^{1, 2, 3}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

In this paper, the Lotka-Volterra 3-species mutualism models with diffusion and delay effects is investigated. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the unique positive steady-state solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition. Our approach to the problem is based on inequality skill and the method of the upper and lower solutions for a more general reaction—diffusion system. Finally, some numerical simulations are given to illustrate our results.

Modeling and analysis of the dynamics of biological populations by means of differential equations is one of the primary concerns in population growth problems. A well-known and extensively studied class of models in population dynamics is the Lotka-Volterra system which models the interaction among various species. In the earlier literature most of the discussions are devoted to coupled systems of two equations (cf. [

Throughout the paper we assumed that the function

This work is motivated from the following two-prey one-predator model:

Based on the above results, we are mainly interested in studying the asymptotic behavior of the solution of (

This paper is arranged as follows. In Section

To prove the main results in this paper we first give some preliminary results for a more general time-delayed parabolic system in the form

By writing

We say that the vector function

For quasimonotone nondecreasing functions

(

It is clear that if there exists a positive constant vector

Let

In system (

Let

To investigate the asymptotic behavior of the solution of (

Let the conditions in Lemma

the sequences

for any

if

In Lemma

Let the conditions in Lemma

To prove the existence and uniqueness of constant positive solution of the steady-state problem (

Let

With the inequalities (

To prove the main results in this paper, we apply Lemmas

Let

The steady-state problem (

For any nontrivial nonnegative

The positive constant solution

(i)Since the boundary conditions are

(ii)It is obvious that the pairs

(iii)Lemma

All the results of Theorem

This follows from the argument in the proof of Theorem

Our model and result are different from the existence ones such as those of Pao [

In this section, we give numerical simulations supporting our theoretical analysis. As an example, we consider system (

The solution of system (

The solution of system (

The solution of system (

The solution of system (

It is well known that the analysis of stability for a system of delay reaction-diffusion multispecies model is quite difficult since the reaction among multispecies is more complex. Therefore, the works on this subject are very rare. A detailed analysis on the stability for a two-prey one-predator model, one-prey two-predator model, and three-species food-chain model with delay and diffusion was given by Pao [

In this paper, based on the ideas of Pao [

(a) The steady-state problem (

(b) For any nontrivial nonnegativ

(c) The positive constant solution

The condition

The authors are grateful to the referee for her/his comments. This work is supported by Science and Technology Study Project of Chongqing Municipal Education Commission (Grant no. KJ 080511) of China, Natural Science Foundation Project of CQ CSTC (Grant no. 2008BB7415 and 2007BB2450) of China, Foundation Project of Doctor Graduate Student Innovation of Beijing University of Technology of China, the NSFC (Grant nos.10471009), and BSFC (Grant no. 1052001) of China.