  In this paper, we derive a new chemotaxis model with degenerate diffusion and
density-dependent chemotactic sensitivity, and we provide a more realistic
description of cell migration process for its early and late stages. Different
from the existing studies focusing on the case of non-degenerate diffusion, the
new model with degenerate diffusion causes us some essential difficulty on the
boundedness estimates and the propagation behavior of its compact support. In
the presence of logistic damping, for the early stage before tumour cells
spread to the whole body, we first estimate the expanding speed of tumour
region as $O(t^{\beta})$ for $0<\beta<\frac{1}{2}$. Then, for the late stage of
cell migration, we further prove that the asymptotic profile of the original
system is just its corresponding steady state. The global convergence of the
original weak solution to the steady state with exponential rate $O(e^{-ct})$
for some $c>0$ is also obtained.
