Past exam of the topics in applied mathematics course of the University of Oxford / 2010 / A / 3 / b / ii / Solution

Domain geometry can select biological pattern dimension even when the same reaction parameters and diffusion coefficients are used. A sufficiently wide developing tissue can admit instabilities with variation in two spatial directions; nonlinear development may produce spots, labyrinths or other two-dimensional patterns. A thin strip suppresses transverse unstable modes, allowing primarily bands whose concentration varies along its long axis and is nearly uniform across its width.

This is a mechanism for shape-dependent pattern selection during growth: increasing tissue width can admit additional transverse modes without changing the local chemical interactions. Linear instability predicts initial wavelengths and permitted directions, not the eventual stable spot or stripe pattern; nonlinear saturation and boundary geometry determine that later outcome. Zero-flux boundaries also mean these modes arise through internal redistribution and reaction rather than externally prescribed chemical concentrations.