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 species*y*will 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* *x*1*t*

*b*−*a*11*xt*−

*a*12*yt*
1 *βxt* *γyt*

*,*

*yt* *yt*

*a*21*xt*

1 *βxt* *γyt*−*d*−*a*22*yt*

*.*

1.1

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* *x*1*t*

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:

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, a*11*n* represent the
intrinsic growth rate and density-dependent coeﬃcient of the prey at time *n*, respectively.

*dn, a*22*n*denote the death rate and density-dependent coeﬃcient of the predator at time

*n*, respectively.*a*12*n*denotes the capturing rate of the predator;*a*21*n/a*12*n*represents 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. Let*R* 0*,* ∞,*Z* {1*,*2*, . . .*}and*k*1*, k*2denote the set of integer*k*satisfying

*k*1 ≤ *k* ≤ *k*2*.* 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

*gL*_{}_{inf}

*n*∈*Z* *gn*,*gM*sup*n*∈*Z* *gn.*

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

*xθ, yθ, u*1*θ, u*2*θ*

*φ*1*θ, φ*2*θ, ψ*1*θ, ψ*2*θ*

*,* *φi, ψi*∈*DC* *, φi*0*>*0*, ψi*0*>*0*, i*1*,*2*.*

1.4

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

*xn>*0*,* *yn>*0*,* *uin>*0*,* *n*∈*Z* *,* *i*1*,*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 species*y*will 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 on*Z* such that

*rL>*0*,* *bL*≥0*,* *dL>*0*,* 2.1

H2 *cin*,*ein*,*fin*and*aijn*are nonnegative bounded sequences of real numbers

defined on*Z* such that

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 *xn*exp*gn*−*anxn,* 2.3

where functions*an*,*gn*are bounded and continuous defined on*Z* with*aL*_{,}_{g}L_{>}_{0. We}

have the following result which is given in23.

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

a*there exists a positive constantM >*1*such that*

*M*−1*<*lim inf

*n*→ ∞ *xn*≤lim sup_{n}_{→ ∞} *xn*≤*M* 2.4

*for any positive solutionxnof*2.3*;*

blim*n*→ ∞*x*1*n*−*x*2*n* 0*for any two positive solutionsx*1*nandx*2*nof*2.3*.*

Second, one considers the following nonautonomous linear equation:

Δ*un* 1 *fn*−*enun,* 2.5

where functions*fn*and*en*are bounded and continuous defined on*Z* with*fL>* 0 and
0*< eL*_{≤}_{e}M_{<}_{1}_{.}_{The following Lemma}_{2.2}_{is a direct corollary of Theorem 6}_{.}_{2 of L. Wang and}

M. Q. Wang24, page 125.

**Lemma 2.2.** *Letunbe the nonnegative solution of*2.5*withu*0*>*0, then

a*fL _{/e}M_{<}*

_{lim inf}

*n*→ ∞*un* ≤lim sup*n*→ ∞*un*≤*fM/eL* *for any positive solutionun*

*of* 2.5*;*

blim*n*→ ∞*u*1*n*−*u*2*n* 0*for any two positive solutionsu*1*nandu*2*nof*2.5*.*

Further, considering the following:

Δ*un* 1 *fn*−*enun* *ωn,* 2.6

where functions*fn*and *en* are bounded and continuous defined on *Z* with *fL* _{>}_{0,}

0*< eL*_{≤}_{e}M_{<}_{1 and}_{ω}_{}_{n}_{}_{≥}_{0}_{.}_{The following Lemma}_{2.3}_{is a direct corollary of Lemma 3 of}

Xu and Teng25.

**Lemma 2.3.** *Letun, n*0*, u*0*be the positive solution of* 2.6*withu*0*>*0, then for any constants

* >* 0*andM >* 0, there exist positive constants*δandn, Msuch that for anyn*0 ∈*Z* *and*
|*u*0|*< M,when*|*ωn*|*< δ,one has*

|*un, n*0*, u*0−*u*∗*n, n*0*, u*0|*< * *forn >n* *n*0*,* 2.7

Finally, one considers the following nonautonomous linear equation:

Δ*un* 1 −*enun* *ωn,* 2.8

where functions*en*are bounded and continuous defined on*Z* with 0*< eL*_{≤}_{e}M_{<}_{1 and}

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

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

* >* 0 *andM >* 0, there exist positive constants*δandn, Msuch that for anyn*0 ∈ *Zand*
|*u*0|*< M,whenωn< δ, one has*

*un, n*0*, u*0*< * *forn >n* *n*0*.* 2.9

**3. Main Results**

**Theorem 3.1.** *Suppose that assumptions* *H*1*andH*2*hold, then there exists a constantM >* 0

*such that*

lim sup

*n*→ ∞ *xn< M,* lim sup*n*→ ∞ *yn< M,* lim sup*n*→ ∞ *u*1*n< M,* lim sup*n*→ ∞ *u*2*n< M,*

3.1

*for any positive solutionxn, yn, u*1*n, u*2*nof system*1.3*.*

*Proof.* Given any solution*xn, yn, u*1*n, u*2*n*of system1.3, we have

Δ*u*1*n* 1≤*rn*−*e*1*nu*1*n,* 3.2

for all*n*≥*n*0*,*where*n*0is the initial time.

Consider the following auxiliary equation:

Δ*vn* 1 *rn*−*e*1*nvn,* 3.3

from assumptionsH1*,*H2and Lemma2.2, there exists a constant*M*1*>*0 such that

lim sup

*n*→ ∞ *vn*≤*M*1*,* 3.4

where*vn*is the solution of3.3with initial condition*vn*0 *u*1*n*0*.*By the comparison
theorem, we have

*u*1*n*≤*vn,* ∀*n*≥*n*0*.* 3.5

From this, we further have

lim sup

Then, we obtain that for any constant*ε >*0*,*there exists a constant*n*1*> n*0such that

*u*1*n< M*1 *ε* ∀*n*≥*n*1*.* 3.7

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

*xn*≤*xn*exp{*bn*−*a*11*nxn* *c*1*nM*1 *ε*}*,* 3.8

for all*n*≥*n*1*.*Considering the following auxiliary equation:

*zn* 1 *zn*exp{*bn*−*a*11*nzn* *c*1*nM*1 *ε*}*,* 3.9

thus, as a direct corollary of Lemma2.1, we get that there exists a positive constant*M*2 *>*0
such that

lim sup

*n*→ ∞ *zn*≤*M*2*,* 3.10

where*zn*is the solution of3.9with initial condition*zn*1 *xn*1*.*By the comparison
theorem, we have

*xn*≤*zn,* ∀*n*≥*n*1*.* 3.11

From this, we further have

lim sup

*n*→ ∞ *xn*≤*M*2*.* 3.12

Then, we obtain that for any constant*ε >*0*,*there exists a constant*n*2*> n*1such that

*xn< M*2 *ε,* ∀*n*≥*n*2*.* 3.13

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

*yn* 1≤*yn*exp*a*21*nM*2 *ε*−*dn*−*a*22*nyn*

*,* 3.14

for all*n*≥*n*2 *τ.*Following a similar argument as above, we get that there exists a positive
constant*M*3such that

lim sup

*n*→ ∞

*yn< M*3*.* 3.15

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

lim sup

From3.6and3.12–3.16, we can choose the constant*M*max{*M*1*, M*2*, M*3*, M*4},
such that

lim sup

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

*n*→ ∞ *u*1*n< M,* lim sup*n*→ ∞ *u*2*n< M.*

3.17

This completes the proof of Theorem3.1.

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

H3 *bn* *c*1*nu*∗_{10}*nL>*0*,*where*u*∗_{10}*n*is some positive solution of the following
equation:

Δ*un* 1 *rn*−*e*1*nun.* 3.18

**Theorem 3.2.** *Suppose that assumptionsH*1*–H*3*hold, then there exists a constantηx* *>* 0*such*

*that*

lim inf

*n*→ ∞ *xn> ηx,* 3.19

*for any positive solutionxn, yn, u*1*n, u*2*nof system*1.3*.*

*Proof.* According to assumptionsH1andH3*,*we can choose positive constants*ε*0and*ε*1
such that

*bn*−*a*11*nε*0−

*a*12*nε*1
1 *γnε*1

*c*1*n*

*u*∗_{10}*n*−*ε*1

*L*

*> ε*0*,*

*a*21*nε*0
1 *βnε*0 −

*dn*
*M*

*<*−*ε*0*.*

3.20

Consider the following equation with parameter*α*0:

Δ*vn* 1 *rn*−*e*1*nvn*−*f*1*nα*0*.* 3.21

Let *un* be any positive solution of system 3.18 with initial value *un*0 *v*0*.* By
assumptionsH1–H3and Lemma2.2, we obtain that*un*is globally asymptotically stable
and converges to*u*∗_{10}*n*uniformly for*n* → ∞*.*Further, from Lemma2.3, we obtain that,
for any given*ε*1 *>* 0 and a positive constant*M >* 0*M*is given in Theorem3.1, there exist
constants*δ*1 *δ*1*ε*1*>*0 and*n*∗_{1} *n*∗_{1}*ε*1*, M>*0*,*such that for any*n*0 ∈*Z* and 0≤*v*0 ≤*M,*
when*f*1*nα*0*< δ*1, we have

_{v}_{}_{n, n}_{0}_{, v}_{0}_{}_{−}* _{u}*∗

10*n* *< ε*1*,* ∀*n*≥*n*0 *n*∗1*,* 3.22

Let*α*0 ≤min{*ε*0*, δ*1*/f*_{1}*M* 1}*,*from3.20, we obtain that there exist*α*0and*n*1such
that

*bn*−*a*11*nα*0− *a*12*nε*1
1 *γnε*1

*c*1*n*

*u*∗_{10}*n*−*ε*1

*> α*0*,*

*a*21*nα*0

1 *βnα*0 −*dn<*−*α*0*,* *f*1*n< f*
*M*
1 1*,*

3.23

for all*n > n*1*.*

We first prove that

lim sup

*n*→ ∞ *xn*≥*α*0*,* 3.24

for any positive solution*xn, yn, u*1*n, u*2*n*of system1.3. In fact, if3.24is not true,
then there exists aΦ*θ* *φ*1*θ, φ*2*θ, ψ*1*θ, ψ*2*θ*such that

lim sup

*n*→ ∞

*xn,*Φ*< α*0*,* 3.25

where *xn,*Φ*, yn,*Φ*, u*1*n,*Φ*, u*2*n,*Φ is the solution of system 1.3 with initial
condition*xθ, yθ, u*1*θ, u*2*θ* Φ*θ*,*θ*∈−*τ,*0*.*So, there exists an*n*2*> n*1such that

*xn,*Φ*< α*0 ∀*n > n*2*.* 3.26

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

Δ*u*1*n* 1*> rn*−*e*1*nu*1*n*−*f*_{1}*Mα*0*,* 3.27

for*n > n*2*.*Let *vn* be the solution of3.21 with initial condition *vn*2 *u*1*n*2*,*by the
comparison theorem, we have

*u*1*n*≥*vn,* ∀*n*≥*n*2*.* 3.28

In3.22, we choose*n*0*n*2and*v*0*u*1*n*2*,*since*f*1*nα*0*< δ*1*,*then for given*ε*1*,*we have

*vn* *vn, n*2*, u*1*n*2*> u*∗_{10}*n*−*ε*1*,* 3.29

for all*n*≥*n*2 *n*∗_{1}*.*Hence, from3.28, we further have

*u*1*n> u*∗_{10}*n*−*ε*1*,* ∀*n*≥*n*2 *n*∗_{1}*.* 3.30

From the second equation of system1.3, we have

*yn* 1≤*yn*exp

*a*21*nα*0
1 *βnα*0 −

*dn*

for all*n > n*2 *τ.*Obviously, we have*yn* → 0 as*n* → ∞*.*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*_{}}_{}_{{}_{}_{ϕ}m

2 }of initial functions such that

From the second equation of system1.3, we have

*yn* 1≤*yn*exp

*a*21*nα*0
1 *βnα*0 −

*dn*

*,* 3.47

for*m*≥*m*0,*q*≥*K*_{1}*m*, and*n*∈*sqm* *τ, tqm.*Therefore, we get that

*yn< ε*1*,* 3.48

for any*n*∈*sqm* *τ* *n*∗*, tqm.*Further, from the first equation of systems1.3,3.46, and

3.48, we obtain

*xn* 1*, Zm*≥*xn, Zm*exp

*bn*−*a*11*nα*0−

*a*12*nε*1
1 *γnε*1

*c*1*n*

*u*∗_{10}*n*−*ε*1

≥*xn, Zm*exp*α*0*,*

3.49

for any*m*≥*m*0,*q*≥*K*1*m*, and*n*∈*s*

*m*

*q* 1 *τ* *n*∗*, tqm.*Hence,

*xtqm, Zm*

≥*xtqm*−1*, Zm*

exp*α*0*.* 3.50

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

*α*0

*m* 12 ≥*x*

*tqm, Zm*

≥*xtqm*−1*, Zm*

exp*α*0

≥ *α*0

*m* 12exp*α*0*>*

*α*0

*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 *xn*exp{*bn*−*a*11*nxn* *c*1*nu*1*n*}*,*

Δ*u*1*n* 1 *rn*−*e*1*nu*1*n*−*f*1*nxn.*

3.52

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

H4suppose*λ*max{|1−*aM*11*x*|*,*|1−*aL*11*x*|} *cM*1 *<*1,*δ*1−*eL*1 *f*1*Mx <*1*,*where*x*,*x*
are given in the proof of Lemma3.3.

**Lemma 3.3.** *Suppose that assumptionsH*1*–H*3*hold, then*

a*there exists a constantM >*1*such that*

*M*−1 *<*lim inf

*n*→ ∞ *xn<*lim sup_{n}_{→ ∞} *xn< M,* lim sup_{n}_{→ ∞} *u*1*n< M,* 3.53

*for any positive solutionxn, u*1*nof system*3.52*.*

b*if assumptionH*4*holds, then each fixed positive solutionxn, u*1*nof system*3.52
*is globally uniformly attractive onR*2_{0}*.*

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

Here, we prove conclusion b. Letting *x*∗_{10}*n, u*∗_{10}*n*be some solution of system

3.52, by conclusiona, there exist constants*x*,*x*, and*M >*1, such that

*x*−*ε < xn, x*∗_{10}*n< x* *ε,* *u*1*n, u*∗10*n< M,* 3.54

for any solution*xn, u*1*n*of system3.52and*n > n*∗*.*We make transformation*xn*

*x*∗_{10}*n*exp*v*1*n*and*u*1*n* *u*∗10*n* *v*2*n.*Hence, system3.52is equivalent to

*v*1*n* 1

1−*a*11*nx*∗10*n*exp{*θ*1*nv*1*n*}

*v*1*n* *c*1*nv*2*n,*

Δ*v*2*n* 1 −*e*1*nv*2*n*−*f*1*nx*∗10*n*exp{*θ*2*nv*1*n*}*v*1*n.*

3.55

According to H4, there exists a *ε >* 0 small enough, such that *λε* max{|1 −*aM*_{11}*x*

*ε*|*,*|1−*aL*

11*x*−*ε*|} *c*1*M* *<* 1, *σε* 1 −*eL*1 *f*1*Mx* *ε* *<* 1*.*Noticing that *θin* ∈ 0*,*1

implies that*x*∗_{10}*n*exp*θinv*1*n* *i* 1*,*2lie between*x*∗_{10}*n*and*xn.*Therefore,*x*−*ε <*

*x*∗_{10}*n*exp*θinv*1*n< x* *ε*,*i*1*,*2*.*It follows from3.55that

|*v*1*n* 1| ≤

1−*a*11*nx*∗10*n*exp{*θ*1*nv*1*n*}

*v*1*n* *c*1*nv*2*n,*
|*v*2*n* 1| ≤1−*e*1*nv*2*n*−*f*1*nx*∗10*n*exp{*θ*2*nv*1*n*}*v*1*n.*

3.56

Let*μ*max{*λε _{, σ}ε*

_{}}

_{,}_{then 0}

_{< μ <}_{1. It follows easily from}

_{}

_{3.56}

_{}

_{that}

max{|*v*1*n* 1|*,*|*v*2*n* 1|} ≤*μ*max{|*v*1*n*|*,*|*v*2*n*|}*.* 3.57

Considering the following equations:

*xn* 1 *xn*exp*bn*−*a*11*nxn*−*gn* *c*1*nu*1*n*

*,*

Δ*u*1*n* 1 *rn*−*e*1*nu*1*n*−*f*1*nxn,*

3.58

then we have the following result.

**Lemma 3.4.** *Suppose that assumptionsH*1*–H*4*hold, then there exists a positive constantδ*2*such*
*that for any positive solutionxn, u*1*nof system*3.58*, one has*

lim

*n*→ ∞|*xn*−*xn*|0*,* *n*lim→ ∞|*u*1*n*−*un*|0*,* *gn*∈0*, δ*2*,* 3.59

*wherexn,unis the solution of system*3.52*withxn*0 *xn*0*andun*0 *u*1*n*0*.*

The proof of Lemma3.4is similar to Lemma3.3, one omits it here.
Let*x*∗*n, u*∗_{1}*n*be a fixed solution of system3.52defined on*R*2

0*,*one assumes that

H5 −*dn* *a*21*nx*∗*n*−*τ/*1 *βnx*∗*n*−*τL>*0*.*

**Theorem 3.5.** *Suppose that assumptionsH*1*–H*5*hold, then there exists a constantηy* *>* 0*such*

*that*

lim inf

*n*→ ∞ *yn> ηy,* 3.60

*for any positive solutionxn, yn, u*1*n, u*2*nof system*1.3*.*

*Proof.* According to assumptionH5*,*we can choose positive constants*ε*2,*ε*3, and*n*1, such
that for all*n*≥*n*1*,*we have

−*dn* *a*21*nx*

∗_{}_{n}_{−}_{τ}_{}_{−}_{ε}

3

1 *βnx*∗*n*−*τ*−*ε*3 *γnε*2 −*a*22*nε*2−*c*2*nε*3*> ε*2*.* 3.61

Considering the following equation with parameter*α*1:

Δ*vn* 1 −*e*2*nvn* *f*2*nα*1*,* 3.62

by Lemma2.4, for given*ε*3*>*0 and*M >*0*M*is given in Theorem3.1., there exist constants

*δ*3 *δ*3*ε*3 *>* 0 and*n*∗_{3} *n*∗_{3}*ε*3*, M* *>* 0, such that for any*n*0 ∈ *Z* and 0 ≤ *v*0 ≤ *M,*when

*f*2*nα*0*< δ*3*,*we have

We choose*α*1 *<* max{*ε*2*, δ*3*/*1 *f*_{2}*M*}if there exists a constant *n* such that*a*12*n*−

*δ*2*γn* ≡ 0 for all *n > n,* otherwise *α*1 *<* max{*ε*2*, δ*3*/*1 *f*_{2}*M, δ*2*/a*12*n*−*δ*2*γnM*}*.*
Obviously, there exists an*n*2*> n*1, such that

*a*12*nα*1
1 *γnα*1

*< δ*2*,* *f*2*nα*1*< δ*3*,* ∀*n > n*2*.* 3.64

Now, We prove that

lim sup

*n*→ ∞ *yn*≥*α*1*,* 3.65

for any positive solution*xn, yn, u*1*n, u*2*n*of system1.3. In fact, if3.65is not true,
then for*α*1, there exist aΦ*θ* *φ*1*θ, φ*2*θ, ψ*1*θ, ψ*2*θ*and*n*3*> n*2such that for all*n > n*3*,*

*yn,*Φ*< α*1*,* 3.66

where*φi*∈*DC* and*ψi*∈*DC* *i*1*,*2*.*Hence, for all*n > n*3*,*one has

0*<* *a*12*nyn*

1 *βnxn* *γnyn* *<*

*a*12*nα*1

1 *γnα*1 *< δ*2*.* 3.67

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

lim

*n*→ ∞|*xn*−*x*

∗_{}_{n}_{}_{|}_{}_{0}_{,}_{lim}

*n*→ ∞|*u*1*n*−*u*

∗_{}_{n}_{}_{|}_{}_{0}_{,}_{}_{3.68}_{}

for any solution*xn, yn, u*1*n, u*2*n*of system1.3. Therefore, for any small positive
constant*ε*3*>*0*,*there exists an*n*∗_{4}such that for all*n*≥*n*3 *n*∗_{4}*,*we have

*xn*≥*x*∗_{10}*n*−*ε*3*.* 3.69

From the fourth equation of system1.3, one has

Δ*u*2*n* 1≤ −*e*2*nu*2*n* *f*2*Mα*1*.* 3.70

In3.63, we choose*n*0 *n*3 and*v*0 *un*3*.*Since*f*2*nα*1 *< δ*3*,*then for all*n*≥ *n*3 *n*∗_{3}, we
have

*u*2*n*≤*ε*3*.* 3.71

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

*yn* 1≥*yn*exp

*a*21*n*

*x*∗_{10}*n*−*ε*3

1 *βnx*∗_{10}*n*−*ε*3

*γnα*1

−*dn*−*a*22*nα*1−*c*2*nε*3

for all*n > n*3 *τ* *n*∗∗*,*where*n*∗∗max{*n*_{3}∗*, n*∗_{4}}*.*Obviously, we have*yn* → ∞as*n* → ∞*,*
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 sequence*Zm*_{}_{{}_{φ}m

2 }of initial functions, such that

The above inequality implies that

*tqm*−*sqm*≥

ln*m* 1

*θ*2

*,* ∀*q*≥*K*_{1}*m, m*1*,*2*, . . . .* 3.80

Choosing a large enough*m*1*,*such that

*tqm*−*sqm>n*∗∗ *τ* 2*,* ∀*m*≥*m*1*, q*≥*K*_{1}*m,* 3.81

then for*m*≥*m*1*, q*≥*K*_{1}*m,*we have

0*<* *a*12*nyn*

1 *βnxn* *γnyn* *<*

*a*12*nα*1
1 *γnα*1

*< δ*2*,* 3.82

for all*n*∈*sqm* 1*, tqm.*Therefore, it follows from system1.3that

*xn* 1≥*xn*exp*bn*−*a*11*nxn*−*δ*2 *c*1*nu*1*n,*

*u*1*n* 1 *rn*−*e*1*n*−1*u*1*n*−*f*1*nxn,*

3.83

for all*n*∈*sqm* 1*, tqm.*Further, by Lemmas3.3and3.4, we obtain that for any small positive

constant*ε*3*>*0*,*we have

*xn*≥*x*∗_{10}*n*−*ε*3*,* 3.84

for any*m* ≥ *m*1,*q* ≥ *K*1*m*, and*n* ∈ *s*

*m*

*q* 1 *n*∗∗*, tqm.*For any *m* ≥ *m*1,*q* ≥ *K*1*m*, and

*n*∈*sqm* 1*, tqm,*by the first equation of systems1.3and3.77, it follows that

Δ*u*2

*n* 1*, Zm*≤ −*e*2*nu*2

*n, Zm* *f*2*n*

*α*1

*m* 1
≤ −*e*2*nu*2

*n, Zm* *f*2*nα*1*.*

3.85

Assume that*vn*is the solution of3.62with the initial condition*vsqm* 1 *u*2*sqm* 1,

then from comparison theorem and the above inequality, we have

*u*2

*n, Zm*≤*vn,* ∀*n*∈*sqm* 1*, tqm*

*, m*≥*m*1*, q*≥*K*_{1}*m.* 3.86

In3.63, we choose*n*0 *sqm* 1 and*v*0 *u*2*sqm* 1*.*Since 0*< v*0 *< M*and*f*2*nα*1 *< δ*3*,*
then we have

*vn*≤*ε*3*,* ∀*n*∈

*sqm* 1 *n*∗∗*, tqm*

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 control*u*1*n*. So, the control variable*u*1*n*has 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.

*holds, then*

lim

*n*→ ∞*yn* 0*,* 3.93

*for any positive solutionxn, yn, u*1*n, u*2*nof system*1.3*.*

*Proof.* By the condition, for any positive constant*εε < α*1*,*where*α*1is given in Theorem3.5,
there exist constants*ε*1and*n*1*,*such that

−*dn* *a*21*nx*∗*n*−*τ* *ε*1

1 *βnx*∗*n*−*τ* *ε*1−

*a*22*nε <*−*ε*1*,* 3.94

for*n > n*1*.*First, we show that there exists an*n*2 *> n*1*,*such that*yn*2*< ε.*Otherwise, there
exists an*n*∗_{1}, such that

*yn*≥*ε,* ∀*n > n*1 *n*∗1*.* 3.95

Hence, for all*n*≥*n*1 *n*∗_{1}*,*one has

*xn* 1*< xn*exp

*bn*−*a*11*nxn*− *a*12*nε*

1 *γnε* *c*1*nu*1*n*

*,*

Δ*u*1*n* 1 *rn*−*e*1*nu*1*n* *f*1*nxn.*

3.96

Therefore, from Lemma3.3and comparison theorem, it follows that for the above*ε*1*,*there
exists an*n*∗_{2}*>*0, such that

*xn< x*∗*n* *ε*1*,* ∀*n > n*1 *n*∗2*.* 3.97

Hence, for*n > n*1 *n*∗2*,*we have

*ε*≤*yn* 1*< yn*exp

−*dn* *a*21*nx*∗*n*−*τ* *ε*1

1 *βnx*∗*n*−*τ* *ε*1−

*a*22*nε*

≤*yn*1 *n*∗2

exp−*ε*1

*n*−*n*1−*n*∗2

−→0 as*n*−→ ∞*.*

3.98

So,*ε <*0*,*which is a contradiction. Therefor, there exists an*n*2*> n*1*,*such that*yn*2*< ε.*
Second, we show that

*yn< ε*exp*μ,* ∀*n > n*2*,* 3.99

where

*μ*max*n*∈*Z*

*dn* *a*21*nx*∗*n*−*τ* *ε*1

1 *βnx*∗*n*−*τ* *ε*1

*a*22*nε*

is bounded. Otherwise, there exists an*n*3*> n*2*,*such that*yn*3≥*ε*exp{*μ*}*.*Hence, there must
exist an*n*4 ∈*n*2*, n*3−1such that*yn*4*< ε*,*yn*4 1≥ *ε*, and*yn*≥*ε*for*n*∈*n*4 1*, n*3*.*
Let*P*1be a nonnegative integer, such that

*n*3 *n*4 *P*1 1*.* 3.101

It follows from3.101that

*ε*exp*μ*≤*yn*3≤*yn*4exp

which leads to a contradiction. This shows that 3.99 holds. By the arbitrariness of*ε,* it
immediately follows that*yn* → 0 as*n* → ∞*.*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.

**References**

1 R. S. Cantrell and C. Cosner, “On the dynamics of predator-prey models with the
Beddington-DeAngelis functional response,”*Journal of Mathematical Analysis and Applications, vol. 257, no. 1, pp.*
206–222, 2001.

2 N. P. Cosner, D. L. deAngelis, J. S. Ault, and D. B. Olson, “Eﬀects of spatial grouping on the functional
response of predators,”*Theoretical Population Biology, vol. 56, pp. 65–75, 1999.*

3 J. Cui and Y. Takeuchi, “Permanence, extinction and periodic solution of predator-prey system with
Beddington-DeAngelis functional response,”*Journal of Mathematical Analysis and Applications, vol.*
317, no. 2, pp. 464–474, 2006.

4 D. T. Dimitrov and H. V. Kojouharov, “Complete mathematical analysis of predator-prey models
with linear prey growth and Beddington-DeAngelis functional response,”*Applied Mathematics and*
*Computation, vol. 162, no. 2, pp. 523–538, 2005.*

5 H.-F. Huo, W.-T. Li, and J. J. Nieto, “Periodic solutions of delayed predator-prey model with the
Beddington-DeAngelis functional response,”*Chaos, Solitons and Fractals, vol. 33, no. 2, pp. 505–512,*
2007.

6 T.-W. Hwang, “Global analysis of the predator-prey system with Beddington-DeAngelis functional
response,”*Journal of Mathematical Analysis and Applications, vol. 281, no. 1, pp. 395–401, 2003.*

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.

11 J. R. Beddington, “Mutual interference between parasites or predators and its eﬀect on searching
eﬃciency,”*Journal of Animal Ecology, vol. 44, pp. 331–340, 1975.*

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 for*n*-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.