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

2017/11/08 by Ramón G. Plaza, Plaza, Ramon G. · 5 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Micro and Nano Robotics

paper · pdf · doi:10.48550/arxiv.1711.03015

openalex publication_date 2017/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is devoted to the justification of the macroscopic, mean-field\nnutrient taxis system with doubly degenerate cross-diffusion proposed by Leyva\net al. (2013) to model the complex spatio-temporal dynamics exhibited by the\nbacterium B. subtilis during experiments run in vitro. This justification is\nbased on a microscopic description of the movement of individual cells whose\nchanges in velocity (in both speed and orientation) obey a velocity jump\nprocess (Othmer, Dunbar, Alt, 1988), governed by a transport equation of\nBoltzmann type. For that purpose, the asymptotic method introduced by Hillen\nand Othmer (2000, 2002) is applied, which consists of the computation of the\nleading order term in a regular Hilbert expansion for the solution to the\ntransport equation, under an appropriate parabolic scaling and a first order\nperturbation of the turning rate of Schnitzer type (Schnitzer, 1993). The\nresulting parabolic limit equation at leading order for the bacterial cell\ndensity recovers the degenerate nonlinear cross diffusion term and the\nassociated chemotactic drift appearing in the original system of equations.\nAlthough the bacterium B. subtilis is used as a prototype, the method and\nresults apply in more generality.\n

Cited by

• L^∞ bounds and asymptotic behavior in a doubly degenerate chemotaxis system below six dimensions
• Cross-diffusion limits in multispecies kinetic models
• Stability and uniqueness of bounded weak solutions to triangular degenerate cross-diffusion systems
• Global weak solutions to a doubly degenerate nutrient taxis system on the whole real line
• Global solvability for doubly degenerate nutrient taxis system with a wide range of bacterial responses in physical dimension

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