\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 88, pp. 1--7.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/88\hfil Periodic solutions]
{Periodic solutions for a stage-structure ecological model on time
scales}

\author[K. Zhuang\hfil EJDE-2007/88\hfilneg]
{Kejun Zhuang}  

\address{Kejun Zhuang \newline
School of Statistics and Applied Mathematics,
Anhui University of Finance and Economics,
Bengbu, Anhui, 233030, China}
\email{zhkj123@163.com}

\thanks{Submitted April 9, 2007. Published June 15, 2007.}
\subjclass[2000]{92D25, 34C25}
\keywords{Time scales; stage-structure; coincidence degree; periodic solutions}

\begin{abstract}
 In this paper, by using the Mawhin's continuation theorem,
 we prove the existence of periodic solutions for a stage-structure
 ecological model on time scales. This unifies the results for
 differential and difference equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

 Recently, Zheng and Cui  constructed the following
stage-structure population model, with patches and diffusion,
\begin{equation}
   \begin{gathered}
     I'_1(t)=aM_1(t)-bI_1(t)-cI_1(t),  \\
     M'_1(t)=cI_1(t)-\alpha M^2_1(t)+D_{12}(M_2(t)-M_1(t)),  \\
     M'_2(t)=-\beta M_2(t)+D_{21}(M_1(t)-M_2(t)),
   \end{gathered} \label{e1.1}
\end{equation}
where $a, b, c, \alpha, \beta, D_{12}$ and $D_{21}$ are  positive
constants. $I_1(t)$ is the density of immature rana chensinensis in
patch $1$, $M_i(t)$ denotes the density of mature rana chensinensis
in the $i$-patch $(i=1,2)$. $D_{ij}$ is the diffusion coefficient of
the species from patch $j$ to patch $i$. The permanence and
stability of equilibrium of system \eqref{e1.1} were investigated
in \cite{z3}.

Taking into account the periodicity of environment, Zhang and
Zheng reconstructed the model as follows:
\begin{equation}
   \begin{gathered}
     I'_1(t)=a(t)M_1(t)-b(t)I_1(t)-c(t)I_1(t),  \\
     M'_1(t)=c(t)I_1(t)-\alpha (t)M^2_1(t)+D_{12}(t)(M_2(t)-M_1(t)),  \\
     M'_2(t)=-\beta (t)M_2(t)+D_{21}(t)(M_1(t)-M_2(t)),
   \end{gathered}  \label{e1.2}
\end{equation}
where all the coefficients are positive continuous $\omega$-periodic
functions. Based on the theory of coincidence degree, the existence
of positive periodic solution was established in \cite{z2}.
The  corresponding  discrete system
\begin{equation}
   \begin{gathered}
     I_1(k+1)=I_1(k)\exp \big\{ -b(k)-c(k)+a(k)\frac{M_1(k)}{I_1(k)} \big\},  \\
     M_1(k+1)=M_1(k)\exp \big\{ -D_{12}(k)-\alpha(k)M_1(k)+\frac{c(k)I_1(k)
 +D_{12}(k)M_2(k)}{M_1(k)} \big\},  \\
     M_2(k+1)=M_2(k)\exp \big\{ -\beta(k)-D_{21}(k)
+\frac{D_{21}(k)M_1(k)}{M_2(k)} \big\},
   \end{gathered}   \label{e1.3}
\end{equation}
was considered in \cite{z1}. The coefficients are all strictly positive
$\omega$-periodic sequences. The existence of periodic solutions for
\eqref{e1.3} was done. However, the work of \cite{z1,z2} was repeated to
some extent. It is natural to ask whether there is a unified way to
explore such kind of problem. To unify the continuous and discrete
analysis, Stefan Hilger in his Ph.D. Thesis initiated the theory of
calculus on time scales in \cite{h1}. The theme has received much
attention in recent years, such as \cite{b1,b2,b3}. It is true that
unification and extension are the two main features of the calculus
on time scales.

In this paper, we consider the following  ecological model with
stage structure and diffusion  on time scales:
\begin{equation}
   \begin{gathered}
     u_1^\Delta (t)=-b(t)-c(t)+a(t)\frac{e^{u_2(t)}}{e^{u_1(t)}},  \\
     u_2^\Delta (t)=-D_{12}(t)-\alpha(t)e^{u_2(t)}+\frac{c(t)e^{u_1(t)}+D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}},  \\
     u_3^\Delta (t)=-\beta(t)-D_{21}(t)+\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}},
   \end{gathered}   \label{e1.4}
\end{equation}
where $a(t), b(t), c(t), \alpha(t), D_{12}(t), \beta(t)$ and
$D_{21}(t)$ are rd-continuous positive $\omega$-pe\-riodic functions
on time scales $\mathbb{T}$. Set $I_1(t)=e^{u_1(t)}$,
$M_1(t)=e^{u_2(t)}$ and $M_2(t)=e^{u_3(t)}$, if
$\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{Z}$, then \eqref{e1.4} can
be reduced to \eqref{e1.2} and \eqref{e1.3}, respectively. As a result, it is
unnecessary to investigate the periodic solutions of \eqref{e1.2}
and \eqref{e1.3} separately.


Thus, we shall prove the periodicity of system \eqref{e1.4} by Mawhin's
continuation theorem in coincidence degree theory to unify the
results in \cite{z1,z2}. This approach has been widely applied to
deal with the existence of periodic solutions of differential
equations and difference equations but rarely applied to the dynamic
equations on time scales \cite{b2,b3}.

\section{Preliminary results}

 For the convenience of the reader,  we
first present some basic definitions and lemmas about time scales
and the continuation theorem of the coincidence degree theory;
 more details can be found in \cite{b1,g1}. A time scale $\mathbb{T}$ is an
arbitrary nonempty closed subset of
 real numbers $\mathbb{R}$. Throughout this paper, we assume that
the time scale $\mathbb{T}$ is unbounded above and below, such as
$\mathbb{R}$, $\mathbb{Z}$ and $\bigcup_{k\in\mathbb{Z}}[2k,2k+1]$.
The following definitions and lemmas about time scales are from
\cite{b1}.

\begin{definition} \label{def2.1} \rm
 The forward jump operator
$\sigma:\mathbb{T}\to \mathbb{T}$, the backward jump
operator $\rho:\mathbb{T}\to\mathbb{T}$, and the graininess
$\mu:\mathbb{T}\to\mathbb{R}^+=[0,+\infty)$ are defined,
respectively, by
$$
\sigma(t):=\inf\{s\in\mathbb{T}:s>t\}, \quad
\rho(t):=\sup\{s\in\mathbb{T}:s<t\}, \quad
\mu(t)=\sigma(t)-t.
$$
If $\sigma(t)=t$, then $t$ is called right-dense
(otherwise: right-scattered), and if $\rho(t)=t$, then $t$ is called
left-dense(otherwise: left-scattered).
\end{definition}

\begin{definition} \label{def2.2} \rm
Assume $f:\mathbb{T}\to\mathbb{R}$
is a function and let $t\in\mathbb{T}$. Then we define $f^\Delta(t)$
to be the number (provided it exists) with the property that given
any $\varepsilon>0$, there is a neighborhood $U$ of $t$ such that
$$
| f(\sigma(t))-f(s)-f^\Delta(t)(\sigma(t)-s)
|\leq\varepsilon|\sigma(t)-s|\quad \mbox{for all } s\in U.
$$
In this case, $f^\Delta(t)$ is called the delta (or Hilger)
derivative of $f$ at $t$. Moreover, $f$ is said to be delta or
Hilger differentiable on $\mathbb{T}$ if $f^\Delta(t)$ exists for
all $t\in\mathbb{T}$. A function $F:\mathbb{T}\to\mathbb{R}$
is called an antiderivative of $f:\mathbb{T}\to\mathbb{R}$
provided $F^\Delta(t)=f(t)$ for all $t\in \mathbb{T}$. Then we
define
$$
\int_r^sf(t)\Delta t=F(s)-F(r)\quad \mbox{for } r,s\in\mathbb{T}.
$$
\end{definition}

\begin{definition} \label{def2.3} \rm
 A function
$f:\mathbb{T}\to\mathbb{R}$ is said to be rd-continuous if
it is continuous at right-dense points in $\mathbb{T}$ and its
left-sided limits exist(finite) at left-dense points in
$\mathbb{T}$. The set of rd-continuous functions
$f:\mathbb{T}\to\mathbb{R}$ will be denoted by
$C_{rd}(\mathbb{T})$.
\end{definition}

\begin{lemma} \label{lem2.1}
Every rd-continuous function has an antiderivative.
\end{lemma}

\begin{lemma} \label{lem2.2}
If $a,b\in\mathbb{T}$, $\alpha$,
$\beta\in\mathbb{R}$ and $f, g\in C_{rd}(\mathbb{T})$,then
\begin{itemize}
\item[(a)]  $\int_a^b[\alpha f(t)+\beta g(t)]\Delta
t=\alpha\int_a^bf(t)\Delta t+\beta\int_a^b g(t) \Delta t$;
\item[(b)]
 if $f(t)\geq 0$ for all $a\leq t<b$, then $\int_a^bf(t)\Delta
t\geq 0$;
\item[(c)] if $|f(t)|\leq g(t)$ on
$[a,b):=\{t\in\mathbb{T}:a\leq t <b\}$, then
$|\int_a^bf(t)\Delta t |\leq\int_a^bg(t)\Delta t$.
\end{itemize}
\end{lemma}

For simplicity, we use the following notations throughout this
paper. Let $\mathbb{T}$ be $\omega$-periodic, that is
$t\in\mathbb{T}$ implies $t+\omega\in\mathbb{T}$,
\begin{gather*}
k=\min \{\mathbb{R}^+\cap\mathbb{T}\}, \quad
I_\omega=[k,k+\omega]\cap\mathbb{T},\quad
g^L=\inf_{t\in\mathbb{T}}g(t),\\
g^M=\sup_{t\in\mathbb{T}}g(t), \quad
\bar{g}=\frac{1}{\omega}\int_{I_\omega}g(s)\Delta
s=\frac{1}{\omega}\int_k^{k+\omega}g(s)\Delta s,
\end{gather*}
where $g\in C_{rd}(\mathbb{T})$ is an $\omega$-periodic real
function, i.e., $g(t+\omega)=g(t)$ for all $t\in \mathbb{T}$.

Now, we introduce some concepts and a useful result from \cite{g1}.

Let $X, Z$ be normed vector spaces,
$L:\mathop{\rm Dom}L\subset X\to Z$ be a linear mapping,
$N:X \to Z$ be a continuous mapping. The mapping $L$ will
be called a Fredholm mapping of index zero if
$\dim \ker L=\mathop{\rm codim}\mathop{\rm Im}L<+\infty$ and
$\mathop{\rm Im}L$ is closed in $Z$. If $L$ is a Fredholm mapping of index
zero and there exist continuous projections $P:X\to X$ and
$Q:Z\to Z$ such that $\mathop{\rm Im}P=\ker L$,
$\mathop{\rm Im}L=\ker Q=\mathop{\rm Im}(I-Q)$, then it follows that
$L|\mathop{\rm Dom}L\cap\ker P:(I-P)X\to\mathop{\rm Im}L$ is
invertible. We denote the inverse of that map by $K_P$. If $\Omega$
is an open bounded subset of $X$, the mapping $N$ will be called
$L$-compact on $\bar{\Omega}$ if $QN(\bar{\Omega})$ is bounded and
$K_P(I-Q)N:\bar{\Omega}\to X$ is compact. Since $\mathop{\rm Im}Q$
is isomorphic to $\ker L$, there exists an isomorphism
$J:\mathop{\rm Im}Q\to\ker L$.


Next, we state the Mawhin's continuation theorem, which is a main
tool in the proof of our theorem.

\begin{lemma}[Continuation Theorem] \label{lem2.3}
 Let $L$ be a
Fredholm mapping of index zero and $N$ be $L$-compact on
$\bar{\Omega}$. Suppose
\begin{itemize}
\item[(a)] for each $\lambda\in(0,1)$,
every solution $u$ of $Lu=\lambda Nu$ is such that
$u\notin\partial\Omega$;
\item[(b)]  $QNu\neq 0$ for each
$u\in\partial\Omega\cap\ker L$  and the Brouwer degree
$\deg \{JQN,\Omega\cap\ker L,0\}\neq 0$.
\end{itemize}
Then the operator equation $Lu=Nu$ has at least one solution lying in
$\mathop{\rm Dom}L\cap\bar{\Omega}$.
\end{lemma}

\section{Existence of periodic solutions}

 In this section, we shall derive the sufficient conditions
for the global existence of periodic solutions to system \eqref{e1.4}.

\begin{theorem} \label{thm3.1}
 If
$$
\frac{c^La^L}{(b+c)^M}+\frac{D_{12}^LD_{21}^L}{(\beta+D_{21})^M}-D_{12}^M>0,
$$
 then \eqref{e1.4}  has at least one $\omega$-periodic solution.
\end{theorem}


\begin{proof} Let
\begin{gather*}
X=Z=\big\{ (u_1,u_2,u_3)^T\in
C(\mathbb{T},\mathbb{R}^3): u_i(t+\omega)=u_i(t),\;i=1,2,3,
\forall t\in \mathbb{T} \big\},
\\
\| (u_1,u_2,u_3)^T\| =\sum_{i=1}^3 \max_{t\in
I_\omega}|u_i(t)|,\quad (u_1,u_2,u_3)^T\in X \quad(\mbox{or in} Z).
\end{gather*}
Then $X$ and $Z$ are both Banach spaces when they are
endowed with the above norm $\| \cdot \|$. Let
\begin{gather*}
N \begin{bmatrix}
     u_1  \\
     u_2  \\
     u_3  \end{bmatrix}
= \begin{bmatrix}
     N_1  \\
     N_2  \\
     N_3    \end{bmatrix}
= \begin{bmatrix}
     -b(t)-c(t)+a(t)\frac{e^{u_2(t)}}{e^{u_1(t)}}  \\
     -D_{12}(t)-\alpha(t)e^{u_2(t)}+\frac{c(t)e^{u_1(t)}
+D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}}  \\
     -\beta(t)-D_{21}(t)+\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}}
   \end{bmatrix}, \\
L \begin{bmatrix}
     u_1  \\
     u_2  \\
     u_3    \end{bmatrix}
=\begin{bmatrix}
     u_1^\Delta  \\
     u_2^\Delta  \\
     u_3^\Delta   \end{bmatrix},\quad
P \begin{bmatrix}
     u_1  \\
     u_2  \\
     u_3  \end{bmatrix}
=Q \begin{bmatrix}
     u_1  \\
     u_2  \\
     u_3 \end{bmatrix}
= \begin{bmatrix}
     \frac{1}{\omega}\int_k^{k+\omega}u_1(t)\Delta t  \\
     \frac{1}{\omega}\int_k^{k+\omega}u_2(t)\Delta t  \\
     \frac{1}{\omega}\int_k^{k+\omega}u_3(t)\Delta t \end{bmatrix}.
\end{gather*}
Then
\begin{gather*}
\ker L= \big\{ (u_1,u_2,u_3)^T\in X :
(u_1(t),u_2(t),u_3(t))^T=(h_1,h_2,h_3)^T\in\mathbb{R}^3
,t\in\mathbb{T}  \big\}, \\
\mathop{\rm Im}L=\big\{ (u_1,u_2,u_3)^T\in Z :
\bar{u}_1=\bar{u}_2=\bar{u}_3=0,t\in\mathbb{T} \big\}, \\
\dim \ker L=3=\mathop{\rm codim} \mathop{\rm Im}L.
\end{gather*}
Since $\mathop{\rm Im}L$ is closed in $Z$, then $L$ is a
Fredholm mapping of index zero. It is not difficult to prove that
$P$ and $Q$ are continuous projections such that
$\mathop{\rm Im}P=\ker L$ and
$\mathop{\rm Im}L=\ker Q=\mathop{\rm Im}(I-Q)$. Furthermore, the
generalized inverse (of $L$) $K_P:\mathop{\rm Im}L\to
\ker P\cap \mathop{\rm Dom}L$ exists and is given by
$$
K_P \begin{bmatrix}
     u_1  \\
     u_2  \\
     u_3    \end{bmatrix}
= \begin{bmatrix}
     \int_k^tu_1(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega}
\int_k^t u_1(s)\Delta s\Delta t  \\
     \int_k^tu_2(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega}
\int_k^t u_2(s)\Delta s\Delta t  \\
     \int_k^tu_3(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega}
\int_k^t u_3(s)\Delta s\Delta t
   \end{bmatrix} .
$$
Thus
$$
QN\begin{bmatrix}
     u_1  \\
     u_2  \\
     u_3    \end{bmatrix}
= \begin{bmatrix}
     \frac{1}{\omega}\int_k^{k+\omega}\big( -b(t)-c(t)
+a(t)\frac{e^{u_2(t)}}{e^{u_1(t)}}\big)\Delta t  \\
     \frac{1}{\omega}\int_k^{k+\omega}\big( -D_{12}(t)-\alpha(t)e^{u_2(t)}
+\frac{c(t)e^{u_1(t)}+D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}}\big)\Delta t  \\
     \frac{1}{\omega}\int_k^{k+\omega}\big( -\beta(t)-D_{21}(t)
+\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}}\big)\Delta t
   \end{bmatrix},
$$
\begin{align*}
& K_P(I-Q)N \begin{bmatrix}
     u_1  \\
     u_2  \\
     u_3    \end{bmatrix} \\
&= \begin{bmatrix}
     \int_k^tu_1(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega}
\int_k^tu_1(s)\Delta s\Delta t-\left( t-k-\frac{1}{\omega}
\int_k^{k+\omega}(t-k)\Delta t \right)\bar{u}_1  \\
     \int_k^tu_2(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega}
\int_k^tu_2(s)\Delta s\Delta t-\left( t-k-\frac{1}{\omega}
\int_k^{k+\omega}(t-k)\Delta t \right)\bar{u}_2  \\
     \int_k^tu_3(s)\Delta s-\frac{1}{\omega}\int_k^{k+\omega}
\int_k^tu_3(s)\Delta s\Delta t-\left( t-k-\frac{1}{\omega}
\int_k^{k+\omega}(t-k)\Delta t \right)\bar{u}_3
   \end{bmatrix}.
\end{align*}
Obviously, $QN$ and $K_P(I-Q)N$ are continuous. According to
Arzela-Ascoli theorem, it is easy to show that
$K_P(I-Q)N(\bar{\Omega})$ is compact for any open bounded set
$\Omega\subset X$ and $QN(\bar{\Omega})$ is bounded. Thus, $N$ is
$L$-compact on $\bar{\Omega}$.


Now, we shall search an appropriate open bounded subset $\Omega$ for
the application of the continuation theorem, Lemma \ref{lem2.3}. For the
operator equation $Lu=\lambda Nu$, where $\lambda\in (0,1)$, we have
\begin{equation}
\begin{gathered}
     u_1^\Delta (t)=\lambda\left(-b(t)-c(t)
+a(t)\frac{e^{u_2(t)}}{e^{u_1(t)}}\right),  \\
     u_2^\Delta (t)=\lambda\left(-D_{12}(t)
-\alpha(t)e^{u_2(t)}+\frac{c(t)e^{u_1(t)}+D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}}\right),  \\
     u_3^\Delta (t)=\lambda\left(-\beta(t)-D_{21}(t)
+\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}}\right).
   \end{gathered}  \label{e3.1}
\end{equation}
Assume that $(u_1,u_2,u_3)^T\in X$ is a solution of system \eqref{e3.1} for
a certain $\lambda\in (0,1)$. Integrating \eqref{e3.1} on both sides from
$k$ to $k+\omega$, we obtain
\begin{equation}
 \begin{gathered}
     \bar{b}\omega+\bar{c}\omega=\int_k^{k+\omega}a(t)
\frac{e^{u_2(t)}}{e^{u_1(t)}}\Delta t,  \\
     \bar{D}_{12}\omega+\int_k^{k+\omega}\alpha(t)e^{u_2(t)}\Delta t
=\int_k^{k+\omega}\frac{c(t)e^{u_1(t)}+D_{12}(t)e^{u_3(t)}}{e^{u_2(t)}}\Delta t,  \\
     \bar{\beta}\omega+\bar{D}_{21}\omega
=\int_k^{k+\omega}\frac{D_{21}(t)e^{u_2(t)}}{e^{u_3(t)}} \Delta t.
   \end{gathered}\label{e3.2}
\end{equation}
Since $(u_1,u_2,u_3)^T\in X$, there exist
$\xi_i,\eta_i\in[k,k+\omega]$, $i=1,2,3$, such that
\begin{equation}
u_i(\xi_i)=\min_{t\in[k,k+\omega]}\{u_i(t)\},\quad
u_i(\eta_i)=\max_{t\in[k,k+\omega]}\{u_i(t)\},\quad
i=1,2,3.\label{e3.3}
\end{equation}
 From \eqref{e3.1} and \eqref{e3.3}, we have
\begin{gather}
a(\eta_1)e^{u_2(\eta_1)}=(b(\eta_1)+c(\eta_1))e^{u_1(\eta_1)},
\label{e3.4} \\
c(\eta_2)e^{u_1(\eta_2)}+D_{12}(\eta_2)e^{u_3(\eta_2)}
=D_{12}(\eta_2)e^{u_2(\eta_2)}+\alpha(\eta_2)e^{2u_2(\eta_2)},
\label{e3.5} \\
D_{21}(\eta_3)e^{u_2(\eta_3)}=\beta(\eta_3)e^{u_3(\eta_3)}
+D_{21}(\eta_3)e^{u_3(\eta_3)}.
\label{e3.6}
\end{gather}
Thus,
\begin{gather}
(b+c)^Le^{u_1(\eta_1)}\leq a^Me^{u_2(\eta_2)},\label{e3.7} \\
(\beta+D_{21})^Le^{u_3(\eta_3)}\leq D_{21}^Me^{u_2(\eta_2)}.\label{e3.8}
\end{gather}
 From \eqref{e3.5}, \eqref{e3.7} and \eqref{e3.8}, we have
\begin{align*}
\alpha^Le^{2u_2(\eta_2)}+D_{12}^Le^{u_2(\eta_2)}
&\leq c^Me^{u_1(\eta_1)}+D_{12}^Me^{u_3(\eta_3)}\\
&\leq \frac{c^Ma^M}{(b+c)^L}e^{u_2(\eta_2)}
  +\frac{D_{12}^MD_{21}^M}{(\beta+D_{21})^L}e^{u_2(\eta_2)}.
\end{align*}
 From the above inequality, we  get
\begin{equation}
e^{u_2(\eta_2)}\leq \frac{1}{\alpha^L} \big[
\frac{c^Ma^M}{(b+c)^L}+\frac{D_{12}^MD_{21}^M}{(\beta+D_{21})^L}-D_{12}^L
\big]:= L_2.\label{e3.9}
\end{equation}
Substituting \eqref{e3.9} into \eqref{e3.7} and \eqref{e3.8}, respectively,
the following estimates hold:
$$
e^{u_1(\eta_1)}\leq\frac{a^M}{(b+c)^L}L_2:=L_1,\quad
e^{u_3(\eta_3)}\leq\frac{D_{21}^M}{(\beta+D_{21})^L}L_2:=L_3 .
$$
Similarly, we also get the  results:
\begin{gather*}
e^{u_2(\xi_2)}\geq\frac{1}{\alpha^M}\big[
\frac{c^La^L}{(b+c)^M}+\frac{D_{12}^LD_{21}^L}{(\beta+D_{21})^M}-D_{12}^M
\big]:=l_2, \\
e^{u_1(\xi_1)}\geq\frac{a^L}{(b+c)^M}l_2:=l_1  \, , \quad
e^{u_3(\xi_3)}\geq \frac{D_{21}^L}{(\beta+D_{21})^M}l_2:=l_3   .
\end{gather*}
So, we have
\begin{gather*}
\max_{t\in[k,k+\omega]}|u_1(t)|\leq\max\{ | \ln L_1
|,|\ln l_1| \}:=R_1, \\
\max_{t\in[k,k+\omega]}|u_2(t)|\leq\max\{ |\ln L_2
|,  | \ln l_2 | \}:=R_2, \\
\max_{t\in[k,k+\omega]}|u_3(t)|\leq\max \{ | \ln L_3
|,| \ln l_3 | \}:=R_3.
\end{gather*}
Clearly, $R_1, R_2$ and $R_3$ are independent of $\lambda$. Let
$R=R_1+R_2+R_3+R_0$, where $R_0$ is taken sufficiently large such
that $R_0\geq\sum_{i=1}^3|l_i|+\sum_{i=1}^3|L_i|$.

Now, we consider the algebraic equations:
\begin{equation}
   \begin{gathered}
     \bar{b}+\bar{c}-\bar{a}{e^{y-x}}=0,  \\
     \bar{D}_{12}+\bar{\alpha}e^{y}-\bar{c}e^{x-y}-\bar{D}_{12}e^{z-y}=0,  \\
     \bar{\beta}+\bar{D}_{21}-\bar{D}_{21}e^{y-z}=0,
   \end{gathered}  \label{e3.10}
\end{equation}
every solution $(x^\ast,y^\ast,z^\ast)^T$ of \eqref{e3.10} satisfies
$\| (x^\ast,y^\ast,z^\ast)^T \|<R$. Now, we define
$\Omega=\{ (u_1(t),u_2(t),u_3(t))^T\in X,\|
(u_1(t),u_2(t),u_3(t))^T \|<R \}$. Then it is clear that
$\Omega$ verifies the requirement (a) of Lemma \ref{lem2.3}. If
$(u_1(t),u_2(t),u_3(t))^T\in
\partial\Omega\cap\ker L=\partial\Omega\cap\mathbb{R}^3$, then
$(u_1(t),u_2(t),u_3(t))^T$ is a constant vector in $\mathbb{R}^3$
with
$\|(u_1(t),u_2(t),u_3(t))^T\|=|u_1|+|u_2|+|u_3|=R$, so
we have
$$
QN \begin{bmatrix}
     u_1  \\
     u_2  \\
     u_3   \end{bmatrix}
   = \begin{bmatrix}
     \bar{a}{e^{u_2-u_1}}-\bar{b}-\bar{c}  \\
     \bar{c}e^{u_1-u_2}+\bar{D}_{12}e^{u_3-u_2}-\bar{D}_{12}-\bar{\alpha}e^{u_2}  \\
     \bar{D}_{21}e^{u_2-u_3}-\bar{\beta}-\bar{D}_{21}
   \end{bmatrix}
   \neq \begin{bmatrix}
     0  \\
     0  \\
     0   \end{bmatrix}.
$$
Moreover, define
$$
\phi(u_1,u_2,u_3,\mu)= \begin{bmatrix}
     \bar{a}e^{u_2-u_1}-\bar{b}  \\
     \bar{c}e^{u_1-u_2}-\bar{\alpha}e^{u_2}  \\
     -\bar{D}_{21}+\bar{D}_{21}e^{u_2-u_3}
   \end{bmatrix} +\mu \begin{bmatrix}
     -\bar{c} \\
     \bar{D}_{12}(e^{u_3-u_2}-1)  \\
     -\bar{\beta}
   \end{bmatrix} ,
$$
where $\mu\in[0,1]$ is a parameter. If
$(u_1,u_2,u_3)^T\in\partial\Omega\cap\ker L$, then
$\phi(u_1,u_2,u_3,\mu)\neq 0$. In addition, we can easily see that
the algebraic equation $\phi(u_1,u_2,u_3,0)=0$ has a unique solution
in $\mathbb{R}^3$. Thus the invariance of homotopy produces
\begin{align*}
\deg (JQN,\Omega\cap\ker L,0)
&=\deg (QN,\Omega\cap\ker L,0)\\
&=\deg (\phi(u_1,u_2,u_3,1),\Omega\cap\ker L,0)\\
&=\deg (\phi(u_1,u_2,u_3,0),\Omega\cap\ker L,0)\\
&\neq 0.
\end{align*}
By now, we have verified that $\Omega$ fulfills
all requirements of Lemma \ref{lem2.3}; therefore, system \eqref{e1.4} has at least
one $\omega$-periodic solution in $\mathop{\rm Dom}L\cap\bar{\Omega}$. The
proof is complete.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
Note that systems \eqref{e1.2} and \eqref{e1.3} are
special cases of system \eqref{e1.4}, both \eqref{e1.2} and \eqref{e1.3}
have at least one $\omega$-periodic solution when the conditions of
Theorem \ref{thm3.1} hold. In addition, the conditions are weaker than those
in \cite{z1,z2}.
\end{remark}


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\end{document}