Reaction-diffusion fronts and the butterfly set

A single-species reaction-diffusion model is used for studying the coexistence of multiple stable steady states. In these systems, one can define a potential-like functional that contains the stability properties of the states, and the essentials of the motion of wave fronts in one- and two-dimensional space. Using a quintic polynomial for the reaction term and taking advantage of the well-known butterfly bifurcation, we analyze the different scenarios involving the competition of two and three stable steady states, based on equipotential curves and points in parameter space. The predicted behaviors, including a front splitting instability, are contrasted to numerical integrations of reaction fronts in two dimensions.