Role of dimension and geometry: non-uniqueness in the pseudo-one-dimensional limit for Turing patterns on parallelograms

Abstract Turing patterns provide a classical mechanism for understanding self-organisation in reaction-diffusion systems. In one spatial dimension, a critical domain size is derivable, which defines the bifurcation point for patterning, meaning that domain lengths must be larger than a specified critical size for patterns to exist. In this work, using geometric spectral theory, we establish the existence of a critical domain size for Dirichlet boundary conditions in arbitrary dimensions, whereas for Neumann boundary conditions such a critical size cannot be guaranteed even for convex domains when measured by volume or surface area. Nevertheless, analysis of convex domains suggests that alternative measures of domain size may still yield a well-defined Turing bifurcation threshold. Furthermore, using parallelograms as analytically tractable test cases, we investigate the bifurcation structure of Turing patterns, focusing on the limit of reducing the shape from two-dimensional to pseudo-one-dimensional. Through linear stability analysis, numerical calculation of Laplacian eigenvalues and using recent results on analytical bounds, we demonstrate that there is no unique critical length for patterning to occur, rather, the length depends on how the dimension reduction takes place. These findings are supported by numerical simulations of the full nonlinear system, which confirm the analytical predictions. Our results challenge the traditional assumption of a unique threshold for pattern onset, highlighting the fundamental role of domain geometry in the interpretation and modelling of biological pattern formation. This work emphasises the need for careful geometric consideration in both theoretical and experimental studies using reaction-diffusion frameworks.