Pattern formation of a nonlocal, anisotropic interaction model

TL;DR

This paper investigates a class of anisotropic, nonlocal interaction models, including the Kücken-Champod model, analyzing how anisotropy influences pattern formation through analytical and numerical methods.

Contribution

It introduces a generalized anisotropic interaction model with a tensor field, extending previous isotropic models, and studies the impact of anisotropy on pattern formation.

Findings

Anisotropy parameter controls pattern complexity.

Transition from isotropic to anisotropic patterns analyzed.

Numerical simulations confirm analytical predictions.

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 K\"ucken-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.