This article primarily investigates the convergence rates of solutions to the chemotaxis-Stokes system, which consists of two distinct species represented by the following system \(\begin{aligned} {\left\{ \begin{array}{ll} u_{1t}+w\cdot \nabla u_{1}=\Delta u_{1}-\chi \nabla \cdot \bigl (u_{1}\nabla v_{1}\bigl )+u_{1}\bigl (\lambda _{1}-\mu _{1}u_{1}+au_{2}\bigl ),& \quad x\in \Omega ,t>0,\\ u_{2t}+w\cdot \nabla u_{2}=\Delta u_{2}+\xi \nabla \cdot \bigl (u_{2}\nabla v_{2}\bigl )+u_{2}\bigl (\lambda _{2}-\mu _{2}u_{2}-bu_{1}\bigl ),& \quad x\in \Omega ,t>0,\\ v_{1t}+w\cdot \nabla v_{1}=\Delta v_{1}-v_{1}+u_{2},& \quad x\in \Omega ,t>0,\\ w\cdot \nabla v_{2}=\Delta v_{2}-v_{2}+u_{1},& \quad x\in \Omega ,t>0,\\ w_{t}+\nabla P=\Delta w+(u_{1}+u_{2})\nabla \phi ,& \quad x\in \Omega ,t>0,\\ \nabla \cdot w=0,& \quad x\in \Omega ,t>0, \end{array}\right. } \end{aligned}\) subject to homogeneous Neumann boundary conditions within a smooth bounded domain \(\Omega \subset \mathbb {R}^3\) . By formulating a suitable energy functional, the following results are established: \(\bullet \) If \(\lambda _{2}\mu _{1}>b\lambda _{1}\) and both \(\mu _{1}\) and \(\mu _{2}\) are sufficiently large, it is demonstrated that for any global bounded solution originating from adequately regular initial data with \(u_{10}, u_{20}\not \equiv 0\) , \(\begin{aligned} (u_{1},u_{2},v_{1},v_{2},w)(\cdot ,t)\rightarrow (u_{1\star },u_{2\star },u_{2\star },u_{1\star },0) \quad \text{ uniformly } \text{ in }~\Omega ~\text{ as }~t\rightarrow \infty , \end{aligned}\) where \((u_{1\star },u_{2\star },u_{2\star },u_{1\star },0)\) denotes the unique positive spatially homogeneous equilibrium of this system. \(\bullet \) If \(\lambda _{2}\mu _{1}\le b\lambda _{1}\) and \(\mu _{1}\) is sufficiently large, then all global bounded solutions with reasonably smooth initial data satisfying \(u_{10}\not \equiv 0\) exhibit the following behavior: \( (u_{1},u_{2},v_{1},v_{2},w)(\cdot ,t) \rightarrow \left( \frac{\lambda _{1}}{\mu _{1}},0,0,\frac{\lambda _{1}}{\mu _{1}},0\right) \quad \text {uniformly on} \ \Omega \ \text {as} \ t \rightarrow \infty . \)
