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.

本文的目的是研究在活化剂耗尽模型的参数不均匀分布下在一维和二维域中的图灵图案形成。与以前的这种性质的研究不同，通过使用严格的波模式选择来紧密链接和估计参数的异质分布，以便激发域的不同子集中的不同模式。这使我们能够将数值解与理论线性稳定性分析结果联系起来。我们最有说服力的结果表明，相邻子集的波模局部演化，但在整个界面上具有连续性。解决方案的这些局部补丁导致仅在参数的异构分布存在时才稳定的全局异构解决方案。此外，我们的结果表明，初始条件在激发波模式的选择中继续发挥着至关重要的作用，并因此形成了不均匀的模式。特别是，初始条件会影响图案的方向和极性，而对于预图案而言，图案会保留方向和极性。通过使用有限元方法和后向欧拉方案分别处理空间离散和时间离散，获得了数值解。