Turing instability conditions in confined systems with an effective position-dependent diffusion coefficient

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.