Guaranteeing Spatial Uniformity in Reaction-Diffusion Systems Using Weighted L^2 Norm Contractions

We present conditions that guarantee spatial uniformity of the solutions of reaction-diffusion partial differential equations. These equations are of central importance to several diverse application fields concerned with pattern formation. The conditions make use of the Jacobian matrix and Neumann eigenvalues of elliptic operators on the given spatial domain. We present analogous conditions that apply to the solutions of diffusively-coupled networks of ordinary differential equations. We derive numerical tests making use of linear matrix inequalities that are useful in certifying these conditions. We discuss examples relevant to enzymatic cell signaling and biological oscillators. From a systems biology perspective, the paper’s main contributions are unified verifiable relaxed conditions that guarantee spatial uniformity of biological processes.

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