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The current paper is concerned with the spatial spreading speed and minimalwave speed of the following Keller-Segel chemoattraction system,\begin{equation}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-\chi(uv_x)_x+u(a-bu),\quad x\in\R\cr 0=v_{xx}- \lambda v+\mu u,\quad x\in\R, \end{cases}\end{equation} where $\chi$, $a$, $b$, $\lambda$, and $\mu$ are positiveconstants.  Assume $b>\chi\mu$. Then if in addition,$\big(1+\frac{1}{2}\frac{(\sqrt{a}-\sqrt{\lambda})_+}{(\sqrt{a}+\sqrt{\la})}\big)\chi\mu{ \leq} b$ holds, it is proved that $c_0^*=2\sqrt a$ is the spreading speed ofthe solutions of \eqref{abstract-eq1} with nonnegative continuous initialfunction $u_0$ with nonempty compact support, that is,  $$  \limsup_{|x|\ge ct, t\to\infty}u(t,x;u_0)=0\quad \forall\, c>c_0^*  $$  and  $$  \liminf_{|x|\le ct,t\to\infty} u(t,x;u_0)>0\quad \forall \, 0<c<c_0^*,  $$  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 $\lambda \geq a$ holds, then$c_0^*=2\sqrt a$ 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 $c\ge c_0^*$, \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 lessthan $c_0^*$. Note that $c_0^*=2\sqrt a$ is the spatial spreading speed as wellas 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}

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