Pattern Formation and Cross-Diffusion for a Chemotaxis Model

Authors

  • Moustafa Ibrahim* Г‰cole centrale de Nantes
  • Mazen Saad Г‰cole centrale de Nantes

DOI:

https://doi.org/10.11145/85

Abstract

Chemotaxis is the feature movement of cell or an organism along a chemical concentration gradient. The mathematical analysis of chemotaxis modelss how a plenitude of spatial patterns such as the chemotaxis models applied to skin pigmentation patterns, that lead to aggregations of one type of pigment cells into a striped spatial pattern. The analysis of pattern formation can be traced to a seminal paper by Turing [1], who established that an action-diffusion system can generate stable nonuniform patterns in space if the components of the system interact with each other.Our motivation is the numerical simulations of the pattern formation for a volume-filling chemotaxis model. In [2], the effect of volume-filling is expressed through a nonlinear squeezing probability. We investigate pattern formation using Turing's principle and the standard argument used by Murray [3,4]. Next, we introduce an implicit nite volume scheme; it is presented on a general mesh satisfying the orthogonality condition [5,6].The originality of this scheme is the upstream approach to discretize thecross-diffusion term. Finally, we present some numerical results showing the spatial patterns for the chemotaxis model.

References

[1] A. Turing, The chemical basis of morphogenesis, Philosophical Transactions ofthe Royal Society of London. Series B, Biological Sciences 237 37{72, 1952.

[2] Z. Wang and T. Hillen, Classical solutions and pattern formation for a vol-ume lling chemotaxis model, Chaos: An Interdisciplinary Journal of NonlinearScience 17 037108, 2007.

[3] J.D. Murray, Mathematical biology: I. An introduction, Springer, 2002.

[4] J.D. Murray, Mathematical biology II: spatial models and biomedical applica-tions, Springer, 2003.

[5] B. Andreianov, M. Bendahmane and M. Saad, Finite volume methods for de-generate chemotaxis model, Journal of computational and applied mathematics235 4015{4031, 2011.

[6] M. Bendahmane, M. Saad, Mathematical analysis and pattern formation for apartial immune system modeling the spread of an epidemic disease, Acta appli-candae mathematicae 115 17{42, 2011.

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Published

2013-04-22

Issue

Pilot Issue

Section

Conference Contributions

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