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Pattern formation of a nonlocal, anisotropic interaction model

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Motivated by the formation of fingerprint patterns during pregnancy we consider a large system of interacting particles. These interactions among the particles can be described by short-range repulsion forces to prevent collisions as well as long-range attraction forces keeping the particles together, and lead to an isotropic interaction model. To describe the pattern formation accurately we extend this isotropic model and consider an underlying tensor field representing the stress. This leads to an additional anisotropy and the resulting class of models captures more complex, anisotropic patterns than in the isotropic model. Mathematically, this class of interacting particle models is given by a large system of nonlinear ordinary differential equations and can be regarded as a generalization of the isotropic interaction model. Using mathematical analysis and numerical simulations we analyse this class of models for the first time and adapt it in such a way that fingerprint patterns can be obtained as stationary solutions. This is joint work with M. Burger, B. Duering, P. Markowich and C.-B. Schoenlieb.

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