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This paper considers the asymptotic dynamics of steady states to the Lotka-Volterra competition diffusion systems with random perturbations by two-parameter white noise on the whole real line. By the fundamental solution of heat equation, we get the asymptotic fluctuating behaviors near the stable states respectively. That is, near the steady state ( u, v)=(0,1), the mean value Eu( x, t) is shifted above the equilibrium u=0 and Ev( x, t) is shifted below the equilibrium v=1. However, near the steady state ( u, v)=(1,0), the mean value Eu( x, t) is shifted below the equilibrium u =1 and Eu( x, t)=0.

Nonlinear reaction diffusion systems arise in several fields and have been studied by many authors (see [

where

with

function at least twice differentiable at equilibrium.

At present time, it is a well developed area of research which includes qualitative properties of traveling wavefronts for many complex systems. Traveling waves are natural phenomena ubiquitously for reaction diff- usion systems in many scientific areas, such as in biophysics, population genetics, mathematical ecology, chemistry, chemical physics and so on [

Consider the LV competition-diffusion system

where

where

For

ordinary differential equations in the first quadrant, we have the following cases for the system (see [

1) Monostable case:

2) Coexistence case:

3) Bistable case:

Traveling wavefronts of the system (2) have been studied very extensively. We refer readers to the references for traveling wave solutions connecting two equilibria.

1) Conley and Gardner [

2) Tang and Fife [

3) Kanel and Zhou [

4) Fei and Carr [

For instance, we give some results on the traveling wave solutions of system (2).

Theorem 1. [

there exist positive increasing traveling wavefronts

2) There do not exist traveling wavefront

where

Theorem 2. [

In fact, under the conditions

X. X. Bao and Z. C. Wang [

where

We know that in a linear system the noise does not affect the mean value at equilibrium; however, in a nonlinear system, the mean is displaced from an equilibrium. How can one describe this displaced mean value? H. C. Tuckwell [

In this paper, we are interested in calculating the statistical properties of the steady states of the LV competi- tion-diffusion system (2) under the influence of random perturbations by two-parameter white noise

where

where

The initial condition to (8) is

equilibria (0,1) and (1,0), and the boundary conditions of the traveling wavefront are

We present asymptotic representations of steady states of the LV competition diffusion system that it is randomly perturbed by two-parameter white noise

For

We write the solution of the system (2) as

and rewrite the system (8) in the following form

where

We put (11) into (12). Equating coefficients of powers of

As we know, the fundamental solution of the deterministic linear system

is

where

From the sequence of linear SPDEs we have the solutions of initial value problems (15) and (16), respectively

According to the zero-mean property of Itô integral we have

These give the expectation of stochastic process

The equilibrium

it has two negative eigenvalues

such that

thus

Therefore, the solution of (15) is

In order to compute the expectations

Since

so we have

Since

so we have

Therefore, we get

that is,

As complexity of the formula of expectation

the signs of

By the formula

and l’Hôpital’s rule, we have

Denote

and

Similarly, we have

calculating the limits we have

as

that is,

where

since

Therefore, we get the random perturbation of the traveling wave solution of (8) near the equilibrium point

since

these imply that the effect of zero-mean white noise on the system near the lower equilibrium

We now consider another equilibrium

it has two negative eigenvalues

such that

thus

Therefore, the solution of (15) is

so we have

The solution of (16) is

hence we have

Let

Then, we get the random perturbation of the traveling wavefront of (8) near the equilibrium point

From (56),

implies that the effect of zero-mean white noise on the system near the lower equilibrium

Remark 1. In the future paper, we will consider simulation of solutions on bounded domains and compare with the present analytical results. Also, we want to consider the system that the white noise is included in the 2nd component of (8), but according to the complicated calculations in Sections 3 and 4, we must look for a new idea to deal with this coupled problem.

This work was supported by National Natural Sciences Foundation of China (Grant No. 11471129). Corresponding author: Yanbin Tang.