Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model

This work deals with a Neumann initial-boundary value problem for a two-species predator-prey chemotaxis system

$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\chi\nabla\cdot(u\nabla w)+u(\lambda-u+av),\quad &x\in \Omega,\quad t>0,\\ v_t = d_2\Delta v+\xi\nabla\cdot(v\nabla w)+v(\mu-v-bu),\quad &x\in \Omega,\quad t>0,\\ 0 = d_3\Delta w-\alpha w+\beta_1 u+ \beta_2 v,\quad &x\in\Omega,\quad t>0,\\ \end{array} \right. \end{eqnarray*} $

in a bounded domain $ \Omega\subset \mathbb{R}^n \,\,(n = 2,3) $ with smooth boundary $ \partial\Omega $, where the parameters $ d_1, d_2, d_3,\chi, \xi,\lambda,\mu,\alpha,\beta_1,\beta_2, a, b $ are positive. It is shown that for any appropriate regular initial date $ u_0 $, $ v_0 $, the corresponding system possesses a global bounded classical solution in $ n = 2 $, and also in $ n = 3 $ for $ \chi $ being sufficiently small. Moreover, by constructing some suitable functionals, it is proved that if $ b\lambda<\mu $ and the parameters $ \chi $ and $ \xi $ are sufficiently small, then the solution $ (u,v,w) $ of this system converges to $ (\frac{\lambda+a\mu}{1+ab}, \frac{\mu-b\lambda}{1+ab}, \frac{\beta_1(\lambda+a\mu)+\beta_2(\mu-b\lambda)}{\alpha(1+ab)}) $ exponentially as $ t\rightarrow \infty $; if $ b\lambda\geq \mu $ and $ \chi $ is sufficiently small and $ \xi $ is arbitrary, then the solution $ (u,v,w) $ converges to $ (\lambda,0,\frac{\beta_1\lambda}{\alpha}) $ with exponential decay when $ b\lambda> \mu $, and with algebraic decay when $ b\lambda = \mu $.