In this section, we first recall a theorem for the existence of random attractors. Please note that here we omit the basic knowledge about random dynamical systems (RDS) and random attractor. Reader can refer to [1, 11, 17-19] for these knowledge.

*Suppose D is the collection of random subsets of X, and* {*K*(*ω*)}_{ω∈Ω}, ∈ *D is a random absorbing set for RDS ϕ in D and ϕ is D-pullback asymptotically compact in X. Then ϕ has a unique D-random attractor* {*A*(ω)}_{ω∈Ω} *which has the following form*
$$A\left(\omega \right)={\cap}_{\tau \ge 0}\overline{{\cup}_{t\ge \tau}\varphi \left(t,{\theta}_{-t}\omega ,K\left({\theta}_{-t}\omega \right)\right).}$$(9)

Next, we shall show that system (1)-(6) generates a random dynamical system. For our purpose, we first transform this stochastic system into a deterministic dynamical with random attractor. Assume that (Ω, ℱ, ℙ) is the probability space defined in Section 1. Define (*θ*_{t}) _{t∈ℝ} on Ω by
$${\theta}_{t}\omega \left(\cdot \right)=\omega \left(\cdot +t\right)-\omega \left(t\right),\phantom{\rule{thinmathspace}{0ex}}t\in \mathbb{R},$$

then (Ω, ℱ, ℙ, (*θ*_{t}) _{t∈ℝ}) is a metric dynamical system.

Set $\stackrel{~}{g}=(\stackrel{~}{u},\stackrel{~}{v},\stackrel{~}{w},\stackrel{~}{y}{)}^{T}$, then the system (1)-(6) can be rewritten as follows:
$$\frac{d\stackrel{~}{g}}{dt}=A\stackrel{~}{g}+\stackrel{~}{F}\left(\stackrel{~}{g}\right)+\sigma \stackrel{~}{g}\circ \frac{d{W}_{t}}{dt},\phantom{\rule{thinmathspace}{0ex}}t>0$$(10)
$$\stackrel{~}{g}\left(0,x\right)=\stackrel{~}{g}\left(x\right)={\left(\stackrel{~}{u}\left(x\right),\stackrel{~}{v}\left(x\right),\stackrel{~}{w}\left(x\right),\stackrel{~}{y}\left(x\right)\right)}^{T},x\in D,$$(11)

where
$$A=\left(\begin{array}{cccc}{d}_{1}\mathrm{\Delta}& 0& 0& 0\\ 0& {d}_{2}\mathrm{\Delta}& 0& 0\\ 0& 0& {d}_{1}\mathrm{\Delta}& 0\\ 0& 0& 0& {d}_{2}\mathrm{\Delta}\end{array}\right),\phantom{\rule{thinmathspace}{0ex}}\stackrel{~}{F}\left(\stackrel{~}{g}\right)=\left(\begin{array}{c}-\left(F+k\right)\stackrel{~}{u}+{\stackrel{~}{u}}^{2}\stackrel{~}{v}+{D}_{1}\left(\stackrel{~}{w}-\stackrel{~}{u}\right)\\ F\left(1-\stackrel{~}{v}\right)-{\stackrel{~}{u}}^{2}\stackrel{~}{v}+{D}_{2}\left(\stackrel{~}{y}-\stackrel{~}{v}\right)\\ -\left(F+k\right)\stackrel{~}{w}+{\stackrel{~}{w}}^{2}\stackrel{~}{y}+{D}_{1}\left(\stackrel{~}{u}-\stackrel{~}{w}\right)\\ F\left(1-\stackrel{~}{y}\right)-{\stackrel{~}{w}}^{2}\stackrel{~}{y}+{D}_{2}\left(\stackrel{~}{v}-\stackrel{~}{y}\right)\end{array}\right).$$

To transform the stochastic system into a deterministic system with random parameter, we introduce the following one-dimensional Ornstein-Uhlenbeck process:
$$dz+zdt=d{W}_{t}.$$(12)

From [13], we know that the stationary solution of Ornstein-Uhlenbeck process has the following form:
$$z\left({\theta}_{t}\omega \right)\equiv -\underset{-\mathrm{\infty}}{\overset{0}{\int}}{e}^{s}\left({\theta}_{t}\omega \right)\left(s\right)ds,\phantom{\rule{thinmathspace}{0ex}}t\in \mathbb{R}$$.

Moreover, the random variable *z*(*θ*_{t} *ω*) is tempered, and ℙ—a.e. ω ∈ Ω, t ↦ *z*(*θ*_{t} *ω*) is continuous in *t*, and satisfies the properties (see [1, 11, 13]):
$$\begin{array}{cc}\underset{t\to \pm \mathrm{\infty}}{lim}\frac{\left|z\left({\theta}_{t}\omega \right)\right|}{\left|t\right|}=0;& \underset{t\to \pm \mathrm{\infty}}{lim}\frac{1}{t}\underset{0}{\overset{t}{\int}}\phantom{\rule{thinmathspace}{0ex}}z\left({\theta}_{s}\omega \right)ds=0\end{array}$$(13).

Set $(u\left(t\right),v\left(t\right),w\left(t\right),y\left(t\right){)}^{T}={e}^{-\sigma z\left({\theta}_{t}\omega \right)}{\left(\stackrel{~}{u}\left(t\right),\stackrel{~}{v}\left(t\right),\stackrel{~}{w}\left(t\right),\stackrel{~}{y}\left(t\right)\right)}^{T}$ Then, we obtain the equivalent system of (10) and (11) as:
$$\frac{\mathrm{\partial}u}{\mathrm{\partial}t}={d}_{1}\mathrm{\Delta}u-\left(F+k-\sigma z\left({\theta}_{t}\omega \right)\right)u+{e}^{2\sigma z\left({\theta}_{t}\omega \right)}{u}^{2}v+{D}_{1}\left(w-u\right),$$(14)
$$\frac{\mathrm{\partial}v}{\mathrm{\partial}t}={d}_{2}\mathrm{\Delta}v+F{e}^{-\sigma z\left({\theta}_{t}\omega \right)}+\left(\sigma z\left({\theta}_{t}\omega \right)-F\right)v-{e}^{2\sigma z\left({\theta}_{t}\omega \right)}{u}^{2}v+{D}_{2}\left(y-v\right),$$(15)
$$\frac{\mathrm{\partial}w}{\mathrm{\partial}t}={d}_{1}\mathrm{\Delta}w-\left(F+k-\sigma z\left({\theta}_{t}\omega \right)\right)w+{e}^{2\sigma z\left({\theta}_{t}\omega \right)}{w}^{2}y+{D}_{1}\left(u-w\right),$$(16)
$$\frac{\mathrm{\partial}y}{\mathrm{\partial}t}={d}_{2}\mathrm{\Delta}y+F{e}^{-\sigma z\left({\theta}_{t}\omega \right)}+\left(\sigma z\left({\theta}_{t}\omega \right)-F\right)y-{e}^{2\sigma z\left({\theta}_{t}\omega \right)}{w}^{2}y+{D}_{2}\left(v-y\right),$$(17)

that is, *g* = (*u, v, w, y*)^{T} satisfies
$$\frac{dg}{dt}=Ag+F\left(g,\omega \right)+\sigma z\left({\theta}_{t}\omega \right)g,\phantom{\rule{thinmathspace}{0ex}}t>0,$$(18)
$$g\left(0,x\right)={g}_{0}\left(x\right)={e}^{-\sigma z\left(\omega \right)}{\stackrel{~}{g}}_{0}\left(x\right)=({u}_{0}\left(x\right),{v}_{0}\left(x\right),{w}_{0}\left(x\right),{y}_{0}\left(x\right){)}^{T},\phantom{\rule{thinmathspace}{0ex}}x\in D,$$(19)

with
$$F\left(g,w\right)=\left(\begin{array}{c}-\left(F+k\right)u+{e}^{2\sigma z\left({\theta}_{t}\omega \right)}{u}^{2}v+{D}_{1}\left(w-u\right)\\ F{e}^{-\sigma z\left({\theta}_{t}\omega \right)}-Fv-{e}^{2\sigma z\left({\theta}_{t}\omega \right)}{u}^{2}v+{D}_{2}\left(y-v\right)\\ -\left(F+k\right)w+{e}^{2\sigma z\left({\theta}_{t}\omega \right)}{w}^{2}y+{D}_{1}\left(u-w\right)\\ F{e}^{-\sigma z\left({\theta}_{t}\omega \right)}-Fy-{e}^{2\sigma z\left({\theta}_{t}\omega \right)}{w}^{2}y+{D}_{2}\left(v-y\right)\end{array}\right).$$

Notice that for ℙ—*a.e. ω* ∈ Ω, *F*(*g, ω*) is locally Lipschtiz continuous with respect to g. In [10], You proved that the deterministic system has a unique solution by the Galekin method. Similar to deterministic system, by the Galekin method, for ℙ—*a.e. ω* ∈ Ω, we can prove that for g_{0} ∈ [*L*^{2}(*D*)]^{4}, (18)-(19) has a unique solution *g*(·, *ω*, *g*_{0}) ∈ *C*([0, ∞), [*L*^{2}(*D*)]^{4}) ∩ *L*^{2}((0, ∞), [*H*^{1}(*D*)]^{4}) with *g*(0, *ω*, *g*_{0}) = *g*_{0}. Moreover, similarly to Lemma 3 of [10], we can prove that *g*(*t*, *ω*, *g*_{0}) is a unique, global, weak solution with respect to *g*_{0} ∈ [*L*^{2}(*D*)]^{4}, for *t* ∈ [0, ∞). This shows that (18) and (19) generate a continuous random dynamical system (*φ*(*t*))_{t ≥ 0} over (Ω, ℱ, ℙ, (*θ*_{t}) _{t∈ℝ}) with
$$\begin{array}{cc}\phi \left(t,\omega ,{g}_{0}\right)=g\left(t,\omega ,{g}_{0}\right),& \mathrm{\forall}\left(t,\omega ,{g}_{0}\right)\in {\mathbb{R}}^{+}\times \mathrm{\Omega}\times {[{L}^{2}\left(D\right)]}^{4}.\end{array}$$(20)

Now assume that *ϕ* : ℝ^{+} × Ω × [*L*^{2}(*D*)]^{4} → [*L*^{2}(*D*)]^{4} is given by
$$\varphi \left(t,\omega ,{\stackrel{~}{g}}_{0}\right)=\stackrel{~}{g}\left(t,\omega ,{\stackrel{~}{g}}_{0}\right)=g\left(t,\omega ,{e}^{-\sigma z\left(\omega \right)}{g}_{0}\right){e}^{\sigma z\left({\theta}_{t}\omega \right)}.$$(21)

Then *ϕ* is a continuous dynamical system associated to (10)-(11). Notice that two dynamical systems are conjugate to each other. Thus, in the following sections, we consider only the existence of a random attractor of *φ*.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.