In this paper, we consider an indirect pursuit-evasion system with nonlinear signal-dependent diffusion and sensitivity: \(\begin{aligned} \left\{ \begin{aligned}&u_{t}=\nabla \cdot (D_{1}(z)\nabla u)-\chi \nabla \cdot (S_{1}(z)u\nabla z)+u(\lambda _{1}-\mu _{1}u+av), & x\in \Omega ,t>0,\\&v_{t}=\nabla \cdot (D_{2}(w)\nabla v)+\xi \nabla \cdot (S_{2}(w)v\nabla w)+v(\lambda _{2}-\mu _{2} v-bu), & x\in \Omega ,t>0,\\&0=\Delta w+\alpha u-\beta w, & x\in \Omega ,t>0,\\&0=\Delta z+\gamma v-\delta z, & x\in \Omega ,t>0 \end{aligned}\right. \end{aligned}\) in a bounded smooth domain \(\Omega \subset \mathbb {R}^{2}\) with homogeneous Neumann boundary conditions. Here, \(\chi \) , \(\xi \) , \(\lambda _{1}\) , \(\lambda _{2}\) , \(\mu _{1}\) , \(\mu _{2}\) , a, b, \(\alpha \) , \(\beta \) , \(\gamma \) , \(\delta \) are positive constants. Although the global existence results of related fully parabolic models have been proposed (see Wan-Zheng-Shan (Wan in J Evol Equ 23:78, 2023) and Wan-Zheng (Wan in Nonlinear Anal Real World Appl 82:104234, 2024)), there is no result for a parabolic-elliptic system. We first demonstrate that this system possesses a unique global and bounded classical solution if \(\lim \limits _{z\rightarrow \infty }\frac{(S_{1}(z))^{2}}{D_{1}(z)}\) and \(\lim \limits _{w\rightarrow \infty }\frac{(S_{2}(w))^{2}}{D_{2}(w)}\) exist. Notably, this result is particularly superior to the results in Wan (J Evol Equ 23:78, 2023), Xiang (Nonlinear Anal Real World Appl 71:103797, 2023), Zeng (J Nonlinear Math Phys 31:12, 2024). Furthermore, the large-time behavior of solutions to this system is also investigated.
