  The current series of three papers is concerned with the asymptotic dynamics
in the following chemotaxis model $$\partial_tu=\Delta u-\chi\nabla(u\nabla
v)+u(a(x,t)-ub(x,t))\ ,\ 0=\Delta v-\lambda v+\mu u \ \ (1)$$where $\chi,
\lambda, \mu$ are positive constants, $a(x,t)$ and $b(x,t)$ are positive and
bounded. In the first of the series, we investigate the persistence and
asymptotic spreading. Under some explicit condition on the parameters, we show
that (1) has a unique nonnegative time global classical solution
$(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for every
$t_0\in R$ and every $u_0\in C^{b}_{\rm unif}(R^N)$, $u_0\geq 0$. Next we show
the pointwise persistence phenomena in the sense that, for any solution
$(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ of (1) with strictly positive initial
function $u_0$, then$$0<\inf_{t_0\in R, t\geq
0}u(x,t+t_0;t_0,u_0)\le\sup_{t_0\in R, t\geq 0} u(x,t+t_0;t_0,u_0)<\infty$$and
show the uniform persistence phenomena in the sense that there are $0<m<M$ such
that for any strictly positive initial function $u_0$, there is $T(u_0)>0$ such
that$$m\le u(x,t+t_0;t_0,u_0)\le M\ \forall\,t\ge T(u_0),\ x\in R^N.$$We then
discuss the spreading properties of solutions to (1) with compactly supported
initial and prove that there are positive constants $0<c_{-}^{*}\le
c_{+}^{*}<\infty$ such that for every $t_0\in R$ and every $u_0\in C^b_{\rm
unif}(R^N), u_0\ge 0$ with nonempty compact support, we have
that$$\lim_{t\to\infty}\sup_{|x|\ge ct}u(x,t+t_0;t_0,u_0)=0,\ \forall
c>c_+^*,$$and$$\liminf_{t\to\infty}\sup_{|x|\le ct}u(x,t+t_0;t_0,u_0)>0, \
\forall 0<c<c_-^*.$$We also discuss the spreading properties of solutions to
(1) with front-like initial functions. In the second and third of the series,
we will study the existence, uniqueness, and stability of strictly positive
entire solutions and the existence of transition fronts, respectively.
