<p style='text-indent:20px;'>In this paper, we deal with the following indirect pursuit-evasion model</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+ u(\lambda-u+av), \quad x\in \Omega, t&gt;0, \\ v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+ v(\mu-v-bu), \quad x\in \Omega, t&gt;0, \\ { }{0 = \Delta w- w+v}, \quad x\in \Omega, t&gt;0, \\ { }{0 = \Delta z- z+ u}, \quad x\in \Omega, t&gt;0, \\ \end{array}\right. \ \ \ \ \ \ \ \ \ \ (\star) \end{align} $ \end{document} </tex-math></disp-formula></p> <p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^N(N\geq1) $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \chi, \xi, \lambda, \mu $\end{document}</tex-math></inline-formula> as well as <inline-formula><tex-math id="M4">\begin{document}$ a $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ b $\end{document}</tex-math></inline-formula> are positive parameters. This system is used to achieve some insight into possible dynamical properties of pursuit-evasion processes, in which the respective tactic movements are oriented along gradients of some indirectly produced stimuli, rather than following individuals directly. One main purpose of the present paper is to remove the restriction of <inline-formula><tex-math id="M6">\begin{document}$ N\leq3 $\end{document}</tex-math></inline-formula>. Indeed, by using a iteration argument combined with suitable a priori estimates, we conclude that for any <inline-formula><tex-math id="M7">\begin{document}$ N\geq1 $\end{document}</tex-math></inline-formula>, an associated initial-boundary value problem <inline-formula><tex-math id="M8">\begin{document}$ (\star) $\end{document}</tex-math></inline-formula> admits a unique global bounded classical solution. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \chi&lt;\left\{\begin{array}{ll} 4\sqrt{\frac{a(1+ab)}{b(\lambda+a\mu)}}, \quad\; \; \mbox{if}\; \; \lambda&gt;b\mu, \\ 4\sqrt{\frac{a}{b\lambda}}, \quad\; \; \mbox{if}\; \; \lambda\leq b\mu\ \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and <inline-formula><tex-math id="M9">\begin{document}$ \xi&lt;4\sqrt{\frac{b(1+ab)}{a(\mu-b\lambda)_+}} $\end{document}</tex-math></inline-formula>, the corresponding solution <inline-formula><tex-math id="M10">\begin{document}$ (u, v, w, z) $\end{document}</tex-math></inline-formula> of the system decays to <inline-formula><tex-math id="M11">\begin{document}$ (u_*, v_*, v_*, u_*) $\end{document}</tex-math></inline-formula> exponentially (or algebraically), where</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ u_* = \left\{\begin{array}{ll} \frac{\lambda+a\mu}{1+ab}, \quad\; \; \mbox{if}\; \; \lambda&gt;b\mu, \\ \lambda, \quad\; \; \mbox{if}\; \; \lambda\leq b\mu\ \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and <inline-formula><tex-math id="M12">\begin{document}$ v_* = \frac{(\lambda-b\mu)_+}{1+ab} $\end{document}</tex-math></inline-formula>. To the best of our knowledge, there is the first result on convergence rates of solutions of the system.</p>