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

Abstract

<p style='text-indent:20px;'>This work deals with a Neumann initial-boundary value problem for a two-species predator-prey chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^n \,\,(n = 2,3) $\end{document}</tex-math></inline-formula> with smooth boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where the parameters <inline-formula><tex-math id="M3">\begin{document}$ d_1, d_2, d_3,\chi, \xi,\lambda,\mu,\alpha,\beta_1,\beta_2, a, b $\end{document}</tex-math></inline-formula> are positive. It is shown that for any appropriate regular initial date <inline-formula><tex-math id="M4">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ v_0 $\end{document}</tex-math></inline-formula>, the corresponding system possesses a global bounded classical solution in <inline-formula><tex-math id="M6">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, and also in <inline-formula><tex-math id="M7">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> being sufficiently small. Moreover, by constructing some suitable functionals, it is proved that if <inline-formula><tex-math id="M9">\begin{document}$ b\lambda<\mu $\end{document}</tex-math></inline-formula> and the parameters <inline-formula><tex-math id="M10">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> are sufficiently small, then the solution <inline-formula><tex-math id="M12">\begin{document}$ (u,v,w) $\end{document}</tex-math></inline-formula> of this system converges to <inline-formula><tex-math id="M13">\begin{document}$ (\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)}) $\end{document}</tex-math></inline-formula> exponentially as <inline-formula><tex-math id="M14">\begin{document}$ t\rightarrow \infty $\end{document}</tex-math></inline-formula>; if <inline-formula><tex-math id="M15">\begin{document}$ b\lambda\geq \mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> is sufficiently small and <inline-formula><tex-math id="M17">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is arbitrary, then the solution <inline-formula><tex-math id="M18">\begin{document}$ (u,v,w) $\end{document}</tex-math></inline-formula> converges to <inline-formula><tex-math id="M19">\begin{document}$ (\lambda,0,\frac{\beta_1\lambda}{\alpha}) $\end{document}</tex-math></inline-formula> with exponential decay when <inline-formula><tex-math id="M20">\begin{document}$ b\lambda> \mu $\end{document}</tex-math></inline-formula>, and with algebraic decay when <inline-formula><tex-math id="M21">\begin{document}$ b\lambda = \mu $\end{document}</tex-math></inline-formula>.</p>