  This paper is concerned with a class of reaction-diffusion system with
density-suppressed motility \begin{equation*} \begin{cases}
u_{t}=\Delta(\gamma(v) u)+\alpha u F(w), & x \in \Omega, \quad t>0, \\ v_{t}=D
\Delta v+u-v, & x \in \Omega, \quad t>0, \\ w_{t}=\Delta w-u F(w), & x \in
\Omega, \quad t>0, %\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial
\nu}=\frac{\partial w}{\partial \nu}=0, & x \in \partial \Omega, \quad t>0, \\
%(u, v, w)(x, 0)=\left(u_{0}, v_{0}, w_{0}\right)(x), & x \in \Omega,
\end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a
smooth bounded domain $\Omega\subset \mathbb{R}^n~(n\leq 2)$, where $\alpha>0$
and $D>0$ are constants. The random motility function $\gamma$ satisfies
\begin{equation*} \gamma\in C^3((0,+\infty)),\ \gamma>0,\ \gamma'<0\,\
\text{on}\,\ (0,+\infty) \ \ \text{and}\ \
\lim_{v\rightarrow+\infty}\gamma(v)=0. \end{equation*} %and %\begin{equation*}
%\lim_{x\rightarrow+\infty}\gamma(x)=0. %\end{equation*} The intake rate
function $F$ satisfies \begin{equation*} F\in C^1([0,+\infty)),\,F(0)=0\,\
\text{and}\ \,F>0\,\ \text{on}\,\ (0,+\infty). \end{equation*} We show that the
above system admits a unique global classical solution for all non-negative
initial data $$ u_0\in C^0(\overline{\Omega}),\,v_0\in
W^{1,\infty}(\Omega),\,w_0\in W^{1,\infty}(\Omega). $$
  Moreover, if there exist $k>0$ and $\overline{v}>0$ such that
\begin{equation*} \inf_{v>\overline{v}}v^k\gamma(v)>0, \end{equation*} then the
global solution is bounded uniformly in time.
