  In this paper, we study a nutrient-taxis model with porous medium slow
diffusion \begin{align*} \left\{ \begin{aligned} &u_t=\Delta
u^m-\chi\nabla\cdot(u\nabla v)+\xi uv-\rho u, \\ &v_t-\Delta v=-vu+\mu v(1-v),
\end{aligned}\right. \end{align*} in a bounded domain $\Omega\subset \mathbb
R^3$ with zero-flux boundary condition. It is shown that for any
$m>\frac{11}4-\sqrt 3$,the problem admits a global weak solution for any large
initial datum. We divide the study into three cases,(i) $\xi\mu=0, \rho\ge 0$;
(ii) $\xi\mu\rho>0$; (iii) $\xi\mu>0$, $\rho=0$. In particular, for Case (i)
and Case (ii), the global solutions are uniformly bounded. Subsequently, the
large time behavior of these global bounded solutions are also discussed. At
last, we also extend the results to the coupled chemotaxis-Stokes system.
Important progresses for chemotaxis-Stokes system with $m>\frac 76$, $m>\frac
87$ and $m>\frac 98$ have been carried out respectively by \cite{W2, TW2, W3},
but leave a gap for $1<m\le \frac98$. Our result for chemotaxis-Stokes system
supplements part of the gap $(\frac{11}4-\sqrt 3, \frac 98)$. Here
$\frac{11}4-\sqrt 3\approx 1.018$.
