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In this work we first prove, by formal arguments, that the diffusion limit ofnonlinear kinetic equations, where both the transport term and the turningoperator are density-dependent, leads to volume-exclusion chemotacticequations. We generalise an asymptotic preserving scheme for such nonlinearkinetic equations based on a micro-macro decomposition. By properlydiscretizing the nonlinear term implicitly-explicitly in an upwind manner, thescheme produces accurate approximations also in the case of strongchemosensitivity. We show, via detailed calculations, that the scheme presentsthe following properties: asymptotic preserving, positivity preserving andenergy dissipation, which are essential for practical applications. We extendthis scheme to two dimensional kinetic models and we validate its efficiency bymeans of 1D and 2D numerical experiments of pattern formation in biologicalsystems.

Asymptotic preserving schemes for nonlinear kinetic equations leading to volume-exclusion chemotaxis in the diffusive limit