Proof of Theorem 2.
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.