# Note on the Stability Property of a Cooperative System Incorporating Harvesting

Authors: Xiangdong Xie; Fengde Chen; Yalong Xue
Journal: Discrete Dynamics in Nature and Society
 (2014)

---

## Abstract

The stability of a kind of cooperative model incorporating harvesting is revisited in this paper. By using an iterative method, the global attractivity of the interior equilibrium point of the system is investigated. We show that the condition which ensures the existence of a unique positive equilibrium is enough to ensure the global attractivity of the positive equilibrium. Our results significantly improve the corresponding results of Wei and Li (2013).


---

## Body

## 1. Introduction

In [1], Wei and Li proposed and studied the following cooperative system incorporating harvesting:
(1)
x
˙
=
x
(
r
1
-
b
1
x
-
a
1
x
y
+
k
1
)
-
E
q
x
,
y
˙
=
y
(
r
2
-
b
2
y
-
a
2
y
x
+
k
2
)
,
where x and y denote the densities of two populations at time t. The parameters r
1
,
r
2
,
a
1
,
a
2
,
b
1
,
b
2
,
k
1
,
k
2
,
E
,
q are all positive constants. Assume that r
1
>
E
q; then, the equilibria of (1) are
(2)
H
0
(
0,0
)
,
H
1
(
0
,
r
2
k
2
a
2
+
k
2
b
)
,
H
3
(
(
r
1
-
E
q
)
k
1
a
1
+
k
1
b
1
,
0
)
,
H
3
(
x
*
,
y
*
)
,
where
(3)
x
*
=
-
(
k
2
P
-
F
)
+
(
k
2
P
-
F
)
2
+
4
P
Q
M
2
P
,
y
*
=
r
2
(
x
*
+
k
2
)
b
2
x
*
+
a
2
+
k
2
b
2
,
P
=
r
2
b
1
+
k
1
b
1
b
2
+
a
1
b
2
,
Q
=
r
1
-
E
q
,
F
=
r
2
Q
+
b
2
k
1
Q
-
k
1
a
2
b
1
-
a
1
a
2
,
M
=
r
2
k
2
+
k
1
k
2
b
2
+
a
2
k
1
.
Wei and Li had showed that H
0
,
H
1
,
H
2 are unstable and concerned with the persistence and stability property of the system; by applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, they obtained the following results.Theorem A.
Ifr
1
>
E
q, k
1
b
1
>
a
1, k
2
b
2
>
a
2, then the system (1) is persistent. More precisely,
(4)
C
≤
liminf
⁡
t
→
+
∞
x
(
t
)
≤
limsup
⁡
t
→
+
∞
x
(
t
)
≤
A
,
D
≤
liminf
⁡
t
→
+
∞
x
(
t
)
≤
limsup
⁡
t
→
+
∞
y
(
t
)
≤
B
,
where
(5)
A
=
r
1
-
E
q
b
1
,
B
=
r
2
b
2
,
C
=
(
r
1
-
E
q
)
(
k
1
b
1
-
a
1
)
b
1
2
k
1
,
D
=
r
2
(
k
2
b
2
-
a
2
)
b
2
2
k
2
.Theorem B.
Ifr
1
>
E
q, k
1
b
1
>
a
1, k
2
b
2
>
a
2,
(6)
b
1
+
a
1
y
*
+
k
1
>
B
a
2
2
(
C
+
k
2
)
(
x
*
+
k
2
)
+
A
a
1
2
(
D
+
k
1
)
(
y
*
+
k
1
)
,
b
2
+
a
2
x
*
+
k
2
>
B
a
2
2
(
C
+
k
2
)
(
x
*
+
k
2
)
+
A
a
1
2
(
D
+
k
1
)
(
y
*
+
k
1
)
,
where A
,
B
,
C
,
D are defined by Theorem A, then the positive equilibrium point H
3 of system (1) is globally asymptotically stable.Now let us consider the following example.Example 1.
We have(7)
x
˙
=
x
(
2
-
x
-
2
x
y
+
1
)
-
x
,
y
˙
=
y
(
2
-
y
-
2
y
x
+
1
)
.
Here we choose r
1
=
r
2
=
2, b
1
=
b
2
=
1, k
1
=
k
2
=
1, a
1
=
a
2
=
2, E
=
q
=
1, and the parameters r
1
,
r
2
,
a
1
,
a
2
,
b
1
,
b
2
,
k
1
,
k
2
,
E
,
q are all positive constants. Obviously, r
1
=
2
>
1
=
E
q, k
1
b
1
=
1
<
2
=
a
1, k
2
b
2
=
1
<
2
=
a
2. Hence, the conditions of Theorems A and B are not all satisfied; however, numeric simulations (Figure 1) show that the unique positive equilibrium (0.4806248475, 0.8507810594) is globally attractive.Dynamics behaviors of system (7). Here, we take the initial conditions (
x
1
(
0
)
,
x
2
(
0
)
)
=
(
0.5,1.2
), (
1.5,1
), (
0.2,0.5
) and (
1,0.6
), respectively.
(a)
(b)The above example shows that it is possible to obtain some weaker conditions than those of Theorems A and B to ensure the persistent and stability of the system. The aim of this paper is to prove the following result.Theorem 2.
Assume thatr
1
>
E
q holds; then, the unique positive equilibrium E
*
(
x
*
,
y
*
) is globally attractive; that is,
(8)
lim
⁡
t
→
+
∞
x
(
t
)
=
x
*
,
lim
⁡
t
→
+
∞
y
(
t
)
=
y
*
.Concerned with the persistent property of the system, as a direct corollary of Theorem2, we have the following.Corollary 3.
Assume thatr
1
>
E
q holds; then, system (1) is permanent.Remark 4.
A comparison of Theorems A, B, and2 and Corollary 3 shows that k
1
b
1
>
a
1, k
2
a
2
>
b
2, and inequalities (6) are redundant. Therefore, our results significantly improve the corresponding main results of Wei and Li [1].We will prove Theorem2 in the next section. For more works on mutualism system, one could refer to [2–10] and the references cited therein.


## 2. Proof of the Main Results

As a direct corollary of Lemma 2.2 of Chen [11], we have the following.Lemma 5.
Ifa
>
0
,
b
>
0 and x
˙
≥
x
(
b
-
a
x
), when t
≥
0 and x
(
0
)
>
0, we have
(9)
liminf
⁡
t
→
+
∞
x
(
t
)
≥
b
a
.
If a
>
0, b
>
0 and x
˙
≤
x
(
b
-
a
x
), when t
≥
0 and x
(
0
)
>
0, we have
(10)
limsup
⁡
t
→
+
∞
x
(
t
)
≤
b
a
.Proof of Theorem2.
By the first equation of system (1), we have
(11)
x
˙
(
t
)
≤
x
(
t
)
(
r
1
-
E
q
-
b
1
x
(
t
)
)
.
From Lemma 5, it follows that
(12)
limsup
⁡
t
→
+
∞
x
(
t
)
≤
r
1
-
E
q
b
1
.
Hence, for enough small ε
>
0
(
ε
<
min
⁡
{
(
r
1
-
E
q
)
k
1
/
(
k
1
b
1
+
a
1
)
,
r
2
k
2
/
(
k
2
b
2
+
a
2
)
}
), it follows from (12) that there exists a T
1
′
>
0 such that
(13)
x
(
t
)
<
r
1
-
E
q
b
1
+
ε
≝
M
1
(
1
)
∀
t
>
T
1
′
.
Similarly, for the above ε
>
0, it follows from the second equation of system (1) that there exists a T
1
>
T
1
′ such that
(14)
y
(
t
)
<
r
2
b
2
+
ε
≝
M
2
(
1
)
∀
t
>
T
1
.
(14) together with the first equation of system (1) implies that
(15)
x
˙
=
x
(
r
1
-
b
1
x
-
a
1
x
y
+
k
1
)
-
E
q
x
≤
x
(
r
1
-
E
q
-
b
1
x
-
a
1
x
M
2
(
1
)
+
k
1
)
∀
t
>
T
1
.
Therefore, by Lemma 5, we have
(16)
limsup
⁡
t
→
+
∞
x
1
(
t
)
≤
r
1
-
E
q
b
1
+
(
a
1
/
(
M
2
(
1
)
+
k
1
)
)
.
That is, for ε
>
0 to be defined by (12) and (13), there exists a T
2
′
>
T
1 such that
(17)
x
(
t
)
<
r
1
-
E
q
b
1
+
(
a
1
/
(
M
2
(
1
)
+
k
1
)
)
+
ε
2
≝
M
1
(
2
)
>
0
∀
t
>
T
2
′
.
It follows from (13) and the second equation of system (1) that
(18)
y
˙
=
y
(
r
2
-
b
2
y
-
a
2
y
x
+
k
2
)
≤
y
(
r
2
-
b
2
y
-
a
2
y
M
1
(
1
)
+
k
2
)
.
Therefore, by Lemma 5, we have
(19)
limsup
⁡
t
→
+
∞
y
(
t
)
≤
r
2
b
2
+
(
a
2
/
(
M
1
(
1
)
+
k
2
)
)
.
That is, for ε
>
0 to be defined by (13) and (14), there exists a T
2
>
T
2
′ such that
(20)
y
(
t
)
<
r
2
b
2
+
(
a
2
/
(
M
1
(
1
)
+
k
2
)
)
+
ε
2
≝
M
2
(
2
)
>
0
∀
t
>
T
2
.
From the first equation of system (1) and the positivity of y
(
t
), we have
(21)
x
˙
=
x
(
r
1
-
b
1
x
-
a
1
x
y
+
k
1
)
-
E
q
x
≥
x
(
r
1
-
E
q
-
b
1
x
-
a
1
x
k
1
)
∀
t
>
T
2
.
Therefore, by Lemma 5, we have
(22)
liminf
⁡
t
→
+
∞
x
(
t
)
≥
r
1
-
E
q
b
1
+
(
a
1
/
k
1
)
.
Hence, for ε
>
0 to be defined by (12) and (13), there exists a T
3
′
>
T
2 such that
(23)
x
(
t
)
>
r
1
-
E
q
b
1
+
(
a
1
/
k
1
)
-
ε
≝
m
1
(
1
)
,
∀
t
>
T
3
′
.
Similarly, it follows from the second equation of system (1) that there exists a T
3
>
T
3
′ such that
(24)
y
(
t
)
>
r
2
b
2
+
(
a
2
/
k
2
)
-
ε
≝
m
2
(
1
)
,
∀
t
>
T
3
.
(24) together with the first equation of system (1) implies that
(25)
x
˙
=
x
(
r
1
-
b
1
x
-
a
1
x
y
+
k
1
)
-
E
q
x
≥
x
(
r
1
-
E
q
-
b
1
x
-
a
1
x
m
2
(
1
)
+
k
1
)
∀
t
>
T
3
.
Therefore, by Lemma 5, we have
(26)
liminf
⁡
t
→
+
∞
x
(
t
)
≥
r
1
-
E
q
b
1
+
(
a
1
/
(
m
2
(
1
)
+
k
1
)
)
.
That is, for ε
>
0 to be defined by (12) and (13), there exists a T
4
′
>
T
3 such that
(27)
x
(
t
)
>
r
1
-
E
q
b
1
+
(
a
1
/
(
m
2
(
1
)
+
k
1
)
)
-
ε
2
≝
m
1
(
2
)
>
0
,
∀
t
>
T
4
′
.
Similarly, by (23) and the second equation of system (1), for ε
>
0 to be defined by (12) and (13), there exists a T
4
>
T
4
′ such that
(28)
y
(
t
)
>
r
2
b
2
+
(
a
2
/
(
m
1
(
1
)
+
k
2
)
)
-
ε
2
≝
m
2
(
2
)
>
0
,
∀
t
>
T
4
.
Noting that a
1
/
(
M
2
(
1
)
+
k
1
)
>
0, a
2
/
(
M
1
(
1
)
+
k
2
)
>
0, it immediately follows that
(29)
M
1
(
2
)
=
r
1
-
E
q
b
1
+
(
a
1
/
(
M
2
(
1
)
+
k
1
)
)
+
ε
2
<
r
1
-
E
q
b
1
+
ε
=
M
1
(
1
)
;
M
2
(
2
)
=
r
2
b
2
+
(
a
2
/
(
M
1
(
1
)
+
k
2
)
)
+
ε
2
<
r
2
b
2
+
ε
=
M
2
(
1
)
.
Also, since m
1
(
1
)
>
0, m
2
(
1
)
>
0, it follows that a
1
/
(
m
2
(
1
)
+
k
1
)
<
a
1
/
k
1, a
2
/
(
m
1
(
1
)
+
k
2
)
<
a
2
/
k
2, and so
(30)
m
1
(
2
)
=
r
1
-
E
q
b
1
+
(
a
1
/
(
m
2
(
1
)
+
k
1
)
)
-
ε
2
>
r
1
-
E
q
b
1
+
(
a
1
/
k
1
)
-
ε
=
m
1
(
1
)
;
m
2
(
2
)
=
r
2
b
2
+
(
a
2
/
(
m
1
(
1
)
+
k
2
)
)
-
ε
2
>
r
2
b
2
+
(
a
2
/
k
2
)
-
ε
=
m
2
(
1
)
.
Repeating the above procedure, we get four sequences M
i
(
n
), m
i
(
n
), i
=
1,2, n
=
1,2
,
…, such that for n
≥
2
(31)
M
1
(
n
)
=
r
1
-
E
q
b
1
+
(
a
1
/
(
M
2
(
n
-
1
)
+
k
1
)
)
+
ε
n
;
M
2
(
n
)
=
r
2
b
2
+
(
a
2
/
(
M
1
(
n
-
1
)
+
k
2
)
)
+
ε
n
;
m
1
(
n
)
=
r
1
-
E
q
b
1
+
(
a
1
/
(
m
2
(
n
-
1
)
+
k
1
)
)
-
ε
n
;
m
2
(
n
)
=
r
2
b
2
+
(
a
2
/
(
m
1
(
n
-
1
)
+
k
2
)
)
-
ε
n
.
Obviously,
(32)
m
i
(
n
)
<
x
i
(
t
)
<
M
i
(
n
)
∀
t
≥
T
2
n
,
i
=
1,2
.
We claim that sequences M
i
(
n
), i
=
1,2 are strictly decreasing, and sequences m
i
(
n
), i
=
1,2 are strictly increasing. To proof this claim, we will carry them out by induction. Firstly, from (29) and (30) we have
(33)
M
i
(
2
)
<
M
i
(
1
)
,
m
i
(
2
)
>
m
i
(
1
)
,
i
=
1,2
.
Let us assume now that our claim is true for n; that is,
(34)
M
i
(
n
)
<
M
i
(
n
-
1
)
,
m
i
(
n
)
>
m
i
(
n
-
1
)
,
i
=
1,2
.
Then,
(35)
a
1
M
2
(
n
)
+
k
1
>
a
1
M
2
(
n
-
1
)
+
k
1
,
r
2
b
2
+
(
a
2
/
(
M
1
(
n
)
+
k
2
)
)
>
r
2
b
2
+
(
a
2
/
(
M
1
(
n
-
1
)
+
k
2
)
)
.
From (34) and the expression of M
i
(
n
), it immediately follows that
(36)
M
1
(
n
+
1
)
=
r
1
-
E
q
b
1
+
(
a
1
/
(
M
2
(
n
)
+
k
1
)
)
+
ε
n
+
1
M
2
(
n
+
1
)
<
r
1
-
E
q
b
1
+
(
a
1
/
(
M
2
(
n
-
1
)
+
k
1
)
)
+
ε
n
=
M
1
(
n
)
,
M
2
(
n
+
1
)
=
r
2
b
2
+
(
a
2
/
(
M
1
(
n
)
+
k
2
)
)
+
ε
n
+
1
M
2
(
n
+
1
)
<
r
2
b
2
+
(
a
2
/
(
M
1
(
n
-
1
)
+
k
2
)
)
+
ε
n
=
M
2
(
n
)
.
Also, it follows from (34) that m
i
(
n
)
≥
m
i
(
n
-
1
), i
=
1,2. Then,
(37)
a
1
m
2
(
n
)
+
k
1
<
a
1
m
2
(
n
-
1
)
+
k
1
,
a
2
m
1
(
n
)
+
k
2
<
a
2
m
1
(
n
-
1
)
+
k
2
.
From (37) and the expression of m
i
(
n
), it immediately follows that
(38)
m
1
(
n
+
1
)
=
r
1
-
E
q
b
1
+
(
a
1
/
(
m
2
(
n
)
+
k
1
)
)
-
ε
n
+
1
m
1
(
n
-
1
)
>
r
1
-
E
q
b
1
+
(
a
1
/
(
m
2
(
n
-
1
)
+
k
1
)
)
-
ε
n
=
m
1
(
n
)
,
m
2
(
n
+
1
)
=
r
2
b
2
+
(
a
2
/
(
m
1
(
n
)
+
k
2
)
)
-
ε
n
+
1
m
1
(
n
-
1
)
>
r
2
b
2
+
(
a
2
/
(
m
1
(
n
-
1
)
+
k
2
)
)
-
ε
n
=
m
2
(
n
)
.
Therefore,
(39)
lim
⁡
t
→
+
∞
M
1
(
n
)
=
x
¯
,
lim
⁡
t
→
+
∞
M
2
(
n
)
=
y
¯
,
lim
⁡
t
→
+
∞
m
1
(
n
)
=
x
_
,
lim
⁡
t
→
+
∞
m
2
(
n
)
=
y
_
.
Letting n
→
+
∞ in (31), we obtain
(40)
b
1
x
¯
+
a
1
x
¯
y
¯
+
k
1
=
r
1
-
E
q
,
b
2
y
¯
+
a
2
y
¯
x
¯
+
k
2
=
r
2
;
b
1
x
_
+
a
1
x
_
y
_
+
k
1
=
r
1
-
E
q
,
b
2
y
_
+
a
2
y
_
x
_
+
k
2
=
r
2
.
(40) shows that (
x
¯
,
y
¯
) and (
x
_
,
y
_
) are positive solutions of the equations
(41)
b
1
x
+
a
1
x
y
+
k
1
=
r
1
-
E
q
,
b
2
y
+
a
2
y
x
+
k
2
=
r
2
.
Wei and Li [1] had already showed that, under the assumption that r
1
>
E
q holds, (41) has a unique positive solution E
*
(
x
*
,
y
*
). Hence, we conclude that
(42)
x
¯
=
x
_
=
x
*
,
y
¯
=
y
_
=
y
*
;
that is,
(43)
lim
⁡
t
→
+
∞
x
(
t
)
=
x
*
,
lim
⁡
t
→
+
∞
y
(
t
)
=
y
*
.
Thus, the unique interior equilibrium E
*
(
x
*
,
y
*
) is globally attractive. This completes the proof of Theorem 2.Proof of Corollary3.
Noting thatM
1
(
1
), M
2
(
1
), m
1
(
1
), m
2
(
1
) are only dependent on the coefficients of the system (1) and independent of the solution of system (1), hence, (13), (14), (23), and (24) show that the system is permanent. This ends the proof of Corollary 3.


## 3. Discussion

In this paper, we revisited the stability property of a cooperative system incorporating harvesting which was proposed by Wei and Li [1]; by using the iterative method, we show that the condition which ensures the existence of a unique positive equilibrium is enough to ensure the global attractivity of the positive equilibrium. The numeric simulation of Example 1 shows the feasibility of our results. It seems interesting to investigate the stability property of the corresponding discrete type model of system (1); we leave this for future study.



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*Source: 327823-2014-04-02.xml*