  In this paper, we study the relationship between a diffusive model and a
non-diffusive model which are both derived from the well-known Keller-Segel
model, as a coefficient of diffusion $\varepsilon$ goes to zero. First, we
establish the global well-posedness of classical solutions to the Cauchy
problem for the diffusive model with smooth initial data which is of small
$L^2$ norm, together with some {\it a priori} estimates uniform for $t$ and
$\varepsilon$. Then we investigate the zero-diffusion limit, and get the global
well-posedness of classical solutions to the Cauchy problem for the
non-diffusive model. Finally, we derive the convergence rate of the diffusive
model toward the non-diffusive model. It is shown that the convergence rate in
$L^\infty$ norm is of the order $O(\varepsilon^{1/2})$. It should be noted that
the initial data is small in $L^2$-norm but can be of large oscillations with
constant state at far field. As a byproduct, we improve the corresponding
result on the well-posedness of the non-difussive model which requires small
oscillations.
