Boundary conditions influence on Turing patterns under anomalous diffusion: A numerical exploration

Abstract

In this study, it was investigated numerically how boundary conditions influence the formation of Turing-like patterns under various diffusion conditions in complex media. It was found that Dirichlet boundary conditions can induce their symmetry in the patterns once the boundary concentrations of morphogens reach critical thresholds that depend on the diffusion regime and the domain size. We find that anomalous diffusion, characterized in our model by the parameter $\lambda$, can expand or contract the Turing instability region. Then, since superdiffusive conditions lead to a larger instability window, we conjecture that a possible explanation for the emergence of self-similarity in our system may be associated with the excitation of different scales. Our findings generally offer insights into reaction–diffusion systems’ pattern orientation and selection mechanisms.