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Global attractivity of a discrete Lotka-Volterra competition system with infinite delays and feedback controls

Abstract

In this paper, we propose a discrete Lotka-Volterra competition system with infinite delays and feedback controls. Sufficient conditions which ensure the global attractivity of the system are obtained. An example together with its numerical simulation shows the feasibility of the main results.

1 Introduction

For the last decades, the ecological competition systems governed by differential equations of Lotka-Volterra type have been investigated extensively. Many interesting results concerned with the global existence and attractivity of periodic solution, persistence and extinction of the population, etc. have been obtained; we refer to [14] and the references therein. Already, many authors [521] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Particularly, the persistence, permanence, extinction, local and global stability and the existence of positive periodic solutions, etc., for discrete competitive systems are studied in [5, 9, 12, 13, 1518, 21]. Chen and Zhou [9] discussed the following discrete Lotka-Volterra competition system:

{ x 1 ( n + 1 ) = x 1 ( n ) exp [ r 1 ( n ) ( 1 x 1 ( n ) K 1 ( n ) μ 1 ( n ) x 2 ( n ) ) ] , x 2 ( n + 1 ) = x 2 ( n ) exp [ r 2 ( n ) ( 1 x 2 ( n ) K 2 ( n ) μ 2 ( n ) x 1 ( n ) ) ] .
(1.1)

They obtained sufficient conditions which guarantee the persistence of system (1.1). Also, for the periodic case, they obtained sufficient conditions for the existence of a globally stable periodic solution.

Chen [12] studied the following nonautonomous two-species discrete competitive systems with deviating arguments:

{ x 1 ( n + 1 ) = x 1 ( n ) exp [ r 1 ( n ) ( 1 x 1 ( n ) K 1 ( n ) μ 1 ( n ) s = n H 1 ( n s ) x 2 ( s ) ) ] , x 2 ( n + 1 ) = x 2 ( n ) exp [ r 2 ( n ) ( 1 x 2 ( n ) K 2 ( n ) μ 2 ( n ) s = n H 2 ( n s ) x 1 ( s ) ) ] .
(1.2)

They obtained sufficient conditions for the permanence of system (1.2).

On the other hand, feedback control is the basic mechanism by which systems, whether mechanical, electrical or biological, maintain their equilibrium or homeostasis. In the higher life forms, the conditions under which life can continue are quite narrow. A change in body temperature of half a degree is generally a sign of illness. The homeostasis of the body is maintained through the use of feedback control [22]. A primary contribution of C.R. Darwin during the last century was the theory that feedback over long time periods is responsible for the evolution of species. In 1931 Volterra [23] explained the balance between two populations of fish in a closed pond using the theory of feedback. Later, a series of mathematical models have been established to describe the dynamics of feedback control systems; see [14, 1620, 2325] and the references therein.

The purpose of this paper is to study the global attractivity of the following discrete Lotka-Volterra competition system with infinite delays and feedback controls:

{ x 1 ( n + 1 ) = x 1 ( n ) exp [ r 1 a 11 x 1 ( n ) a 12 s = n H 1 ( n s ) x 2 ( s ) x 1 ( n + 1 ) = c 1 s = n H 2 ( n s ) μ 1 ( s ) ] , x 2 ( n + 1 ) = x 2 ( n ) exp [ r 2 a 22 x 2 ( n ) a 21 s = n H 3 ( n s ) x 1 ( s ) x 2 ( n + 1 ) = c 2 s = n H 4 ( n s ) μ 2 ( s ) ] , μ 1 ( n + 1 ) = ( 1 a 1 ) μ 1 ( n ) + b 1 s = n H 5 ( n s ) x 1 ( s ) , μ 2 ( n + 1 ) = ( 1 a 2 ) μ 2 ( n ) + b 2 s = n H 6 ( n s ) x 2 ( s ) ,
(1.3)

where 0< a i <1, r i , b i , c i , a i j (0,), i=1,2, j=1,2, x i (n) (i=1,2) are the density of the i species at time n and μ i (n) (i=1,2) are the control variables at time n. H i (n) (i=1,2,,6) are bounded nonnegative sequences such that n = 0 H i (n)=1.

By the biological meaning, we focus our discussion on the positive solutions of (1.3). So, it is assumed that the initial conditions of (1.3) are of the form

x i (s)= Φ i (s)0, Φ i (0)>0, μ i (s)= Ψ i (s)0, Ψ i (0)>0,i=1,2,
(1.4)

where s=,n,n+1,,1,0. One can easily show that the solutions of (1.3) with (1.4) remain positive for all n Z + , where Z + ={0,1,2,}.

Further, assume

( r 1 a 21 r 2 d 1 )( r 2 a 12 r 1 d 2 )>0,
(1.5)

where d 1 = 1 a 1 ( a 1 a 11 + b 1 c 1 ), d 2 = 1 a 2 ( a 2 a 22 + b 2 c 2 ). Then system (1.3) has a unique positive equilibrium ( x 1 , x 2 , μ 1 , μ 2 ) with

x 1 = r 2 a 12 r 1 d 2 a 12 a 21 d 1 d 2 , x 2 = r 1 a 21 r 2 d 1 a 12 a 21 d 1 d 2 , μ 1 = a 1 b 1 x 1 , μ 2 = a 2 b 2 x 2 .

The aim of this paper is, by developing the analysis technique of Chen [12], Liao and Yu [14], Chen and Teng [15], to obtain a set of sufficient conditions for the global attractivity of system (1.3). The paper is organized as follows. In Section 2, as preliminaries, some useful lemmas are given. In Section 3, we study the global attractivity of positive equilibrium of system (1.3). In Section 4, the numerical simulations on the global attractivity of equilibrium are given.

2 Preliminaries

In this section, we introduce some auxiliary lemmas which will be useful in the following.

Lemma 1 (see [15])

Let the function f(u)=uexp(αβu), where α and β are positive constants. Then f(u) is nondecreasing on u(0, 1 β ].

Lemma 2 (see [15])

Assume that the sequence {u(n)} satisfies

u(n+1)=u(n)exp ( α β u ( n ) ) ,n=1,2,,

where α and β are positive constants and u(0)>0. We have

  1. (i)

    if α<2, then lim n u(n)= α β .

  2. (ii)

    if α1, then u(n) 1 β for all n=2,3, .

Lemma 3 (see [8])

Suppose that functions f,g: Z + ×[0,)[0,) satisfy f(n,x)g(n,x) (f(n,x)g(n,x)) for n Z + and x[0,) and g(n,x) is nondecreasing with respect to x>0. If sequences {x(n)} and {u(n)} are the nonnegative solutions of the following difference equations:

x(n+1)=f ( n , x ( n ) ) ,u(n+1)=g ( n , u ( n ) ) ,n=0,1,2,,

respectively, and x(0)u(0) (x(0)u(0)), then for all n0, we have

x(n)u(n) ( x ( n ) u ( n ) ) .

Lemma 4 (see [12])

Let x:ZR be a nonnegative bounded sequence, and let H: Z + R be a nonnegative sequence such that n = 0 H(n)=1, where Z={0,±1,±2,}, R=(,). Then

lim inf n + x(n) lim inf n + s = n H(ns)x(s) lim sup n + s = n H(ns)x(s) lim sup n + x(n).

We further consider the following discrete linear equation:

u(n+1)=(1 γ 1 )u(n)+ γ 2 s = n H(ns)x(s),n Z + ,
(2.1)

where 0< γ 1 <1, γ 2 (0,). H(n) is a nonnegative sequence defined on Z + such that n = 0 H(n)=1 and x(n) is a nonnegative bounded sequence defined on Z with

x lim inf n x(n) lim sup n x(n) x ,

where x , x are nonnegative constants.

Lemma 5 Any solution of system (2.1) with u(0)>0 satisfies

γ 2 γ 1 x lim inf n u(n) lim sup n u(n) γ 2 γ 1 x .

Proof From Lemma 4,

lim sup n + s = n H(ns)x(s) lim sup n + x(n) x .

Hence, for each ε>0, there exists an enough large integer n 0 such that for n n 0 ,

s = n H(ns)x(n) x +ε.

By system (2.1), we can obtain

u ( n ) = u ( n 0 ) ( 1 γ 1 ) n n 0 + γ 2 i = n 0 n 1 [ ( 1 γ 1 ) n i 1 s = i H ( i s ) x ( s ) ] u 0 ( 1 γ 1 ) n n 0 + γ 2 ( x + ε ) i = n 0 n 1 ( 1 γ 1 ) n i 1 = u 0 ( 1 γ 1 ) n n 0 + γ 2 ( x + ε ) 1 ( 1 γ 1 ) n n 0 γ 1 γ 2 ( x + ε ) γ 1 , n .

Thus,

lim sup n u(n) γ 2 ( x + ε ) γ 1 .

By the arbitrariness of ε, we can obtain

lim sup n u(n) γ 2 γ 1 x .

We can prove lim inf n u(n) γ 2 γ 1 x in a similar way. Thus, we complete the proof. □

Lemma 6 Assume lim n x(n)= x ¯ . For every solution u(n) of equation (2.1), we have

lim n u(n)= γ 2 γ 1 x ¯ .

By Lemma 5, the proof of Lemma 6 is obtained easily. Hence, we omit it here.

3 Global attractivity

In this section, we derive sufficient conditions which guarantee that the positive equilibrium of system (1.3) is globally attractive. The technique of proofs is to use an iteration scheme.

Theorem 1 Assume

( r 1 a 21 r 2 d 1 )( r 2 a 12 r 1 d 2 )>0

and

r 2 a 12 a 22 + b 1 c 1 r 1 a 1 a 11 < r 1 1, r 1 a 21 a 11 + b 2 c 2 r 2 a 2 a 22 < r 2 1.

Then equilibrium ( x 1 , x 2 , μ 1 , μ 2 ) of system (1.3) with (1.4) is globally attractive.

Proof Let ( x 1 (n), x 2 (n), μ 1 (n), μ 2 (n)) be any solution of system (1.3) with (1.4). Denote

U i = lim sup n x i (n), V i = lim inf n x i (n),i=1,2,

and

P i = lim sup n μ i (n), Q i = lim inf n μ i (n),i=1,2.

We now claim that U i = V i = x i , P i = Q i = μ i , i=1,2.

From the first equation of system (1.3), we obtain

x 1 (n+1) x 1 (n)exp [ r 1 a 11 x 1 ( n ) ] ,n=0,1,2,.

Consider the auxiliary equation

p(n+1)=p(n)exp [ r 1 a 11 p ( n ) ] .
(3.1)

From r 1 1, by the conclusion (ii) of Lemma 2, we have that p(n) 1 a 11 for all n2, where p(n) is any solution of equation (3.1) with initial value p(0)>0. From Lemma 1, we have f(p)=pexp( r 1 a 11 p) is nondecreasing for p(0, 1 a 11 ].

Hence, from Lemma 3, we obtain x 1 (n)p(n) for all n2, where p(n) is the solution of equation (3.1) with p(2)= x 1 (2). Further, combining it with the conclusion (i) of Lemma 2, we obtain

U 1 = lim sup n x 1 (n) lim n p(n)= r 1 a 11 := M 1 1 .

From the second equation of system (1.3), we obtain

x 2 (n+1) x 2 (n)exp [ r 2 a 22 x 2 ( n ) ] ,n=0,1,2,.

By a similar argument as that above, we have

U 2 = lim sup n x 2 (n) r 2 a 22 := M 1 2 .

By Lemma 4 and Lemma 5, we obtain

P i = lim sup n μ i (n) b i a i M 1 i ,i=1,2.

Then, for any constant ε>0 sufficiently small, there is an integer n 1 >2 such that if n n 1 , then

x i (n) M 1 i +ε, μ i (n) b i a i M 1 i +ε,i=1,2.

Further, from Lemma 4 and the first equation of system (1.3), we have

x 1 (n+1) x 1 (n)exp [ r 1 a 11 x 1 ( n ) a 12 ( M 1 2 + ε ) c 1 ( b 1 a 1 M 1 1 + ε ) ] ,n n 1 .

Consider the auxiliary equation

p(n+1)=p(n)exp [ r 1 a 11 p ( n ) a 12 ( M 1 2 + ε ) c 1 ( b 1 a 1 M 1 1 + ε ) ] ,n n 1 .
(3.2)

From r 2 a 12 a 22 + b 1 c 1 r 1 a 1 a 11 < r 1 1 and the arbitrariness of ε>0, we have

0< r 1 a 12 ( M 1 2 + ε ) c 1 ( b 1 a 1 M 1 1 + ε ) <1.

By the conclusion (ii) of Lemma 2, we have that p(n) 1 a 11 for all n n 1 , where p(n) is any solution of equation (3.2) with initial value p( n 1 )>0. From Lemma 1, we have

f(p)=exp [ r 1 a 11 p a 12 ( M 1 2 + ε ) c 1 ( b 1 a 1 M 1 1 + ε ) ]

is nondecreasing for p(0, 1 a 11 ].

Hence, from Lemma 3, we have x 1 (n)p(n) for all n n 1 , where p(n) is the solution of equation (3.2) with p( n 1 )= x 1 ( n 1 ). Combining it with the conclusion (i) of Lemma 2, we obtain

V 1 = lim inf n x 1 (n) lim n p(n)= 1 a 11 [ r 1 a 12 ( M 1 2 + ε ) c 1 ( b 1 a 1 M 1 1 + ε ) ] .

From the arbitrariness of ε>0, we conclude V 1 m 1 1 , where

m 1 1 = 1 a 11 [ r 1 a 12 M 1 2 c 1 b 1 a 1 M 1 1 ] .

From Lemma 4 and the second equation of system (1.3), we further have

x 2 (n+1) x 2 (n)exp [ r 2 a 22 x 2 ( n ) a 21 ( M 1 1 + ε ) c 2 ( b 2 a 2 M 1 2 + ε ) ] ,n n 1 .

By a similar argument as that above, we can obtain

V 2 = lim inf n x 2 (n) 1 a 22 [ r 2 a 21 M 1 2 c 1 b 1 a 1 M 1 1 ] := m 1 2 .

By Lemma 4 and Lemma 5, we further obtain

Q i = lim inf n μ i (n) b i a i m 1 i ,i=1,2.

Hence, for ε>0 sufficiently small, there is an n 2 > n 1 such that if n n 2 , then

x i (n) m 1 i ε, μ i (n) b i a i m 1 i ε,i=1,2.

From Lemma 4 and the first equation of system (1.3), we further have

x 1 (n+1) x 1 (n)exp [ r 1 a 11 x 1 ( n ) a 12 ( m 1 2 ε ) c 1 ( b 1 a 1 m 1 1 ε ) ] ,n n 2 .

Consider the auxiliary equation

p(n+1)p(n)exp [ r 1 a 11 p ( n ) a 12 ( m 1 2 ε ) c 1 ( b 1 a 1 m 1 1 ε ) ] ,n n 2 .

From r 2 a 12 a 22 + b 1 c 1 r 1 a 1 a 11 < r 1 1 and the arbitrariness of ε>0, we have

0< r 1 a 12 ( m 1 2 ε ) c 1 ( b 1 a 1 m 1 1 ε ) <1.

Similarly to the above discussion, we can obtain

U 1 = lim inf n x 1 (n) lim n p(n)= 1 a 11 [ r 1 a 12 ( m 1 2 ε ) c 1 ( b 1 a 1 m 1 1 ε ) ] .

From the arbitrariness of ε>0, we conclude U 1 M 2 1 , where

M 2 1 = 1 a 11 [ r 1 a 12 m 1 2 c 1 b 1 a 1 m 1 1 ] .

From Lemma 4 and the second equation of system (1.3), we further have

x 2 (n+1) x 2 (n)exp [ r 2 a 22 x 2 ( n ) a 21 ( m 1 1 ε ) c 2 ( b 2 a 2 m 1 2 ε ) ] ,n n 2 .

By a similar argument as that above, we can obtain

U 2 = lim inf n x 2 (n) 1 a 22 [ r 2 a 21 m 1 1 c 2 b 2 a 2 m 1 2 ] := M 2 2 .

By Lemma 4 and Lemma 5, we obtain

P i = lim sup n μ i (n) b i a i M 2 i ,i=1,2.

Hence, for ε>0 sufficiently small, there is an n 3 such that if n n 3 ,

x i (n) M 2 i +ε, μ i (n) b i a i M 2 i +ε,i=1,2.

From Lemma 4 and the first equation of system (1.3), we have

x 1 (n+1) x 1 (n)exp [ r 1 a 11 x 1 ( n ) a 12 ( M 2 2 + ε ) c 1 ( b 1 a 1 M 2 1 + ε ) ] ,n n 3 .

Consider the auxiliary equation

p(n+1)=p(n)exp [ r 1 a 11 p ( n ) a 12 ( M 2 2 + ε ) c 1 ( b 1 a 1 M 2 1 + ε ) ] ,n n 3 .

Since

0< r 1 a 12 ( M 2 2 + ε ) c 1 ( b 1 a 1 M 2 1 + ε ) <1,

similarly to the above discussion, we obtain

V 1 = lim inf n x 1 (n) lim n p(n)= 1 a 11 [ r 1 a 12 ( M 2 2 + ε ) c 1 ( b 1 a 1 M 2 1 + ε ) ] .

From the arbitrariness of ε>0, we conclude V 1 m 2 1 , where

m 2 1 = 1 a 11 [ r 1 a 12 M 2 2 c 1 b 1 a 1 M 2 1 ] .

From Lemma 4 and the second equation of system (1.3), we further have

x 2 (n+1) x 2 (n)exp [ r 2 a 22 x 2 ( n ) a 21 ( M 2 1 + ε ) c 2 ( b 2 a 2 M 2 2 + ε ) ] ,n n 3 .

By a similar argument as that above, we can obtain

V 2 = lim inf n x 2 (n) 1 a 22 [ r 2 a 21 M 2 1 c 2 b 2 a 2 M 2 2 ] := m 2 2 .

Continuing the above process, we can obtain four sequences { M n i }, { m n i }, i=1,2 such that

M n + 1 1 = 1 a 11 [ r 1 a 12 m n 2 c 1 b 1 a 1 m n 1 ] , M n + 1 2 = 1 a 22 [ r 2 a 21 m n 1 c 2 b 2 a 2 m n 2 ]
(3.3)

and

m n 1 = 1 a 11 [ r 1 a 12 M n 2 c 1 b 1 a 1 M n 1 ] , m n 2 = 1 a 22 [ r 2 a 21 M n 1 c 2 b 2 a 2 M n 2 ] .
(3.4)

Clearly, we have

m n i V i U i M n i ,i=1,2.

Now, by means of the inductive method, we prove { M n i } is monotonically decreasing, { m n i } is monotonically increasing, i=1,2.

Firstly, it is clear that M 2 i M 1 i , m 2 i m 1 i , i=1,2. For n=k (k2), we assume M k i M k 1 i and m k i m k 1 i , i=1,2, then we have

M k + 1 1 = 1 a 11 [ r 1 a 12 m k 2 c 1 b 1 a 1 m k 1 ] 1 a 11 [ r 1 a 12 m k 1 2 c 1 b 1 a 1 m k 1 1 ] = M k 1 , M k + 1 2 = 1 a 22 [ r 2 a 21 m k 1 c 2 b 2 a 2 m k 2 ] 1 a 22 [ r 2 a 21 m k 1 1 c 2 b 2 a 2 m k 1 2 ] = M k 2

and

m k + 1 1 = 1 a 11 [ r 1 a 12 M k + 1 2 c 1 b 1 a 1 M k + 1 1 ] 1 a 11 [ r 1 a 12 M k 2 c 1 b 1 a 1 M k 1 ] = m k 1 , m k + 1 2 = 1 a 22 [ r 2 a 21 M k + 1 1 c 2 b 2 a 2 M k + 1 2 ] 1 a 22 [ r 2 a 21 M k 1 c 2 b 2 a 2 M k 2 ] = m k 2 .

Therefore, { M n i } is monotonically decreasing, { m n i } is monotonically increasing, i=1,2. Consequently, lim n M n i and lim n m n i both exist, i=1,2. Let

lim n M n i = x ¯ i , lim n m n i = y ¯ i ,i=1,2.

From (3.3) and (3.4), we obtain

{ a 11 x ¯ 1 + a 12 y ¯ 2 + b 1 c 1 a 1 y ¯ 1 = r 1 , a 22 x ¯ 2 + a 21 y ¯ 1 + b 2 c 2 a 2 y ¯ 2 = r 2 , a 11 y ¯ 1 + a 12 x ¯ 2 + b 1 c 1 a 1 x ¯ 1 = r 1 , a 22 y ¯ 2 + a 21 x ¯ 1 + b 2 c 2 a 2 x ¯ 2 = r 2 .
(3.5)

It is clear that ( x 1 , x 2 , μ 1 , μ 2 ) is a unique solution of equations (3.5). Therefore,

U i = V i = lim n x i (n)= x i ,i=1,2.

Further, by Lemma 6, we can obtain lim n μ i (n)= μ i , i=1,2. Thus, we complete the proof of Theorem 1. □

4 Example

The following example shows the feasibility of the main results.

Example 1 Choose r 1 =0.6, r 2 =0.4, a 12 =0.3, a 21 =0.2, a 11 =0.5, a 22 =0.4, a 1 =0.8, a 2 =1.2, b 1 =0.1, b 2 =0.2, c 1 =0.2, c 2 =0.1, H 1 (n)= H 4 (n)= e 1 e e n , H 2 (n)= H 3 (n)= e 2 1 e 2 e 2 n , H 5 (n)= H 6 (n)= e 4 1 e 4 e 4 n in system (1.3). By calculating, we have that positive equilibrium (0.8189,0.5669,0.1024,0.0945), ( r 1 a 21 r 2 d 1 )( r 2 a 12 r 1 d 2 )=0.0117, r 1 ( r 2 a 12 a 22 + b 1 c 1 r 1 a 1 a 11 )=0.27, r 2 ( r 1 a 21 a 11 + b 2 c 2 r 2 a 2 a 22 )=0.1433. Since

( r 1 a 21 r 2 d 1 ) ( r 2 a 12 r 1 d 2 ) > 0 , ( r 2 a 12 a 22 + b 1 c 1 r 1 a 1 a 11 ) < r 1 1 , ( r 1 a 21 a 11 + b 2 c 2 r 2 a 2 a 22 ) < r 2 1 ,

the conditions of Theorem 1 hold. So, equilibrium (0.8189,0.5669,0.1024,0.0945) is globally attractive.

Choose initial values ( x 1 (s), x 2 (s), μ 1 (s), μ 2 (s))=(0.9,0.7,0.2,0.1), s=,n,n+1,,1,0.

By the numerical simulation (see Figure 1), we find that the solution ( x 1 (n), x 2 (n), μ 1 (n), μ 2 (n)) turns to equilibrium (0.8189,0.5669,0.1024,0.0945) as n.

Figure 1
figure1

Dynamical behaviors of system ( 1.3 ) with r 1 =0.6 , r 2 =0.4 , a 12 =0.3 , a 21 =0.2 , a 11 =0.5 , a 22 =0.4 , a 1 =0.8 , a 2 =1.2 , b 1 =0.1 , b 2 =0.2 , c 1 =0.2 , c 2 =0.1 , H 1 (n)= H 4 (n)= e 1 e e n , H 2 (n)= H 3 (n)= e 2 1 e 2 e 2 n , H 5 (n)= H 6 (n)= e 4 1 e 4 e 4 n .

Author’s contributions

The author carried out the proof of the theorem and approved the final manuscript.

References

  1. 1.

    Ahmad S: On the nonautonomous Lotka-Volterra competition equation. Proc. Am. Math. Soc. 1993, 117: 199-204. 10.1090/S0002-9939-1993-1143013-3

    Article  Google Scholar 

  2. 2.

    Ahmad S, Lazer AC: Average growth and extinction in a Lotka-Volterra system. Nonlinear Anal. 2005, 62: 545-557. 10.1016/j.na.2005.03.069

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ahmad S, Lazer AC: Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system. Nonlinear Anal. 2000, 40: 37-49. 10.1016/S0362-546X(00)85003-8

    MathSciNet  Article  Google Scholar 

  4. 4.

    Tang X, Zou X: On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments. Proc. Am. Math. Soc. 2006, 134: 2967-2974. 10.1090/S0002-9939-06-08320-1

    MathSciNet  Article  Google Scholar 

  5. 5.

    Liao X, Zhou S, Chen Y: On permanence and global stability in a general Gilpin-Ayala competition predator-prey discrete system. Appl. Math. Comput. 2007, 190: 500-509. 10.1016/j.amc.2007.01.041

    MathSciNet  Article  Google Scholar 

  6. 6.

    Zhou Z, Zou X: Stable periodic solutions in a discrete periodic logistic equation. Appl. Math. Lett. 2003, 16(2):165-171. 10.1016/S0893-9659(03)80027-7

    MathSciNet  Article  Google Scholar 

  7. 7.

    Liu X: A note on the existence of periodic solution in discrete predator-prey models. Appl. Math. Model. 2010, 34: 2477-2483. 10.1016/j.apm.2009.11.012

    MathSciNet  Article  Google Scholar 

  8. 8.

    Wang L, Wang M: Ordinary Difference Equation. Xinjing Univ. Press, Xinjiang; 1989.

    Google Scholar 

  9. 9.

    Chen YM, Zhou Z: Stable periodic solution of a discrete periodic Lotka-Volterra competition system. J. Math. Appl. 2003, 277(1):358-366.

    Google Scholar 

  10. 10.

    Yang XT: Uniform persistence and periodic solutions for a discrete predator-prey system with delays. J. Math. Anal. Appl. 2006, 316(1):161-177. 10.1016/j.jmaa.2005.04.036

    MathSciNet  Article  Google Scholar 

  11. 11.

    Wang K: Permanence and global asymptotical stability of a predator-prey model with mutual interference. Nonlinear Anal., Real World Appl. 2011, 12: 1062-1071. 10.1016/j.nonrwa.2010.08.028

    MathSciNet  Article  Google Scholar 

  12. 12.

    Chen FD: Permanence in a discrete Lotka-Volterra competition model with deviating arguments. Nonlinear Anal., Real World Appl. 2008, 9: 2150-2155. 10.1016/j.nonrwa.2007.07.001

    MathSciNet  Article  Google Scholar 

  13. 13.

    Xiu JB, Teng ZD, Jiang HJ: Permanence and global attractivity for discrete nonautonomous two-species Lotka-Volterra competitive system with delays and feedback controls. Period. Math. Hung. 2011, 63(1):19-45. 10.1007/s10998-011-7019-2

    Article  Google Scholar 

  14. 14.

    Liao LS, Yu JS, Wang L: Global attractivity in a logistic difference model with a feedback control. Comput. Math. Appl. 2002, 44: 1403-1411. 10.1016/S0898-1221(02)00265-1

    MathSciNet  Article  Google Scholar 

  15. 15.

    Chen GY, Teng ZD: On the stability in a discrete two-species competition system. J. Appl. Math. Comput. 2012, 38: 25-36. 10.1007/s12190-010-0460-1

    MathSciNet  Article  Google Scholar 

  16. 16.

    Kong XZ, Chen LP, Yang WS: Note on the persistent property of a discrete Lotka-Volterra competition system with delays and feedback controls. Adv. Differ. Equ. 2010., 2: Article ID 9

    Google Scholar 

  17. 17.

    Liao X, Zhou S, Chen Y: Permanence and global stability in a discrete n -species competition system with feedback controls. Nonlinear Anal., Real World Appl. 2008, 9: 1661-1671. 10.1016/j.nonrwa.2007.05.001

    MathSciNet  Article  Google Scholar 

  18. 18.

    Liao X, Ouyang Z, Zhou S: Permanence of species in nonautonomous discrete Lotka-Volterra competition system with delays and feedback controls. J. Comput. Appl. Math. 2008, 211: 1-10. 10.1016/j.cam.2006.10.084

    MathSciNet  Article  Google Scholar 

  19. 19.

    Li YK, Zhang TW: Permanence and almost periodic sequence solution for a discrete delay logistic equation with feedback control. Nonlinear Anal., Real World Appl. 2011, 12: 1850-1864. 10.1016/j.nonrwa.2010.12.001

    MathSciNet  Article  Google Scholar 

  20. 20.

    Li QY, Liu HW, Zhang FQ: The permanence and extinction of a discrete predator-prey system with time delay and feedback controls. Adv. Differ. Equ. 2010., 2: Article ID 20

    Google Scholar 

  21. 21.

    Yu ZX, Li Z: Permanence and global attractivity of a discrete two-prey one-predator model with infinite delay. Discrete Dyn. Nat. Soc. 2009. doi:10.1155/2009/732510

    Google Scholar 

  22. 22.

    Wiener N: Cybernetics or Control and Communication in the Animal and the Machine. MIT Press, Cambridge; 1948.

    Google Scholar 

  23. 23.

    Gopalsamy K: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic, Boston; 1992.

    Google Scholar 

  24. 24.

    Gopalsamy K, Weng PX: Feedback regulation of logistic growth. Int. J. Math. Math. Sci. 1993, 16: 177-192. 10.1155/S0161171293000213

    MathSciNet  Article  Google Scholar 

  25. 25.

    Liao LS: Feedback regulation of a logistic growth variable coefficients. J. Math. Appl. 2001, 259(2):489-500.

    Google Scholar 

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Acknowledgements

The author would like to thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Scientific Research Programmes of Colleges in Anhui (KJ2011A197).

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Wu, D. Global attractivity of a discrete Lotka-Volterra competition system with infinite delays and feedback controls. Adv Differ Equ 2013, 14 (2013). https://doi.org/10.1186/1687-1847-2013-14

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Keywords

  • competition system
  • feedback control
  • discrete
  • global attractivity
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