# Almost Periodic Solution of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response and Feedback Controls

Authors: Kerong Zhang; Jianli Li; Aiwen Yu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We consider a modified Leslie-Gower predator-prey model with the Beddington-DeAngelis functional response and feedback controls as follows:x
˙
t
=
x
t
a
1
t
-
b
t
x
t
-
c
t
y
t
/
α
t
+
β
t
x
t
+
γ
t
y
t
-
e
1
t
u
t, u
˙
t
=
-
d
1
t
u
t
+
p
1
t
x
t
-
τ, y
˙
t
=
y
t
a
2
t
-
r
t
y
t
/
x
t
+
k
t
-
e
2
t
ν
t, and ν
˙
(
t
)
=
-
d
2
(
t
)
ν
(
t
)
+
p
2
(
t
)
y
(
t
-
τ
). Sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained.


---

## Body

## 1. Introduction

In recent years, the modified predator-prey systems with periodic or almost periodic coefficients have been studied extensively.Leslie [1] proposed the famous Leslie predator-prey system as follows:
(1)
x
˙
(
t
)
=
x
(
a
-
b
x
)
-
p
(
x
)
y
,
y
˙
=
y
(
e
-
f
y
x
)
,
where x and y stand for the population of the prey and the predator at time t, respectively, and p
(
x
) is the so-called predator functional response to the prey. The term y
/
x is the Leslie-Gower term which measures the loss in the predator population due to rarity of its favorite food.Global stability of the positive locally asymptotically stable equilibrium in a class of predator-prey systems has been introduced by Hsu and Huang [2], and the system is as follows:
(2)
d
x
d
t
=
r
x
(
1
-
x
k
-
y
p
(
x
)
)
,
d
y
d
x
=
y
[
s
(
1
-
h
y
s
)
]
,
x
(
0
)
>
0
,
y
(
0
)
>
0
,
r
,
s
,
k
,
h
>
0
.
When the functional response p
(
x
) equals m
x, then (2) turns into a Leslie-Gower system [3].On the other hand, the periodic solution (almost periodic solution) and some other properties of Leslie-Gower predator-prey models were studied (see [4–9]). In particular, Zhang [10] discussed the almost periodic solution of a modified Leslie-Gower predator-prey model with the Beddington-DeAngelis function response as follows:
(3)
x
˙
(
t
)
=
x
(
t
)
(
h
y
s
r
1
(
t
)
-
b
(
t
)
x
(
t
)
x
˙
(
t
)
=
x
(
t
)
x
-
c
(
t
)
y
(
t
)
α
(
t
)
+
β
(
t
)
x
(
t
)
+
γ
(
t
)
y
(
t
)
)
,
y
˙
(
t
)
=
y
(
t
)
(
r
2
(
t
)
-
d
(
t
)
y
(
t
)
x
(
t
)
+
k
(
t
)
)
,
where x
(
t
) is the size of prey population and y
(
t
) is the size of predator population.Stimulated by the above reasons, in this paper, we incorporate the feedback control into model (3) and consider the following model:
(4)
x
˙
(
t
)
=
x
(
t
)
(
h
y
s
a
1
(
t
)
-
b
(
t
)
x
(
t
)
x
˙
(
t
)
=
x
(
t
)
x
-
c
(
t
)
y
(
t
)
α
(
t
)
+
β
(
t
)
x
(
t
)
+
γ
(
t
)
y
(
t
)
-
e
1
(
t
)
u
(
t
)
)
,
u
˙
(
t
)
=
-
d
1
(
t
)
u
(
t
)
+
p
1
(
t
)
x
(
t
-
τ
)
,
y
˙
(
t
)
=
y
(
t
)
(
a
2
(
t
)
-
r
(
t
)
y
(
t
)
x
(
t
)
+
k
(
t
)
-
e
2
(
t
)
ν
(
t
)
)
,
ν
˙
(
t
)
=
-
d
2
(
t
)
ν
(
t
)
+
p
2
(
t
)
y
(
t
-
τ
)
,
where τ
>
0 and all the coefficients b
(
t
), c
(
t
), r
(
t
), k
(
t
), α
(
t
), β
(
t
), γ
(
t
), a
i
(
t
), d
i
(
t
), p
i
(
t
), and e
i
(
t
)
(
i
=
1,2
) are all continuous, almost periodic functions on R.Associated with (4), we consider a group of initial conditions with the following form (we assume, without loss of generality, that the initial time t
0
=
0):
(5)
x
(
s
)
=
ϕ
(
s
)
≥
0
,
s
∈
[
-
τ
,
0
]
,
ϕ
(
0
)
>
0
,
y
(
s
)
=
φ
(
s
)
≥
0
,
s
∈
[
-
τ
,
0
]
,
φ
(
0
)
>
0
,
u
(
0
)
>
0
,
ν
(
0
)
>
0
.Letf be a continuous bounded function on R and we set
(6)
f
l
=
inf
⁡
t
∈
R
f
(
t
)
,
f
u
=
sup
⁡
t
∈
R
f
(
t
)
.
Throughout this paper, we assume that the coefficients of the almost periodic system (4) satisfy
(7)
min
⁡
i
=
1,2
{
b
l
,
c
l
,
α
l
,
β
l
,
γ
l
,
r
l
,
k
l
,
a
i
l
,
d
i
l
,
p
i
l
,
e
i
l
}
>
0
,
max
⁡
i
=
1,2
{
b
u
,
c
u
,
α
u
,
β
u
,
γ
u
,
r
u
,
k
u
,
a
i
u
,
d
i
u
,
p
i
u
,
e
i
u
}
<
+
∞
.
By constructing a suitable Lyapunov functional, we obtain some sufficient conditions for the existence of a globally attractive positive almost periodic solution of system (4) with initial conditions (5).


## 2. Permanence

In this section, we give some definitions and results that we will use in the rest of the paper.Lemma 1 (see [11]).
Ifa
>
0, b
>
0, and x
˙
≥
(
≤
)
x
(
b
-
a
x
), when t
≥
0 and x
(
0
)
>
0, one has
(8)
liminf
⁡
t
→
+
∞
x
(
t
)
≥
b
a
,
(
limsup
⁡
t
→
+
∞
x
(
t
)
≤
b
a
)
.Lemma 2 (see [11]).
Ifa
>
0, b
>
0, and x
˙
≥
(
≤
)
b
-
a
x, when t
≥
0 and x
(
0
)
>
0, one has
(9)
liminf
⁡
t
→
+
∞
x
(
t
)
≥
b
a
,
(
limsup
⁡
t
→
+
∞
x
(
t
)
≤
b
a
)
.Set the following:(10)
M
1
=
a
1
u
b
l
,
L
1
=
p
1
u
M
1
d
1
l
,
M
2
=
a
2
u
(
M
1
+
k
u
)
r
l
,
L
2
=
p
2
u
M
2
d
2
l
,
m
1
=
a
1
l
-
c
u
/
r
l
-
e
1
u
L
1
b
u
,
l
1
=
p
1
l
m
1
d
1
u
,
m
2
=
1
r
u
(
a
2
l
-
e
2
u
L
2
)
(
m
1
+
k
l
)
,
l
2
=
p
2
l
m
2
d
2
u
.Theorem 3.
Suppose that system (4) with initial condition (5) satisfies the following condition:
(11)
a
1
l
-
c
u
r
l
-
e
1
u
L
1
>
0
,
a
2
l
-
e
2
u
L
2
>
0
.
Then system (4) is permanent; that is, any positive solution (
x
(
t
)
,
u
(
t
)
,
y
(
t
)
,
ν
(
t
)
)
T of the system (4) satisfies
(12)
0
<
m
1
≤
liminf
⁡
t
→
+
∞
x
(
t
)
≤
limsup
⁡
t
→
+
∞
x
(
t
)
≤
M
1
,
0
<
l
1
≤
liminf
⁡
t
→
+
∞
u
(
t
)
≤
limsup
⁡
t
→
+
∞
u
(
t
)
≤
L
1
,
0
<
m
2
≤
liminf
⁡
t
→
+
∞
y
(
t
)
≤
limsup
⁡
t
→
+
∞
y
(
t
)
≤
M
2
,
0
<
l
2
≤
liminf
⁡
t
→
+
∞
ν
(
t
)
≤
limsup
⁡
t
→
+
∞
ν
(
t
)
≤
L
2
.Proof.
From the first equation of (4), we have the following:
(13)
x
˙
(
t
)
≤
x
(
t
)
(
a
1
u
-
b
l
x
(
t
)
)
.
Applying Lemma 1 to (13) leads to
(14)
limsup
⁡
t
→
+
∞
x
(
t
)
≤
a
1
u
b
l
=
M
1
.
From (14), we know that there exists an enough large T
1
>
0 such that
(15)
x
(
t
)
≤
M
1
,
t
≥
T
1
>
0
,
so there exists an enough large T
2
=
T
1
+
τ such that
(16)
x
(
t
-
τ
)
≤
M
1
,
t
≥
T
2
>
0
.
It follows from (16) and the second equation of system (4) that, for t
≥
T
2,
(17)
u
˙
(
t
)
≤
-
d
l
u
(
t
)
+
p
1
u
M
1
.
Applying Lemma 2 to (17) leads to
(18)
limsup
⁡
t
→
+
∞
u
(
t
)
≤
p
1
u
M
1
d
1
l
=
L
1
.
By using a similar argument as that in the proof of (14) and (18), we can get the following:
(19)
limsup
⁡
t
→
+
∞
y
(
t
)
≤
a
2
u
(
M
1
+
k
u
)
r
l
=
M
2
,
limsup
⁡
t
→
+
∞
ν
(
t
)
≤
p
2
u
M
2
d
2
l
=
L
2
.
From (18) and the first equation of system (4) we know
(20)
x
˙
(
t
)
≥
x
(
t
)
(
a
1
l
-
c
u
γ
l
-
e
1
u
L
1
-
b
u
x
(
t
)
)
.
Applying Lemma 1 and (11) to the above leads to
(21)
liminf
⁡
t
→
+
∞
x
(
t
)
≥
a
1
l
-
c
u
/
r
l
-
e
1
u
L
1
b
u
=
m
1
.
Therefore, we know that there exists an enough large T
3 such that
(22)
x
(
t
)
≥
m
1
,
t
≥
T
3
>
0
.
From the second equation of system (4) we have the following:
(23)
u
˙
(
t
)
≥
-
d
1
u
u
(
t
)
+
p
1
l
m
1
.
Applying Lemma 2 to the above, we obtain the following:
(24)
liminf
⁡
t
→
+
∞
u
(
t
)
≥
p
1
l
m
1
d
1
u
=
l
1
.
By using a similar method as that in the proof of (21) and (24), it follows that
(25)
liminf
⁡
t
→
+
∞
y
(
t
)
≥
1
r
u
(
a
2
l
-
e
2
u
L
2
)
(
m
1
+
k
l
)
=
m
2
liminf
⁡
t
→
+
∞
ν
(
t
)
≥
p
2
l
m
2
d
2
u
=
l
2
.
This completes the proof.We denote byΩ the set of all solutions z
(
t
)
=
(
x
(
t
)
,
u
(
t
)
,
y
(
t
)
,
ν
(
t
)
)
T of system (4) satisfying m
1
≤
x
(
t
)
≤
M
1, l
1
≤
u
(
t
)
≤
L
1, m
2
≤
y
(
t
)
≤
M
2, and l
2
≤
ν
(
t
)
≤
L
2 for all t
>
0.Theorem 4.
Consider the following:Ω
≠
∅.Proof.
From the properties of almost periodic function there exists a sequence{
t
n
} with t
n
→
+
∞ as n
→
+
∞ such that
(26)
a
i
(
t
+
t
n
)
⟶
a
i
(
t
)
,
d
i
(
t
+
t
n
)
⟶
d
i
(
t
)
,
e
i
(
t
+
t
n
)
⟶
e
i
(
t
)
,
p
i
(
t
+
t
n
)
⟶
p
i
(
t
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
i
=
1,2
)
,
b
(
t
+
t
n
)
⟶
b
(
t
)
,
c
(
t
+
t
n
)
⟶
c
(
t
)
,
r
(
t
+
t
n
)
⟶
r
(
t
)
,
k
(
t
+
t
n
)
⟶
k
(
t
)
,
α
(
t
+
t
n
)
⟶
α
(
t
)
,
β
(
t
+
t
n
)
⟶
β
(
t
)
,
γ
(
t
+
t
n
)
⟶
γ
(
t
)
,
as n
→
∞ uniformly on R. Let z
(
t
)
=
(
x
(
t
)
,
u
(
t
)
,
y
(
t
)
,
ν
(
t
)
)
T be a solution of system (4) satisfying m
1
≤
x
(
t
)
≤
M
1, l
1
≤
u
(
t
)
≤
L
1, m
2
≤
y
(
t
)
≤
M
2, and l
2
≤
ν
(
t
)
≤
L
2 for t
>
T. Clearly, the sequence z
(
t
+
t
n
) is uniformly bounded and equicontinuous on each bounded subset of R. Therefore, by the Arzelà-Ascoli theorem, there exists a subsequence z
(
t
+
t
k
) which converges to a continuous function z
*
(
t
)
=
(
x
*
(
t
)
,
u
*
(
t
)
,
y
*
(
t
)
,
ν
*
(
t
)
)
T as k
→
+
∞ uniformly on each bounded subset of R. Let T
0
∈
R be given. We may assume that t
k
+
T
0
≥
T for all k. For t
≥
0, we have the following:
(27)
x
(
t
+
t
k
+
T
0
)
=
x
(
t
k
+
T
0
)
+
∫
T
0
t
+
T
0
‍
x
(
s
+
t
k
)
(
a
1
(
s
+
t
k
)
-
b
(
s
+
t
k
)
x
(
s
+
t
k
)
+
∫
T
0
t
+
T
0
‍
x
(
s
+
t
k
)
a
-
-
(
c
(
s
+
t
k
)
y
(
s
+
t
k
)
)
+
∫
T
0
t
+
T
0
‍
x
(
s
+
t
k
)
a
-
×
(
α
(
s
+
t
k
)
+
β
(
s
+
t
k
)
x
(
s
+
t
k
)
+
∫
T
0
t
+
T
0
‍
x
(
s
+
t
k
)
a
-
×
×
+
γ
(
s
+
t
k
)
y
(
s
+
t
k
)
)
-
1
+
∫
T
0
t
+
T
0
‍
x
(
s
+
t
k
)
a
-
-
e
1
(
s
+
t
k
)
u
(
s
+
t
k
)
)
d
s
,
u
(
t
+
t
k
+
T
0
)
=
u
(
t
k
+
T
0
)
-
∫
T
0
t
+
T
0
‍
d
1
(
s
+
t
k
)
u
(
s
+
t
k
)
+
p
1
(
s
+
t
k
)
x
(
s
+
t
k
-
τ
)
d
s
,
y
(
t
+
t
k
+
T
0
)
=
y
(
t
k
+
T
0
)
+
∫
T
0
t
+
T
0
‍
y
(
s
+
t
k
)
(
a
2
(
s
+
t
k
)
-
r
(
s
+
t
k
)
y
(
s
+
t
k
)
x
(
s
+
t
k
)
+
k
(
s
+
t
k
)
+
∫
T
0
t
+
T
0
‍
y
(
s
+
t
k
)
-
-
e
2
(
s
+
t
k
)
ν
(
s
+
t
k
)
r
(
s
+
t
k
)
y
(
s
+
t
k
)
x
(
s
+
t
k
)
+
k
(
s
+
t
k
)
)
d
s
,
ν
˙
(
t
+
t
k
+
T
0
)
=
ν
(
t
k
+
T
0
)
+
∫
T
0
t
+
T
0
‍
-
d
2
(
s
+
t
k
)
ν
(
s
+
t
k
)
+
p
2
(
s
+
t
k
)
y
(
s
+
t
k
-
τ
)
d
s
.
Applying Lebesgue’s dominated convergence theorem and letting k
→
+
∞ in (27), we obtain the following:
(28)
x
*
(
t
+
T
0
)
=
x
*
(
T
0
)
+
∫
T
0
t
+
T
0
‍
x
*
(
s
)
(
a
1
(
s
)
-
b
(
s
)
x
*
(
s
)
x
*
(
t
+
-
T
0
)
=
-
c
(
s
)
y
*
(
s
)
α
(
s
)
+
β
(
s
)
x
*
(
s
)
+
γ
(
s
)
y
*
(
s
)
x
*
(
t
+
-
T
0
)
=
-
e
1
(
s
)
u
*
(
s
)
)
d
s
,
u
*
(
t
+
T
0
)
=
u
*
(
T
0
)
u
*
(
t
+
T
0
)
=
-
∫
T
0
t
+
T
0
‍
d
1
(
s
)
u
*
(
s
)
+
p
1
(
s
)
x
*
(
s
-
τ
)
d
s
,
y
*
(
t
+
T
0
)
=
y
*
(
T
0
)
y
*
(
t
+
T
0
)
=
+
∫
T
0
t
+
T
0
‍
y
*
(
s
)
(
a
2
(
s
)
-
r
(
s
)
y
*
(
s
)
x
*
(
s
)
+
k
(
s
)
y
*
(
t
+
T
0
)
=
+
∫
T
0
t
+
T
0
‍
y
*
(
s
)
-
r
(
s
)
y
*
(
s
)
x
*
(
s
)
+
k
(
s
)
-
e
2
(
s
)
ν
*
(
s
)
)
d
s
,
ν
*
(
t
+
T
0
)
=
ν
*
(
T
0
)
ν
*
(
t
+
T
0
)
=
+
∫
T
0
t
+
T
0
‍
-
d
2
(
s
)
ν
*
(
s
)
+
p
2
(
s
)
y
*
(
s
-
τ
)
d
s
.
Since T
0
∈
R is arbitrarily given, z
*
(
t
)
=
(
x
*
(
t
)
,
u
*
(
t
)
,
y
*
(
t
)
,
ν
*
(
t
)
)
T is a solution of system (4) on R. It is clear that m
1
≤
x
*
(
t
)
≤
M
1, l
1
≤
u
*
(
t
)
≤
L
1, m
2
≤
y
*
(
t
)
≤
M
2, l
2
≤
ν
*
(
t
)
≤
L
2 for t
∈
R. Thus z
*
(
t
)
∈
Ω. This completes the proof.


## 3. Existence of a Unique Almost Periodic Solution

Now let us state several definitions and lemmas which will be useful in the proving of the main result of this section.Definition 5 (see [12]).
A functionf
(
t
,
x
), where f is an m
-vector, t is a real scalar, and x is an n
-vector, is said to be almost periodic in t uniformly with respect to x
∈
S
⊂
R
n, if f
(
t
,
x
) is continuous in t
∈
R and x
∈
S and if, for any ε
>
0, there is a constant l
(
ε
)
>
0 such that in any interval of length l
(
ε
) there exists a ς such that the inequality
(29)
|
f
(
t
+
ς
,
x
)
-
f
(
t
,
x
)
|
<
ε
is satisfied for all t
∈
(
-
∞
,
+
∞
), x
∈
S. The number ς is called an ε
-
t
r
a
n
s
l
a
t
i
o
n number of f
(
t
,
x
).Definition 6 (see [12]).
A functionf
:
R
→
R is said to be asymptotically almost periodic function, if there exists an almost periodic function q
(
t
) and a continuous function r
(
t
) such that f
(
t
)
=
q
(
t
)
+
r
(
t
), t
∈
R and r
(
t
)
→
0 as t
→
∞.Lemma 7 (see [13]).
Letf be a nonnegative, integral, and uniformly continuous function defined on [
0
,
+
∞
); then lim
⁡
t
→
+
∞
f
(
t
)
=
0.Theorem 8.
Suppose that all conditions of Theorem3 hold; furthermore assume that(H)
Θ
>
0, where Θ
=
min
⁡
{
Θ
1
,
Θ
2
,
Θ
3
,
Θ
4
},
(30)
Θ
1
=
b
l
m
1
-
p
1
u
M
1
-
c
u
β
u
M
1
M
2
(
α
l
+
β
l
m
1
+
γ
l
m
2
)
2
Θ
1
=
-
r
u
M
1
M
2
(
m
1
+
k
l
)
2
>
0
,
Θ
2
=
γ
l
M
1
+
k
u
-
c
l
m
2
α
u
+
β
u
M
1
+
γ
u
M
2
Θ
2
=
-
c
u
γ
u
M
2
2
(
α
l
+
β
l
m
1
+
γ
l
m
2
)
2
-
p
2
u
M
2
>
0
,
Θ
3
=
d
1
l
-
e
1
u
,
Θ
4
=
d
2
l
-
e
2
u
.Then system (4) with initial conditions (5) is globally attractive.Proof.
Letx
(
t
)
=
e
x
1
(
t
), y
(
t
)
=
e
y
1
(
t
), and then system (4) is transformed into
(31)
x
˙
1
(
t
)
=
a
1
(
t
)
-
b
(
t
)
e
x
1
(
t
)
x
˙
1
(
t
)
=
-
c
(
t
)
e
y
1
(
t
)
α
(
t
)
+
β
(
t
)
e
x
1
(
t
)
+
γ
(
t
)
e
y
1
(
t
)
-
e
1
(
t
)
u
(
t
)
,
u
˙
(
t
)
=
-
d
1
(
t
)
u
(
t
)
+
p
1
(
t
)
e
x
1
(
t
-
τ
)
,
y
˙
1
(
t
)
=
a
2
(
t
)
-
r
(
t
)
e
y
1
(
t
)
e
x
1
(
t
)
+
k
(
t
)
-
e
2
(
t
)
ν
(
t
)
,
ν
˙
(
t
)
=
-
d
2
(
t
)
ν
(
t
)
+
p
2
(
t
)
e
y
1
(
t
-
τ
)
.
Suppose that z
1
(
t
)
=
(
x
1
(
t
)
,
u
(
t
)
,
y
1
(
t
)
,
ν
(
t
)
)
T and z
1
*
(
t
)
=
(
x
1
*
(
t
)
,
u
*
(
t
)
,
y
1
*
(
t
)
,
ν
*
(
t
)
)
T are any two positive solutions of (31).
LetV
(
t
)
=
V
1
(
t
)
+
V
2
(
t
)
+
V
3
(
t
)
+
V
4
(
t
), where
(32)
V
1
(
t
)
=
|
x
1
(
t
)
-
x
1
*
(
t
)
|
,
V
2
(
t
)
=
|
u
(
t
)
-
u
*
(
t
)
|
+
p
1
u
∫
t
-
τ
t
‍
|
e
x
1
(
t
)
-
e
x
1
*
(
t
)
|
d
s
,
V
3
(
t
)
=
|
y
1
(
t
)
-
y
1
*
(
t
)
|
,
V
4
(
t
)
=
|
ν
(
t
)
-
ν
*
(
t
)
|
+
p
2
u
∫
t
-
τ
t
‍
|
e
y
1
(
t
)
-
e
y
1
*
(
t
)
|
d
s
.
Calculating the right derivative D
+
V
1
(
t
) of V
1
(
t
) along the solution of (31), we have the following:
(33)
D
+
V
1
(
t
)
=
sgn
⁡
(
x
1
(
t
)
-
x
1
*
(
t
)
)
D
+
V
1
(
t
)
=
×
[
-
b
(
t
)
(
e
x
1
(
t
)
-
e
x
1
*
(
t
)
)
D
+
V
1
(
t
)
=
-
-
-
c
(
t
)
e
y
1
(
t
)
α
(
t
)
+
β
(
t
)
e
x
1
(
t
)
+
γ
(
t
)
e
y
1
(
t
)
D
+
V
1
(
t
)
=
-
-
+
c
(
t
)
e
y
1
*
(
t
)
α
(
t
)
+
β
(
t
)
e
x
1
*
(
t
)
+
γ
(
t
)
e
y
1
*
(
t
)
D
+
V
1
(
t
)
=
-
-
-
e
1
(
t
)
(
u
(
t
)
-
u
*
(
t
)
)
(
e
x
1
(
t
)
-
e
x
1
*
(
t
)
)
]
D
+
V
1
(
t
)
=
sgn
⁡
(
x
1
(
t
)
-
x
1
*
(
t
)
)
D
+
V
1
(
t
)
=
×
[
-
b
(
t
)
e
ξ
(
t
)
(
x
1
(
t
)
-
x
1
*
(
t
)
)
D
+
V
1
(
t
)
=
-
-
-
c
(
t
)
e
y
1
(
t
)
α
(
t
)
+
β
(
t
)
e
x
1
(
t
)
+
γ
(
t
)
e
y
1
(
t
)
D
+
V
1
(
t
)
=
-
-
+
c
(
t
)
e
y
1
*
(
t
)
α
(
t
)
+
β
(
t
)
e
x
1
(
t
)
+
γ
(
t
)
e
y
1
(
t
)
D
+
V
1
(
t
)
=
-
-
-
c
(
t
)
e
y
1
*
(
t
)
α
(
t
)
+
β
(
t
)
e
x
1
(
t
)
+
γ
(
t
)
e
y
1
(
t
)
D
+
V
1
(
t
)
=
-
-
+
c
(
t
)
e
y
1
*
(
t
)
α
(
t
)
+
β
(
t
)
e
x
1
*
(
t
)
+
γ
(
t
)
e
y
1
*
(
t
)
D
+
V
1
(
t
)
=
-
-
-
e
1
(
t
)
(
u
(
t
)
-
u
*
(
t
)
)
e
ξ
(
t
)
(
x
1
(
t
)
-
x
1
*
(
t
)
)
]
D
+
V
1
(
t
)
≤
sgn
⁡
(
x
1
(
t
)
-
x
1
*
(
t
)
)
D
+
V
1
(
t
)
=
×
[
-
b
l
m
1
(
x
1
(
t
)
-
x
1
*
(
t
)
)
]
D
+
V
1
(
t
)
=
-
c
(
t
)
α
(
t
)
+
β
(
t
)
e
x
1
(
t
)
+
γ
(
t
)
e
y
1
(
t
)
D
+
V
1
(
t
)
=
·
e
η
(
t
)
(
y
1
(
t
)
-
y
1
*
(
t
)
)
D
+
V
1
(
t
)
=
-
+
(
c
(
t
)
e
y
1
*
(
t
)
[
β
(
t
)
(
e
x
1
(
t
)
-
e
x
1
*
(
t
)
)
D
+
V
1
(
t
)
=
xxxxxxxxxxxx
+
γ
(
t
)
(
e
y
1
(
t
)
-
e
y
1
*
(
t
)
)
]
)
D
+
V
1
(
t
)
=
-
×
(
(
α
(
t
)
+
β
(
t
)
e
x
1
*
(
t
)
+
γ
(
t
)
e
y
1
*
(
t
)
)
D
+
V
1
(
t
)
=
-
xxx
×
(
α
(
t
)
+
β
(
t
)
e
x
1
*
(
t
)
+
γ
(
t
)
e
y
1
*
(
t
)
)
)
-
1
D
+
V
1
(
t
)
=
-
-
e
1
(
t
)
(
u
(
t
)
-
u
*
(
t
)
)
D
+
V
1
(
t
)
≤
(
c
u
β
u
M
1
M
2
(
α
l
+
β
l
m
1
+
γ
l
m
2
)
2
-
b
l
m
1
)
D
+
V
1
(
t
)
=
×
|
x
1
(
t
)
-
x
1
*
(
t
)
|
D
+
V
1
(
t
)
=
+
(
c
u
γ
u
M
2
2
(
α
l
+
β
l
m
1
+
γ
l
m
2
)
2
D
+
V
1
(
t
)
=
x
x
x
+
c
l
m
2
α
u
+
β
u
M
1
+
γ
u
M
2
c
u
γ
u
M
2
2
(
α
l
+
β
l
m
1
+
γ
l
m
2
)
2
)
D
+
V
1
(
t
)
=
×
|
y
1
(
t
)
-
y
1
*
(
t
)
|
D
+
V
1
(
t
)
=
+
e
1
u
|
u
(
t
)
-
u
*
(
t
)
|
.
Further, it follows that
(34)
D
+
V
2
(
t
)
=
sgn
⁡
(
u
(
t
)
-
u
*
(
t
)
)
×
(
e
x
1
*
(
t
-
τ
)
-
d
1
(
t
)
(
u
(
t
)
-
u
*
(
t
)
)
D
D
D
+
p
1
(
t
)
(
e
x
1
(
t
-
τ
)
-
e
x
1
*
(
t
-
τ
)
)
D
D
D
+
p
1
u
(
e
x
1
(
t
)
-
e
x
1
*
(
t
)
)
D
D
D
-
p
1
(
t
)
(
e
x
1
(
t
-
τ
)
-
e
x
1
*
(
t
-
τ
)
)
)
≤
-
d
1
l
|
u
(
t
)
-
u
*
(
t
)
|
+
p
1
u
M
1
|
x
1
(
t
)
-
x
1
*
(
t
)
|
,
D
+
V
3
(
t
)
=
sgn
⁡
(
y
1
(
t
)
-
y
1
*
(
t
)
)
×
[
-
r
(
t
)
e
y
1
(
t
)
e
x
1
(
t
)
+
k
(
t
)
+
r
(
t
)
e
y
1
*
(
t
)
e
x
1
*
(
t
)
+
k
(
t
)
DDDD
-
e
2
(
t
)
(
ν
(
t
)
-
ν
*
(
t
)
)
r
(
t
)
e
y
1
(
t
)
e
x
1
(
t
)
+
k
(
t
)
]
≤
-
r
l
m
2
M
1
+
k
u
|
y
1
(
t
)
-
y
1
*
(
t
)
|
+
r
u
M
1
M
1
(
m
1
+
k
l
)
2
|
x
1
(
t
)
-
x
1
*
(
t
)
|
+
e
2
u
|
ν
(
t
)
-
ν
*
(
t
)
|
,
D
+
V
4
(
t
)
≤
-
d
2
l
|
ν
(
t
)
-
ν
*
(
t
)
|
+
p
2
u
M
2
|
y
1
(
t
)
-
y
1
*
(
t
)
|
.
Therefore, we have the following:
(35)
D
+
V
(
t
)
=
D
+
V
1
(
T
)
+
D
+
V
2
(
T
)
+
D
+
V
3
(
T
)
+
D
+
V
4
(
T
)
≤
-
(
b
l
m
1
-
p
1
u
M
1
-
c
u
β
u
M
1
M
2
(
α
l
+
β
l
m
1
+
γ
l
m
2
)
2
-
r
u
M
1
M
1
(
m
1
+
k
l
)
2
)
×
|
x
1
(
t
)
-
x
1
*
(
t
)
|
-
(
c
u
γ
u
M
2
2
(
α
l
+
β
l
m
1
+
γ
l
m
2
)
2
r
l
m
2
M
1
+
k
u
-
c
l
m
2
α
u
+
β
u
M
1
+
γ
u
M
2
x
x
-
c
u
γ
u
M
2
2
(
α
l
+
β
l
m
1
+
γ
l
m
2
)
2
-
p
2
u
M
2
)
|
y
1
(
t
)
-
y
1
*
(
t
)
|
-
(
d
1
l
-
e
1
u
)
|
u
(
t
)
-
u
*
(
t
)
|
-
(
d
2
l
-
e
2
u
)
|
ν
(
t
)
-
ν
*
(
t
)
|
≤
-
Θ
(
|
x
1
(
t
)
-
x
1
*
(
t
)
|
+
|
y
1
(
t
)
-
y
1
*
(
t
)
|
≤
-
Θ
2
+
|
u
(
t
)
-
u
*
(
t
)
|
+
|
ν
(
t
)
-
ν
*
(
t
)
|
)
.
Integrating the above inequality on internal [
0
,
t
], it follows that, for t
≥
0,
(36)
V
(
t
)
+
Θ
∫
0
t
‍
|
x
1
(
t
)
-
x
1
*
(
t
)
|
+
|
y
1
(
t
)
-
y
1
*
(
t
)
|
V
(
t
)
+
Θ
∫
0
t
‍
+
|
u
(
t
)
-
u
*
(
t
)
|
+
|
ν
(
t
)
-
ν
*
(
t
)
|
d
s
≤
V
(
0
)
<
+
∞
.
Then, for t
>
0, we obtain that
(37)
∫
0
t
‍
|
x
1
(
t
)
-
x
1
*
(
t
)
|
+
|
y
1
(
t
)
-
y
1
*
(
t
)
|
∫
0
t
‍
+
|
u
(
t
)
-
u
*
(
t
)
|
+
|
ν
(
t
)
-
ν
*
(
t
)
|
d
s
≤
V
(
0
)
Θ
<
+
∞
.
By Lemma 7, we obtain
(38)
lim
⁡
t
→
+
∞
|
x
1
(
t
)
-
x
1
*
(
t
)
|
=
0
,
lim
⁡
t
→
+
∞
|
y
1
(
t
)
-
y
1
*
(
t
)
|
=
0
,
lim
⁡
t
→
+
∞
|
u
(
t
)
-
u
*
(
t
)
|
=
0
,
lim
⁡
t
→
+
∞
|
ν
(
t
)
-
ν
*
(
t
)
|
=
0
.
Then the solution of systems (4) and (5) is globally attractive.Theorem 9.
Suppose that all conditions of Theorem8 hold; then there exists a unique almost periodic solution of systems (4) and (5).Proof.
According to Theorem4, there exists a bounded positive solution W
(
t
)
=
(
w
1
(
t
)
,
w
2
(
t
)
,
w
3
(
t
)
,
w
4
(
t
)
)
T of (4) and (5). Then there exists a sequence {
t
k
′
}, t
k
′
→
∞ as k
→
∞, such that (
w
1
(
t
+
t
k
′
)
,
w
2
(
t
+
t
k
′
)
,
w
3
(
t
+
t
k
′
)
,
w
4
(
t
+
t
k
′
)
)
T is a solution of the following system:
(39)
x
˙
(
t
)
=
x
(
t
)
(
a
1
(
t
+
t
k
′
)
-
b
(
t
+
t
k
′
)
x
(
t
)
xxxxx
-
c
(
t
+
t
k
′
)
y
(
t
)
α
(
t
+
t
k
′
)
+
β
(
t
+
t
k
′
)
x
(
t
)
+
γ
(
t
+
t
k
′
)
y
(
t
)
xxxxx
-
e
1
(
t
+
t
k
′
)
u
(
t
)
)
,
u
˙
(
t
)
=
-
d
1
(
t
+
t
k
′
)
u
(
t
)
+
p
1
(
t
+
t
k
′
)
x
(
t
-
τ
)
,
y
˙
(
t
)
=
y
(
t
)
(
a
2
(
t
+
t
k
′
)
-
r
(
t
+
t
k
′
)
y
(
t
)
x
(
t
)
+
k
(
t
+
t
k
′
)
xxxxx
-
e
2
(
t
+
t
k
′
)
ν
(
t
)
r
(
t
+
t
k
′
)
y
(
t
)
x
(
t
)
+
k
(
t
+
t
k
′
)
)
,
ν
˙
(
t
)
=
-
d
2
(
t
+
t
k
′
)
ν
(
t
)
+
p
2
(
t
+
t
k
′
)
y
(
t
-
τ
)
.
According to Theorem 3, we get that not only {
(
w
1
(
t
+
t
k
′
)
,
w
2
(
t
+
t
k
′
)
,
w
3
(
t
+
t
k
′
)
,
w
4
(
t
+
t
k
′
)
)
T
} but also {
(
w
˙
1
(
t
+
t
k
′
)
,
w
˙
2
(
t
+
t
k
′
)
,
w
˙
3
(
t
+
t
k
′
)
,
w
˙
4
(
t
+
t
k
′
)
)
T
} are uniformly bounded and equicontinuous. By Ascoli’s theorem there exists a uniformly convergent subsequence w
i
(
t
+
t
k
)
⊆
w
i
(
t
+
t
k
′
)
(
i
=
1,2
,
3,4
) such that, for any ε
>
0, there exists a K
(
ε
)
>
0 with the property that if m
,
k
≥
K
(
ε
), then
(40)
|
w
i
(
t
+
t
m
)
-
w
i
(
t
+
t
k
)
|
<
ε
,
(
i
=
1,2
,
3,4
)
.
This is to say, w
i
(
t
+
t
k
)
(
i
=
1,2
,
3,4
) are asymptotically almost periodic functions; hence there exist four almost periodic functions P
i
(
t
+
t
k
)
(
i
=
1,2
,
3,4
) and four continuous functions F
i
(
t
+
t
k
)
(
i
=
1,2
,
3,4
) such that
(41)
w
i
(
t
+
t
k
)
=
P
i
(
t
+
t
k
)
+
F
i
(
t
+
t
k
)
,
t
∈
R
,
i
=
1,2
,
3,4
,
where
(42)
lim
⁡
k
→
+
∞
P
i
(
t
+
t
k
)
=
P
i
(
t
)
,
lim
⁡
k
→
+
∞
⁡
F
i
(
t
+
t
k
)
=
0
,
i
=
1,2
,
3,4
,
P
i
(
t
)
(
i
=
1,2
,
3,4
) are an almost periodic function.
Therefore,(43)
lim
⁡
k
→
+
∞
w
i
(
t
+
t
k
)
=
P
i
(
t
)
,
(
i
=
1,2
,
3,4
)
.
On the other hand,
(44)
lim
⁡
k
→
+
∞
w
˙
i
(
t
+
t
k
)
=
lim
⁡
k
→
+
∞
lim
⁡
h
→
0
w
i
(
t
+
t
k
+
h
)
-
w
i
(
t
+
t
k
)
h
lim
⁡
k
→
+
∞
w
˙
i
(
t
+
t
k
)
=
lim
⁡
h
→
+
∞
lim
⁡
k
→
0
w
i
(
t
+
t
k
+
h
)
-
w
i
(
t
+
t
k
)
h
lim
⁡
k
→
+
∞
w
˙
i
(
t
+
t
k
)
=
lim
⁡
h
→
0
P
i
(
t
+
h
)
-
P
i
(
t
)
h
,
(
i
=
1,2
,
3,4
)
.
So P
˙
i
(
t
)
(
i
=
1,2
,
3,4
) exist. Now we will prove that (
P
1
(
t
)
,
P
2
(
t
)
,
P
3
(
t
)
,
P
4
(
t
)
)
T is an almost periodic solution of system (4).
From properties of almost periodic function, there exits a sequence{
t
n
}, {
t
n
}
→
∞ as n
→
∞, such that
(45)
a
i
(
t
+
t
n
)
⟶
a
i
(
t
)
,
d
i
(
t
+
t
n
)
⟶
d
i
(
t
)
,
e
i
(
t
+
t
n
)
⟶
e
i
(
t
)
,
p
i
(
t
+
t
n
)
⟶
p
i
(
t
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
i
=
1,2
)
,
b
(
t
+
t
n
)
⟶
b
(
t
)
,
c
(
t
+
t
n
)
⟶
c
(
t
)
,
r
(
t
+
t
n
)
⟶
r
(
t
)
,
k
(
t
+
t
n
)
⟶
k
(
t
)
,
α
(
t
+
t
n
)
⟶
α
(
t
)
,
β
(
t
+
t
n
)
⟶
β
(
t
)
,
γ
(
t
+
t
n
)
⟶
γ
(
t
)
,
as n
→
∞ uniformly on R.
It is easy to know thatw
i
(
t
+
t
n
)
→
P
i
(
t
)
(
i
=
1,2
,
3,4
) as n
→
∞, and then we have the following:
(46)
P
˙
1
(
t
)
=
lim
⁡
n
→
+
∞
w
˙
1
(
t
+
t
n
)
=
lim
⁡
n
→
+
∞
[
+
γ
(
t
+
t
n
)
w
3
(
t
+
t
n
)
)
-
1
(
t
+
t
n
)
w
2
(
t
+
t
n
)
-
e
1
(
t
+
t
n
)
w
2
(
t
+
t
n
)
)
)
w
1
(
t
+
t
n
)
(
a
1
(
t
+
t
n
)
-
b
(
t
+
t
n
)
w
1
(
t
+
t
n
)
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
-
(
c
(
t
+
t
n
)
w
3
(
t
+
t
n
)
)
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
×
(
α
(
t
+
t
n
)
+
β
(
t
+
t
n
)
w
1
(
t
+
t
n
)
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
+
γ
(
t
+
t
n
)
w
3
(
t
+
t
n
)
)
-
1
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
-
e
1
(
t
+
t
n
)
w
2
(
t
+
t
n
)
)
+
γ
(
t
+
t
n
)
w
3
(
t
+
t
n
)
)
-
1
]
=
P
1
(
t
)
(
c
(
t
)
P
3
(
t
)
α
(
t
)
+
β
(
t
)
P
1
(
t
)
+
γ
(
t
)
P
3
(
t
)
a
1
(
t
)
-
b
(
t
)
P
1
(
t
)
xxxxxxxx
-
c
(
t
)
P
3
(
t
)
α
(
t
)
+
β
(
t
)
P
1
(
t
)
+
γ
(
t
)
P
3
(
t
)
-
e
1
(
t
)
P
2
(
t
)
)
.
By using a similar argument as that in the above, we have the following:
(47)
P
˙
2
(
t
)
=
-
d
1
(
t
)
P
2
(
t
)
+
p
1
(
t
)
P
1
(
t
-
τ
)
,
P
˙
3
(
t
)
=
P
3
(
t
)
(
a
2
(
t
)
-
r
(
t
)
P
3
(
t
)
P
1
(
t
)
+
k
(
t
)
-
e
2
(
t
)
P
4
(
t
)
)
,
P
˙
4
(
t
)
=
-
d
2
(
t
)
P
4
(
t
)
+
p
2
(
t
)
P
3
(
t
-
τ
)
.
This proves thatP
i
(
t
)
(
i
=
1,2
,
3,4
) is a nonnegative almost periodic solution of systems (4) and (5); by Theorem 8, it follows that there exists a globally asymptotically stable nonnegative almost periodic solution of system (4). The proof is complete.


## 4. An Example

Consider the following system:(48)
x
˙
(
t
)
=
x
(
t
)
(
4
-
2
x
(
t
)
-
10
y
(
t
)
2
+
20
x
(
t
)
+
20
y
(
t
)
-
2
u
(
t
)
)
,
u
˙
(
t
)
=
-
3
u
(
t
)
+
1
5
x
(
t
-
τ
)
,
y
˙
(
t
)
=
y
(
t
)
(
1
10
-
20
y
(
t
)
x
(
t
)
+
23
-
2
5
ν
(
t
)
)
,
ν
˙
(
t
)
=
-
2
ν
(
t
)
+
2
y
(
t
-
τ
)
.By a simple calculation, we check that all conditions in Theorems8 and 9 are fulfilled. Therefore, by Theorems 8 and 9, system (48) has a unique globally asymptotically stable nonnegative almost periodic solution (see Figure 1).Figure 1
Dynamic behavior of system (48) with the initial (
x
(
0
)
,
y
(
0
)
,
u
(
0
)
,
v
(
0
)
)
T
=
(
0.7,1.5,1.0,1.1
)
T, for τ
=
0, t
∈
[
0,5
]. From the figure, we could easily see that the solution (
x
(
t
)
,
y
(
t
)
,
u
(
t
)
,
v
(
t
)
)
T is asymptotic to the unique almost periodic solution of the system (48).



---

*Source: 252579-2014-04-22.xml*