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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 60, pp. 1--11.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/60\hfil Permanence in logistic systems]
{Permanence in logistic and Lotka-Volterra \\
systems with dispersal and time delays}
\author[J. Cui, M. Guo\hfil EJDE-2005/60\hfilneg]
{Jingan Cui, Mingna Guo }  % in alphabetical order

\address{Department of Mathematics\\
Nanjing Normal University\\
Nanjing 210097, China}
\email{cuija@njnu.edu.cn}

\date{}
\thanks{Submitted November 24, 2004. Published June 10, 2005.}
\thanks{Supported by grants 10471066 from the NNSF of China
 and 02KJB110004 from the \hfill\break\indent
Jiangsu Provincial Department of Education}
\subjclass[2000]{92D25, 34C60, 34K20} 
\keywords{Logistic equation; Lotka-Volterra system; dispersal;
 permanence; \hfill\break\indent
 extinction; time delay}

\begin{abstract}
 In this paper, we consider the effect of dispersal on the
 permanence of single and interacting populations modelled by
 systems of integro differential equations. Different from
 former studies, our discussion here includes the important
 situation when species live in a weak  patchy environment;
 i.e., species in some isolated patches will  become extinct
 without the contribution from other patches.
 For the single population model considered in this paper,
 we show that the same species can persist for some
 dispersal rates and the species will vanish in some isolated
 patches. Based on the results for a single population model, we
 derive sufficient conditions for the permanence of two interacting
 competitive and predator-prey dispersing systems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\section{Introduction}

As discussed by several authors, for many species spatial factors are
important in population dynamics.  Such theoretical studies of spatial
distributions can be traced back at least as far as Skellem \cite{s1},
and has been extensively considered in many papers;
see for example in \cite{a1,a2,f1,f2,f3,f4,f5,h2,h3,k1,l1,v1,t1,t2,t3}
and references cited therein.
Most of the previous papers focused on the coexistence of populations
modelled by systems of ordinary differential equations and the
stability (local and global) of equilibria. Many existing models
deal with a single population dispersing among patches. Some of
them deal with competition and predator-prey interactions in
patchy environments.

Recently persistence and stability of population dynamical systems
involving time delays have been discussed by some authors; see for
example \cite{b1,b2,b3,s3,z1} and references cited therein. All of
these studies assume that the intrinsic growth rates are all
positive (this means that populations live in a suitable
environment). They obtained some sufficient conditions that
guarantee permanence of population or stability of  positive
equilibria or positive periodic solutions.


However, the actual living environments of some endangered and
rare species are not always like this. Because of the ecological
effects of the human activities and industry, e.g. the location of
manufacturing industries, pollution of the atmosphere,
rivers, soil, etc., more and more habitats were broken into
patches and some of the patches were polluted.
In some of these patches the species will
go extinct without the contribution from other patches, and hence
the species lived in a weak patchy environment. The living
environments of some endangered and rare species such as giant
 panda \cite{y2} and alligator sinensis \cite{z2} are some convincing
examples.

In order to protect endangered and rare species, we should
consider theoretically the effect of dispersal on the permanence
of single and multiple species living in weak environments. The
present paper consider the following interesting problem: How does
dispersal lead to the permanence of endangered single
and multiple species which could not persist within isolated
patches?


The organization of this paper is as follows.
 In the next section, we introduce  notation, give some definitions
and state a lemma which will be essential for our proofs.
In section 3, a dispersing single species model is given to consider its
permanence.  We obtain that the dispersal system can be made permanent under
different appropriate dispersal conditions,
 even if the  endangered species become extinct in some isolated patches
without the  contribution from other patches
(Theorems \ref{thm3.2}, \ref{thm3.3}).
In section 4, by using the main results in section 3,
we consider the effect of dispersal  on the survival of
competitive species. In section 5, we consider the effect of
dispersal on the permanence of prey-predator system. We can choose
appropriate dispersal rates making  the dispersing system
permanent even if the prey species has negative intrinsic growth
rates in some patches. Finally, the main results obtained in this paper
are biologically discussed in section 6.

\section{Preliminaries}

In this section we introduce some notations and state a lemma
which will be useful in the subsequent sections.
Let $f(t)$ be a continuous $\omega$-periodic function defined on
$\mathbb{R}=(-\infty,+\infty)$, and set
$$
A_{\omega}(f)=\omega^{-1}\int_0^\omega f(t)dt.
$$
Let $\mathbb{R}_+$ be the cone of nonnegative vectors in $\mathbb{R}^n$. If
$x,y\in \mathbb{R}^n$, we write $x\leq y$ if $x_i\leq y_i$ for $i=1,2,\dots,n$;
similarly $x<y$ if $x_i<y_i$, for $i=1,2,\dots,n$.
Let $C=C([-\tau,0],\mathbb{R}^n)$ be the Banach space
of continuous functions mapping the interval $[-\tau,0]$ into
$\mathbb{R}^n$ with supremum norm. If $\varphi ,\psi \in C$, we write
$\varphi \leq \psi$ ($\varphi <\psi$) in case the indicated
inequality holds at each point of $[\tau,0]$. Denote
$C^+=\{\varphi \in C: \varphi \geq 0,\varphi (0)>0\}$.

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 $p(1<p<\min _t
\{\frac{a_1(t)}{c_1(t)},\frac{a_2(t)}{c_2(t)}\})$ and $H>1$ 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 $M(m<M)$,such that for
given $0<\delta <\iota $, there is a constant $T=T(\delta,\iota)>0$
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 $a_i(t)<c_i(t)(i=1,\dots,n)$ holds. From
Theorem \ref{thm3.2}, the positive solutions of \eqref{e3.3} with \eqref{e3.4} are
uniformly ultimately bounded. We know that there exists a constant
$M>0$, 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 $m(0<m<M)$ and $T_2\geq 0$, such that
positive solutions of \eqref{e3.13} satisfies
\[
u_i(t)\geq m \quad for \quad i=1,\dots,n \quad and \quad t\geq
\sigma +T_2, %\label{e3.14}
\]
where $m$ dependent on assumptions (H3.1) and (H3.2).
Taking $T=\max \{T_1,T_2\}$, by Lemma \ref{lem2.1}, \eqref{e3.10} holds for
$t>\sigma +T$. 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
$$
0<x_i(t)\leq N_{xi},0<y_i(t)\leq N_{yi},
$$
for $i=1,\dots,n$ and $t\geq \tau _1$. This completes the proof.
\end{proof}
For the nest theorem, let
\begin{itemize}
\item[(H4.1)] There exists $i_0(1\leq i_0 \leq n)$, such that
$A_{\omega}(\theta_1)>0$, 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
$$
0<x_i(t)\leq \bar{N}_{xi},\quad i=1,\dots,n,
$$
for $t\geq \bar{\tau _1}$. Moreover, for $t\geq \bar{\tau _1}$ we
have
\begin{align*}
\dot y_i(t)&\leq  y_i(t)[e_i(t)\bar{N}_{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*}
Let $(v_1(t),\dots,v_n(t))$ be the solution of the initial-value
problem
\begin{equation}
\begin{aligned}
\dot v_i(t)&= v_i(t)[e_i(t)\bar{N}_{xi}-q_i(t)v_i(t)+p_i(t)
\int_{-\tau}^0\bar{K_i}(s)v_i(t+s)ds]\\
&\quad + \sum _{j=1}^n \lambda _{ij}(t)(v_j(t)- v_i(t))\\
v_i(s)&= y_i(s)>0,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
$$
0<y_i(t)\leq \bar{N}_{yi},\quad i=1,\dots,n,
$$
for $t\geq \bar{\tau _2}$. This completes the proof.
\end{proof}
For the next theorem let
\begin{itemize}
\item[(H5.1)] There exists $i_0(1\leq i_0 \leq n)$, such that
$A_{\omega}(\bar{\theta}_1)>0$, 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.

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\end{document}