Propagation reversal for bistable differential equations on trees
Vladimir Svigler
University of West Bohemia, Faculty of Applies Sciences, Department of Mathematics and NTIS Czech Rep
Co-Author(s): Hermen Jan Hupkes, Mia Juki\`{c} (Mathematisch Instituut, Universiteit Leiden) and Petr Stehl\`{i}k (Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia)
Abstract:
We study traveling wave solutions to the bistable differential equations on infinite $k$-ary trees in the form
\begin{align*}
\dot{u}_i &= d(k u_{i+1}-(k+1)u_i+u_{i-1}) + g(u_i;a),
\end{align*}
in which $i \in \mathbb{Z}$, $d>0$ and $g:\mathbb{R} \to \mathbb{R}$ is a bistable nonlinearity of the Nagumo type, e.g.,
\[
g(s;a) = s(1-s)(s-a), \quad a\in(0,1).
\]
In this talk, we discuss how comparison principles and construction of explicit lower and upper solution can be used to obtain information about the dependence of the wave speed $c \in \mathbb{R}$ on the parameters $a,d,k$.
We show that wave-solutions are pinned provided the diffusion parameter $d$ is small. Upon increasing the diffusion $d$, the wave starts to travel with non-zero speed $c \neq 0$, in a direction that depends on the detuning parameter $a$. However, once the diffusion is sufficiently strong, the wave propagates in a single direction up the tree irrespective of the detuning parameter $a$.
As a consequence, we show that for certain range of the detuning parameter $a$ the changes to the diffusion parameter $d$ lead to a reversal of the propagation direction. This is in stark contrast to the behaviour of the standard lattice equation with $k=1$.
