1. Introduction
In the recent monograph [1, p.129], Kulenovic and Glass give an open problem as follows:
Open problem 6.10.16 (A population model).
Assume that
and
. Investigate the global character of all positive solutions of the systems:
(1)
where
, which may be viewed as a population model.
To this end, we consider Equation (1) and obtain some interesting results about the positive solutions of Equation (1).
2. Basic Lemma
Lemma 1 Assume that
,
. Then the following statements are true:
1) If
, then Equation (1) has a unique nonegative equilibrium solution as follows:

2) If
, then Equation (1) has two no-negative equilibrium solutions as follows:

where
,
such that
(2)
Proof: The equilibrium equations about Equation (1) can be written as follows:
(3)
It is easy to see that
,
is a group solutions of Equation (3).
By (3) we obtain
(4)
Thus
(5)
Noting that (3) and (4) we get:

Changing (5) to (6)
(6)
Set


Observing that



So, by the convex functions properties, if
, then we can obtain Equation (6) has a unique positive solution
.
In fact, by the continuous of
, we can get

Hence, we complete the proof.
3. Main Results
Theorem 3.1 Assume that
and
.
Then every positive solutions
and
of Equation (1) have the following properties:
1)
;
2)
.
Proof: By Equation (1) we have

It is to say that
.
By Equation (1) we also get

Thus
,
.
This completes the proof.
Theorem 3.2 Assume that
, 
and
. Then every positive solutions of Equation (1) convergences to the unique no-negative equilibrium solution
.
Proof: By Theorem 3.1, we have that there exists a nature number n0 such that
for
.
Hence, by Equation (1) we get

Thus
is decreasing.
Suppose that
(7)
Then by Equation (1) we have

By induction we obtain

Thus
. Hence there exists a
such that
for
.
Noting that Equation (1)

By induction,

It is to see that
. This is a contradiction with (7), then
.
Noting that Equation (1) we have

i.e.

Let
,
. Then

By induction we obtain

as
, then
(8)
Because of
, we obtain that
.
Hence
(9)
By (9) we get
.
We complete the proof.
Theorem 3.3 Assume that
,
and
. Then Equation (1) is permanent.
Proof: By Equation (1) we obtain

There exists two positive constants
and
such that

Hence
.
Using Theorem 3.1, we complete the proof.
NOTES