Global existence and stability in a two-species chemotaxis system

Abstract

<p style='text-indent:20px;'>This paper deals with the following two-species chemotaxis system <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\left\{ \begin{array}{*{35}{l}} \ \ {{u}_{t}}=\Delta u-{{\chi }_{1}}\nabla \cdot (u\nabla v)+{{\mu }_{1}}u(1-u-{{a}_{1}}w), &amp; x\in \Omega ,t&gt;0, &amp; \\ \ \ {{v}_{t}}=\Delta v-v+h(w), &amp; x\in \Omega ,t&gt;0, &amp; \\ \ \ {{w}_{t}}=\Delta w-{{\chi }_{2}}\nabla \cdot (w\nabla z)+{{\mu }_{2}}w(1-w-{{a}_{2}}u), &amp; x\in \Omega ,t&gt;0, &amp; \\ \ \ {{z}_{t}}=\Delta z-z+h(u),&amp; x\in \Omega ,t&gt;0, &amp; \\\end{array} \right.$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded domain <inline-formula><tex-math id="M393">\begin{document}$Ω\subset\mathbb{R}^{n}$\end{document}</tex-math></inline-formula> with smooth boundary. The parameters in the system are positive and the signal production function <i>h</i> is a prescribed <i>C</i>¹-regular function. The main objectives of this paper are two-fold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium <inline-formula><tex-math id="M394">\begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document}</tex-math></inline-formula> may be globally attractive in the weak competition case (i.e., <inline-formula><tex-math id="M395">\begin{document}$0 &lt; a_{1}, a_{2} &lt; 1$\end{document}</tex-math></inline-formula>), while the constant stationary solution (0, <i>h</i>(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e., <inline-formula><tex-math id="M396">\begin{document}$a_{1}&gt;1&gt;a_{2}&gt;0$\end{document}</tex-math></inline-formula>). In the fully strong competition case (i.e. <inline-formula><tex-math id="M397">\begin{document}$a_{1}, a_{2}&gt;1$\end{document}</tex-math></inline-formula>), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, <i>h</i>(1), 1, 0) and (1, 0, 0, <i>h</i>(1)) and the instability of the positive spatially homogeneous <inline-formula><tex-math id="M398">\begin{document}$(u_{*}, v_{*}, w_{*}, z_{*})$\end{document}</tex-math></inline-formula>. The matter which species ultimately wins out depends crucially on the starting advantage each species has.