Analysis of Anisotropic Interaction Equations for Fingerprint Simulations

Motivated by the simulation of fingerprints we consider a class of interaction models with anisotropic, repulsive-attractive interaction forces whose orientations depend on an underlying tensor field. 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. These anisotropic forces in our class of models cannot be derived from a potential and 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 resulting stationary solutions both analytically and numerically by considering the particle model and its continuum counterpart.