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, [email protected]
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:
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
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
n
u
i
n
⎫
⎬
⎭
,
Δu
i
n
r
i
n
− e
i
n
u
i
−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,
i
exp
b
u
i
− 1
a
l
ii
exp
−b
u
i
τ
ii
·
a
u
ii
exp
τ
ii
− 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
τ
ii
> 0,a
u
ii
exp
⎧
⎨
⎩
τ
ii
⎛
⎝
N
j1
a
u
ij
M
j
W
i
d
u
i
− b
l
i
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
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
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
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
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
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
− 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
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
. 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
exp
⎧
⎨
⎩
b
i
n
−
N
j1
a
ij
M
j
ε
− d
i
n
W
i
ε
i
s
−
N
j1
a
ij
s
M
j
ε
− d
i
s
W
i
ε
⎞
⎠
⎫
D
i
s
N
j1
a
ij
s
M
j
ε
d
i
s
W
i
ε
− b
i
n − σ
i
r
i
n
≤
1 − e
l
i
u
i
n
c
i
n
x
i
n − σ
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
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
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
⎤
⎦
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
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
n
−
N
j1,j
/
i
a
ij
n
M
j
−
r
u
i
d
u
i
1 − A
i
−
d
u
u
i
H
i
x
i
n
⎫
⎬
⎭
≥ x
i
n
exp
⎧
⎨
⎩
b
i
n
−
N
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
ii
D
u
i
d
u
i
H
i
· exp
⎧
⎨
⎩
b
l
i
−
N
j1,j
/
i
a
u
ij
M
j
−
r
⎬
⎭
.
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
l
i
−
N
j1,j
/
i
a
u
ij
M
j
−
r
u
i
d
u
i
e
l
i
−
a
u
ii
exp
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
ij
M
j
−
r
u
i
d
u
i
e
l
i
−
a
u
ii
exp
τ
ii
D
u
i
d
u
u
i
n
r
i
n
c
i
n
x
i
n − σ
i
≥ r
l
i
c
l
i
m
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 →∞
−
1
2
x
2
n − 3
−
1
2
u
1
n
,
x
2
n 1
x
2
n
exp
1
8
−
1
2
u
1
n
x
1
n − 4
,
Δu
2
n 1
1
8
−
1
2
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
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
2
e
l
2
7
8
. 3.4
Thus 1.9 does not hold.
References
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Volterra competitive system with delays and feedback controls,” Journal of Computational and Applied
Mathematics, vol. 211, no. 1, pp. 1–10, 2008.
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3 S. Ahmad and A. C. Lazer, “On the nonautonomous N-competing species problems,” Applicable
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8 Z. D. Teng, “Permanence and extinction in nonautonomous Lotka-Volterra competitive systems with
delays,” Acta Mathematica Sinica, vol. 44, no. 2, pp. 293–306, 2001.
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10 Y H. Fan and L L. Wang, “Permanence for a discrete model with feedback control and delay,”
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