Dynamical analysis of a heterogeneous spatial diffusion Zika model with vector-bias and environmental transmission

Liping Wang; Xinyu Wang; Dajun Liu; Xuekang Zhang; Peng Wu; Liping Wang; Xinyu Wang; Dajun Liu; Xuekang Zhang; Peng Wu

doi:10.3934/era.2024061

Electronic Research Archive

2024, Volume 32, Issue 2: 1308-1332. doi: 10.3934/era.2024061

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Research article

Dynamical analysis of a heterogeneous spatial diffusion Zika model with vector-bias and environmental transmission

1.
School of Mathematics-Physics and Finance, Anhui Polytechnic University, Wuhu 241000, China
2.
School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China

Academic Editor: Pengcheng Xiao

Received: 05 November 2023 Revised: 03 January 2024 Accepted: 26 January 2024 Published: 31 January 2024

In this study, we formulate a reaction-diffusion Zika model which incorporates vector-bias, environmental transmission and spatial heterogeneity. The main question of this paper is the analysis of the threshold dynamics. For this purpose, we establish the mosquito reproduction number $R_{1}$ and basic reproduction number $R_{0}$ . Then, we analyze the dynamical behaviors in terms of $R_{1}$ and $R_{0}$ . Numerically, we find that the ignorance of the vector-bias effect will underestimate the infection risk of the Zika disease, ignorance of the spatial heterogeneity effect will overestimate the infection risk, and the environmental transmission is indispensable.

Keywords:

Citation: Liping Wang, Xinyu Wang, Dajun Liu, Xuekang Zhang, Peng Wu. Dynamical analysis of a heterogeneous spatial diffusion Zika model with vector-bias and environmental transmission[J]. Electronic Research Archive, 2024, 32(2): 1308-1332. doi: 10.3934/era.2024061

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Abstract

1. Introduction

Lattice dynamical systems arise from a variety of applications in electrical engineering, biology, chemical reaction, pattern formation and so on, see, e.g., ^{[4,7,14,19,33]}. Many researchers have discussed broadly the deterministic models in ^[6,12,34,39], etc. Stochastic lattice equations, driven by additive independent white noise, was discussed for the first time in ^[2], followed by extensions in ^{[8,13,15,16,21,23,27,32,35,36,37,38,40]}.

In this paper, we will study the long term behavior of the following second order non-autonomous stochastic lattice system driven by additive white noise: for given $\tau\in\mathbb{R}$ , $t > \tau$ and $i\in\mathbb{Z}$ ,

$\begin{equation} \begin{cases} \ddot{u}+\nu A\dot{u}+h(\dot{u})+Au+\lambda u+f(u) = g(t)+a\dot{\omega}(t),\\ u(\tau) = (u_{\tau i})_{i\in\mathbb{Z}} = u_{\tau},\; \; \dot{u}(\tau) = (u_{\tau i}^{1})_{i\in\mathbb{Z}} = u_{\tau}^{1}, \end{cases} \end{equation}$

(1.1)

where $u = (u_{i})_{i\in\mathbb{Z}}$ is a sequence in $l^{2}$ , $\nu$ and $\lambda$ are positive constants, $\dot{u} = (\dot{u}_{i})_{i\in\mathbb{Z}}$ and $\ddot{u} = (\ddot{u}_{i})_{i\in\mathbb{Z}}$ denote the fist and the second order time derivatives respectively, $Au = ((Au)_{i})_{i\in\mathbb{Z}}$ , $A\dot{u} = ((A\dot{u})_{i})_{i\in\mathbb{Z}}$ , $A$ is a linear operators defined in (2.2), $a = (a_{i})_{i\in\mathbb{ Z}}\in l^{2}$ , $f(u) = \big(f_{i}(u_{i})\big)_{i\in\mathbb{Z}}$ and $h(\dot{u}) = \big(h_{i}(\dot{u}_{i})\big)_{i\in\mathbb{Z}}$ satisfy certain conditions, $g(t) = \big(g_{i}(t)\big)_{i\in\mathbb{Z}}\in L^{2}_{loc}(\mathbb{R}, l^{2})$ is a given time dependent sequence, and $\omega(t) = W(t, \omega)$ is a two-sided real-valued Wiener process on a probability space.

The approximation we use in the paper was first proposed in ^[18,22] where the authors investigated the chaotic behavior of random equations driven by $\mathcal{G}_{\delta}(\theta_{t}\omega)$ . Since then, their work was extended by many scholars. To the best of my knowledge, there are three forms of Wong-Zakai approximations $\mathcal{G}_{\delta}(\theta_{t}\omega)$ used recenly, Euler approximation of Brownian ^{[3,10,17,20,25,28,29,30]}, Colored noise ^[5,11,26,31] and Smoothed approximation of Brownian motion by mollifiers ^[9]. In this paper, we will focus on Euler approximation of Brownian and compare the long term behavior of system (1.1) with pathwise deterministic system given by

$\begin{equation} \begin{cases} \ddot{u}^{\delta}+\nu A\dot{u}^{\delta}+h(\dot{u}^{\delta})+Au^{\delta}+\lambda u^{\delta}+f(u^{\delta}) = g(t)+a\mathcal{G}_{\delta}(\theta_{t}\omega),\\ u^{\delta}(\tau) = (u^{\delta}_{\tau i})_{i\in\mathbb{Z}} = u^{\delta}_{\tau},\; \; \dot{u}^{\delta}(\tau) = (u^{\delta,1}_{\tau i})_{i\in\mathbb{Z}} = u^{\delta,1}_{\tau}, \end{cases} \end{equation}$

(1.2)

for $\delta\in\mathbb{R}$ with $\delta\neq0$ , $\tau\in\mathbb{R}$ , $t > \tau$ and $i\in\mathbb{Z}$ , where $\mathcal{G}_{\delta}(\theta_{t}\omega)$ is defined in (3.2). Note that the solution of system (1.2) is written as $u^{\delta}$ to show its dependence on $\delta$ .

This paper is organized as follows. In Section 2, we prove the existence and uniqueness of random attractors of system (1.1). Section 3 is devoted to consider the the Wong-Zakai approximations associated with system (1.1). In Section 4, we establish the convergence of solutions and attractors for approximate system (1.2) when $\delta\rightarrow0$ .

Throughout this paper, the letter $c$ and $c_{i}(i = 1, 2, \ldots)$ are generic positive constants which may change their values from line to line.

2. Stochastic lattice system with additive white noise

In this section, we will define a continuous cocycle for second order non-autonomous stochastic lattice system (1.1), and then prove the existence and uniqueness of pullback attractors.

A standard Brownian motion or Wiener process $(W_{t})_{t\in\mathbb{R}}$ (i.e., with two-sided time) in $\mathbb{R}$ is a process with $W_{0} = 0$ and stationary independent increments satisfying $W_{t}-W_{s}\sim \mathcal{N}(0, |t-s|I)$ . $\mathcal{F}$ is the Borel $\sigma$ -algebra induced by the compact-open topology of $\Omega$ , and $P$ is the corresponding Wiener measure on $(\Omega, \mathcal{F})$ , where

$\Omega = \{\omega\in C(\mathbb{R},\mathbb{R}):\omega(0) = 0\},$

the probability space $(\Omega, \mathcal{F}, P)$ is called Wiener space. Define the time shift by

$\begin{equation} \begin{split}\nonumber \theta_{t}\omega(\cdot) = \omega(\cdot+t)-\omega(t),\quad \omega \in \Omega,\ t \in\mathbb{R}.\\ \end{split} \end{equation}$

Then $(\Omega, \mathcal{F}, P, \{\theta_{t}\}_{t\in\mathbb{R}})$ is a metric dynamical system (see ^[1]) and there exists a $\{\theta_{t}\}_{t\in\mathbb{R}}$ -invariant subset $\tilde{\Omega}\subseteq\Omega$ of full measure such that for each $\omega\in\Omega$ ,

$\begin{equation} \begin{split} \frac{\omega(t)}{t}\rightarrow0\; \; \text{as}\; \; t\rightarrow \pm\infty. \end{split} \end{equation}$

(2.1)

For the sake of convenience, we will abuse the notation slightly and write the space $\tilde{\Omega}$ as $\Omega$ .

We denote by

$l^{p} = \{u|u = (u_{i})_{i\in \mathbb{Z}},u_{i}\in \mathbb{R} ,\ \sum\limits_{i\in \mathbb{Z}}|u_{i}|^{p} < +\infty\},$

with the norm as

$\|u\|_{p}^{p} = \sum\limits_{i\in \mathbb{Z}}|u_{i}|^{p}.$

In particular, $l^{2}$ is a Hilbert space with the inner product $(\cdot, \cdot)$ and norm $\|\cdot\|$ given by

$(u,v) = \sum\limits_{i\in \mathbb{Z}}u_{i}v_{i},\qquad \|u\|^{2} = \sum\limits_{i\in \mathbb{Z}}|u_{i}|^{2},$

for any $u = (u_{i})_{i\in \mathbb{Z}}, \ v = (v_{i})_{i\in \mathbb{Z}}\in l^{2}$ .

Define linear operators $B$ , $B^{*}$ , and $A$ acting on $l^{2}$ in the following way: for any $u = (u_{i})_{i\in\mathbb{Z}}\in l^{2}$ ,

$(Bu)_{i} = u_{i+1}-u_{i},\qquad(B^{*}u)_{i} = u_{i-1}-u_{i},$

and

$\begin{equation} \begin{split} (Au)_{i} = 2u_{i}-u_{i+1}-u_{i-1}. \end{split} \end{equation}$

(2.2)

Then we find that $A = BB^{*} = B^{*}B$ and $(B^{*}u, v) = (u, Bv)$ for all $u, v\in l^{2}$ .

Also, we let $F_{i}(s) = \int_{0}^{s}f_{i}(r)dr$ , $h(\dot{u}) = (h_{i}(\dot{u}_{i}))_{i\in\mathbb{Z}}$ , $f(u) = (f_{i}(u_{i}))_{i\in\mathbb{Z}}$ with $f_{i}, h_{i}\in C^{1}(\mathbb{R}, \mathbb{R})$ satisfy the following assumptions:

$\begin{equation} |f_{i}(s)|\leq \alpha_{1}(|s|^{p}+|s|), \end{equation}$

(2.3)

$\begin{equation} sf_{i}(s)\geq \alpha_{2}F_{i}(s)\geq\alpha_{3}|s|^{p+1}, \end{equation}$

(2.4)

and

$\begin{equation} h_{i}(0) = 0,\; \; 0 < h_{1}\leq h_{i}'(s)\leq h_{2},\; \; \forall s\in\mathbb{R}, \end{equation}$

(2.5)

where $p > 1$ , $\alpha_{i}$ and $h_{j}$ are positive constants for $i = 1, 2, 3$ and $j = 1, 2$ .

In addition, we let

$\begin{equation} \begin{split} \beta = \frac{h_{1}\lambda}{4\lambda+h_{2}^{2}},\; \; \beta < \frac{1}{\nu}, \end{split} \end{equation}$

(2.6)

and

$\begin{equation} \begin{split} \sigma = \frac{h_{1}\lambda}{\sqrt{4\lambda+h_{2}^{2}}(h_{2}+\sqrt{4\lambda+h_{2}^{2}})}. \end{split} \end{equation}$

(2.7)

For any $u, v\in l^{2}$ , we define a new inner product and norm on $l^{2}$ by

$(u,v)_{\lambda} = (1-\nu\beta)(Bu,Bv)+\lambda(u,v),\; \; \|u\|^{2}_{\lambda} = (u,u)_{\lambda} = (1-\nu\beta)\|Bu\|^{2}+\lambda\|u\|^{2}.$

Denote

$l^{2} = (l^{2},(\cdot,\cdot),\|\cdot\|),\qquad l^{2}_{\lambda} = (l^{2},(\cdot,\cdot)_{\lambda},\|\cdot\|_{\lambda}).$

Then the norms $\|\cdot\|$ and $\|\cdot\|_{\lambda}$ are equivalent to each other.

Let $E = l^{2}_{\lambda}\times l^{2}$ endowed with the inner product and norm

$(\psi_{1},\psi_{2})_{E} = (u^{(1)},u^{(2)})_{\lambda}+(v^{(1)},v^{(2)}),\qquad \|\psi\|_{E}^{2} = \|u\|^{2}_{\lambda}+\|v\|^{2},$

for $\psi_{j} = (u^{(j)}, v^{(j)})^{T} = \big((u^{(j)}_{i}), (v^{(j)}_{i})\big)^{T}_{i\in \mathbb{Z}}\in E, \ j = 1, 2, \quad \psi = (u, v)^{T} = \big((u_{i}), (v_{i})\big)^{T}_{i\in \mathbb{Z}}\in E.$

A family $D = \{D(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}$ of bounded nonempty subsets of $E$ is called tempered (or subexponentially growing) if for every $\epsilon > 0$ , the following holds:

$\lim\limits_{t\rightarrow -\infty}e^{\epsilon t}\|D(\tau+t,\theta_{t}\omega)\|^{2} = 0,$

where $\|D\| = \mathop{\sup}\limits_{x\in D}\|x\|_{E}$ . In the sequel, we denote by $\mathcal{D}$ the collection of all families of tempered nonempty subsets of $E$ , i.e.,

$\begin{equation} \nonumber \mathcal{D} = \{D = \{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\}:D\; \text{is tempered in}\; E\}. \end{equation}$

The following conditions will be needed for $g$ when deriving uniform estimates of solutions, for every $\tau\in\mathbb{R}$ ,

$\begin{equation} \int_{-\infty}^{\tau}e^{\gamma s}\|g(s)\|^{2}ds < \infty, \end{equation}$

(2.8)

and for any $\varsigma > 0$

$\begin{equation} \lim\limits_{t\rightarrow -\infty}e^{\varsigma t}\int_{-\infty}^{0}e^{ \gamma s}\|g(s+t)\|^{2}ds = 0, \end{equation}$

(2.9)

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$ .

Let $\bar{v} = \dot{u}+\beta u$ and $\bar{\varphi} = (u, \bar{v})^{T}$ , then system (1.1) can be rewritten as

$\begin{eqnarray} \dot{\bar{\varphi}}+L_{1}(\bar{\varphi}) = H_{1}(\bar{\varphi})+G_{1}(\omega), \end{eqnarray}$

(2.10)

with initial conditions

$\bar{\varphi}_{\tau} = (u_{\tau},\bar{v}_{\tau})^{T} = (u_{\tau},u_{\tau}^{1}+\beta u_{\tau})^{T},$

where

$L_{1}(\bar{\varphi}) = \left( \begin{array}{ccc} \beta u-\bar{v} \\ (1-\nu\beta)Au+\nu A\bar{v}+\lambda u +\beta^{2} u-\beta\bar{v} \end{array} \right) + \left( \begin{array}{ccc} 0 \\ h(\bar{v}-\beta u) \end{array} \right) ,$

$H_{1}(\bar{\varphi}) = \left( \begin{array}{ccc} 0 \\ -f(u)+g(t) \end{array} \right),\; \; \; \; G_{1}(\omega) = \left( \begin{array}{ccc} 0 \\ a\dot\omega(t) \end{array} \right) .$

Denote

$v(t) = \bar{v}(t)-a\omega(t) \; \; \text{and}\; \; \varphi = (u,v)^{T}.$

By (2.10) we have

$\begin{eqnarray} \dot{\varphi}+L(\varphi) = H(\varphi)+G(\omega), \end{eqnarray}$

(2.11)

with initial conditions

$\varphi_{\tau} = (u_{\tau},v_{\tau})^{T} = (u_{\tau},u_{\tau}^{1}+\beta u_{\tau}-a\omega(\tau))^{T},$

where

$L(\varphi) = \left( \begin{array}{ccc} \beta u-v \\ (1-\nu\beta) Au+\nu Av+\lambda u+\beta^{2} u-\beta v \end{array} \right) + \left( \begin{array}{ccc} 0 \\ h(v-\beta u+a\omega(t)) \end{array} \right) ,$

$H(\varphi) = \left( \begin{array}{ccc} 0 \\ -f(u)+g(t) \end{array} \right),\; \; \; \; G(\omega) = \left( \begin{array}{ccc} a\omega(t)\\ \beta a\omega(t)-\nu Aa\omega(t) \end{array} \right) .$

Note that system (2.11) is a deterministic functional equation and the nonlinearity in (2.11) is locally Lipschitz continuous from $E$ to $E$ . Therefore, by the standard theory of functional differential equations, system (2.11) is well-posed. Thus, we can define a continuous cocycle $\Phi_{0}:\mathbb{R}^{+}\times\mathbb{R}\times \Omega\times E\rightarrow E$ associated with system (2.10), where for $\tau\in\mathbb{R}$ , $t\in\mathbb{R}^{+}$ and $\omega\in\Omega$

$\begin{equation} \begin{split}\nonumber \Phi_{0}(t,\tau,\omega,\bar{\varphi}_{\tau})& = \bar{\varphi}(t+\tau,\tau,\theta_{-\tau}\omega,\bar{\varphi}_{\tau})\\ & = (u(t+\tau,\tau,\theta_{-\tau}\omega,u_{\tau}),\bar{v}(t+\tau,\tau,\theta_{-\tau}\omega,\bar{v}_{\tau}))^{T}\\ & = (u(t+\tau,\tau,\theta_{-\tau}\omega,u_{\tau}),v(t+\tau,\tau,\theta_{-\tau}\omega,v_{\tau})+a(\omega(t)-\omega(-\tau)))^{T}\\ & = \varphi(t+\tau,\tau,\theta_{-\tau}\omega,\varphi_{\tau})+(0,a(\omega(t)-\omega(-\tau)))^{T}, \end{split} \end{equation}$

where $v_{\tau} = \bar{v}_{\tau}+a\omega(-\tau)$ .

Lemma 2.1. Suppose that (2.3)–(2.8) hold. Then for every $\tau\in\mathbb{R}$ , $\omega\in\Omega$ , and $T > 0$ , there exists $c = c(\tau, \omega, T) > 0$ such that for all $t\in[\tau, \tau+T]$ , the solution $\varphi$ of system (2.11) satisfies

$\begin{equation} \begin{split}\nonumber \|\varphi(t,\tau,\omega,\varphi_{\tau})\|^{2}_{E} +\int_{\tau}^{t}\|\varphi(s,\tau,\omega,\varphi_{\tau})\|^{2}_{E}ds \leq& c\int^{t}_{\tau}\big(\|g(s)\|^{2}+|\omega(s)|^{2}+| \omega(s)|^{p+1}\big)ds\\ &+c\big(\|\varphi_{\tau}\|^{2}_{E}+2\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau,i})\big). \end{split} \end{equation}$

Proof. Taking the inner product $(\cdot, \cdot)_{E}$ on both side of the system (2.11) with $\varphi$ , it follows that

$\begin{equation} \begin{split} &\frac{1}{2}\frac{d}{dt}\|\varphi\|^{2}_{E}+(L(\varphi),\varphi)_{E} = (H(\varphi),\varphi)_{E}+(G(\omega),\varphi)_{E}. \end{split} \end{equation}$

(2.12)

For the second term on the left-hand side of (2.12), we have

$\begin{equation} \begin{split}\nonumber (L(\varphi),\varphi)_{E} = \beta\|u\|^{2}_{\lambda}+\beta^{2}(u,v)-\beta\|v\|^{2}+\nu(Av,v)+(h(v-\beta u+a\omega(t)),v). \end{split} \end{equation}$

By the mean value theorem and (2.5), there exists $\xi_{i}\in(0, 1)$ such that

$\begin{equation} \begin{split}\nonumber &\beta^{2}(u,v)+(h(v-\beta u+a\omega(t)),v)\\ & = \beta^{2}(u,v) +\sum\limits_{i\in\mathbb{Z}}h'_{i}(\xi_{i}(v_{i}-\beta u_{i}+a_{i}\omega(t)))(v_{i}-\beta u_{i}+a_{i}\omega(t))v_{i}\\ &\geq (\beta^{2}-h_{2}\beta)\|u\|\|v\|+h_{1}\|v\|^{2}-h_{2}|(a\omega(t),v)|. \end{split} \end{equation}$

Then

$\begin{equation} \begin{split}\nonumber (L(\varphi),\varphi)_{E}-\sigma\|\varphi\|^{2}_{E}-\frac{h_{1}}{2}\|v\|^{2} \geq &(\beta-\sigma)\|u\|^{2}_{\lambda}+(\frac{h_{1}}{2}-\beta-\sigma)\|v\|^{2}\\&- \frac{\beta h_{2}}{\sqrt{\lambda}}\|u\|_{\lambda}\|v\|-h_{2}|(a\omega(t),v)|, \end{split} \end{equation}$

which along with (2.6) and (2.7) implies that

$\begin{equation} \begin{split} (L(\varphi),\varphi)_{E}\geq\sigma\|\varphi\|^{2}_{E}+\frac{h_{1}}{2}\|v\|^{2} -\frac{\sigma+h_{1}}{6}\|v\|^{2}-c|\omega(t)|^{2}\|a\|^{2}. \end{split} \end{equation}$

(2.13)

As to the first term on the right-hand side of (2.12), by (2.3) and (2.4) we get

$\begin{equation} \begin{split} (H(\varphi),\varphi)_{E}& = (-f(u),\dot{u}+\beta u-a\omega(t))+(g(t),v)\\ &\leq-\frac{d}{dt}\big(\sum\limits_{i\in\mathbb{Z}}F_{i}(u_{i})\big)-\alpha_{2}\beta\sum\limits_{i\in\mathbb{Z}}F_{i}(u_{i}) +\alpha_{1}\sum\limits_{i\in\mathbb{Z}}(|u_{i}|^{p}+|u_{i}|)|a_{i}\omega(t)|+(g(t),v)\\ &\leq-\frac{d}{dt}\big(\sum\limits_{i\in\mathbb{Z}}F_{i}(u_{i})\big)-\frac{\alpha_{2}\beta}{p+1}\sum\limits_{i\in\mathbb{Z}}F_{i}(u_{i}) +c|\omega(t)|^{p+1}\|a\|^{p+1}\\ &\quad+\frac{\sigma\lambda}{4}\|u\|^{2}+c\|a\|^{2}|\omega(t)|^{2}+\frac{\sigma+h_{1}}{6}\|v\|^{2}+c\|g(t)\|^{2}. \end{split} \end{equation}$

(2.14)

The last term of (2.12) is bounded by

$\begin{equation} \begin{split} (G(\omega),\varphi)_{E}& = \omega(t)(a,u)_{\lambda}+\beta \omega(t)(a,v)-\nu\omega(t)(Aa,v)\\ &\leq \frac{\sigma}{4}\|u\|^{2}_{\lambda}+\frac{1}{\sigma}\|a\|^{2}_{\lambda}|\omega(t)|^{2}+\frac{\sigma+h_{1}}{6}\|v\|^{2}+c|\omega(t)|^{2}\|a\|^{2}. \end{split} \end{equation}$

(2.15)

It follows from (2.12)–(2.15) that

$\begin{equation} \begin{split} &\frac{d}{dt}\Big(\|\varphi\|_{E}^{2}+2\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{i})\Big) +\gamma\Big(\|\varphi\|_{E}^{2}+2\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{i})\Big)+\gamma\|\varphi\|_{E}^{2}\\ &\leq c\Big(\|g(t)\|^{2}+|\omega(t)|^{2}+|\omega(t)|^{p+1}\Big), \end{split} \end{equation}$

(2.16)

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$ . Multiplying (2.16) by $e^{\gamma t}$ and then integrating over $(\tau, t)$ with $t\geq\tau$ , we get for every $\omega\in\Omega$

$\begin{equation} \begin{split} &\|\varphi(t,\tau,\omega,\varphi_{\tau})\|_{E}^{2} +\gamma\int^{t}_{\tau}e^{\gamma(s-t)}\|\varphi(s,\tau,\omega,\varphi_{\tau})\|_{E}^{2}ds\\ &\leq e^{\gamma(\tau-t)}\Big(\|\varphi_{\tau}\|_{E}^{2}+2\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau,i})\Big)+c\int^{t}_{\tau}e^{\gamma(s-t)}\Big(\|g(s)\|^{2}+|\omega(s)|^{2}+|\omega(s)|^{p+1}\Big)ds, \end{split} \end{equation}$

(2.17)

which implies desired result.

Lemma 2.2. Suppose that (2.3)–(2.9) hold. Then the continuous cocycle $\Phi_{0}$ associated with system (2.10) has a closed measurable $\mathcal{D}$ -pullback absorbing set $K_{0} = \{K_{0}(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$ , where for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$

$\begin{equation} \begin{split} K_{0}(\tau,\omega) = \{\bar{\varphi}\in E:\|\bar{\varphi}\|^{2}_{E}\leq R_{0}(\tau,\omega)\}, \end{split} \end{equation}$

(2.18)

where $\bar{\varphi}_{\tau-t}\in D(\tau-t, \theta_{-t}\omega)$ and $R_{0}(\tau, \omega)$ is given by

$\begin{equation} \begin{split} R_{0}(\tau,\omega) = c+c|\omega(-\tau)|^{2}+c\int^{0}_{-\infty}e^{\gamma s}\Big(\|g(s+\tau)\|^{2}+|\omega(s)-\omega(-\tau)|^{2}+| \omega(s)-\omega(-\tau)|^{p+1}\Big)ds, \end{split} \end{equation}$

(2.19)

where $c$ is a positive constant independent of $\tau$ , $\omega$ and $\mathcal{D}$ .

Proof. By (2.17), we get for every $\tau\in\mathbb{R}$ , $t\in\mathbb{R}^{+}$ and $\omega\in\Omega$

$\begin{equation} \begin{split} &\|\varphi(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})\|^{2}_{E}+\gamma\int_{\tau-t}^{\tau}e^{\gamma(s-\tau)}\|\varphi(s,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})\|^{2}_{E}ds\\ &\leq e^{-\gamma t}\Big(\|\varphi_{\tau-t}\|^{2}_{E}+2\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau-t,i})\Big)\\ &\quad +c\int^{\tau}_{\tau-t}e^{\gamma(s-\tau)}\Big(\|g(s)\|^{2}+|\omega(s-\tau)-\omega(-\tau)|^{2}+|\omega(s-\tau)-\omega(-\tau)|^{p+1}\Big)ds\\ &\leq e^{-\gamma t}\Big(\|\varphi_{\tau-t}\|^{2}_{E}+2\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau-t,i})\Big)\\ &\quad +c\int^{0}_{-t}e^{\gamma s}\Big(\|g(s+\tau)\|^{2}+|\omega(s)-\omega(-\tau)|^{2}+| \omega(s)-\omega(-\tau)|^{p+1}\Big)ds. \end{split} \end{equation}$

(2.20)

By (2.1) and (2.8), the last integral on the right-hand side of (2.20) is well defined. For any $s\geq \tau-t$ ,

$\begin{equation} \begin{split}\nonumber &\bar{\varphi}(s,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t}) = \varphi(s,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})+(0,a(\omega(s-\tau)-\omega(-\tau)))^{T}, \end{split} \end{equation}$

which along with (2.20) implies that

$\begin{equation} \begin{split} &\|\bar{\varphi}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t})\|^{2}_{E} +\gamma\int_{\tau-t}^{\tau}e^{\gamma(s-\tau)}\|\bar{\varphi}(s,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t})\|^{2}_{E}ds\\ &\leq2\|\varphi(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})\|^{2}_{E} +2\gamma\int_{\tau-t}^{\tau}e^{\gamma(s-\tau)}\|\varphi(s,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})\|^{2}_{E}ds\\ &\quad+2\|a\|^{2}\Big(|\omega(-\tau)|^{2}+\gamma\int_{\tau-t}^{\tau}e^{\gamma(s-\tau)}|\omega(s-\tau)-\omega(-\tau)|^{2}ds\Big)\\ &\leq 4e^{-\gamma t}\Big(\|\bar{\varphi}_{\tau-t}\|^{2}_{E}+\|a\|^{2}|\omega(-t)-\omega(-\tau)|^{2}+\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau-t,i})\Big)+c|\omega(-\tau)|^{2}\\ &\quad +c\int^{0}_{-\infty}e^{\gamma s}\Big(\|g(s+\tau)\|^{2}+|\omega(s)-\omega(-\tau)|^{2}+| \omega(s)-\omega(-\tau)|^{p+1}\Big)ds.\\ \end{split} \end{equation}$

(2.21)

By (2.3) and (2.4) we have

$\begin{equation} \begin{split} \sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau-t,i})\leq \frac{1}{\alpha_{2}}\sum\limits_{i\in\mathbb{Z}} f_{i}(u_{\tau-t,i})u_{\tau-t, i} \leq\frac{1}{\alpha_{2}}\max\limits_{-\|u_{\tau-t}\|\leq s\leq\|u_{\tau-t}\|}|f'_{i}(s)|\|u_{\tau-t}\|^{2}. \end{split} \end{equation}$

(2.22)

Using $\bar{\varphi}_{\tau-t}\in D(\tau-t, \theta_{-t}\omega)$ , (2.1) and (2.22), we find

$\begin{equation} \begin{split} \limsup\limits_{t\rightarrow +\infty}4e^{-\gamma t}\Big(\|\bar{\varphi}_{\tau-t}\|^{2}_{E}+\|a\|^{2}|\omega(-t)-\omega(-\tau)|^{2}+\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau-t,i})\Big) = 0, \end{split} \end{equation}$

(2.23)

which along with (2.21) implies that there exists $T = T(\tau, \omega, D) > 0$ such that for all $t\geq T$ ,

(2.24)

where $c$ is a positive constant independent of $\tau$ , $\omega$ and $D$ . Note that $K_{0}$ given by (2.18) is closed measurable random set in $E$ . Given $\tau\in\mathbb{R}$ , $\omega\in\Omega$ , and $D\in\mathcal{D}$ , it follows from (2.24) that for all $t\geq T$ ,

$\begin{equation} \begin{split} \Phi_{0}(t,\tau-t,\theta_{-t}\omega,D(\tau-t,\theta_{-t}\omega))\subseteq K_{0}(\tau,\omega), \end{split} \end{equation}$

(2.25)

which implies that $K_{0}$ pullback attracts all elements in $\mathcal{D}$ . By (2.1) and (2.9), one can easily check that $K_{0}$ is tempered, which along with (2.25) completes the proof.

Next, we will get uniform estimates on the tails of solutions of system (2.10).

Lemma 2.3. Suppose that (2.3)–(2.9) hold. Then for every $\tau\in\mathbb{R}$ , $\omega\in\Omega$ , $D = \{D(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$ and $\varepsilon > 0$ , there exist $T = T(\tau, \omega, D, \varepsilon) > 0$ and $N = N(\tau, \omega, \varepsilon) > 0$ such that for all $t\geq T$ , the solution $\bar{\varphi}$ of system (2.10) satisfies

$\begin{equation} \begin{split}\nonumber \sum\limits_{|i|\geq N}|\bar{\varphi}_{i}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t,i})|_{E}^{2}\leq\varepsilon,\\ \end{split} \end{equation}$

where $\bar{\varphi}_{\tau-t}\in D(\tau-t, \theta_{-t}\omega)$ and $|\bar{\varphi}_{i}|_{E}^{2} = (1-\nu\beta)|Bu|_{i}^{2}+\lambda |u_{i}|^{2}+|\bar{v}_{i}|^{2}$ .

Proof. Let $\eta$ be a smooth function defined on $\mathbb{R}^{+}$ such that $0\leq\eta(s)\leq1$ for all $s\in\mathbb{R}^{+}$ , and

$\begin{equation} \nonumber \eta(s) = \left\{\begin{array}{l} 0,\; \; 0\leq s\leq1;\\ 1,\; \; s\geq2. \end{array}\right. \end{equation}$

Then there exists a constant $C_{0}$ such that $|\eta'(s)|\leq C_{0}$ for $s\in\mathbb{R}^{+}$ . Let $k$ be a fixed positive integer which will be specified later, and set $x = (x_{i})_{i\in\mathbb{Z}}$ , $y = (y_{i})_{i\in\mathbb{Z}}$ with $x_{i} = \eta(\frac{|i|}{k})u_{i}$ , $y_{i} = \eta(\frac{|i|}{k})v_{i}$ . Note $\psi = (x, y)^{T} = ((x_{i}), (y_{i}))^{T}_{i\in\mathbb{Z}}$ . Taking the inner product of system (2.11) with $\psi$ , we have

$\begin{equation} \begin{split} (\dot{\varphi},\psi)_{E}+(L(\varphi),\psi)_{E} = (H(\varphi),\psi)_{E}+(G,\psi)_{E}. \end{split} \end{equation}$

(2.26)

For the first term of (2.26), we have

$\begin{equation} \begin{split} (\dot{\varphi},\psi)_{E}& = (1-\nu\beta)\sum\limits_{i\in\mathbb{Z}}(B\dot{u})_{i}(Bx)_{i}+\lambda\sum\limits_{i\in\mathbb{Z}}\dot{u}_{i}x_{i} +\sum\limits_{i\in\mathbb{Z}}\dot{v}_{i}y_{i}\\ & = \frac{1}{2}\frac{d}{dt}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{i}|^{2}_{E} +(1-\nu\beta)\sum\limits_{i\in\mathbb{Z}}(B\dot{u})_{i}\Big((Bx)_{i}-\eta(\frac{|i|}{k})(Bu)_{i}\Big)\\ &\geq\frac{1}{2}\frac{d}{dt}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{i}|^{2}_{E} -\frac{(1-\nu\beta)C_{0}}{k}\sum\limits_{i\in\mathbb{Z}}|(B(v-\beta u+a\omega(t))_{i}||u_{i+1}|\\ &\geq\frac{1}{2}\frac{d}{dt}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{i}|^{2}_{E} -\frac{c}{k}\|\varphi\|^{2}_{E}-\frac{c}{k}|\omega(t)|^{2}\|a\|^{2}, \end{split} \end{equation}$

(2.27)

where $|\varphi_{i}|_{E}^{2} = (1-\nu\beta)|Bu|_{i}^{2}+\lambda |u_{i}|^{2}+|v_{i}|^{2}$ . As to the second term on the left-hand side of (2.26), we get

$\begin{equation} \begin{split}\nonumber (L(\varphi),\psi)_{E} = &\beta(1-\nu\beta)(Au,x)+(1-\nu\beta)((Au,y)-(Av,x))+\nu(Av,y)+\lambda\beta(u,x)\\ &+\beta^{2}(u,y)-\beta(v,y)+(h(v-\beta u+a\omega(t)),y). \end{split} \end{equation}$

It is easy to check that

$\begin{equation} \begin{split}\nonumber (Au,x) = \sum\limits_{i\in\mathbb{Z}}(Bu)_{i}\Big(\eta(\frac{|i|}{k})(Bu)_{i}+(Bx)_{i}-\eta(\frac{|i|}{k})(Bu)_{i}\Big) \geq\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|Bu|_{i}^{2}-\frac{2C_{0}}{k}\|u\|^{2}, \end{split} \end{equation}$

$\begin{equation} \begin{split}\nonumber (Av,y) = \sum\limits_{i\in\mathbb{Z}}(Bv)_{i}\Big(\eta(\frac{|i|}{k})(Bv)_{i}+(By)_{i}-\eta(\frac{|i|}{k})(Bv)_{i}\Big) \geq\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|Bv|_{i}^{2}-\frac{2C_{0}}{k}\|v\|^{2}, \end{split} \end{equation}$

and

$\begin{equation} \begin{split}\nonumber (Au,y)-(Av,x)\geq-\frac{C_{0}}{k}\sum\limits_{i\in\mathbb{Z}}|(Bu)_{i}||v_{i+1}|-\frac{C_{0}}{k}\sum\limits_{i\in\mathbb{Z}}|(Bv)_{i}||u_{i+1}| \geq-\frac{2 C_{0}}{k}(\|u\|^{2}+\|v\|^{2}). \end{split} \end{equation}$

By the mean value theorem and (2.5), there exists $\xi_{i}\in(0, 1)$ such that

$\begin{equation} \begin{split}\nonumber &\beta^{2}(u,y)+(h(v-\beta u+a\omega(t)),y)\\ & = \beta^{2}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})u_{i}v_{i} +\sum\limits_{i\in\mathbb{Z}}h'_{i}(\xi_{i}(v_{i}-\beta u_{i}+a_{i}\omega(t)))(v_{i}-\beta u_{i}+a_{i}\omega(t))\eta(\frac{|i|}{k})v_{i}\\ &\geq \beta(\beta-h_{2})\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|u_{i}v_{i}|+h_{1}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}|^{2} -h_{2}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}a_{i}\omega(t)|. \end{split} \end{equation}$

Then

$\begin{equation} \begin{split}\nonumber &(L(\varphi),\varphi)_{E}-\sigma\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{i}|^{2}_{E} -\frac{h_{1}}{2}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}|^{2}\\ &\geq (\beta-\sigma)\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\Big((1-\nu\beta)|Bu|_{i}^{2}+\lambda u_{i}^{2}\Big)+(\frac{h_{1}}{2}-\beta-\sigma)\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}|^{2}\\ &\quad-\frac{\beta h_{2}}{\sqrt{\lambda}}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}|\Big((1-\nu\beta)(Bu)_{i}^{2}+\lambda| u_{i}|^{2}\Big)^{\frac{1}{2}}-h_{2}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}a_{i}\omega(t)|-\frac{c}{k}\|\varphi\|^{2}_{E}, \end{split} \end{equation}$

which along with (2.6) and (2.7) implies that

$\begin{equation} \begin{split} (L(\varphi),\varphi)_{E}&\geq\sigma\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{i}|^{2}_{E}+ \frac{h_{1}}{2}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}|^{2}-\frac{c}{k}\|\varphi\|^{2}_{E}-h_{2}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}a_{i}\omega(t)|\\ &\geq\sigma\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{i}|^{2}_{E}+ \frac{h_{1}}{6}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}|^{2}-\frac{c}{k}\|\varphi\|^{2}_{E} -c\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|a_{i}|^{2}|\omega(t)|^{2}. \end{split} \end{equation}$

(2.28)

As to the first term on the right-hand side of (2.26), by (2.3) and (2.4)we get

$\begin{equation} \begin{split} (H(\varphi),\psi)_{E}& = -\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})f_{i}(u_{i})(\dot{u_{i}}+\beta u_{i}-a_{i}\omega(t))+\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})g_{i}(t)v_{i}\\ &\leq-\frac{d}{dt}\big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})F_{i}(u_{i})\big) -\frac{\alpha_{2}\beta}{p+1}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})F_{i}(u_{i})\\ &\quad+c\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\omega(t)|^{p+1}|a_{i}|^{p+1} +\frac{\sigma\lambda}{4}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|u_{i}|^{2}\\ &\quad+c\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|a_{i}|^{2}|\omega(t)|^{2} +\frac{\sigma}{6} \sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}|^{2} +c\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|g_{i}(t)|^{2}. \end{split} \end{equation}$

(2.29)

For the last term of (2.26), we have

$\begin{equation} \begin{split} (G,\psi)_{E}& = \omega(t)(a,x)_{\lambda}+\beta \omega(t)(a,y)-\nu\omega(t)(Aa,y)\\ & = \omega(t)(1-\nu\beta)(Ba,Bx)-\nu\omega(t)(Ba,By)+\lambda\omega(t)(a,x)+\beta \omega(t)(a,y), \end{split} \end{equation}$

(2.30)

As to the first two terms on the right-hand side of (2.30), we get

$\begin{equation} \begin{split} \omega(t)(1-\nu\beta)(Ba,Bx)& = \omega(t)(1-\nu\beta)\sum\limits_{i\in\mathbb{Z}}(a_{i+1}-a_{i})\Big(\eta(\frac{|i+1|}{k})u_{i+1}-\eta(\frac{|i|}{k})u_{i}\Big)\\ &\leq\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i+1|}{k})u_{i+1}^{2}\Big)^{\frac{1}{2}}\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i+1|}{k})\big(\omega(t)(1-\nu\beta)(a_{i+1}-a_{i})\big)^{2}\Big)^{\frac{1}{2}}\\ &\quad+\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})u_{i}^{2}\Big)^{\frac{1}{2}}\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\big(\omega(t)(1-\nu\beta)(a_{i+1}-a_{i})\big)^{2}\Big)^{\frac{1}{2}}\\ &\leq \frac{\sigma\lambda}{8}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})u_{i}^{2}+c|\omega(t)|^{2}\sum\limits_{|i|\geq k}a_{i}^{2}, \end{split} \end{equation}$

(2.31)

and

$\begin{equation} \begin{split} -\nu\omega(t)(Ba,By)& = -\nu\omega(t)\sum\limits_{i\in\mathbb{Z}}(a_{i+1}-a_{i})\Big(\eta(\frac{|i+1|}{k})v_{i+1}-\eta(\frac{|i|}{k})v_{i}\Big)\\ &\leq \frac{\sigma}{6}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})v_{i}^{2}+c|\omega(t)|^{2}\sum\limits_{|i|\geq k}a_{i}^{2}. \end{split} \end{equation}$

(2.32)

The last two terms of (2.30) is bounded by

$\begin{equation} \begin{split} \lambda\omega(t)(a,x)+\beta \omega(t)(a,y)\leq \frac{\sigma\lambda}{8}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})u_{i}^{2}+\frac{\sigma+h_{1}}{6}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})v_{i}^{2} +c|\omega(t)|^{2}\sum\limits_{|i|\geq k}a_{i}^{2}. \end{split} \end{equation}$

(2.33)

It follows from (2.26)–(2.33) that

$\begin{equation} \begin{split} &\frac{d}{dt}\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\big(|\varphi_{i}|_{E}^{2} +2 F_{i}(u_{i})\big)\Big) +\gamma\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\big(|\varphi_{i}|_{E}^{2} +2 F_{i}(u_{i})\big)\Big)+\gamma\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi|_{E}^{2}\\ &\leq\frac{c}{k}\|\varphi\|^{2}_{E}+\frac{c}{k}|\omega(t)|^{2} +c\sum\limits_{|i|\geq k}|a_{i}|^{p+1}|\omega(t)|^{p+1}+c\sum\limits_{|i|\geq k}|g_{i}(t)|^{2}+c\sum\limits_{|i|\geq k}|a_{i}|^{2}|\omega(t)|^{2}, \end{split} \end{equation}$

(2.34)

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$ . Multiplying (2.34) by $e^{\gamma t}$ , replacing $\omega$ by $\theta_{-\tau}\omega$ and integrating on $(\tau-t, \tau)$ with $t\in\mathbb{R}^{+}$ , we get for every $\omega\in\Omega$

$\begin{equation} \begin{split} &\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\Big(|\varphi_{i}(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t,i})|^{2}_{E} +2F_{i}(u_{i}(\tau,\tau-t,\theta_{-\tau}\omega,u_{\tau-t,i}))\Big)\\ &\leq e^{-\gamma t}\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\big(|\varphi_{\tau-t,i}|^{2}_{E}+2 F_{i}(u_{\tau-t,i})\big)\Big) +\frac{c}{k}\int^{\tau}_{\tau-t}e^{\gamma(s-\tau)}\|\varphi(s,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t})\|^{2}_{E}ds\\ &\quad+\frac{c}{k}\int^{0}_{-\infty}e^{\gamma s}|\omega(s)-\omega(-\tau)|^{2}ds +c\sum\limits_{|i|\geq k}|a_{i}|^{p+1}\int_{-\infty}^{0}e^{\gamma s}|\omega(s)-\omega(-\tau)|^{p+1}ds\\ &\quad+c\sum\limits_{|i|\geq k}|a_{i}|^{2}\int_{-\infty}^{0}e^{\gamma s}|\omega(s)-\omega(-\tau)|^{2}ds+c\int_{-\infty}^{0}e^{\gamma s}\sum\limits_{|i|\geq k}|g_{i}(s+\tau)|^{2}ds. \end{split} \end{equation}$

(2.35)

For any $s\geq \tau-t$ ,

which along with (2.35) implies that

$\begin{equation} \begin{split} &\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\Big(|\bar{\varphi}_{i}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t,i})|^{2}_{E} +2F_{i}(u_{i}(\tau,\tau-t,\theta_{-\tau}\omega,u_{\tau-t,i}))\Big)\\ &\leq 4e^{-\gamma t}\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\big(|\bar{\varphi}_{\tau-t,i}|^{2}_{E}+|a_{i}|^{2}|\omega(-t)-\omega(-\tau)|^{2}+ F_{i}(u_{\tau-t,i})\big)\Big)\\ &\quad+\frac{c}{k}\int^{\tau}_{\tau-t}e^{\gamma(s-\tau)}\|\bar{\varphi}(s,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t})\|^{2}_{E}ds+\frac{c}{k}\int^{0}_{-\infty}e^{\gamma s}|\omega(s)-\omega(-\tau)|^{2}ds\\ &\quad +c\sum\limits_{|i|\geq k}|a_{i}|^{p+1}\int_{-\infty}^{0}e^{\gamma s}|\omega(s)-\omega(-\tau)|^{p+1}ds+c\sum\limits_{|i|\geq k}|a_{i}|^{2}\int_{-\infty}^{0}e^{\gamma s}|\omega(s)-\omega(-\tau)|^{2}ds\\ &\quad+c\int_{-\infty}^{0}e^{\gamma s}\sum\limits_{|i|\geq k}|g_{i}(s+\tau)|^{2}ds+2\sum\limits_{|i|\geq k}|a_{i}|^{2}|\omega(-\tau)|^{2}. \end{split} \end{equation}$

(2.36)

By (2.1) and (2.8), the last four integrals in (2.36) are well defined. By (2.3) and (2.4), we obtain

$\begin{equation} \begin{split}\nonumber \sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k}) F_{i}(u_{i,\tau-t})\leq \frac{1}{\alpha_{2}}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k}) f_{i}(u_{\tau-t,i})u_{\tau-t,i} \leq\frac{1}{\alpha_{2}}\max\limits_{-\|u_{\tau-t}\|\leq s\leq\|u_{\tau-t}\|}|f'_{i}(s)|\|u_{\tau-t}\|^{2}, \end{split} \end{equation}$

which along with $\bar{\varphi}_{\tau-t}\in D(\tau-t, \theta_{-t}\omega)$ and (2.1) implies that

$\begin{equation} \begin{split} \label{127}\nonumber \limsup\limits_{t\rightarrow +\infty}4e^{-\gamma t}\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\big(|\bar{\varphi}_{\tau-t,i}|^{2}_{E}+|a_{i}|^{2}|\omega(-t)-\omega(-\tau)|^{2}+ F_{i}(u_{\tau-t,i})\big)\Big) = 0. \end{split} \end{equation}$

Then there exists $T_{1} = T_{1}(\tau, \omega, D, \varepsilon) > 0$ such that for all $t\geq T_{1}$ ,

$\begin{equation} \begin{split} 4e^{-\gamma t}\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\big(|\bar{\varphi}_{\tau-t,i}|^{2}_{E}+|a_{i}|^{2}|\omega(-t)-\omega(-\tau)|^{2}+ F_{i}(u_{\tau-t,i})\big)\Big)\leq\frac{\varepsilon}{4}. \end{split} \end{equation}$

(2.37)

By (2.1) and (2.24), there exist $T_{2} = T_{2}(\tau, \omega, D, \varepsilon) > T_{1}$ and $N_{1} = N_{1}(\tau, \omega, \varepsilon) > 0$ such that for all $t\geq T_{2}$ and $k\geq N_{1}$

$\begin{equation} \begin{split} \frac{c}{k}\int^{\tau}_{\tau-t}e^{\gamma(s-\tau)}\|\bar{\varphi}(s,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t})\|^{2}_{E}ds+\frac{c}{k}\int^{0}_{-\infty}e^{\gamma s}|\omega(s)-\omega(-\tau)|^{2}ds\leq\frac{\varepsilon}{4}. \end{split} \end{equation}$

(2.38)

By (2.8), there exists $N_{2} = N_{2}(\tau, \omega, \varepsilon) > N_{1}$ such that for all $k\geq N_{2}$ ,

$\begin{equation} \begin{split} 2\sum\limits_{|i|\geq k}|a_{i}|^{2}|\omega(-\tau)|^{2}+c\int_{-\infty}^{0}e^{\gamma s}\sum\limits_{|i|\geq k}|g_{i}(s+\tau)|^{2}ds\leq\frac{\varepsilon}{4}. \end{split} \end{equation}$

(2.39)

By (2.1) again, we find that there exists $N_{3} = N_{3}(\tau, \omega, \varepsilon) > N_{2}$ such that for all $k\geq N_{3}$ ,

$\begin{equation} \begin{split} c\sum\limits_{|i|\geq k}|a_{i}|^{p+1}\int_{-\infty}^{0}e^{\gamma s}|\omega(s)-\omega(-\tau)|^{p+1}ds +c\sum\limits_{|i|\geq k}|a_{i}|^{2}\int_{-\infty}^{0}e^{\gamma s}|\omega(s)-\omega(-\tau)|^{2}ds \leq\frac{\varepsilon}{4}. \end{split} \end{equation}$

(2.40)

Then it follows from (2.36)–(2.40) that for all $t\geq T_{2}$ and $k\geq N_{3}$

$\begin{equation} \begin{split}\nonumber \sum\limits_{|i|\geq 2k}|\bar{\varphi}_{i}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t,i})|^{2}_{E} \leq\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\bar{\varphi}_{i}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t,i})|^{2}_{E} \leq\varepsilon. \end{split} \end{equation}$

This concludes the proof.

As a consequence of Lemma 2.2 and Lemma 2.3, we get the existence of $\mathcal{D}$ -pullback attractors for $\Phi_{0}$ immediately.

Theorem 2.1. Suppose that (2.3)–(2.9) hold. Then the continuous cocycle $\Phi_{0}$ associated with system (2.10) has a unique $\mathcal{D}$ -pullback attractors $\mathcal{A}_{0} = \{\mathcal{A}_{0}(\tau, \omega):\tau\in\mathbb{R}$ , $\omega\in\Omega\}\in \mathcal{D}$ in $E$ .

3. Wong-Zakai approximation of second order lattice system

In this section, we will approximate the solutions of system (1.1) by the pathwise Wong-Zakai approximated system (1.2). Given $\delta\neq0$ , define a random variable $\mathcal{G}_{\delta}$ by

$\begin{equation} \begin{split} \mathcal{G}_{\delta}(\omega) = \frac{\omega(\delta)}{\delta},\; \; \text{for all}\; \omega\in\Omega. \end{split} \end{equation}$

(3.1)

From (3.1) we find

$\begin{equation} \begin{split} \mathcal{G}_{\delta}(\theta_{t}\omega) = \frac{\omega(t+\delta)-\omega(t)}{\delta}\; \; \text{and}\; \int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds = \int_{t}^{t+\delta}\frac{\omega(s)}{\delta}ds+\int_{\delta}^{0}\frac{\omega(s)}{\delta}ds. \end{split} \end{equation}$

(3.2)

By (3.2) and the continuity of $\omega$ we get for all $t\in\mathbb{R}$ ,

$\begin{equation} \begin{split} \lim\limits_{\delta\rightarrow0}\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds = \omega(t). \end{split} \end{equation}$

(3.3)

Note that this convergence is uniform on a finite interval as stated below.

Lemma 3.1. (^[17]). Let $\tau\in\mathbb{R}$ , $\omega\in\Omega$ and $T > 0$ . Then for every $\varepsilon > 0$ , there exists $\delta_{0} = \delta_{0}(\varepsilon, \tau, \omega, T) > 0$ such that for all $0 < |\delta| < \delta_{0}$ and $t\in[\tau, \tau+T]$ ,

$\Big|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)\Big| < \varepsilon.$

By Lemma 3.1, we find that there exist $c = c(\tau, \omega, T) > 0$ and $\tilde{\delta}_{0} = \tilde{\delta}_{0}(\tau, \omega, T) > 0$ such that for all $0 < |\delta| < \tilde{\delta}_{0}$ and $t\in[\tau, \tau+T]$ ,

$\begin{equation} \begin{split} \Big|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds\Big|\leq c. \end{split} \end{equation}$

(3.4)

By (3.3) we find that $\mathcal{G}_{\delta}(\theta_{t}\omega)$ is an approximation of the white noise in a sense. This leads us to consider system (1.2) as an approximation of system (1.1).

Let $\bar{v}^{\delta} = \dot{u}^{\delta}+\beta u^{\delta}$ and $\bar{\varphi}_{\delta} = (u^{\delta}, \bar{v}^{\delta})$ , the system (1.2) can be rewritten as

$\begin{eqnarray} \dot{\bar{\varphi}}_{\delta}+L_{\delta,1}(\bar{\varphi}_{\delta}) = H_{\delta,1}(\bar{\varphi}_{\delta})+G_{\delta,1}(\omega), \end{eqnarray}$

(3.5)

with initial conditions

$\bar{\varphi}_{\delta,\tau} = (u^{\delta}_{\tau},\bar{v}^{\delta}_{\tau})^{T} = (u_{\tau}^{\delta},u_{\tau}^{\delta,1}+\beta u_{\tau}^{\delta})^{T},$

where

$L_{\delta,1}(\bar{\varphi}) = \left( \begin{array}{ccc} \beta u^{\delta}-\bar{v}^{\delta} \\ (1-\nu\beta)Au^{\delta}+\nu A\bar{v}^{\delta}+\lambda u^{\delta} +\beta^{2} u^{\delta}-\beta\bar{v}^{\delta} \end{array} \right) + \left( \begin{array}{ccc} 0 \\ h(\bar{v}^{\delta}-\beta u^{\delta}) \end{array} \right) ,$

$H_{\delta,1}(\bar{\varphi_{\delta}}) = \left( \begin{array}{ccc} 0 \\ -f(u^{\delta})+g(t) \end{array} \right),\; \; \; \; G_{\delta,1}(\omega) = \left( \begin{array}{ccc} 0 \\ a\mathcal{G}_{\delta}(\theta_{t}\omega) \end{array} \right) .$

Denote

$v^{\delta}(t) = \bar{v}^{\delta}(t)-a\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds \; \; \text{and}\; \; \varphi_{\delta} = (u^{\delta},v^{\delta})^{T}.$

By (3.5) we have

$\begin{eqnarray} \dot{\varphi}_{\delta}+L_{\delta}(\varphi_{\delta}) = H_{\delta}(\varphi_{\delta})+G_{\delta}(\omega), \end{eqnarray}$

(3.6)

with initial conditions

$\varphi_{\delta,\tau} = (u^{\delta}_{\tau},v^{\delta}_{\tau})^{T} = (u^{\delta}_{\tau},u_{\tau}^{\delta,1}+\beta u^{\delta}_{\tau}-a\int_{0}^{\tau}\mathcal{G}_{\delta}(\theta_{s}\omega)ds)^{T},$

where

$L_{\delta}(\varphi_{\delta}) = \left( \begin{array}{ccc} \beta u^{\delta}-v^{\delta} \\ (1-\nu\beta) Au^{\delta}+\nu Av^{\delta}+\lambda u^{\delta}+\beta^{2} u^{\delta}-\beta v^{\delta} \end{array} \right) + \left( \begin{array}{ccc} 0 \\ h(v^{\delta}-\beta u^{\delta}+a\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds) \end{array} \right) ,$

$H_{\delta}(\varphi_{\delta}) = \left( \begin{array}{ccc} 0 \\ -f(u^{\delta})+g(t) \end{array} \right),\; \; \; \; G_{\delta}(\omega) = \left( \begin{array}{ccc} a\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds\\ \beta a\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\nu Aa\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds \end{array} \right) .$

Note that system (3.6) is a deterministic functional equation and the nonlinearity in (3.6) is locally Lipschitz continuous from $E$ to $E$ . Therefore, by the standard theory of functional differential equations, system (3.6) is well-posed. Thus, we can define a continuous cocycle $\Phi_{\delta}:\mathbb{R}^{+}\times\mathbb{R}\times \Omega\times E\rightarrow E$ associated with system (3.5), where for $\tau\in\mathbb{R}$ , $t\in\mathbb{R}^{+}$ and $\omega\in\Omega$

$\begin{equation} \begin{split}\nonumber \Phi_{\delta}(t,\tau,\omega,\bar{\varphi}_{\delta,\tau})& = \bar{\varphi}_{\delta}(t+\tau,\tau,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau})\\ & = (u^{\delta}(t+\tau,\tau,\theta_{-\tau}\omega,u^{\delta}_{\tau}),\bar{v}^{\delta}(t+\tau,\tau,\theta_{-\tau}\omega,\bar{v}^{\delta}_{\tau}))^{T}\\ & = \Big(u^{\delta}(t+\tau,\tau,\theta_{-\tau}\omega,u^{\delta}_{\tau}),v^{\delta}(t+\tau,\tau,\theta_{-\tau}\omega,v^{\delta}_{\tau})+a\int_{-\tau}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds\Big)^{T}\\ & = \varphi_{\delta}(t+\tau,\tau,\theta_{-\tau}\omega,\varphi_{\delta,\tau})+\Big(0,a\int_{-\tau}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds\Big)^{T}, \end{split} \end{equation}$

where $v_{\tau}^{\delta} = \bar{v}_{\tau}^{\delta}-a\int_{-\tau}^{0}\mathcal{G}_{\delta}(\theta_{s}\omega)ds$ .

For later purpose, we now show the estimates on the solutions of system (3.6) on a finite time interval.

Lemma 3.2. Suppose that (2.3)–(2.8) hold. Then for every $\tau\in\mathbb{R}$ , $\omega\in\Omega$ , and $T > 0$ , there exist $\delta_{0} = \delta_{0}(\tau, \omega, T) > 0$ and $c = c(\tau, \omega, T) > 0$ such that for all $0 < |\delta| < \delta_{0}$ and $t\in[\tau, \tau+T]$ , the solution $\varphi_{\delta}$ of system (3.6) satisfies

$\begin{equation} \begin{split}\nonumber &\|\varphi_{\delta}(t,\tau,\omega,\varphi_{\delta,\tau})\|^{2}_{E} +\int_{\tau}^{t}\|\varphi_{\delta}(s,\tau,\omega,\varphi_{\delta,\tau})\|^{2}_{E}ds\leq c\Big(\|\varphi_{\delta,\tau}\|^{2}_{E}+2\sum\limits_{i\in\mathbb{Z}}F_{i}(u^{\delta}_{\tau,i})\Big)\\ &\quad+ c\int^{t}_{\tau}\Big(\|g(s)\|^{2}+|\int_{0}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}+|\int_{0}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{p+1}|\Big)ds. \end{split} \end{equation}$

Proof. Taking the inner product $(\cdot, \cdot)_{E}$ on both side of the system (3.6) with $\varphi_{\delta}$ , it follows that

$\begin{equation} \begin{split} &\frac{1}{2}\frac{d}{dt}\|\varphi_{\delta}\|^{2}_{E}+(L_{\delta}(\varphi_{\delta}),\varphi_{\delta})_{E} = (H_{\delta}(\varphi_{\delta}),\varphi_{\delta})_{E}+(G_{\delta}(\omega),\varphi_{\delta})_{E}. \end{split} \end{equation}$

(3.7)

By the similar calculations in (2.13)–(2.15), we get

$\begin{equation} \begin{split} (L_{\delta}(\varphi_{\delta}),\varphi_{\delta})_{E}\geq\sigma\|\varphi_{\delta}\|^{2}_{E}+\frac{h_{1}}{2}\|v^{\delta}\|^{2} -\frac{\sigma+h_{1}}{6}\|v^{\delta}\|^{2}-c|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2}\|a\|^{2}, \end{split} \end{equation}$

(3.8)

$\begin{equation} \begin{split} (H_{\delta}(\varphi_{\delta}),\varphi_{\delta})_{E} &\leq-\frac{d}{dt}(\sum\limits_{i\in\mathbb{Z}}F_{i}(u^{\delta}_{i}))-\frac{\alpha_{2}\beta}{p+1}\sum\limits_{i\in\mathbb{Z}}F_{i}(u^{\delta}_{i}) +c|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{p+1}\|a\|^{p+1}\\ &\quad+\frac{\sigma\lambda}{4}\|u^{\delta}\|^{2}+c\|a\|^{2}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2}+c\|g(t)\|^{2}+\frac{\sigma+h_{1}}{6}\|v^{\delta}\|^{2}, \end{split} \end{equation}$

(3.9)

and

$\begin{equation} \begin{split} (G_{\delta}(\omega),\varphi_{\delta})_{E}\leq \frac{\sigma}{4}\|u^{\delta}\|^{2}_{\lambda} +c\|a\|^{2}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2}+\frac{\sigma+h_{1}}{6}\|v^{\delta}\|^{2}. \end{split} \end{equation}$

(3.10)

It follows from (3.7)–(3.10) that

$\begin{equation} \begin{split} &\frac{d}{dt}\Big(\|\varphi_{\delta}\|_{E}^{2}+2\sum\limits_{i\in\mathbb{Z}}F_{i}(u_{i}^{\delta})\Big) +\gamma\Big(\|\varphi_{\delta}\|_{E}^{2}+2\sum\limits_{i\in\mathbb{Z}}F_{i}(u_{i}^{\delta})\Big) +\gamma\|\varphi_{\delta}\|_{E}^{2}\\ &\leq c\Big(\|g(t)\|^{2}+|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2}+|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{p+1}\Big), \end{split} \end{equation}$

(3.11)

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$ . Multiplying (3.11) by $e^{\gamma t}$ and integrating on $(\tau, t)$ with $t\geq\tau$ , we get for every $\omega\in\Omega$

$\begin{equation} \begin{split}\nonumber &\|\varphi_{\delta}(t,\tau,\omega,\varphi_{\delta,\tau})\|_{E}^{2} +\gamma\int^{t}_{\tau}e^{\gamma(s-t)}\|\varphi_{\delta}(s,\tau,\omega,\varphi_{\delta,\tau})\|_{E}^{2}ds\leq e^{\gamma(\tau-t)}\Big(\|\varphi_{\delta,\tau}\|_{E}^{2}+2\sum\limits_{i\in\mathbb{Z}}F_{i}(u_{\tau,i}^{\delta})\Big)\\ &\quad+c\int_{\tau}^{t}e^{\gamma(s-t)}\Big(\|g(s)\|^{2}+|\int_{0}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}+|\int_{0}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{p+1}\Big)ds, \end{split} \end{equation}$

which implies the desired result.

In what follows, we derive uniform estimates on the solutions of system (3.5) when $t$ is sufficiently large.

Lemma 3.3. Suppose that (2.3)–(2.8) hold. Then for every $\delta\neq0$ , $\tau\in\mathbb{R}$ , $\omega\in\Omega$ , and $D = \{D(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$ , there exists $T = T(\tau, \omega, D, \delta) > 0$ such that for all $t\geq T$ , the solution $\bar{\varphi}_{\delta}$ of system (3.5) satisfies

$\begin{equation} \begin{split}\nonumber &\|\bar{\varphi}_{\delta}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t})\|^{2}_{E} +\gamma\int_{\tau-t}^{\tau}e^{\gamma(s-\tau)}\|\bar{\varphi}_{\delta}(s,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t})\|^{2}_{E}ds\leq R_{\delta}(\tau,\omega), \end{split} \end{equation}$

where $\bar{\varphi}_{\delta, \tau-t}\in D(\tau-t, \theta_{-t}\omega)$ and $R_{\delta}(\tau, \omega)$ is given by

$\begin{equation} \begin{split} R_{\delta}(\tau,\omega) = &c\int^{0}_{-\infty}e^{\gamma s}\Big(\|g(s+\tau)\|^{2}+|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2} +|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{p+1}\Big)ds\\ &+c+c|\int_{-\tau}^{0}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}, \end{split} \end{equation}$

(3.12)

where $c$ is a positive constant independent of $\tau$ , $\omega$ and $\delta$ .

Proof. Multiplying (3.11) by $e^{\gamma t}$ , replacing $\omega$ by $\theta_{-\tau}\omega$ and integrating on $(\tau-t, \tau)$ with $t\in\mathbb{R}^{+}$ , we get for every $\omega\in\Omega$

$\begin{equation} \begin{split} &\|\varphi_{\delta}(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\delta,\tau-t})\|^{2}_{E}+2\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{i}^{\delta}(\tau,\tau-t,\theta_{-\tau}\omega,u^{\delta}_{\tau-t,i}))\\ &\quad+\gamma\int_{\tau-t}^{\tau}e^{\gamma(s-\tau)}\|\varphi_{\delta}(s,\tau-t,\theta_{-\tau}\omega,\varphi_{\delta,\tau-t})\|^{2}_{E}ds\\ &\leq e^{-\gamma t}\Big(\|\varphi_{\delta,\tau-t}\|^{2}_{E}+2\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau-t,i}^{\delta})\Big)\\ &\quad +c\int^{0}_{-\infty}e^{\gamma s}\Big(\|g(s+\tau)\|^{2}+|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}+|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{p+1}\Big)ds. \end{split} \end{equation}$

(3.13)

By (2.1), (2.8) and (3.2), the last integral on the right-hand side of (3.13) is well defined. For any $s\geq \tau-t$ ,

$\begin{equation} \begin{split}\nonumber &\bar{\varphi}_{\delta}(s,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t}) = \varphi_{\delta}(s,\tau-t,\theta_{-\tau}\omega,\varphi_{\delta,\tau-t})+\Big(0,a\int_{0}^{s}\mathcal{G}_{\delta}(\theta_{l-\tau}\omega)dl\Big)^{T}, \end{split} \end{equation}$

which along with (3.13) shows that

$\begin{equation} \begin{split} &\|\bar{\varphi}_{\delta}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t})\|^{2}_{E} +\gamma\int_{\tau-t}^{\tau}e^{\gamma(s-\tau)}\|\bar{\varphi}_{\delta}(s,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\tau-t})\|^{2}_{E}ds\\ &\leq 4e^{-\gamma t}\Big(\|\bar{\varphi}_{\delta,\tau-t}\|^{2}_{E}+\|a\|^{2}|\int_{-\tau}^{-t}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}+\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau-t,i})\Big)+c|\int_{-\tau}^{0}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}\\ &\quad+c\int^{0}_{-\infty}e^{\gamma s}\Big(\|g(s+\tau)\|^{2}+|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2} +| \int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{p+1}\Big)ds, \end{split} \end{equation}$

(3.14)

Note that (2.3) and (2.4) implies that

$\begin{equation} \begin{split}\nonumber \sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau-t,i}^{\delta})\leq \frac{1}{\alpha_{2}}\sum\limits_{i\in\mathbb{Z}} f_{i}(u_{\tau-t,i}^{\delta})u_{\tau-t,i}^{\delta} \leq\frac{1}{\alpha_{2}}\max\limits_{-\|u_{\tau-t}^{\delta}\|\leq s\leq\|u_{\tau-t}^{\delta}\|}|f'_{i}(s)|\|u_{\tau-t}^{\delta}\|^{2}, \end{split} \end{equation}$

which along with $\bar{\varphi}_{\delta, \tau-t}\in D(\tau-t, \theta_{-t}\omega)$ , (2.1) and (3.2) implies that

$\begin{equation} \begin{split} \limsup\limits_{t\rightarrow +\infty}4e^{-\gamma t}\Big(\|\bar{\varphi}_{\delta,\tau-t}\|^{2}_{E}+\|a\|^{2}|\int_{-\tau}^{-t}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}+\sum\limits_{i\in\mathbb{Z}} F_{i}(u_{\tau-t,i})\Big) = 0. \end{split} \end{equation}$

(3.15)

Then (3.14) and (3.15) can imply the desired estimates.

Next, we show that system (3.5) has a $\mathcal{D}$ -pullback absorbing set.

Lemma 3.4. Suppose that (2.3)–(2.9) hold. Then the continuous cocycle $\Phi_{\delta}$ associated with system (3.5) has a closed measurable $\mathcal{D}$ -pullback absorbing set $K_{\delta} = \{K_{\delta}(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$ , where for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$

$\begin{equation} \begin{split} K_{\delta}(\tau,\omega) = \{\bar{\varphi}_{\delta}\in E:\|\bar{\varphi}_{\delta}\|^{2}_{E}\leq R_{\delta}(\tau,\omega)\}, \end{split} \end{equation}$

(3.16)

where $R_{\delta}(\tau, \omega)$ is given by (3.12).In addition, we have for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$

$\begin{equation} \begin{split} \lim\limits_{\delta\rightarrow0}R_{\delta}(\tau,\omega) = R_{0}(\tau,\omega), \end{split} \end{equation}$

(3.17)

where $R_{0}(\tau, \omega)$ is defined in (2.19).

Proof. Note $K_{\delta}$ given by (3.16) is closed measurable random set in $E$ . Given $\tau\in\mathbb{R}$ , $\omega\in\Omega$ , and $D\in\mathcal{D}$ , it follows from Lemma 3.3 that there exists $T_{0} = T_{0}(\tau, \omega, D, \delta)$ such that for all $t\geq T_{0}$ ,

$\begin{equation} \begin{split}\nonumber \Phi_{\delta}(t,\tau-t,\theta_{-t}\omega,D(\tau-t,\theta_{-t}\omega))\subseteq K_{\delta}(\tau,\omega), \end{split} \end{equation}$

which implies that $K_{\delta}$ pullback attracts all elements in $\mathcal{D}$ . By (2.1), (2.8) and (3.2), we can prove $K_{\delta}(\tau, \omega)$ is tempered. The convergence (3.17) can be obtained by Lebesgue dominated convergence as in ^[17].

We are now in a position to derive uniform estimates on the tail of solutions of system (3.5).

Lemma 3.5. Suppose that (2.3)–(2.8) hold. Then for every $\tau\in\mathbb{R}$ , $\omega\in\Omega$ and $\varepsilon > 0$ , there exist $\delta_{0} = \delta_{0}(\omega) > 0$ , $T = T(\tau, \omega, \varepsilon) > 0$ and $N = N(\tau, \omega, \varepsilon) > 0$ such that for all $t\geq T$ and $0 < |\delta| < \delta_{0}$ , the solution $\bar{\varphi}_{\delta}$ of system (3.5) satisfies

$\begin{equation} \begin{split}\nonumber \sum\limits_{|i|\geq N}|\bar{\varphi}_{\delta,i}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t,i})|_{E}^{2}\leq\varepsilon,\\ \end{split} \end{equation}$

where $\bar{\varphi}_{\delta, \tau-t}\in K_{\delta}(\tau-t, \theta_{-t}\omega)$ and $|\bar{\varphi}_{\delta, i}|_{E}^{2} = (1-\nu\beta)|Bu^{\delta}|_{i}^{2}+\lambda |u^{\delta}_{i}|^{2}+|\bar{v}^{\delta}_{i}|^{2}$ .

Proof. Let $\eta$ be a smooth function defined in Lemma 2.3, and set $x = (x_{i})_{i\in\mathbb{Z}}$ , $y = (y_{i})_{i\in\mathbb{Z}}$ with $x_{i} = \eta(\frac{|i|}{k})u^{\delta}_{i}$ , $y_{i} = \eta(\frac{|i|}{k})v^{\delta}_{i}$ . Note $\psi = (x, y)^{T} = ((x_{i}), (y_{i}))^{T}_{i\in\mathbb{Z}}$ . Taking the inner product of system (3.6) with $\psi$ , we have

$\begin{equation} \begin{split} (\dot{\varphi}_{\delta},\psi)_{E}+(L_{\delta}(\varphi_{\delta}),\psi)_{E} = (H_{\delta}(\varphi_{\delta}),\psi)_{E}+(G_{\delta},\psi)_{E}. \end{split} \end{equation}$

(3.18)

For the first term of (3.18), we have

$\begin{equation} \begin{split} (\dot{\varphi}_{\delta},\psi)_{E}& = (1-\nu\beta)\sum\limits_{i\in\mathbb{Z}}(B\dot{u}^{\delta})_{i}(Bx)_{i}+\lambda\sum\limits_{i\in\mathbb{Z}}\dot{u}_{i}^{\delta}x_{i} +\sum\limits_{i\in\mathbb{Z}}\dot{v}_{i}^{\delta}y_{i}\\ & = \frac{1}{2}\frac{d}{dt}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{\delta,i}|^{2}_{E} +(1-\nu\beta)\sum\limits_{i\in\mathbb{Z}}(B\dot{u}^{\delta})_{i}\Big((Bx)_{i}-\eta(\frac{|i|}{k})(Bu^{\delta})_{i}\Big)\\ &\geq\frac{1}{2}\frac{d}{dt}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{\delta,i}|^{2}_{E} -\frac{(1-\nu\beta)C_{0}}{k}\sum\limits_{i\in\mathbb{Z}}|B(v^{\delta}-\beta u^{\delta}+a\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds)|_{i}|u_{i+1}^{\delta}|\\ &\geq\frac{1}{2}\frac{d}{dt}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{\delta,i}|^{2}_{E} -\frac{c}{k}\|\varphi_{\delta}\|^{2}_{E}-\frac{c}{k}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2}\|a\|^{2}, \end{split} \end{equation}$

(3.19)

where $|\varphi_{\delta, i}|_{E}^{2} = (1-\nu\beta)|Bu^{\delta}|_{i}^{2}+\lambda |u^{\delta}_{i}|^{2}+|v^{\delta}_{i}|^{2}$ . By the similar calculations in (2.28)–(2.33), we get

$\begin{equation} \begin{split} (L_{\delta}(\varphi_{\delta}),\psi)_{E} \geq&\sigma\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{\delta,i}|^{2}_{E}+ \frac{h_{1}}{6}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v_{i}^{\delta}|^{2}-\frac{c}{k}\|\varphi_{\delta}\|^{2}_{E}\\ &-c\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|a_{i}|^{2}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2}, \end{split} \end{equation}$

(3.20)

$\begin{equation} \begin{split} (H_{\delta}(\varphi_{\delta}),\psi)_{E} &\leq-\frac{d}{dt}(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})F_{i}(u^{\delta}_{i})) -\frac{\alpha_{2}\beta}{p+1}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})F_{i}(u^{\delta}_{i})\\ &\quad +\frac{\sigma\lambda}{4}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|u_{i}^{\delta}|^{2}+\frac{\sigma}{6} \sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v^{\delta}_{i}|^{2} +c\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|g_{i}(t)|^{2}\\ &\quad +c\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|a_{i}|^{2}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2} +c\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|a_{i}|^{p+1}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{p+1}, \end{split} \end{equation}$

(3.21)

and

$\begin{equation} \begin{split} (G_{\delta},\psi)_{E}& = (1-\nu\beta)\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds(Bx,Ba)_{\lambda}+\beta \int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds(y,a)\\ &\leq \frac{\sigma\lambda}{4}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|u^{\delta}_{i}|^{2} +\Big(\frac{h_{1}}{6}+\frac{\sigma}{3}\Big)\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|v^{\delta}_{i}|^{2} +c|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|a_{i}|^{2}. \end{split} \end{equation}$

(3.22)

It follows from (3.18)–(3.22) that

$\begin{equation} \begin{split} &\frac{d}{dt}\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\big(|\varphi_{\delta,i}|_{E}^{2}+2 F_{i}(u_{i}^{\delta})\big)\Big) +\gamma\Big(\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\big(|\varphi_{\delta,i}|_{E}^{2}+2 F_{i}(u_{i}^{\delta})\big)\Big)+\gamma\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{\delta,i}|_{E}^{2}\\ &\leq\frac{c}{k}\|\varphi_{\delta}\|^{2}_{E}+\frac{c}{k}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2}+c\sum\limits_{|i|\geq k}|g_{i}(t)|^{2} +c\sum\limits_{|i|\geq k}|a_{i}|^{p+1}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{p+1}\\ &\quad+ c\sum\limits_{|i|\geq k}|a_{i}|^{2}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds|^{2}, \end{split} \end{equation}$

(3.23)

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$ . Multiplying (3.23) by $e^{\gamma t}$ , replacing $\omega$ by $\theta_{-\tau}\omega$ and integrating on $(\tau-t, \tau)$ with $t\in\mathbb{R}^{+}$ , we get for every $\omega\in\Omega$

$\begin{equation} \begin{split} &\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\Big(|\varphi_{\delta,i}(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\delta,\tau-t,i})|^{2}_{E} +2F_{i}(u_{i}^{\delta}(\tau,\tau-t,\theta_{-\tau}\omega,u^{\delta}_{\tau-t,i}))\Big)\\ &\leq e^{-\gamma t}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\Big(|\varphi_{\delta,\tau-t,i}|^{2}_{E}+2 F_{i}(u_{\tau-t,i}^{\delta})\Big) +\frac{c}{k}\int^{\tau}_{\tau-t}e^{\gamma(s-\tau)}\|\varphi_{\delta}(s,\tau-t,\theta_{-\tau}\omega,\varphi_{\delta,\tau-t})\|^{2}_{E}ds\\ &\quad+\frac{c}{k}\int^{0}_{-\infty}e^{\gamma s}|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}ds +c\int_{-\infty}^{0}e^{\gamma s}\sum\limits_{|i|\geq k}|g_{i}(s+\tau)|^{2}ds\\ &\quad+c\sum\limits_{|i|\geq k}|a_{i}|^{2}\int_{-\infty}^{0}e^{\gamma s}|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}ds+c\sum\limits_{|i|\geq k}|a_{i}|^{p+1}\int_{-\infty}^{0}e^{\gamma s}|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{p+1}ds. \end{split} \end{equation}$

(3.24)

For any $s\geq\tau-t$ ,

which along with (3.24) shows that

$\begin{equation} \begin{split} &\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\bar{\varphi}_{\delta,i}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t,i})|^{2}_{E}\\ &\leq2\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\varphi_{\delta,i}(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\delta,\tau-t,i})|^{2}_{E} +2\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|a_{i}\int_{0}^{\tau}\mathcal{G}_{\delta}(\theta_{l-\tau}\omega)dl|^{2}\\ &\leq 2\sum\limits_{|i|\geq k}|a_{i}\int_{-\tau}^{0}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2} +4e^{-\gamma t}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\Big(|\bar{\varphi}_{\delta,\tau-t,i}|^{2}_{E}+|a_{i}\int_{-\tau}^{-t}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}+ F_{i}(u_{\tau-t,i}^{\delta})\Big)\\ &\quad+\frac{c}{k}\int^{\tau}_{\tau-t}e^{\gamma(s-\tau)}\|\bar{\varphi}_{\delta}(s,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t})\|^{2}_{E}ds +\frac{c}{k}\int^{0}_{-\infty}e^{\gamma s}|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}ds\\ &\quad +c\int_{-\infty}^{0}e^{\gamma s}\sum\limits_{|i|\geq k}|g_{i}(s+\tau)|^{2}ds+c\sum\limits_{|i|\geq k}|a_{i}|^{2}\int_{-\infty}^{0}e^{\gamma s}|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}ds\\ &\quad+c\sum\limits_{|i|\geq k}|a_{i}|^{p+1}\int_{-\infty}^{0}e^{\gamma s}|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{p+1}ds. \end{split} \end{equation}$

(3.25)

By (2.1) and (2.8), the last four integrals on the right-hand side of (3.24) are well defined. Note that (2.3) and (2.4) implies that

$\begin{equation} \begin{split}\nonumber \sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k}) F_{i}(u_{\tau-t,i}^{\delta})\leq \frac{1}{\alpha_{2}}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k}) f_{i}(u_{\tau-t,i}^{\delta})u_{\tau-t,i}^{\delta} \leq\frac{1}{\alpha_{2}}\max\limits_{-\|u_{\tau-t}^{\delta}\|\leq s\leq\|u_{\tau-t}^{\delta}\|}|f'_{i}(s)|\|u_{\tau-t}^{\delta}\|^{2}. \end{split} \end{equation}$

Since $\bar{\varphi}_{\delta, \tau-t}\in K_{\delta}(\tau-t, \theta_{-t}\omega)$ , we find

$\begin{equation} \begin{split}\nonumber \limsup\limits_{t\rightarrow +\infty}e^{-\gamma t}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\bar{\varphi}_{\delta,\tau-t,i}|^{2}_{E}\leq\limsup\limits_{t\rightarrow +\infty}e^{-\gamma t}\|K_{\delta}(\tau-t,\theta_{-t}\omega)\|^{2}_{E} = 0, \end{split} \end{equation}$

which along with (2.1) and (3.2) shows that there exist $T_{1} = T_{1}(\tau, \omega, \varepsilon) > 0$ and $\delta_{0} > 0$ such that for all $t\geq T_{1}$ and $0 < |\delta| < \delta_{0}$ ,

$\begin{equation} \begin{split} 4e^{-\gamma t}\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})\Big(|\bar{\varphi}_{\delta,\tau-t,i}|^{2}_{E}+|a_{i}\int_{-\tau}^{-t}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}+ F_{i}(u_{\tau-t,i}^{\delta})\Big)\leq\frac{\varepsilon}{4}. \end{split} \end{equation}$

(3.26)

By Lemma 3.3, (2.1) and (3.2), there exist $T_{2} = T_{2}(\tau, \omega, \varepsilon) > T_{1}$ and $N_{1} = N_{1}(\tau, \varepsilon) > 0$ such that for all $t\geq T_{2}$ , $k\geq N_{1}$ and $0 < |\delta| < \delta_{0}$

$\begin{equation} \begin{split} \frac{c}{k}\int^{\tau}_{\tau-t}e^{\gamma(s-\tau)}\|\bar{\varphi}_{\delta}(s,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t})\|^{2}_{E}ds+\frac{c}{k}\int^{0}_{-\infty}e^{\gamma s}|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}ds\leq\frac{\varepsilon}{4}. \end{split} \end{equation}$

(3.27)

By (2.8), there exists $N_{2} = N_{2}(\tau, \varepsilon) > N_{1}$ such that for all $k\geq N_{2}$ ,

$\begin{equation} \begin{split} 2\sum\limits_{|i|\geq k}|a_{i}\int_{-\tau}^{0}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}+c\int_{-\infty}^{0}e^{\gamma s}\sum\limits_{|i|\geq k}|g_{i}(s+\tau)|^{2}ds\leq\frac{\varepsilon}{4}. \end{split} \end{equation}$

(3.28)

By (2.1) and (3.2) again, we find that there exists $N_{3} = N_{3}(\tau, \varepsilon) > N_{2}$ such that for all $k\geq N_{3}$ and $0 < |\delta| < \delta_{0}$ ,

$\begin{equation} \begin{split} c\sum\limits_{|i|\geq k}|a_{i}|^{p+1}\int_{-\infty}^{0}e^{\gamma s}|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{p+1}ds +c\sum\limits_{|i|\geq k}|a_{i}|^{2}\int_{-\infty}^{0}e^{\gamma s}|\int_{-\tau}^{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|^{2}ds \leq\frac{\varepsilon}{4}. \end{split} \end{equation}$

(3.29)

Then it follows from (3.25)–(3.29) that for all $t\geq T_{2}$ , $k\geq N_{3}$ and $0 < |\delta| < \delta_{0}$ ,

$\begin{equation} \begin{split}\nonumber \sum\limits_{|i|\geq 2k}|\bar{\varphi}_{\delta,i}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t,i})|^{2}_{E} \leq\sum\limits_{i\in\mathbb{Z}}\eta(\frac{|i|}{k})|\bar{\varphi}_{\delta,i}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t,i})|^{2}_{E} \leq\varepsilon. \end{split} \end{equation}$

This concludes the proof.

By Lemma 3.4, $\Phi_{\delta}$ has a closed $\mathcal{D}$ -pullback absorbing set, and Lemma 3.5 shows that $\Phi_{\delta}$ is asymptotically null in $E$ with respect to $\mathcal{D}$ . Therefore, we get the existence of $\mathcal{D}$ -pullback attractors for $\Phi_{\delta}$ .

Lemma 3.6. Suppose that (2.3)–(2.9) hold. Then the continuous cocycle $\Phi_{\delta}$ associated with (3.5) has a unique $\mathcal{D}$ -pullback attractors $\mathcal{A}_{\delta} = \{\mathcal{A}_{\delta}(\tau, \omega):\tau\in\mathbb{R}$ , $\omega\in\Omega\}\in \mathcal{D}$ in $E$ .

For the attractor $\mathcal{A}_{\delta}$ of $\Phi_{\delta}$ , we have the uniform compactness as showed below.

Lemma 3.7. Suppose that (2.3)–(2.9) hold. Then for every $\tau\in\mathbb{R}$ , $\omega\in\Omega$ , there exists $\delta_{0} = \delta_{0}(\omega) > 0$ such that $\mathop{\bigcup}\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau, \omega)$ is precompact in $E$ .

Proof. Given $\varepsilon > 0$ , we will prove that $\mathop{\bigcup}\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau, \omega)$ has a finite covering of balls of radius less than $\varepsilon$ . By (3.2) we have

$\begin{equation} \begin{split} \int^{0}_{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl = -\int^{s+\delta}_{s}\frac{\omega(l)}{\delta}dl+\int^{\delta}_{0}\frac{\omega(l)}{\delta}dl. \end{split} \end{equation}$

(3.30)

By $\lim_{\delta\rightarrow0}\int^{\delta}_{0}\frac{\omega(r)}{\delta}dr = 0$ , there exists $\delta_{1} = \delta_{1}(\omega) > 0$ such that for all $0 < |\delta| < \delta_{1}$ ,

$\begin{equation} \begin{split} |\int^{\delta}_{0}\frac{\omega(l)}{\delta}dl|\leq1. \end{split} \end{equation}$

(3.31)

Similarly, there exists $l_{1}$ between $s$ and $s+\delta$ such that $\int^{s+\delta}_{s}\frac{\omega(l)}{\delta}dl = \omega(l_{1})$ , which along with (2.1) implies that there exists $T_{1} = T_{1}(\omega) < 0$ such that for all $s\leq T_{1}$ and $|\delta|\leq1$ ,

$\begin{equation} \begin{split} |\int^{s+\delta}_{s}\frac{\omega(l)}{\delta}dl|\leq 1-s. \end{split} \end{equation}$

(3.32)

Let $\delta_{2} = \min\{\delta_{1}, 1\}$ . By (3.30)–(3.32) we get for all $0 < |\delta| < \delta_{2}$ and $s\leq T_{1}$ ,

$\begin{equation} \begin{split} |\int^{0}_{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl| < 2-s. \end{split} \end{equation}$

(3.33)

By (3.4), there exist $\delta_{0} = \delta_{0}(\omega)\in(0, \delta_{2})$ and $c_{1}(\omega) > 0$ such that for all $0 < |\delta|\leq\delta_{0}$ and $T_{1}\leq s\leq0$ ,

$\begin{equation} \begin{split}\nonumber |\int^{0}_{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|\leq c_{1}(\omega), \end{split} \end{equation}$

which along with (3.33) implies that for all $0 < |\delta| < \delta_{0}$ and $s\leq0$ ,

$\begin{equation} \begin{split} |\int^{0}_{s}\mathcal{G}_{\delta}(\theta_{l}\omega)dl|\leq -s+c_{2}(\omega), \end{split} \end{equation}$

(3.34)

where $c_{2}(\omega) = 2+c_{1}(\omega)$ . Denote by

$\begin{equation} \begin{split}\nonumber B(\tau,\omega) = \{\bar{\varphi}_{\delta}\in E:\|\bar{\varphi}_{\delta}\|^{2}\leq R(\tau,\omega)\}, \end{split} \end{equation}$

and

$\begin{equation} \begin{split} R(\tau,\omega) = &c\int_{-\infty}^{0}e^{\gamma s}\Big(\|g(s+\tau)\|^{2}+2(c_{2}-s)^{2}+2(|\tau|+c_{2})^{2}+2^{p}(c_{2}-s)^{p+1}+2^{p}(|\tau|+c_{2})^{p+1}\Big)ds\\ &\quad+c+2c(|\tau|+c_{2})^{2}, \end{split} \end{equation}$

(3.35)

with $c$ and $c_{2}$ being as in (3.12) and (3.34). By (3.12) and (3.35) we find that for all $0 < |\delta| < \delta_{0}$ ,

$\begin{equation} \begin{split} R_{\delta}(\tau,\omega)\leq R(\tau,\omega). \end{split} \end{equation}$

(3.36)

By (3.35) and (3.36), we find that $K_{\delta}(\tau, \omega)\subseteq B(\tau, \omega)$ for all $0 < |\delta| < \delta_{0}$ , $\tau\in\mathbb{R}$ and $\omega\in\Omega$ . Therefore, for every $\tau\in\mathbb{R}$ , $\omega\in\Omega$ ,

$\begin{equation} \begin{split} \bigcup\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau,\omega)\subseteq\bigcup\limits_{0 < |\delta| < \delta_{0}}K_{\delta}(\tau,\omega) \subseteq B(\tau,\omega). \end{split} \end{equation}$

(3.37)

By Lemma 3.5, there exist $T = T(\tau, \omega, \varepsilon) > 0$ and $N = N(\tau, \omega, \varepsilon) > 0$ such that for all $t\geq T$ and $0 < |\delta| < \delta_{0}$ ,

$\begin{equation} \begin{split} \sum\limits_{|i|\geq N}|\bar{\varphi}_{\delta,i}(\tau,\tau-t,\theta_{-\tau}\omega,\bar{\varphi}_{\delta,\tau-t,i})|^{2}_{E}\leq\frac{\varepsilon}{4}, \end{split} \end{equation}$

(3.38)

for any $\bar{\varphi}_{\delta, \tau-t}\in K_{\delta}(\tau-t, \theta_{-t}\omega)$ . By (3.38) and the invariance of $\mathcal{A}_{\delta}$ , we obtain

$\begin{equation} \begin{split} \sum\limits_{|i|\geq N}|\bar{\varphi}_{i}|^{2}_{E}\leq\frac{\varepsilon}{4},\; \; \text{for all}\; \bar{\varphi} = (\bar{\varphi}_{i})_{i\in\mathbb{Z}}\in\bigcup\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau,\omega). \end{split} \end{equation}$

(3.39)

We find that (3.37) implies the set $\{(\bar{\varphi}_{i})_{|i| < N}:\bar{\varphi}\in \mathop{\bigcup}\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau, \omega)\}$ is bounded in a finite dimensional space and hence is precompact. This along with (3.39) implies $\mathop{\bigcup}\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau, \omega)$ has a finite covering of balls of radius less than $\varepsilon$ in $E$ . This completes the proof.

4. Upper semicontinuity of pullback attractors

In this section, we will study the limiting of solutions of (3.5) as $\delta\rightarrow0$ . Hereafter, we need an additional condition on $f$ : For all $i\in\mathbb{Z}$ and $s\in\mathbb{R}$ ,

$\begin{equation} \begin{split} |f'_{i}(s)|\leq\alpha_{4}|s|^{p-1}+\kappa_{i}, \end{split} \end{equation}$

(4.1)

where $\alpha_{4}$ is a positive constant, $\kappa = (\kappa_{i})_{i\in\mathbb{Z}}\in l^{2}$ and $p > 1$ .

Lemma 4.1. Suppose that (2.3)–(2.7) and (4.1) hold. Let $\bar{\varphi}$ and $\bar{\varphi}_{\delta}$ are the solutions of (2.10) and (3.5), respectively. Then for every $\tau\in\mathbb{R}$ , $\omega\in\Omega$ , $T > 0$ and $\varepsilon\in(0, 1)$ , there exist $\delta_{0} = \delta_{0}(\tau, \omega, T, \varepsilon) > 0$ and $c = c(\tau, \omega, T) > 0$ such that for all $t\in[\tau, \tau+T]$ and $0 < |\delta| < \delta_{0}$ ,

$\begin{equation} \begin{split}\nonumber \|\bar{\varphi}_{\delta}(t,\tau,\omega,\bar{\varphi}_{\delta,\tau})-\bar{\varphi}(t,\tau,\omega,\bar{\varphi}_{\tau})\|^{2}_{E} \leq 2e^{c(t-\tau)}\|\bar{\varphi}_{\delta,\tau}-\bar{\varphi}_{\tau}\|^{2}_{E}+c\varepsilon. \end{split} \end{equation}$

Proof. Let $\tilde{\varphi} = \varphi_{\delta}-\varphi$ and $\tilde{\varphi} = (\tilde{u}, \tilde{v})^{T}$ , where $\tilde{u} = u^{\delta}-u$ , $\tilde{v} = v^{\delta}-v$ , $\varphi$ and $\varphi_{\delta}$ are the solutions of (2.11) and (3.6), respectively. By (2.11) and (3.6) we get

$\begin{equation} \begin{split} \dot{\tilde{\varphi}}+\tilde{L}(\tilde{\varphi}) = \tilde{H}(\tilde{\varphi})+\tilde{G}(\omega), \end{split} \end{equation}$

(4.2)

where

$\begin{equation} \begin{aligned}\nonumber \tilde{L}(\tilde{\varphi})& = \left( \begin{array}{ccc} \beta \tilde{u}-\tilde{v}\\ (1-\nu\beta)A\tilde{u}+\nu A\tilde{v}+\lambda \tilde{u} +\beta^{2} \tilde{u}-\beta\tilde{v} \end{array} \right)\\ &\quad+ \left( \begin{array}{ccc} 0 \\ h\big(v^{\delta}-\beta u^{\delta}+a\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds\big)-h\big(v-\beta u+a\omega(t)\big) \end{array} \right), \end{aligned} \end{equation}$

$\begin{equation} \begin{aligned}\nonumber \tilde{H}(\tilde{\varphi}) = \left( \begin{array}{ccc} 0 \\ -f(u^{\delta})+f(u) \end{array} \right),\; \; \; \; \tilde{G}(\omega) = \left( \begin{array}{ccc} a\big(\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)\big)\\ (\beta a-\nu Aa)\big(\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)\big) \end{array} \right). \end{aligned} \end{equation}$

Taking the inner product of (4.2) with $\tilde{\varphi}$ in $E$ , we have

$\begin{equation} \begin{split} \frac{1}{2}\frac{d}{dt}\|\tilde{\varphi}\|^{2}_{E}+(\tilde{L}(\tilde{\varphi}),\tilde{\varphi})_{E} = (\tilde{H}(\tilde{\varphi}),\tilde{\varphi})_{E}+(\tilde{G}(\omega),\tilde{\varphi})_{E}. \end{split} \end{equation}$

(4.3)

For the second term on the left-hand side of (4.3), using the similar calculations in (2.13) we have

$\begin{equation} \begin{split} (\tilde{L}(\tilde{\varphi}),\tilde{\varphi})_{E}&\geq\sigma\|\tilde{\varphi}\|_{E}^{2}+\frac{h_{1}}{2}\|\tilde{v}\|^{2} -h_{2}|\big(a(\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)),\tilde{v}\big)|\\ &\geq\sigma\|\tilde{\varphi}\|_{E}^{2}+\frac{h_{1}}{4}\|\tilde{v}\|^{2} -c|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t))|^{2}\|a\|^{2}. \end{split} \end{equation}$

(4.4)

For the first term on the right-hand side of (4.3), by (4.1) we get

$\begin{equation} \begin{split} (f(u)-f(u^{\delta}),\tilde{v})& = \sum\limits_{i\in\mathbb{Z}}(f_{i}(u_{i})-f_{i}(u^{\delta}_{i}))\tilde{v}_{i} = \frac{1}{h_{1}}\sum\limits_{i\in\mathbb{Z}}|f_{i}(u_{i})-f_{i}(u^{\delta}_{i})|^{2}+\frac{h_{1}}{4}\sum\limits_{i\in\mathbb{Z}}|\tilde{v}_{i}|^{2}\\ &\leq c(\|\varphi\|^{2p-2}_{E}+\|\varphi_{\delta}\|^{2p-2}_{E})\|\tilde{\varphi}\|^{2}_{E}+\frac{h_{1}}{4}\|\tilde{v}\|^{2}+\frac{2\|\kappa\|^{2}}{h_{1}\lambda}\|\tilde{\varphi}\|^{2}_{E}. \end{split} \end{equation}$

(4.5)

As to the last term of (4.3), we have

$\begin{equation} \begin{split} &\big(a(\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)),\tilde{u}\big)_{\lambda} +\big((\beta a-\nu Aa)(\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)),\tilde{v}\big)\\ &\leq\sigma\|\tilde{u}\|^{2}_{\lambda}+\frac{1}{4\sigma}|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)|^{2}\|a\|^{2}_{\lambda} +\sigma\|\tilde{v}\|^{2}+c|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)|^{2}\|a\|^{2}. \end{split} \end{equation}$

(4.6)

It follows from (4.3)–(4.6) that

$\begin{equation} \begin{split} \frac{d}{dt}\|\tilde{\varphi}\|^{2}_{E}\leq c(\|\varphi\|^{2p-2}_{E}+\|\varphi_{\delta}\|^{2p-2}_{E}+1)\|\tilde{\varphi}\|^{2}_{E}+c|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)|^{2}. \end{split} \end{equation}$

(4.7)

By Lemma 2.1 and Lemma 3.2, there exists $\delta_{1} = \delta_{1}(\tau, \omega, T) > 0$ and $c_{1} = c_{1}(\tau, \omega, T) > 0$ such that for all $0 < |\delta| < \delta_{1}$ and $t\in[\tau, \tau+T]$ ,

$\begin{equation} \begin{split}\nonumber \|\varphi_{\delta}(t,\tau,\omega,\varphi_{\delta,\tau})\|^{2}_{E}+\|\varphi(t,\tau,\omega,\varphi_{\tau})\|^{2}_{E}\leq c_{1}, \end{split} \end{equation}$

which along with (4.7) shows that for all $0 < |\delta| < \delta_{1}$ and $t\in[\tau, \tau+T]$

$\begin{equation} \begin{split} \frac{d}{dt}\|\tilde{\varphi}\|^{2}_{E}\leq c\|\tilde{\varphi}\|^{2}_{E}+c|\int_{0}^{t}\mathcal{G}_{\delta}(\theta_{s}\omega)ds-\omega(t)|^{2}. \end{split} \end{equation}$

(4.8)

Applying Gronwall's inequality and Lemma 3.1 to (4.8), we see that for every $\varepsilon\in(0, 1)$ , there exists $\delta_{0} = \delta_{0}(\tau, \omega, T, \varepsilon)\in(0, \delta_{1})$ such that for all $0 < |\delta| < \delta_{0}$ and $t\in[\tau, \tau+T]$

$\begin{equation} \begin{split} \|\tilde{\varphi}(t,\tau,\omega,\tilde{\varphi}_{\tau})\|^{2}_{E}\leq e^{c(t-\tau)}\|\tilde{\varphi}_{\tau}\|^{2}_{E}+c\varepsilon. \end{split} \end{equation}$

(4.9)

On the other hand, we have

$\begin{equation} \begin{split}\nonumber \bar{\varphi}_{\delta}(t,\tau,\omega,\bar{\varphi}_{\delta,\tau})-\bar{\varphi}(t,\tau,\omega,\bar{\varphi}_{\tau}) = \tilde{\varphi}+\big(0,a(\int^{t}_{0}\mathcal{G}_{\delta}(\theta_{s})ds-\omega(t))\big)^{T}, \end{split} \end{equation}$

which along with (4.9) implies the desired result.

Finally, we establish the upper semicontinuity of random attractors as $\delta\rightarrow0$ .

Theorem 4.1. Suppose that (2.3)–(2.9) and (4.1) hold. Then for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$ ,

$\begin{equation} \begin{split} \lim\limits_{\delta\rightarrow0}d_{E}(\mathcal{A}_{\delta}(\tau,\omega),\mathcal{A}_{0}(\tau,\omega)) = 0, \end{split} \end{equation}$

(4.10)

where $d_{E}(\mathcal{A}_{\delta}(\tau, \omega), \mathcal{A}_{0}(\tau, \omega)) = \mathop{\sup}\limits_{x\in\mathcal{A}_{\delta}(\tau, \omega)}\mathop{\inf}\limits_{y\in\mathcal{A}_{0}(\tau, \omega)}\|x-y\|_{E}$ .

Proof. Let $\delta_{n}\rightarrow0$ and $\bar{\varphi}_{\delta_{n}, \tau}\rightarrow \bar{\varphi}_{\tau}$ in $E$ . Then by Lemma 4.1, we find that for all $\tau\in\mathbb{R}$ , $t\geq0$ and $\omega\in\Omega$ ,

$\begin{equation} \begin{split} \Phi_{\delta_{n}}(t,\tau,\omega,\bar{\varphi}_{\delta_{n},\tau})\rightarrow \Phi_{0}(t,\tau,\omega,\bar{\varphi}_{\tau}) \; \; \text{in}\; \; E. \end{split} \end{equation}$

(4.11)

By (3.16)–(3.17) we have, for all $\tau\in\mathbb{R}$ and $\omega\in\Omega$ ,

$\begin{equation} \begin{split} \lim\limits_{\delta\rightarrow0}\|K_{\delta}(\tau,\omega)\|_{E}^{2}\leq R_{0}(\tau,\omega). \end{split} \end{equation}$

(4.12)

Then by (4.11), (4.12) and Lemma 3.7, (4.10) follows from Theorem 3.1 in ^[24] immediately.

5. Conclusions

In this paper we use similar idea in ^[30] but apply to second order non-autonomous stochastic lattice dynamical systems with additive noise. we establish the convergence of solutions of Wong-zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the step-length of the Wiener shift approaches zero. In addition, as to the second order non-autonomous stochastic lattice dynamical systems with multiplicative noise, we can use the similar method in ^[29] to get the corresponding results.

Acknowledgements

The authors would like to thank anonymous referees and editors for their valuable comments and constructive suggestions.

Conflict of interest

The authors declare no conflict of interest.

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