  In this paper, we consider the following parabolic-parabolic-elliptic system
} \begin{align*} \left\{\aligned & u_t=\Delta u-\nabla\cdot(u\nabla
v)+\xi\nabla\cdot(u\nabla w)+au-\mu u^{\alpha}, && x\in\Omega, t>0,\\ &
v_t=\Delta v+\nabla\cdot(v\nabla w)-v+u,&& x\in\Omega, t>0,\\ & 0=\Delta
w-w+u,&& x\in\Omega, t>0\\ \endaligned\right. \end{align*} on a bounded domain
$\Omega\subset \mathbb{R}^{N}$ ($N\geq1$) with smooth boundary $\partial
\Omega$, where $\mu$, $a$, $\alpha$ are positive constants and
$\xi\in\mathbb{R}$. If one of the following cases holds:\\ (i) $N\geq4$ and
$\alpha>\frac{4N-4+N\sqrt{2N^2-6N+8}}{2N}$;\\ (ii) $N=3$, $\alpha>2$, for any
$\mu>0$ or $\alpha=2$, the index $\mu$ should be suitably big;\\ (iii) $N=2$,
$\alpha\geq2$, for any $\mu>0$.\\ Without any restriction on the index $\xi$,
for any given suitably regular initial data, the corresponding Neumann
initial-boundary problem admits a unique global and bounded classical solution.
