Finsler geometry modeling for anisotropic diffusion in Turing patterns

Abstract

Abstract Turing patterns are known to be described by diffusion reaction (DR) equations, and the patterns become zebra-like anisotropic if the diffusion constant is directionally dependent. However, the origin of this dependence is unclear. In this study, we report that Turing patterns can be studied with the Finsler geometry (FG) modeling technique by applying a hybrid numerical technique, which combines DR equations and the Monte Carlo (MC) technique. In the DR equations and the Hamiltonian for MC, the Finsler metric is introduced using internal degrees of freedom. We numerically show that anisotropic patterns appear according to a constraint given by some external forces applied to the internal degrees of freedom. In this FG modeling technique, direction-dependent diffusion constants are unnecessary, and these constants automatically or effectively become anisotropic. We consider that the internal degrees of freedom introduced for the Finsler metric play an essential role in the anisotropic patterns.