Turing patterns in domains with periodic inhomogeneities; a homogenization approach

Abstract

In this work, we study reaction–diffusion equations with space-dependent, differentiable, and ɛ-periodic diffusion coefficients, using the asymptotic homogenization technique. The importance of this problem is that experimental evidence suggests that in some biological systems, spatial inhomogeneities are important to regulate spatial patterns, and several pattern-forming systems are regulated by a stratified domain that determines the transport properties. Here, we consider the case when the stratification is periodic with a period ɛ; consequently, the diffusion coefficient is a space-dependent ɛ-periodic function. A methodology based on the asymptotic homogenization technique was developed to solve this problem, and the conditions for Turing instability were found in terms of the effective coefficients of the equivalent homogenized domain. Additionally, extensive numerical calculations were performed to validate the results predicted by the homogenization procedure; one- and two-dimensional examples are presented. We conclude that the homogenization method is useful in the field of pattern formation to handle problems involving spatial inhomogeneities or stratified domains. In addition, the proposed generalization of Turing analysis for a homogenized problem enriches the field of pattern formation.