  This paper is devoted to the justification of the macroscopic, mean-field
nutrient taxis system with doubly degenerate cross-diffusion proposed by Leyva
et al. (2013) to model the complex spatio-temporal dynamics exhibited by the
bacterium B. subtilis during experiments run in vitro. This justification is
based on a microscopic description of the movement of individual cells whose
changes in velocity (in both speed and orientation) obey a velocity jump
process (Othmer, Dunbar, Alt, 1988), governed by a transport equation of
Boltzmann type. For that purpose, the asymptotic method introduced by Hillen
and Othmer (2000, 2002) is applied, which consists of the computation of the
leading order term in a regular Hilbert expansion for the solution to the
transport equation, under an appropriate parabolic scaling and a first order
perturbation of the turning rate of Schnitzer type (Schnitzer, 1993). The
resulting parabolic limit equation at leading order for the bacterial cell
density recovers the degenerate nonlinear cross diffusion term and the
associated chemotactic drift appearing in the original system of equations.
Although the bacterium B. subtilis is used as a prototype, the method and
results apply in more generality.
