Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 358594, 31 pages
doi:10.1155/2012/358594
Research Article
Permanence and Almost Periodic Solutions of a
Discrete Ratio-Dependent Leslie System with Time
Delays and Feedback Controls
Gang Yu and Hongying Lu
School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics,
Dalian 116025, China
Correspondence should be addressed to Hongying Lu, hongyinglu543@163.com
Received 10 July 2012; Accepted 2 September 2012
Academic Editor: Wolfgang Ruess
Copyright q 2012 G. Yu and H. Lu. 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.
We consider a discrete almost periodic ratio-dependent Leslie system with time delays and
feedback controls. Sufficient conditions are obtained for the permanence and global attractivity
of the system. Furthermore, by using an almost periodic functional Hull theory, we show that the
almost periodic system has a unique globally attractive positive almost periodic solution.
1. Introduction
Among the relationships between the species living in the same outer environment, the
predator-prey theory plays an important and fundamental role. The predator-prey models
have been extensively studied by many scholars 1–4. Under the assumption that reduction
in a predator population has a reciprocal relationship with per capita availability of its
preferred food, Leslie 5 proposed the famous Leslie-Gower predator-prey model where the
carry capacity of the predator’s environment is proportional to the number of prey:
ẋ1 x1 b − ax1 − cx1 x2 ,
x2
,
ẋ2 x2 g − f
x1
1.1
where x1 and x2 represent prey and predator densities at time t, respectively. cx1 is the
predator’s rate of feeding upon prey, that is, the so-called predator’s functional response.
2
Abstract and Applied Analysis
If we assume that the predator consumes the prey according to the functional response px1 ,
then system 1.1 formulates as the following:
ẋ1 x1 b − ax1 − px1 x2 ,
x2
,
ẋ2 x2 g − f
x1
1.2
where px1 is prey-dependent functional responses. Owing to its theoretical and practical
significance, system 1.2 and its various generalized forms have been studied extensively
and seen great progress see, for example, 6–9.
However, in the study of the dynamic behaviors of predator-prey system, many
scholars argued that the ratio-dependent predator-prey systems are more realistic 10, 11.
A ratio-dependent predator-prey system with Leslie-Gower term takes the form of
x1
x2 ,
x2
x2
ẋ2 x2 g − f
.
x1
ẋ1 x1 b − ax1 − p
1.3
Though much progress has been seen in the asymptotic convergence of solutions
of population systems, such systems are not well studied in the sense that most results
are continuous-time cases related. Already, many scholars have paid attention to the
nonautonomous discrete population models, since the discrete time models governed by
difference equation are more appropriate than the continuous ones when the populations
have a short life expectancy, nonoverlapping generations in the real world see 12–23.
Furthermore, the asymptotic convergence of solutions of difference equations with delay is
one of the most important topics in the study of population dynamics, and many excellent
results have already been obtained and seen great progress see 24–26 and the references
cited therein.
Since time delays occur so often in nature, a number of ecological systems can be
described as systems with time delays see 14, 15, 19, 21–27. One of the most important
problems for this type of system is to analyze the effect of time delays on the stability
of the system. Furthermore, as we know, ecological systems in the real world are often
distributed by unpredictable forces which can result in changes in biological parameters such
as survival rates, so it is reasonable to study models with control variables which are so-called
disturbance functions 22, 23.
So it is very interesting to study dynamics of the following discrete ratio-dependent
Leslie system with time delays and feedback controls:
ckx1 kx2 k
− dku1 k − σ1 ,
x1 k 1 x1 k exp bk − akx1 k − τ1 −
h2 x22 k x12 k
x2 k − τ2 x2 k 1 x2 k exp gk − fk
− pku2 k − σ2 ,
x1 k − τ2 Δui k −αi kui k βi kxi k − ρi , i 1, 2,
1.4
Abstract and Applied Analysis
3
where xi k, i 1, 2 stand for the density of the prey and the predator at time k, respectively.
ui k, i 1, 2 are the control variables at time k. h2 is a positive constant, denoting the
constant of capturing half-saturation.
In this paper, we are concerned with the effects of the almost periodicity of ecological
and environmental parameters and time delays on the global dynamics of the discrete ratiodependent Leslie systems with feedback controls. To do so, for system 1.4 we always
assume that for i 1, 2, k ∈ Z
H1 ak, bk, ck, dk, gk, fk, pk, αi k, βi k are all bounded nonnegative almost periodic sequences such that
0 < al ≤ au ,
0 < bl ≤ bu ,
0 < cl ≤ cu ,
0 < dl ≤ du ,
0 < f l ≤ f u,
0 < pl ≤ pu ,
0 < αli ≤ αui < 1,
0 < gl ≤ gu,
0 < βil ≤ βiu .
1.5
Here, we let Z, Z denote the sets of all integers, nonnegative integers, respectively, and
use the notations: f u supk∈Z {fk}, f l infk∈Z {fk}, for any bounded sequence {fk}
defined on Z.
Let τ max{τi , σi , ρi , i 1, 2}, we consider system 1.4 with the following initial
conditions:
xi θ φi θ,
ui θ ϕi θ,
θ ∈ −τ, 0 ∩ Z, φi 0 > 0,
θ ∈ −τ, 0 ∩ Z, ϕi 0 > 0, i 1, 2.
1.6
One can easily show that the solutions of system 1.4 with initial condition 1.5 are
defined and remain positive for k ∈ Z .
The principle aim of this paper is to study the dynamic behaviors of system 1.4,
such as permanence, global attractivity, and existence of a unique globally attractive positive
almost periodic solution of the system. To the best of our knowledge, no work has been done
for the nonautonomous difference system 1.4.
The organization of this paper is as follows. In the next section, we introduce some
definitions and several useful lemmas. In Section 3, we explore the permanent property of
system 1.4. We study globally attractive property of system 1.4 in Section 4 and the almost
periodic property of system 1.4 in Section 5. Finally, the conclusion ends with brief remarks.
2. Preliminaries
In this section, we will introduce some basic definitions and several useful lemmas.
Definition 2.1. System 1.4 is said to be permanent, if there are positive constants mi and Mi ,
such that for each positive solution x1 k, x2 k, u1 k, u2 kT of system 1.4 satisfies
mi ≤ lim inf xi k ≤ lim sup xi k ≤ Mi ,
k → ∞
k → ∞
ni ≤ lim inf ui k ≤ lim sup ui k ≤ Ni ,
k → ∞
k → ∞
i 1, 2.
2.1
4
Abstract and Applied Analysis
Definition 2.2. Suppose that Xk x1 k, x2 k, u1 k, u2 kT is any solution of system
1.4. Xk is said to be a strictly positive solution in Z if for k ∈ Z and i 1, 2 such that
0 < inf xi k ≤ sup xi k < ∞,
k∈Z
0 < inf ui k ≤ sup ui k < ∞.
k∈Z
k∈Z
k∈Z
2.2
Definition 2.3 see 16. A sequence x : Z → R is called an almost periodic sequence if the
ε-translation set of x
E{ε, x} {τ ∈ Z : |xk τ − xk| < ε, ∀k ∈ Z}
2.3
is a relatively dense set in Z for all ε > 0; that is, for any given ε > 0, there exists an integer
lε > 0 such that each interval of length lε contains an integer τ ∈ E{ε, x} with
|xk τ − xk| < ε,
∀k ∈ Z,
2.4
τ is called an ε-translation number of xk.
Definition 2.4 see 17. The hull of f, denoted by Hf, is defined by
H f gk, x : lim fk τn , x gk, x uniformly on Z × S ,
n→∞
2.5
for some sequence {τn }, where S is any compact set in D.
Lemma 2.5 see 20. Assume that {yk} satisfies yk1 > 0 and
yk 1 ≤ yk exp rk 1 − ayk ,
2.6
for k ∈ k1 , ∞, where a is a positive constant and k1 ∈ Z . Then
lim sup yk ≤
k → ∞
1
expr u − 1.
aru
2.7
Lemma 2.6 see 20. Assume that {yk} satisfies yk2 > 0 and
yk 1 ≥ yk exp rk 1 − ayk ,
2.8
for k ∈ k2 , ∞, lim supk → ∞ yk ≤ M, where a is a constant such that aM > 1 and k2 ∈ Z .
Then
lim inf yk ≥
k → ∞
1
expr u 1 − aM.
a
2.9
Abstract and Applied Analysis
5
Lemma 2.7 see 22. Assume that A > 0 and y0 > 0. Suppose that
yk 1 ≤ Ayk Bk,
n 1, 2, . . . .
2.10
Then for any integer m ≤ k,
yk ≤ Am yk − m m−1
Aj B k − j − 1 .
2.11
j0
Especially, if A < 1 and B is bounded above with respect to M, then
M
.
1−A
lim sup yk ≤
k → ∞
2.12
Lemma 2.8 see 22. Assume that A > 0 and y0 > 0. Suppose that
yk 1 ≥ Ayk Bk,
n 1, 2, . . . .
2.13
Then for any integer m ≤ k,
yk ≥ Am yk − m m−1
Aj B k − j − 1 .
2.14
j0
Especially, if A < 1 and B is bounded below with respect to N, then
lim inf yk ≥
k → ∞
N
.
1−A
2.15
3. Permanence
In this section, we establish a permanent result for system 1.4.
Theorem 3.1. Assume that H1 holds; assume further that
H2 bl > cu /2|h|
holds. Then system 1.4 is permanent.
Proof. Let Xk x1 k, x2 k, u1 k, u2 kT be any positive solution of system 1.4, from
the first equation of system 1.4, it follows that
x1 k 1 ≤ x1 k expbk ≤ x1 k expbu .
3.1
6
Abstract and Applied Analysis
It follows from 3.1 that
k−1
k−1
x1 j 1
expbu ≤ expbu τ1 ,
≤
x
j
1
jk−τ1
jk−τ1
3.2
x1 k − τ1 ≥ x1 k exp−bu τ1 .
3.3
which implies that
Substituting 3.3 into the first equation of 1.4, it immediately follows that
x1 k 1 ≤ x1 k expbk − akx1 k − τ1 ≤ x1 k exp bk − ak exp−bu τ1 x1 k .
3.4
By applying Lemma 2.5 to 3.4, we have
lim sup x1 k ≤
k → ∞
1
expbu τ1 1 − 1 : M1 .
al
3.5
For any ε > 0 small enough, it follows from 3.5 that there exists enough large K1 such that
for k ≥ K1 ,
x1 k ≤ M1 ε.
3.6
From the second equation of system 1.4 it follows that
x2 k 1 ≤ x2 k exp gk ≤ x2 k exp g u ,
fk
x2 k − τ2 , for k ≥ K1 τ.
x2 k 1 ≤ x2 k exp gk −
M1 ε
3.7
From 3.7, similar to the argument of 3.1, one has
fk
exp −g u τ2 x2 k .
x2 k 1 ≤ x2 k exp gk −
M1 ε
3.8
By applying Lemma 2.5 to 3.8 again, we have
lim sup x2 k ≤
k → ∞
M1 ε
exp g u τ2 1 − 1 .
l
f
3.9
Setting ε → 0 in the above inequality, we have
lim sup x2 k ≤
k → ∞
M1
exp g u τ2 1 − 1 : M2 .
l
f
3.10
Abstract and Applied Analysis
7
For any ε > 0 small enough, it follows from 3.5 and 3.10 that there exists enough
large K2 > K1 τ such that for i 1, 2 and k ≥ K2
xi k ≤ Mi ε.
3.11
For k > K2 τ, 3.11 combining with the third and fourth equations of system 1.4 leads to
Δui k ≤ −αi kui k βi kMi ε,
i 1, 2,
3.12
ui k 1 ≤ 1 − αli ui k βiu Mi ε,
i 1, 2.
3.13
that is,
By applying Lemma 2.7, it follows from 3.13 that
lim sup ui k ≤
βiu Mi ε
,
i 1, 2.
3.14
: Ni ,
i 1, 2.
3.15
αli
k → ∞
Letting ε → 0 in the above inequality, we have
lim sup ui k ≤
βiu Mi
k → ∞
αli
For any ε > 0 small enough, it follows from 3.11 and 3.15 that there exists enough
large K3 > K2 τ such that for i 1, 2 and k ≥ K3
xi k ≤ Mi ε,
ui k ≤ Ni ε.
3.16
Thus, from 3.16 and the first equation of system 1.4, it follows that
cu
− du N1 ε
x1 k 1 ≥ x1 k exp bl − au M1 ε −
2|h|
: x1 k expD1ε ,
for k ≥ K3 τ,
where
D1ε bl − au M1 ε −
≤ bl − au M1
cu
− du N1 ε
2|h|
3.17
8
Abstract and Applied Analysis
≤ bl −
au expbu − 1
al exp−bu τ1 ≤ bl − bu
≤ 0.
3.18
For any integer η ≤ k, it follows from 3.17 that
k−1
k−1
x1 j 1
expD1ε ≥ exp D1ε η ,
≥
jk−η x1 j
jk−η
3.19
x1 η ≤ x1 k exp − k − η D1ε .
3.20
and hence
From the third equation of system 1.4, we have
u1 k 1 ≤ 1 − αl1 u1 k β1u x1 k − ρ1
≤ 1 − αl1 u1 k β1u exp −ρ1 D1ε x1 k
3.21
A1 u1 k B1 k,
where A1 1 − αl1 and B1 k β1u exp−ρ1 D1ε x1 k. Therefore, for any m ≤ k − τ, by
Lemma 2.7, 3.20, and 3.21, one has
u1 k ≤ Am
1 u1 k − m m−1
j
A1 B 1 k − j − 1
j0
≤ Am
1 u1 k − m m−1
j
A1 β1u exp −ρ1 D1ε x1 k − j − 1
3.22
j0
≤ Am
1 u1 k − m x1 k
m−1
j
A1 β1u exp − j 1 ρ1 D1ε .
j0
Since α1 k ∈ 0, 1, we have 0 < A1 < 1. So
m
0 ≤ Am
1 u1 k − m ≤ A1 N1 ε −→ 0,
for m −→ ∞.
3.23
Abstract and Applied Analysis
9
By the conditions H1 , for any solution x1 k, x2 k, u1 k, u2 kT of system 1.4,
l
u
there exists a positive integer M such that du Am
1 u1 k − m < 1/2b − c /2|h| for m > M.
l
u
u
In fact, we can choose M max{1, logA1 b − c /2|h|/2d N1 1}. Then we get
u1 k ≤ AM
1 N1 ε x1 k
M−1
j
A1 β1u exp − j 1 ρ1 D1ε
3.24
j0
where F1ε M−1
j0
AM
1 N1
ε F1ε x1 k,
j
A1 β1u exp−j 1 ρ1 D1ε .
Substituting 3.20 and 3.24 into the first equation of system 1.4, one has
ck
x1 k 1 ≥ x1 k exp bk − akx1 k − τ1 −
− dku1 k − σ1 2|h|
ck
− dkAM
≥ x1 k exp bk −
1 N1 ε
2|h|
− ak exp−τ1 D1ε dkF1ε exp−σ1 D1ε x1 k
3.25
S2ε k
x1 k ,
: x1 k exp S1ε k 1 −
S1ε k
where
S1ε k bk −
ck
− dkAM
1 N1 ε,
2|h|
3.26
S2ε k ak exp−τ1 D1ε dkF1ε exp−σ1 D1ε .
In particular, we have
Su2ε
Sl1ε
M1 au exp−τ1 D1ε du F1ε exp−σ1 D1ε bl − cu /2|h| − du AM
1 N1 ε
au exp−τ1 D1ε expbu τ1 1 − 1
bl
al
au exp −τ1 bl exp bl τ1 1 − 1
≥
au
bl
exp bl − 1
≥
bl
M1
≥
> 1,
3.27
10
Abstract and Applied Analysis
where we use the inequality expx − 1 > x for x > 0. By applying Lemma 2.6 to 3.25, it
follows that
Sl1ε
Su2ε
u
.
lim inf x1 k ≥ u exp S1ε 1 − l M1
k → ∞
S2ε
S1ε
3.28
Setting ε → 0 in the above inequality, we have
Sl1
Su2
u
lim inf x1 k ≥ u exp S1 1 − l M1
: m1 .
k → ∞
S2
S1
3.29
From 3.29 we know that there exists enough large K4 > K3 τ such that for k ≥ K4 ,
x1 k ≥ m1 − ε.
3.30
From 3.30 and the second equation of system 1.4, it follows that
f u M2 ε
− pu N2 ε
x2 k 1 ≥ x2 k exp g l −
m1 − ε
: x2 k expD2ε ,
3.31
for k ≥ K4 τ,
where D2ε g l − f u M2 ε/m1 − ε − pu N2 ε ≤ 0.
This implies for any integer η ≤ k
x2 η ≤ x2 k exp − k − η D2ε .
3.32
From 3.32 and the fourth equation of system 1.4, by a procedure similar to the discussion
of 3.21–3.24, we can verify that
3.33
u2 k ≤ AN
2 N2 ε F2ε x2 k,
where
A2 1 − αl2 ,
F2ε N−1
j
A2 β2u exp − j 1 ρ2 D2ε ,
j0
N max 1, logA2
l
g
2pu N2
1 .
3.34
Abstract and Applied Analysis
11
Substituting 3.32 and 3.33 into the second equation of system 1.4, one has
fk
x2 k − τ2 − pku2 k − σ2 x2 k 1 ≥ x2 k exp gk −
m1 − ε
≥ x2 k exp gk − pkAN
2 N2 ε
fk
−
exp−τ2 D2ε pkF2ε exp−σ2 D2ε x2 k
m1
S4ε k
: x2 k exp S3ε k 1 −
x2 k ,
S3ε k
3.35
where
S3ε k gk − pkAN
2 N2 ε,
fk
exp−τ2 D2ε pkF2ε exp−σ2 D2ε .
m1
S4ε k 3.36
In particular, we have
Su4ε
Sl3ε
f u /m1 exp−τ2 D2ε pu F2 exp−σ2 D2ε M2 ≥
M2
g l − p u AN
2 N2 ε
u
f /m1 exp−τ2 D2ε M1 exp g u τ2 1 − 1
gl
fl
f u /m1 exp −τ2 g l M1 exp g l τ2 1 − 1
≥
fu
gl
exp g l − 1
≥
gl
3.37
> 1.
By applying Lemma 2.6 to 3.35 again, it follows that
Su4ε
Sl3ε
u
.
lim inf x2 k ≥ u exp S3ε 1 − l M2
k → ∞
S4ε
S3ε
3.38
Setting ε → 0 in the above inequality, we have
Sl3
Su4
u
: m2 .
lim inf x2 k ≥ u exp S3 1 − l M2
k → ∞
S4
S3
3.39
12
Abstract and Applied Analysis
From 3.29 and 3.39 we know that there exists enough large K5 > K4 τ such that for
k ≥ K5 ,
xi k ≥ mi − ε.
3.40
For k > K5 τ, 3.40 combining with the third and fourth equations of system 1.4 produces
ui k 1 ≥ 1 − αui ui k βil mi − ε,
i 1, 2.
3.41
By applying Lemma 2.8, it follows from 3.41 that
lim inf ui k ≥
k → ∞
βil mi − ε
,
αui
i 1, 2.
3.42
i 1, 2.
3.43
Setting ε → 0 in the above inequality, we have
lim inf ui k ≥
k → ∞
βil mi
: ni ,
αui
Consequently, combining 3.5, 3.10, 3.15, 3.29, 3.39 with 3.43, system 1.4 is
permanent. This completes the proof of Theorem 3.1.
4. Global Attractivity
Firstly, we prove two lemmas which will be useful to our main result.
Lemma 4.1. For any two positive solutions x1 k, x2 k, u1 k, u2 kT and y1 k, y2 k, v1 k,
v2 kT of system 1.4, one has
ln
x1 k 1
y1 k 1
ln
x1 k
− ak x1 k − y1 k
y1 k
− dku1 k − σ1 − v1 k − σ1 E1 k x1 k − y1 k E2 k x2 k − y2 k
Abstract and Applied Analysis
k−1
P1 s bs − asy1 s − τ1 ak
13
sk−τ1
−
csy1 sy2 s
h2 y22 s
y12 s
− dsv1 s − σ1 x1 s − y1 s
x1 sP2 s −as x1 s − τ1 − y1 s − τ1 − dsu1 s − σ1 − v1 s − σ1 ,
E1 s x1 s − y1 s E2 s x2 s − y2 s
4.1
where
csy1 sy2 s x1 s y1 s
csx2 s
E1 s ,
− 2 2
h x2 s x12 s
h2 x22 s x12 s h2 y22 s y12 s
h2 csy1 sy2 s x2 s y2 s
csy1 s
E2 s ,
2 2
− 2 2
2
2
2
2
h x2 s x12 s
h x2 s x1 s h y2 s y1 s
csy1 sy2 s
P1 s exp θ1 s bs − asy1 s − τ1 −
− dsv1 s − σ1 ,
h2 y22 s y12 s
csx1 sx2 s
P2 s exp θ2 s bs − asx1 s − τ1 −
− dsu1 s − σ1 h2 x22 s x12 s
csy1 sy2 s
1 − θ2 s bs − asy1 s − τ1 −
− dsv1 s − σ1 ,
h2 y22 s y12 s
θ1 s, θ2 s ∈ 0, 1.
4.2
Proof. It follows from the first equation of system 1.4 that
ln
x1 k 1
x1 k
− ln
y1 k 1
y1 k
ln
y1 k 1
x1 k 1
− ln
x1 k
y1 k
ckx1 kx2 k
bk − akx1 k − τ1 −
− dku1 k − σ1 h2 x22 k x12 k
− bk − aky1 k − τ1 −
cky1 ky2 k
h2 y22 k y12 k
− dkv1 k − σ1 14
Abstract and Applied Analysis
−ak x1 k − τ1 − y1 k − τ1 − dku1 k − σ1 − v1 k − σ1 cky1 ky2 k
ckx1 kx2 k
2
2
2
h x2 k x1 k h2 y22 k y12 k
−ak x1 k − y1 k − dku1 k − σ1 − v1 k − σ1 E1 k x1 k − y1 k E2 k x2 k − y2 k
ak x1 k − y1 k − x1 k − τ1 − y1 k − τ1 ,
−
4.3
where
cky1 ky2 k x1 k y1 k
ckx2 k
,
E1 k 2 2
− 2 2
2
2
2
2
h x2 k x12 k
h x2 k x1 k h y2 k y1 k
h2 cky1 ky2 k x2 k y2 k
cky1 k
E2 k .
2 2
− 2 2
2
2
2
2
h x2 k x12 k
h x2 k x1 k h y2 k y1 k
4.4
Hence
ln
x1 k
x1 k 1
ln
− ak x1 k − y1 k − dku1 k − σ1 − v1 k − σ1 y1 k 1
y1 k
E1 k x1 k − y1 k E2 k x2 k − y2 k
ak x1 k − x1 k − τ1 − y1 k − y1 k − τ1 .
4.5
Since
x1 k − x1 k − τ1 − y1 k − y1 k − τ1 k−1
x1 s 1 − x1 s −
sk−τ1
k−1
k−1
y1 s 1 − y1 s
sk−τ1
x1 s 1 − y1 s 1 − x1 s − y1 s ,
sk−τ1
x1 s 1 − y1 s 1 − x1 s − y1 s
csx1 sx2 s
x1 s exp bs − asx1 s − τ1 −
− dsu1 s − σ1 h2 x22 s x12 s
csy1 sy2 s
−y1 s exp bs − asy1 s − τ1 −
− dsv1 s − σ1 h2 y22 s y12 s
− x1 s − y1 s
4.6
Abstract and Applied Analysis
15
csy1 sy2 s
x1 s − y1 s exp bs − asy1 s − τ1 −
− dsv1 s − σ1 − 1
h2 y22 s y12 s
csx1 sx2 s
x1 s exp bs − asx1 s − τ1 −
− dsu1 s − σ1 h2 x22 s x12 s
csy1 sy2 s
− exp bs − asy1 s − τ1 −
− dsv1 s − σ1 .
h2 y22 s y12 s
4.7
By the mean value theorem, one has
x1 s 1 − y1 s 1 − x1 s − y1 s
csy1 sy2 s
x1 s − y1 s P1 s bs − asy1 s − τ1 −
− dsv1 s − σ1 h2 y22 s y12 s
x1 sP2 s − as x1 s − τ1 − y1 s − τ1 − dsu1 s − σ1 − v1 s − σ1 csy1 sy2 s
csx1 sx2 s
−
h2 x22 s x12 s h2 y22 s y12 s
csy1 sy2 s
x1 s − y1 s P1 s bs − asy1 s − τ1 −
− dsv1 s − σ1 h2 y22 s y12 s
x1 sP2 s −as x1 s − τ1 − y1 s − τ1 − dsu1 s − σ1 − v1 s − σ1 E1 s x1 s − y1 s E2 s x2 s − y2 s .
4.8
Combining 4.6 with 4.8, we can easily obtain 4.1. The proof of Lemma 4.1 is completed.
Lemma 4.2. For any two positive solutions x1 k, x2 k, u1 k, u2 kT and y1 k, y2 k, v1 k,
v2 kT of system 1.4, one has
ln
x2 k 1
y2 k 1
ln
fk fky2 k − τ2 x2 k
−
x2 k − y2 k x1 k − τ2 − y1 k − τ2 y2 k x1 k − τ2 x1 k − τ2 y1 k − τ2 − pku2 k − σ2 − v2 k − σ2 16
Abstract and Applied Analysis
k−1 fk
y2 s − τ2 − psv2 s − σ2 x2 s − y2 s
Q1 s gs − fs
x1 k − τ2 sk−τ
y1 s − τ2 2
x2 sQ2 s −
fs x2 s − τ2 − y2 s − τ2 x1 s − τ2 fsy2 s − τ2 x1 s − τ2 − y1 s − τ2 x1 s − τ2 y1 s − τ2 − pku2 s − σ2 − v2 s − σ2 ,
4.9
where
y2 s − τ2 − psv2 s − σ2 ,
Q1 s exp ϕ1 s gs − fs
y1 s − τ2 x2 s − τ2 − psu2 s − σ2 Q2 s exp ϕ2 s gs − fs
x1 s − τ2 y2 s − τ2 1 − ϕ2 s gs − fs
− psv2 s − σ2 ,
y1 s − τ2 ϕ1 s, ϕ2 s ∈ 0, 1.
4.10
Proof. The proofs of Lemma 4.2 are very similar to those of Lemma 4.1. So we omit the detail
here.
Now we are in the position of stating the main result on the global attractivity of
system 1.4.
Theorem 4.3. In addition to H1 -H2 , assume further that
H3 there exist positive constants si , ωi , i 1, 2 such that
α : min s1 μ1 − s2 μ2 − ω1 β1u , s2 ν1 − s1 ν2 − ω2 β2u , ω1 αl1 − s1 1 , ω2 αl2 − s2 2 > 0
4.11
holds, where μ1 , μ2 , ν1 , ν2 , 1 , 2 are defined by 4.37. Then for any two positive solutions x1 k,
x2 k, u1 k, u2 kT and y1 k, y2 k, v1 k, v2 kT of system 1.4, one has
lim xi k − yi k 0,
k → ∞
lim |ui k − vi k| 0,
k → ∞
i 1, 2.
4.12
Abstract and Applied Analysis
17
Proof. For two arbitrary nontrivial solutions x1 k, x2 k, u1 k, u2 kT and y1 k, y2 k,
v1 k, v2 kT of system 1.4, we have from Theorem 3.1 that there exist positive constants
k0 and Mi , Ni , mi , ni i 1, 2 such that for all k ≥ k0 and i 1, 2
mi ≤ xi k ≤ Mi ,
ni ≤ ui k ≤ Ni ,
mi ≤ yi k ≤ Mi ,
ni ≤ vi k ≤ Ni .
4.13
Firstly, we define
V11 k ln x1 k − ln y1 k.
4.14
From 4.1, we have
x1 k 1 ln
y k 1 ≤
1
x1 k
ln
dk|u1 k − σ1 − v1 k − σ1 |
−
ak
x
−
y
k
k
1
1
y k
1
E1 kx1 k − y1 k E2 kx2 k − y2 k
ak
k−1
P1 sG1 s x1 sP2 sE1 sx1 s − y1 s
sk−τ1
4.15
x1 sP2 s asx1 s − τ1 − y1 s − τ1 ds|u1 s − σ1 − v1 s − σ1 |
E2 sx2 s − y2 s ,
where
G1 s bs asy1 s − τ1 csy1 sy2 s
h2 y22 s y12 s
dsv1 s − σ1 .
4.16
By the mean value theorem, we have
x1 k
,
x1 k − y1 k expln x1 k − exp ln y1 k ξ1 k ln
y1 k
4.17
that is,
ln
x1 k
1 x1 k − y1 k ,
y1 k ξ1 k
4.18
18
Abstract and Applied Analysis
where ξ1 k lies between x1 k and y1 k. So, we have
x1 k
ln
−
ak
x
−
y
k
k
1
1
y k
1
x1 k x1 k x1 k
− ln
ln
ln
−
ak
x
−
y
k
k
1
1
y1 k y1 k y1 k
4.19
x1 k 1
1
−
ln
−
− ak x1 k − y1 k.
y1 k ξ1 k ξ1 k
For k > k0 τ, it follows that
ΔV11 ≤ −
1
1
− − ak x1 k − y1 k
ξ1 k
ξ1 k
dk|u1 k − σ1 − v1 k − σ1 |
E1 kx1 k − y1 k E2 kx2 k − y2 k
ak
k−1
P1 sG1 s M1 P2 sE1 sx1 s − y1 s
4.20
sk−τ1
M1 P2 s asx1 s − τ1 − y1 s − τ1 ds|u1 s − σ1 − v1 s − σ1 |
E2 sx2 s − y2 s .
Secondly, we define
V12 k k−1
ds σ1 |u1 s − v1 s|
sk−σ1
k−1τ
1
sk
as
k−1
P1 uG1 u M1 P2 uE1 ux1 u − y1 u
us−τ1
M1 P2 u aux1 u − τ1 − y1 u − τ1 du|u1 u − σ1 − v1 u − σ1 |
E2 ux2 u − y2 u .
4.21
Abstract and Applied Analysis
19
Then
ΔV12 dk σ1 |u1 k − v1 k| − dk|u1 k − σ1 − v1 k − σ1 |
kτ
1
as P1 kG1 k M1 P2 kE1 kx1 k − y1 k
sk1
M1 P2 k akx1 k − τ1 − y1 k − τ1 dk|u1 k − σ1 − v1 k − σ1 | E2 kx2 k − y2 k
− ak
k−1
P1 uG1 u M1 P2 uE1 ux1 u − y1 u
us−τ1
M1 P2 u aux1 u − τ1 − y1 u − τ1 du|u1 u − σ1 − v1 u − σ1 | E2 ux2 u − y2 u .
4.22
Thirdly, we define
V13 k M1
k−1
1
l2τ
P2 l τ1 al τ1 x1 l − y1 l
as
slτ1 1
lk−τ1
k−1
M1
P2 l σ1 dl σ1 |u1 l − v1 l|
lk−σ1
lσ
1 τ1
4.23
as.
slσ1 1
By a simple calculation, it follows that
ΔV13 k2τ
1
asM1 P2 k τ1 ak τ1 x1 k − y1 k
skτ1 1
−
kτ
1
asM1 P2 kakx1 k − τ1 − y1 k − τ1 sk1
kσ
1 τ1
4.24
asM1 P2 k σ1 dk σ1 |u1 k − v1 k|
skσ1 1
−
kτ
1
asM1 P2 kdk|u1 k − σ1 − v1 k − σ1 |.
sk1
We now define
V1 k V11 k V12 k V13 k.
4.25
20
Abstract and Applied Analysis
Then for all k > k0 τ, it follows from 4.14–4.24 that
ΔV1 ≤ −
1
1
− − ak x1 k − y1 k dk σ1 |u1 k − v1 k|
ξ1 k
ξ1 k
E1 kx1 k − y1 k E2 kx2 k − y2 k
kτ
1
as P1 kG1 k M1 P2 kE1 kx1 k − y1 k
sk1
M1 P2 kE2 kx2 k − y2 k
k2τ
1
4.26
asM1 P2 k τ1 ak τ1 x1 k − y1 k
skτ1 1
kσ
1 τ1
asM1 P2 k σ1 dk σ1 |u1 k − v1 k|.
skσ1 1
Similarly, we define
V2 k V21 k V22 k V23 k,
4.27
V21 k ln x2 k − ln y2 k,
4.28
where
V22 k k−1
k−1
M2
fs τ2 x1 s − y1 s ps σ2 |u2 s − v2 s|
2
sk−τ2 m1
sk−σ2
k−1τ
2
sk
k−1 fs Q1 uG2 ux2 u − y2 u
m1 us−τ2
M2 Q2 u
fu x2 u − τ2 − y2 u − τ2 m1
M2
fux1 u − τ2 − y1 u − τ2 2
m1
p u|u2 u − σ2 − v2 u − σ2 | ,
4.29
Abstract and Applied Analysis
21
k−1
2 fs
l2τ
M2 Q2 l τ2 fl τ2 x2 l − y2 l
m1 lk−τ
m1
slτ 1
V23 k 2
k−1
M22 m21
lk−τ2
k−1
M2
2
2 fs
l2τ
Q2 l τ2 fl τ2 x1 l − y1 l
m1
slτ 1
4.30
2
Q2 l σ2 pl σ2 |u2 l − v2 l|
lk−σ2
G2 s gs fs
lσ
2 τ2
fs
,
m1
slσ 1
2
y2 s − τ2 psv2 s − σ2 .
y1 s − τ2 4.31
Thus for all k > k0 τ, it follows from 4.27–4.30 that
ΔV2 ≤ −
1
fk 1
x2 k − y2 k
− −
ξ2 k
ξ2 k x1 k − τ2 M2
fk τ2 x1 k − y1 k pk σ2 |u2 k − v2 k|
2
m1
kτ
2
fs
Q1 kG2 kx2 k − y2 k
m1
sk1
k2τ
2
fs M2
Q2 k τ2 fk τ2 x2 k − y2 k
m1 m1
skτ 1
2
k2τ
2
fs M22
Q2 k τ2 fk τ2 x1 k − y1 k
2
m1 m1
skτ 1
2
kσ
2 τ2
fs
M2 Q2 k σ2 pk σ2 |u2 k − v2 k|.
m1
skσ 1
2
4.32
For i 1, 2, we define
k−1
βi s ρi xi s − yi s.
4.33
ΔVi ≤ −αi k|ui k − vi k| βi k ρi xi k − yi k.
4.34
Vi k |ui k − vi k| sk−ρi
By calculation, it derives that
We now define a Lyapunov function as
V k s1 V1 k s2 V2 k ω1 V1 k ω2 V2 k.
4.35
22
Abstract and Applied Analysis
It is easy to see that V k > 0 and V k0 τ < ∞. Calculating the difference of V along the
solution of system 1.4, we have that for k ≥ k0 τ,
u u
2
l
u
u
u
u u
u u
ΔV ≤ s1 − min a ,
− a E1 τ1 a P1 G1 M1 P2 E1 M1 P2 a
M1
M2 u
M2 u u
u f Q2 ω1 β1 x1 k − y1 k
s2 2 f 1 τ2
m1
m1
fl
fu
fu
2
M2 u u
u u
s2 − min
τ2
,
−
Q f
Q G M1 M2 m1
m1 1 2 m1 2
u
u
u
u x2 k − y2 k
s1 E 1 τ1 a M1 P ω2 β
2
2
2
−ω1 αl1 s1 du 1 τ1 au M1 P2u |u1 k − v1 k|
fu
−ω2 αl2 s2 pu 1 τ2
M2 Q2u |u2 k − v2 k|
m1
≤ − s1 μ1 − s2 μ2 − ω1 β1u x1 k − y1 k − s2 ν1 − s1 ν2 − ω2 β2u x2 k − y2 k
− ω1 αl1 − s1 1 |u1 k − v1 k| − ω2 αl2 − s2 2 |u2 k − v2 k|
≤ −α
2 xi k − yi k |ui k − vi k| ,
i1
4.36
where
2
μ1 min al ,
− au − E1u − τ1 au P1u Gu1 M1 P2u E1u M1 P2u au ,
M1
M2 u u
M2
Q2 f ,
μ2 2 f u 1 τ2
m1
m1
fu
fl
fu
2
M2 u u
− τ2
Q1u Gu2 ,
−
Q2 f ,
ν1 min
M1 M2 m1
m1
m1
ν2 E2u 1 τ1 au M1 P2u ,
1 du 1 τ1 au M1 P2u ,
fu
u
u
M2 Q2 .
2 p 1 τ2
m1
4.37
Abstract and Applied Analysis
23
Summing both sides of 4.36 from k0 τ to k, it derives that
k
2 k
xi s − yi s |ui s − vi s| .
V s 1 − V s ≤ −α
sk0 τ
4.38
sk0 τ i1
It then follows from 4.38 that for k > k0 τ,
V k 1 α
k
2 xi s − yi s |ui s − vi s| ≤ V k0 τ,
4.39
sk0 τ i1
that is
2 k
xi s − yi s |ui s − vi s| ≤ V k0 τ .
α
sk τ i1
4.40
2 ∞ xi s − yi s |ui s − vi s| ≤ V k0 τ < ∞,
α
kk τ i1
4.41
0
Then
0
from which we see that
lim xi k − yi k 0,
k → ∞
lim |ui k − vi k| 0,
k → ∞
i 1, 2.
4.42
This completes the proof of Theorem 4.3.
5. Almost Periodic Solutions
In this section, we consider the almost periodic property of system 1.4.
We assume that a∗ k ∈ Hak, b∗ k ∈ Hbk, c∗ k ∈ Hck, d∗ k ∈
Hdk, f ∗ k ∈ Hfk, g ∗ k ∈ Hgk, p∗ k ∈ Hpk, α∗i k ∈ Hαi k, βi∗ k ∈
Hβi k, i 1, 2. By Definitions 2.3 and 2.4, there exists an integer valued sequence γn with
γn → ∞ as n → ∞ such that for i 1, 2, k ∈ Z
a k γn −→ a∗ k,
d k γn −→ d∗ k,
p k γn −→ p∗ k,
b k γn −→ b∗ k,
f k γn −→ f ∗ k,
αi k γn −→ α∗i k,
c k γn −→ c∗ k,
g k γn −→ g ∗ k,
βi k γn −→ βi∗ k.
5.1
24
Abstract and Applied Analysis
Then we have the following Hull equation of system 1.4
c∗ kx1 kx2 k
∗
x1 k 1 x1 k exp b k − a kx1 k − τ1 −
− d ku1 k − σ1 ,
h2 x22 k x12 k
5.2
x2 k − τ2 ∗
∗
∗
x2 k 1 x2 k exp g k − f k
− p ku2 k − σ2 ,
x1 k − τ2 Δui k −α∗i kui k βi∗ kxi k − ρi , i 1, 2.
∗
∗
According to the almost periodic theory, we know that if system 1.4 satisfies
H1 –H3 , then Hull equation 5.2 also satisfies H1 –H3 .
The following Lemma is Lemma 4.1 in 15.
Lemma 5.1. If each hull equation of system 1.4 has a unique strictly positive solution, then the
almost periodic difference system 1.4 has a unique strictly positive almost periodic solution.
Theorem 5.2. Assume that H1 –H3 holds. Then the almost periodic difference system 1.4 has a
unique strictly positive almost periodic solution which is globally attractive.
Proof. We divided the proof into two steps.
Step 1. We prove the existence of a strictly positive solution of each Hull equation 5.2.
By the almost periodicity of the parameters of system 1.4, there exists an integervalued sequence γn with γn → ∞ as n → ∞ such that for i 1, 2, k ∈ Z
a∗ k γn −→ a∗ k,
d∗ k γn −→ d∗ k,
p∗ k γn −→ p∗ k,
b∗ k γn −→ b∗ k,
f ∗ k γn −→ f ∗ k,
α∗i k γn −→ α∗i k,
c∗ k γn −→ c∗ k,
g ∗ k γn −→ g ∗ k,
βi∗ k γn −→ βi∗ k.
5.3
Suppose that Xk x1 k, x2 k, u1 k, u2 kT is any positive solution of Hull
equation 5.2. Since H1 , H2 hold, according to Theorem 3.1, we obtain
mi ≤ lim inf xi k ≤ lim sup xi k ≤ Mi ,
k → ∞
k → ∞
ni ≤ lim inf ui k ≤ lim sup ui k ≤ Ni ,
k → ∞
k → ∞
i 1, 2.
5.4
Therefore, one has
0 < inf xi k ≤ sup xi k < ∞,
k∈Z
k∈Z
0 < inf ui k ≤ sup ui k < ∞,
k∈Z
k∈Z
i 1, 2.
5.5
Abstract and Applied Analysis
25
For any ε > 0 small enough, it follows from Theorem 3.1 that there exists enough large
K0 > K5 τ such that for i 1, 2 and k ≥ K0
mi − ε ≤ xi k ≤ Mi ε,
ni − ε ≤ ui k ≤ Ni ε.
5.6
Let xn k xn k γn and un k un k γn for k ≥ K0 τ − γn , n 1, 2, . . .. For
any positive integer q and i 1, 2, it is not difficult to show that there are sequences {xin k :
n ≥ q} and {uin k : n ≥ q} such that the sequences {xin k} and {uin k} have subsequences,
denoted by {xin k} and {uin k} again, converging on any finite interval of Z as n → ∞,
respectively. So we have sequences {yi k} and {vi k} i 1, 2 such that for k ∈ Z, n → ∞
xin k −→ yi k,
uin k −→ vi k,
x1 k 1 x1 k exp b∗ k γn − a∗ k γn x1 k − τ1 −
c∗ k γn x1 kx2 k
h2 x22 k x12 k
∗
5.7
− d k γn u1 k − σ1 ,
x2 k − τ2 x2 k 1 x2 k exp g ∗ k γn − f ∗ k γn
− p∗ k γn u2 k − σ2 ,
x1 k − τ2 Δui k −α∗i k γn ui k βi∗ k γn xi k − ρi , i 1, 2,
which implies
∗
∗
y1 k 1 y1 k exp b k − a ky1 k − τ1 −
c∗ ky1 ky2 k
h2 y22 k y12 k
∗
− d kv1 k − σ1 ,
y2 k − τ2 ∗
∗
∗
y2 k 1 y2 k exp g k − f k
− p kv2 k − σ2 ,
y1 k − τ2 Δvi k −α∗i kvi k βi∗ kyi k − ρi , i 1, 2.
5.8
Thus we can easily see that Y k y1 k, y2 k, v1 k, v2 kT is a solution of Hull
equation 5.2 and mi − ε ≤ yi k ≤ Mi ε, ni − ε ≤ vi k ≤ Ni ε, for i 1, 2, k ∈ Z. Since ε
is an arbitrary small positive number, it follows that
0 < inf yi k ≤ sup yi k < ∞,
k∈Z
k∈Z
0 < inf vi k ≤ sup vi k < ∞,
k∈Z
k∈Z
i 1, 2.
5.9
This completes the proof of Step 1.
Step 2. We show the uniqueness of the strictly positive solution of each Hull equation
5.2.
26
Abstract and Applied Analysis
Suppose that the Hull equation 5.2 has two arbitrary strictly positive solutions
X ∗ k x1∗ k, x2∗ k, u∗1 k, u∗2 kT and Y ∗ k y1∗ k, y2∗ k, v1∗ k, v2∗ kT . Now we define
a Lyapunov function V ∗ k on k ∈ Z as follows:
V ∗ k 2
2
si Vi1∗ k Vi2∗ k Vi3∗ k ωi Vi ∗ k,
i1
5.10
i1
where
Vi1∗ k ln xi∗ k − ln yi∗ k,
∗
V12
k k−1
i 1, 2,
ds σ1 u∗1 s − v1∗ s
sk−σ1
k−1τ
1
as
k−1
P1 uG1 uM1 P2 uE1 ux1∗ u − y1∗ u
us−τ1
sk
M1 P2 u aux1∗ u − τ1 − y1∗ u − τ1 duu∗1 u − σ1 − v1∗ u − σ1 E2 ux2∗ u − y2∗ u ,
∗
V13
k M1
k−1
1
l2τ
P2 l τ1 al τ1 x1∗ l − y1∗ l
as
slτ1 1
lk−τ1
M1
k−1
1 τ1
lσ
P2 l σ1 dl σ1 u∗1 l − v1∗ l
as,
slσ1 1
lk−σ1
∗
V22
k k−1
k−1
∗
M2
x s − y∗ s fs
τ
ps σ2 u∗2 s − v2∗ s
2
1
1
2
sk−τ2 m1
sk−σ2
k−1τ
k−1 2 fs Q1 uG2 ux2∗ u − y2∗ u
m
1 us−τ2
sk
M2 Q2 u
fu ∗
x2 u − τ2 − y2∗ u − τ2 m1
∗
M2
x u − τ2 − y∗ u − τ2 fu
1
1
m21
,
puu∗2 u − σ2 − v2∗ u − σ2 Abstract and Applied Analysis
∗
V23
k 27
k−1
2 fs
l2τ
M2 Q2 l τ2 fl τ2 x2∗ l − y2∗ l
m1 lk−τ
m1
slτ 1
2
2
k−1
M22 m21
lk−τ2
k−1
M2
lk−σ2
2 fs
l2τ
Q2 l τ2 fl τ2 x1∗ l − y1∗ l
m1
slτ 1
2
2 τ2
lσ
fs
Q2 l σ2 pl σ2 u∗2 l − v2∗ l
,
m1
slσ 1
2
k−1
βi s ρi xi∗ s − yi∗ s,
Vi ∗ k u∗i k − vi∗ k sk−ρi
5.11
where Ei , Gi , Pi , Qi , i 1, 2 are defined by Theorem 4.3.
Similar to the discussion of 3.32, calculating the difference of V ∗ k along the solution
of the Hull equation 5.2, one has
ΔV ∗ ≤ −α
2 ∗
x k − y∗ k u∗ k − v∗ k ,
i
i
i
i
for k ∈ Z.
5.12
i1
It follows from 5.12 we know that V ∗ k is a nonincreasing function on Z. Summing both
sides of the above inequalities from k to 0, we get
α
2 0 ∗
x s − y∗ s u∗ s − v∗ s ≤ V ∗ k − V ∗ 0,
i
i
i
i
for k < 0.
5.13
sk i1
So we have
2
0
sk
∗
i1 |xi s
− yi∗ s| |u∗i s − vi∗ s| < ∞, k → −∞, thus we obtain
lim xi∗ k − yi∗ k 0,
lim u∗i k − vi∗ k 0,
k → −∞
k → −∞
i 1, 2.
5.14
From 5.14, for a positive integer K, we have
∗
x k − y∗ k < ε,
i
i
∗
u k − v∗ k < ε,
i
i
for k < −K, i 1, 2.
5.15
28
Abstract and Applied Analysis
It follows from 5.10 that
1
1 ∗
xi k − yi∗ k ≤
ε,
mi
mi
∗
V12
k σ1 du maxu∗1 s − v1∗ s
Vi1∗ k ≤
s≤k
τ1 2 au
u u
P1 G1 M1 P2u E1u maxx1∗ s − y1∗ s
s≤k
M1 P2u au maxx1∗ s − y1∗ s du maxu∗1 s − v1∗ s
s≤k
s≤k
E2u maxx2∗ s − y2∗ s
s≤k
≤ σ1 du τ1 2 au P1u Gu1 M1 P2u E1u M1 P2u au du E2u
ε,
∗
V13
k ≤ τ1 M1 P2u au τ1 au maxx1∗ s − y1∗ s σ1 du maxu∗1 s − v1∗ s
s≤k
s≤k
≤ τ1 M1 P2u au τ1 au σ1 du ε,
M2 u
f maxx1∗ s − y1∗ s σ2 pu maxu∗2 s − v2∗ s
2
s≤k
s≤k
m1
fu
τ2 2
Q1u Gu2 maxx2∗ s − y2∗ s
m1
s≤k
u
M2
u f
M2 Q2
maxx2∗ s − y2∗ s 2 f u maxx1∗ s − y1∗ s
m1 s≤k
s≤k
m1
∗
u
∗
,
p max u s − v s
∗
V22
k ≤ τ2
s≤k
2
2
≤
∗
V23
k
u
f u M2 u
M2 u
2f
u
u u
u
u
τ2 2 f σ2 p τ2 ε,
Q1 F2 M2 Q2
f p
m1
m1 m21
m1
fu
fu
M2
u
M2 Q2 τ2
maxx2∗ s − y2∗ s τ2 2 f u maxx1∗ s − y1∗ s
≤ τ2
m1
m1 s≤k
s≤k
m1
σ2 pu maxu∗ s − v∗ s
s≤k
2
2
fu
fu
M2 u
u
u
≤ τ2
M2 Q2 τ2
τ2 2 f σ2 p ε,
m1
m1
m1
Vi ∗ k ≤ maxu∗i s − vi∗ s ρi βiu maxxi∗ s − yi∗ s ≤ 1 ρi βiu ε.
s≤k
s≤k
5.16
Abstract and Applied Analysis
29
Let
Q s1
1
σ1 du τ1 2 au P1u Gu1 M1 P2u E1u M1 P2u au du E2u
m1
τ1 M1 P2u au τ1 au σ1 du s2
u
f u M2 u
1
M2 u
2f
u
u u
u
u
Q1 G2 M2 Q2
τ2 2 f σ2 p τ2 f p
m2
m1
m1 m21
m1
5.17
2
fu
fu
M2 u
u
u
M2 Q2 τ2
τ2 2 f σ2 p
ωi 1 ρi βiu .
τ2
m1
m1
m1
i1
It follows from 5.10 and the above inequalities that V ∗ k ≤ Qε for k < −K, hence
limk → −∞ V ∗ k 0. Furthermore, V ∗ k is a nonincreasing function on Z, thus V ∗ k ≡ 0, that
is, xi∗ k yi∗ k, u∗i k vi∗ k, i 1, 2 for all k ∈ Z. This ends the proof of Step 2.
By the above discussion and Theorem 4.3, we conclude that any hull equation of
system 1.4 has a unique strictly positive solution. Then the almost periodic system 1.4
has a unique strictly positive almost periodic solution which is globally attractive. The proof
of Theorem 5.2 is completed.
6. Concluding Remarks
In this paper, a discrete almost periodic ratio-dependent Leslie system with time delays
and feedback controls is considered. By applying the difference inequality, some sufficient
conditions are established, which are independent of feedback control variables, to ensure
the permanence of system 1.4. By constructing the suitable Lyapunov function, we show
that system 1.4 is globally attractive under some appropriate conditions. Furthermore, by
using an almost periodic functional hull theory, we show that the almost periodic system
1.4 has a unique strictly positive almost periodic solution which is globally attractive.
We would like to mention here the question of whether the feedback control variables
have the influence on the stability property of the system or not. It is, in fact, a very challenging problem, and we leave it for our future work.
Acknowledgments
The authors are grateful to the editor, Wolfgang Ruess, and referees for a number of
helpful suggestions that have greatly improved our original submission. This work is
supported by the National Natural Science Foundation of China nos. 71171035, 71173029,
China Postdoctoral Science Foundation no. 2012M511598, and the Teaching and Research
Foundation of Dongbei University of Finance and Economics nos. YY11001, GW11003.
30
Abstract and Applied Analysis
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