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