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Vietnam Journal of Mathematics

, Volume 45, Issue 1–2, pp 127–152 | Cite as

A Hydrodynamic Limit for Chemotaxis in a Given Heterogeneous Environment

  • Stefan Grosskinsky
  • Daniel Marahrens
  • Angela StevensEmail author
Article

Abstract

In this paper, the first equation within a class of well-known chemotaxis systems is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells are assumed to interact with attractive chemical molecules on a finite number of lattice sites, but they only directly interact among themselves on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean, which results in a non-trivial macroscopic chemotaxis equation for the cells. Methodologically, the limiting procedure and its proofs are based on results by Koukkous (Stoch. Process. Appl. 84, 297–312, cite.Kou99) and Kipnis and Landim (Scaling limits of interacting particle systems, cite.KL99). Numerical simulations extend and illustrate the theoretical findings.

Keywords

Chemotaxis Interacting stochastic many particle system Hydrodynamic limit Stochastic lattice gas Block estimates 

Mathematics Subject Classification (2010)

60K35 60F25 35K55 35Q82 82C22 92B05 

Notes

Acknowledgments

Major parts of this work were done while D. Marahrens and A. Stevens were working at the University of Heidelberg. This work was finished while A. Stevens took part in the CGP-program, 2015 at the Isaac Newton Institute, Cambridge.

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Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  • Stefan Grosskinsky
    • 1
  • Daniel Marahrens
    • 2
  • Angela Stevens
    • 3
    Email author
  1. 1.University of Warwick, Mathematics InstituteCoventryUK
  2. 2.Max-Planck-Institute for Mathematics in the Sciences (MPI MIS)LeipzigGermany
  3. 3.Westfälische Wilhelms-Universität MünsterApplied MathematicsMünsterGermany

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