CRM: Centro De Giorgi
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Mathematical Modelling in Life Sciences

poster: Pattern Formation of a Nonlocal, Anisotropic Interaction Model for Simulating Fingerprints

speaker: Lisa Maria Kreusser (DAMTP, Centre for Mathematical Sciences)

abstract: 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 in general. 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. Based on these theoretical and numerical results we adapt the forces in the Kücken-Champod model in such a way that we can model fingerprint patterns (and more general any desired pattern) as stationary solutions. Besides, we investigate the dependence of the model parameters on the distances of the fingerprint lines. This is joint work with Martin Burger, Bertram Düring, Peter A. Markowich and Carola-Bibiane Schönlieb.

Wed 13 Sep, 19:01 - 21:00, Corridoio piano terra
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