On the spatial structure of stationary patterns in a class of reaction-diffusion systems

We study the stationary patterns of a class of reaction-diffusion systems on 1-dimensional space with no-flux boundary conditions and homogeneous Dirichlet boundary conditions. We show that the only possible stationary pattern for such systems is the spatially periodic pattern with ''simple'' and symmetric waves. We obtain bounds on the number of inflection points per spatial period for these stationary patterns. In the case of scalar reaction-diffusion equations, our results are sharper: there is exactly one internal extrema per spatial period. A model, the Brussellator, is studied as an example. Numerical results are presented as verifications to our analysis.