Realistic examples of reaction–diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of ‘open’ reaction–diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction–diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction–diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

控制空间和时空模式形成的反应扩散现象的现实例子很少是孤立的系统，无论是化学上还是热力学上。然而，即使是“开放”反应-扩散系统的公式也常常忽略域边界的作用。封闭反应扩散系统的大多数理想化都采用无通量边界条件，并且通常会在这些边界上或沿这些边界形成模式。受与胚胎发育中空间形式出现相关的模式场边界的驱动，我们为两个物种的反应 - 扩散系统提出了一组混合边界条件，该系统在远离域边界的情况下为各种不同的反应动力学，在边界附近具有规定的均匀状态。我们表明，这些边界条件可以来自更大的异质场，这表明如果细胞信号或介质的其他特性在空间上发生变化，这些条件可以自然出现。我们解释了这种模式定位背后的基本机制，并证明它可以在一个、两个和三个维度上捕获大范围的本地化模式，并且该框架可以应用于涉及两个以上物种的系统。此外，所提出的边界条件导致域内部更对称的模式，并可能在发展系统中捕获更现实的边界。最后，我们表明这些孤立的模式对初始条件的波动更稳健，并且它们允许通过几何进行模式选择的有趣可能性，