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

In a bounded domain $ \Omega\subset \mathbb{R}^n $ with smooth boundary, this work considers the indirect pursuit-evasion model

$ \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*} $

with positive parameters $ \chi, \xi, \lambda, \mu $, $ a $ and $ b $.

It is firstly asserted that when $ n\le 3 $, 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 $ b\lambda<\mu $ and under some explicit smallness conditions on $ \chi $ and $ \xi $, any nontrival bounded classical solution converges to the spatially homogeneous coexistence state in the large time limit; if $ b\lambda>\mu $, however, then under an explicit smallness assumption on $ \chi $ but without any restriction on $ \xi $, any bounded classical solution $ (u, v) $ with $ u\not\equiv 0 $ stabilizes to $ (\lambda, 0) $ as $ t\to \infty $.