In this paper, a fully parabolic predator–prey chemotaxis model with indirect signal production \(\begin{aligned} {\left\{ \begin{array}{ll}u_t=d_1\Delta u+\chi _1\nabla \cdot (u\nabla z)+\mu _{1}u(1-u-e_{1}v),& x\in \Omega ,t>0,\\ v_t=d_2\Delta v-\chi _2\nabla \cdot (v\nabla z)+\mu _{2}v(1+e_{2}u-v),& x\in \Omega ,t>0,\\ w_t=\Delta w-w+u+v,& x\in \Omega ,t>0,\\ z_t=\Delta z-z+w,& x\in \Omega ,t>0 \end{array}\right. } \end{aligned}\) under the homogeneous Neumann boundary conditions in a bounded smooth domain \(\Omega \subset \mathbb {R}^n(n\ge 1)\) with smooth boundary \(\partial \Omega \) is examined, where \(d_1,d_2,\chi _1,\chi _2 > 0\) , \(\mu _{1},\mu _{2},e_{1},e_{2}\ge 0\) . When \(n \le 3\) , by the method of some priori estimates and semigroup technique, we demonstrate that the model has a globally bounded classical solution for all appropriate regular initial data. Additionally, the convergence of the solution is asserted by constructing Lyapunov functions. (i) If \(e_{1}<1\) , \(\frac{\mu _{1}}{\chi _1^{2}}\) and \(\frac{\mu _{2}}{\chi _2^{2}}\) are sufficiently large, then the solution \(\left( u,v,w,z\right) \) with \(u, v \ge (\not \equiv ) 0\) exponentially converges to a unique positive equilibrium point; (ii) If \( e_{1}>1 \) , \(\frac{\mu _{2}}{\chi _2^{2}}\) is sufficiently large, then the solution \(\left( u,v,w,z\right) \) satisfying \(v \ge (\not \equiv ) 0\) exponentially converges to the semi-trivial equilibrium point; (iii) If \(e_{1}=1\) , \(\frac{\mu _{2}}{\chi _2^{2}}\) is sufficiently large, then the solution \(\left( u,v,w,z\right) \) algebraically converges to the semi-trivial equilibrium point.
