Existence and stability of traveling waves in buffered systems

We study wave propagation in the buffered bistable equation, i.e., the bistable equation where the diffusing species reacts with immobile buffers that restrict its diffusion. Such a model describes wave front propagation in excitable systems where the diffusing species is buffered; in particular, the study of the propagation of waves of increased calcium concentration in a variety of cell types depends directly upon the analysis of such buffered excitability. However, despite the biological importance of these types of equations, there have been almost no analytical studies of their properties. Here, we study the question of whether or not the inclusion of multiple buffers can eliminate propagated waves. First, we prove that a unique (up to translation) traveling wave front exists. Moreover, the wave speed is also unique. Then we prove that this traveling wave front is stable, i.e., that any initial condition which vaguely resembles a traveling wave front (in a way we make precise) evolves to the unique wave front. We thus prove that multiple stationary buffers cannot prevent the existence of a traveling wave front in the buffered bistable equation and may not eliminate stable wave fronts. This suggests (although we do not prove) that the same result is true for more complex and realistic models of calcium wave propagation, a result of direct physiological relevance.