# Research Article Uniformly Asymptotic Stability of Positive Almost Periodic

```Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 182158, 9 pages
http://dx.doi.org/10.1155/2013/182158
Research Article
Uniformly Asymptotic Stability of Positive Almost Periodic
Solutions for a Discrete Competitive System
Qinglong Wang and Zhijun Liu
Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China
Correspondence should be addressed to Zhijun Liu; zhijun [email protected]
Received 11 February 2013; Accepted 26 April 2013
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is devoted to the study of almost periodic solutions of a discrete two-species competitive system. With the help of the
methods of the Lyapunov function, some analysis techniques, and preliminary lemmas, we establish a criterion for the existence,
uniqueness, and uniformly asymptotic stability of positive almost periodic solution of the system. Numerical simulations are
presented to illustrate the analytical results.
1. Introduction
In recent years, many works have been done for the difference
system (see [1–14] and the references cited therein) since the
discrete time models governed by the difference equation are
more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world. In particular, Qin et al. [1] introduced
the following discrete Lotka-Volterra competitive system:
π₯1 (π + 1)
= π₯1 (π) exp [π1 (π) − π1 (π) π₯1 (π) −
π2 (π) π₯2 (π)
],
1 + π₯2 (π)
π₯2 (π + 1)
= π₯2 (π) exp [π2 (π) − π2 (π) π₯2 (π) −
π1 (π) π₯1 (π)
],
1 + π₯1 (π)
π = 0, 1, 2 . . . ,
investigated the permanence and global asymptotic stability
of positive periodic solutions of system (1).
Notice that the investigation of almost periodic solutions
for difference equations is one of most important topics in
the qualitative theory of difference equations due to the
applications in biology, ecology, neural network, and so forth
(see [10–14] in detail), and few work has been done previously
on an almost periodic version which is corresponding to periodic system (1). In this paper, we will further investigate the
existence, uniqueness, and uniformly asymptotic stability of
positive almost periodic solution of the above almost periodic
version. To this end, we assume that the coefficients of system
(1) {ππ (π)}, {ππ (π)} and {ππ (π)} are bounded nonnegative almost
periodic sequences.
For the sake of simplicity and convenience in the following discussion, the notations below will be used throughout
this paper:
ππ = sup {π (π)} ,
π∈Z+
(1)
where π₯π (0) &gt; 0, π₯π (π) stand for the densities of species π₯π
at the πth generation, ππ (π) represent the natural growth rates
of species π₯π at the πth generation, ππ (π) are the intraspecific
effects of the πth generation of species π₯π on own population, and ππ (π) measure the interspecific effects of the πth
generation of species π₯π on species π₯π (π, π = 1, 2; π =ΜΈ π). They
ππΏ = inf+ {π (π)} ,
π∈Z
(2)
where {π(π)} is a bounded sequence and Z+ = {0, 1, 2, . . .}.
The remaining part of this paper is organized as follows.
In the next section, we introduce some notations, definitions,
and lemmas which are available for our main results. In
Section 3, sufficient conditions for the existence, uniqueness,
and uniformly asymptotic stability of positive almost periodic
solution of system (1) are given. Numerical simulations are
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π₯1∗ (π)
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(b)
π₯2∗ (π)
π₯2∗ (π)
(a)
1
760
π
π
40
60
80
100
1
700
720
740
π
π
(c)
(d)
π₯1∗ (π)
Figure 1: Positive almost periodic solution of system (13). (a), (c) Time-series
and
with initial values π₯1∗ (0) = 1.19, π₯2∗ (0) = 1.18
∗
∗
for π ∈ [0, 100], respectively. (b), (d) Time-series π₯1 (π) and π₯2 (π) with the above initial values for π ∈ [700, 800], respectively.
carried out to substantiate the above analytical results in
Section 4. Finally, we give some proofs of theorems in the
appendices for convenience in reading this paper.
π₯2∗ (π)
is a relatively dense set in Z for all π &gt; 0; that is, for any given
π &gt; 0, there exists an integer π(π) &gt; 0 such that each discrete
interval of length π(π) contains a π = π(π) ∈ πΈ{π, π₯} such that
|π₯ (π + π) − π₯ (π)| &lt; π,
2. Preliminaries
∀π ∈ Z.
In this section, we will need some preparations and give some
notations, definitions, and lemmas which will be useful for
our main results.
Denote by R, R+ , Z, and Z+ the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers,
respectively. R2 and Rπ denote the cone of 2-dimensional and
π-dimensional real Euclidean space, respectively.
π is called the π-translation number of π₯(π).
Definition 1 (see [13]). A sequence π₯ : Z → Rπ is called an
almost periodic sequence if the following π-translation set of
π₯
σ΅¨
σ΅¨σ΅¨
σ΅¨σ΅¨π (π + π, π₯) − π (π, π₯)σ΅¨σ΅¨σ΅¨ &lt; π,
πΈ {π, π₯} := {π ∈ Z : |π₯ (π + π) − π₯ (π)| &lt; π, ∀π ∈ Z}
(3)
(4)
Definition 2 (see [13]). Let π : Z &times; D → Rπ , where D is
an open set in Rπ . π(π, π₯) is said to be almost periodic in π
uniformly for π₯ ∈ D, or uniformly almost periodic for short,
if for any π &gt; 0 and any compact set S in D there exists a
positive integer π(π, S) such that any interval of length π(π, S)
contains an integer π for which
(5)
for all π ∈ Z and all π₯ ∈ S. π is called the π-translation number
of π(π, π₯).
3
1.22
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1.2
1.18
1.18
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1.16
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1.14
π₯2∗ (π)
π₯2∗ (π)
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(b)
π₯2∗ (π)
π₯2∗ (π)
(a)
1.14
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π₯1∗ (π)
1
1.25
1.2
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π₯1∗(
π)
1.1
1.05 0
20
40
60
80
100
1.2
1.15
π₯∗
1 (π
)
π
(c)
1.1
1.05 700
720
740
760
780
800
π
(d)
Figure 2: Phase portrait. (a), (b) 2-dimensional phase portrait of almost periodic system (13). Time-series π₯1∗ (π) and π₯2∗ (π) with initial values
π₯1∗ (0) = 1.19, π₯2∗ (0) = 1.18 for π ∈ [0, 100] and π ∈ [700, 800], respectively. (c), (d) 3-dimensional phase portrait of almost periodic system
(13). Time-series π₯1∗ (π) and π₯2∗ (π) with the above initial values for π ∈ [0, 100] and π ∈ [700, 800], respectively.
Lemma 3 (see [13]). {π₯(π)} is an almost periodic sequence if
and only if for any sequence {βπσΈ  } ⊂ Z there exists a subsequence
{βπ } ⊂ {βπσΈ  } such that π₯(π + βπ ) converges uniformly on π ∈ Z
as π → ∞. Furthermore, the limit sequence is also an almost
periodic sequence.
Consider the following almost periodic difference system:
π₯ (π + 1) = π (π, π₯ (π)) ,
+
π
π ∈ Z+ ,
(6)
π
where π : Z &times; Sπ΅ → R , Sπ΅ = {π₯ ∈ R : βπ₯β &lt; π΅}, and
π(π, π₯) is almost periodic in π uniformly for π₯ ∈ Sπ΅ and is
continuous in π₯. The product system of (6) is the following
system:
π₯ (π + 1) = π (π, π₯ (π)) ,
π¦ (π + 1) = π (π, π¦ (π)) , (7)
and Zhang [14] obtained the following lemma.
Lemma 4 (see [14]). Suppose that there exists a Lyapunov
function π(π, π₯, π¦) defined for π ∈ Z+ , βπ₯β &lt; π΅, βπ¦β &lt; π΅
satisfying the following conditions:
(i) π(βπ₯ − π¦β) ≤ π(π, π₯, π¦) ≤ π(βπ₯ − π¦β), where π, π ∈
πΎ with πΎ = {π ∈ πΆ(R+ , R+ ) : π(0) = 0 and π is
increasing};
(ii) |π(π, π₯1 , π¦1 ) − π(π, π₯2 , π¦2 )| ≤ πΏ(βπ₯1 − π₯2 β + βπ¦1 − π¦2 β),
where πΏ &gt; 0 is a constant;
(iii) Δπ(2.2) (π, π₯, π¦) ≤ −π½π(π, π₯, π¦), where 0 &lt; π½ &lt; 1 is a
constant and
Δπ(2.2) (π, π₯, π¦) = π (π + 1, π (π, π₯) , π (π, π¦)) − π (π, π₯, π¦) .
(8)
Moreover, if there exists a solution π(π) of system (6) such that
βπ(π)β ≤ π΅∗ &lt; π΅ for π ∈ Z+ , then there exists a unique uniformly asymptotically stable almost periodic solution π(π) of
system (6) which satisfies βπ(π)β ≤ π΅∗ . In particular, if π(π, π₯)
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π₯2∗ (π), π₯2 (π)
780
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π₯1 (π)
π₯1∗ (π)
π₯1 (π)
1.1
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1
760
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π
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π
π
π₯2∗ (π)
π₯2 (π)
π₯2∗ (π)
π₯2 (π)
(c)
(d)
π₯1∗ (π)
Figure 3: Uniformly asymptotic stability. (a), (c) Time-series
π₯2 (π) with initial values π₯1 (0) = 1.06, π₯2 (0) = 1.03 for π ∈ [0, 100],
above initial values for π ∈ [700, 800], respectively.
is periodic of period π, then there exists a unique uniformly
asymptotically stable periodic solution of system (6) of periodic
π.
Lemma 5 (see [1]). Any positive solution (π₯1 (π), π₯2 (π)) of system (1) satisfies
lim sup π₯π (π) ≤
π → +∞
π
def [exp (ππ
ππ =
πππΏ
− 1)]
π2πΏ &gt; π1π.
and
with initial values π₯1∗ (0) = 1.19, π₯2∗ (0) = 1.18
respectively. (b), (d) Time-series π₯1∗ (π), π₯2∗ (π), π₯1 (π), and
and π₯1 (π) and
π₯2 (π) with the
Then, any positive solution (π₯1 (π), π₯2 (π)) of system (1) satisfies
def
lim inf π₯π (π) ≥ ππ =
π → +∞
πππΏ − πππ
πππ
exp (πππΏ − πππππ − πππ) ,
π =ΜΈ π; π, π = 1, 2.
(11)
,
π = 1, 2.
(9)
Lemma 6 (see [1]). Suppose that system (1) satisfies the following assumptions:
π1πΏ &gt; π2π,
π₯2∗ (π)
(10)
3. Main Result
From (9) and (11), we denote by Ω the set of all solutions
(π₯1 (π), π₯2 (π)) of system (1) satisfying ππ ≤ π₯π (π) ≤ ππ , π =
1, 2 for all π ∈ Z+ . According to Lemma 4, we first prove that
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there is a bounded solution of system (1), and then structure
a suitable Lyapunov function for system (1).
Theorem 7. If the assumptions in (10) hold, then Ω =ΜΈ Φ.
A computation shows that
π1πΏ − π2π = 1.1530 &gt; 0,
Theorem 8. If the assumptions in (10) are satisfied, furthermore, 0 &lt; π½ &lt; 1, where π½ = min{π 1 , π 2 }, and
−
2
(1 + π2 )
2
(1 + π1 )
π 2 = 2π2πΏ π2 − π2π2 π22 −
−
(1 + π1ππ1 ) π2ππ2
(1 + π2ππ2 ) π1ππ1
−
π1π2 π12
4
(1 + π1 )
,
(1 + π2ππ2 ) π1ππ1
2
(1 + π2 )
(12)
2
(1 + π1 )
(1 + π1ππ1 ) π2ππ2
−
π2π2 π22
(14)
and moreover, we have
The proof of Theorem 7 is given in Appendix A.
π 1 = 2π1πΏ π1 − π1π2 π12 −
π2πΏ − π1π = 1.0900 &gt; 0,
π 1 = 0.3515,
π 2 = 0.2502,
(15)
that is, 0 &lt; π½ = min{π 1 , π 2 } = 0.2502 &lt; 1. It is easy to see
that the assumptions of Theorem 8 are satisfied. Hence, in
system (13) there exists a unique uniformly asymptotically
stable positive almost periodic solution. From Figure 1, it is
easy to see that there exists a positive almost periodic solution
(π₯1∗ (π‘), π₯2∗ (π‘)), and the 2-dimensional and 3-dimensional
phase portraits of almost periodic system (13) are revealed
in Figure 2, respectively. Figure 3 shows that any positive
solution (π₯1 (π), π₯2 (π)) tends to the almost periodic solution
(π₯1∗ (π), π₯2∗ (π)).
Appendices
4
(1 + π2 )
A. Proof of Theorem 7
,
then there exists a unique uniformly asymptotically stable
almost periodic solution of system (1) which is bounded by Ω
for all π ∈ Z+ .
The proof of Theorem 8 is given in Appendix B.
Clearly, by an inductive argument we have from system (1)
that
π−1
π₯1 (π) = π₯1 (0) exp ∑ [π1 (π) − π1 (π) π₯1 (π) −
π=0
π−1
π₯2 (π) = π₯2 (0) exp ∑ [π2 (π) − π2 (π) π₯2 (π) −
π=0
4. Numerical Simulations
In this section, we give the following example to check the
feasibility of the assumptions of Theorem 8.
Example 9. Consider the following discrete system:
π1 (π) π₯1 (π)
].
1 + π₯1 (π)
(A.1)
According to Lemmas 5 and 6, for any solution (π₯1 (π), π₯2 (π))
of system (1) and an arbitrarily small constant π &gt; 0, there
exists π0 sufficiently large such that
π1 − π ≤ π₯1 (π) ≤ π1 + π,
π₯1 (π + 1)
π2 (π) π₯2 (π)
],
1 + π₯2 (π)
π2 − π ≤ π₯2 (π) ≤ π2 + π,
∀π ≥ π0 .
= π₯1 (π) exp [1.20 − 0.02 sin (√2ππ)
(A.2)
− (1.05 + 0.01 sin (√2ππ)) π₯1 (π)
−
(0.025 + 0.002 cos (√2ππ)) π₯2 (π)
1 + π₯2 (π)
],
π₯2 (π + 1)
= π₯2 (π) exp [1.15 − 0.02 cos (√2ππ)
− (1.02 + 0.02 cos (√2ππ)) π₯2 (π)
−
(0.035 + 0.005 sin (√3ππ)) π₯1 (π)
1 + π₯1 (π)
].
(13)
Set {ππ } be any positive integer sequence such that ππ → +∞
as π → +∞, we can show that there exists a subsequence of
{ππ } still denoted by {ππ }, such that π₯π (π+ππ ) → π₯π∗ (π), π = 1, 2
uniformly in π on any finite subset πΆ of Z+ as π → +∞,
where πΆ = {π1 , π2 ⋅ ⋅ ⋅ ππ }, πβ ∈ Z+ (β = 1, 2 ⋅ ⋅ ⋅ π), and π is a
finite number.
As a matter of fact, for any finite subset πΆ ⊂ Z+ , ππ + πβ &gt;
π0 , β = 1, 2 ⋅ ⋅ ⋅ π, when π is large enough. Therefore, ππ − π ≤
π₯π (π + ππ ) ≤ ππ + π, π = 1, 2; that is, {π₯π (π + ππ )} are uniformly
bounded for π large enough.
Now, for π1 ∈ πΆ, we can choose a subsequence {ππ(1) } of
{ππ } such that {π₯1 (π1 +ππ(1) )} and {π₯2 (π1 +ππ(1) )} uniformly converge on Z+ for π large enough.
Analogously, for π2 ∈ πΆ, we can also choose a subsequence {ππ(2) } of {ππ(1) } such that {π₯1 (π2 + ππ(2) )} and {π₯2 (π2 +
ππ(2) )} uniformly converge on Z+ for π large enough.
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Repeating the above process, for ππ ∈ πΆ, we get a subsequence {ππ(π) } of {ππ(π−1) } such that {π₯1 (ππ +ππ(π) )} and {π₯2 (ππ +
ππ(π) )} uniformly converge on Z+ for π large enough.
Now, we choose the sequence {ππ(π) } which is a subsequence of {ππ } denoted by {ππ }; then, for all π ∈ πΆ, we obtain
that π₯π (π + ππ ) → π₯π∗ (π), π = 1, 2 uniformly in π ∈ πΆ as
π → +∞. Hence, the conclusion is valid by the arbitrary of
πΆ.
Recall the almost periodicity of {ππ (π)}, {ππ (π)} and {ππ (π)},
π = 1, 2, for the above sequence {ππ }, ππ → +∞ as π → +∞,
there exists a subsequence denoted by {ππ } such that
ππ (π + ππ ) σ³¨→ ππ (π) ,
ππ (π + ππ ) σ³¨→ ππ (π) ,
(A.3)
Let π → +∞, we have
π₯1∗ (π + πΌ)
π+πΌ−1
= π₯1∗ (πΌ) exp ∑ [π1 (π) − π1 (π) π₯1∗ (π) −
π=πΌ
π2 (π) π₯2∗ (π)
],
1 + π₯2∗ (π)
π₯2∗ (π + πΌ)
π+πΌ−1
= π₯2∗ (πΌ) exp ∑ [π2 (π) − π2 (π) π₯2∗ (π) −
π=πΌ
π1 (π) π₯1∗ (π)
].
1 + π₯1∗ (π)
(A.6)
Since πΌ is arbitrary, we know that (π₯1∗ (π), π₯2∗ (π)) is a solution
of system (1) on Z+ , and
ππ (π + ππ ) σ³¨→ ππ (π) ,
0 &lt; π1 − π ≤ π₯1∗ (π) ≤ π1 + π,
as π → +∞ uniformly on Z+ .
For any πΌ ∈ Z+ , we can assume that ππ + πΌ ≥ π0 for π large
enough. Let π ∈ Z+ , by an inductive argument of system (1)
from ππ + πΌ to π + ππ + πΌ, we obtain
π₯1 (π + ππ + πΌ)
0 &lt; π2 − π ≤ π₯2∗ (π) ≤ π2 + π,
π+ππ +πΌ−1
π=ππ +πΌ
0 &lt; π2 ≤ π₯2∗ (π) ≤ π2 ,
π2 (π) π₯2 (π)
],
1 + π₯2 (π)
π₯2 (π + ππ + πΌ)
(A.7)
Notice that π is an arbitrarily small positive constant; it follows
that
0 &lt; π1 ≤ π₯1∗ (π) ≤ π1 ,
= π₯1 (ππ +πΌ) exp ∑ [π1 (π)−π1 (π) π₯1 (π)−
∀π ∈ Z+ .
∀π ∈ Z+ .
(A.8)
Thus, Ω =ΜΈ Φ. This completes the proof.
π+ππ +πΌ−1
π (π) π₯1 (π)
= π₯2 (ππ +πΌ) exp ∑ [π2 (π)−π2 (π) π₯2 (π)− 1
].
1 + π₯1 (π)
π=π +πΌ
π
(A.4)
B. Proof of Theorem 8
Denote π1 (π) = ln π₯1 (π), π2 (π) = ln π₯2 (π). It follows from
system (1) that
Thus, it derives that
π₯1 (π + ππ + πΌ)
π+πΌ−1
= π₯1 (ππ + πΌ) exp ∑ [π1 (π + ππ ) − π1 (π + ππ ) π₯1 (π + ππ )
π1 (π + 1) = π1 (π) + π1 (π) − π1 (π) ππ1 (π) −
π2 (π) ππ2 (π)
,
1 + ππ2 (π)
π2 (π + 1) = π2 (π) + π2 (π) − π2 (π) ππ2 (π) −
π1 (π) ππ1 (π)
.
1 + ππ1 (π)
(B.1)
π=πΌ
−
π2 (π + ππ ) π₯2 (π + ππ )
],
1 + π₯2 (π + ππ )
According to Theorem 7, we can see that the system (B.1) has
a bounded solution (π1 (π), π2 (π)) satisfying
π₯2 (π + ππ + πΌ)
π+πΌ−1
= π₯2 (ππ + πΌ) exp ∑ [π2 (π + ππ ) − π2 (π + ππ ) π₯2 (π + ππ )
π=πΌ
−
π1 (π + ππ ) π₯1 (π + ππ )
].
1 + π₯1 (π + ππ )
ln π1 ≤ π1 (π) ≤ ln π1 ,
ln π2 ≤ π2 (π) ≤ ln π2 ,
(A.5)
π ∈ Z+ .
(B.2)
Thus, |π1 (π)| ≤ π΄, |π2 (π)| ≤ π΅, where π΄ = max{| ln π1 |,
| ln π1 |}, π΅ = max{| ln π2 |, | ln π2 |}. Define the norm
Journal of Applied Mathematics
7
σ΅¨ σ΅¨
σ΅¨ σ΅¨
σ΅¨ σ΅¨
σ΅¨
σ΅¨
≤ (σ΅¨σ΅¨σ΅¨π1 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π1 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨πΜ1 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨πΜ1 (π)σ΅¨σ΅¨σ΅¨)
β(π1 (π), π2 (π))β = |π1 (π)| + |π2 (π)|, where (π1 (π), π2 (π)) ∈
R2 . Consider the product system of system (B.1) as follow:
π1 (π + 1) = π1 (π) + π1 (π) − π1 (π) ππ1 (π) −
σ΅¨
σ΅¨ σ΅¨
σ΅¨
⋅ (σ΅¨σ΅¨σ΅¨π1 (π) − πΜ1 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π1 (π) − πΜ1 (π)σ΅¨σ΅¨σ΅¨)
π2 (π) ππ2 (π)
,
1 + ππ2 (π)
σ΅¨
σ΅¨ σ΅¨
σ΅¨ σ΅¨
σ΅¨ σ΅¨
σ΅¨
+ (σ΅¨σ΅¨σ΅¨π2 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π2 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨πΜ2 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨πΜ2 (π)σ΅¨σ΅¨σ΅¨)
σ΅¨
σ΅¨ σ΅¨
σ΅¨
⋅ (σ΅¨σ΅¨σ΅¨π2 (π) − πΜ2 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π2 (π) − πΜ2 (π)σ΅¨σ΅¨σ΅¨)
π (π) ππ1 (π)
π2 (π + 1) = π2 (π) + π2 (π) − π2 (π) ππ2 (π) − 1
,
1 + ππ1 (π)
π1 (π + 1) = π1 (π) + π1 (π) − π1 (π) ππ1 (π) −
π2 (π) ππ2 (π)
,
1 + ππ2 (π)
π2 (π + 1) = π2 (π) + π2 (π) − π2 (π) ππ2 (π) −
π1 (π) ππ1 (π)
.
1 + ππ1 (π)
(B.3)
We assume that π = (π1 (π), π2 (π)), π = (π1 (π), π2 (π)) are
any two solutions of system (B.1) defined on S; then, βπβ ≤ π·,
βπβ ≤ π·, where π· = π΄ + π΅, and S = {(π1 (π), π2 (π)) | ln ππ ≤
ππ (π) ≤ ln ππ , π = 1, 2, π ∈ Z+ }.
Let us construct a Lyapunov function defined on Z+ &times; S &times;
S as follows:
2
2
π (π, π, π) = (π1 (π) − π1 (π)) + (π2 (π) − π2 (π)) .
(B.4)
It is obvious that the norm βπ−πβ = |π1 (π)−π1 (π)|+|π2 (π)−
π2 (π)| is equivalent to βπ − πβ∗ = [(π1 (π)−π1 (π))2 +(π2 (π)−
π2 (π))2 ]1/2 ; that is, there are two constants πΆ1 &gt; 0, πΆ2 &gt; 0,
such that
πΆ1 βπ − πβ ≤ βπ − πβ∗ ≤ πΆ2 βπ − πβ ,
(B.5)
then,
2
2
(πΆ1 βπ − πβ) ≤ π (π, π, π) ≤ (πΆ2 βπ − πβ) .
(B.6)
Let π, π ∈ πΆ(R+ , R+ ), π(π₯) = πΆ12 π₯2 , π(π₯) = πΆ22 π₯2 ; then,
condition (i) of Lemma 4 is satisfied.
Μ π)
Μ ∈ Z+ &times; S &times; S, we
Moreover, for any (π, π, π), (π, π,
have
σ΅¨σ΅¨
Μ π)
Μ σ΅¨σ΅¨σ΅¨σ΅¨
σ΅¨σ΅¨π (π, π, π) − π (π, π,
σ΅¨
σ΅¨
σ΅¨σ΅¨
2
2
= σ΅¨σ΅¨σ΅¨(π1 (π) − π1 (π)) + (π2 (π) − π2 (π))
2
2σ΅¨
−(πΜ1 (π) − πΜ1 (π)) − (πΜ2 (π) − πΜ2 (π)) σ΅¨σ΅¨σ΅¨σ΅¨
σ΅¨
2
2σ΅¨
≤ σ΅¨σ΅¨σ΅¨σ΅¨(π1 (π) − π1 (π)) − (πΜ1 (π) − πΜ1 (π)) σ΅¨σ΅¨σ΅¨σ΅¨
σ΅¨
2
2σ΅¨
+ σ΅¨σ΅¨σ΅¨σ΅¨(π2 (π) − π2 (π)) − (πΜ2 (π) − πΜ2 (π)) σ΅¨σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨
= σ΅¨σ΅¨σ΅¨(π1 (π) − π1 (π)) + (πΜ1 (π) − πΜ1 (π))σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨
⋅ σ΅¨σ΅¨σ΅¨(π1 (π) − π1 (π)) − (πΜ1 (π) − πΜ1 (π))σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨
+ σ΅¨σ΅¨σ΅¨(π2 (π) − π2 (π)) + (πΜ2 (π) − πΜ2 (π))σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨
⋅ σ΅¨σ΅¨σ΅¨(π2 (π) − π2 (π)) − (πΜ2 (π) − πΜ2 (π))σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨ σ΅¨
σ΅¨
≤ πΏ {σ΅¨σ΅¨σ΅¨π1 (π) − πΜ1 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π2 (π) − πΜ2 (π)σ΅¨σ΅¨σ΅¨
σ΅¨
σ΅¨ σ΅¨
σ΅¨
+ σ΅¨σ΅¨σ΅¨π1 (π) − πΜ1 (π)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π2 (π) − πΜ2 (π)σ΅¨σ΅¨σ΅¨}
(B.7)
Μ = (πΜ1 (π), πΜ2 (π)), π
Μ = (Μ
where π
π1 (π), πΜ2 (π)), and πΏ =
4 max{π΄, π΅}. Thus, condition (ii) of Lemma 4 is satisfied.
Finally, calculating the Δπ(π) of π(π) along the solutions
of system (B.3), we have
Δπ(π΅.3) (π)
= π (π + 1) − π (π)
2
= (π1 (π + 1) − π1 (π + 1))
2
+ (π2 (π + 1) − π2 (π + 1))
2
2
− (π1 (π) − π1 (π)) − (π2 (π) − π2 (π))
2
2
= [(π1 (π + 1) − π1 (π + 1)) − (π1 (π) − π1 (π)) ]
2
2
+ [(π2 (π + 1) − π2 (π + 1)) − (π2 (π) − π2 (π)) ]
= [ (π1 (π) − π1 (π)) − π1 (π) (ππ1 (π) − ππ1 (π) )
2
−π2 (π) (
ππ2 (π)
ππ2 (π)
2
−
)] −(π1 (π)−π1 (π))
π
(π)
1+π 2
1 + ππ2 (π)
+ [ (π2 (π) − π2 (π)) − π2 (π) (ππ2 (π) − ππ2 (π) )
2
−π1 (π) (
ππ1 (π)
ππ1 (π)
2
−
)] −(π2 (π)−π2 (π))
π
(π)
1+π 1
1 + ππ1 (π)
= −2π1 (π) (π1 (π) − π1 (π)) (ππ1 (π) − ππ1 (π) )
− 2π2 (π) (π1 (π) − π1 (π)) (
+ 2π1 (π) π2 (π) (
ππ2 (π)
ππ2 (π)
−
) (ππ1 (π) − ππ1 (π) )
1 + ππ2 (π) 1 + ππ2 (π)
+ π12 (π) (ππ1 (π) − ππ1 (π) )
+ π22 (π) (
ππ2 (π)
ππ2 (π)
−
)
1 + ππ2 (π) 1 + ππ2 (π)
2
2
ππ2 (π)
ππ2 (π)
−
)
1 + ππ2 (π) 1 + ππ2 (π)
8
Journal of Applied Mathematics
− 2π2 (π) (π2 (π) − π2 (π)) (ππ2 (π) − ππ2 (π) )
− 2π1 (π) (π2 (π) − π2 (π)) (
+ 2π2 (π) π1 (π) (
π1 (π)
π1 (π)
π
π
−
)
π
(π)
1
1+π
1 + ππ1 (π)
2
ππ1 (π)
ππ1 (π)
−
) (ππ2 (π) − ππ2 (π) )
1 + ππ1 (π) 1 + ππ1 (π)
2
2
+ 2[
π2 (π) π1 (π) π2 (π) π1 (π)
2
(1 + π1 (π))
+
By the mean value theorem, it derives that
πππ (π) − πππ (π) = ππ (π) (ππ (π) − ππ (π)) ,
ππ (π)
πππ (π)
πππ (π)
−
=
(ππ (π) − ππ (π)) ,
2
π
(π)
π
(π)
π
π
1+π
1+π
(1 + ππ (π))
(B.9)
π = 1, 2, where ππ (π) and ππ (π) lie between πππ (π) and πππ (π) ,
respectively. Substituting (B.9) into (B.8), we get
2
= −2π1 (π) π1 (π) (π1 (π) − π1 (π))
−
4
2π2 (π) π2 (π)
(1 + π2 (π))
2
2
(π2 (π) − π2 (π))
(π1 (π) − π1 (π)) (π2 (π) − π2 (π))
2π1 (π) π2 (π) π1 (π) π2 (π)
2
(1 + π2 (π))
(π1 (π) − π1 (π))
− 2π2 (π) π2 (π) (π2 (π) − π2 (π))
+
−
4
(1 + π1 (π))
2π1 (π) π1 (π)
(1 + π1 (π))
2
2
π1 (π) π2 (π) π1 (π) π2 (π)
(1 + π2 (π))
−
2
2
π2 (π) π2 (π)
2
(1 + π2 (π))
]
≤ [−2π1πΏ π1 + π1π2 π12 +
π1π2 π12
(1 + π1 )
+ [−2π2πΏ π2 + π2π2 π22 +
+[
π2ππ1ππ1 π2
2
(1 + π1 )
π2π2 π22
2
4
(1 + π2 )
π1ππ1
+
2
] (π1 (π) − π1 (π))
4
+
2
(1 + π1 )
] (π2 (π) − π2 (π))
π1ππ2ππ1 π2
+
2
(1 + π2 )
2
π2ππ2
2
(1 + π2 )
]
2
&times; {(π1 (π) − π1 (π)) + (π2 (π) − π2 (π)) }
= − [2π1πΏ π1 − π1π2 π12 −
−
π1ππ1
(1 + π1 )
2
π1π2 π12
π1ππ2ππ1 π2
−
4
(1 + π1 )
−
2
(1 + π2 )
−
π2ππ1ππ1 π2
2
(1 + π1 )
π2ππ2
]
2
(1 + π2 )
− [2π2πΏ π2 − π2π2 π22 −
−
π1ππ1
2
(1 + π1 )
−
π2π2 π22
4
(1 + π2 )
π1ππ2ππ1 π2
2
(1 + π2 )
−
−
π2ππ1ππ1 π2
2
(1 + π1 )
π2ππ2
2
(1 + π2 )
]
2
2
(π1 (π) − π1 (π))
(π2 (π) − π2 (π)) (π1 (π) − π1 (π))
2
+ π22 (π) π22 (π) (π2 (π) − π2 (π))
+
(1 + π1 (π))
&times; (π1 (π) − π1 (π))
&times; (π2 (π) − π2 (π))
π12 (π) π12 (π)
π1 (π) π1 (π)
−
2
2
+ π12 (π) π12 (π) (π1 (π) − π1 (π))
+
]
&times; (π1 (π) − π1 (π)) (π2 (π) − π2 (π))
Δπ(π΅.3) (π)
(1 + π2 (π))
4
(1 + π2 (π))
2
(B.8)
+
π22 (π) π22 (π)
+ [−2π2 (π) π2 (π) + π22 (π) π22 (π) +
&times; (π2 (π) − π2 (π))
ππ1 (π)
ππ1 (π)
−
).
π
(π)
1+π 1
1 + ππ1 (π)
π22 (π) π22 (π)
]
4
(1 + π1 (π))
&times; (π1 (π) − π1 (π))
+ π22 (π) (ππ2 (π) − ππ2 (π) )
+ π12 (π) (
π12 (π) π12 (π)
= [−2π1 (π) π1 (π) + π12 (π) π12 (π) +
2π2 (π) π1 (π) π2 (π) π1 (π)
2
(1 + π1 (π))
&times; (π1 (π) − π1 (π))
(π2 (π) − π2 (π))
&times; (π2 (π) − π2 (π))
2
2
= − {π 1 (π1 (π) − π1 (π)) + π 2 (π2 (π) − π2 (π)) }
2
2
≤ −π½ {(π1 (π) − π1 (π)) + (π2 (π) − π2 (π)) }
= −π½π (π) ,
(B.10)
where π½ = min{π 1 , π 2 }. By the conditions of Theorem 8, we
have 0 &lt; π½ &lt; 1, and hence, condition (iii) of Lemma 4 is
satisfied. So, it follows from Lemma 4 that there exists a unique uniformly asymptotically stable almost periodic solution
Journal of Applied Mathematics
(π1∗ (π), π2∗ (π)) of system (B.1) which is bounded by S for all
π ∈ Z+ ; that is, there exists a unique uniformly asymptotically
stable almost periodic solution (π₯1∗ (π), π₯2∗ (π)) of system (1)
which is bounded by Ω for all π ∈ Z+ . This completed the
proof.
Acknowledgments
This work is supported by the National Natural Science
Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (nos. 210134, 212111), the Project
of Key Laboratory of Biological Resources Protection and
Utilization of Hubei Province, and the Key Subject of Hubei
Province (Forestry).
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