Propagation failure in discrete reaction-diffusion system based on the butterfly bifurcation

Reaction-diffusion systems are used in biology, chemistry, and physics to model the interaction of spatially distributed species. Particularly of interest is the spatial replacement of one equilibrium state by another, depicted as traveling waves or fronts. Their profiles and traveling velocity depend on the nonlinearities in the reaction term and on spatial diffusion. If the reaction occurs at regularly spaced points, the velocities also depend on lattice structures and the orientation of the traveling front. Interestingly, there is a wide region of parameters where the speeds become zero and the fronts do not propagate. In this paper, we focus on systems with three stable coexisting equilibrium states that are described by the butterfly bifurcation and study to what extent the three possible 1D traveling fronts suffer from propagation failure. We demonstrate that discreteness of space affects the three fronts differently. Regions of propagation failure add a new layer of complexity to the butterfly diagram. The analysis is extended to planar fronts traveling through different orientations in regular 2D lattices. Both propagation failure and the existence of preferred orientations play a role in the transient and long-time evolution of 2D patterns.