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doi:10.1155/2010/738306

Research Article

The Permanence and Extinction of a Discrete

Predator-Prey System with Time Delay and

Feedback Controls

Qiuying Li, Hanwu Liu, and Fengqin Zhang

Department of Mathematics, Yuncheng University, Yuncheng 044000, China

Correspondence should be addressed to Qiuying Li,liqy-123@163.com

Received 23 May 2010; Revised 4 August 2010; Accepted 7 September 2010

Academic Editor: Yongwimon Lenbury

Copyrightq2010 Qiuying Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A discrete predator-prey system with time delay and feedback controls is studied. Sufficient conditions which guarantee the predator and the prey to be permanent are obtained. Moreover, under some suitable conditions, we show that the predator speciesywill be driven to extinction. The results indicate that one can choose suitable controls to make the species coexistence in a long term.

1. Introduction

The dynamic relationship between predator and its prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. The traditional predator-prey models have been studied extensivelye.g., see1–10and references cited therein, but they are questioned by several biologists. Thus, the Lotka-Volterra type predator-prey model with the Beddington-DeAngelis functional response has been proposed and has been well studied. The model can be expressed as follows:

xt x1t

ba11xt

a12yt 1 βxt γyt

,

yt yt

a21xt

1 βxt γytda22yt

.

1.1

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et al.12. It is similar to the well-known Holling type II functional response but has an extra term γy in the denominator which models mutual interference between predators. It can be derived mechanistically from considerations of time utilization 11 or spatial limits on predation. But few scholars pay attention to this model. Hwang6showed that the system has no periodic solutions when the positive equilibrium is locally asymptotical stability by using the divergency criterion. Recently, Fan and Kuang9further considered the nonautonomous case of system1.1, that is, they considered the following system:

xt x1t

For the general nonautonomous case, they addressed properties such as permanence, extinction, and globally asymptotic stability of the system. For the periodicalmost periodic case, they established sufficient criteria for the existence, uniqueness, and stability of a positive periodic solution and a boundary periodic solution. At the end of their paper, numerical simulation results that complement their analytical findings were present.

However, we note that ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecosystem is the question of whether an ecosystem can withstand those unpredictable forces which persist for a finite period of time or not. In the language of control variables, we call the disturbance functions as control variables. In 1993, Gopalsamy and Weng 13 introduced a control variable into the delay logistic model and discussed the asymptotic behavior of solution in logistic models with feedback controls, in which the control variables satisfy certain differential equation. In recent years, the population dynamical systems with feedback controls have been studied in many papers, for example, see13–22and references cited therein.

It has been found that discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. It is reasonable to study discrete models governed by difference equations. Motivated by the above works, we focus our attention on the permanence and extinction of species for the following nonautonomous predator-prey model with time delay and feedback controls:

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where xn, yn are the density of the prey species and the predator species at time n, respectively. uin i 1,2 are the feedback control variables. bn, a11n represent the intrinsic growth rate and density-dependent coefficient of the prey at time n, respectively.

dn, a22ndenote the death rate and density-dependent coefficient of the predator at time

n, respectively.a12ndenotes the capturing rate of the predator;a21n/a12nrepresents the rate of conversion of nutrients into the reproduction of the predator. Further,τ is a positive integer.

For the simplicity and convenience of exposition, we introduce the following notations. LetR 0, ∞,Z {1,2, . . .}andk1, k2denote the set of integerksatisfying

k1 ≤ kk2. We denote DC : −τ,0 → R to be the space of all nonnegative and bounded discrete time functions. In addition, for any bounded sequence gn,we denote

gLinf

nZ gn,gMsupnZ gn.

Given the biological sense, we only consider solutions of system 1.3 with the following initial condition:

xθ, yθ, u1θ, u2θ

φ1θ, φ2θ, ψ1θ, ψ2θ

, φi, ψiDC , φi0>0, ψi0>0, i1,2.

1.4

It is not difficult to see that the solutions of system1.3with the above initial condition are well defined for alln≥0 and satisfy

xn>0, yn>0, uin>0, nZ , i1,2. 1.5

The main purpose of this paper is to establish a new general criterion for the permanence and extinction of system1.3, which is dependent on feedback controls. This paper is organized as follows. In Section 2, we will give some assumptions and useful lemmas. In Section3, some new sufficient conditions which guarantee the permanence of all positive solutions of system1.3are obtained. Moreover, under some suitable conditions, we show that the predator speciesywill be driven to extinction.

2. Preliminaries

In this section, we present some useful assumptions and state several lemmas which will be useful in the proving of the main results.

Throughout this paper, we will have both of the following assumptions:

H1 rn, bn, dn, βn and γn are nonnegative bounded sequences of real numbers defined onZ such that

rL>0, bL≥0, dL>0, 2.1

H2 cin,ein,finandaijnare nonnegative bounded sequences of real numbers

defined onZ such that

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Now, we state several lemmas which will be used to prove the main results in this paper.

First, we consider the following nonautonomous equation:

xn 1 xnexpgnanxn, 2.3

where functionsan,gnare bounded and continuous defined onZ withaL,gL >0. We

have the following result which is given in23.

Lemma 2.1. Letxnbe the positive solution of 2.3withx0>0, then

athere exists a positive constantM >1such that

M−1<lim inf

n→ ∞ xn≤lim supn→ ∞ xnM 2.4

for any positive solutionxnof2.3;

blimn→ ∞x1nx2n 0for any two positive solutionsx1nandx2nof2.3.

Second, one considers the following nonautonomous linear equation:

Δun 1 fnenun, 2.5

where functionsfnandenare bounded and continuous defined onZ withfL> 0 and 0< eLeM<1.The following Lemma2.2is a direct corollary of Theorem 6.2 of L. Wang and

M. Q. Wang24, page 125.

Lemma 2.2. Letunbe the nonnegative solution of2.5withu0>0, then

afL/eM<lim inf

n→ ∞un ≤lim supn→ ∞unfM/eL for any positive solutionun

of 2.5;

blimn→ ∞u1nu2n 0for any two positive solutionsu1nandu2nof2.5.

Further, considering the following:

Δun 1 fnenun ωn, 2.6

where functionsfnand en are bounded and continuous defined on Z with fL > 0,

0< eLeM<1 andωn0.The following Lemma2.3is a direct corollary of Lemma 3 of

Xu and Teng25.

Lemma 2.3. Letun, n0, u0be the positive solution of 2.6withu0>0, then for any constants

> 0andM > 0, there exist positive constantsδandn, Msuch that for anyn0 ∈Z and |u0|< M,when|ωn|< δ,one has

|un, n0, u0−un, n0, u0|< forn >n n0, 2.7

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Finally, one considers the following nonautonomous linear equation:

Δun 1 −enun ωn, 2.8

where functionsenare bounded and continuous defined onZ with 0< eLeM <1 and

ωn≥0.In25, the following Lemma2.4has been proved.

Lemma 2.4. Letunbe the nonnegative solution of 2.8withu0 > 0, then, for any constants

> 0 andM > 0, there exist positive constantsδandn, Msuch that for anyn0 ∈ Zand |u0|< M,whenωn< δ, one has

un, n0, u0< forn >n n0. 2.9

3. Main Results

Theorem 3.1. Suppose that assumptions H1andH2hold, then there exists a constantM > 0

such that

lim sup

n→ ∞ xn< M, lim supn→ ∞ yn< M, lim supn→ ∞ u1n< M, lim supn→ ∞ u2n< M,

3.1

for any positive solutionxn, yn, u1n, u2nof system1.3.

Proof. Given any solutionxn, yn, u1n, u2nof system1.3, we have

Δu1n 1≤rne1nu1n, 3.2

for allnn0,wheren0is the initial time.

Consider the following auxiliary equation:

Δvn 1 rne1nvn, 3.3

from assumptionsH1,H2and Lemma2.2, there exists a constantM1>0 such that

lim sup

n→ ∞ vnM1, 3.4

wherevnis the solution of3.3with initial conditionvn0 u1n0.By the comparison theorem, we have

u1nvn,nn0. 3.5

From this, we further have

lim sup

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Then, we obtain that for any constantε >0,there exists a constantn1> n0such that

u1n< M1 εnn1. 3.7

According to the first equation of system1.3, we have

xnxnexp{bna11nxn c1nM1 ε}, 3.8

for allnn1.Considering the following auxiliary equation:

zn 1 znexp{bna11nzn c1nM1 ε}, 3.9

thus, as a direct corollary of Lemma2.1, we get that there exists a positive constantM2 >0 such that

lim sup

n→ ∞ znM2, 3.10

whereznis the solution of3.9with initial conditionzn1 xn1.By the comparison theorem, we have

xnzn,nn1. 3.11

From this, we further have

lim sup

n→ ∞ xnM2. 3.12

Then, we obtain that for any constantε >0,there exists a constantn2> n1such that

xn< M2 ε,nn2. 3.13

Hence, from the second equation of system1.3, we obtain

yn 1≤ynexpa21nM2 εdna22nyn

, 3.14

for allnn2 τ.Following a similar argument as above, we get that there exists a positive constantM3such that

lim sup

n→ ∞

yn< M3. 3.15

By a similar argument of the above proof, we further obtain

lim sup

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From3.6and3.12–3.16, we can choose the constantMmax{M1, M2, M3, M4}, such that

lim sup

n→ ∞ xn< M, lim supn→ ∞ yn< M, lim sup

n→ ∞ u1n< M, lim supn→ ∞ u2n< M.

3.17

This completes the proof of Theorem3.1.

In order to obtain the permanence of system1.3, we assume that

H3 bn c1nu10nL>0,whereu10nis some positive solution of the following equation:

Δun 1 rne1nun. 3.18

Theorem 3.2. Suppose that assumptionsH1–H3hold, then there exists a constantηx > 0such

that

lim inf

n→ ∞ xn> ηx, 3.19

for any positive solutionxn, yn, u1n, u2nof system1.3.

Proof. According to assumptionsH1andH3,we can choose positive constantsε0andε1 such that

bna110−

a121 1 γnε1

c1n

u10nε1

L

> ε0,

a210 1 βnε0 −

dn M

<ε0.

3.20

Consider the following equation with parameterα0:

Δvn 1 rne1nvnf10. 3.21

Let un be any positive solution of system 3.18 with initial value un0 v0. By assumptionsH1–H3and Lemma2.2, we obtain thatunis globally asymptotically stable and converges tou10nuniformly forn → ∞.Further, from Lemma2.3, we obtain that, for any givenε1 > 0 and a positive constantM > 0Mis given in Theorem3.1, there exist constantsδ1 δ1ε1>0 andn1 n1ε1, M>0,such that for anyn0 ∈Z and 0≤v0 ≤M, whenf10< δ1, we have

vn, n0, v0u

10n < ε1,nn0 n∗1, 3.22

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Letα0 ≤min{ε0, δ1/f1M 1},from3.20, we obtain that there existα0andn1such that

bna110− a121 1 γnε1

c1n

u10nε1

> α0,

a210

1 βnα0 −dn<α0, f1n< f M 1 1,

3.23

for alln > n1.

We first prove that

lim sup

n→ ∞ xnα0, 3.24

for any positive solutionxn, yn, u1n, u2nof system1.3. In fact, if3.24is not true, then there exists aΦθ φ1θ, φ2θ, ψ1θ, ψ2θsuch that

lim sup

n→ ∞

xn,Φ< α0, 3.25

where xn,Φ, yn,Φ, u1n,Φ, u2n,Φ is the solution of system 1.3 with initial conditionxθ, yθ, u1θ, u2θ Φθ,θ∈−τ,0.So, there exists ann2> n1such that

xn,Φ< α0 ∀n > n2. 3.26

Hence,3.26together with the third equation of system1.3lead to

Δu1n 1> rne1nu1nf10, 3.27

forn > n2.Let vn be the solution of3.21 with initial condition vn2 u1n2,by the comparison theorem, we have

u1nvn,nn2. 3.28

In3.22, we choosen0n2andv0u1n2,sincef10< δ1,then for givenε1,we have

vn vn, n2, u1n2> u10nε1, 3.29

for allnn2 n1.Hence, from3.28, we further have

u1n> u10nε1,nn2 n1. 3.30

From the second equation of system1.3, we have

yn 1≤ynexp

a210 1 βnα0 −

dn

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for alln > n2 τ.Obviously, we haveyn → 0 asn → ∞.Therefore, we get that there

Now, we prove the conclusion of Theorem3.2. In fact, if it is not true, then there exists a sequence{Zm}{ϕm

2 }of initial functions such that

(10)
(11)

From the second equation of system1.3, we have

yn 1≤ynexp

a210 1 βnα0 −

dn

, 3.47

formm0,qK1m, andnsqm τ, tqm.Therefore, we get that

yn< ε1, 3.48

for anynsqm τ n, tqm.Further, from the first equation of systems1.3,3.46, and

3.48, we obtain

xn 1, Zmxn, Zmexp

bna110−

a121 1 γnε1

c1n

u10nε1

xn, Zmexpα0,

3.49

for anymm0,qK1m, andns

m

q 1 τ n, tqm.Hence,

xtqm, Zm

xtqm−1, Zm

expα0. 3.50

In view of3.37and3.38, we finally have

α0

m 12 ≥x

tqm, Zm

xtqm−1, Zm

expα0

α0

m 12expα0>

α0

m 12,

3.51

which is a contradiction. Therefore, the conclusion of Theorem3.2holds. This completes the proof of Theorem3.2.

In order to obtain the permanence of the component yn of system1.3, we next consider the following single-specie system with feedback control:

xn 1 xnexp{bna11nxn c1nu1n},

Δu1n 1 rne1nu1nf1nxn.

3.52

For system3.52, we further introduce the following assumption:

H4supposeλmax{|1−aM11x|,|1−aL11x|} cM1 <1,δ1−eL1 f1Mx <1,wherex,x are given in the proof of Lemma3.3.

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Lemma 3.3. Suppose that assumptionsH1–H3hold, then

athere exists a constantM >1such that

M−1 <lim inf

n→ ∞ xn<lim supn→ ∞ xn< M, lim supn→ ∞ u1n< M, 3.53

for any positive solutionxn, u1nof system3.52.

bif assumptionH4holds, then each fixed positive solutionxn, u1nof system3.52 is globally uniformly attractive onR20.

Proof. Based on assumptionsH1–H3, conclusionacan be proved by a similar argument as in Theorems3.1and3.2.

Here, we prove conclusion b. Letting x10n, u10nbe some solution of system

3.52, by conclusiona, there exist constantsx,x, andM >1, such that

xε < xn, x10n< x ε, u1n, u∗10n< M, 3.54

for any solutionxn, u1nof system3.52andn > n.We make transformationxn

x10nexpv1nandu1n u∗10n v2n.Hence, system3.52is equivalent to

v1n 1

1−a11nx∗10nexp{θ1nv1n}

v1n c1nv2n,

Δv2n 1 −e1nv2nf1nx∗10nexp{θ2nv1n}v1n.

3.55

According to H4, there exists a ε > 0 small enough, such that λε max{|1 −aM11x

ε|,|1−aL

11xε|} c1M < 1, σε 1 −eL1 f1Mx ε < 1.Noticing that θin ∈ 0,1

implies thatx10nexpθinv1n i 1,2lie betweenx10nandxn.Therefore,xε <

x10nexpθinv1n< x ε,i1,2.It follows from3.55that

|v1n 1| ≤

1−a11nx∗10nexp{θ1nv1n}

v1n c1nv2n, |v2n 1| ≤1−e1nv2nf1nx∗10nexp{θ2nv1n}v1n.

3.56

Letμmax{λε, σε},then 0< μ <1. It follows easily from3.56that

max{|v1n 1|,|v2n 1|} ≤μmax{|v1n|,|v2n|}. 3.57

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Considering the following equations:

xn 1 xnexpbna11nxngn c1nu1n

,

Δu1n 1 rne1nu1nf1nxn,

3.58

then we have the following result.

Lemma 3.4. Suppose that assumptionsH1–H4hold, then there exists a positive constantδ2such that for any positive solutionxn, u1nof system3.58, one has

lim

n→ ∞|xnxn|0, nlim→ ∞|u1nun|0, gn∈0, δ2, 3.59

wherexn,unis the solution of system3.52withxn0 xn0andun0 u1n0.

The proof of Lemma3.4is similar to Lemma3.3, one omits it here. Letxn, u1nbe a fixed solution of system3.52defined onR2

0,one assumes that

H5 −dn a21nxnτ/1 βnxnτL>0.

Theorem 3.5. Suppose that assumptionsH1–H5hold, then there exists a constantηy > 0such

that

lim inf

n→ ∞ yn> ηy, 3.60

for any positive solutionxn, yn, u1n, u2nof system1.3.

Proof. According to assumptionH5,we can choose positive constantsε2,ε3, andn1, such that for allnn1,we have

dn a21nx

nτε

3

1 βnxnτε3 γnε2 −a222−c23> ε2. 3.61

Considering the following equation with parameterα1:

Δvn 1 −e2nvn f21, 3.62

by Lemma2.4, for givenε3>0 andM >0Mis given in Theorem3.1., there exist constants

δ3 δ3ε3 > 0 andn3 n3ε3, M > 0, such that for anyn0 ∈ Z and 0 ≤ v0 ≤ M,when

f20< δ3,we have

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We chooseα1 < max{ε2, δ3/1 f2M}if there exists a constant n such thata12n

δ2γn ≡ 0 for all n > n, otherwise α1 < max{ε2, δ3/1 f2M, δ2/a12nδ2γnM}. Obviously, there exists ann2> n1, such that

a121 1 γnα1

< δ2, f21< δ3,n > n2. 3.64

Now, We prove that

lim sup

n→ ∞ ynα1, 3.65

for any positive solutionxn, yn, u1n, u2nof system1.3. In fact, if3.65is not true, then forα1, there exist aΦθ φ1θ, φ2θ, ψ1θ, ψ2θandn3> n2such that for alln > n3,

yn,Φ< α1, 3.66

whereφiDC andψiDC i1,2.Hence, for alln > n3,one has

0< a12nyn

1 βnxn γnyn <

a121

1 γnα1 < δ2. 3.67

Therefore, from system1.3, Lemmas3.3and3.4, it follows that

lim

n→ ∞|xnx

n|0, lim

n→ ∞|u1nu

n|0, 3.68

for any solutionxn, yn, u1n, u2nof system1.3. Therefore, for any small positive constantε3>0,there exists ann4such that for allnn3 n4,we have

xnx10nε3. 3.69

From the fourth equation of system1.3, one has

Δu2n 1≤ −e2nu2n f21. 3.70

In3.63, we choosen0 n3 andv0 un3.Sincef21 < δ3,then for allnn3 n3, we have

u2nε3. 3.71

Equations3.69,3.71together with the second equation of system1.3lead to

yn 1≥ynexp

a21n

x10nε3

1 βnx10nε3

γnα1

dna221−c23

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for alln > n3 τ n∗∗,wheren∗∗max{n3, n4}.Obviously, we haveyn → ∞asn → ∞, which is contradictory to the boundedness of solution of system1.3. Therefore,3.65holds. Now, we prove the conclusion of Theorem3.5. In fact, if it is not true, then there exists a sequenceZm{φm

2 }of initial functions, such that

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The above inequality implies that

tqmsqm

lnm 1

θ2

,qK1m, m1,2, . . . . 3.80

Choosing a large enoughm1,such that

tqmsqm>n∗∗ τ 2,mm1, qK1m, 3.81

then formm1, qK1m,we have

0< a12nyn

1 βnxn γnyn <

a121 1 γnα1

< δ2, 3.82

for allnsqm 1, tqm.Therefore, it follows from system1.3that

xn 1≥xnexpbna11nxnδ2 c1nu1n,

u1n 1 rne1n−1u1nf1nxn,

3.83

for allnsqm 1, tqm.Further, by Lemmas3.3and3.4, we obtain that for any small positive

constantε3>0,we have

xnx10nε3, 3.84

for anymm1,qK1m, andns

m

q 1 n∗∗, tqm.For any mm1,qK1m, and

nsqm 1, tqm,by the first equation of systems1.3and3.77, it follows that

Δu2

n 1, Zm≤ −e2nu2

n, Zm f2n

α1

m 1 ≤ −e2nu2

n, Zm f21.

3.85

Assume thatvnis the solution of3.62with the initial conditionvsqm 1 u2sqm 1,

then from comparison theorem and the above inequality, we have

u2

n, Zmvn,nsqm 1, tqm

, mm1, qK1m. 3.86

In3.63, we choosen0 sqm 1 andv0 u2sqm 1.Since 0< v0 < Mandf21 < δ3, then we have

vnε3,n

sqm 1 n∗∗, tqm

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Equation3.86together with3.87lead to

which is a contradiction. Therefore, the conclusion of Theorem3.5holds.

Remark 3.6. In Theorems 3.2 and 3.5, we note thatH1–H3 are decided by system1.3, which is dependent on the feedback controlu1n. So, the control variableu1nhas impact on the permanence of system1.3. That is, there is the permanence of the species as long as feedback controls should be kept beyond the range. If not, we have the following result.

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holds, then

lim

n→ ∞yn 0, 3.93

for any positive solutionxn, yn, u1n, u2nof system1.3.

Proof. By the condition, for any positive constantεε < α1,whereα1is given in Theorem3.5, there exist constantsε1andn1,such that

dn a21nxnτ ε1

1 βnxnτ ε1−

a22nε <ε1, 3.94

forn > n1.First, we show that there exists ann2 > n1,such thatyn2< ε.Otherwise, there exists ann1, such that

ynε,n > n1 n∗1. 3.95

Hence, for allnn1 n1,one has

xn 1< xnexp

bna11nxna12

1 γnε c1nu1n

,

Δu1n 1 rne1nu1n f1nxn.

3.96

Therefore, from Lemma3.3and comparison theorem, it follows that for the aboveε1,there exists ann2>0, such that

xn< xn ε1,n > n1 n∗2. 3.97

Hence, forn > n1 n∗2,we have

εyn 1< ynexp

dn a21nxnτ ε1

1 βnxnτ ε1−

a22

yn1 n∗2

exp−ε1

nn1−n∗2

−→0 asn−→ ∞.

3.98

So,ε <0,which is a contradiction. Therefor, there exists ann2> n1,such thatyn2< ε. Second, we show that

yn< εexpμ,n > n2, 3.99

where

μmaxnZ

dn a21nxnτ ε1

1 βnxnτ ε1

a22

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is bounded. Otherwise, there exists ann3> n2,such thatyn3≥εexp{μ}.Hence, there must exist ann4 ∈n2, n3−1such thatyn4< ε,yn4 1≥ ε, andynεfornn4 1, n3. LetP1be a nonnegative integer, such that

n3 n4 P1 1. 3.101

It follows from3.101that

εexpμyn3≤yn4exp

which leads to a contradiction. This shows that 3.99 holds. By the arbitrariness ofε, it immediately follows thatyn → 0 asn → ∞.This completes the proof of Theorem3.7.

Acknowledgments

This work was supported by the National Sciences Foundation of Chinano. 11071283and the Sciences Foundation of Shanxino. 2009011005-3.

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7 T. K. Kar and U. K. Pahari, “Modelling and analysis of a prey-predator system with stage-structure and harvesting,”Nonlinear Analysis. Real World Applications, vol. 8, no. 2, pp. 601–609, 2007.

8 Z. Li, W. Wang, and H. Wang, “The dynamics of a Beddington-type system with impulsive control strategy,”Chaos, Solitons and Fractals, vol. 29, no. 5, pp. 1229–1239, 2006.

9 M. Fan and Y. Kuang, “Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response,”Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 15–39, 2004.

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12 D. L. deAngelis, R. A. Goldstein, and R. V. O’Neill, “A model for trophic interaction,”Ecology, vol. 56, pp. 881–892, 1975.

13 K. Gopalsamy and P. X. Weng, “Feedback regulation of logistic growth,” International Journal of Mathematics and Mathematical Sciences, vol. 16, no. 1, pp. 177–192, 1993.

14 F. Chen, “Positive periodic solutions of neutral Lotka-Volterra system with feedback control,”Applied Mathematics and Computation, vol. 162, no. 3, pp. 1279–1302, 2005.

15 H.-F. Huo and W.-T. Li, “Positive periodic solutions of a class of delay differential system with feedback control,”Applied Mathematics and Computation, vol. 148, no. 1, pp. 35–46, 2004.

16 P. Weng, “Existence and global stability of positive periodic solution of periodic integrodifferential systems with feedback controls,”Computers & Mathematics with Applicationsl, vol. 40, no. 6-7, pp. 747– 759, 2000.

17 K. Wang, Z. Teng, and H. Jiang, “On the permanence forn-species non-autonomous Lotka-Volterra competitive system with infinite delays and feedback controls,”International Journal of Biomathematics, vol. 1, no. 1, pp. 29–43, 2008.

18 F. Chen, J. Yang, L. Chen, and X. Xie, “On a mutualism model with feedback controls,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 581–587, 2009.

19 Y.-H. Fan and L.-L. Wang, “Permanence for a discrete model with feedback control and delay,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 945109, 8 pages, 2008.

20 Y. Li and L. Zhu, “Existence of positive periodic solutions for difference equations with feedback control,”Applied Mathematics Letters, vol. 18, no. 1, pp. 61–67, 2005.

21 L. Chen and Z. Li, “Permanence of a delayed discrete mutualism model with feedback controls,” Mathematical and Computer Modelling, vol. 50, no. 7-8, pp. 1083–1089, 2009.

22 F. Chen, “The permanence and global attractivity of Lotka-Volterra competition system with feedback controls,”Nonlinear Analysis, vol. 7, no. 1, pp. 133–143, 2006.

23 Z. Zhou and X. Zou, “Stable periodic solutions in a discrete periodic logistic equation,” Applied Mathematics Letters, vol. 16, no. 2, pp. 165–171, 2003.

24 L. Wang and M. Q. Wang,Ordinary Difference Equation, Xinjiang University Press, Urumqi, China, 1991.

References

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