Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process. (English) Zbl 1415.92039
Summary: This paper is devoted to the justification of the macroscopic, mean-field nutrient taxis system with doubly degenerate cross-diffusion proposed by J. F. Leyva et al. [Physica A 392, No. 22, 5644–5662 (2013; Zbl 1395.92023)] to model the complex spatio-temporal dynamics exhibited by the bacterium Bacillus 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 governed by a transport equation of Boltzmann type. For that purpose, the asymptotic method introduced by T. Hillen and H. G. Othmer [SIAM J. Appl. Math. 61, No. 3, 751–775 (2000; Zbl 1002.35120); ibid. 62, No. 4, 1222–1250 (2002; Zbl 1103.35098)] 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 [M. J. Schnitzer, “Theory of continuum random walks and application to chemotaxis”, Phys. Rev. E (3) 48, No. 4, 2553–2568 (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.
MSC:
| 92C17 | Cell movement (chemotaxis, etc.) |
| 60J75 | Jump processes (MSC2010) |
| 35K65 | Degenerate parabolic equations |
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