AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2012.32029AM-17397ArticlesPhysics&Mathematics On a Population Model of Systems ecunZhang1*LiyingWang1*JieHuang1WenqiangJi1Institute of Applied Mathematics, Naval Aeronautical and Astronautical University Yantai, China* E-mail:dczhang1967@tom.com(EZ);ytliyingwang@163.com(LW);230220120302185187November 26, 2011February 2, 2012 February 10, 2012© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Population Model; Global Attractor; Difference Equations
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

Thus

Noting that (3) and (4) we get: Changing (5) to (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

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

Because of , we obtain that .

Hence

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.

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