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

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. (Phys A 392:5644–5662, 2013) 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 Hillen and Othmer (SIAM J Appl Math 61:751–775, 2000; SIAM J Appl Math 62:1222–1250, 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 in Phys Rev E 48:2553–2568, 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.

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Fig. 1

Notes

  1. 1.

    Notice that when the diffusion coefficient is constant, \(D_u \equiv 1 > 0\) (after normalizations), this rule implies that the bacterial response function should be \(\zeta = u\) and the chemotactic flux reads \(\varvec{J}_c = - u \chi (v) \nabla v\), recovering the standard Keller–Segel model (Keller and Segel 1971a, b).

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Acknowledgements

The author warmly thanks Thomas Hillen and Michael Winkler for enlightening conversations. This research was partially supported by DGAPA-UNAM, program PAPIME, Grant PE-104116.

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Correspondence to Ramón G. Plaza.

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Plaza, R.G. Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process. J. Math. Biol. 78, 1681–1711 (2019). https://doi.org/10.1007/s00285-018-1323-x

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Keywords

  • Chemotaxis
  • Degenerate diffusion
  • Velocity jump processes
  • Transport equations

Mathematics Subject Classification

  • 92C17
  • 60J75
  • 35K65