Multiscale analysis for diffusion-driven neutrally stable states

The Turing instabilities for reaction-diffusion systems are studied from the Fourier normal modes which appear by searching the solution obtained from linearization of the reaction-diffusion system at the spatially homogeneous steady state. The linear stability analysis is only appropriate when the temporal eigenvalues associated to every given spatial eigenvalue have non-zero real part. If the real part of the temporal eigenvalue in a normal mode is equal to zero there is no enough information coming from the linearized system. Given an arbitrary spatial eigenvalue, by equating to zero the real part of the corresponding temporal eigenvalue will lead to a neutral stability manifold in the parameter space. If for a given spatial eigenvalue the other parameters in the reaction-diffusion process drive the system to the neutral manifold, then neither stability nor instability can be warranted by the usual linear analysis. In order to give a sketch of the nonlinear analysis we use a multiple scales method. As an application, we analyze the behavior of solutions to the Schnakenberg trimolecular reaction kinetics in the presence of diffusion.