Turing patterns with space varying diffusion coefficients: Eigenfunctions satisfying the Legendre equation

The problem of pattern formation in reaction–diffusion systems with space varying diffusion is studied. On this regard, it has already been shown that the necessary conditions for Turing instability are the same as for the case of homogeneous diffusion, but the sufficient conditions depend on the specific space dependence of the diffusion coefficient. In this work we consider the particular case when the operator of the spectral Sturm–Liouville problem associated with the general reaction–diffusion system has the Legendre polynomials as eigenfunctions. We then take a step forward, generalizing the standard weakly nonlinear analysis for these eigenfunctions, instead of the eigenfunctions of the Laplace operator. With the proposed generalization, conditions can be established for the formation of stripped or spotted patterns, which are verified numerically, and compared with the case of homogeneous diffusion, using the Schnakenberg reaction–diffusion system. Our results enrich the field of pattern formation and the generalization of the nonlinear analysis developed here can also be of interest in other fields as well as motivate further generalization by using general orthogonal functions.