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Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 249364, 9 pages
doi:10.1155/2010/249364
Research Article
Note on the Persistent Property of a Discrete
Lotka-Volterra Competitive System with Delays
and Feedback Controls
Xiangzeng Kong,
1, 2
Liping Chen,
1, 2
and Wensheng Yang
1, 2
1
Key Lab of Network Security and Cryptology, Fujian Normal University, Fuzhou 350007, China
2
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China
Correspondence should be addressed to Xiangzeng Kong,
Received 26 June 2010; Accepted 12 September 2010
Academic Editor: P. J. Y. Wong
Copyright q 2010 Xiangzeng Kong 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 nonautonomous N-species discrete Lotka-Volterra competitive system with delays and feedback
controls is considered in this work. Sufficient conditions on the coefficients are given to guarantee
that all the species are permanent. It is shown that these conditions are weaker than those of Liao
et al. 2008.
1. Introduction
Traditional Lotka-Volterra competitive systems have been extensively studied by many
authors 1–7.The autonomous model can be expressed as follows:
u
i
t
b
i
u
i
t
⎡
⎣
1 −
N
j1
a
ij
u
j
t
⎤
⎦
,i 1, ,N, 1.1
where b
i
> 0, a
ii
> 0, a
ij
≥ 0 i
/
j, u
i
tdenoting the density of the ith species at time t.Montes
de Oca and Zeeman 6 investigated the general nonautonomous N-species Lotka-Volterra
competitive system
u
i
t
u
i
t
⎡
⎣
b
i
t
−
N
j1
c
ij
t
u
j
t
⎤
⎦
,c
ij
≥ 0,i 1, ,N, 1.2
2 Advances in Difference Equations
and obtained that if the coefficients are continuous and bounded above and below by positive
constants, and if for each i 2, ,N,there exists an integer k
i
<isuch that
b
i
c
ij
<
b
ki
c
k
i
j
,j 1, ,i, 1.3
then u
i
→ 0 exponentially for 2 ≤ i ≤ N, and u
i
t → X
∗
, where X
∗
is a certain solution of
a logistic equation. Teng 8 and Ahmad and Stamova 9 also studied the coexistence on a
nonautonomous Lotka-Volterra competitive system. They obtained the necessary or sufficient
conditions for the permanence and the extinction. For more works relevant to system 1.1,
one could refer to 1–9 and the references cited therein.
However, to the best of the authors’ knowledge, to this day, still less scholars
consider the general nonautonomous discrete Lotka-Volterra competitive system with
delays and feedback controls. Recently, in 1 Liao et al. considered the following general
nonautonomous discrete Lotka-Volterra competitive system with delays and feedback
controls:
x
i
n 1
x
i
n
exp
⎧
⎨
⎩
b
i
n
−
N
j1
a
ij
n
x
j
n − τ
ij
− d
i
n
u
i
n
⎫
⎬
⎭
,
Δu
i
n
r
i
n
− e
i
n
u
i
n
c
i
n
x
i
n − σ
i
,i 1, 2, ,N,
x
i
θ
φ
i
θ
≥ 0,θ∈ N
−τ,0
:
{
−τ,−τ 1, ,−1, 0
}
,
1.4
where x
i
ni 1, 2, ,N is the density of competitive species; u
i
n is the control variable;
e
i
n : Z → 0, 1; bounded sequences r
i
n, c
i
n, b
i
n, a
ij
n,andd
i
n : Z → R
; τ
ij
and σ
i
are positive integer; Z, R
denote the sets of all integers and all positive real numbers,
respectively; Δ is the first-order forward difference operator Δu
i
nu
i
n 1 − u
i
n; τ
max{max
1≤i,j≤N
τ
ij
, max
1≤i≤N
σ
i
} > 0.
In 1, Liao et al. obtained sufficient conditions for permanence of the system 1.4.
They obtained what follows.
Lemma 1.1. Assume that
min
1≤i≤N
M
i
Δ
i
> 1 1.5
hold, then system 1.4 is permanent, where
M
i
Δ
i
exp
b
u
i
− 1
a
l
ii
exp
−b
u
i
τ
ii
·
a
u
ii
exp
τ
ii
N
j1
a
u
ij
M
j
W
i
d
u
i
− b
l
i
b
l
i
−
N
j1,j
/
i
a
u
ij
M
j
− d
u
i
W
i
,
W
i
r
u
i
c
u
i
M
i
e
l
i
,M
i
exp
b
u
i
− 1
a
l
ii
exp
−b
u
i
τ
ii
.
1.6
Advances in Difference Equations 3
Since
exp
b
u
i
− 1
> 0,a
l
ii
exp
−b
u
i
τ
ii
> 0,a
u
ii
exp
⎧
⎨
⎩
τ
ii
⎛
⎝
N
j1
a
u
ij
M
j
W
i
d
u
i
− b
l
i
⎞
⎠
⎫
⎬
⎭
> 0.
1.7
Hence, the above inequality 1.5 implies
b
l
i
−
N
j1,j
/
i
a
u
ij
M
j
− d
u
i
W
i
> 0. 1.8
That is
b
l
i
>
N
j1,j
/
i
a
u
ij
M
j
d
u
i
W
i
N
j1,j
/
i
a
u
ij
M
j
d
u
i
r
u
i
c
u
i
M
i
e
l
i
N
j1,j
/
i
a
u
ij
M
j
d
u
i
r
u
i
e
l
i
d
u
i
c
u
i
M
i
e
l
i
.
1.9
It was shown that in [1] Liao et al. considered system 1.4 where all coefficients r
i
n, c
i
n, d
i
n,
a
ij
n, e
i
n, and b
i
n were assumed to satisfy conditions 1.9.
In this work, we shall study system 1.4 and get the same results as 1 do under the
weaker assumption that
b
l
i
>
N
j1,j
/
i
a
u
ij
M
j
d
u
i
r
u
i
e
l
i
. 1.10
Our main results are the following Theorem 1.2.
Theorem 1.2. Assume that 1.10 holds, then system 1.4 is permanent.
Remark 1.3. The inequality 1.9 implies 1.10, but not conversely, for
N
j1,j
/
i
a
u
ij
M
j
d
u
i
r
u
i
e
l
i
≤
N
j1,j
/
i
a
u
ij
M
j
d
u
i
r
u
i
e
l
i
d
u
i
c
u
i
M
i
e
l
i
. 1.11
Therefore, we have improved the permanence conditions of 1 for system 1.4.
Theorem 1.2 will be proved in Section 2.InSection 3, an example will be given to
illustrate that 1.10 does not imply 1.9; that is, the condition 1.10 is better than 1.9.
4 Advances in Difference Equations
2. Proof of Theorem 1.2
The following lemma can be found in 10.
Lemma 2.1. Assume that A>0 and y0 > 0, and further suppose that
(1)
y
n 1
≤ Ay
n
B
n
,n 1, 2, 2.1
Then for any integer k ≤ n,
y
n
≤ A
k
y
n − k
k−1
i0
A
i
B
n − i − 1
. 2.2
Especially, if A<1 and B is bounded above with respect to M,then
lim
n →∞
sup y
n
≤
M
1 − A
. 2.3
2
y
n 1
≥ Ay
n
B
n
,n 1, 2, 2.4
Then for any integer k ≤ n,
y
n
≥ A
k
y
n − k
k−1
i0
A
i
B
n − i − 1
. 2.5
Especially, if A<1 and B is bounded below with respect to m
∗
,then
lim
n →∞
inf y
n
≥
m
∗
1 − A
. 2.6
Following comparison theorem of difference equation is Theorem 2.1 of [11 , page 241].
Lemma 2.2. Let n ∈ N
n
0
{n
0
,n
0
1, ,n
0
l, }, r ≥ 0. For any fixed n, gn, r is a
nondecreasing function with respect to r, and for n ≥ n
0
, following inequalities hold: yn 1 ≤
gn, yn, un 1 ≥ gn, un. If gn
0
≤ un
0
,thenyn ≤ un for all n ≥ n
0
.
Now let us consider the following single species discrete model:
N
n 1
N
n
exp
{
a
n
− b
n
N
n
}
, 2.7
where {an} and {bn} are strictly positive sequences of real numbers defined for n ∈ N
{0, 1, 2, } and 0 <a
l
≤ a
u
,0<b
l
≤ b
u
. Similarly to the proof of Propositions 1 and 3 in 12,
we can obtain the following.
Advances in Difference Equations 5
Lemma 2.3. Any solution of system 2.7 with initial condition N0 > 0 satisfies
m ≤ lim
n →∞
inf N
n
≤ lim
n →∞
sup N
n
≤ M, 2.8
where
M
1
b
l
exp
{
a
u
− 1
}
,m
a
l
b
u
exp
a
l
− b
u
M
. 2.9
The following lemma is direct conclusion of 1.
Lemma 2.4. Let xnx
1
n,x
2
n, ,x
N
n,u
1
n,u
2
n, ,u
N
n denote any positive
solution of system 1.4.Then there exist positive constants M
i
,W
i
i 1, 2, ,N such that
lim
n →∞
sup x
i
n
≤ M
i
, lim
n →∞
sup u
i
n
≤ W
i
,i 1, 2, ,N, 2.10
where
M
i
exp
b
u
i
− 1
a
l
ii
exp
−b
u
i
τ
ii
,W
i
r
u
i
c
u
i
M
i
e
l
i
i 1, 2, ,N
. 2.11
Proposition 2.5. Suppose assumption 1.10 holds, then there exist positive constant m
i
and w
i
such
that
lim
n →∞
inf x
i
n
≥ m
i
, lim
n →∞
inf u
i
n
≥ w
i
. 2.12
Proof. We first prove lim
n →∞
inf x
i
n ≥ m
i
.
By Lemma 2.4 and by the first equation of system 1.4, we have
x
i
n 1
x
i
n
exp
⎧
⎨
⎩
b
i
n
−
N
j1
a
ij
n
x
j
n − τ
ij
− d
i
n
u
i
n
⎫
⎬
⎭
≥ x
i
n
exp
⎧
⎨
⎩
b
i
n
−
N
j1
a
ij
M
j
ε
− d
i
n
W
i
ε
⎫
⎬
⎭
2.13
for n sufficiently large, then
n−1
sn−τ
ii
x
i
s 1
x
i
s
≥ exp
⎧
⎨
⎩
n−1
sn−τ
ii
⎛
⎝
b
i
s
−
N
j1
a
ij
s
M
j
ε
− d
i
s
W
i
ε
⎞
⎠
⎫
⎬
⎭
. 2.14
6 Advances in Difference Equations
Thus
x
i
n − τ
ii
≤ x
i
n
exp
n−1
sn−τ
ii
D
i
s
, 2.15
where
D
i
s
N
j1
a
ij
s
M
j
ε
d
i
s
W
i
ε
− b
i
s
. 2.16
From the second equation of system 1.4, we have
u
i
n
1 − e
i
n
u
i
n
c
i
n
x
i
n − σ
i
r
i
n
≤
1 − e
l
i
u
i
n
c
i
n
x
i
n − σ
i
r
i
n
: A
i
u
i
n
B
i
n
.
2.17
Then, Lemma 2.1 implies that for any k ≤ n − τ
ii
,
u
i
n
≤ A
k
i
u
i
n − k
k−1
j0
A
j
i
B
i
n − j − 1
A
k
i
u
i
n − k
k−1
j0
A
j
i
r
i
n − j − 1
c
i
n − j − 1
x
i
n − j − 1 − σ
i
≤ A
k
i
u
i
n − k
k−1
j0
A
j
i
r
i
n − j − 1
c
u
i
exp
j 1 σ
i
D
u
i
x
i
n
≤ A
k
i
u
i
n − k
k−1
j0
A
j
i
r
u
i
k−1
j0
A
j
i
c
u
i
c
u
i
exp
j 1 σ
i
D
u
i
x
i
n
≤ A
k
i
W
i
1 − A
k
i
1 − A
i
r
u
i
H
i
x
i
n
,
2.18
where
H
i
⎡
⎣
k−1
j0
A
j
i
c
u
i
c
u
i
exp
j 1 σ
i
D
u
i
⎤
⎦
u
. 2.19
For any small positive constant ε>0, there exists a K>0 such that
d
u
i
W
i
−
r
u
i
d
u
i
1 − A
i
A
k
i
<ε ∀k>K. 2.20
Advances in Difference Equations 7
From the first equation of system 1.4, 2.18,and2.20, we have
x
i
n 1
≥ x
i
n
exp
⎧
⎨
⎩
b
i
n
−
N
j1,j
/
i
a
ij
n
M
j
− a
u
ii
exp
τ
ii
D
u
i
x
i
n
−d
u
i
W
i
A
k
i
−
1 − A
k
i
1 − A
i
r
u
i
d
u
i
− d
u
i
H
i
x
i
n
⎫
⎬
⎭
x
i
n
exp
⎧
⎨
⎩
b
i
n
−
N
j1,j
/
i
a
ij
n
M
j
−
r
u
i
d
u
i
1 − A
i
−
d
u
i
W
i
−
r
u
i
d
u
i
1 − A
i
A
k
i
−
a
u
ii
exp
τ
ii
D
u
i
d
u
i
H
i
x
i
n
⎫
⎬
⎭
≥ x
i
n
exp
⎧
⎨
⎩
b
i
n
−
N
j1,j
/
i
a
ij
n
M
j
−
r
u
i
d
u
i
1 − A
i
− ε −
a
u
ii
exp
τ
ii
D
u
i
d
u
i
H
i
x
i
n
⎫
⎬
⎭
.
2.21
By Lemmas 2.2 and 2.3, we have
lim
n →∞
inf x
i
n
≥
b
l
i
−
N
j1,j
/
i
a
u
ij
M
j
−
r
u
i
d
u
i
/e
l
i
− ε
a
u
ii
exp
τ
ii
D
u
i
d
u
i
H
i
· exp
⎧
⎨
⎩
b
l
i
−
N
j1,j
/
i
a
u
ij
M
j
−
r
u
i
d
u
i
e
l
i
− ε −
a
u
ii
exp
τ
ii
D
u
i
d
u
i
H
i
M
i
⎫
⎬
⎭
.
2.22
Setting ε → 0in2.22 leads to
lim
n →∞
inf x
i
n
≥
b
l
i
−
N
j1,j
/
i
a
u
ij
M
j
−
r
u
i
d
u
i
/e
l
i
a
u
ii
exp
τ
ii
D
u
i
d
u
i
H
i
· exp
⎧
⎨
⎩
b
l
i
−
N
j1,j
/
i
a
u
ij
M
j
−
r
u
i
d
u
i
e
l
i
−
a
u
ii
exp
τ
ii
D
u
i
d
u
i
H
i
M
i
⎫
⎬
⎭
.
2.23
Thus,
lim
n →∞
inf x
i
n
≥ m
i
, 2.24
8 Advances in Difference Equations
where
m
i
b
l
i
−
N
j1,j
/
i
a
u
ij
M
j
−
r
u
i
d
u
i
/e
l
i
a
u
ii
exp
τ
ii
D
u
i
d
u
i
H
i
· exp
⎧
⎨
⎩
b
l
i
−
N
j1,j
/
i
a
u
ij
M
j
−
r
u
i
d
u
i
e
l
i
−
a
u
ii
exp
τ
ii
D
u
i
d
u
i
H
i
M
i
⎫
⎬
⎭
.
2.25
Second, we prove lim
n →∞
inf u
i
n ≥ w
i
. For enough small ε>0, from the second equation of
system 1.4, we have
u
i
n 1
1 − e
i
n
u
i
n
r
i
n
c
i
n
x
i
n − σ
i
≥ r
l
i
c
l
i
m
i
− ε
1 − e
u
i
u
i
n
2.26
for sufficient large n. Hence
u
i
n
≥
1 − e
u
i
n
u
i
0
1 −
1 − e
u
i
e
u
i
r
l
i
c
l
i
m
i
− ε
. 2.27
Thus, we obtain
lim
n →∞
inf u
i
n
≥ w
i
. 2.28
This completes the proof.
3. An Example
In this section, we give an example to illustrate that 1.10 does not imply 1.9. Consider the
two-species system with delays and feedback controls for t ∈ −∞, ∞
x
1
n 1
x
1
n
exp
1
2
− 2x
1
n − 1
−
1
2
x
2
n − 3
−
1
2
u
1
n
,
x
2
n 1
x
2
n
exp
1
2
−
1
2
x
1
n − 3
− 2x
2
n − 1
−
1
2
u
2
n
,
Δu
1
n 1
1
8
−
1
2
u
1
n
x
1
n − 4
,
Δu
2
n 1
1
8
−
1
2
u
2
n
x
2
n − 8
.
3.1
We have
b
l
1
b
l
2
1
2
,M
1
M
2
1
2
,a
u
12
M
2
d
u
1
r
u
1
e
l
1
3
8
,a
u
21
M
1
d
u
2
r
u
2
e
l
2
3
8
. 3.2
Advances in Difference Equations 9
So
b
l
1
>a
u
12
M
2
d
u
1
r
u
1
e
l
1
,b
l
2
>a
u
21
M
1
d
u
2
r
u
2
e
l
2
. 3.3
Therefore 1.10 holds.
But
1
2
b
l
1
<a
u
12
M
2
d
u
1
r
u
1
c
u
1
M
1
e
l
1
7
8
,
1
2
b
l
2
<a
u
21
M
1
d
u
2
r
u
2
c
u
2
M
2
e
l
2
7
8
. 3.4
Thus 1.9 does not hold.
References
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