Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process

TL;DR

This paper rigorously derives a macroscopic bacterial nutrient-taxis model with degenerate cross-diffusion from a microscopic velocity jump process, validating the continuum description of B. subtilis movement.

Contribution

It provides a mathematical derivation of a complex macroscopic model from microscopic cell movement processes, connecting individual behavior to population-level equations.

Findings

Derivation of the nutrient-taxis system as a parabolic limit

Validation of the degenerate cross-diffusion term

Applicability to general bacterial movement models

Abstract

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.