Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 872738, 18 pages
doi:10.1155/2011/872738
Research Article
Feedback Control in a Periodic Delay
Single-Species Difference System
Yi Yang1, 2 and Zhijun Liu1, 2
1
Key Laboratory of Biological Resources Protection and Utilization of Hubei Province,
Hubei University for Nationalities, Enshi, Hubei 445000, China
2
Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China
Correspondence should be addressed to Zhijun Liu, zhijun liu47@hotmail.com
Received 19 October 2010; Accepted 8 January 2011
Academic Editor: Li Xian Zhang
Copyright q 2011 Y. Yang and Z. Liu. 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 periodic delay single-species difference system with feedback control is established. With the
help of analysis method and Lyapunov function, a good understanding of the permanence and
global attractivity of the system is gained. Numerical simulations are presented to verify the
validity of the proposed criteria. Our results show that feedback control has no influence on the
permanence while it has influence on the global attractivity of the system.
1. Introduction
In 1978, Ludwig et al. 1 considered a single-species system which is modeled by
x t xta − bxt − hx,
1.1
where xt is the density of species x at time t, a is the intrinsic growth rate, b is the competing
rate, and hx-term represents predation. To be specific, Murray 2 took hx in the form of
cx2 t/d x2 t, and the dynamic behavior of xt is then governed by
x t xta − bxt −
cx2 t
,
d x2 t
1.2
where cx2 t/d x2 t is an S-shaped function and a, b, c, and d are positive constants. For
the relevant ecology sense of system 1.2, we refer the readers to 1, 2 and the references
cited therein.
2
Discrete Dynamics in Nature and Society
It seems reasonable to assume that the reproduction of species x will not be
instantaneous, but mediated by a delay required for gestation of species x. Thus, a revised
version is
x t xta − bxt − τ −
cx2 t
,
d x2 t
1.3
where the constant delay τ is positive. It is evident that all the coefficients of system 1.3 are
assumed to be constant. However, in the real world, the coefficients are not fixed constants
owing to the variation of environments. The influence of a varying environment is important
for evolutionary theory as the selective forces on systems in such a fluctuating medium differ
from those in a stable environment. In addition, as we know, ecosystems are often disturbed
by unpredictable forces and, so, species population may experience changes. In ecology,
an interesting issue is whether or not an ecosystem can withstand those unpredictable
disturbances which persist for a finite period of time. In the language of control, we call the
disturbance functions control variables. For more discussion on this direction, we refer the
readers to 3–8.
Considering the possible effects of fluctuating environment and feedback control on
system 1.3, we obtain the following periodic system:
x t xtat − btxt − τ −
ctx2 t
− gtxtut,
dt x2 t
1.4
u t −etut htxt − τ,
where ut is the control variable. We assume that coefficients at, bt, ct, dt, gt, et,
and ht are continuous and bounded above and below by positive constants and et ∈ 0, 1.
Following the same idea and method in 9–12, one can easily derive the discrete
analogue of system 1.4, which takes the form of
xk 1 xk exp ak − bkxk − l −
ckxk
− gkuk ,
dk x2 k
1.5
Δuk −ekuk hkxk − l,
where k ∈ {0, 1, 2, . . .}, l is a positive integer, all the coefficients ak, bk, ck, dk, gk,
ek, and hk are positive bounded sequences and ek ∈ 0, 1. Δ is the first forward
difference operator Δuk uk 1 − uk. For biological reasons, we only consider
the solution xk, uk of system 1.5 with initial value x−l, x−l 1, . . . , x−1 ≥ 0,
x0, u0 > 0. The principle aim of this paper is to explore the permanence and global
attractivity of system 1.5. To the best of our knowledge, no work has been done for system
1.5.
For the sake of simplicity and convenience, the notations and definitions below will
be used through this paper: Z and denote the set of nonnegative integers and the
greatest integer function, respectively. We denote f U supk∈Z fk, f L infk∈Z fk for
any bounded nonnegative sequence {fk}. Meanwhile, we denote the product of fk from
kβ
kβ
k α to k β by kα fk with the understanding that kα fk 1 for all α > β.
Discrete Dynamics in Nature and Society
3
Definition 1.1. System 1.5 is said to be permanent if there exist positive constants x, x and u,
u such that
x lim inf xk lim sup xk x,
k → ∞
k → ∞
u lim inf uk lim sup uk u.
k → ∞
1.6
k → ∞
Definition 1.2. The positive solutions of system 1.5 are globally attractive if any two positive
solutions xk, uk and x∗ k, u∗ k of system 1.5 satisfy
lim |xk − x∗ k| 0,
k → ∞
lim |uk − u∗ k| 0.
k → ∞
1.7
The organization of this paper is as follows. In the next Sections 2 and 3, two
main results on permanence and global attractivity of system 1.5 are given, respectively.
Numerical simulations are present to illustrate the validity of our main results in Section 4,
and a brief conclusion is provided to summarize the paper in the final section.
2. Permanence
This section is concerned with the permanence of system 1.5. We first introduce the
following lemmas which are useful for establishing our result.
Lemma 2.1 see 4. Assume the constant A > 0 and y0 > 0, and further suppose that
1 if
yk 1 ≤ Ayk Bk,
k 1, 2, . . . ,
2.1
then for any integer p ≤ k,
p−1
yk ≤ Ap y k − p Ai Bk − i − 1.
2.2
i0
Especially, if A < 1 and Bk is bounded above with respect to M, then
lim sup yk ≤
k → ∞
M
.
1−A
2.3
2 If
yk 1 ≥ Ayk Bk,
k 1, 2, . . . ,
2.4
4
Discrete Dynamics in Nature and Society
then for any integer p ≤ k,
p−1
yk ≥ Ap y k − p Ai Bk − i − 1.
2.5
i0
Especially, if A < 1 and Bk is bounded below with respect to m, then
lim inf yk ≥
k → ∞
m
.
1−A
2.6
Lemma 2.2 see 10. Assume that yk satisfies yn1 > 0 and
yk 1 ≤ yk exp rk 1 − αyk
2.7
for k ∈ n1 , ∞, where α is a positive constant and n1 ∈ Z . Then
lim sup yk ≤
k → ∞
1
U
exp
r
−
1
.
αr U
2.8
Lemma 2.3 see 10. Assume that yk satisfies yn2 > 0 and
yk 1 ≥ yk exp rk 1 − αyk
2.9
for k ∈ n2 , ∞, limk → ∞ sup yk ≤ M, where α is a positive constant such that αM > 1 and
n2 ∈ Z . Then
lim inf yk ≥
k → ∞
1
exp r U 1 − αM .
α
2.10
Now, we state the permanence of system 1.5.
Theorem 2.4. System 1.5 is permanent provided that
2aL dL > cU .
2.11
Proof. It follows from the first equation of system 1.5 that
xk 1 ≤ xk exp{ak}.
2.12
Let xk exp{yk}, then 2.12 can be rewritten as
yk 1 − yk ≤ ak.
2.13
Discrete Dynamics in Nature and Society
5
Summing both sides of 2.13 from k − l to k − 1 yields
k−1
k−1
ai ≤ aU l,
yi 1 − yi ≤
ik−l
2.14
ik−l
which implies that
yk − l ≥ yk − aU l,
2.15
xk − l ≥ xk exp −aU l .
2.16
and hence
Following the above result, again from the first equation of system 1.5, we have
xk 1 ≤ xk exp ak − bk exp −aU l xk .
2.17
By applying Lemma 2.2 to 2.17, we can obtain that
exp aU l 1 − 1 def
lim sup xk ≤
x.
k → ∞
bL
2.18
For any constant ε > 0, it follows from 2.18 that there exists a n1 ∈ Z large enough such
that
xk ≤ x ε,
for k ≥ n1 ,
2.19
which, together with the second equation of system 1.5, leads to
uk ≤ −ekuk hkx ε,
for k ≥ n1 l,
2.20
that is,
uk 1 ≤ 1 − eL uk hU x ε,
for k ≥ n1 l.
2.21
Combing 2.21 with Lemma 2.11 and setting ε → 0 in 2.21, one has
lim sup uk ≤
k → ∞
hU x def
u.
eL
2.22
For any sufficient small constant ε > 0, it follows from 2.18 and 2.22 that there exists
n2 > n1 l such that
xk ≤ x ε,
uk ≤ u ε,
for k ≥ n2 .
2.23
6
Discrete Dynamics in Nature and Society
Thus, by 2.23 and the first equation of system 1.5, we have
xk 1 xk exp ak − bkxk − l −
ckxk
− gkuk
dk x2 k
ck
− gku
≥ xk exp ak − bkx −
dk/xk xk
ck
≥ xk exp ak − bkx − − gku
2 dk
2.24
for k ≥ n2 . Let xk exp{yk}, then 2.24 is equivalent to
yk 1 − yk ≥
ck
ak − bkx − − gku .
2 dk
2.25
For convenience of exposition, we set Rk {ak−bkx −ck/2 dk−gku}. Summing
both sides of 2.25 from k − l to k − 1 leads to
k−1
k−1
Ri.
yi 1 − yi ≥
ik−l
2.26
ik−l
This implies that
k−1
yk − l ≤ yk −
2.27
Ri,
ik−l
and hence,
k−1
xk − l ≤ xk exp −
Ri .
2.28
ik−l
From the second equation of system 1.5, we have
uk 1 1 − ekuk hkxk − l
≤ 1 − eL uk hkxk − l
def
Auk Bk,
2.29
Discrete Dynamics in Nature and Society
7
where A 1 − eL and Bk hkxk − l. Then, for any integer p ≤ k − l, combing 2.28,
2.29 with Lemma 2.1 1, we have
p−1
uk ≤ Ap u k − p Ai Bk − i − 1
i0
p−1
Ap u k − p Ai hk − i − 1xk − i − 1 − l
2.30
i0
⎧
⎫
k−1
⎨
⎬
.
≤ Ap u k − p Ai hU xk exp −
R j
⎩ jk−i1l
⎭
i0
p−1
Since ek ∈ 0, 1, we obtain that 0 < A < 1. So
0 ≤ Ap u k − p ≤ uAp −→ 0,
for p −→ ∞.
2.31
By the assumption of Theorem 2.4, for any solution xk, uk
√ of system 1.5, there exists
U p
L
U
an integer q > 0 such that g A uk −√p < 1/2a − c /2 dL for p > q. In fact, we can
choose q max{1, logA aL − cU /2 dL /2g U u 1}. Then
⎧
⎫
q−1
k−1
⎬
⎨
R j
uk ≤ Aq u k − q Ai hU xk exp −
⎩ jk−i1l
⎭
i0
q−1
q
i U
L
xk,
A h exp −i 1 lR
≤ uA 2.32
for k > q.
i0
q−1
q−1
Since i0 Ai hU exp−i1lRL is bounded above, let K1 { i0 Ai hU exp−i1lRL },
then we have
uk ≤ uAq K1 xk,
for k > q,
2.33
which, together with 2.24 and 2.28, leads to
xk 1 xk exp ak − bkxk − l −
ckxk
−
gkuk
dk x2 k
⎫
⎧⎡
⎤
⎪
⎪
⎬
⎨
ck
⎢
⎥
≥ xk exp ⎣ak − − uAq gk⎦ − K1 gk exp −lRL bk xk
⎪
⎪
⎭
⎩
2 dk
#
$
!
"
S2 k
≥ xk exp S1 k 1 −
xk ,
S1 k
2.34
8
Discrete Dynamics in Nature and Society
where S1 k ak − ck/2 dk − uAq gk, S2 k K1 gk exp−lRL bk. Note that
g U K1 bU exp −lRL
x
×x
√ SL1
aL − cU / 2 dL − uAq g U
SU
2
bU exp −lRL
exp aU l 1 − 1
×
aL
bL
bU exp −laL
exp aL l 1 − 1
≥
×
aL
bU
L
exp a − 1
≥
aL
≥
2.35
> 1,
where we use the inequality expx − 1 > x for x > 0. Therefore, by Lemma 2.3, it follows that
lim inf xk ≥
k → ∞
SL1
SU
2
&
%
exp
SU
1
1−
SU
2
SL1
'(
x
def
x.
2.36
For any constant ε > 0, by 2.36 we know that there exists a sufficiently large integer n3 >
n2 l such that
xk ≥ x − ε,
for k ≥ n3 .
2.37
Hence, it follows from 2.37 and the second equation of system 1.5 that
uk 1 ≥ 1 − eU uk hL x − ε ,
for k ≥ n3 .
2.38
Applying Lemma 2.12 and setting ε → 0 in 2.38, we have
lim inf uk ≥
k → ∞
hL def
x u.
eU
2.39
Consequently, combing 2.18, 2.22, and 2.36 with 2.39, system 1.5 is permanent. This
completes the proof.
3. Global Attractivity
On the basis of permanence, in this section we further provide sufficient conditions that
guarantee the positive solutions of system 1.5 are globally attractive. To do so, we first give
the following lemma.
Discrete Dynamics in Nature and Society
9
Lemma 3.1. For any two positive solutions xk, uk and x∗ k, u∗ k of system 1.5, one has
ln
xk
xk 1
ln ∗
− bkxk − x∗ k − ckJk − gkuk − u∗ k
x∗ k 1
x k
k−1
csx∗ s
∗
∗
P s as − bsx s − l −
− gsu s
bk
ds x∗ s2
sk−l
× xs − x∗ s − Qsxs{bsxs − l − x∗ s − l
)
csJs gsus − u s ,
∗
3.1
where
{ds − xsx∗ s × xs − x∗ s}
Js ,
ds xs2 × ds x∗ s2
%
P s exp ηs as − bsx s − l −
∗
ds x∗ s2
%
Qs exp ξs as − bsxs − l −
(
csx∗ s
∗
csxs
ds xs2
%
∗
1 − ξs as − bsx s − l −
− gsu s
(
3.2
− gsus
(
csx∗ s
∗
ds x∗ s2
− gsu s
ηs, ξs ∈ 0, 1.
Proof. It follows from system 1.5 that we have
ln
xk 1
x∗ k 1
xk
xk 1
− ln
− ln ∗
ln
∗
x k 1
x k
xk
x∗ k
(
%
ckxk
ak − bkxk − l −
− gkuk
dk xk2
%
(
ckx∗ k
∗
∗
− ak − bkx k − l −
− gku k
dk x∗ k2
−bkxk − l − x∗ k − l − ckJk − gkuk − u∗ k.
3.3
10
Discrete Dynamics in Nature and Society
Hence,
ln
xk
xk 1
ln ∗
− ckJk − gkuk − u∗ k
x∗ k 1
x k
− bk{xk − x∗ k − xk − xk − l x∗ k − x∗ k − l}
3.4
xk
ln ∗
− ckJk − gkuk − u∗ k
x k
− bkxk − x∗ k bk{xk − xk − l − x∗ k − x∗ k − l}.
Note that
k−1
xk − xk − l − x∗ k − x∗ k − l k−1
xs 1 − xs −
sk−l
k−1
x∗ s 1 − x∗ s
sk−l
{xs 1 − x∗ s 1 − xs − x∗ s},
sk−l
xs 1 − x∗ s 1 − xs − x∗ s
%
xs exp as − bsxs − l −
%
∗
csxs
ds xs2
%
xs exp as − bsxs − l −
%
− gsus
csxs
− exp as − bsx s − l −
%
− gsu s − xs − x∗ s
csx∗ s
ds x∗ s2
3.6
(
∗
− gsu s
(
csx∗ s
∗
xs − x s exp as − bsx s − l −
(
− gsus
ds xs2
∗
(
∗
ds x∗ s2
∗
(
csx∗ s
∗
− x s exp as − bsx s − l −
3.5
∗
ds x∗ s2
− gsu s − 1 .
By the mean value theorem, one has
xs 1 − x∗ s 1 − xs − x∗ s
∗
P s as − bsx s − l −
csx∗ s
ds x∗ s2
∗
− gsu s xs − x∗ s
*
)
− Qsxs bsxs − l − x∗ s − l csJs gsus − u∗ s .
3.7
Discrete Dynamics in Nature and Society
11
Then we can easily obtain 3.1 by substituting 3.5 and 3.7 into 3.4. The proof is
complete.
Now, we state our main result on the global attractivity of system 1.5.
Theorem 3.2. If the assumption of Theorem 2.4 holds and, further, suppose there exist positive
constants ρ and δ such that
def
λ min ρφ − δhU , δeL − ρϕ > 0,
3.8
where φ, ϕ are defined by 3.27 and 3.28, respectively. Then the positive solutions of system 1.5
are globally attractive.
Proof. Let xk, uk and x∗ k, u∗ k be any two positive solutions of system 1.5. To
prove Theorem 3.2, we first consider the following three steps for the first equation of system
1.5.
Step 1. Let V11 k | ln xk − ln x∗ k|. It follows from 3.1 that
+
+ +
+
+ xk 1 + + xk
+
∗
+ln
+ ≤ +ln
+ ck|Jk| gk|uk − u∗ k|
−
bkxk
−
x
k
+ x∗ k 1 + + x∗ k
+
bk
k−1
*
P sβs|xs − x∗ s| csQsxs|Js|
sk−l
)
Qsxs bs|xs − l − x∗ s − l| gs|us − u∗ s|
+
+
+ xk
+
− bkxk − x∗ k++ γk|xk − x∗ k| gk|uk − u∗ k|
≤ ++ln ∗
x k
bk
k−1
*
sk−l
P sβs γsQsxs |xs − x∗ s|
)
Qsxs bs|xs − l − x∗ s − l| gs|us − u∗ s| ,
3.9
where
cs
gsu∗ s,
βs as bsx∗ s − l 2 ds
γs csds xsx∗ s
ds2
3.10
.
By the mean value theorem, we get
xk − x∗ k expln xk − expln x∗ k θk ln
xk
,
x∗ k
3.11
12
Discrete Dynamics in Nature and Society
that is,
ln
1
xk
xk − x∗ k,
x∗ k θk
3.12
where θk lies between xk and x∗ k, then
+
+
+ xk
+
∗
+ln
+
+ x∗ k − bkxk − x k+
+
+ +
+ +
+
+ xk + + xk + + xk
+
∗
+
+
+
+
+
− ln
ln
+ln ∗
− bkxk − x k++
x k + + x∗ k + + x∗ k
+
+
+
+
+ xk +
+ 1
+
1
∗
∗
∗
+
+
+
−
+ln ∗
xk − x k − bkxk − x k++
|xk − x k| +
x k + θk
θk
+
+
+
+
+ xk +
+
+ 1
1
∗
+
+
+
−
+ln ∗
− bk++|xk − x∗ k|
|xk − x k| +
x k + θk
θk
+
+$
+ !
+
+ 1
+
+ xk +
1
+
+
+
−
−
− bk++ |xk − x∗ k|.
+ln ∗
x k +
θk + θk
3.13
According to Theorem 2.4, there exists a n0 ∈ Z such that {xk, x∗ k} ≤ x and {uk,
u∗ k} ≤ u for all k ≥ n0 . Therefore, for all k ≥ n0 , we can obtain that
ΔV 11 V11 k 1 − V11 k
+
+$
!
+ 1
+
1
≤−
− ++
− bk++ |xk − x∗ k| γk|xk − x∗ k|
θk
θk
gk|uk − u∗ k|
bk
3.14
k−1
*
P sβs γsQsx |xs − x∗ s|
sk−l
)
bsQsx|xs − l − x∗ s − l| gsQsx|us − u∗ s| .
Step 2. Let
V12 k k−1l
sk
bs
k−1
*
is−l
P iβi γiQix |xi − x∗ i|
)
biQix|xi − l − x∗ i − l| giQix|ui − u∗ i| .
3.15
Discrete Dynamics in Nature and Society
13
Then
ΔV 12 V12 k 1 − V12 k
kl
*
bs P kβk γkQkx |xk − x∗ k|
sk1
bkQkx|xk − l − x∗ k − l| gkQkx|uk − u∗ k|
− bk
)
3.16
k−1
*
P iβi γiQix |xi − x∗ i|
ik−l
)
biQix|xi − l − x∗ i − l| giQix|ui − u∗ i| .
Step 3. Let
V13 k x
k−1
bw lQw l|xw − x∗ w|
wk−l
w2l
bs.
3.17
swl1
By a simple calculation, it derives that
V13 V13 k 1 − V13 k
k2l
bsbk lQk lx|xk − x∗ k|
skl1
−
kl
3.18
bsbkQkx|xk − l − x∗ k − l|.
sk1
Now, we can define
V1 k V11 k V12 k V13 k.
3.19
Then for all k ≥ n0 , it follows from 3.14–3.18 that
V1 V1 k 1 − V1 k
+
+$
!
+ 1
+
1
≤−
− ++
− bk++ |xk − x∗ k| γk|xk − x∗ k| gk|uk − u∗ k|
θk
θk
kl
bsgkQkx|uk − u∗ k| sk1
kl
k2l
bsbk lQk lx|xk − x∗ k|
skl1
*
)
bs P kβk γkQkx |xk − x∗ k| .
sk1
3.20
14
x
Discrete Dynamics in Nature and Society
0.0276
0.0275
0.0274
0.0274
0.0272
0.0274
0.027
x 0.0273
0.0268
0.0273
0.0266
0.0273
0.0264
0
200
400
600
800
1000
0.0272
980
985
990
n
a
u
×10−3
1.4
1.3
1.3
1.2
1.2
u
1.1
1
0.9
0.9
200
995
1000
1.1
1
0
1000
b
×10−3
1.4
0.8
995
n
400
600
800
1000
0.8
980
985
990
n
n
c
d
Figure 1: Permanence of system 4.1 with x−1 0.04 and {x0, u0} {0.0265, 0.0008}. a and b
show x for k ∈ 0, 1000 and x for k ∈ 980, 1000, respectively. c and d show u for k ∈ 0, 1000 and u
for k ∈ 980, 1000, respectively.
For the second equation of system 1.5, we will consider the following two steps.
Step 1. Let V21 k |uk − u∗ k|. Then
V21 V21 k 1 − V21 k
|uk 1 − u∗ k 1| − |uk − u∗ k|
|1 − ekuk − u∗ k hkxk − l − x∗ k − l| − |uk − u∗ k|
≤ 1 − ek|uk − u∗ k| − |uk − u∗ k| hk|xk − l − x∗ k − l|
≤ −ek|uk − u∗ k| hk|xk − l − x∗ k − l|.
3.21
15
0.0276
0.0275
0.0274
0.0274
0.0272
0.0274
x, x∗
x, x∗
Discrete Dynamics in Nature and Society
0.027
0.0273
0.0268
0.0273
0.0266
0.0273
0.0264
0
200
400
600
800
0.0272
980
1000
985
990
n
∗
x
995
1000
x∗
a
b
×10−3
1.4
×10−3
1.4
1.3
1.3
1.2
1.2
u, u∗
u, u∗
1000
x
x
1.1
1.1
1
1
0.9
0.9
0.8
995
n
0
200
400
600
800
1000
0.8
0
985
990
n
n
u
u
∗
u∗
u
c
d
Figure 2: Global attractivity of system 4.1 with different initial values. x with x−1 0.04, {x0, u0} {0.0265, 0.0008} and x∗ with x∗ −1 0.05, {x∗ 0, u∗ 0} {0.0274, 0.0014}. a and b show x, x∗ for
k ∈ 0, 1000 and for k ∈ 980, 1000, respectively. c and d show u, u∗ for k ∈ 0, 1000 and for
k ∈ 980, 1000, respectively.
Step 2. Let V22 k k−1
wk−l
hw l|xw − x∗ w|. Then
V22 V22 k 1 − V22 k
k
wk1−l
hw l|xw − x∗ w| −
k−1
hw l|xw − x∗ w|
wk−l
hk l|xk − x∗ k| − hk|xk − l − x∗ k − l|.
3.22
16
Discrete Dynamics in Nature and Society
So we can define
V2 k V21 k V22 k.
3.23
Then it follows from 3.21 and 3.22 that
V2 V2 k 1 − V2 k
≤ hk l|xk − x∗ k| − ek|uk − u∗ k|.
3.24
Now, we could define
V k ρV1 k δV2 k,
3.25
where ρ, δ are mentioned in 3.8.
Obviously, V k ≥ 0 for all k ∈ Z and V n0 l < ∞. Therefore, combining 3.20
and 3.24, for all k ≥ n0 l, we have
ΔV ρΔV 1 δΔV 2
!
+
+$
k2l
+ 1
+
1
− ++
− bk++ γk ≤ρ −
bsbk lQk lx
θk
θk
skl1
kl
bs P kβk γkQkx
sk1
ρ gk kl
|xk − x∗ k| δhk l|xk − x∗ k|
bsgkQkx |uk − u∗ k| − δek|uk − u∗ k|
sk1
"
#
2
≤ −ρ min bL , − bU − γ U − lbU |xk − x∗ k| δhU |xk − x∗ k|
x
− δeL − ρ g U lbU g U QU x |uk − u∗ k|
≤ − ρφ − δhU |xk − x∗ k| − δeL − ρϕ |uk − u∗ k|
≤ −λ{|xk − x∗ k| |uk − u∗ k|},
3.26
where
bU QU x P U βU γ U QU x,
#
"
L 2
U
− γ U − lbU ,
φ min b , − b
x
3.27
Discrete Dynamics in Nature and Society
17
ϕ g U lbU g U QU x.
3.28
Summing both sides of 3.26 from n0 l to k, it derives that
k
V s 1 − V s ≤ −λ
sn0 l
k
{|xs − x∗ s| |us − u∗ s|},
3.29
sn0 l
which implies
V k 1 λ
k
{|xs − x∗ s| |us − u∗ s|} ≤ V n0 l ≤ V 0.
3.30
sn0 l
It follows from the above inequality that
∞
{|xs − x∗ s| |us − u∗ s|} ≤
sn0 l
V 0
< ∞,
λ
3.31
that is,
lim {|xk − x∗ k| |uk − u∗ k|} 0,
k → ∞
3.32
and we can easily obtain that
lim |xk − x∗ k| 0,
k → ∞
lim |uk − u∗ k| 0,
k → ∞
3.33
which implies that the positive solutions of system 1.5 are globally attractive, this completes
the proof.
4. Numerical Simulations
To verify the feasibilities of our main results, we consider a specific example:
xk 1 xk exp 0.0069 0.00069 sink − 0.25 − 0.0025 sinkxk − 1
0.0011 0.00011 sinkxk
−
− 0.034 − 0.0034 sinkuk ,
4.02 0.402 sink x2 k
uk 1 {1 − 0.99 0.0099 sink}uk 0.042 − 0.0042 sinkxk − 1.
4.1
√
√
Obviously, 2aL dL ≈ 0.0236, cU ≈ 0.0012, that is, 2aL dL > cU . Then the assumption
of Theorem 2.4 is satisfied, which indicates that system 4.1 is permanent see Figure 1.
18
Discrete Dynamics in Nature and Society
We assume ρ 20 and δ 1.5, by a simple computation, we have φ ≈ 0.0049,
ϕ ≈ 0.0584, λ ≈ 0.0286, then the sufficient conditions of Theorem 3.2 are satisfied. Thus, the
positive solutions of system 4.1 are globally attractive. From Figure 2, we can see that xk
and uk tend to x∗ k and u∗ k, respectively.
5. Conclusion
We conclude with a brief discussion of our results. The sufficient condition required for the
result of Theorem 2.4 does not depend on the size of the feedback control. However, the
sufficient conditions required for the result of Theorem 3.2 show that the feedback control
has an effect on the global attractivity of system 1.5.
Acknowledgments
The paper is supported by the Key Project of Chinese Ministry of Educationno. #210134, the
Innovation Term of Educational Department of Hubei Province of China no. #T200804, and
the Scientific Research Foundation of the Education Department of Hubei Province of China
no. #D20101902.
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