We study a reaction-diffusion system within a long channel in the regime in which the projected Fick-Jacobs-Zwanzig operator for confined diffusion can be used. We found that under this approximation, Turing instability conditions can be modified due to the channel geometry. The dispersion relation, range of unstable modes where pattern formation occurs, and spatial structure of the patterns itself change as functions of the geometric parameters of the channel. This occurs for the three channels analyzed, for which the values of the projected operators can be found analytically. For the reaction term, we use the well-known Schnakenberg kinetics.

中文翻译：

---

具有有效的位置相关扩散系数的受限系统中的图灵不稳定性条件。

我们研究了一种长扩散通道内的反应扩散系统，该系统中可以使用投影的Fick-Jacobs-Zwanzig算子进行受限扩散。我们发现，在这种近似下，由于通道的几何形状，可以修改图灵不稳定性条件。色散关系，发生图案形成的不稳定模式的范围以及图案本身的空间结构根据通道的几何参数而变化。对于分析的三个通道都会发生这种情况，可以通过分析找到这些通道的预计算子值。对于反应项，我们使用众所周知的Schnakenberg动力学。