The Permanence and Extinction of a Discrete Predator-Prey System with Time Delay and Feedback Controls

Abstract

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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Yongwimon Lenbury

Department of Mathematics, Yuncheng University, Yuncheng 044000, China

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

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 extensively (e.g., see [1-10] and 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: [figure omitted; refer to PDF] The functional response in system (1.1) was introduced by Beddington [11] and DeAngelis 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. Hwang [6] showed that the system has no periodic solutions when the positive equilibrium is locally asymptotical stability by using the divergency criterion. Recently, Fan and Kuang [9] further considered the nonautonomous case of system (1.1), that is, they considered the following system: [figure omitted; refer to PDF] For the general nonautonomous case, they addressed properties such as permanence, extinction, and globally asymptotic stability of the system. For the periodic (almost 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, see [13-22] and 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: [figure omitted; refer to PDF] where x(n) , y(n) are the density of the prey species and the predator species at time n , respectively. _ui (n) (i=1 ,2) are the feedback control variables. b(n),_a11 (n) represent the intrinsic growth rate and density-dependent coefficient of the prey at time n , respectively. d(n),_a22 (n) denote the death rate and density-dependent coefficient of the predator at time n , respectively. _a12 (n) denotes the capturing rate of the predator; _a21 (n)/_a12 (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 [_k1 ,_k2 ] denote the set of integer k satisfying _k1 ≤k≤_k2 . We denote D_C+ : [-τ,0][arrow right]_R+ to be the space of all nonnegative and bounded discrete time functions. In addition, for any bounded sequence g(n), we denote ^gL =_{inf n∈Z+} g(n) , ^gM =_{sup n∈Z+} g(n).

Given the biological sense, we only consider solutions of system (1.3) with the following initial condition: [figure omitted; refer to PDF]

It is not difficult to see that the solutions of system (1.3) with the above initial condition are well defined for all n≥0 and satisfy [figure omitted; refer to PDF]

The main purpose of this paper is to establish a new general criterion for the permanence and extinction of system (1.3), which is dependent on feedback controls. This paper is organized as follows. In Section 2, we will give some assumptions and useful lemmas. In Section 3, some new sufficient conditions which guarantee the permanence of all positive solutions of system (1.3) are 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 ) r(n) , b(n) , d(n) , β(n) and γ(n) are nonnegative bounded sequences of real numbers defined on _Z+ such that [figure omitted; refer to PDF]

(_H2 ) _ci (n) , _ei (n) , _fi (n) and _aij (n) are nonnegative bounded sequences of real numbers defined on _Z+ such that [figure omitted; refer to PDF]

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

First, we consider the following nonautonomous equation: [figure omitted; refer to PDF] where functions a(n) , g(n) are bounded and continuous defined on _Z+ with ^aL , ^gL >0 . We have the following result which is given in [23].

Lemma 2.1.

Let x(n) be the positive solution of (2.3) with x(0)>0 , then

(a) there exists a positive constant M>1 such that

[figure omitted; refer to PDF] for any positive solution x(n) of (2.3);

(b) _{lim n[arrow right]∞} (^x(1) (n)-^x(2) (n))=0 for any two positive solutions ^x(1) (n) and ^x(2) (n) of (2.3).

Second, one considers the following nonautonomous linear equation: [figure omitted; refer to PDF] where functions f(n) and e(n) are bounded and continuous defined on _Z+ with ^fL >0 and 0<^eL ≤^eM <1. The following Lemma 2.2 is a direct corollary of Theorem 6.2 of L. Wang and M. Q. Wang [24, page 125].

Lemma 2.2.

Let u(n) be the nonnegative solution of (2.5) with u(0)>0 , then

(a) ^fL /^eM <_{lim inf n[arrow right]∞} u(n) ≤_{lim sup n[arrow right]∞} u(n)≤^fM /^eL for any positive solution u(n) of (2.5);

(b) _{lim n[arrow right]∞} (^u(1) (n)-^u(2) (n))=0 for any two positive solutions ^u(1) (n) and ^u(2) (n) of (2.5).

Further, considering the following: [figure omitted; refer to PDF] where functions f(n) and e(n) are bounded and continuous defined on _Z+ with ^fL >0 , 0<^eL ≤^eM <1 and ω(n)≥0. The following Lemma 2.3 is a direct corollary of Lemma 3 of Xu and Teng [25].

Lemma 2.3.

Let u(n,_n0 ,_u0 ) be the positive solution of (2.6) with u(0)>0 , then for any constants ...>0 and M>0 , there exist positive constants δ(...) and n...(...,M) such that for any _n0 ∈_Z+ and |_u0 |<M, when |ω(n)|<δ, one has [figure omitted; refer to PDF] where ^u* (n,_n0 ,_u0 ) is a positive solution of (2.5) with ^u* (_n0 ,_n0 ,_u0 )=_u0 .

Finally, one considers the following nonautonomous linear equation: [figure omitted; refer to PDF] where functions e(n) are bounded and continuous defined on _Z+ with 0<^eL ≤^eM <1 and ω(n)≥0. In [25], the following Lemma 2.4 has been proved.

Lemma 2.4.

Let u(n) be the nonnegative solution of (2.8) with u(0)>0 , then, for any constants ...>0 and M>0 , there exist positive constants δ(...) and n...(...,M) such that for any _n0 ∈Z and |_u0 |<M, when ω(n)<δ , one has [figure omitted; refer to PDF]

3. Main Results

Theorem 3.1.

Suppose that assumptions (_H1 ) and (_H2 ) hold, then there exists a constant M>0 such that [figure omitted; refer to PDF] for any positive solution (x(n),y(n),_u1 (n),_u2 (n)) of system (1.3).

Proof.

Given any solution (x(n),y(n),_u1 (n),_u2 (n)) of system (1.3), we have [figure omitted; refer to PDF] for all n≥_n0 , where _n0 is the initial time.

Consider the following auxiliary equation: [figure omitted; refer to PDF] from assumptions (_H1 ), (_H2 ) and Lemma 2.2, there exists a constant _M1 >0 such that [figure omitted; refer to PDF] where v(n) is the solution of (3.3) with initial condition v(_n0 )=_u1 (_n0 ). By the comparison theorem, we have [figure omitted; refer to PDF] From this, we further have [figure omitted; refer to PDF] Then, we obtain that for any constant [straight epsilon]>0, there exists a constant _n1 >_n0 such that [figure omitted; refer to PDF]

According to the first equation of system (1.3), we have [figure omitted; refer to PDF] for all n≥_n1 . Considering the following auxiliary equation: [figure omitted; refer to PDF] thus, as a direct corollary of Lemma 2.1, we get that there exists a positive constant _M2 >0 such that [figure omitted; refer to PDF] where z(n) is the solution of (3.9) with initial condition z(_n1 )=x(_n1 ). By the comparison theorem, we have [figure omitted; refer to PDF] From this, we further have [figure omitted; refer to PDF] Then, we obtain that for any constant [straight epsilon]>0, there exists a constant _n2 >_n1 such that [figure omitted; refer to PDF]

Hence, from the second equation of system (1.3), we obtain [figure omitted; refer to PDF] for all n≥_n2 +τ. Following a similar argument as above, we get that there exists a positive constant _M3 such that [figure omitted; refer to PDF]

By a similar argument of the above proof, we further obtain [figure omitted; refer to PDF]

From (3.6) and (3.12)-(3.16), we can choose the constant M=max {_M1 ,_M2 ,_M3 ,_M4 } , such that [figure omitted; refer to PDF] This completes the proof of Theorem 3.1.

In order to obtain the permanence of system (1.3), we assume that

(_H3 ) ^{[b(n)+_c1 (n)u10* (n)]L} >0, where ^u10* (n) is some positive solution of the following equation: [figure omitted; refer to PDF]

Theorem 3.2.

Suppose that assumptions (_H1 )-(_H3 ) hold, then there exists a constant _ηx >0 such that [figure omitted; refer to PDF] for any positive solution (x(n),y(n),_u1 (n),_u2 (n)) of system (1.3).

Proof.

According to assumptions (_H1 ) and (_H3 ), we can choose positive constants _{[straight epsilon]0} and _{[straight epsilon]1} such that [figure omitted; refer to PDF] Consider the following equation with parameter _α0 : [figure omitted; refer to PDF] Let u(n) be any positive solution of system (3.18) with initial value u(_n0 )=_v0 . By assumptions (_H1 )-(_H3 ) and Lemma 2.2, we obtain that u(n) is globally asymptotically stable and converges to ^u10* (n) uniformly for n[arrow right]+∞. Further, from Lemma 2.3, we obtain that, for any given _{[straight epsilon]1} >0 and a positive constant M>0 (M is given in Theorem 3.1), there exist constants _δ1 =_δ1 (_{[straight epsilon]1} )>0 and ^n1* =^n1* (_{[straight epsilon]1} ,M)>0, such that for any _n0 ∈_Z+ and 0≤_v0 ≤M, when _f1 (n)_α0 <_δ1 , we have [figure omitted; refer to PDF] where v(n,_n0 ,_v0 ) is the solution of (3.21) with initial condition v(_n0 ,_n0 ,_v0 )=_v0 .

Let _α0 ≤min {_{[straight epsilon]0} ,_δ1 /(^f1M +1)}, from (3.20), we obtain that there exist _α0 and _n1 such that [figure omitted; refer to PDF] for all n>_n1 .

We first prove that [figure omitted; refer to PDF] for any positive solution (x(n),y(n),_u1 (n),_u2 (n)) of system (1.3). In fact, if (3.24) is not true, then there exists a Φ(θ)=(_[varphi]1 (θ),_[varphi]2 (θ),_ψ1 (θ),_ψ2 (θ)) such that [figure omitted; refer to PDF] where (x(n,Φ),y(n,Φ),_u1 (n,Φ),_u2 (n,Φ)) is the solution of system (1.3) with initial condition (x(θ),y(θ),_u1 (θ),_u2 (θ))=Φ(θ) , θ∈[-τ,0]. So, there exists an _n2 >_n1 such that [figure omitted; refer to PDF] Hence, (3.26) together with the third equation of system (1.3) lead to [figure omitted; refer to PDF] for n>_n2 . Let v(n) be the solution of (3.21) with initial condition v(_n2 )=_u1 (_n2 ), by the comparison theorem, we have [figure omitted; refer to PDF] In (3.22), we choose _n0 =_n2 and _v0 =_u1 (_n2 ), since _f1 (n)_α0 <_δ1 , then for given _{[straight epsilon]1} , we have [figure omitted; refer to PDF] for all n≥_n2 +^n1* . Hence, from (3.28), we further have [figure omitted; refer to PDF] From the second equation of system (1.3), we have [figure omitted; refer to PDF] for all n>_n2 +τ. Obviously, we have y(n)[arrow right]0 as n[arrow right]+∞. Therefore, we get that there exists an ^n2* such that [figure omitted; refer to PDF] for any n>_n2 +τ+^n2* . Hence, by (3.26), (3.30), and (3.32), it follows that [figure omitted; refer to PDF] for any n>_n2 +τ+^n...* , where ^n...* =max {^n1* ,^n2* }. Thus, from (3.23) and (3.33), we have _{lim n[arrow right]+∞} x(n)=+∞, which leads to a contradiction. Therefore, (3.24) holds.

Now, we prove the conclusion of Theorem 3.2. In fact, if it is not true, then there exists a sequence {^Z(m) }={(^{[straight phi]1(m)} ,^{[straight phi]2(m)} ,^ψ1(m) ,^ψ2(m) )} of initial functions such that [figure omitted; refer to PDF] On the other hand, by (3.24), we have [figure omitted; refer to PDF] Hence, there are two positive integer sequences {^sq(m) } and {^tq(m) } satisfying [figure omitted; refer to PDF] and _{lim q[arrow right]∞} ^sq(m) =∞, such that [figure omitted; refer to PDF] [figure omitted; refer to PDF] By Theorem 3.1, for any given positive integer m , there exists a ^K(m) such that x(n,^Z(m) )<M , y(n,^Z(m) )<M , _u1 (n,^Z(m) )<M , and _u2 (n,^Z(m) )<M for all n>^K(m) . Because of ^sq(m) [arrow right]+∞ as q[arrow right]+∞, there exists a positive integer ^K1(m) such that ^sq(m) >^K(m) +τ and ^sq(m) >_n1 as q>^K1(m) . Let q≥^K1(m) , for any n∈[^sq(m) ,^tq(m) ] , we have [figure omitted; refer to PDF] where _θ1 =_{sup n∈Z+} {b(n)+_a11 (n)M+_a12 (n)M/(1+γ(n)M)+_c1 (n)M}. Hence, [figure omitted; refer to PDF] The above inequality implies that [figure omitted; refer to PDF] So, we can choose a large enough _m...0 such that [figure omitted; refer to PDF] From the third equation of system (1.3) and (3.38), we have [figure omitted; refer to PDF] for any m≥_m...0 , q≥^K1(m) , and n∈[^sq(m) +1,^tq(m) ]. Assume that v(n) is the solution of (3.21) with the initial condition v(^sq(m) +1)=_u1 (^sq(m) +1) , then from comparison theorem and the above inequality, we have [figure omitted; refer to PDF] In (3.22), we choose _n0 =^sq(m) +1 and _v0 =_u1 (^sq(m) +1) , since 0<_v0 <M and _f1 (n)_α0 <_δ1 , then for all n∈[^sq(m) +1,^tq(m) ] , we have [figure omitted; refer to PDF] Equation (3.44) together with (3.45) lead to [figure omitted; refer to PDF] for all n∈[^sq(m) +1+^n...* ,^tq(m) ] , q≥^K1(m) , and m≥_m...0 .

From the second equation of system(1.3), we have [figure omitted; refer to PDF] for m≥_m...0 , q≥^K1(m) , and n∈[^sq(m) +τ,^tq(m) ]. Therefore, we get that [figure omitted; refer to PDF] for any n∈[^sq(m) +τ+^n...* ,^tq(m) ]. Further, from the first equation of systems (1.3), (3.46), and (3.48), we obtain [figure omitted; refer to PDF] for any m≥_m...0 , q≥^K1(m) , and n∈[^sq(m) +1+τ+^n...* ,^tq(m) ]. Hence, [figure omitted; refer to PDF] In view of (3.37) and (3.38), we finally have [figure omitted; refer to PDF] which is a contradiction. Therefore, the conclusion of Theorem 3.2 holds. This completes the proof of Theorem 3.2.

In order to obtain the permanence of the component y(n) of system (1.3), we next consider the following single-specie system with feedback control: [figure omitted; refer to PDF]

For system (3.52), we further introduce the following assumption:

(_H4 ) suppose λ=max {|1-^a11M x¯|,|1-^a11L x...|}+^c1M <1 , δ=1-^e1L +^f1M x¯<1, where x¯ , x... are given in the proof of Lemma 3.3.

For system(3.52), we have the following result.

Lemma 3.3.

Suppose that assumptions (_H1 )-(_H3 ) hold, then

(a) there exists a constant M>1 such that

[figure omitted; refer to PDF] for any positive solution (x(n),_u1 (n)) of system (3.52).

(b) if assumption (_H4 ) holds, then each fixed positive solution (x(n),_u1 (n)) of system (3.52) is globally uniformly attractive on ^R+02 .

Proof.

Based on assumptions (_H1 )-(_H3 ) , conclusion (a) can be proved by a similar argument as in Theorems 3.1 and 3.2.

Here, we prove conclusion (b). Letting (^x10* (n),^u10* (n)) be some solution of system (3.52), by conclusion (a), there exist constants x¯ , x... , and M>1 , such that [figure omitted; refer to PDF] for any solution (x(n),_u1 (n)) of system (3.52) and n>^n* . We make transformation x(n)=^x10* (n)exp (_v1 (n)) and _u1 (n)=^u10* (n)+_v2 (n). Hence, system (3.52) is equivalent to [figure omitted; refer to PDF] According to (_H4 ) , there exists a [straight epsilon]>0 small enough, such that ^{λ[straight epsilon]} =max {|1-^a11M (x¯+[straight epsilon])|,|1-^a11L (x...-[straight epsilon])|}+^c1M <1 , ^{σ[straight epsilon]} =1-^e1L +^f1M (x¯+[straight epsilon])<1. Noticing that _θi (n)∈[0,1] implies that ^x10* (n)exp (_θi (n)_v1 (n)) (i=1,2) lie between ^x10* (n) and x(n). Therefore, x...-[straight epsilon]<^x10* (n)exp (_θi (n)_v1 (n))<x¯+[straight epsilon] , i=1,2. It follows from (3.55) that [figure omitted; refer to PDF] Let μ=max {^{λ[straight epsilon]} ,^{σ[straight epsilon]} }, then 0<μ<1 . It follows easily from (3.56) that [figure omitted; refer to PDF] Therefore, _{lim sup n[arrow right]∞} max {|_v1 (n+1)|,|_v2 (n+1)|}[arrow right]0 , as n[arrow right]+∞, and we can easily obtain that _{lim sup n[arrow right]∞} |_v1 (n+1)|=0 and _{lim sup n[arrow right]∞} |_v2 (n+1)|=0. The proof is completed.

Considering the following equations: [figure omitted; refer to PDF] then we have the following result.

Lemma 3.4.

Suppose that assumptions (_H1 )-(_H4 ) hold, then there exists a positive constant _δ2 such that for any positive solution (x(n),_u1 (n)) of system (3.58), one has [figure omitted; refer to PDF] where (x...(n),u...(n)) is the solution of system (3.52) with x...(_n0 )=x(_n0 ) and u...(_n0 )=_u1 (_n0 ).

The proof of Lemma 3.4 is similar to Lemma 3.3, one omits it here.

Let (^x* (n),^u1* (n)) be a fixed solution of system (3.52) defined on ^R+02 , one assumes that

(_H5 ) ^{(-d(n)+(_a21 (n)x* (n-τ)/(1+β(n)x* (n-τ)))L} >0.

Theorem 3.5.

Suppose that assumptions (_H1 )-(_H5 ) hold, then there exists a constant _ηy >0 such that [figure omitted; refer to PDF] for any positive solution (x(n),y(n),_u1 (n),_u2 (n)) of system (1.3).

Proof.

According to assumption (_H5 ), we can choose positive constants _{[straight epsilon]2} , _{[straight epsilon]3} , and _n1 , such that for all n≥_n1 , we have [figure omitted; refer to PDF] Considering the following equation with parameter _α1 : [figure omitted; refer to PDF] by Lemma 2.4, for given _{[straight epsilon]3} >0 and M>0 (M is given in Theorem 3.1.), there exist constants _δ3 =_δ3 (_{[straight epsilon]3} )>0 and ^n3* =^n3* (_{[straight epsilon]3} ,M)>0 , such that for any _n0 ∈_Z+ and 0≤_v0 ≤M, when _f2 (n)_α0 <_δ3 , we have [figure omitted; refer to PDF]

We choose _α1 <max {_{[straight epsilon]2} ,_δ3 /(1+^f2M )} if there exists a constant n' such that _a12 (n)-_δ2 γ(n)≡0 for all n>^{n[variant prime]} , otherwise _α1 <max {_{[straight epsilon]2} ,_δ3 /(1+^f2M ),_δ2 /(_a12 (n)-_δ2 γ(n^)M )}. Obviously, there exists an _n2 >_n1 , such that [figure omitted; refer to PDF]

Now, We prove that [figure omitted; refer to PDF] for any positive solution (x(n),y(n),_u1 (n),_u2 (n)) of system (1.3). In fact, if (3.65) is not true, then for _α1 , there exist a Φ(θ)=(_[varphi]1 (θ),_[varphi]2 (θ),_ψ1 (θ),_ψ2 (θ)) and _n3 >_n2 such that for all n>_n3 , [figure omitted; refer to PDF] where _[varphi]i ∈D_C+ and _ψi ∈D_C+ (i=1,2). Hence, for all n>_n3 , one has [figure omitted; refer to PDF] Therefore, from system (1.3), Lemmas 3.3 and 3.4, it follows that [figure omitted; refer to PDF] for any solution (x(n),y(n),_u1 (n),_u2 (n)) of system (1.3). Therefore, for any small positive constant _{[straight epsilon]3} >0, there exists an ^n4* such that for all n≥_n3 +^n4* , we have [figure omitted; refer to PDF] From the fourth equation of system (1.3), one has [figure omitted; refer to PDF] In (3.63), we choose _n0 =_n3 and _v0 =u(_n3 ). Since _f2 (n)_α1 <_δ3 , then for all n≥_n3 +^n3* , we have [figure omitted; refer to PDF] Equations (3.69), (3.71) together with the second equation of system (1.3) lead to [figure omitted; refer to PDF] for all n>_n3 +τ+^n...** , where ^n...** =max {^n3* ,^n4* }. Obviously, we have y(n)[arrow right]+∞ as n[arrow right]+∞, which is contradictory to the boundedness of solution of system (1.3). Therefore, (3.65) holds.

Now, we prove the conclusion of Theorem 3.5. In fact, if it is not true, then there exists a sequence ^Z(m) ={^[varphi]1(m) ,^[varphi]2(m) ,^ψ1(m) ,^ψ2(m) } of initial functions, such that [figure omitted; refer to PDF] where (x(n,^Z(m) ),y(n,^Z(m) ),_u1 (n,^Z(m) ),_u2 (n,^Z(m) )) is the solution of system (1.3) with initial condition (x(θ),y(θ),_u1 (θ),_u2 (θ))=^Z(m) (θ) for all θ∈[-τ,0]. On the other hand, it follows from (3.65) that [figure omitted; refer to PDF] Hence, there are two positive integer sequences {^sq(m) } and {^tq(m) } satisfying [figure omitted; refer to PDF] and _{lim q[arrow right]∞} ^sq(m) =∞, such that [figure omitted; refer to PDF] [figure omitted; refer to PDF]

By Theorem 3.1, for given positive integer m , there exists a ^K(m) such that x(n,^Z(m) )<M , y(n,^Z(m) )<M , _u1 (n,^Z(m) )<M , and _u2 (n,^Z(m) )<M for all n>^K(m) . Because that ^sq(m) [arrow right]+∞ as q[arrow right]+∞, there is a positive integer ^K1(m) such that ^sq(m) >^K(m) +τ and ^sq(m) >_n2 as q>^K1(m) . Let q≥^K1(m) , for any n∈[^sq(m) ,^tq(m) ] , we have [figure omitted; refer to PDF] where _θ2 =_{sup n∈N} {d(n)+_a21 (n)M+_a22 (n)M+_c2 (n)M}. Hence, [figure omitted; refer to PDF] The above inequality implies that [figure omitted; refer to PDF] Choosing a large enough _m...1 , such that [figure omitted; refer to PDF] then for m≥_m...1 , q≥^K1(m) , we have [figure omitted; refer to PDF] for all n∈[^sq(m) +1,^tq(m) ]. Therefore, it follows from system (1.3) that [figure omitted; refer to PDF] for all n∈[^sq(m) +1,^tq(m) ]. Further, by Lemmas 3.3 and 3.4, we obtain that for any small positive constant _{[straight epsilon]3} >0, we have [figure omitted; refer to PDF] for any m≥_m...1 , q≥^K1(m) , and n∈[^sq(m) +1+^n** ,^tq(m) ]. For any m≥_m...1 , q≥^K1(m) , and n∈[^sq(m) +1,^tq(m) ], by the first equation of systems (1.3) and (3.77), it follows that [figure omitted; refer to PDF] Assume that v(n) is the solution of (3.62) with the initial condition v(^sq(m) +1)=_u2 (^sq(m) +1) , then from comparison theorem and the above inequality, we have [figure omitted; refer to PDF] In (3.63), we choose _n0 =^sq(m) +1 and _v0 =_u2 (^sq(m) +1). Since 0<_v0 <M and _f2 (n)_α1 <_δ3 , then we have [figure omitted; refer to PDF] Equation (3.86) together with (3.87) lead to [figure omitted; refer to PDF] for all n∈[^sq(m) +1+^n...** ,^tq(m) ] , q≥^K1(m) , and m≥_m...1 .

So, for any m≥_m...1 , q≥^K1(m) , and n∈[^sq(m) +τ+1+^n...** ,^tq(m) ], from the second equation of systems (1.3), (3.61), (3.77), (3.84), and (3.88), it follows that [figure omitted; refer to PDF]

Hence, [figure omitted; refer to PDF]

In view of (3.76) and (3.77), we finally have [figure omitted; refer to PDF]

which is a contradiction. Therefore, the conclusion of Theorem 3.5 holds.

Remark 3.6.

In Theorems 3.2 and 3.5, we note that (_H1 )-(_H3 ) are decided by system(1.3), which is dependent on the feedback control _u1 (n) . So, the control variable _u1 (n) has impact on the permanence of system (1.3). 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.

Theorem 3.7.

Suppose that assumption [figure omitted; refer to PDF]

holds, then [figure omitted; refer to PDF] for any positive solution (x(n),y(n),_u1 (n),_u2 (n)) of system (1.3).

Proof.

By the condition, for any positive constant [straight epsilon] ([straight epsilon]<_α1 , where _α1 is given in Theorem 3.5), there exist constants _{[straight epsilon]1} and _n1 , such that [figure omitted; refer to PDF] for n>_n1 . First, we show that there exists an _n2 >_n1 , such that y(_n2 )<[straight epsilon]. Otherwise, there exists an ^n1* , such that [figure omitted; refer to PDF] Hence, for all n≥_n1 +^n1* , one has [figure omitted; refer to PDF] Therefore, from Lemma 3.3 and comparison theorem, it follows that for the above _{[straight epsilon]1} , there exists an ^n2* >0 , such that [figure omitted; refer to PDF] Hence, for n>_n1 +^n2* , we have [figure omitted; refer to PDF] So, [straight epsilon]<0, which is a contradiction. Therefor, there exists an _n2 >_n1 , such that y(_n2 )<[straight epsilon].

Second, we show that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is bounded. Otherwise, there exists an _n3 >_n2 , such that y(_n3 )≥[straight epsilon]exp {μ}. Hence, there must exist an _n4 ∈[_n2 ,_n3 -1] such that y(_n4 )<[straight epsilon] , y(_n4 +1)≥[straight epsilon] , and y(n)≥[straight epsilon] for n∈[_n4 +1,_n3 ]. Let _P1 be a nonnegative integer, such that [figure omitted; refer to PDF] It follows from (3.101) that [figure omitted; refer to PDF] which leads to a contradiction. This shows that (3.99) holds. By the arbitrariness of [straight epsilon], it immediately follows that y(n)[arrow right]0 as n[arrow right]+∞. This completes the proof of Theorem 3.7.

Acknowledgments

This work was supported by the National Sciences Foundation of China (no. 11071283) and the Sciences Foundation of Shanxi (no. 2009011005-3).

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