^{1}

We consider an almost periodic discrete Schoener’s competition model with delays. By means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique strictly positive almost periodic solution which is globally attractive. An example together with numerical simulation indicates the feasibility of the main result.

In 2009, Wu et al. [

By the biological meaning, the system (

Schoener’s competition system has been studied by many scholars. Topics such as existence, uniqueness, and global attractivity of positive periodic solutions of the system were extensively investigated, and many excellent results have been derived (see [

Notice that the investigation of almost periodic solutions for difference equations is one of most important topics in the qualitative theory of difference equations due to the applications in biology, ecology, neural network, and so forth (see [

Denote by

Throughout this paper, we assume the following.

The remaining part of this paper is organized as follows. In Section

In this section, we give the definitions and lemmas of the terminologies involved.

A sequence

Let

The hull of

Suppose that

Assume that

Assume that

Now we state several lemmas which will be useful in proving our main result.

Any solution

Assume that

For any small positive constant

It follows from (

Thus, by using (

When

Letting

Similar to the analysis of (

Note that condition (H2) of Proposition

Assume that (H2) holds; system (

Assume that (H2) holds and further that there exist positive constants

In this section, by means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, we will study the existence of a globally attractive almost periodic solution of system (

Let

By the almost periodic theory, we can conclude that if system (

By Theorem 3.4 in [

If each hull equation of system (

Now we investigate a globally attractive almost periodic solution of system (

If the almost periodic difference system (

By Lemma

Now we prove the existence of a strictly positive solution of any hull equation (

Let

In fact, for any finite subset

Now, for

Similarly, for

Repeating this procedure, for

Now pick the sequence

By the arbitrary of

Combining with

Now we prove the uniqueness of the strictly positive solution of each hull equation (

Calculating the difference of

Define

Let

It follows from (

In view of the above discussion, any hull equation of system (

Let

Let

In this section, we give the following example to check the feasibility of our result.

Consider the following almost periodic discrete Schoener’s competition model with delays

By simple computation, we derive

Also it is easy to see that conditions (H2) and (H3) are verified. Therefore, system (

Dynamic behavior of the first component

Dynamic behavior of the second component

Dynamic behavior of the first component

Dynamic behavior of the second component

There are no financial interest conflicts between the authors and the commercial identity.

This work is supported by National Natural Science Foundation of China (no. 61132008) and Scientific Research Program Funded by Shaanxi Provincial Education Department of China (no. 2013JK1098).