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In this article we propose a unified framework in order to studyreaction-diffusion systems containing self- and cross-diffusion using a freeenergy approach. This framework naturally leads to the formulation of an energylaw, and to a numerical method respecting a discrete version of the latter. Itconstitutes an alternative method and complements the standard linear stabilityanalysis, as it allows for the numerical study of nonlinear patterns, whilemonitoring the energy evolution, even in complex geometries. As an application,we propose and study a modified Gray-Scott system augmented with self- andcross-diffusion terms. Numerical simulations unveil original patterns, clearlydistinct from those obtained with linear diffusion only

On pattern formation in reaction-diffusion systems containing self- and cross-diffusion