Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source

Abstract

In this paper, we study traveling wave solutions of the chemotaxis system \begin{document}$ \begin{matrix} \left\{ \begin{array}{*{35}{l}} {{u}_{t}} = \Delta u-{{\chi }_{1}}\nabla (u\nabla {{v}_{1}})+{{\chi }_{2}}\nabla (u\nabla {{v}_{2}})+u(a-bu),\qquad x\in \mathbb{R} \tau {{\partial }_{t}}{{v}_{1}} = (\Delta -{{\lambda }_{1}}I){{v}_{1}}+{{\mu }_{1}}u,\qquad x\in \mathbb{R}, \tau \partial {{v}_{2}} = (\Delta -{{\lambda }_{2}}I){{v}_{2}}+{{\mu }_{2}}u,\qquad x\in \mathbb{R}, \\\end{array} \right. & (0.1) \\\end{matrix} $\end{document} where \begin{document}$ \tau>0,\chi_{i}> 0,\lambda_i> 0, \mu_i>0 $\end{document} ( \begin{document}$ i = 1,2 $\end{document} ) and \begin{document}$ a>0, b> 0 $\end{document} are constants. Under some appropriate conditions on the parameters, we show that there exist two positive constant \begin{document}$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) such that for every \begin{document}$ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c , (0.1) has a traveling wave solution \begin{document}$ (u,v_1,v_2)(x,t) = (U,V_1,V_2)(x-ct) $\end{document} connecting \begin{document}$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $\end{document} and \begin{document}$ (0,0,0) $\end{document} satisfying \begin{document}$ \lim\limits_{z\to \infty}\frac{U(z)}{e^{-\mu z}} = 1, $\end{document} where \begin{document}$ \mu\in (0,\sqrt a) $\end{document} is such that \begin{document}$ c = c_\mu: = \mu+\frac{a}{\mu} $\end{document} . Moreover, \begin{document}$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = \infty $\end{document} and \begin{document}$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = c_{\tilde{\mu}^*}, $\end{document} where \begin{document}$ \tilde{\mu}^* = {\min\{\sqrt{a}, \sqrt{\frac{\lambda_1+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_2+\tau a}{(1-\tau)_{+}}}\}} $\end{document} . We also show that (1) has no traveling wave solution connecting \begin{document}$ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $\end{document} and \begin{document}$ (0,0,0) $\end{document} with speed \begin{document}$ c .