This paper is to study the following chemotaxis model \(\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta (\varphi (v)u)+ru-\mu u^\alpha ,&x\in \Omega ,t>0,\\&v_t=\Delta v-u^\beta v,&x\in \Omega ,t>0, \end{aligned} \right. \end{aligned}\) under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^n (n\ge 1)\) , where the parameters \(\mu \) , \(\beta >0\) , \(r\in \mathbb {R}\) and \(\alpha >1\) . The motility function \(\varphi \) satisfies \(\varphi \in C^3([0,\infty ))\) with \(\varphi (s)>0\) for all \(s\ge 0\) , and the purpose of this paper is to improve the conditions regarding \(\alpha \) and \(\beta \) used in [28] to ensure the existence of global bounded classical solutions for the system. The results indicate that when \(\beta <\min \bigl \{\alpha -1, \frac{n+1}{n+2}\alpha \bigr \}\) , the system has a global bounded classical solution. In addition, it was found that in the critical case where \(\beta \) satisfies the above condition, the global bounded classical solution can be guaranteed by a sufficiently large \(\mu \) . Finally, we obtained that the solution will converge to a constant equilibrium as time approaches infinity.
