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

In this paper, we study traveling wave solutions of the chemotaxis system

$ \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} $

where $ \tau>0,\chi_{i}> 0,\lambda_i> 0,\ \mu_i>0 $ ($ i = 1,2 $) and $ \ a>0,\ b> 0 $ are constants. Under some appropriate conditions on the parameters, we show that there exist two positive constant $ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) $ such that for every $ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2, \lambda_2) $, (0.1) has a traveling wave solution $ (u,v_1,v_2)(x,t) = (U,V_1,V_2)(x-ct) $ connecting $ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $ and $ (0,0,0) $ satisfying

$ \lim\limits_{z\to \infty}\frac{U(z)}{e^{-\mu z}} = 1, $

where $ \mu\in (0,\sqrt a) $ is such that $ c = c_\mu: = \mu+\frac{a}{\mu} $. Moreover,

$ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = \infty $

and

$ \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}^*}, $

where $ \tilde{\mu}^* = {\min\{\sqrt{a}, \sqrt{\frac{\lambda_1+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_2+\tau a}{(1-\tau)_{+}}}\}} $. We also show that (1) has no traveling wave solution connecting $ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $ and $ (0,0,0) $ with speed $ c<2\sqrt{a} $.