# The Stability of Solutions for a Fractional Predator-Prey System

Authors: Yingjia Guo
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study a class of fractional predator-prey systems with Holling II functional response. A unique positive solution of this system is obtained. In order to prove the asymptotical stability of positive equilibrium for this system, we study the Lyapunov stability theory of a fractional system.


---

## Body

## 1. Introduction

We consider the following fractional predator-prey model with Holling Type II functional response:(1)
d
α
x
(
t
)
=
x
(
t
)
(
a
-
b
x
(
t
)
-
γ
y
(
t
)
1
+
β
x
(
t
)
)
d
t
α
,
d
α
y
(
t
)
=
y
(
t
)
(
-
e
+
k
γ
x
(
t
)
1
+
β
x
(
t
)
)
d
t
α
,
(
0
<
α
≤
1
)
,
where x
(
t
) and y
(
t
) represent the population densities of prey and predator at time t, respectively. The parameters a, b
/
a, γ, β, e, and k are positive constants that stand for prey intrinsic growth rate, carrying capacity, the maximum ingestion rate, half-saturation constant, predator death rate, and the conversion factor, respectively. One of the most popular predator-prey models with Holling Type II functional response is established in [1, 2]. The asymptotic behavior of a stochastic predator-prey system with Holling Type II functional response is studied in [3]. The existence and asymptotical stability of equilibria and limit cycles for predator-prey systems with Holling II are obtained in [4]. A more complicated case about predator-prey systems is studied in [5, 6]. White noise is always present in natural world; Liu et al. studied the asymptotic behavior of a stochastic predator-prey system with Holling II functional response.However, in the real world, there are still many problems that cannot be solved by usual prey-predator model. Some complexity on multiscale analysis can be simplified by the fractional order calculus. Fractional differential equations have been studied in many other fields, for example, economic [7], physics [8–10], material [11], and so forth. In this paper, we present a fractional prey-predator model (1) to describe the ecosystem which performs well in the practical problem.We will study long time behavior of system (1). If a
k
γ
>
a
e
β
+
b
e, then system (1) has a unique positive equilibrium:
(2)
x
*
=
e
k
γ
-
e
β
,
y
*
=
k
γ
(
a
k
γ
-
a
e
β
-
b
e
)
(
k
γ
-
e
β
)
2
.
The positive equilibrium is asymptotically stable, which is proved in Section 5. The fractional derivative of (1) is modified Riemann-Liouville derivative, which is established in [12, 13]. There are some good properties on the fractional derivative to study fractional system, such as the chain rule and fractional Taylor series. The details of modified Riemann-Liouville derivative are given in Section 2. We show that there is a unique nonnegative solution of (1) in Section 3. In order to prove that the positive equilibrium is stable, we give a Lyapunov stability theorem of the fractional system in Section 4.


## 2. Preliminaries

### 2.1. Fractional Derivative via Fractional Difference

For an introduction to the classical fractional calculus we refer the reader to [14–17].Definition 1.
Letf
:
ℜ
→
ℜ, x
→
f
(
x
), be a continuous function, and let h
>
0 be a constant discretization span. Define the forward operator F
W
(
h
); that is, (the symbol := means that the left side is defined by the right one)
(3)
F
W
(
h
)
·
f
(
x
)
:
=
f
(
x
+
h
)
.
Then the fractional difference of order α, 0
<
α
<
1, of f
(
x
) is defined by the expression
(4)
Δ
α
f
(
x
)
:
=
(
F
W
-
1
)
α
f
(
x
)
iiiiiiiiiiii
=
∑
k
=
0
∞
‍
(
-
1
)
k
(
α
k
)
f
(
x
+
(
α
-
k
)
h
)
,
and its derivative of fractional order is defined by the expression
(5)
f
(
α
)
(
x
)
=
lim
⁡
h
↓
0
Δ
α
f
(
x
)
h
α
,
0
<
α
≤
1
.


### 2.2.  Modified Riemann-Liouville Derivative

In this section we briefly review the main notions and results from the recent fractional calculus proposed by Jumarie [12, 18, 19].Definition 2 (Riemann-Liouville definition revisited).
Refer to the function of Definition1. Then its fractional derivative of order α is defined by the expression
(6)
f
(
α
)
(
x
)
:
=
1
Γ
(
-
α
)
∫
0
x
‍
(
x
-
ξ
)
-
α
-
1
(
f
(
ξ
)
-
f
(
0
)
)
d
ξ
,
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
α
<
0
.
For positiveα, one will set
(7)
f
(
α
)
(
x
)
=
(
f
(
α
-
1
)
(
x
)
)
′
,
0
<
α
<
1
iiiiiiiiiii
=
1
Γ
(
1
-
α
)
d
d
x
∫
0
x
‍
(
x
-
ξ
)
-
α
(
f
(
ξ
)
-
f
(
0
)
)
d
ξ
,
f
(
α
)
(
x
)
:
=
(
f
(
n
)
(
x
)
)
(
α
-
n
)
,
n
≤
α
<
n
+
1
,
n
≥
1
.


### 2.3. Useful Relations

Here we give some properties of the modified Riemann-Liouville derivative (see [13]) which are used further in this paper.(i)
Consider(8)
Γ
(
1
+
α
k
)
=
:
(
α
k
)
!
.
(ii)
Useful differential relation:(9)
d
α
f
≅
Γ
(
1
+
α
)
d
f
,
0
<
α
≤
1
.
(iii)
Consider(10)
D
α
x
γ
=
Γ
(
1
+
γ
)
Γ
-
1
(
γ
+
1
-
α
)
x
γ
-
α
,
γ
>
0
.
(iv)
The chain rule:(11)
(
u
(
x
)
v
(
x
)
)
(
α
)
=
u
(
α
)
(
x
)
v
(
x
)
+
u
(
x
)
v
(
α
)
(
x
)
,
i
i
(
f
[
u
(
x
)
]
)
(
α
)
=
f
u
′
(
u
)
u
(
α
)
(
x
)
.


### 2.4. Integration with respect tod
t
α

The solution of the equation(12)
i
d
α
x
=
f
(
t
)
d
t
α
,
t
≥
0
,
x
(
0
)
=
x
0
,
is defined by the following result (see [13]).Lemma 3.
Letf
(
x
) denote a continuous function; then the solution of (12) is defined by the equality
(13)
∫
0
t
‍
f
(
ξ
)
d
ξ
α
=
α
∫
0
t
‍
(
t
-
ξ
)
α
-
1
f
(
ξ
)
d
ξ
,
0
<
ξ
≤
1
.


## 2.1. Fractional Derivative via Fractional Difference

For an introduction to the classical fractional calculus we refer the reader to [14–17].Definition 1.
Letf
:
ℜ
→
ℜ, x
→
f
(
x
), be a continuous function, and let h
>
0 be a constant discretization span. Define the forward operator F
W
(
h
); that is, (the symbol := means that the left side is defined by the right one)
(3)
F
W
(
h
)
·
f
(
x
)
:
=
f
(
x
+
h
)
.
Then the fractional difference of order α, 0
<
α
<
1, of f
(
x
) is defined by the expression
(4)
Δ
α
f
(
x
)
:
=
(
F
W
-
1
)
α
f
(
x
)
iiiiiiiiiiii
=
∑
k
=
0
∞
‍
(
-
1
)
k
(
α
k
)
f
(
x
+
(
α
-
k
)
h
)
,
and its derivative of fractional order is defined by the expression
(5)
f
(
α
)
(
x
)
=
lim
⁡
h
↓
0
Δ
α
f
(
x
)
h
α
,
0
<
α
≤
1
.


## 2.2.  Modified Riemann-Liouville Derivative

In this section we briefly review the main notions and results from the recent fractional calculus proposed by Jumarie [12, 18, 19].Definition 2 (Riemann-Liouville definition revisited).
Refer to the function of Definition1. Then its fractional derivative of order α is defined by the expression
(6)
f
(
α
)
(
x
)
:
=
1
Γ
(
-
α
)
∫
0
x
‍
(
x
-
ξ
)
-
α
-
1
(
f
(
ξ
)
-
f
(
0
)
)
d
ξ
,
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
α
<
0
.
For positiveα, one will set
(7)
f
(
α
)
(
x
)
=
(
f
(
α
-
1
)
(
x
)
)
′
,
0
<
α
<
1
iiiiiiiiiii
=
1
Γ
(
1
-
α
)
d
d
x
∫
0
x
‍
(
x
-
ξ
)
-
α
(
f
(
ξ
)
-
f
(
0
)
)
d
ξ
,
f
(
α
)
(
x
)
:
=
(
f
(
n
)
(
x
)
)
(
α
-
n
)
,
n
≤
α
<
n
+
1
,
n
≥
1
.


## 2.3. Useful Relations

Here we give some properties of the modified Riemann-Liouville derivative (see [13]) which are used further in this paper.(i)
Consider(8)
Γ
(
1
+
α
k
)
=
:
(
α
k
)
!
.
(ii)
Useful differential relation:(9)
d
α
f
≅
Γ
(
1
+
α
)
d
f
,
0
<
α
≤
1
.
(iii)
Consider(10)
D
α
x
γ
=
Γ
(
1
+
γ
)
Γ
-
1
(
γ
+
1
-
α
)
x
γ
-
α
,
γ
>
0
.
(iv)
The chain rule:(11)
(
u
(
x
)
v
(
x
)
)
(
α
)
=
u
(
α
)
(
x
)
v
(
x
)
+
u
(
x
)
v
(
α
)
(
x
)
,
i
i
(
f
[
u
(
x
)
]
)
(
α
)
=
f
u
′
(
u
)
u
(
α
)
(
x
)
.


## 2.4. Integration with respect tod
t
α

The solution of the equation(12)
i
d
α
x
=
f
(
t
)
d
t
α
,
t
≥
0
,
x
(
0
)
=
x
0
,
is defined by the following result (see [13]).Lemma 3.
Letf
(
x
) denote a continuous function; then the solution of (12) is defined by the equality
(13)
∫
0
t
‍
f
(
ξ
)
d
ξ
α
=
α
∫
0
t
‍
(
t
-
ξ
)
α
-
1
f
(
ξ
)
d
ξ
,
0
<
ξ
≤
1
.


## 3.  Existence and Uniqueness of the Nonnegative Solution

Theorem 4.
For any initial value(
x
(
0
)
,
y
(
0
)
)
∈
R
+
2, there is a unique global solution (
x
(
t
)
,
y
(
t
)
) of system (1) on t
≥
0.Proof.
Note that the coefficients of system (1) are locally Lipschitz continuous for the given initial value (
x
(
0
)
,
y
(
0
)
)
∈
R
2; there is a unique local solution (
x
(
t
)
,
y
(
t
)
) on t
∈
[
0
,
τ
e
), where τ
e is the explosion time. Hence, we know that (
x
(
t
)
,
y
(
t
)
), t
∈
[
0
,
τ
e
) is a unique positive local solution of system (1). To show that this solution is global, we need to show that τ
e
=
∞. Let m
0
≥
1 be sufficiently large so that x
(
0
), y
(
0
) all lie within the interval [
1
/
m
0
,
m
0
]. For each integer m
≥
m
0, define
(14)
τ
m
=
inf
⁡
{
1
m
t
∈
[
0
,
τ
e
)
:
min
⁡
{
x
(
t
)
,
y
(
t
)
}
iiiiiiiiiii
iiii
≤
1
m
or
max
⁡
{
x
(
t
)
,
y
(
t
)
}
≥
m
}
.
Clearly,τ
m is increasing as m
→
∞. Set τ
∞
=
lim
⁡
m
→
∞
τ
m, where τ
∞
≤
τ
e. If we can show that τ
∞
=
∞, then τ
e
=
∞ and (
x
(
t
)
,
y
(
t
)
)
∈
R
+
2 for all t
≥
0. In other words, to complete the proof all we need to show is that τ
∞
=
∞. If this statement is false, then there is a constant T
>
0 such that
(15)
τ
∞
≤
T
.
Hence there is an integer m
1
≥
m
0 such that
(16)
τ
m
≤
T
,
∀
m
≥
m
1
.
Define a C
2-function V
:
R
+
2
→
R
¯
+ by
(17)
V
(
x
,
y
)
=
(
x
-
c
-
c
log
⁡
x
c
)
+
1
k
(
y
-
1
-
log
⁡
y
)
,
where c is a positive constant to be determined later. Using the chain rule, we get
(18)
d
α
V
d
t
α
=
∂
V
∂
x
d
α
x
d
t
α
+
∂
V
∂
y
d
α
y
d
t
α
=
(
x
-
c
)
(
a
-
b
x
-
γ
y
1
+
β
x
)
+
1
k
(
y
-
1
)
(
-
e
+
k
γ
x
1
+
β
x
)
≤
-
a
c
+
e
k
+
(
a
+
b
c
)
x
-
b
x
2
-
(
e
k
-
γ
c
)
y
.
Choose c
=
e
/
k
γ such that (
e
/
k
)
-
γ
c
=
0; then
(19)
d
α
V
d
t
α
≤
-
a
c
+
e
k
+
(
a
+
b
c
)
x
-
b
x
2
=
-
b
(
x
-
a
+
b
c
2
b
)
2
+
(
a
+
b
c
)
2
4
b
+
e
k
-
a
c
≤
(
a
+
b
c
)
2
4
b
+
e
k
=
K
,
where K is a positive constant. Then
(20)
d
α
V
≤
K
d
t
α
.
Therefore, on the one hand,(21)
∫
0
τ
m
∧
T
‍
d
α
V
(
x
(
t
)
,
y
(
t
)
)
≤
∫
0
τ
m
∧
T
‍
K
d
t
α
≤
K
∫
0
T
‍
d
t
α
.
Using the equality (13), we have
(22)
∫
0
T
‍
d
t
α
=
α
∫
0
T
‍
(
T
-
t
)
α
-
1
d
t
,
0
<
α
≤
1
,
which implies that
(23)
∫
0
τ
m
∧
T
‍
d
α
V
(
x
(
t
)
,
y
(
t
)
)
≤
K
α
∫
0
T
‍
(
T
-
t
)
α
-
1
d
t
=
K
T
α
.
On the other hand, using the differential relation (9),
(24)
∫
0
τ
m
∧
T
‍
d
α
V
=
∫
0
τ
m
∧
T
‍
Γ
(
1
+
α
)
d
V
=
Γ
(
1
+
α
)
×
[
V
(
x
(
τ
m
∧
T
)
,
y
(
τ
m
∧
T
)
)
-
V
(
x
(
0
)
,
y
(
0
)
)
]
.
Therefore,
(25)
V
(
x
(
τ
m
∧
T
)
,
y
(
τ
m
∧
T
)
)
≤
V
(
x
(
0
)
,
y
(
0
)
)
+
K
T
α
Γ
(
1
+
α
)
.
Set Ω
m
=
{
τ
m
≤
T
} for m
≥
m
1. Note that there is at least one of x
(
τ
m
) and y
(
τ
m
) that equals either m or 1
/
m; then
(26)
V
(
x
(
τ
m
)
,
y
(
τ
m
)
)
≥
(
m
-
c
-
c
log
⁡
m
c
)
∧
(
1
m
-
c
+
c
log
⁡
(
c
m
)
)
∧
1
k
(
m
-
1
-
log
⁡
m
)
∧
1
k
(
1
m
-
1
+
log
⁡
m
)
.
Hence,
(27)
V
(
x
(
0
)
,
y
(
0
)
)
+
1
Γ
(
1
+
α
)
K
T
α
≥
(
m
-
c
-
c
log
⁡
m
c
)
∧
(
1
m
-
c
+
c
log
⁡
(
c
m
)
)
∧
1
k
(
m
-
1
-
log
⁡
m
)
∧
1
k
(
1
m
-
1
+
log
⁡
m
)
.
Let m
→
∞ lead to the contradiction that ∞
>
V
(
x
(
0
), y
(
0
)
)
+
(
1
/
Γ
(
1
+
α
)
)
K
T
α
=
∞. So we must therefore have τ
∞
=
∞.


## 4. Lyapunov Stability Theory

Consider the equation(28)
d
α
x
d
t
α
=
f
(
x
)
.
Here, x and f
(
x
) are n-dimensional column vectors. Suppose that f
(
0
)
=
0 and f
(
x
) is continuous in G
:
|
x
|
<
H and satisfies local Lipschitz condition.Definition 5 (Lyapunov Stability).
x
=
φ
(
t
,
τ
,
ξ
0
) is said to be stable (or Lyapunov stable) if, given ɛ
>
0, there exists a δ
>
0 such that, for any ξ satisfying |
ξ
-
ξ
0
|
<
δ, the solution of (28) is defined for t
≥
τ and has the following inequality:
(29)
|
φ
(
t
,
τ
,
ξ
)
-
φ
(
t
,
τ
,
ξ
0
)
|
<
ɛ
,
t
≥
τ
.We remark that a solution which is not stable is said to be unstable.Definition 6 (asymptotic stability).
x
=
φ
(
t
,
τ
,
ξ
0
) is said to be asymptotically stable if it is Lyapunov stable and there exists δ
0
>
0 such that, for any ξ satisfying |
ξ
-
ξ
0
|
≤
δ
0, lim
⁡
t
→
∞
(
φ
(
t
,
τ
,
ξ
)
-
φ
(
t
,
τ
,
ξ
0
)
)
=
0.Define aC
1-scalar function V
(
x
) in
(30)
|
x
|
≤
h
<
H
.Using the chain rule, we have(31)
d
α
V
d
t
α
=
∑
i
=
1
n
‍
∂
V
∂
x
i
d
α
x
i
d
t
α
=
∑
i
=
1
n
‍
∂
V
∂
x
i
f
i
(
x
)
.Theorem 7.
LetV
(
x
) be C
1 function satisfying V
(
0
)
=
0, V
(
x
)
>
0.(i)
Ifd
α
V
/
d
t
α
≤
0, then system (28) has a stable null solution;
(ii)
ifd
α
V
/
d
t
α
<
0, then system (28) has an asymptotically stable null solution;
(iii)
the null solution of system (28) is unstable provided that d
α
V
/
d
t
α
>
0.Proof.
Consider the following.
Case 1 (
(
d
α
V
/
d
t
α
)
≤
0
)
. Consider
(32)
∀
ɛ
>
0
(
ɛ
≤
h
)
,
let
r
ɛ
=
min
⁡
ɛ
≤
|
x
|
≤
h
V
(
x
)
,
r
ɛ
>
0
.
Obviously,
(33)
∃
δ
=
δ
(
ɛ
)
>
0
,
s
.
t
.
|
V
(
x
)
-
V
(
0
)
|
=
V
(
x
)
<
ɛ
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
for
|
x
|
<
δ
.
Let ɛ
=
r
ɛ; then V
(
x
)
<
r
ɛ.
Let|
ξ
|
<
δ
(
ɛ
); the right maximal interval of existence of x
=
φ
(
t
,
τ
,
ξ
) for (30) is τ
≤
t
<
t
1.
That is to say, forτ
≤
t
<
t
1,
(34)
d
α
V
d
t
α
(
φ
(
t
,
τ
,
ξ
)
)
=
∑
i
=
1
n
‍
∂
V
∂
x
i
(
φ
(
t
,
τ
,
ξ
)
)
·
f
i
(
φ
(
t
,
τ
,
ξ
)
)
≤
0
.
Then d
α
V
≤
0.
Using (9),
(35)
d
α
V
d
t
α
=
Γ
(
1
+
α
)
d
V
d
t
≤
0
.
Therefore,
(36)
V
(
φ
(
t
,
τ
,
ξ
)
)
≤
V
(
ξ
)
<
r
ɛ
,
τ
≤
t
<
t
1
.
Thus, from (32), we obtain
(37)
|
φ
(
t
,
τ
,
ξ
)
|
<
ɛ
,
τ
≤
t
<
t
1
.
Case  2
(
(
d
α
V
/
d
t
α
)
<
0
)
. We can easily know that the system (28) has a stable null solution. We will consider that there exists δ
0
>
0 such that for any ξ satisfying |
ξ
|
<
δ
0 then lim
⁡
t
→
+
∞
φ
(
t
,
τ
,
ξ
)
=
0.
Letδ
0
>
0; then |
φ
(
t
,
τ
,
ξ
)
|
<
ɛ
≤
h for |
ξ
|
<
δ
0, t
≥
τ. φ
(
t
,
τ
,
ξ
) is bounded for t
≥
τ; then there exists increasing t
k which converges to +
∞ such that
(38)
lim
⁡
k
→
+
∞
φ
(
t
k
,
τ
,
ξ
)
=
ξ
¯
.
Suppose ξ
¯
≠
0. One obtains
(39)
φ
(
t
,
τ
,
ξ
)
≠
0
,
t
≥
τ
.
Since d
α
V
/
d
t
α
<
0, then
(40)
d
α
V
(
φ
(
t
,
τ
,
ξ
¯
)
)
d
t
α
<
0
,
t
≥
τ
.
Using (9) yields
(41)
d
α
V
(
φ
(
t
,
τ
,
ξ
¯
)
)
d
t
α
=
Γ
(
1
+
α
)
d
V
(
φ
(
t
,
τ
,
ξ
¯
)
)
d
t
<
0
.
Therefore, we have
(42)
V
(
φ
(
t
,
τ
,
ξ
¯
)
)
<
V
(
ξ
¯
)
,
t
>
τ
.
Besides, for any t satisfying t
>
τ,
(43)
d
α
V
d
t
α
<
0
,
which implies that
(44)
V
(
φ
(
t
k
,
τ
,
ξ
)
)
<
V
(
φ
(
t
,
τ
,
ξ
)
)
,
t
k
>
t
.
Then according to (38),
(45)
lim
⁡
k
→
∞
V
(
φ
(
t
k
,
τ
,
ξ
)
)
=
V
(
ξ
¯
)
<
V
(
φ
(
t
,
τ
,
ξ
)
)
,
t
>
τ
.
It is easy to verify that, for any k,
(46)
φ
(
t
+
t
k
,
τ
,
ξ
)
=
φ
(
t
+
τ
,
τ
,
φ
(
t
k
,
τ
,
ξ
)
)
.
Note that
(47)
φ
(
1
+
t
k
,
τ
,
ξ
)
=
φ
(
1
+
τ
,
τ
,
φ
(
t
k
,
τ
,
ξ
)
)
,
when t
=
1, which implies that
(48)
V
(
φ
(
1
+
t
k
,
τ
,
ξ
)
)
=
V
(
φ
(
1
+
τ
,
τ
,
φ
(
t
k
,
τ
,
ξ
)
)
)
.
Let k
→
+
∞ and note that V
(
x
) is continuous and φ
(
t
,
τ
,
ξ
) is continuous for initial value; we have
(49)
V
(
φ
(
1
+
t
k
,
τ
,
ξ
)
)
=
V
(
φ
(
1
+
τ
,
τ
,
φ
(
t
k
,
τ
,
ξ
)
)
)
<
V
(
ξ
¯
)
,
which contradicts (45).
Thus,ξ
¯
=
0.
Case 3 (
(
d
α
V
/
d
t
α
)
>
0
)
. Suppose system (28) has a stable null solution; then ∀
ɛ
>
0, ∃
δ
(
ɛ
)
>
0, |
ξ
|
<
δ
(
ɛ
); we have |
φ
(
t
,
τ
,
ξ
)
|
<
ɛ, t
≥
τ.
Letξ
≠
0, |
ξ
|
<
δ
(
ɛ
); according to condition (iii) in Theorem 7, we know
(50)
V
(
φ
(
t
,
τ
,
ξ
)
)
>
V
(
ξ
)
>
0
,
t
>
τ
.
Then there exists α
>
0 satisfying |
φ
(
t
,
τ
,
ξ
)
|
≥
α.
Next, we showd
α
V
(
φ
(
t
,
τ
,
ξ
)
)
/
d
t
α
>
0; then there exists M
>
0, such that
(51)
d
α
V
(
φ
(
t
,
τ
,
ξ
)
)
d
t
α
≥
M
.
Thus,
(52)
V
(
φ
(
t
,
τ
,
ξ
)
)
>
V
(
φ
(
t
,
τ
,
ξ
)
)
-
V
(
φ
(
τ
,
τ
,
ξ
)
)
=
V
(
φ
(
t
,
τ
,
ξ
)
)
-
V
(
ξ
)
>
M
Γ
(
1
+
α
)
(
t
-
τ
)
α
,
t
≥
τ
.
Noting that V
(
φ
(
t
,
τ
,
ξ
)
) is bounded and (
M
/
Γ
(
1
+
α
)
)
(
t
-
τ
)
α is unbounded when t
≥
τ, this contradiction shows that the null solution of system (28) is unstable when d
α
V
/
d
t
α
>
0.Example 8.
Consider the stability of the null solution of system:(53)
d
α
x
d
t
α
=
y
2
p
-
1
,
d
α
y
d
t
α
=
-
x
2
p
-
1
,
where p
∈
N. Define a function V
(
x
,
y
)
=
(
x
2
p
/
2
p
)
+
(
y
2
p
/
2
p
). Then V
(
0,0
)
=
0, V
(
x
,
y
)
>
0, (
x
,
y
)
≠
(
0,0
), d
α
V
/
d
t
α
=
0. Here by Theorem 7 the system has a stable solution.Example 9.
Consider the stability of the null solution of system:(54)
d
α
x
d
t
α
=
2
y
+
y
z
-
x
3
,
d
α
y
d
t
α
=
-
x
-
x
z
-
y
3
,
d
α
z
d
t
α
=
x
y
-
z
3
.
Define a function V
(
x
,
y
,
z
)
=
p
x
2
+
q
y
2
+
r
z
2, where p
,
q
,
r
>
0. Obviously, V
(
0,0
,
0
)
=
0, V
(
x
,
y
,
z
)
>
0, (
x
,
y
,
z
)
≠
(
0,0
,
0
), d
α
V
/
d
t
α
=
2
(
2
p
-
q
)
x
y
+
2
(
p
-
q
+
r
)
x
y
z
-
2
(
p
x
4
+
q
y
4
+
r
z
4
); we can choose the constants p
,
q
,
r such that (
2
p
-
q
)
=
0, p
-
q
+
r
=
0. Then d
α
V
/
d
t
α
<
0 and by Theorem 7 the system has an asymptotically stable null solution.


## 5. The Stability of the Solution

Sincea
k
γ
>
a
e
β
+
b
e, then there is a positive equilibrium (
x
*
,
y
*
) of system (1), and
(55)
a
=
b
x
*
+
γ
y
*
1
+
β
x
*
,
e
=
k
γ
x
*
1
+
β
x
*
.
Let
(56)
V
1
(
x
,
y
)
=
(
x
-
x
*
-
x
*
log
⁡
x
x
*
)
+
a
1
(
y
-
y
*
-
y
*
log
⁡
y
y
*
)
,
where a
1 is a positive constant to be determined later. Then using the chain rule, we have
(57)
d
α
V
1
d
t
α
=
∂
V
1
∂
x
d
α
x
d
t
α
+
∂
V
1
∂
y
d
α
y
d
t
α
=
(
x
-
x
*
)
(
a
-
b
x
-
γ
y
1
+
β
x
)
+
a
1
(
y
-
y
*
)
(
-
e
+
k
γ
x
1
+
β
x
)
=
-
b
(
x
-
x
*
)
2
+
γ
β
y
*
(
1
+
β
x
*
)
(
1
+
β
x
)
(
x
-
x
*
)
2
-
γ
1
+
β
x
(
x
-
x
*
)
(
y
-
y
*
)
+
a
1
k
γ
(
1
+
β
x
*
)
(
1
+
β
x
)
(
x
-
x
*
)
(
y
-
y
*
)
≤
-
(
b
-
γ
β
y
*
1
+
β
x
*
)
(
x
-
x
*
)
2
-
γ
1
+
β
x
(
1
-
a
1
k
1
+
β
x
*
)
(
x
-
x
*
)
(
y
-
y
*
)
.
Choose a
1
=
(
1
+
β
x
*
)
/
k such that 1
-
(
a
1
k
/
(
1
+
β
x
*
)
)
=
0 yields
(58)
d
α
V
1
d
t
α
≤
-
(
b
-
γ
β
y
*
1
+
β
x
*
)
(
x
-
x
*
)
2
.Let(59)
V
2
(
x
,
y
)
=
1
2
[
(
x
-
x
*
)
+
1
k
(
y
-
y
*
)
]
2
,
V
2
~
(
x
,
y
)
=
(
x
-
x
*
)
+
1
k
(
y
-
y
*
)
.Note that(60)
d
α
V
2
~
d
t
α
=
∂
V
2
~
∂
x
d
α
x
d
t
α
+
∂
V
2
~
∂
y
d
α
y
d
t
α
=
x
(
a
-
b
x
-
γ
y
1
+
β
x
)
+
y
k
(
-
e
+
k
γ
x
1
+
β
x
)
=
-
b
x
(
x
-
x
*
)
+
γ
y
*
(
x
-
x
*
)
-
x
*
(
y
-
y
*
)
1
+
β
x
*
.
Then
(61)
d
α
V
2
d
t
α
=
[
(
x
-
x
*
)
+
1
k
(
y
-
y
*
)
]
d
α
V
2
~
d
t
α
=
[
(
x
-
x
*
)
+
1
k
(
y
-
y
*
)
]
×
[
-
b
x
(
x
-
x
*
)
+
γ
y
*
(
x
-
x
*
)
-
x
*
(
y
-
y
*
)
1
+
β
x
*
]
=
-
b
x
(
x
-
x
*
)
2
+
(
γ
y
*
1
+
β
x
*
+
b
y
*
k
)
(
x
-
x
*
)
2
-
b
k
(
x
-
x
*
)
2
y
-
γ
x
*
k
(
1
+
β
x
*
)
(
y
-
y
*
)
2
+
[
γ
y
*
k
(
1
+
β
x
*
)
-
γ
x
*
1
+
β
x
*
-
b
x
*
k
]
×
(
x
-
x
*
)
(
y
-
y
*
)
≤
(
γ
y
*
1
+
β
x
*
+
b
y
*
k
)
(
x
-
x
*
)
2
-
γ
x
*
k
(
1
+
β
x
*
)
(
y
-
y
*
)
2
+
[
γ
y
*
k
(
1
+
β
x
*
)
-
γ
x
*
1
+
β
x
*
-
b
x
*
k
]
×
(
x
-
x
*
)
(
y
-
y
*
)
.
Note that
(62)
[
γ
y
*
k
(
1
+
β
x
*
)
-
γ
x
*
1
+
β
x
*
-
b
x
*
k
]
(
x
-
x
*
)
(
y
-
y
*
)
≤
[
(
γ
y
*
/
k
(
1
+
β
x
*
)
)
-
(
γ
x
*
/
(
1
+
β
x
*
)
)
-
(
b
x
*
/
k
)
]
2
2
(
γ
x
*
/
k
(
1
+
β
x
*
)
)
×
(
x
-
x
*
)
2
+
1
2
γ
x
*
k
(
1
+
β
x
*
)
(
y
-
y
*
)
2
:
=
δ
(
x
-
x
*
)
2
+
1
2
γ
x
*
k
(
1
+
β
x
*
)
(
y
-
y
*
)
2
.
Then
(63)
d
α
V
2
d
t
α
≤
(
γ
y
*
1
+
β
x
*
+
b
y
*
k
+
δ
)
(
x
-
x
*
)
2
-
1
2
γ
x
*
k
(
1
+
β
x
*
)
(
y
-
y
*
)
2
.Consider a Lyapunov functionV
(
x
,
y
) defined by
(64)
V
(
x
,
y
)
=
V
1
(
x
,
y
)
+
a
2
V
2
(
x
,
y
)
,
where a
2 is a positive constant to be determined later. Then
(65)
d
α
V
d
t
α
=
d
α
V
1
d
t
α
+
a
2
d
α
V
2
d
t
α
≤
-
(
b
-
γ
β
y
*
1
+
β
x
*
)
(
x
-
x
*
)
2
+
a
2
(
γ
y
*
1
+
β
x
*
+
b
y
*
k
+
δ
)
(
x
-
x
*
)
2
-
a
2
2
γ
x
*
k
(
1
+
β
x
*
)
(
y
-
y
*
)
2
=
-
[
b
-
γ
β
y
*
1
+
β
x
*
-
a
2
(
γ
y
*
1
+
β
x
*
+
b
y
*
k
+
δ
)
]
×
(
x
-
x
*
)
2
-
a
2
2
γ
x
*
k
(
1
+
β
x
*
)
(
y
-
y
*
)
2
.
Choose a
2
>
0 such that
(66)
b
-
γ
β
y
*
1
+
β
x
*
-
a
2
(
γ
y
*
1
+
β
x
*
+
b
y
*
k
+
δ
)
=
1
2
(
b
-
γ
β
y
*
1
+
β
x
*
)
.
Then it follows from (65) that
(67)
d
α
V
d
t
α
≤
-
1
2
(
b
-
γ
β
y
*
1
+
β
x
*
)
(
x
-
x
*
)
2
-
a
2
2
γ
x
*
k
(
1
+
β
x
*
)
(
y
-
y
*
)
2
.
According to (66),
(68)
b
-
γ
β
y
*
1
+
β
x
*
=
2
a
2
(
γ
y
*
1
+
β
x
*
+
b
y
*
k
+
δ
)
>
0
.
Thus,
(69)
d
α
V
d
t
α
<
0
.Hence, from these arguments, we get the following result.Theorem 10.
Ifa
k
γ
>
a
e
β
+
b
e such that system (1) has a positive equilibrium (
x
*
,
y
*
) and a
2 is defined as in the proof, then system (1) is asymptotically stable.



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*Source: 124145-2014-04-16.xml*