  The current paper is concerned with the spreading speeds of the following
parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$,
\begin{equation} \begin{cases} u_t=\Delta u-\chi\nabla\cdot ( u\nabla v) +
u(a-bu),\quad x\in\mathbb{R}^{N},\cr {v_t}=\Delta v -\lambda v+\mu u,\quad x\in
\mathbb{R}^{N}. \end{cases}(1) \end{equation} where $\chi, \ a,\ b,\ \lambda,\
\mu$ are positive constants. Assume $b>\frac{N\mu\chi}{4}$. Among others, it is
proved that $2\sqrt{a}$ is the spreading speed of the global classical
solutions of (1) with nonempty compactly supported initial functions, that is,
$$ \lim_{t\to\infty}\sup_{|x|\geq ct}u(x,t;u_0,v_0)=0\quad \forall\,\,
c>2\sqrt{a} $$ and $$ \liminf_{t\to\infty}\inf_{|x|\leq ct}u(x,t;u_0,v_0)>0
\quad \forall\,\, 0<c<2\sqrt{a}. $$ where $(u(x,t;u_0,v_0), v(x,t;u_0,v_0))$ is
the unique global classical solution of (1) with $u(x,0;u_0,v_0)=u_0$,
$v(x,0;u_0,v_0)=v_0$, and ${\rm supp}(u_0)$, ${\rm supp}(v_0)$ are nonempty and
compact. It is well known that $2\sqrt{a}$ is the spreading speed of the
following Fisher-KPP equation, $$ u_t=\Delta u+u(a-bu),\quad \forall\,\
x\in\mathbb{R}^{N}. $$ Hence, if $b>\frac{N\mu\chi}{4}$, the chemotaxis neither
speeds up nor slows down the spatial spreading in the Fisher-KPP equation.
