Propagation Reversal for Bistable Differential Equations on Trees

.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 \(\mathbb{Z}^2_{\times }\). 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 nonzero 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.Keywordsreaction-diffusion equationslattice differential equationstraveling wavespropagation reversalwave pinningtree graphsMSC codes34A3337L6039A1265M22