Pattern formation of a nonlocal, anisotropic interaction model

Martin Burger, Bertram Düring, Lisa Maria Kreusser, Peter A. Markowich, Carola Bibiane Schönlieb

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9 Citations (SciVal)
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We consider a class of interacting particle models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. An example of this class of models is the so-called Kücken-Champod model describing the formation of fingerprint patterns. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In contrast to isotropic interaction models the anisotropic forces in our class of models cannot be derived from a potential. The underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. This anisotropy is characterized by one parameter in the model. We study the variation of this parameter, describing the transition between the isotropic and the anisotropic model, analytically and numerically. We analyze the equilibria of the corresponding mean-field partial differential equation and investigate pattern formation numerically in two dimensions by studying the dependence of the parameters in the model on the resulting patterns.

Original languageEnglish
Pages (from-to)409-451
Number of pages43
JournalMathematical Models and Methods in Applied Sciences
Issue number3
Early online date29 Dec 2017
Publication statusPublished - 1 Mar 2018


  • dynamical systems
  • Nonlocal interactions
  • pattern formation

ASJC Scopus subject areas

  • Modelling and Simulation
  • Applied Mathematics


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