In this work we first prove, by formal arguments, that the diffusion limit of
nonlinear kinetic equations, where both the transport term and the turning
operator are density-dependent, leads to volume-exclusion chemotactic
equations. We generalise an asymptotic preserving scheme for such nonlinear
kinetic equations based on a micro-macro decomposition. By properly
discretizing the nonlinear term implicitly-explicitly in an upwind manner, the
scheme produces accurate approximations also in the case of strong
chemosensitivity. We show, via detailed calculations, that the scheme presents
the following properties: asymptotic preserving, positivity preserving and
energy dissipation, which are essential for practical applications. We extend
this scheme to two dimensional kinetic models and we validate its efficiency by
means of 1D and 2D numerical experiments of pattern formation in biological
systems.Comment: 30 pages, 8 figure