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,

$$\begin{aligned} \left\{ \begin{aligned}&u_t=u_{xx}-\chi (uv_x)_x +u(a-bu),\quad x\in {{\mathbb {R}}},\\&0=v_{xx}- \lambda v+\mu u,\quad x\in {{\mathbb {R}}}, \ \end{aligned}\right. \end{aligned}$$

(0.1)

where $\chi$, a, b, $\lambda$, and $\mu$ are positive constants. Assume $b>\chi \mu$ . Then if in addition $\big (1+\frac{1}{2}\frac{(\sqrt{a}-\sqrt{\lambda })_+}{(\sqrt{a}+\sqrt{\lambda })}\big )\chi \mu { \le } b$ holds, it is proved that $c_0^*=2\sqrt{a}$ is the spreading speed of the solutions of (0.1) with nonnegative continuous initial function $u_0$ with nonempty compact support, that is,

$$\begin{aligned} \limsup _{|x|\ge ct, t\rightarrow \infty }u(t,x;u_0)=0\quad \forall \, c>c_0^* \end{aligned}$$

and

$$\begin{aligned} \liminf _{|x|\le ct,t\rightarrow \infty } u(t,x;u_0)>0\quad \forall \, 0<c<c_0^*, \end{aligned}$$

where $(u(t,x;u_0),v(t,x;u_0))$ is the unique global classical solution of (0.1) 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 $c_0^*=2\sqrt{a}$ is the minimal speed of the traveling wave solutions of (0.1) connecting (0, 0) and $(\frac{a}{b},\frac{\mu }{\lambda }\frac{a}{b})$, that is, for any $c\ge c_0^*$, (0.1) has a traveling wave solution connecting (0, 0) and $(\frac{a}{b},\frac{\mu }{\lambda }\frac{a}{b})$ with speed c, and (0.1) has no such traveling wave solutions with speed less than $c_0^*$. Note that $c_0^*=2\sqrt{a}$ is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation,

$$\begin{aligned} u_t=u_{xx}+u(a-bu),\quad x\in {{\mathbb {R}}}. \end{aligned}$$

(0.2)

Hence, if $\lambda \ge a$ and $b>\chi \mu$, or $\lambda <a$ and $b\ge \big (1+\frac{1}{2}\frac{(\sqrt{a}-\sqrt{\lambda })}{(\sqrt{a}+\sqrt{\lambda })}\big )\chi \mu$, then the chemotaxis neither speeds up nor slows down the spatial spreading in (0.1).