Turing Pattern Formation Under Heterogeneous Distributions of Parameters for an Activator-Depleted Reaction Model

Abstract

The purpose of this article is to study Turing pattern formation in one- and two-dimensional domains under heterogeneous distributions of the parameters for an activator-depleted model. Unlike previous studies of this nature, the choice of the heterogeneous distributions of the parameters is closely linked and estimated by use of rigorous wave mode selection in order to excite different modes in different subsets of the domains. This allows us to relate the numerical solutions with theoretical linear stability analytical results. Our most revealing results show that the wave modes of adjacent subsets evolve locally and yet possess continuity across the interface. These local patches of the solutions result in a globally heterogeneous solution stable only in the presence of heterogeneous distributions of the parameters. Furthermore, our results show that initial conditions continue to play a crucial role in the selection of excitable wave modes and consequently the formation of the inhomogeneous patterns formed. In particular, initial conditions influence pattern orientation and polarity, and yet with a prepattern, the patterns conserved orientation and polarity. Numerical solutions are obtained by the use of the finite element method and the backward Euler scheme to deal with the spatial and the time discretisations, respectively.