  We introduce a numerical scheme to approximate a quasi-linear hyperbolic
system which models the movement of cells under the influence of chemotaxis.
Since we expect to find solutions which contain vacuum parts, we propose an
upwinding scheme which handles properly the presence of vacuum and, besides,
which gives a good approximation of the time asymptotic states of the system.
For this scheme we prove some basic analytical properties and study its
stability near some of the steady states of the system. Finally, we present
some numerical simulations which show the dependence of the asymptotic behavior
of the solutions upon the parameters of the system.

Numerical Analysis