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

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  • 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 $.

    Mathematics Subject Classification: Primary: 5B40, 35B65, 35K57, 35Q92, 92C17.


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