Global stability in a multi-dimensional predator-prey system with prey-taxis
Dan Li
Abstract
<p style='text-indent:20px;'>This paper studies the predator-prey systems with prey-taxis <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} { \label{1.1}} \left\{ \begin{array}{llll} u_{t} = \Delta u-\chi\nabla\cdot(u\nabla v)+\gamma uv-\rho u, \\ v_{t} = \Delta v-\xi uv+\mu v(1-v), \ \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>in a bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^{n} $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M2">\begin{document}$ (n = 2, 3) $\end{document}</tex-math></inline-formula> with Neumann boundary conditions, where the parameters <inline-formula><tex-math id="M3">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \rho $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> are positive. It is shown that the two-dimensional system possesses a unique global-bounded classical solution. Furthermore, we use some higher-order estimates to obtain the classical solutions with uniform-in-time bounded for suitably small initial data. Finally, we establish that the solution stabilizes towards the prey-only steady state <inline-formula><tex-math id="M8">\begin{document}$ (0, 1) $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M9">\begin{document}$ \rho>\gamma $\end{document}</tex-math></inline-formula> and towards the co-existence steady state <inline-formula><tex-math id="M10">\begin{document}$ (\frac{\mu(\gamma-\rho)}{\xi\rho}, \frac{\rho}{\gamma}) $\end{document}</tex-math></inline-formula> if <inline-formula><tex-math id="M11">\begin{document}$ \gamma>\rho $\end{document}</tex-math></inline-formula> under some conditions in the norm of <inline-formula><tex-math id="M12">\begin{document}$ L^{\infty}(\Omega) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M13">\begin{document}$ t\rightarrow\infty $\end{document}</tex-math></inline-formula>.