Existence of Traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

TL;DR

This paper proves the existence of traveling wave solutions in a chemotaxis system with logistic growth, identifying minimal wave speeds and analyzing how these speeds depend on system parameters.

Contribution

It establishes the existence of traveling wave solutions for speeds above a critical value and characterizes how this critical speed varies with chemotactic sensitivities and other parameters.

Findings

Existence of traveling waves for speeds greater than a critical threshold

Critical wave speed approaches specific limits as chemotactic sensitivities tend to zero

Wave profile asymptotically matches exponential decay with rate depending on system parameters

Abstract

We study traveling wave solutions of the following chemotaxis systems,$$\begin{cases}u_t=\Delta u-\chi_1\nabla(u\nabla v_1)+\chi_2\nabla(u\nabla v_2)+u(a-bu),\ x\in\mathbb{R}^N\\ 0=\Delta v_1-\lambda_1v_1+\mu_1u,\ x\in\mathbb{R}^N,\\ 0=\Delta v_2-\lambda_2v_2+\mu_2u,\ x\in\mathbb{R}^N,\end{cases}$$where $u(x,t), v_1(x,t)$ and $v_2(x,t)$ represent the population densities of a mobile species, a chemoattractant, and a chemo-repulsion, respectively. In an earlier work, we proved that there is a constant $K\geq0$ such that if $b+\chi_2\mu_2>\chi_1\mu_1+K$, then the steady solution $(\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2})$ is asymptotically stable with respect to positive perturbations. In this paper, we prove that if $b+\chi_2\mu_2>\chi_1\mu_1+K$, then there exist a number $c^*(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\geq 2\sqrt a$ such that for every $c\in ( c^*(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) , \infty)$ and $\xi\in S^{N-1}$, the system has a traveling wave solution $(u,v_1,v_2)=(U(x\cdot\xi-ct),V_1(x\cdot\xi-ct),V_2(x\cdot\xi-ct))$ with speed $c$ connecting the constant solutions $(\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2})$ and $(0,0,0)$, and it does not have such traveling wave solutions of speed less than $2\sqrt a$. Moreover we prove that$$\lim_{(\chi_1,\chi_2)\to(0^+,0^+)}c^{*}(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)=\begin{cases}2\sqrt a\ \text{if}\ a\leq \min\{\lambda_1, \lambda_2\}\\ \frac{a+\lambda_1}{\sqrt{\lambda_1}}\ \text{if}\ \lambda_1\leq \min\{a, \lambda_2\}\\ \frac{a+\lambda_2}{\sqrt{\lambda_2}}\ \text{if}\ \lambda_2\leq \min\{a, \lambda_1\}\end{cases},\forall \lambda_1,\lambda_2,\mu_1,\mu_2>0,$$and$$\lim_{x\to\infty}\frac{U(x)}{e^{-\sqrt{a}\mu x}}=1,$$ where $\mu$ solves $\sqrt a(\mu+\frac{1}{\mu})=c$ in the interval $(0 , \min\{1,\sqrt{\frac{\lambda_1}{a}},\sqrt{\frac{\lambda_2}{a}}\})$.