  The diffusive Lotka-Volterra predator-prey model \begin{eqnarray*} \left\{
\begin{array}{rcll} u_t &=& \nabla\cdot \left[ d_1\nabla u + \chi v^2 \nabla
\Big(\dfrac{u}{v}\Big)\right] +u(m_1-u+av), \qquad & x\in\Omega, \ t>0, \\ v_t
&=& d_2\Delta v+v(m_2-bu-v), \qquad & x\in\Omega, \ t>0, \end{array} \right.
\end{eqnarray*} is considered in a bounded domain $\Omega\subset\mathbb{R}^n$,
$n \in\{2,3\}$, under Neumann boundary condition, where $d_1, d_2, m_1, \chi,
a, b$ are positive constants and $m_2$ is a real constant. The purpose of this
paper is to establish global existence and boundedness of classical solutions
in the case $n=2$ and global existence of weak solutions in the case $n=3$ as
well as show long-time stabilization. More precisely, we prove that the
solutions $(u(\cdot,t), v(\cdot,t))$ converge to the constant steady state
$(u_*, v_*)$ as $t \to \infty$, where $u_*, v_*$ solves
$u_*(m_1-u_*+av_*)=v_*(m_2-bu_*-v_*)=0$ with $u_* > 0$ (covering both
coexistence as well as prey-extinction cases).
