We consider the dynamic behaviors of a discrete competitive system. A good understanding of the permanence, existence, and global stability of positive periodic solutions is gained. Numerical simulations are also presented to substantiate the analytical results.

In biomathematics, one of the most challenging aspects of mathematical biology is competition modeling. Although the mathematical idea is simple [

Although much progress has been seen for Lotka-Volterra competitive systems, such systems are not well studied in the sense that most results are continuous time versions related. Many authors [

In this paper, we will consider the dynamic behavior of a discrete-time competitive system. Let us first introduce its continuous time version which is motivated in [

Following the same idea and method in [

It is well known that, compared to the continuous time systems, the discrete-time ones are more difficult to deal with. The principle aim of this paper is to explore the permanence, existence, and global stability of positive periodic solutions of system (

For the sake of simplicity and convenience in the following discussion, the notations below will be used through this paper:

For biological reasons, in system (

The organization of this paper is as follows. In the next section, we establish the permanence of system (

In this section, we will establish sufficient conditions for the permanence of system (

System (

Any positive solution

To prove Proposition

Assume that there exists an

We claim that

Suppose that

Suppose that system (

By Proposition

Assume that there exists an

In the following we will prove

Assume that

Now, we are in a position to state Theorem

If the inequalities in (

In this section, we suppose system (

If the inequalities in (

We know that

Next, we derive sufficient conditions which guarantee that the positive periodic solution of system (

A positive periodic solution

Now, we give the main result in this section.

In addition to (

Let

Now, by (

Theorem

In this paper, we have investigated the permanence and global stability of positive periodic solutions of a discrete competitive system. Each species is not isolated from its living environment, but competes with the other for the same resource. Sufficient conditions which guarantee the permanence, existence and global stability of positive periodic solutions are established, respectively. The theoretical results are confirmed by the following examples and their numerical results.

To verify the sufficient conditions for permanence of system (

Permanence of system (

Now, we further verify the sufficient conditions for the existence and global stability of positive periodic solutions of periodic system (

The coefficient values when

Odd number | 0.76 | 1.58 | 1.25 | 1.80 | 0.03 | 0.02 |

Even number | 1.26 | 0.98 | 1.65 | 1.98 | 0.02 | 0.01 |

Besides, we choose the positive periodic solution with initial values

(a) Time series of

(a) Time series of

(a) Phase portrait of

The work is supported by the Innovation Term of Educational Department of Hubei Province in China (T200804), the National Science Foundation of Hubei Province in China (2008CDB068), and the Innovation Project of Hubei Institute for Nationalities for postgraduate students. We would like to thank the Editor Professor A. Vecchio and the referee for careful reading of the original manuscript and valuable comments and suggestions that greatly improved the presentation of this work.