Applied Mathematics
Vol. 3  No. 2 (2012) , Article ID: 17397 , 3 pages DOI:10.4236/am.2012.32029

On a Population Model of Systems *

Decun Zhang, Liying Wang, Jie Huang, Wenqiang Ji

Institute of Applied Mathematics, Naval Aeronautical and Astronautical University, Yantai, China


Received November 26, 2011; revised February 2, 2012; accepted February 10, 2012

Keywords: Population Model; Global Attractor; Difference Equations


In this paper, we investigate the global character of all positive solutions of a population model of systems. Some interesting convergence properties of the solution are given, and lastly, we obtain that the solution is permanent under some conditions.

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:


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


Proof: The equilibrium equations about Equation (1) can be written as follows:


It is easy to see that, is a group solutions of Equation (3).

By (3) we obtain




Noting that (3) and (4) we get:

Changing (5) to (6)



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:



Proof: By Equation (1) we have

It is to say that .

By Equation (1) we also get


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


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


Let,. Then

By induction we obtain

as, then


Because of, we obtain that.



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


Using Theorem 3.1, we complete the proof.


  1. M. R. S. Kulenovic and G. Ladas, “Dynamics of Second Order Rational Difference Equations,” Chapman & Hall/ CRC, Boca Raton, 2002.


*Research supported by Distinguished Expert Foundation and Youth Science Foundation of Naval Aeronautical and Astronautical University.