s} $,
such that $\lambda (\tau _{s})=\lambda _{s}$ and
$W _{0}(\lambda (\tau ),\tau )=0$ and
\begin{equation}
\lambda '(\tau )_{\ }=-\frac{\partial W _{0}(\lambda,\tau
)/\partial \tau }{\partial W _{0}(\lambda ,\tau )/\partial \lambda },
\label{54}
\end{equation}
for $\tau$ in a neighborhood of $\tau_{s}$.
Letting, $\lambda (\tau )=p(\tau )+iq(\tau )$, from \eqref{54} we have
\[
p'(\tau )_{/\tau =\tau _{s}}=\frac{q}{2(1-s\tau_{s})}\,.
\]
From the hypothesis $0<\omega/\beta<\alpha$, we conclude
that
\[
p'(\tau )_{/\tau =\tau _{s}}>0,
\]
which completes the proof.
\end{proof}
\section{Hopf bifurcation occurrence for large delays}\label{s6}
For large delays $\tau$, let $\lambda=\kappa+i \zeta$. According
to the Hopf bifurcation theorem \cite{h93}, we come to the main
result of this paper for large time delays.
\begin{theorem}\label{t61}
Assume $0<\omega/\beta<\alpha$, $\alpha\delta>\sigma$,
$\alpha>0$, $\beta>0$ and $\delta$ close to $0$.
There exists $\varepsilon _{1}>0$ such that, for each
$0\leq \varepsilon <\varepsilon _{1}$, equation \eqref{53} has a
family of periodic solutions $p_{l}(\varepsilon )$ with period
$T_{l}=T_{l}(\varepsilon )$, for the parameter values
$\tau =\tau(\varepsilon )$ such that $p_{l}(0)=P_{2} $,
$T_{l}(0)=\frac{2\pi}{\zeta_{l}}$ and $\tau (0)=\tau _{l}$,
where $\tau_{l}$ and $\zeta_{l}$ are given respectively in
equations \eqref{45} and \eqref{46}.
\end{theorem}
\begin{proof}
As in the previous section, we apply the Hopf bifurcation
theorem introduced in \cite{h93}. From the expression of $f$ in
\eqref{53}, we have,
\[
f(0,\tau )=0\quad \text{and}\quad \frac{\partial f(0,\tau )}{%
\partial \varphi }=0,\quad \text{for all }\tau >0
\]
From equation \eqref{44} and theorem \ref{t2}, the characteristic equation
\eqref{44} has a pair of simple imaginary roots $\lambda _{l}=i\zeta_{l} $
and $\overline{\lambda}_{l}=-i\zeta_{l}$ at $\tau=\tau_{l}$.
From equation \eqref{44}, $W(\lambda _{l},\tau _{l})=0$ and
$\frac{\partial }{\partial \lambda }W(\lambda _{l},\tau _{l})=
2i\zeta_{l}+p+(s-\tau(is\zeta_{l}+q))e^{-i\zeta_{l}\tau_{l}}\neq
0$. According to the implicit function theorem, there exists a
complex function $\lambda=\lambda(\tau)$ defined in a neighborhood
of $\tau _{l} $, such that $\lambda (\tau _{l})=\lambda _{l}$ and
$W(\lambda (\tau ),\tau )=0$ and
\begin{equation}
\lambda '(\tau )=-\frac{\partial W(\lambda ,\tau
)/\partial \tau }{\partial W(\lambda ,\tau )/\partial \lambda }, \label{61}
\end{equation}
for $\tau$ in a neighborhood of $\tau_{l}$.
Then
\begin{equation} \label{62}
\lambda '(\tau)=-\lambda\frac{s\lambda^{3}+(s^{2}p+q)
\lambda^{2}+(sr+pq)\lambda+qr}{\tau
s\lambda^{3}+(s+\tau(sp+q))\lambda^{2}+(2q+\tau(sr+pq))\lambda+pq-sr+qr}
\end{equation}
From equation \eqref{62} we have
\begin{equation} \label{63}
\begin{aligned}
&\kappa'(\tau )\big|_{\tau =\tau_l}=\\
&\zeta_{l}^{2}\frac{s^{2}\zeta_{l}^{4}
+(sqr(\tau-1)+2q^{2})\zeta_{l}^{2}+sr^{2}(q-sr)
+pq^{2}(p+r)-qr(2q+\tau(sr+pq))}{A^{2}+B^{2}},
\end{aligned}
\end{equation}
where
\begin{gather*}
A=-(s+\tau(sp+q))\zeta_{l}^{2}+pq-sr+qr, \\
B=-\tau s\zeta_{l}^{2}+(2q+\tau(sr+pq))\zeta_{l}.
\end{gather*}
From the expression of $r$, when $\delta$ is close to $0$,
then $r$ is very small.
From equation \eqref{63}, we conclude that,
$$
\kappa'(\tau )_{/\tau =\tau _{l}}>0
$$
Therefore, the transversality condition is verified, which
completes the proof.
\end{proof}
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\end{document}