  In this paper we study the zero-flux chemotaxis-system \begin{equation*}
\begin{cases} u_{ t}=\nabla \cdot ((u+1)^{m-1} \nabla u-(u+1)^\alpha
\chi(v)\nabla v) + ku-\mu u^2 & x\in \Omega, t>0, \\ v_{t} = \Delta v-vu & x\in
\Omega, t>0,\\ \end{cases} \end{equation*} $\Omega$ being a bounded and smooth
domain of $\mathbb{R}^n$, $n\geq 1$, and where $m,k \in \mathbb{R}$, $\mu>0$
and $\alpha < \frac{m+1}{2}$. For any $v\geq 0$ the chemotactic sensitivity
function is assumed to behave as the prototype $\chi(v) =
\frac{\chi_0}{(1+av)^2}$, with $a\geq 0$ and $\chi_0>0$. We prove that for
nonnegative and sufficiently regular initial data $u(x,0)$ and $v(x,0),$ the
corresponding initial-boundary value problem admits a global bounded classical
solution provided $\mu$ is large enough.
