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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 22, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/22\hfil Permanence for a competition and cooperation model]
{Permanence for a competition and cooperation model of enterprise
 cluster with delays and feedback controls}

\author[P. Liu, Y. Li \hfil EJDE-2013/22\hfilneg]
{Ping Liu, Yongkun Li } 

\address{Ping Liu  \newline
Department of Mathematics, Yunnan University,
Kunming, Yunnan 650091,  China}
\email{liuping@ynu.edu.cn}

\address{Yongkun Li \newline
Department of Mathematics, Yunnan University,
Kunming, Yunnan 650091,  China}
\email{yklie@ynu.edu.cn}

\thanks{Submitted October 4, 2012. Published January 25, 2013.}
\thanks{Supported by grant 10971183 from the National Natural
Sciences Foundation of China}
\subjclass[2000]{34C11, 91B55, 92D25, 93B52}
\keywords{Enterprise cluster; permanence;
 competition and cooperation model; \hfill\break\indent
time delay; feedback control}

\begin{abstract}
 In this article, based on population ecology theory, we present a
 competition and cooperation system of the enterprise cluster with
 time delays and feedback controls. By using  differential inequalities,
 we obtain sufficient conditions for the permanence of the system,
 which shows that the time delay, feedback control and initial production
 have an influence on the persistent properties of the system.
 We further interpret our result from the economic point of view.
 Some suggestions about enterprise cluster are put forward through
 the analysis of our results.
\end{abstract}

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

\section{Introduction}

    Enterprise cluster refers to the concentration of similar or related
enterprises in a specific area, which form fixed economic outputs and have
 certain economic influence on outsides \cite{p1}.
After a large number of observations, it is found that there is a similarity
between the enterprise cluster and the ecological population system.
 Recently, some researchers have presented some models about enterprise
clusters based on ecology theory, which arouse growing interest in applying
the methods of ecology and dynamic system theory to study enterprise clusters,
for example 
\cite{c3,g1,l1,l2,t1,x1,x2,y1,y2} and references cited therein.
For an example, in \cite{t1}, the authors  considered  the following competition
and cooperation of enterprise cluster based on the ecosystem
\begin{gather*}
   x_1'(t)=r_1x_1(t)[1-\frac{1}{K}x_1(t)-\frac{1}{K}\alpha(x_2(t)-b_2)^{2}],\\
   x_2'(t)=r_2x_2(t)[1-\frac{1}{K}x_2(t)+\frac{1}{K}\beta(x_1(t)-b_1)^{2}],
\end{gather*}
where $x_1(t)$, $x_2(t)$ represent the outputs of enterprise A and enterprise B,
respectively, $r_i, b_i, K, \alpha, \beta$ are positive constants, $i=1,2$.
 $r_1,r_2$ are the intrinsic growth rates, $K$ denotes the carrying capacity
of market under the natural conditions, $\alpha, \beta$ are the competitive
power coefficients of the two enterprises, and $b_1, b_2$ are the initial
 productions of the enterprises, respectively.

      Let  $a_1=\frac{r_1}{K}$, $a_2=\frac{r_2}{K}$, $c_1=\frac{\alpha}{K}$,
$c_2=\frac{\beta}{K}$, the system above becomes
\begin{gather*}
     x_1'(t)=x_1(t)[r_1-a_1x_1(t)-c_1(x_2-b_2)^{2}],\\
       x_2'(t)=x_2(t)[r_2-a_2x_2(t)+c_2(x_1-b_1)^{2}].
\end{gather*}
In real world, competitors always invade the core assets of enterprises
by counterplans and bring the actual loss, which is
not transient happened, there is a time delay.
On the other hand, enterprises  in the real world are continuously distributed
 by unpredictable forces which can result in
changes in the economic parameters such as intrinsic growth rates.
Of practical interest in economics is the question of whether or not
an  enterprise cluster  can withstand those unpredictable disturbances
which persist for a finite period of time. In the language of
control variables, we call the disturbance functions as control variables.

 Motivated by   above, in this paper, we propose a competitive and cooperation
model of $n$ satellite enterprises and a dominant enterprise
under center halfback model with time-varying delays and feedback controls
as follows:
\begin{equation} \label{e1.1}
\begin{aligned}
 \frac{\mathrm{d}x_1(t)}{\mathrm{d}t}
 &= x_1(t)\Big[r_1(t)-\sum_{i=0}^{m}a_1^{i}(t)x_1(t-i\tau)
 -\gamma_1(t)(x_2(t)-b_2)^{2}\\
&\quad -q_1(t)\int_{-\delta_1}^{0}F_1(s)u_1(t+s)\mathrm{d}s\Big],\\
\frac{\mathrm{d}x_2(t)}{\mathrm{d}t}
 &= x_2(t)\Big[r_2(t)-\sum_{j=0}^{n}a_2^{j}(t)x_2(t-j\tau)
 +\gamma_2(t)\int_{-\sigma}^{0}H(s)(x_1(t+s)-b_1)^{2}\mathrm{d}s\\
 &\quad -q_2(t)\int_{-\delta_2}^{0}F_2(s)u_2(t+s)\mathrm{d}s\Big],\\
 \frac{\mathrm{d}u_k(t)}{\mathrm{d}t}
&= -d_k(t)u_k(t)+e_k(t)x_k(t)+f_k(t)\int_{-\eta_k}^{0}G_k(s)x_k(t+s)
 \mathrm{d}s,\;k=1,2
\end{aligned}
\end{equation}
with initial conditions
\begin{equation} \label{e1.2}
\begin{gathered}
x_1(t)=\phi_1(t)\geq 0,\quad\text{for $t\in[-\gamma,0)$ and }\phi_1(0)>0,\\
x_2(t)=\phi_2(t)\geq 0,\quad\text{for $t\in[-\gamma,0)$ and }  \phi_2(0)>0,\\
u_k(t)=\phi_{k+2}(t)\geq 0,\quad\text{for $t\in[-\gamma,0)$ and }
 \phi_{k+2}(0)>0,\; k=1,2,
\end{gathered}
\end{equation}
where $\xi=\max\{\delta_1,\delta_2,\eta_1,\eta_2,\sigma,m\tau,n\tau\}$,
 $\phi_1(t),\phi_2(t), \phi_{k+2}(t)(k=1,2)$ are continuous on $[-\xi,0]$,
 $x_1(t)$ and $x_2(t)$ denote the outputs of enterprises A and B in cluster
respectively, $r_1(t)$ and $r_2(t)$ are their intrinsic growth rates at time
$t$, $a_1^{i}(t)$ and $a_2^{j}(t)$ account for their self-regulation
coefficients, $\gamma_1(t)$ and $\gamma_2(t)$ represent their contribution
coefficients to the other, $b_1, b_2$ are the initial productions of the
enterprises respectively, $\delta_k,\eta_k,\sigma,\tau,m,n$ are positive
constants, $F_k(s),G_k(s),H(s)$ are all nonnegative continuous functions
such that
$$
\int_{-\delta_k}^{0}F_k(s)\mathrm{d}s=1, \quad
\int_{-\eta_k}^{0}G_k(s)\mathrm{d}s=1, \quad
\int_{-\sigma}^{0}H(s)\mathrm{d}s=1\; (k=1,2),
$$
 the above two equations describe the process of interactions
between enterprises A and B, the latter two equations are control equations,
 $u_1(t)$ and $u_2(t)$ are feedback control variables,
 $a_1^{i}(t)$, $a_2^{j}(t)$ ($i=0,1,\dots,m$; $j=0,1,\dots,n$),
$r_k(t)$, $\gamma_k(t)$, $q_k(t)$, $d_k(t)$, $e_k(t)$,
$f_k(t)$ $(k=1,2)$
are continuous, bounded and positive real-valued functions on $[0,+\infty)$.

Since the competition and cooperation   among inter-members in a cluster
is the driving force for the evolution of enterprise cluster, a nature
question is that under what conditions  an  enterprise cluster can attain
the goal of co-existence, co-evolution and common prosperity?
Our main purpose of this paper is to study the permanence
of \eqref{e1.1}.
Our result shows that not only the time delay and feedback control but also the initial production have influence on the permanence of  system \eqref{e1.1}.

\section{Main results}

In this section, we establish the permanence   of   system \eqref{e1.1}.
It is not difficult to see that solutions of system \eqref{e1.1} and \eqref{e1.2} are well defined for all $t\geq 0$ and satisfy
$$
x_k(t)>0,\quad u_k(t)>0,\quad t\geq0,\;k=1,2.
$$
For convenience, we shall introduce some notations, definition and lemmas
which will be useful for our main results. For a continuous bounded function
$g(t)$ defined on $[0,+\infty)$, we denote
$$
g^{M}=\sup_{0\leq t<+\infty}g(t),\quad  g^{L}=\inf_{0\leq t<+\infty}g(t).
$$

\begin{definition}[\cite{w1}] \label{def2.1} \rm
 System \eqref{e1.1} is said to be permanent if there exists two positive
 constants $m,M$ such that
\begin{gather*}
m\leq\liminf_{t\to\infty}x_i(t)\leq\limsup_{t\to\infty}x_i(t)\leq
M,\quad i=1,2; \\
m\leq\liminf_{t\to\infty}u_i(t)\leq\limsup_{t\to\infty}u_i(t)\leq
M,\quad i=1,2;
\end{gather*}
for any solution $(x_1(t),x_2(t),u_1(t),u_2(t))^{T}$ of system \eqref{e1.1}.
\end{definition}

As a direct consequence of \cite[Lemma 2.2]{w1}, we have the following result.

\begin{lemma} \label{lem2.1}
 Assume that for $y(t)>0$, it holds that
$$
\frac{\mathrm{d}y(t)}{\mathrm{d}t}\leq y(t)
\Big[\lambda-\sum_{l=0}^{k}\mu^{l}y(t-l\tau)\Big]
$$
with initial conditions $y(t)=\phi(t)\geq 0$ for $t\in[-k\tau,0)$
and $\phi(0)>0$, where
$$
\lambda > 0,\quad \mu^{l}\geq 0,\quad l=0,1,\dots,k,\quad
\mu=\sum_{l=0}^{k}\mu^{l}>0,
$$
are constants. Then there exists a positive constant $K_{y}<+\infty$ such that
\begin{equation} \label{e2.1}
\limsup_{t\to +\infty}y(t)\leq K_{y}=\frac{\lambda}{\mu}\exp\{\lambda k\tau\}<+\infty.
\end{equation}
\end{lemma}

\begin{lemma}[\cite{n1}] \label{lem2.2}
Assume that for $y(t)>0$, it holds that
$$
\frac{\mathrm{d}y(t)}{\mathrm{d}t}\geq y(t)
\Big[\lambda-\sum_{l=0}^{k}\mu^{l}y(t-l\tau)\Big].
$$
If \eqref{e2.1} holds, then there exists a positive constant $k_{y}>0$
such that
\begin{equation} \label{e2.2}
\liminf_{t\to +\infty}y(t)\geq k_{y}=\frac{\lambda}{\mu}
\exp\{(\lambda-\mu K_{y})k\tau\}>0,
\end{equation}
where $\mu=\sum_{l=0}^{k}\mu^{l}>0$, $\lambda>0$.
\end{lemma}

\begin{lemma}[\cite{c1}] \label{lem2.3}
Let $a>0,b>0$,
\begin{itemize}
  \item [$(I)$] If $\frac{\mathrm{d}x}{\mathrm{d}t}\geq b-ax$, then $\liminf_{t\to +\infty}x(t) \geq \frac{b}{a}$ for $t\geq0$ and $x(0)>0$.
  \item [$(II)$] If $\frac{\mathrm{d}x}{\mathrm{d}t}\leq b-ax$, then $\limsup_{t\to +\infty}x(t) \leq \frac{b}{a}$ for $t\geq0$ and $x(0)>0$.
\end{itemize}
\end{lemma}

\begin{lemma}\cite{c2} \label{lem2.4}
 Assume that $a>0$, $b(t)>0$ is a bounded continuous function and $x(0)>0$.
Further suppose that
$$
\frac{\mathrm{d}x(t)}{\mathrm{d}t}\leq b(t)-ax(t),
$$
then for all $t\geq s\geq 0$,
$$
x(t)\leq x(t-s)\exp\{-as\}+\int_{t-s}^{t}b(r)\exp\{a(r-t)\}\mathrm{d}r.
$$
\end{lemma}

\begin{lemma}\label{lem2.5}
 Assume that $a_1^{iL}>0, a_2^{jL}>0$ $(i=0,1,\dots,m;j=0,1,\dots,n)$,
$d_k^{L}>0$ $(k=1,2)$. Let $(x(t),u(t))^{T}=(x_1(t),x_2(t),u_1(t),u_2(t))^{T}$
 be any positive solution of system \eqref{e1.1}, then there exists a
positive constant $\overline{M}$  which is independent of the solution of
 system \eqref{e1.1}  such that
$$
\limsup_{t\to +\infty}x_k(t)\leq \overline{M},\quad
\limsup_{t\to +\infty}u_k(t)\leq \overline{M},\quad k=1,2.
$$
\end{lemma}

 \begin{proof}
 Let $(x(t),u(t))^{T}=(x_1(t),x_2(t),u_1(t),u_2(t))^{T}$ be a solution of
system \eqref{e1.1} satisfying the initial condition \eqref{e1.2}.
For $t\geq 0$, from the first equation of system \eqref{e1.1}, it follows that
\begin{equation} \label{e2.3}
\frac{\mathrm{d}x_1(t)}{\mathrm{d}t}\leq x_1(t)\Big[r_1^{M}-\sum_{i=0}^{m}a_1^{iL}x_1(t-i\tau)\Big],\quad t\geq 0.
\end{equation}
Applying Lemma \ref{lem2.1} to \eqref{e2.3} leads to
\begin{equation} \label{e2.4}
\limsup_{t\to +\infty}x_1(t)\leq \frac{r_1^{M}}{\sum_{i=0}^{m}a_1^{iL}}\exp\{r_1^{M}m\tau\}:=M_1.
\end{equation}

Next, we show that $x_2(t)$ is bounded above. By \eqref{e2.4}, there exists
a positive constant $T_1>0$ such that $x_1(t)\leq 2M_1$ for $t>T_1$.
Then, from the second equation of system \eqref{e1.1}, we obtain
$$
\frac{\mathrm{d}x_2(t)}{\mathrm{d}t}\leq x_2(t)\Big[r_2^{M}
+\gamma_2^{M}(2M_1-b_1)^{2}-\sum_{j=0}^{n}a_2^{jL}x_2(t-j\tau)\Big],\quad
t\geq T_1.
$$
By Lemma \ref{lem2.1}, it follows that
\begin{equation} \label{e2.5}
\limsup_{t\to +\infty}x_2(t)\leq \frac{r_2^{M}+\gamma_2^{M}
(2M_1-b_1)^{2}}{\sum_{j=0}^{n}a_2^{jL}}
\exp\{(r_2^{M}+\gamma_2^{M}(2M_1-b_1)^{2})n\tau\}:=M_2.
\end{equation}
Thus, there exists a $T_2>T_1+\xi$ such that $x_1(t)\leq 2M_1,x_2(t)\leq 2M_2$
for $t\geq T_2$. It follows from system \eqref{e1.1} that
$$
\frac{\mathrm{d}u_k(t)}{\mathrm{d}t}\leq 2(e_k^{M}
+f_k^{M})M_k-d_k^{L}u_k(t),\quad k=1,2,\; t\geq T_2,
$$
applying Lemma \ref{lem2.3} (II) to the differential inequalities above, we obtain
\begin{equation} \label{e2.6}
\limsup_{t\to +\infty}u_k(t)\leq \frac{2(e_k^{M}+f_k^{M})M_k}{d_k^{L}}
:=M_{k+2},\quad k=1,2.
\end{equation}
Combined with \eqref{e2.4},\eqref{e2.5} and \eqref{e2.6}, we set
\begin{equation} \label{e2.7}
\overline{M}:=\max\{M_1,M_2,M_3,M_4\}.
\end{equation}
Obviously, $\overline{M}$ is independent of the solution of\eqref{e1.1} and
$$
\limsup_{t\to +\infty}x_k(t)\leq \overline{M},\quad
\limsup_{t\to +\infty}u_k(t)\leq \overline{M},\quad k=1,2.
$$
The proof is complete.
\end{proof}

\begin{lemma}  \label{lem2.6}
Assume that $r_2^{L}>0$, $\gamma_k^{L}>0$, $d_k^{L}>0$,
$e_k^{L}>0,f_k^{L}>0$ $(k=1,2)$,
 $2q_2^{M}\overline{M}<\frac{1}{2}r_2^{L}$,
$\gamma_1^{M}(2\overline{M}-b_2)^{2}<\frac{r_1^{L}}{2}$.
Let $(x(t),u(t))^{T}=(x_1(t),x_2(t),u_1(t),u_2(t))^{T}$ be any positive
solution of  \eqref{e1.1}, then there exists a positive constant
$\overline{m}$, which is independent of the solution of \eqref{e1.1}  such that
$$
\liminf_{t\to +\infty}x_k(t)\geq \overline{m},\quad
\liminf_{t\to +\infty}u_k(t)\geq \overline{m},\quad k=1,2,
$$
where $\overline{M}$ is defined by \eqref{e2.7}.
\end{lemma}

 \begin{proof}
Let $(x(t),u(t))^{T}=(x_1(t),x_2(t),u_1(t),u_2(t))^{T}$ be a solution of
 \eqref{e1.1} satisfying the initial condition \eqref{e1.2}.
From the first equation of system \eqref{e1.1} and Lemma \ref{lem2.5}, there exists a
positive constant $T_3>T_2+\xi$ such that
$x_k(t)\leq 2\overline{M}, u_k(t)\leq 2\overline{M},k=1,2$ for $t\geq T_3$,
then we have
\begin{equation} \label{e2.8}
\begin{aligned}
\frac{\mathrm{d}x_1(t)}{\mathrm{d}t}
&\geq x_1(t)\Big[r_1^{L}-\sum_{i=0}^{m}a_1^{iM}2\overline{M}
 -\gamma_1^{M}(2\overline{M}-b_2)^{2}-2q_1^{M}\overline{M}\Big],\quad  t>T_3\\
&\geq x_1(t)\Big[-\sum_{i=0}^{m}a_1^{iM}2\overline{M}
 -\gamma_1^{M}(2\overline{M}-b_2)^{2}-2q_1^{M}\overline{M}\Big]
 =x_1(t)\cdot\theta,
\end{aligned}
\end{equation}
where $\theta=-2\sum_{i=0}^{m}a_1^{iM}\overline{M}
-\gamma_1^{M}(2\overline{M}-b_2)^{2}-2q_1^{M}\overline{M}<0$.
Integrating \eqref{e2.8} from $\alpha$ to $t(\alpha\leq t)$, we obtain
\begin{equation} \label{e2.9}
x_1(\alpha)\leq x_1(t)\exp\{-\theta(t-\alpha)\},
\end{equation}
then
\begin{equation} \label{e2.10}
x_1(t+s)\leq x_1(t)\exp\{\theta s\},\quad s\leq 0.
\end{equation}
By the third equation of system \eqref{e1.1}, we obtain
\begin{equation} \label{e2.11}
\begin{aligned}
\frac{\mathrm{d}u_1(t)}{\mathrm{d}t}
&\leq -d_1^{L}u_1(t)+e_1^{M}x_1(t)+f_1^{M}
 \int_{-\eta_1}^{0}G_1(s)x_1(t+s)\mathrm{d}s\\
&\leq -d_1^{L}u_1(t)+e_1^{M}x_1(t)+f_1^{M}
 \int_{-\eta_1}^{0}G_1(s)x_1(t)\exp\{\theta s\}\mathrm{d}s\\
&\leq -d_1^{L}u_1(t)+e_1^{M}x_1(t)+f_1^{M}\exp\{-\theta\eta_1\}x_1(t)\\
&= (e_1^{M}+f_1^{M}\exp\{-\theta\eta_1\})x_1(t)-d_1^{L}u_1(t).
\end{aligned}
\end{equation}
Applying Lemma \ref{lem2.4} to \eqref{e2.11}, for $t\geq\alpha>T_3+\xi$, we have
\begin{equation} \label{e2.12}
\begin{aligned}
u_1(t)&\leq  u_1(t-\alpha)\exp\{-d_1^{L}\alpha\}+\int_{t-\alpha}^{t}(e_1^{M}
 +f_1^{M}\exp\{-\theta\eta_1\})x_1(r)\exp\{d_1^{L}(r-t)\}\mathrm{d}r\\
        &\leq  u_1(t-\alpha)\exp\{-d_1^{L}\alpha\}+(e_1^{M}+f_1^{M}
 \exp\{-\theta\eta_1\})\\
        &\quad \times\int_{t-\alpha}^{t}x_1(t)\exp\{-\theta(t-r)\}
 \exp\{d_1^{L}(r-t)\}\mathrm{d}r\\
        &\leq  u_1(t-\alpha)\exp\{-d_1^{L}\alpha\}+(e_1^{M}+f_1^{M}
 \exp\{-\theta\eta_1\})\frac{1}{\theta}(1-\exp\{-\theta\alpha\})x_1(t)\\
        &=  u_1(t-\alpha)\exp\{-d_1^{L}\alpha\}+\rho x_1(t),
\end{aligned}
\end{equation}
where $\rho=\frac{1}{\theta}(e_1^{M}+f_1^{M}\exp\{-\theta\eta_1\})
(1-\exp\{-\theta\alpha\})>0$.
 Notice that for large enough $t,\alpha$ and $t-\alpha>T_3$,
then $u_1(t-\alpha)\leq 2\overline{M}$. Thus, for $t>T_3+\alpha$, we obtain
$$
u_1(t)\leq 2\overline{M}\exp\{-d_1^{L}\alpha\}+\rho x_1(t).
$$
Combined with \eqref{e2.10}, for $t>T_3+\alpha+\xi$, we have
\begin{equation}  \label{e2.13}
\begin{aligned}
u_1(t+s)&\leq 2\overline{M}\exp\{-d_1^{L}\alpha\}+\rho x_1(t+s),\quad s\leq 0\\
         &\leq 2\overline{M}\exp\{-d_1^{L}\alpha\}+\rho x_1(t)\exp\{\theta s\}.
\end{aligned}
\end{equation}
Take \eqref{e2.13} into the first equation of system \eqref{e1.1},
for all $t>T_3+\alpha+2\xi$, we obtain
\begin{align*}
\frac{\mathrm{d}x_1(t)}{\mathrm{d}t}
&\geq  x_1(t)\Big[r_1^{L}-\sum_{i=0}^{m}a_1^{iM}x_1(t-i\tau)
 -\gamma_1^{M}(2\overline{M}-b_2)^{2}\\
&\quad - q_1^{M}\int_{-\delta_1}^{0}F_1(s)\Big(2\overline{M}\exp\{-d_1^{L}\alpha\}+\rho x_1(t)\exp\{\theta s\}\Big)\mathrm{d}s\Big]\\
&\geq  x_1(t)\Big[r_1^{L}-\sum_{i=0}^{m}a_1^{iM}x_1(t-i\tau)
 -\gamma_1^{M}(2\overline{M}-b_2)^{2}\\
&\quad - q_1^{M}\Big(2\overline{M}\exp\{-d_1^{L}\alpha\}+\rho\exp\{-\theta\delta_1\}x_1(t)\Big)\Big]\\
&=  x_1(t)\Big[r_1^{L}-(q_1^{M}\rho\exp\{-\theta\delta_1\})x_1(t)
 -\sum_{i=0}^{m}a_1^{iM}x_1(t-i\tau)\\
&\quad -\gamma_1^{M}(2\overline{M}-b_2)^{2}-2q_1^{M}\overline{M}
 \exp\{-d_1^{L}\alpha\}\Big].
\end{align*}
Notice that for large enough $t, \exp\{-d_1^{L}\alpha\}\to 0$ as
$\alpha\to +\infty$. Then, there exists a positive constant
$\alpha_0=\max\{\frac{1}{d_1^{L}}ln\frac{8q_1^{M}\overline{M}}{r_1^{L}}+1,
T_3+\xi\}$ such that
$$
2q_1^{M}\overline{M}\exp\{-d_1^{L}\alpha\}<\frac{r_1^{L}}{4}\quad\text{for }\quad \alpha\geq\alpha_0,
$$
then, for $t>T_3+\alpha_0+2\xi=T_4$, we have
\begin{equation} \label{e2.14}
\begin{aligned}
\frac{\mathrm{d}x_1(t)}{\mathrm{d}t}
&\geq  x_1(t)\Big[\frac{r_1^{L}}{4}-(q_1^{M}\rho'\exp\{-\theta\delta_1\}+a_1^{0M})x_1(t)\\
&\quad -a_1^{1M}x_1(t-\tau)-\cdots--a_1^{mM}x_1(t-m\tau)\Big],
\end{aligned}
\end{equation}
where $\rho'=\frac{1}{\theta}(e_1^{M}+f_1^{M}\exp\{-\theta\eta_1\})
 (1-\exp\{-\theta\alpha_0\})>0$.
Applying  Lemma \ref{lem2.2} to the differential inequality \eqref{e2.14},
it follows that
\begin{equation} \label{e2.15}
\liminf_{t\to +\infty}x_1(t)\geq m_1
=\frac{\frac{1}{4}r_1^{L}}{\mu}\exp\{(\frac{1}{4}r_1^{L}-\mu k_1)m\tau\}>0,
\end{equation}
where
\begin{eqnarray*}
\mu= q_1^{M}\rho'\exp\{-\theta\delta_1\}+\sum_{i=0}^{m}a_1^{iM}>0,\quad
k_1=\frac{r_1^{L}}{4\mu}\exp\{\frac{r_1^{L}}{4}m\tau\}>0.
\end{eqnarray*}
 From \eqref{e2.15}, there exists a positive constant $T_{5}>T_4+\xi$
such that $x_1(t)\geq \frac{m_1}{2}$ for $t\geq T_{5}$.
Then, by the second equation of system \eqref{e1.1}, we have
$$
\frac{\mathrm{d}x_2(t)}{\mathrm{d}t}\geq x_2(t)\Big[r_2^{L}
+\gamma_2^{L}(\frac{m_1}{2}-b_1)^{2}-2q_2^{M}\overline{M}
-\sum_{j=0}^{n}a_2^{jM}x_2(t-j\tau)\Big],\quad  t\geq T_{5}.
$$
Applying Lemma \ref{lem2.2} to the inequality above, we have
\begin{equation} \label{e2.16}
\begin{aligned}
&\liminf_{t\to +\infty}x_2(t)\geq m_2\\
&= \frac{\frac{1}{2}r_2^{L}+\gamma_2^{L}
 (\frac{m_1}{2}-b_1)^{2}}{\sum_{j=0}^{n}a_2^{jM}}
\exp\Big\{\Big[\frac{1}{2}r_2^{L}+\gamma_2^{L}
(\frac{m_1}{2}-b_1)^{2}-\sum_{j=0}^{n}a_2^{jM}k_2\Big]n\tau\Big\},
\end{aligned}
\end{equation}
where $k_2=\frac{\frac{1}{2}r_2^{L}+\gamma_2^{L}
(\frac{m_1}{2}-b_1)^{2}}{\sum_{j=0}^{n}a_2^{jM}}\exp\{[\frac{1}{2}r_2^{L}
+\gamma_2^{L}(\frac{m_1}{2}-b_1)^{2}]n\tau\}$.
 From the above discussion, there exists a $T_6>T_5+\xi$ such that
$$
x_k(t)\geq\frac{1}{2}m_k,\quad k=1,2,\text{ for }t\geq T_6.
$$
By system \eqref{e1.1}, we obtain
$$
\frac{\mathrm{d}u_k(t)}{\mathrm{d}t}
\geq \frac{1}{2}(e_k^{L}+f_k^{L})m_k-d_k^{M}u_k(t),\quad
k=1,2,\; t\geq T_6.
$$
Applying Lemma \ref{lem2.3}(I) to the above differential inequalities, we obtain
\begin{equation} \label{e2.17}
\liminf_{t\to +\infty}u_k(t)\geq m_{k+2}
=\frac{(e_k^{L}+f_k^{L})m_k}{2d_k^{M}}>0,\quad k=1,2.
\end{equation}
Combined with \eqref{e2.15},\eqref{e2.16} and \eqref{e2.17}, we set
 $\overline{m}:=\min\{m_1,m_2,m_3,m_4\}$. Then,
$$
\liminf_{t\to +\infty}x_k(t)\geq \overline{m},\quad
\liminf_{t\to +\infty}u_k(t)\geq \overline{m},\quad k=1,2.
$$
The proof is complete.
\end{proof}

\begin{theorem}  \label{thm2.1}
Assume that $a_1^{iL}>0$, $a_2^{jL}>0$ $(i=0,1,\dots,m;j=0,1,\dots,n)$,
 $r_2^{L}>0$, $\gamma_k^{L}>0$, $d_k^{L}>0$,
 $e_k^{L}>0$, $f_k^{L}>0$ $(k=1,2)$,
$2q_2^{M}\overline{M}<\frac{1}{2}r_2^{L}$,
$\gamma_1^{M}(2\overline{M}-b_2)^{2}<\frac{r_1^{L}}{2}$.
Let $(x(t),u(t))^{T}=(x_1(t),x_2(t),u_1(t),u_2(t))^{T}$ be any positive
solution of system \eqref{e1.1}, then system \eqref{e1.1} is permanent,
where $\overline{M}$ is defined by \eqref{e2.7}.
\end{theorem}

\begin{proof}
Combining Lemma \ref{lem2.5} and Lemma \ref{lem2.6}, the conclusion is obvious.
\end{proof}


\section*{Conclusion}
In economic phenomena, time delays and feedback controls can not be
ignored due to the effect of factors such as   information, technology,
patent protection, institutional arrangement and so on. In this paper,
 our result shows that time delays and feedback controls have an influence
on the permanence of corporation and competition system \eqref{e1.1}
if $2q_2^{M}\overline{M}<\frac{1}{2}r_2^{L}$. The limited time delay
is one of the most important conditions to guarantee the permanence of
the system. Furthermore, the persistent property of the system not only
relies on time delays and feedback controls, but also relies on the
initial production of enterprise if
$\gamma_1^{M}(2\overline{M}-b_2)^{2}<\frac{r_1^{L}}{2}$, which means that
one enterprise with small initial production in a cluster should
expand its competitiveness and stimulate its production. Otherwise,
the other enterprise in the cluster will search for a new co-operative
 enterprise which will add the new transaction cost and will be not good
for its long term development. On the other hand, One enterprise with
large initial production in the cluster should reduce its competitiveness.
 Otherwise, it will annex the other enterprise in the cluster leading to
the extinction.

This prompts us that enterprises in a cluster should keep the moderate
competition, moreover, share resource, improve the efficiency of cooperation,
develop jointly and confront the change of external factors of enterprise
cluster, which will guide enterprise cluster to become a healthy ecosystem of
reciprocal regulation and mutual dependence.


\begin{thebibliography}{00}
\bibitem{c1} F. D. Chen, Z. Li, Y. J. Huang;
\emph{Note on the permanence of a competitive system with infinite
delay and feedback controls}, Nonlinear Anal. Real world Appl. 8 (2007) 680-687.

\bibitem{c2} F. D. Chen, J. H. Yang, L. J. Chen;
\emph{Note on the Persistent property of a feedback control system with delays},
 Nonlinear Anal. Real world Appl. 11 (2010) 1061-1066.

\bibitem{c3} J. Cheng, J. Zhao, Q. Y. Peng;
\emph{Interact mechanism analysis between high-speed railway and conventional
railway based on the compound cooperation and competition model},
Journal of Transportation Engineering and Information, (in Chinese)
 10(1) (2012) 31-37.

\bibitem{g1} C. Y. Gao, P. Du;
\emph{The cooperation and competition model of inter-members in high-tech
virtual industrial cluster based on Lotka-Volterra, Science Technology
Progress and Policy}, (in Chinese) 26(23) (2009) 72-75.

\bibitem{l1} F. Li;
\emph{Research on competition and cooperation mode in technological enterprise
cluster}, Journal of Wuhan University of Technology, (in Chinese)
31(14) (2009) 161-164.

\bibitem{l2}  Y. K. Li, T. W. Zhang;
\emph{Global asymptotical stability of a unique almost periodic solution
for enterprise clusters based on ecology theory with time-varying delays
and feedback controls}, Commun. Nonlinear Sci. Numer. Simul. 17 (2012)  904-913.

\bibitem{n1} Y. Nakata, Y. Muroya;
\emph{Permance for nonautonomous Lotka-Votella cooperative systems with delays},
Nonlinear Anal. Real world Appl. 11 (2010) 528-534.

\bibitem{p1} M. E. Porter;
\emph{Cluster and the new economics of competition}, Harvard Business Review,
1998 (November-December) 77-90.

\bibitem{t1} X. H. Tian, Q .K. Nie;
\emph{On model construction of enterprises' interactive relationship
from the perspective of business ecosystem},
Southern Economics Journal, 4 (2006) 50-57.

\bibitem{w1} L. L. Wang, Y. H. Fan;
\emph{Permanence and existence of periodic solutions for a generalized
system with feedback control}. Appl. Math. Comput. 216 (2010) 902-910.

\bibitem{x1} C. J. Xu;
\emph{Positive periodic solutions of corporation and competition dynamical
model of two enterprises}, Journal of Huaqiao University (Natural Science Ed.),
33(3) (2012) 337-341.

\bibitem{x2} C. J. Xu, Y. F. Shao;
\emph{Existence and global attractivity of periodic solution for enterprise
clusters based on ecology theory with impulse}, Journal of Applied Mathematics
and Computing, 39 (2012) 367-384.

\bibitem{y1} R. Yan, Y. S. Wang;
\emph{Global qualitative analysis of a class of Composite corporation and
competition model}, Journal of Wuyi University, (in Chinese) 23(3) (2009) 73-78.

\bibitem{y2} X. J. Yu, J. Tian;
\emph{Competition and cooperation of enterprise cluster based on ecosystem},
System Engineering, (in Chinese) 25(7) (2007) 108-111.

\end{thebibliography}

\end{document}