Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 340789, 17 pages

doi:10.1155/2012/340789

Research Article

Random Attractors for Stochastic Three-Component Reversible Gray-Scott System on Infinite Lattices

Anhui Gu,¹ Zhaojuan Wang,² and Shengfan Zhou³

¹College of Science, Guilin University of Technology, Guilin 541004, China

²School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

³Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received 13 April 2012; Accepted 29 May 2012

Academic Editor: Xiaohui Liu

Copyright © 2012 Anhui Gu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices.

1. Introduction

Consider the following stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices:

where (the set of integers), , and ; all the parameters are positive constants; is a Brownian motion on and denotes the Stratonovich sense of the stochastic term.

System (1.1) can be considered as a discrete model of stochastic three-component reversible Gray-Scott system in which the existence of a random attractor has been established [1]. When there is no stochastic term, system (1.1) can be considered as a discrete analogue of the following three-component reversible Gray-Scott system in :

which was firstly introduced by Mahara et al. [2], then it was reduced to system (1.2) under some nondimensional transformations in You [3]. Also, the existence of a global attractor for the solution semiflow of (1.2) with Neumann boundary condition on a bounded domain of space dimension was proved in [3].

When and , system (1.2) becomes the two-components Gray-Scott equations which was one of the models signified the seminal work of the Brussell school. The model originated from describing an isothermal, cubic autocatalytic, continuously fed and diffusive reactions of two chemicals (see [4–8]), but neglected the reversible factors. Indeed, the reversibility in the interactions of multispecies is an indispensable factor in many processes in natural and social sciences. If we take the reversibility into account, it yields system (1.2).

Stochastic lattice differential equations have discrete spatial structures and take random influences into account. These random effects are not only introduced to compensate for the defects in some deterministic models, but are also rather intrinsic phenomena. Bates et al. [9] initiated the consideration of stochastic lattice dynamical systems with additive noises and Caraballo and Lu [10] was the first to consider the stochastic lattice dynamical systems with a multiplicative noise, and Han et al. [11] generalized the results of [9, 10] to a more general space. For more details and the quite recent results, we can refer to, for example, [12–15].

Just like the models considered in biology, the discrete time models governed by difference equations are more appropriate than the continuous ones; we can also deal with the chemical and biochemical reactions in the same manner, see, for example, [16, 17] and the references therein. However, very few investigations are on this topic, especially for the stochastic three-component reversible Gray-Scott system on infinite lattices, is widely open, to the best of our knowledge.

The paper is organized as follows. In Section 2, we present some preliminaries and definitions. Section 3 is devoted to the existence of a random attractor.

2. Preliminaries

Let be a probability space, where is a subset of , which endowed with the compact open topology (see [18]), is the Borel -algebra, and is the corresponding Wiener measure on . Let , , then is an ergodic metric dynamical system. Throughout the paper, we denote

the Hilbert space equipped with the usual inner product and norm:

For the reader's convenience, we introduce some basic concepts related to random dynamical systems and random attractor, which are taken from [11, 18, 19]. Let be a separable Hilbert space and a probability space.

Definition 2.1. A stochastic process is a continuous random dynamical system over if is -measurable, and for all ,

(i)the mapping , is continuous for every ,(ii) is the identity on ,(iii)(cocycle property) for all .

Definition 2.2. (i) A set-valued mapping (we may write it as for short) is said to be a random set if the mapping dist is measurable for any , where dist is the distant in between the element and the set .

(ii) A random set is said to be bounded if there exist and a random variable such that for all .

(iii) A random set is called a compact random set if is compact for all .

(iv) A random bounded set is called tempered with respect to if for a.e. ,   for  all  . A random variable is said to be tempered with respect to if for a.e. ,   for  all  .

We consider a continuous random dynamical system (RDS) over and the set of all tempered random sets of .

Definition 2.3. A random set is called an absorbing set in if for all and a.e. there exists such that

Definition 2.4. A random set is called a global random attractor (pullback attractor) for if the following hold:

(i) is a random compact set, that is, is measurable for every and is compact for a.e. ;(ii) is strictly invariant, that is, for and all , ;(iii) attracts all sets in , that is, for all and a.e. , we have

where is the Hausdorff semimetric ().

Proposition 2.5 2.5 (see [11]). Suppose that

(a)there exists a random bounded absorbing set , , such that for any and all , there exists yielding for all ;(b)the RDS is random asymptotically null on , that is, for any , there exist and such that

Then the RDS possesses a unique global random attractor given by

3. Existence of a Random Attractor

In this section, we will derive the random attractor of the stochastic three-component reversible Gray-Scott lattice system (1.1) with a multiplicative white noise.

For , we define to be linear operators from to for , as follows:

It is easy to show that , for all , which implies that .

In the sequel, we rewrite the system (1.1) with initial values as the following integral equations in for

where is a two-sided Brownian motion on the same probability space . To prove that this system (3.2) generates a random dynamical system, we will transform it into a random differential equation system in .

Before performing this transformation, we need to recall some properties of the Ornstein-Uhlenbeck processes. Let

We know that is an Ornstein-Uhlenbeck process on and solves the following one-dimensional stochastic differential equation (see [20] for details):

where for . In fact, we have the following.

Lemma 3.1 3.1 (see [10, 18]). There exists a -variant set of of full measure such that, for , one has

(i)the random variable is tempered;(ii)the mapping

is a stationary solution of Ornstein-Uhlenbeck equation (3.4) with continuous trajectories;(iii)

Obviously, is clearly a homeomorphism in , and the inverse operator is well defined. It easily follows from (3.6) that and have subexponential growth as for all , which implies that they are tempered. Since the mapping of on has the same properties as the original one if we choose the trace -algebra with respect to to be denoted also by , we can change our metric dynamical system with respect to , and still denoted by the symbols .

Let

where , and is a solution of (3.2). Then system (1.1) can be written as the following random system with random coefficients but without white noise:

and an initial condition

Now we establish the following result.

Theorem 3.2. Let and be fixed. Then the following properties hold:

(i)for every , system (3.9) admits a unique solution ,(ii)the solution of system (3.9) depends continuously on the initial data , that is, for each , the mapping is continuous.

Proof. (1) Denote

Then system (3.9) can be written as

Since is continuous with respect to , define

then

For any , ,

For any bounded set with , define a random variable by

Then

and, for any , we have

Then we obtain that is locally Lipschitz in from to . By Proposition in [19], problem (3.12) possesses a unique local solution , where is the maximal interval of existence of the solution of (3.12). Now, we will show that the local solution is a global one. Define

then system (3.9) can be written as

Taking the inner products of (3.20) with , and , respectively, and adding up the resulting equalities, we get

which implies that

Setting

then (3.22) yields

Applying Gronwall's lemma to (3.24), we obtain that, for ,

Denoting

we get

which implies that the solution is defined in any interval .

(2) Let and

be the corresponding solutions of (3.12). Then, denoting − , we have

Taking the inner product of (3.29) with and , respectively, it yields

Due to (3.27), we have

Denoting , we obtain that

By Gronwall's lemma, for , we have

where . According to (3.25), for ,

where . Obviously, is continuous -a.s. Also, we have , which implies that is a tempered random variable. Then by Proposition , [18], for given , there is an -slowly varying random variable for which

where satisfies

Combining with (3.34) and (3.35), we easily conclude that for ,

which implies

where . If and , then the above inequality shows the uniqueness and continuous dependence on the initial data of the solution of (3.12). So the both results of the theorem hold.

Theorem 3.3. System (3.12) generates a continuous random dynamical system over , where for , and for all . Moreover, if one defines by

for , and for all , then is another random dynamical system for which the process solves (3.2) for any initial condition .

Proof. The fact that is continuous random dynamical system follows from Theorem 3.2. The measurability of follows from the properties of the transformation (see [18, 19]). It follows directly the other statements.

Note that the two random dynamical systems are equivalent. It is easy to check that has a random attractor provided possesses a random attractor. Then, we only need to consider the random dynamical system .

Now, we are in the position to study the existence of tempered random bounded absorbing set and global random attractor for the RDS in .

Lemma 3.4. There exists a random bounded ball centered at 0 with random radius such that is a random absorbing set for in ; that is, for any and , there exists yielding , for all .

Proof. By substituting by in (3.25), we have

By the properties of the Ornstein-Uhlenbeck process,

Notice that is tempered, then for ,

We can choose

then is a random absorbing set for in , and . Here, we remain only to check that

Indeed, obviously we have

Lemma 3.5. The RDS generated by (3.9) is random asymptotically null on ; that is, for any , there exist , , and such that

Proof. Choose a smooth cut-off function satisfying for and for , for . Suppose there exists a constant such that for .

Set , and . By taking the inner product of (3.20) with , , and , respectively, we get

Due to [9, 10], we have

Combining with (3.47) to (3.48), we obtain

that is,

Denote

then (3.50) yields

By using Gronwall's lemma, for , we have

Replace by . We then estimate each term on the right-hand side of (3.53). From (3.25) with replaced by and by , respectively, it then follows that

Hence, by using (3.6), there is a , such that if ,

Next, we estimate

By using (3.7), there exist and , such that if and , then

By using (3.7) again, there exists , such that if , we have

Therefore, by letting

we obtain, for and ,

which implies

The proof is completed.

Now, we have the main result.

Theorem 3.6. The random dynamical system generated by system (3.9) has a unique global random attractor.

Proof. It is obvious from Lemmas 3.4, and 3.5, and Proposition 2.5.

Acknowledgments

This work was supported by National Natural Science Foundation of China under Grant 11071165 and Guangxi Provincial Department of Research Project under Grant 201010LX166.

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