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In a bounded domain   \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document}   with smooth boundary, this work considers the indirect pursuit-evasion model\begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{l}u_t = \Delta u -\chi\nabla \cdot(u \nabla w) +u(\lambda -u+a v), \\v_t = \Delta v +\xi \nabla \cdot(v\nabla z) +v(\mu-v-b u), \\0 = \Delta w -w +v, \\0 = \Delta z -z +u, \end{array} \right. \end{eqnarray*} $\end{document}with positive parameters   \begin{document}$ \chi, \xi, \lambda, \mu $\end{document}  ,   \begin{document}$ a $\end{document}   and   \begin{document}$ b $\end{document}  .    It is firstly asserted that when   \begin{document}$ n\le 3 $\end{document}  , for any given suitably regular initial data the corresponding homogeneous Neumann initial-boundary problem admits a global and bounded smooth solution. Moreover, it is shown that if   \begin{document}$ b\lambda   and under some explicit smallness conditions on   \begin{document}$ \chi $\end{document}   and   \begin{document}$ \xi $\end{document}  , any nontrival bounded classical solution converges to the spatially homogeneous coexistence state in the large time limit; if   \begin{document}$ b\lambdau003e\mu $\end{document}  , however, then under an explicit smallness assumption on   \begin{document}$ \chi $\end{document}   but without any restriction on   \begin{document}$ \xi $\end{document}  , any bounded classical solution   \begin{document}$ (u, v) $\end{document}   with   \begin{document}$ u\not\equiv 0 $\end{document}   stabilizes to   \begin{document}$ (\lambda, 0) $\end{document}   as   \begin{document}$ t\to \infty $\end{document}  .

Large time behavior in a predator-prey system with indirect pursuit-evasion interaction