Traveling waves of a full parabolic attraction-repulsion chemotaxis systems with logistic sources

Abstract

In this paper, we study traveling wave solutions of the chemotaxis systems \begin{equation} \begin{cases} u_{t}=Δu -χ_1\nabla( u\nabla v_1)+χ_2 \nabla(u\nabla v_2 )+ u(a -b u), \qquad \ x\in\mathbb{R} \\ τ\partial_tv_1=(Δ- λ_1 I)v_1+ μ_1 u, \qquad \ x\in\mathbb{R}, \\ τ\partial v_2=(Δ- λ_2 I)v_2+ μ_2 u, \qquad \ \ x\in\mathbb{R}, \end{cases} (0.1) \end{equation} where $\tau >0,{\chi }_i>0,{\lambda }_i>0,{\mu }_i>0$ ($i=1,2$) and $a>0,b>0$ are constants, and $N$ is a positive integer. Under some appropriate conditions on the parameters, we show that there exist two positive constant $0<c^{\ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)<c^{\ast \ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)$ such that for every $c^{\ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)\le c<c^{\ast \ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)$, $(0.1)$ 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 $$\underset{z\to \infty }{\lim }\frac{U(z)}{e^{-\mu z}}=1,$$ where $\mu \in (0,\sqrt{a})$ is such that $c=c_{\mu }:=\mu +\frac{a}{\mu }$. Moreover, $$\underset{({\chi }_1,{\chi }_2)\to (0^{+},0^{+}))}{\lim }{c}^{\ast \ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=\infty$$ and $$\underset{({\chi }_1,{\chi }_2)\to (0^{+},0^{+}))}{\lim }{c}^{\ast }(\tau ,{\chi }_1,{\mu }_1,{\lambda }_1,{\chi }_2,{\mu }_2,{\lambda }_2)=c_{{\tilde{\mu }}^{\ast }},$$ where ${\tilde{\mu }}^{\ast }=\min \{\sqrt{a},\sqrt{\frac{{\lambda }_1+\tau a}{(1-\tau {)}_{+}}},\sqrt{\frac{{\lambda }_2+\tau a}{(1-\tau {)}_{+}}}\}$. We also show that $(0.1)$ 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 $c<2\sqrt{a}$.

arXiv admin note: substantial text overlap with arXiv:1610.05215, arXiv:1701.02633, arXiv:1609.05387