  We consider the initial-boundary value problem of a system of
reaction-diffusion equations with density-dependent motility
\begin{equation*}\label{e1}\tag{$\ast$} \begin{cases}
u_t=\Delta(\gamma(v)u)+\alpha u F(w) -\theta u, &x\in \Omega, ~~t>0,\\
v_t=D\Delta v+u-v,& x\in \Omega, ~~t>0,\\ w_t=\Delta w-uF(w),& x\in \Omega,
~~t>0,
  \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=
\frac{\partial w}{\partial \nu}=0,&x\in \partial\Omega, ~~t>0,\\
(u,v,w)(x,0)=(u_0,v_0,w_0)(x), & x\in\Omega, \end{cases} \end{equation*} in a
bounded domain $\Omega\subset\R^2$ with smooth boundary, $\alpha$ and $\theta$
are non-negative constants and $\nu$ denotes the outward normal vector of
$\partial \Omega$. The random motility function $\gamma(v)$ and functional
response function $F(w)$ satisfy the following assumptions: \begin{itemize}
\item $\gamma(v)\in C^{3}([0,\infty)),~0<\gamma_{1}\leq\gamma(v)\leq \gamma_2,
\ |\gamma'(v)|\leq \eta$ for all $v\geq0$; \item $F(w)\in C^1([0,\infty)),
F(0)=0,F(w)>0 \ \mathrm{in}~(0,\infty)~\mathrm{and}~F'(w)>0 \ \mathrm{on}\ \
[0,\infty)$ \end{itemize} for some positive constants $\gamma_1, \gamma_2$ and
$\eta$. Based on the method of weighted energy estimates and Moser iteration,
we prove that the problem \eqref{e1} has a unique classical global solution
uniformly bounded in time. Furthermore we show that if $\theta>0$, the solution
$(u,v,w)$ will converge to $(0,0,w_*)$ in $L^\infty$ with some $w_*>0$ as time
tends to infinity, while if $\theta=0$, the solution $(u,v,w)$ will
asymptotically converge to $(u_*,u_*,0)$ in $L^\infty$ with
$u_*=\frac{1}{|\Omega|}(\|u_0\|_{L^1}+\alpha\|w_0\|_{L^1})$ if $D>0$ is
suitably large.
