Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?

Abstract

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, \begin{equation}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-χ(uv_x)_x +u(a-bu),\quad x\in\R\cr 0=v_{xx}- λv+μu,\quad x\in\R, \end{cases} \end{equation} where $\chi$, $a$, $b$, $\lambda$, and $\mu$ are positive constants. Assume $b>\chi \mu$. Then if in addition, $\big(1+\frac{1}{2}\frac{(\sqrt{a}-\sqrtλ)_+}{(\sqrt{a}+\sqrt{\la})}\big)χμ{ \leq} b$ holds, it is proved that $c_0^{\ast }=2\sqrt{a}$ is the spreading speed of the solutions of \eqref{abstract-eq1} with nonnegative continuous initial function $u_0$ with nonempty compact support, that is, $$\underset{|x|\ge ct,t\to \infty }{\mathrm{limsup}}u(t,x;u_0)=0\forall c>c_0^{\ast }$$ and $$\underset{|x|\le ct,t\to \infty }{\mathrm{liminf}}u(t,x;u_0)>0\forall 0<c<c_0^{\ast },$$ where $(u(t,x;u_0),v(t,x;u_0))$ is the unique global classical solution of \eqref{abstract-eq1} with $u(0,x;u_0)=u_0(x)$. It is also proved that, if $b>2\chi \mu$ and $\lambda \ge a$ holds, then $c_0^{\ast }=2\sqrt{a}$ is the minimal speed of the traveling wave solutions of \eqref{abstract-eq1} connecting $(0,0)$ and $(\frac{a}{b},\frac{\mu }{\lambda }\frac{a}{b})$, that is, for any $c\ge c_0^{\ast }$, \eqref{abstract-eq1} has a traveling wave solution connecting $(0,0)$ and $(\frac{a}{b},\frac{\mu }{\lambda }\frac{a}{b})$ with speed $c$, and \eqref{abstract-eq1} has no such traveling wave solutions with speed less than $c_0^{\ast }$. Note that $c_0^{\ast }=2\sqrt{a}$ is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation, \begin{equation} \label{abstract-eq2} u_t=u_{xx}+u(a-bu),\quad x\in\R. \end{equation}

References in corpus (1)

• Propagation in a Fisher-KPP equation with non-local advection *

Cited by in corpus (1)

• On traveling wave solutions in full parabolic Keller-Segel chemotaxis systems with logistic source