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A discrete mutualism model is studied in this paper. By using the linear approximation method, the local stability of the interior equilibrium of the system is investigated. By using the iterative method and the comparison principle of difference equations, sufficient conditions which ensure the global asymptotical stability of the interior equilibrium of the system are obtained. The conditions which ensure the local stability of the positive equilibrium is enough to ensure the global attractivity are proved.

There are many examples where the interaction of two or more species is to the advantage of all; we call such a situation the mutualism. For example, cellulose of white ants’ gut provides nutrients for flagellates, while flagellates provide nutrients for white ants through the decomposition of cellulose to glucose. As was pointed out by Chen et al. [

The following model was proposed by Chen et al. [

Li [

It is well known that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations, and discrete time models can also provide efficient computational models of continuous models for numerical simulations. Corresponding to system (

It brings to our attention that neither Li [

Throughout this paper, we assume that the coefficients of system (

The aim of this paper is, by further developing the analysis technique of [

In addition to

The rest of the paper is arranged as follows. In Section

In view of the actual ecological implications of system (

We determine the positive equilibrium of the system (

Following we will discuss the local stability of equilibrium

Let

Let

Assume that

Since (

We will give a strict proof of Theorem

Let

Assume that sequence

if

if

Suppose that functions

Let

From the first equation of system (

Now, we will prove

First of all, it is clear that

From Theorem

In this section, we will give an example to illustrate the feasibility of the main result.

Dynamic behaviors of the solution

It is well known [

Recently, by using the iterative method, Xie et al. [

In this paper, by using the linear approximation, comparison principle of difference equations, and method of iteration scheme, we showed that the conditions which ensure the local stability property of the positive equilibrium (

At the end of this paper, we would like to mention here that, for the Lotka-Volterra type mutualism system with time delay, delay is one of the most important factors to influence the dynamic behaviors of the system [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research was supported by the Natural Science Foundation of Fujian Province (2013J01011, 2013J01010) and the Foundation of Fujian Education Bureau (JA13361).