This paper explores the long-time asymptotic behavior of solutions for the following system when \(N = 2,\kappa \in {\mathbb {R}}\) or \(N = 3,\kappa = 0\) , * \(\begin{aligned} \left\{ \begin{array}{l} n_{t}+u\cdot \nabla n =\Delta n -\chi \nabla \cdot ( n \nabla c)+\rho n -\mu n^2,\quad x\in \Omega , t>0, \\ u\cdot \nabla c=\Delta c-c+n,\quad x\in \Omega , t>0, \\ u_t+\nabla p+\kappa (u\cdot \nabla )u=\Delta u+n\nabla \phi ,\quad x\in \Omega , t>0, \\ \nabla \cdot u=0,\quad x\in \Omega , t>0, \end{array}\right. \end{aligned}\) with no-flux boundary conditions for n and c, and no-slip boundary condition for u, in a bounded domain \(\Omega \subseteq \mathbb {R}^N(2\le N\le 3)\) with a smooth boundary, where \( \phi \in W^{2,\infty }(\Omega ),\) and \(\chi > 0, \rho \in \mathbb {{R}},\) \(\mu > 0\) are given parameters. We conclude that when \(\rho >0\) and \(\mu >\frac{\chi \sqrt{\rho }}{4},\) the corresponding solution of the system \((*)\) decays to \((\frac{\rho }{\mu }, \frac{\rho }{\mu }, 0)\) exponentially; however, if \(\rho \le 0\) , we can ensure that the components n and c of any non-trivial global bounded solution to system \((*)\) approach zero in either of the spaces \(L^1(\Omega )\) and \(L^\infty (\Omega )\) , which can be controlled by appropriate multiples of \(\frac{1}{t+1}\) and \(e^{\rho t}\) from above and below, respectively. To the best of our knowledge, this is the first attempt to study the precise asymptotic behavior of the system.
