This paper deals with a fully parabolic predator–prey chemotaxis model with indirect signal consumption \( {\left\{ \begin{array}{ll} u_t = d_1\Delta u + \chi _1\nabla \cdot (u\nabla z) + \mu _1 u(1 - u - e_1 v), & x \in \Omega , t> 0, \\ v_t = d_2\Delta v - \chi _2\nabla \cdot (v\nabla z) + \mu _2 v(1 - v + e_2 u ), & x \in \Omega , t> 0, \\ w_t = \Delta w - w + u + v, & x \in \Omega , t> 0, \\ z_t = \Delta z - wz, & x \in \Omega , t > 0 \end{array}\right. } \) under homogeneous Neumann boundary conditions in a bounded smooth domain \(\Omega \subset \mathbb {R}^n (n \ge 1)\) , where \(d_1, d_2, \chi _1, \chi _2,\) \(\mu _1, \mu _2, e_1, e_2 > 0\) . For spatial dimensions \(n \le 3\) , based on some a priori estimates and semigroup techniques, we prove that the model possesses a global bounded classical solution for all sufficiently regular initial data. Furthermore, the convergence of the solution is asserted by constructing appropriate Lyapunov functionals. (i) if \( e_1, e_2 < 1 \) , then the global bounded classical solution (u, v, w, z) exponentially converges to \(\left( \frac{1 - e_1}{1 + e_1 e_2}, \frac{1 + e_2}{1 + e_1 e_2}, \frac{2 - e_1 + e_2}{1 + e_1 e_2}, 0 \right) \) ; (ii) if \( e_2< 1 < e_1\) and \(e_1 e_2 <1\) , then the global bounded classical solution (u, v, w, z) exponentially converges to (0, 1, 1, 0) ; (iii) if \( e_2 < 1 = e_1\) , then the global bounded classical solution (u, v, w, z) algebraically converges to (0, 1, 1, 0) .
