The
effects of dispersal
on
population
dynamics
1Jingan
Cui
Department of Mathematics, Nanjing Normal University,
Nanjing 210097 China
Email [email protected]
Abstract In this $\mathrm{p}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{r},\mathrm{w}\mathrm{e}$ consider the effect of dispersal on the permanence and extinction of
single axtd multiple endangered species that live in changing patches environment. Different from the
former $\mathrm{s}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{e}\mathrm{s},\mathrm{o}\mathrm{u}\mathrm{r}$discussion include the
more
importantsituation
in conservation biology that specieslive in
a
weak patchyenvironment
in thesense
that species will become extinct insome
oftheisolatedpatches. For single population $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l},\mathrm{w}\mathrm{e}$ show that the identical species can persist for
some
dispersal $\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s},\mathrm{a}\mathrm{n}\mathrm{d}$can
also vanish for another set of restrictionon
dispersal $\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s},\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}$ the endangered singlespecieswill vanish in
some
isolatedpatcheswithout the contribution from other patches. Furthermorewe
consider the$\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e},\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}$and global stability ofthepositive periodic solution. For prey-predator $\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m},\mathrm{w}\mathrm{e}$ can make both prey and predator species to be permanent by choosing the dispersal rates
appropriatelyeven if the prey species has negative intrinsic growth rate in
some
patches. Particulafor a prey-predator system, we provide a sufficient and necessary condition to guarantee the prey and
predator species to be permanent.
Key words. Logistic equation,Lotffi– げ夙 errasystem, diffusion, $\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e},\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$, periodic solution, stability.
1
Introduction
Sincethe poineering theoretical work bySkellem $[9],\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{y}$works havefocused
on
theeffectof spatialfac-torswhich play
a
crucial rule in persistence and stability of population [1-13]. Most of the previous papersdeal withautonomouspopulation systems and indicate that adispersal process in
an
ecological system isoften considered to have
a
stabilizing influenceon
the system [12],but is also probably destabilizingthesystem[8].
Recently, someauthors have alsostudied the influenceofdispersal on thetimedependent population
models (see [13]). The authors always
assume
that the intrinsic growth rates are all continuous andbounded above and below by positive constants(this
means
thateveryspecieslived ina
suitableenviron-ment). Theyobtained some sufficient conditionsthat guarantee permanenceofeveryspeciesand global
stability ofa uniquepositive periodic solution.
$\mathrm{H}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r},\mathrm{t}\mathrm{h}\mathrm{e}$actual livingenvironment ofendangered species is not always like this. Because ofthe
ecological effects of the human
activities
and $\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{y},\mathrm{e}.\mathrm{g}$.
the location of manufacturing industries, thepollutionof the$\mathrm{a}\mathrm{t}\mathrm{m}\mathrm{o}\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e},\mathrm{o}\mathrm{f}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r},\mathrm{o}\mathrm{f}\mathrm{s}\mathrm{o}\mathrm{i}\mathrm{l},\mathrm{e}\mathrm{t}\mathrm{c}$ ,
more
andmore
habitatswere
brokeninto patches andsome
of the patches
were
polluted. Insome
ofthese $\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{s},\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}$ in every patches the species will go extinctwithout the contribution from other $\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{s},\mathrm{a}\mathrm{n}\mathrm{d}$ hence the species live in a weak patchy environment.
The living environments ofsome endangered andrare species such as giant panda [2931] and alligator
sinensis [34]
are some
convincingexamples.In order to protect the endangered and
rare
$\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{s},\mathrm{w}\mathrm{e}$ have to consider the effects ofhabitatffag-mentation anddiffusion
on
thepermanence and extinctionof single attd multiple species living in weakenvironments. The present paper consider the following interesting$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{m}:\mathrm{t}\mathrm{o}$what extent does dispersal
leadto the permanence
or
extinction ofendangered single and multiple species which could not persistwithin
some
isolated patches.Let $C$ denotesthespaceof all bounded continuous functions $f$: $Rarrow R,C_{+}^{0}$ is the setofnonnegative
$f\in C$and$C_{+}$is the set of all$f\in C$suchthat $f$is bounded below byapositiveconstant. Given$f\in$C,we
denote
$f^{\mathrm{A}\mathrm{f}}=\mathrm{s}\mathrm{u}t\geq \mathrm{p}$$f(t)$, $f^{L}=\mathrm{i}\mathrm{n}t\geq \mathrm{f}$ $f(t)$
$1\mathrm{I}$thank Department of Mathematics, Nanjing Normal University, which enabled
me to visit in Japan and also the hospitalityof Department of Systems Engineering, Shizuoka Universityduringmy stay in Japan.
58
and define the lower average $Al(/)$ and upperaverage $A_{\Lambda I}(f)$ of$f$ by
$A_{L}(f)= \lim_{rarrow\infty}\inf_{t-s\geq r}(t-s)^{-1}\int_{s}^{t}\mathrm{f}(\mathrm{r})\mathrm{d}\mathrm{r}$
and
$A_{\Lambda \mathrm{f}}(f)= \lim_{rarrow\infty}\sup_{t-\mathit{8}\geq r}(t-s)^{-1}$
$\int_{s}^{t}\mathrm{f}(\mathrm{r})\mathrm{d}\mathrm{r}$
respectively. If$f\in C$is$\mathrm{a};-\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{c},\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$the average\^A (/) of$f$mustbe equal to$A_{L}(f)$ and$A_{M}(f),\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}$
is
$A_{\omega}(f)=A_{L}(f)=$ Ai$( \mathrm{f})=\omega^{-1}\int_{0}^{\omega}f(r)dr$
Definition. The system ofdifferentialequations
$\dot{x}=F(t, x)$, $x\in R^{n}$
is saidtobe permanent if thereexists a compact set $K$ in the interior of$R_{+}^{n}=\{(x_{1}, x_{2}, \cdot \cdot\cdot, x_{n})\in R^{n}|$ $x_{i}\geq 0$,$\mathrm{i}$$=1,2$,
$\cdots$ ,$n$
},
such that all solutionsstarting in the interior of$R_{+}^{n}$ ultimately enter $K$.
2
The
effect of habitat ffagmentation
on
single
species
In this section,
we
consider the systemas
composed of patches connected by discrete and linear diffusions,each patch is assumed to beoccupied bya single species
as
follows$\dot{x}_{i}$
$=x_{i}[b:(t)-a_{i}(t)x_{i}]+ \sum_{J=1}^{n}D_{ij}(t)$($x_{j}-$xn) $(i=1,2, \cdots, n)$ (2.1)
where $x_{i}(i=1,2, \cdots, n)$ denotes the species $x$ in patch $i$
.
$b_{i}(t)\in C$,$\mathrm{a}\mathrm{i}(\mathrm{t})$$D\{j(t)\in C_{+}(i\neq j)$ and$D_{ii}(t)\equiv 0.$ $b_{i}(t)$ is the intrinsic growth ratefor species $x$ in patch $\mathrm{i};a_{i}(t)$ represents the self-inhibition
coefficient; and$D\{j(t)$ is the dispersalcoefficient of species $x$ from patch$j$ to patch$i$
.
If$a_{i}(t)$,$b_{\iota}(t)$ arecontinuousand bounded above and below by positive$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{s},\mathrm{W}\mathrm{a}\mathrm{n}\mathrm{g}$and Chen [13] showed that the systemispermanent forany$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s},\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}$ and bounded dispersal rates $D_{ij}(t)$
.
$\mathrm{H}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r},\mathrm{i}\mathrm{n}$the process thatthe endangered species be goingto$\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n},\mathrm{i}\mathrm{t}\mathrm{s}$birth rate is less than thedeath
rate.
In this $\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{e},\mathrm{w}\mathrm{e}$ will indicate that humancan rescue
the endangered speciesfrom extinctionby controlling dispersal rates.
Theorem [5]. Given any$\xi_{i}>0(i=1,2, \cdots, n)$, the initial value problem
$i\mathrm{j}$
$= \tau_{i},[b_{i}(t)-a_{i}(t)x_{i}]+\sum_{j=1}^{n}Dij(t)$
.
$-x_{i})$(2.2)
$x_{i}(0)=\xi_{i},i=1,2$,$\cdots$ ,$n$
has a unique solution $x(t)=(x_{1}(t), x_{2}(t),$$\cdots$ ,$x_{n}(t))$ which exists for all $t\geq 0.$ Moreover, there exists
$M>0$,$\tau>0,$ such that
$0<x:(t)\leq M$
for
$t\geq\tau$, (2.3)theregion$D=$ $\{(x_{1}, x_{2}, \cdots, x_{n})|0<x_{i}\leq M, i=1,2, \cdots, n\}$ being positively invariantwithrespect to
(2.1).
A consequence ofTheorem 2.1 is that for $\xi_{i}>0(i=1,2, \cdots, n)$ the solution of (2.2) is ultimately
bounded above. We will showthatthis solution is also ultimately bounded below
away
fromzero
providedthat
one
ofthe following conditions issatisfied.$n$
(H2.1) There exists$i\circ(1\leq i_{0}\leq n)$, such that $A_{L}(\theta)>0,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\theta(t)=b_{i_{0}}(t)-i$$D_{i_{0}j}(t)$
.
$j=1$
Theorem 2.2[5]. Suppose that (H2.1) or (H2.2) holds, then there exists $\delta_{i}$,$0<\delta_{i}<M$ alud$\tau\geq 0,$
such that the solution of (2.2) satisfies
$x_{i}(t)\geq\delta_{i}$, $t\geq\tau$,$i=1,2$,$\cdots$ ,$n$ (2.4)
where $\delta_{i}(i=1,2, \cdot\cdot. , n)$ depend
on
the variousassumptions (H2.1) and (H2.2).Theorem 2.1 and 2.2 have established that under the
one
ofthe assumptions (H2.1)or
$(\mathrm{H}2.2),\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$exist positive constants $m$ and $M,\mathrm{t}\mathrm{h}\mathrm{e}$ solution of (2.1) with positive initial
values ultimately enter the
rectangular region $\Omega=$ $\{(x_{1}, x_{2}, \cdots, x_{n})|m\leq x_{i}\leq M, i=1,2, \cdots, n\}$, therefore the population is
permanent.
Remark 2.1. According to the proof of Theorem 2.2, if species $x$ ispermanent in
a
fixed patch $i$,thenspecies$x$ isalso permanent in other patches for anydispersalrates $Dji(i)(iJ=1,2, \cdots, n)$
.
Next
we
will consider the extinctionof system (2.1). Denote$\psi(t)=\max\{b_{i}(t)-\sum_{\mathrm{j}=1}^{n}1\leq i\leq nDis(t)+\sum_{j=1}^{n}D_{ji}(t)\}$
Theorem 2.3[5]. Supposethat$\int_{0}^{+\infty}\mathrm{i}\mathrm{p}(\mathrm{t})\mathrm{d}\mathrm{t}$ $=$
$-\mathrm{o}\mathrm{o}$
,
then the solution of(2.1)satisfies$X:(t)arrow 0$ $i=1,$2,$\cdots$ ,$n,tarrow l- cx$
Next
we
assume
that thefunctions$b_{i}(t)$,$a_{i}(t)$,$D_{ij}(t)$,$(i,j=1,2, \cdots, n)$ in system (2.1)are
all periodicfunctionswith
common
period $\mathrm{a};,\mathrm{a}\mathrm{n}\mathrm{d}$ considerthe positive periodicsolutionof (2.1).Theorem 2.4[4]. Supposethat the assumption (H2.1) or (H2.2) holds, then system (2.1) has atleast
one
positive$\omega$-periodic solution which is globallyasymptotically stable.3
Permanence in
dispersal prey-predator
system
We introduce
an
exotic predator species $y$ intosome
patches whichwere
occupied by native species $x$.
Assumethat species$x$ and$y$ obey followingLotka-Volterra dispersal model
$i_{i}=x_{i}[b_{i}(t)-a:(t)x_{i}-c_{\mathrm{i}}(t)y_{i}]1$$\sum_{j=1}^{n}D_{\dot{l}j}(t)(x_{j}-x_{i})$
$j.i$ $=y_{\mathrm{i}}[-d_{i}(t)+e_{i}(t)x_{i}-f_{i}(t)y_{i}]+ \sum_{j=1}^{n}\lambda_{ij}(t)(y_{j}- )_{i})$ (3.1)
$i=1,2$,$\cdots$ ,$n$
.
where $y_{i}$ is the density ofspecies $y$ in patch$i$; the coefficients $d_{:}(t)$,$e_{i}(t)$,$\mathrm{c}_{t}(t)$
are
all nonnegative andbounded continuous functions.$f_{i}(t)$,Dis(t) $\in C_{+}(i\neq j)$ and Dis$(t)\equiv 0.$
Theorem 3.1$[5].(\mathrm{A})$.Supposethatfollowing assumption (H3.1)or (H3.2) besatisfied,
(H3.1) There exists $\mathrm{i}\mathrm{o}(1\leq i_{0}\leq n)$such that$A_{L}(\theta_{1})>0,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$ $i (t) $=$bio$( \mathrm{t})-c_{i_{0}}(t)N_{y}-\sum_{j=1}^{n}D_{\dot{w}j}(t)$,
(H3.2) $A_{L}(\gamma_{1})>0,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$ $i(t)
$= \min_{1\leq\dot{|}\leq n}\{b_{i}(t)-c_{t}(t)N_{y}-\sum_{j=1}^{n}D_{ij}(t)+\sum_{j=1}^{n}D_{\mathrm{j}j}(t)\}$
.
where$N_{y}$ is the upper bound of$y_{i}(t)$
.
Thenprey
speciesispermanent.$(\mathrm{B}).\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{e}$further that following assumption (H3.3) or (H3.4) besatisfied,
(H3.3)There exists$\mathrm{i}\mathrm{o}(1\leq i_{0}\leq n)$ suchthat $A_{L}(\theta_{2})>0,\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$$i(t)
$=e_{i_{0}}(t) \zeta_{x\dot{\eta}}-d_{\dot{1}0}(t)-\sum_{j=1}^{n}\lambda_{\dot{*}\mathrm{o}j}(t)$,
(H3.4) $A_{L}(\gamma_{2})>$0,where $i(t)
$= \min_{1\leq i\leq n}$
{
$e:(t)\zeta_{xi}$ -$i$(t)- \sum_{j=1}^{n}\lambda_{\dot{\iota}j}(t)+\sum_{j=1}^{n}\lambda_{j:}(t)$}.
81
Example 3.1. Consider the followingperiodicand patchy predator-prey system
$\dot{x}_{1}$
$=x_{1}$ $(12+ \mathrm{z} \sin t-3x_{1}-y\mathrm{l})$ $\mathrm{f}$$D_{12}(t)(x_{2}-x_{1})$
$i_{2}=x_{2}(-1+ \frac{1}{4}\sin t-x_{2}-y_{2})+$D2i$\{\mathrm{t})(\mathrm{x}\mathrm{i}-x_{2})$
$\dot{y}_{1}=y_{1}(-1+\sin t+x_{1}-y_{1})+$Xi2(t)(y2 $-y_{1}$) (3.2)
$\mathrm{j}_{1}$$=y_{2}(-1+ \sin t+x_{2}-y_{2})$ $+$Xi2$(\mathrm{t})(\mathrm{y}2-y_{2})$.
Where $D_{12}(t)$,$D_{21}(t)$
,
$\lambda_{12}(t)\mathrm{a}\mathrm{n}\mathrm{d}\lambda_{21}(t)$ are all positive continuous and periodic functions withcommon
period$2\pi$
.
Note that if the patches are isolated from each other($D_{12}(t)=D_{21}(t)=$
xi{t)
$=\lambda_{21}(t)=0),\mathrm{i}\mathrm{t}$ isclearthatspecies $x$and $y$will go extinct in patch 2.
Given
anypositive solution $(x_{1}(t), x_{2}(t)$,$\mathrm{j}\mathrm{j}\mathrm{l}$$(t)$,112(t)$)$ of (3.2),wehave$1_{2}^{1} \leq x_{2}(-1+\frac{\mathrm{i}}{4}\sin t-x_{2})+D_{21}(t)(x_{1}-x_{2})\leq x_{1}(12+\frac{1}{\mathrm{f}}\sin t-3x_{1})+D_{12}(t)(x_{2}-x_{1})$
. From the proof of Theorem$3.1,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$exists$\tau_{1}>0,\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}$that
$0<x_{i}(t)\leq$25/6,$0<y_{i}(t)\leq$ 53/12$(i=1,2)$
for
$t\geq\tau_{1}$.
(3.3)Consequently,
$i_{1}\geq x_{1}$($\frac{91}{12}+\frac{1}{4}\sin t-$3xi)$+D_{12}(t)(x_{2}-x_{1})$
$i_{2} \geq x_{2}(-\frac{65}{12}+\frac{1}{4}\sin t-x_{2})+$D2i$\{\mathrm{t})(\mathrm{x}\mathrm{i}-x_{2})$
for $t\geq\tau_{1}$
.
Furthermore,$\dot{x}_{1}\geq x_{1}(\frac{22}{3}-D_{12}^{M}-3x_{1})$
thereexists $\tau_{2}$(r2 $\geq\tau_{1}$), such that
$x_{1}(t)> \frac{\frac{22}{3}-D_{12}^{\mathrm{A}\mathrm{f}}}{3}-\epsilon$
for
$t\geq\tau_{2}$provided $D_{12}^{M}<$ 22/3,where $\epsilon$ be anypositive $\mathrm{n}$umber. $\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y},\mathrm{w}\mathrm{e}$call choose$D_{12}^{M}\leq 1$,$\epsilon=$ l/9,such
that$x_{1}(t)>2=\zeta_{x1}$ for$t\geq\tau_{2}$
.
According tothe proof of Theorem $2.2,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$exist positiveconst-ants $\zeta_{x2}$ and $\tau_{3}(\tau_{3}\geq\tau_{2}),\mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}$that
$x_{2}(t)>\zeta_{x2}$ for $t\geq\tau_{3}$
.
Finally,
$\dot{y}_{1}\geq y_{1}(1+\sin t+x_{1}-y_{1})+\lambda_{12}(t)(y_{2}-y_{1})$
for $t\geq\tau_{3}$
.
By Theorem $2.2,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$exist positiveconstants $\zeta_{y1}$,$\zeta_{y2}$ and T4(r4 $\geq\tau_{3}$) such that $y_{i}(t)\geq$ ;yifor
$t\geq\tau_{4},i=1,2$provided$A_{2\pi}(\lambda_{12})<1.$
To
sum
$\mathrm{u}\mathrm{p},\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}$ the assumptions$D_{12}^{M}\leq 1$ and $A_{2\pi}(\lambda_{12})<1$ (3.4)
the species$x$ and$y$
are
permanent.Accordingto above$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{u}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n},\mathrm{w}\mathrm{e}$knowthat people
can
avoid thelocal extinction ofthe endangeredspecies$x$ and$y$ in patch2by controlling the dispersal rates.
Next we study the followingsystem
$\dot{x}_{1}=x_{1}$[$b_{1}(t)-a_{1}(t)x_{1}$ -Cl$(\mathrm{t})\mathrm{y}$]$+D_{12}(t)x_{2}-$D2i$(\mathrm{t})\mathrm{x}\mathrm{l}$
$\dot{x}_{2}=x_{2}$[$b_{2}(t)-$a2$(t)x_{2}$]$+D_{21}(t)x_{1}-D_{12}(t)x_{2}$ (3.5)
$\dot{y}=y[-d(t)+e(t)x_{1}- (t)’-q(t)y(t-\tau)]$
.
$\mathrm{r}$ is
a
positive constant. For (3.5)we
make the followingassumptions (3.3) $A_{\omega}[b_{1}(t)-D_{21}(t)]>0.$Theorem 3.1.[6] Under theassumption (H3.5),system (3.5) ispermanent if and only if
$(\# 3.6)$ $A_{\omega}[-d(t)+e(t)x_{1}^{*}(t)]>0.$
where $(x_{1}^{*}(t), x_{2}^{*}(t))$ be the positive periodicsolutionof the system
$i_{1}=x_{1}[b_{1}(t)-a_{1}(t)x_{1}]$$+D_{12}(t)x_{2}-D_{21}(t)x_{1}$ $\dot{x}_{2}=x_{2}[b_{2}(t)-a_{2}(t)x_{\mathit{2}}]$ $1$ $D_{21}(t)x_{1}-D_{12}(t)x_{2}$.
References
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where $(x_{1}^{*}(t), x_{2}^{*}(t))$ be the positive periodicsolutionof the system
$\dot{x}_{1}=x_{1}[b_{1}(t)-a_{1}(t)x_{1}]+D_{12}(t)x_{\mathit{2}}-D_{21}(t)x_{1}$ $\dot{x}_{2}=x_{2}[b_{2}(t)-a_{2}(t)x_{\mathit{2}}]+D_{21}(t)x_{1}-D_{12}(t)x_{2}$.
References
[1] L.J.S. Allen, Persistence,extinction,andcriticalpatchnumber for island populations,J. Math. Biol.,
24(1987), 617-625.
[2] E.Beretta and Y.Takeuchi,Global stability ofsingle-speciesdiffusion Volterramodels with
continu-ous
time delays, $\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{l}.\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{B}\mathrm{i}\mathrm{o}\mathrm{l}.,49(1987),431- 448$.
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model,Syst.Sci.&Math.Sci.,ll (1998),107-111.
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on
the time varying Logistic population growth,Com-puters $\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{A}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}.,36(1998),1- 9$
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J. Math. Anal. Appl., 258(2001), 512-535.
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on
permanence in apredator-prey population growthmodel,Com-puters $\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}.\mathrm{A}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}.,44(2002),1085$
-1097.
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