0\}$.
Consider the functional differential equation
\begin{equation}
\dot x(t)=f(t,x_t) \label{e2.1}
\end{equation}
where $f:C\mapsto \mathbb{R}^n$ and $x_t$ denote the element of $C$ given
by $x_t(\theta)=x(t+\theta)$, $-\tau\leq \theta \leq 0$. We assume
that $f(t,\varphi)$ is continuously differentiable in
$\varphi,f(t+\omega,\varphi)=f(t,\varphi)$ for all
$(t,\varphi)\in \mathbb{R}\times C^+$ and $\omega >0$. Then by \cite{h1}, there exists a unique
solution of \eqref{e2.1} through $(t_0,\varphi)$ for
$t_0\in \mathbb{R},\varphi\in C^+$.
We write $x(t,t_0,\varphi )(x_t(t_0,\varphi))$ for the
solution of the initial value problem. By \cite{h1}, $x(t,t_0,\varphi)$
is continuously differential in $\varphi$. In the following the
notation $x_{t_0}=\varphi$ will be used as the condition of the
initial value of \eqref{e2.1}, by which we mean that we consider the
solution $x(t)$ of \eqref{e2.1} which satisfies
$x(t_0+\theta)=\varphi(\theta)$, $\theta\in [-\tau,0]$. Consider the
hypothesis
\begin{itemize}
\item[(H2.1)] If $\varphi,\psi \in C^+$, $\varphi\leq \psi$ and
$\varphi _i(0)=\psi _i(0)$ for some $i$, then $f_i(t,\varphi)\leq
f_i(t,\psi)$.
\end{itemize}
Under the assumption (H2.1), system \eqref{e2.1} exhibits the following
property.
\begin{lemma}[\cite{s2,w1}] \label{lem2.1}
Under assumption (H2.1), we have
\begin{itemize}
\item[(i)] If $\varphi,\psi\in C^+$ with $\varphi \leq \psi$, then
$$
x(t,t_0,\varphi)\leq x(t,t_0,\psi)
$$
for all $t\geq t_0$ for which both are defined.
\item[(ii)] Assume that $y(t)$ is continuously differentiable. If
$\dot y(t)\leq f(t,y_t)$ and $y_{t_0}\leq \varphi$, we have
$$
y(t)\leq x(t,t_0,\varphi)
$$
for all $t\geq t_0$ for which both are defined. If $\dot y(t)\geq
f(t,y_t)$ and $y_{t_0}\geq \varphi$,we have
$$
y(t)\geq x(t,t_0,\varphi)
$$
for all $t\geq t_0$ for which both are defined.
\end{itemize}
\end{lemma}
\noindent\textbf{Definition.}
System \eqref{e2.1} is said to be permanent if there exists a compact set $K$
in the interior of $\mathbb{R}_+$, such
that all solutions with $x_{t_0}\in C^+$ ultimately enter $K$.
\section{Dispersing Logistic system with time delay}
Many authors have studied the stability of positive periodic solution of
the following type population dynamical system with time delay
\begin{equation}
\begin{gathered}
\dot x(t)=x(t)[b(t)-a(t)x(t)+c(t)\int ^0_{-\tau}x(t+\theta )d\mu (\theta )]\\
x(\theta)=\varphi (\theta)\geq 0,\theta \in [-\tau,0],\varphi (0)>0,
\varphi \in C([-\tau,0],\mathbb{R}_+),
\end{gathered}\label{e3.1}
\end{equation}
where $\mu (\theta)$ is nondecreasing and
\begin{equation}
\int _{-\tau}^0 d\mu (\theta )=\mu (0^+)-\mu (-\tau )=1,
\label{e3.2}
\end{equation}
the intrinsic growth rate is $b(t)$, the self-inhibition $a(t)$ and
the reproduction rate $c(t)$ are continuously $\omega$-periodic
functions, and $a(t)>0$, $b(t)>0$, $c(t)\geq 0$ for $t\in \mathbb{R}$.
Generally $c(t)$, which weights the effect of the past history on
the present population density $x(t)$, will make a superposition
of positive and negative effects and here we consider a population
$x$ at time $t$ has benefit from the resources accumulated by the
population itself in the past (D'Ancona \cite{d1}), hence establishing
for $c(t)$ in \eqref{e3.2} a positive sign.
For some particularly endangered species that live in weak
environment, the intrinsic growth rate $b(t)$ may become negative
for some time $t$. We have the following extinction result.
\begin{theorem} \label{thm3.1}
If $b(t)<0$ and $a(t)-c(t)>0$, then species $x$
become extinct eventually.
\end{theorem}
\begin{proof} Obviously, solutions of system \eqref{e3.1} are defined on
$[0,+\infty)$ and remain positive for $t>0$. Let
$V(x(t))=\frac{1}{2}x^2(t)$. If $|x(t)| \geq |x(t+\theta )|$,
$\theta\in [-\tau,0]$, then $\dot V(x(t)) \leq 2b(t)V(x(t))$. Hence, if
$b(t)<0$, then $V(x(t))$ is a Liapunov function for \eqref{e3.1}, and
$x=0$ is globally asymptotically stable. This means species $x$
become extinct eventually. This completes the proof.
\end{proof}
To study the effect of dispersal on the permanence of
system \eqref{e3.1}, we introduce the following system as composed of
multiple heterogeneous patches connected by discrete dispersal.
\begin{equation}
\begin{aligned}
\dot x_i(t)&=x_i(t)[b_i(t)-a_i(t)x_i(t)+c_i(t)\int_{-\tau}^0K_i(s)x_i(t+s)ds]\\
&\quad + \sum _{j=1}^n D_{ij}(t)(x_j(t)-x_i(t)),\quad
i=1,2,\dots,n,
\end{aligned}
\label{e3.3}
\end{equation}
with the initial condition
\begin{equation}
x_i(\theta )=\varphi _i(\theta )\geq 0,\quad
-\tau\leq \theta \leq 0,\quad \varphi _i(0)>0, \quad i=1,2,\dots,n,
\label{e3.4}
\end{equation}
where $x_i(t)$ is the density of species $x$ in patch
$i$; $K_i(s)(i=1,2,\dots,n)$ denote nonnegative piecewise
continuous functions defined on $[-\tau,0]$ and normalized such
that $\int _{-\tau}^0K_i(s)ds=1$. We assume
$b_i(t),a_i(t),c_i(t)$ and $D_{ij}(t)(i,j=1,2,\dots,n)$ are continuous
$\omega$-periodic functions defined on $\mathbb{R}$ and
\begin{equation}
a_i(t),c_i(t)>0,D_{ij}(t)\geq 0\quad \mbox{and} \quad
D_{ii}(t)\equiv 0 \quad \mbox{for} \quad t\in \mathbb{R},i,j=1,2,\dots, n,
\label{e3.5}
\end{equation}
where $b_i(t)$ is the intrinsic growth rate, $a_i(t)$ represents
the intraspecfic relationship, $D_{ij}(t)$ is the dispersal
coefficient for the species from patch $j$ to patch $i(i\neq j)$.
\begin{theorem} \label{thm3.2}
Suppose that $a_i(t)>c_i(t)$ holds for $i=1,2,\dots,n$, then
solutions of \eqref{e3.3} with initial condition \eqref{e3.4} are uniformly bounded and
uniformly ultimately bounded.
\end{theorem}
\begin{proof}
Similar to the proof in \cite{z1}, we know that $\mathbb{R}^n_{+}$ is
positively invariant with respect system \eqref{e3.3}. By
$a_i(t)>c_i(t)$,there exist $p(11$ such
that
\begin{equation}
{b_i}(t)-(a_i(t)-pc_i(t))H<-1 \quad for \quad i=1,2,\dots,n.
\label{e3.6}
\end{equation}
Define
$$
V(t)=V(x_1(t),\dots,x_n(t))=\max _{1\leq i\leq n}\{x_i(t)\}=\|x(t)\|.
$$
If $V(t+\theta)=\|x(t+\theta \|\geq H,V(t+\theta) \leq
pV(t),\theta \in [-\tau ,0]$, calculating the upper right
derivative of $V(t)$ along solutions of \eqref{e3.3}, we have
$$
D^+V(t)\leq V(t)\max _i \{ {b_i}(t)-(a_i(t)-pc_i(t))H\}<-H<-1.
$$
It follows from the theorem of Lyapunove-Razumikhim type
\cite{d1,h1,s2,w1,y1} that positive
solutions of \eqref{e3.3} are uniformly ultimately bounded.
Fix $\tilde{H}>H$. Let $x(t)=(x_1(t),\dots,x_n(t))$ denote the
solution of \eqref{e3.1} through $(\sigma ,\varphi)$ at $t=\sigma$, where
$\varphi =(\varphi_1,\dots, \varphi_n) \in C^+$ and
$0\leq \varphi_i(\theta )\leq \tilde{H}$ on $[-\tau,0]$ for
$i=1,\dots,n$. we claim that $\|x(t)\|\leq \tilde{H}$ for all
$t\geq \sigma$. Otherwise, there exists a $\tilde{t}>\sigma$ such
that
\begin{gather}
\|x(t)\| \leq \tilde{H} \quad for \quad \sigma -\tau \leq
t<\tilde{t}, \label{e3.7}\\
\|x(\tilde{t})\|=\tilde{H}, \label{e3.8} \\
D^+V(\tilde{t})\geq 0. \label{e3.9}
\end{gather}
Using \eqref{e3.7} and \eqref{e3.8}, we have from \eqref{e3.3}
$$
D^+V(\tilde{t})\geq \tilde{H}\max _{1\leq i \leq n} \{
{b_i}(\tilde{t})-(a_i(\tilde{t})-pc_i(\tilde{t}))\tilde{H}\}.
$$
It follows from \eqref{e3.6} that $D^+V(\tilde{t})<0$,which contradicts
\eqref{e3.9} and therefore, the uniform boundedness of the positive
solutions of \eqref{e3.3} with \eqref{e3.4} follows; i.e. , positive solution of
\eqref{e3.3} are uniformly bounded. This completes the proof.
\end{proof}
For the next theorem we us the following hypotheses:
\begin{itemize}
\item[(H3.1)] There exists $i_0(1\leq i_0 \leq n)$, such that
$A_{\omega}(\bar {\theta})>0$,where $ \bar{\theta} (t)=
{b}_{i_0}(t)-\sum^n_{j=1}D_{i_0j}(t)$
\item[(H3.2)] $A_{\omega}( {\phi} )>0$,where $ {\phi}
(t)=\min_{1 \leq i \leq n}\{{b_i}(t)-\sum^n_{j=1}D_{ij}(t)
+\sum^n_{j=1}D_{ji}(t)\}$
\end{itemize}
\begin{theorem} \label{thm3.3}
Assume that $a_i(t)>c_i(t)$ ($i=1,2,\dots,n$).
If one of the the assumption (H3.1) or (H3.2) holds,
then there exist positive constants $m$ and $M(m0$
such that
\begin{equation}
m\leq x_i(\sigma,\phi )\leq M , i=1,\dots,n, \label{e3.10}
\end{equation}
for $t\geq \sigma +T,\sigma \in \mathbb{R}$ and
$\phi \in C^+[\delta,\iota]=
\{\phi \in C^+:\delta\leq \phi (\theta)\leq \iota \}$.
\end{theorem}
\begin{proof}
Suppose that $a_i(t)0$, such that for given $0<\delta <\iota $, there is a constant
$T_1(\delta,\iota)>0$ such that
\begin{equation}
x_i(\sigma,\phi)(t)\leq M,\quad i=1,\dots,n, \label{e3.11}
\end{equation}
for $t\geq \sigma +T_1,\sigma \in \mathbb{R}$ and $\phi \in C^+[\delta,\iota]$.
On the other hand,
\begin{equation}
\dot x_i(t)\geq x_i(t)[ {b_i}(t)-a_i(t)x_i(t)] +
\sum _{j=1}^n D_{ij}(t)(x_j(t)-x_i(t)),
\label{e3.12}
\end{equation}
$i=1,2,\dots,n$.
Applying \cite[Theorem 2]{c1} to the auxiliary system
\begin{equation}
\dot u_i(t)=u_i(t)[ b_i(t)-a_i(t)u_i(t)] + \sum
_{j=1}^n D_{ij}(t)(u_j(t)-u_i(t)), u_i(0)=x_i(0),
\label{e3.13}
\end{equation}
$i=1,2,\dots,n$,
we obtained that there exist $m(0\sigma +T$. This completes the proof.
\end{proof}
Applying the above theorem to a two-patch system, we obtain the
following result. Let
\begin{itemize}
\item[(A3.1)] $A_{\omega}(b_1(t)-D_{12}(t))>0$
\item[(A3.2)] $A_{\omega}(b_2(t)-D_{21}(t))>0$
\item[(A3.3)] $b_1(t)+D_{21}(t)-D_{12}(t)\geq b_2(t)+D_{12}(t)-D_{21}(t)$
and $A_{\omega}(b_2(t)+D_{12}(t)-D_{21}(t))>0$
\item[(A3.4)] $b_1(t)+D_{21}(t)-D_{12}(t)\leq b_2(t)+D_{12}(t)-D_{21}(t)$ and
$A_{\omega}(b_1(t)+D_{21}(t)-D_{12}(t))>0$.
\end{itemize}
\begin{corollary} \label{coro3.1}
If $a_i(t)>c_i(t)$ ($i=1,2$) and one of the
conditions (A3.1)--(A3.4) holds, then the result of Theorem \ref{thm3.3}
holds for $i=1,2$.
\end{corollary}
\section{Permanence in dispersing competitive system}
In this section we consider the competitive
Lotka-Volterra dispersal model
\begin{equation}
\begin{aligned}
\dot x_i(t)&= x_i(t)[b_i(t)-a_i(t)x_i(t)+c_i(t)\int_{-\tau}^0K_i(s)x_i(t+s)ds
-f_i(t)y_i(t)]\\
&\quad + \sum _{j=1}^n D_{ij}(t)(x_j(t)- x_i(t))\\
\dot y_i(t)&=
y_i(t)[d_i(t)-e_i(t)x_i(t)-q_i(t)y_i(t)+p_i(t)\int_{-\tau}^0
{K_i}(s) y_i(t+s)ds]\\
&\quad + \sum _{j=1}^n \lambda _{ij}(t)(y_j(t)- y_i(t))
\end{aligned} \label{e4.1}
\end{equation}
$i=1,2,\dots,n$, with initial conditions
\begin{equation}
\begin{gathered}
x_i(\theta )=\varphi _i(\theta )\geq 0,\quad
y_i(\theta )=\psi _i(\theta )\geq 0, \\
-\tau\leq \theta \leq 0,\quad \varphi _i(0)>0,\quad
\psi _i(0)>0,\; i=1,2,\dots,n,
\end{gathered}\label{e4.2}
\end{equation}
where $y_i(t)$ is the density of species $y$ in patch $i$;
${K_i}(s)$ ($i=1,2,\dots,n$) denote nonnegative piecewise continuous
functions defined on $[-\tau,0]$ and normalized such that
$\int_{-\tau}^0 {K_i}(s)ds=1$. We assume
$e_i(t),f_i(t),p_i(t),q_i(t),\lambda _{ij}(t)(i,j=1,2,\dots,n)$
are continuous $\omega$-periodic functions defined on $\mathbb{R}$ and
\begin{equation}
p_i(t),q_i(t)>0,\lambda _{ij}(t), \geq 0\quad and \quad \lambda
_{ii}(t)\equiv 0
\label{e4.3}
\end{equation}
for $t\in \mathbb{R}$, $i,j=1,2,\dots, n$,
where $d_i(t)$ is the intrinsic growth rate,$e_i(t)$ represents
the intraspecfic relationship, $\lambda _{ij}(t)$ is the dispersal
coefficient for the species $y$ from patch $j$ to patch
$i(i\neq j)$.
\begin{theorem} \label{thm4.1}
Suppose that $a_i(t)>c_i(t)$ and $q_i(t)>p_i(t)$ hold for $t\geq 0$. Let
$(x_1(t),\dots,x_n(t), y_1(t),\dots,y_n(t))$
denote any solution of \eqref{e4.1} with initial conditions \eqref{e4.2}.
Then there exist positive constants $N_{xi},N_{yi}$ and $\tau _1$
such that
\begin{equation}
x_i(t)\leq N_{xi},y_i(t)\leq N_{yi}\quad \mbox{for } i=1,\dots,n\;\mbox{and }
t\geq \tau _1.
\label{e4.4}
\end{equation}
\end{theorem}
\begin{proof} Obviously solutions of system \eqref{e4.1} and \eqref{e4.2} are defined on
$[0,+\infty)$ and remain positive for all $t\geq 0$. It follows from \eqref{e4.1}
and the nonnegativity of the initial values,
\begin{align*}
\dot x_i(t)&\leq x_i(t)[ {b_i}(t)-a_i(t)x_i(t)+c_i(t)
\int_{-\tau}^0K_i(s)x_i(t+s)ds]\\
&\quad + \sum _{j=1}^n D_{ij}(t)(x_j(t)-x_i(t)) \\
\dot y_i(t)&\leq y_i(t)[
{d_i}(t)-q_i(t)y_i(t)+p_i(t)\int_{-\tau}^0 {K_i}(s) y_i(t+s)ds]\\
&\quad + \sum _{j=1}^n \lambda _{ij}(t)(y_j(t)-y_i(t))
\end{align*}
$i=1,2,\dots,n$.
By Lemma \ref{lem2.1} and Theorem \ref{thm3.2}, there exist positive constants
$N_{xi},N_{yi}$ and $\tau _1$ such that
$$
00$, where $\theta_1(t)=
{b}_{i_0}(t)-f_{i_0}(t)N_{yi_0}-\sum^n_{j=1}D_{i_0j}(t)$
\item[(H4.2)] $A_{\omega}(\phi_1 )>0$,where $\phi_1
(t)=\min_{1 \leq i \leq n}\{{b}_i(t)-f_i(t)N_{yi}-\sum^n_{j=1}D_{ij}(t)
+\sum^n_{j=1}D_{ji}(t)\}$
\item[(H4.3)] There exists $i_0(1\leq i_0 \leq n)$, such that
$A_{\omega}(\theta_2)>0$, where $\theta_2(t)=
{d}_{i_0}(t)-e_{i_0}(t)N_{xi_0}-\sum^n_{j=1}\lambda_{i_0j}(t)$
\item[(H4.4)] $A_{\omega}(\phi_2 )>0$, where $\phi_2
(t)=\min_{1 \leq i \leq n}\{{d}_i(t)-e_i(t)N_{xi}-\sum^n_{j=1}\lambda_{ij}(t)
+\sum^n_{j=1}\lambda_{ji}(t)\}$.
\end{itemize}
\begin{theorem} \label{thm4.2}
Assume that $a_i(t)>c_i(t)$ and $q_i(t)>p_i(t)$.
\begin{itemize}
\item[(I)] If one of the assumption (H4.1) or (H4.2) holds,
then there exist $\zeta _{xi}(0<\zeta_{xi}< N_{xi})$ and
$\tau_2\geq \tau_1$, such that
\begin{equation}
x_i(t)\geq \zeta_{xi} \quad \mbox{for }i=1,2,\dots,n, \; t\geq \tau_2
\label{e4.5}
\end{equation}
\item[(II)] If $\lambda _{ij}(t)(i\neq j)$ is positive and one of the
assumption (H4.3) or (H4.4) holds,
then there exist $\zeta _{yi}(0<\zeta_{yi}< N_{yi})$ and
$\tau_3\geq \tau_2$, such that
\begin{equation}
y_i(t)\geq \zeta_{yi} \quad \mbox{for } i=1,2,\dots,n, \; t\geq \tau_3
\label{e4.6}
\end{equation}
\end{itemize}
\end{theorem}
\begin{proof} By Theorem \ref{thm4.1}, there exists $\tau_1>0$ such that
for $i=1,\dots,n$,
\begin{align*}
\dot x_i(t)&\geq x_i(t)[ {b_i}(t)-f_i(t)N_{yi}-a_i(t)x_i(t)
+c_i(t)\int_{-\tau}^0K_i(s)x_i(t+s)ds]\\
&\quad + \sum _{j=1}^n D_{ij}(t)(x_j(t)-x_i(t))
\end{align*}
Let $(u_1(t),\dots,u_n(t))$ be the solution of the initial-value
problem
\begin{align*}
\dot u_i(t)
&=u_i(t)[ {b_i}(t)-f_i(t)N_{yi}-a_i(t)u_i(t)+c_i(t)
\int_{-\tau}^0K_i(s)u_i(t+s)ds]\\
&\quad + \sum _{j=1}^n D_{ij}(t)(u_j(t)-u_i(t))\\
u_i(s)&= x_i(s),s\in [\tau _1-\tau,\tau _1],i=1,\dots,n.
\end{align*}
By Theorem \ref{thm3.3} and Lemma \ref{lem2.1}, there exist
$\zeta _{xi}(0<\zeta_{xi}< N_{xi})$ and $\tau_2\geq \tau_1$, such that
$$
x_i(t)\geq \zeta_{xi} \quad \mbox{for } i=1,2,\dots,n, \; t\geq \tau_2,
$$
provided condition (H4.1) or (H4.2) hold.
We omit the proof of part (II) since it is entirely similar to that of (I).
\end{proof}
\section{Permanence in dispersing predator-prey system}
In this section we consider the dispersing predator-prey model
\begin{equation}
\begin{aligned}
\dot x_i(t)&=x_i(t)[b_i(t)-a_i(t)x_i(t)+c_i(t)\int_{-\tau}^0K_i(s)x_i(t+s)ds
-f_i(t)y_i(t)]\\
&\quad + \sum _{j=1}^n D_{ij}(t)(x_j(t)- x_i(t))\\
\dot y_i(t)&=y_i(t)[-d_i(t)+e_i(t)x_i(t)-q_i(t)y_i(t)+p_i(t)\int_{-\tau}^0
\bar{K_i}(s)y_i(t+s)ds]\\
&\quad+ \sum _{j=1}^n \lambda _{ij}(t)(y_j(t)- y_i(t))
\end{aligned} \label{e5.1}
\end{equation}
$i=1,2,\dots,n$, with initial conditions
\begin{equation}
\begin{gathered}
x_i(\theta )=\varphi _i(\theta )\geq 0,\quad
y_i(\theta )=\psi _i(\theta )\geq 0,\\
-\tau\leq \theta \leq 0,\quad \varphi _i(0)>0,\quad
\psi_i(0)>0,\; i=1,2,\dots,n,
\end{gathered}\label{e5.2}
\end{equation}
where $y_i(t)$ is the density of species $y$ in patch
$i$, $\bar{K_i}(s)$ ($i=1,2,\dots,n$) denote nonnegative piecewise
continuous functions defined on $[-\tau,0]$ and normalized such
that $\int _{-\tau}^0\bar{K_i}(s)ds=1$. All coefficients in system
\eqref{e5.1} are bounded continuous and $\omega$-periodic functions. In
addition, $a_i(t)$, $c_i(t)$, $d_i(t),q_i(t)$ and $p_i(t)$ are positive
for all $t\in [0,\omega]$ and $D_{ii}(t)\equiv 0$.
\begin{theorem} \label{thm5.1}
Suppose $a_i(t)>c_i(t)$ and $q_i(t)>p_i(t)$ hold for $t\geq 0$.
Let $(x_1(t),\dots,x_n(t),y_1(t),\dots,y_n(t))$
denote any solution of \eqref{e5.1} with initial conditions \eqref{e5.2}.
Then there exist positive constants $\bar{N}_{xi},\bar{N}_{yi}$ and
$\bar{\tau} _1$ such that
\begin{equation}
x_i(t)\leq \bar{N}_{xi},y_i(t)\leq \bar{N}_{yi}\quad \mbox{for }
i=1,\dots,n\; t\geq \bar{\tau} _1.
\label{e5.3}
\end{equation}
\end{theorem}
\begin{proof} Obviously, solutions of system \eqref{e5.1} and \eqref{e5.2}
are defined on $[0,+\infty)$ and remain positive for all $t\geq 0$. It follows
from \eqref{e5.1} and the positivity of the initial values,
$$
\dot x_i(t)\leq
x_i(t)[b_i(t)-a_i(t)x_i(t)+c_i(t)\int_{-\tau}^0K_i(s)x_i(t+s)ds]
+ \sum _{j=1}^n D_{ij}(t)(x_j(t)- x_i(t)).
$$
By Theorem \ref{thm3.3} and Lemma \ref{lem2.1}, there exist positive constants
$\bar{N}_{xi}$ and $\bar{\tau _1}$ such that
$$
00,s\in [\bar{\tau}_1-\tau,\bar{\tau}_1],\quad i=1,\dots,n.
\end{aligned}
\label{e5.4}
\end{equation}
By Theorem \ref{thm3.3} and Lemma \ref{lem2.1}, there exist positive constants
$\bar{N}_{yi}$ and $\bar{\tau}_2>\bar{\tau}_1$ such that
$$
00$, where
$\bar{\theta}_1(t)=b_{i_0}(t)-f_{i_0}(t)\bar{N}_{yi_0}-\sum^n_{j=1}D_{i_0j}(t)$
\item[(H5.2)] $A_{\omega}(\bar{\phi}_1 )>0$, where
$\bar{\phi}_1 (t)=\min_{1 \leq i \leq n}\{b_i(t)-f_i(t)\bar{N}_{yi}
-\sum^n_{j=1}D_{ij}(t) +\sum^n_{j=1}D_{ji}(t)\}$.
\item[(H5.3)] There exists $i_0(1\leq i_0 \leq n)$, such that
$A_{\omega}(\bar{\theta}_2)>0$, where
$\bar{\theta}_2(t)=e_{i_0}(t)\bar{\zeta}_{xi_0}-d_{i_0}(t)
-\sum^n_{j=1}\lambda_{i_0j}(t)$,
\item[(H5.4)] $A_{\omega}(\bar{\phi}_2 )>0$, where $\bar{\phi}_2
(t)=\min_{1 \leq i \leq n}\{e_i(t)\bar{\zeta}_{xi}
-d_i(t)-\sum^n_{j=1}\lambda_{ij}(t)+\sum^n_{j=1}\lambda_{ji}(t)\}$.
\end{itemize}
\begin{theorem} \label{thm5.2}
Suppose that $a_i(t)>c_i(t)$ and $q_i(t)>p_i(t)$ hold.
\begin{itemize}
\item[(I)] If one of the assumption (H5.1) or (H5.2) holds,
then there exist $\bar{\zeta} _{xi}(0<\bar{\zeta}_{xi}< \bar{N}_{xi})$
and $\bar{\tau}_3\geq \bar{\tau}_2$ such that
\begin{equation}
x_i(t)\geq \bar{\zeta}_{xi} \quad \mbox{for }i=1,2,\dots,n, \;
t\geq \bar{\tau}_3
\label{e5.5}
\end{equation}
\item[(II)] Suppose further that one of the assumption
(H5.3) or (H5.4) holds.
Then there exist $\bar{\zeta }_{yi}(0<\bar{\zeta}_{yi}<\bar{N}_{yi})$
and $\bar{\tau}_4\geq \bar{\tau}_3$ such that
\begin{equation}
y_i(t)\geq \bar{\zeta}_{yi} \quad \mbox{for }i=1,2,\dots,n \;
t\geq \bar{\tau}_4
\label{e5.6}
\end{equation}
\end{itemize}
\end{theorem}
\begin{proof} Suppose that condition (H5.1) or (H5.2) holds.
By Theorem \ref{thm5.1}, there exists $\bar{\tau}_2$ such that
\begin{align*}
\dot x_i(t)&\geq x_i(t)[b_i(t)-f_i(t)\bar{N}_{yi}(t)-a_i(t)x_i(t)+c_i(t)
\int_{-\tau}^0K_i(s)x_i(t+s)ds]\\
&\quad + \sum _{j=1}^n D_{ij}(t)(x_j(t)- x_i(t)),
\end{align*}
for $t\geq \bar{\tau}_2$. By Theorem \ref{thm3.3} and Lemma \ref{lem2.1}, there
exist
$\bar{\zeta} _{xi}(0<\bar{\zeta}_{xi}< \bar{N}_{xi})$ and
$\bar{\tau}_3\geq \bar{\tau}_2$ such that
$$
x_i(t)\geq \bar{\zeta}_{xi} \quad \mbox{for }i=1,2,\dots,n, \;
t\geq \bar{\tau}_3
$$
provided condition (H5.1) or (H5.2) holds.
Furthermore, suppose that (H5.3) or (H5.4) be satisfied
\begin{align*}
\dot y_i(t)&\geq y_i(t)[-d_i(t)+e_i(t)\bar{\zeta}_{xi}
-q_i(t)y_i(t)+p_i(t)\int_{-\tau}^0\bar{K_i}(s) y_i(t+s)ds]\\
&\quad + \sum _{j=1}^n \lambda _{ij}(t)(y_j(t)- y_i(t))
\end{align*}
for $t\geq \bar{\tau }_2$. Similar to the above discussion, there
exist
$\bar{\zeta} _{yi}(0<\bar{\zeta}_{yi}<\bar{N}_{yi})$ and
$\bar{\tau}_4\geq \bar{\tau}_3$ such that
$$
y_i(t)\geq \bar{\zeta}_{yi} \quad \mbox{for } i=1,2,\dots,n, \;
t\geq \bar{\tau}_4.
$$
This completes the proof.
\end{proof}
\section{Discussion}
Zhang and Chen \cite{z1} showed that in a nonautonomous system composed
of two patches connected by random dispersal and occupied by a
single species, if the species is able to survive then it
continues to do so for any dispersal rate (see \cite[Theorem 3.1]{z1}).
Different from above consideration, in section 3 of the present
paper we focus on the more interesting cases in biology that the
species living in a weak environment in the sense that species
$x$ in some of the isolated patches will be extinct without the
contribution from other patches. By the main results in this
section, dispersing species $x$ becomes permanent in every patches
depending on the choice of the dispersal rates (see Theorem 3.3).
But in \cite{z1}, the authors assumed that $b_i(t)>0$. We find
that this condition does not hold for a weak patchy environment in
the sense that the intrinsic growth rate $b_i(t)$ may become
negative on some time intervals.
By using the results obtained in section 3 and Lemma \ref{lem2.1}, we also
considered the effect of dispersal on the permanence of
competitive and predator-prey systems.
Within the context of the mathematical models used here, the main
results of this paper imply that some endangered species can avoid
extinction by choosing suitable dispersal rates. Hence dispersal
is a major factor on the determination of the permanence or
extinction of the endangered species.
\begin{thebibliography}{00}
\bibitem{a1} L. J. S. Allen;
\emph{Persistence and extinction in single-species reaction-diffusion
models}, Bull. Math. Biol., 45, 209-227 (1983).
\bibitem {a2} L. J. S. Allen;
\emph{Persistence, extinction and critical patch number
for island populations}, J. Math. Biol., 24, 617-625 (1987).
\bibitem{b1} E. Beretta and F. Solimano;
\emph{Global stability and periodic orbits for two
patch predator-prey diffusion delay models}, Math. Biosci., 85, 153-183 (1987).
\bibitem{b2} E. Beretta and Y. Takeuchi;
\emph{Global stability of single-species diffusion Volterra models with
continuous time delays}, Bull. Math. Biol., 49, 431-448 (1987).
\bibitem{b3} E. Beretta and Y. Takeuchi;
\emph{Global asymptotic stability of Lotka-Volterra
diffusion models with continuous time delays}, SIAM J. Appl. Math.,
48, 627-651 (1988).
\bibitem{c1} J. Cui and L. Chen;
\emph{The effect of diffusion on the time varying logistic
population growth}, Computers Math. Applic., 36(3), 1-9 (1998).
\bibitem{d1} U. D'Ancona;
\emph{The struggle for existence}, Leiden, Netherlands: E.J. Brill, 1954.
\bibitem{f1} H. I. Freedman, B. Rai and P. Waltman;
\emph{Mathematical models of population
interactions with dispersal II: Differential survival in a change
of habitat}, J. Math. Anal. Appl., 115, 140-154 (1986).
\bibitem{f2} H. I. Freedman and P. Waltman;
\emph{Mathematical models of population
interactions with dispersal I: Stability of two habitats with and
without a predator}, SIAM J. Appl. Math., 32, 631-648 (1977).
\bibitem{f3} H. I. Freedman and Y. Takeuchi;
\emph{Predator survival versus extinction as a
function of dispersal in a predator-prey model with patchy
environment}, Appl. Anal., 31, 247-266 (1989).
\bibitem {f4} H. I. Freedman and Y.Takeuchi, Global stability and predator dynamics
in a model of prey dispersal in a patchy environment, Nonlin.
Anal. T.M.A., 13, 993-1002 (1989).
\bibitem {f5} H. I. Freedman;
\emph{Single species migration in two habitats: persistence
and extinction}, Math. Model., 8, 778-780 (1987).
\bibitem{h1} J. K. Hale; \emph{Theory of functional differential equations},
Springer-Verlag, Berlin, 1977.
\bibitem {h2} A. Hastings;
\emph{Dynamics of a single species in a spatially varying environment:
The stabilizing role of high dispersal rates}. J. Math. Biol., 16,
49-55 (1982).
\bibitem{h3} R. D. Holt;
\emph{Population dynamics in two-patch environments:some anomalous
consequences of an optimal habitat distribution}. Theoret. Pop.
Bio. 28, 181-208 (1985).
\bibitem {k1} Y. Kuang and Y. Takeuchi;
\emph{Predator-prey dynamics in models of prey dispersal
in two-patch environments}, Math. Biosci., 120, 77-98 (1994).
\bibitem{l1} S. A. Levin;
\emph{Dispersion and population interactions}, Amer. Nat. , 108, 207-228 (1974).
\bibitem{s1} J. D. Skellem;
\emph{Random dispersal in theoretical population}, Biometrika, 38, 196-216 (1951).
\bibitem{s2} H. L. Smith;
\emph{Monotone semiflows generated by functional differential
equations}, J. Differential Equations, 66, 420-442 (1987).
\bibitem{s3} X. Song and L. Chen;
\emph{Persistence and global stability for nonautonomous
predator-prey system with diffusion and time delay}, Computers
Math. Applic., 35 (6), 33-40 (1998).
\bibitem{t1} Y. Takeuchi;
\emph{Cooperative system theory and global stability of
diffusion models}, Acta Appl. Math., 14, 49-57 (1989).
\bibitem{t2} Y. Takeuchi; \emph{Diffusion-mediated persistence in two-species
competition Lotka-Volterra model}, Math. Biosci., 95, 65-83 (1989).
\bibitem{t3} Y. Takeuchi; \emph{Conflict between the need to forage
and the need to aviod competition: persistence of two-species model}, Math.
Biosci., 99, 181-194 (1990).
\bibitem{v1} R. R. Vance;
\emph{The effect of dispersal on population stability in one-species
,discrete space population growth models}. The American Naturalist,
123, 230-254 (1984).
\bibitem{w1} W. Wang, P. Fergola and C. Tenneriello;
\emph{Global attractivity of periodic solutions of population models},
J. Math. Anal. Appl., 211, 498-511 (1997).
\bibitem{y1} K. Yang;
\emph{Delay differential equations with applications in population
dynamics}, Academic Press, INC, 1993.
\bibitem{y2} Y. Yange et al.; \emph{Giant panda's moving habit in Poping},
Acta Theridogica Sinica, 14(1), 9-14 (1994).
\bibitem{z1} J. Zhang and L. Chen;
\emph{Periodic solutions of Single-Species Nonautonomous
Diffusion Models with continuous Time Delays}, Math. Comput.
Modelling, 23, 17-27 (1996).
\bibitem{z2} Y. Zhou;
\emph{Analysis on decline of wild Alligator Sinensis population},
Sichuan Journal of Zoology, 16, 137-139 (1997).
\end{thebibliography}
\end{document}