Propagation reversal for bistable differential equations on trees

Abstract

We study traveling wave solutions to bistable differential equations on infinite $k$-ary trees. These graphs generalize the notion of classical square infinite lattices and our results complement those for bistable lattice equations on $\mathbb{Z}$. Using comparison principles and explicit lower and upper solutions, we show that wave-solutions are pinned for small diffusion parameters. Upon increasing the diffusion, the wave starts to travel with non-zero speed, in a direction that depends on the detuning parameter. However, once the diffusion is sufficiently strong, the wave propagates in a single direction up the tree irrespective of the detuning parameter. In particular, our results imply that changes to the diffusion parameter can lead to a reversal of the propagation direction.