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In this paper, we consider a three-dimensional chemotaxis-Stokes system 0.1 $$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),&\quad&x\in \Omega , t>0,\\&c_t+u\cdot \nabla c=\Delta c-nf(c),&\quad&x\in \Omega ,t>0,\\&u_t+\nabla P=\Delta u+n\nabla \phi ,&\quad&x\in \Omega ,t>0,\\&\nabla \cdot u=0,&\quad&x\in \Omega ,t>0,\\&(\nabla n^m- n S(x,n,c)\cdot \nabla c)\cdot \nu =\frac{\partial c}{\partial \nu }=0,u=0,&\quad&x\in \partial \Omega ,t>0,\\&n(x,0)=n_0(x),c(x,0)=c_0(x),u(x,0)=u_0(x),&\quad&x\in \Omega \end{aligned} \right. \end{aligned}$$ in a bounded domain $$\Omega \subset {\mathbb {R}}^3$$ with smooth boundary, where $$\phi $$ , f and S are given functions with values in $$\Omega $$ , $$[0,\infty )$$ and $${\mathbb {R}}^{3\times 3}$$ , respectively, under the no-flux boundary condition for n, c and Dirichlet boundary condition for u. It is also required that $$f\in C^1([0,\infty ))$$ is locally bounded in $$[0,\infty )$$ , that S satisfies $$|S(x,n,c)|\le n^{l-2}S_0(c)$$ with some nondecreasing $$S_0:[0,\infty )\rightarrow [0,\infty )$$ , and that m fulfills 0.2 $$\begin{aligned} m>\frac{5}{3}l-\frac{20}{9}\quad \text {and}\quad m>-\frac{3}{4}l+\frac{21}{8}\quad \text {with}\quad \frac{25}{12}\ge l>2. \end{aligned}$$ Then, with any reasonably regular initial data, the corresponding initial-boundary problem for (0.1) processes a global in time and bounded solution. This result extends the previous global boundedness result with $$m>m_*(l)$$ , where $$\begin{aligned} m_*(l)=\left\{ \begin{aligned}&l-\frac{5}{6},&\quad&\text {if }\,\frac{31}{12}\ge l>2,\\&\frac{7}{5}l-\frac{28}{15},&\quad&\text {if }\,l>\frac{31}{12}, \end{aligned} \right. \end{aligned}$$ ([15]) since both $$\begin{aligned} l-\frac{5}{6}\ge \frac{5}{3}l-\frac{20}{9} \end{aligned}$$ and $$\begin{aligned} l-\frac{5}{6}\ge -\frac{3}{4}l+\frac{21}{8} \end{aligned}$$ hold on $$\frac{25}{12}\ge l>2$$ .

This work considers a pursuit-evasion model\begin{document}$\begin{equation} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &w_t = \Delta w-w+v,\\ &z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1) \end{equation}$\end{document}with positive parameters \begin{document}$ \chi $\end{document}, \begin{document}$ \xi $\end{document}, \begin{document}$ \mu $\end{document}, \begin{document}$ \lambda $\end{document}, \begin{document}$ a $\end{document} and \begin{document}$ b $\end{document} in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} (\begin{document}$ N $\end{document} is the dimension of the space) with smooth boundary. We prove that if \begin{document}$ a and \begin{document}$ \frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}>\max\{\chi,\xi\} $\end{document}, (1) possesses a global bounded classical solution with a positive constant \begin{document}$ C_{\frac{N}{2}+1} $\end{document} corresponding to the maximal Sobolev regularity. Moreover, it is shown that if \begin{document}$ b\mu, the solution (\begin{document}$ u,v,w,z $\end{document}) converges to a spatially homogeneous coexistence state with respect to the norm in \begin{document}$ L^\infty(\Omega) $\end{document} in the large time limit under some exact smallness conditions on \begin{document}$ \chi $\end{document} and \begin{document}$ \xi $\end{document}. If \begin{document}$ b\mu>\lambda $\end{document}, the solution converges to (\begin{document}$ \mu,0,0,\mu $\end{document}) with respect to the norm in \begin{document}$ L^\infty(\Omega) $\end{document} as \begin{document}$ t\rightarrow \infty $\end{document} under some smallness assumption on \begin{document}$ \chi $\end{document} with arbitrary \begin{document}$ \xi $\end{document}.