Propagation Reversal for Bistable Differential Equations on Trees

SIAM Journal on Applied Dynamical Systems, vol. 22, pp. 1906–1944

Abstract

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 [Formula: see text]. 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.

Authors 4

1. Hermen Jan Hupkes Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, 2300 RA Leiden, The Netherlands

2. Mia Jukić Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden, 2300 RA Leiden, The Netherlands

3. Petr Stehlík

   University of West Bohemia in Pilsen

   Affiliation as printed

   Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 306 14 Plzeň, Czech Republic

4. Vladimír Švígler

   University of West Bohemia in Pilsen

   Affiliation as printed

   Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 306 14 Plzeň, Czech Republic