This paper discusses a discrete multispecies Lotka-Volterra mutualism system. We first obtain the permanence of the system. Assuming that the coefficients in the system are almost periodic sequences, we obtain the sufficient conditions for the existence of a unique almost periodic solution which is globally attractive. In particular, for the discrete two-species Lotka-Volterra mutualism system, the sufficient conditions for the existence of a unique uniformly asymptotically stable almost periodic solution are obtained. An example together with numerical simulation indicates the feasibility of the main result.

In this paper, we consider a discrete multispecies Lotka-Volterra mutualism system:

A number of scholars have studied the difference system (see [

Recently, as far as the multispecies Lotka-Volterra ecosystem is concerned, Wendi and Zhengyi [

Chen [

At the same time, a few scholars have investigated the mutualism system (see [

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

Denote as

Throughout this paper, we assume the following.

From the point of view of biology, in the sequel, we assume that

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

Firstly, we give the definitions of the terminologies involved.

A sequence

A sequence

A solution

Now, we present some results which will play an important role in the proof of the main result.

If

Suppose that

Assume that

Assume that

Consider the following almost periodic difference system:

Suppose that there exists a Lyapunov function

In this section, we establish a permanence result for system (

Assume that (H1) holds. Then, any positive solution

Assume that (H1) holds; then system (

It should be noticed that, from Proposition

The next result tells us that there exist solutions of system (

System (

By the almost periodicity of

The main result of this paper concerns the existence of a globally attractive almost periodic solution of system (

Assume that (H1) and

(H2)

It follows from Proposition

Since

This combined with (

Define

Let

Assume that

Let

Notice that

In view of (

In particular, if

In the following, the main result concerns the existence of a uniformly asymptomatically stable almost periodic solution of system (

From Proposition

Assume that (H1) holds. Then,

By an inductive argument, we have from system (

As a matter of fact, for any finite subset

Now, for

Analogously, for

Repeating the above process, for

Now, we choose the sequence

Recall the almost periodicity of

For any

Assume that (H1) holds; furthermore,

Denote

Define the norm

Let us construct a Lyapunov function defined on

Moreover, for any

Finally, calculating the

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

Consider the discrete multispecies Lotka-Volterra mutualism system:

A computation shows that

Dynamic behavior of the first component

Dynamic behavior of the second component

Dynamic behavior of the third component

The authors declare that there is no conflict of interests regarding the publication of this paper, and there is no financial conflict of interests between the authors and the commercial identity.

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