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 Yang1, 2
1
2
Key Lab of Network Security and Cryptology, Fujian Normal University, Fuzhou 350007, China
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China
Correspondence should be addressed to Xiangzeng Kong, xzkong@fjnu.edu.cn
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:
⎡
ui t
⎤
N
bi ui t⎣1 − aij uj t⎦,
i 1, . . . , N,
1.1
j1
where bi > 0, aii > 0, aij ≥ 0 i /
j, ui 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
⎡
⎤
N
ui t ui t⎣bi t − cij tuj t⎦,
j1
cij ≥ 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 ki < i such that
b
bi
< ki ,
cij
c ki j
j 1, . . . , i,
1.3
then ui → 0 exponentially for 2 ≤ i ≤ N, and ui 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:
⎧
⎨
xi n 1 xi n exp bi n −
⎩
N
j1
⎫
⎬
aij nxj n − τij − di nui n ,
⎭
Δui n ri n − ei nui n ci nxi n − σi ,
xi θ φi θ ≥ 0,
i 1, 2, . . . , N,
1.4
θ ∈ N−τ, 0 : {−τ, −τ 1, . . . , −1, 0},
where xi n i 1, 2, . . . , N is the density of competitive species; ui n is the control variable;
ei n : Z → 0, 1; bounded sequences ri n, ci n, bi n, aij n, and di n : Z → R ; τij
and σi 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 Δui n ui n 1 − ui n; τ max{max1≤i,j≤N τij , max1≤i≤N σi } > 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 Mi Δi > 1
1≤i≤N
1.5
hold, then system 1.4 is permanent, where
N
u
u
l
auii exp τii
j1 aij Mj Wi di − bi
exp biu − 1
,
Mi Δi l
·
u
u
aii exp −biu τii
bil − N
a
M
−
d
W
j
i
j1,j i
ij
i
/
Wi riu
ciu Mi
,
eil
exp biu − 1
Mi l
.
aii exp −biu τii
1.6
Advances in Difference Equations
3
Since
exp biu − 1 > 0,
⎧ ⎛
⎞⎫
N
⎬
⎨
auii exp τii ⎝ auij Mj Wi diu − bil ⎠ > 0.
⎭
⎩
j1
alii exp −biu τii > 0,
1.7
Hence, the above inequality 1.5 implies
bil −
N
j1,j /i
auij Mj − diu Wi > 0.
1.8
That is
bil >
N
j1,j /
i
N
j1,j /
i
N
j1,j /
i
auij Mj diu Wi
auij Mj diu
riu ciu Mi
diu riu
auij Mj eil
1.9
eil
diu ciu Mi
eil
.
It was shown that in [1] Liao et al. considered system 1.4 where all coefficients ri n, ci n, di n,
aij n, ei n, and bi 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
bil >
N
j1,j /i
auij Mj diu riu
eil
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
auij Mj j1,j /
i
diu riu
eil
≤
N
auij Mj j1,j /
i
diu riu
eil
diu ciu Mi
eil
.
1.11
Therefore, we have improved the permanence conditions of 1 for system 1.4.
Theorem 1.2 will be proved in Section 2. In Section 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)
yn 1 ≤ Ayn Bn,
n 1, 2, . . . .
2.1
Ai Bn − i − 1.
2.2
Then for any integer k ≤ n,
yn ≤ Ak yn − k k−1
i0
Especially, if A < 1 and B is bounded above with respect to M, then
M
.
1−A
lim sup yn ≤
n→∞
2.3
2
yn 1 ≥ Ayn Bn,
n 1, 2, . . . .
2.4
Ai Bn − i − 1.
2.5
Then for any integer k ≤ n,
yn ≥ Ak yn − k k−1
i0
Especially, if A < 1 and B is bounded below with respect to m∗ , then
lim inf yn ≥
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 ∈ Nn0 {n0 , n0 1, . . . , n0 l, . . .}, r ≥ 0. For any fixed n, gn, r is a
nondecreasing function with respect to r, and for n ≥ n0 , following inequalities hold: yn 1 ≤
gn, yn, un 1 ≥ gn, un. If gn 0 ≤ un0 , then yn ≤ un for all n ≥ n0 .
Now let us consider the following single species discrete model:
Nn 1 Nn exp{an − bnNn},
2.7
where {an} and {bn} are strictly positive sequences of real numbers defined for n ∈ N {0, 1, 2, . . .} and 0 < al ≤ au , 0 < bl ≤ bu . 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 inf Nn ≤ lim sup Nn ≤ M,
n→∞
n→∞
2.8
where
M
1
exp{au − 1},
bl
m
al
l
u
exp
a
−
b
M
.
bu
2.9
The following lemma is direct conclusion of 1.
Lemma 2.4. Let xn x1 n, x2 n, . . . , xN n, u1 n, u2 n, . . . , uN n denote any positive
solution of system 1.4.Then there exist positive constants Mi , Wi i 1, 2, . . . , N such that
lim sup xi n ≤ Mi ,
n→∞
lim sup ui n ≤ Wi ,
n→∞
i 1, 2, . . . , N,
2.10
i 1, 2, . . . , N.
2.11
where
exp biu − 1
Mi l
,
aii exp −biu τii
Wi riu ciu Mi
eil
Proposition 2.5. Suppose assumption 1.10 holds, then there exist positive constant mi and wi such
that
lim inf xi n ≥ mi ,
n→∞
lim inf ui n ≥ wi .
n→∞
2.12
Proof. We first prove lim inf xi n ≥ mi .
n→∞
By Lemma 2.4 and by the first equation of system 1.4, we have
⎧
⎫
N
⎨
⎬
xi n 1 xi n exp bi n − aij nxj n − τij − di nui n
⎩
⎭
j1
⎫
⎧
N
⎬
⎨
≥ xi n exp bi n − a ij Mj ε − di nWi ε
⎭
⎩
j1
2.13
for n sufficiently large, then
⎧
⎞⎫
⎛
n−1
n−1
N
⎬
⎨
xi s 1
⎝bi s − aij s Mj ε − di sWi ε⎠ .
≥ exp
⎭
⎩sn−τ
sn−τii xi s
j1
ii
2.14
6
Advances in Difference Equations
Thus
xi n − τii ≤ xi n exp
n−1
Di s ,
2.15
sn−τii
where
Di s N
aij s Mj ε di sWi ε − bi s.
2.16
j1
From the second equation of system 1.4, we have
ui n 1 − ei nui n ci nxi n − σi ri n
≤ 1 − eil ui n ci nxi n − σi ri n
2.17
: Ai ui n Bi n.
Then, Lemma 2.1 implies that for any k ≤ n − τii ,
ui n ≤ Aki ui n − k k−1
j Ai B i n − j − 1
j0
Aki ui n − k k−1
j Ai ri n − j − 1 ci n − j − 1 xi n − j − 1 − σi
j0
≤ Aki ui n − k k−1
j Ai ri n − j − 1 ciu exp j 1 σi Diu xi n
2.18
j0
≤ Aki ui n − k k−1
k−1
j
j
Ai riu Ai ciu ciu exp j 1 σi Diu xi n
j0
≤ Aki Wi j0
1 − Aki u
r Hi xi n,
1 − Ai i
where
⎤u
⎡
k−1
j u u
u
Hi ⎣ Ai ci ci exp j 1 σi Di ⎦ .
2.19
j0
For any small positive constant ε > 0, there exists a K > 0 such that
diu Wi −
riu diu
Aki < ε
1 − Ai
∀k > K.
2.20
Advances in Difference Equations
7
From the first equation of system 1.4, 2.18, and 2.20, we have
xi n 1
⎧
⎨
N
≥ xi n exp bi n −
aij nMj − auii exp τii Diu xi n
⎩
j1,j i
/
⎫
k
⎬
1
−
A
i u u
−diu Wi Aki −
ri di − diu Hi xi n
⎭
1 − Ai
⎧
⎨
N
r u du
r u du
aij nMj − i i − diu Wi − i i Aki
xi n exp bi n −
⎩
1 − Ai
1 − Ai
j1,j /
i
2.21
⎫
⎬
u
− aii exp τii Diu diu Hi xi n
⎭
⎧
⎨
⎫
⎬
riu diu
u
≥ xi n exp bi n −
aij nMj −
− ε − aii exp τii Diu diu Hi xi n .
⎩
⎭
1 − Ai
j1,j /i
N
By Lemmas 2.2 and 2.3, we have
u
u u
l
a
M
−
r
d
/e
j
j1,j /
i ij
i i
i −ε
lim inf xi n ≥
n→∞
auii exp τii Diu diu Hi
⎧
⎫
u u
N
⎨
⎬
d
r
i
i
· exp bil −
auij Mj − l − ε − auii exp τii Diu diu Hi Mi .
⎩
⎭
e
j1,j i
bil −
N
/
2.22
i
Setting ε → 0 in 2.22 leads to
u
u u
l
a
M
−
r
d
/e
j
j1,j /
i ij
i i
i
lim inf xi n ≥
n→∞
auii exp τii Diu diu Hi
⎫
⎧
u u
N
⎨
d
r
⎬
· exp bil −
auij Mj − i l i − auii exp τii Diu diu Hi Mi .
⎭
⎩
e
j1,j i
bil −
N
/
2.23
i
Thus,
lim inf xi n ≥ mi ,
n→∞
2.24
8
Advances in Difference Equations
where
auij Mj − riu diu /eil
mi auii exp τii Diu diu Hi
⎧
⎫
u u
N
⎨
⎬
d
r
· exp bil −
auij Mj − i l i − auii exp τii Diu diu Hi Mi .
⎩
⎭
e
j1,j i
bil −
N
j1,j /
i
/
2.25
i
Second, we prove limn → ∞ inf ui n ≥ wi . For enough small ε > 0, from the second equation of
system 1.4, we have
ui n 1 1 − ei nui n ri n ci nxi n − σi ≥ ril cil mi − ε 1 − eiu ui n
2.26
for sufficient large n. Hence
ui n ≥ 1 −
n
eiu ui 0
1 − 1 − eiu l
l
.
r
c
−
ε
m
i
i
i
eiu
2.27
Thus, we obtain
2.28
lim inf ui n ≥ wi .
n→∞
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 ∈ −∞, ∞
!
"
1
1
1
− 2x1 n − 1 − x2 n − 3 − u1 n ,
2
2
2
!
"
1 1
1
− x1 n − 3 − 2x2 n − 1 − u2 n ,
x2 n 1 x2 n exp
2 2
2
x1 n 1 x1 n exp
3.1
1 1
Δu1 n 1 − u1 n x1 n − 4,
8 2
Δu2 n 1 1 1
− u2 n x2 n − 8.
8 2
We have
b1l b2l 1
,
2
M 1 M2 1
,
2
au12 M2 d1u
r1u
e1l
3
,
8
au21 M1 d2u
r2u
e2l
3
.
8
3.2
Advances in Difference Equations
9
So
b1l > au12 M2 d1u
r1u
e1l
,
b2l > au21 M1 d2u
r2u
e2l
.
3.3
Therefore 1.10 holds.
But
r u c1u M1 7
1
b1l < au12 M2 d1u 1
,
2
8
e1l
r u c2u M2 7
1
b2l < au21 M1 d2u 2
.
2
8
e2l
3.4
Thus 1.9 does not hold.
References
1 X. Liao, Z. Ouyang, and S. Zhou, “Permanence of species in nonautonomous discrete LotkaVolterra competitive system with delays and feedback controls,” Journal of Computational and Applied
Mathematics, vol. 211, no. 1, pp. 1–10, 2008.
2 S. Ahmad, “On the nonautonomous Volterra-Lotka competition equations,” Proceedings of the
American Mathematical Society, vol. 117, no. 1, pp. 199–204, 1993.
3 S. Ahmad and A. C. Lazer, “On the nonautonomous N-competing species problems,” Applicable
Analysis, vol. 57, no. 3-4, pp. 309–323, 1995.
4 K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of
Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
5 M. Kaykobad, “Positive solutions of positive linear systems,” Linear Algebra and Its Applications, vol.
64, pp. 133–140, 1985.
6 F. Montes de Oca and M. L. Zeeman, “Extinction in nonautonomous competitive Lotka-Volterra
systems,” Proceedings of the American Mathematical Society, vol. 124, no. 12, pp. 3677–3687, 1996.
7 M. L. Zeeman, “Extinction in competitive Lotka-Volterra systems,” Proceedings of the American
Mathematical Society, vol. 123, no. 1, pp. 87–96, 1995.
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.
9 S. Ahmad and I. M. Stamova, “Almost necessary and sufficient conditions for survival of species,”
Nonlinear Analysis. Real World Applications, vol. 5, no. 1, pp. 219–229, 2004.
10 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.
11 L. Wang and M. Q. Wang, Ordinary Difference Equation, Xinjiang University Press, 1991.
12 F. Chen, “Permanence and global attractivity of a discrete multispecies Lotka-Volterra competition
predator-prey systems,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 3–12, 2006.