Turing patterns resulting from a Sturm-Liouville problem

Pattern formation in reaction-diffusion systems where the diffusion terms correspond to a Sturm-Liouville problem are studied. These correspond to a problem where the diffusion coefficient depends on the spatial variable: $\nabla \cdot \left( \mathcal{D} ( {\mathbf x} ) \nabla {\mathbf u} \right)$. We found that the conditions for Turing instability are the same as in the case of homogeneous diffusion but the nonlinear analysis must be generalized to consider general orthogonal eigenfunctions instead of the standard Fourier approach. The particular case $\mathcal{D} (x)= 1-x^2$, where solutions are linear combinations of Legendre polynomials, is studied in detail. From the developed general nonlinear analysis, conditions for producing stripes and spots are obtained, which are numerically verified using the Schaneknberg system. Unlike to the case with homogeneous diffusion, and due to the properties of the Legendre polynomials, stripped and spotted patterns with variable wavelength are produced, and a change from stripes to spots is predicted when the wavelength increases. The patterns obtained can model biological systems where stripes or spots accumulate close to the boundaries and the theory developed here can be applied to study Turing patterns associated to other eigenfunctions related with Sturm-Liouville problems.