Propagation reversal for bistable differential equations on trees

We study traveling wave solutions to the bistable differential equations on infinite $k$-ary trees in the form $$\dot{u}_i = d(k u_{i+1}-(k+1)u_i+u_{i-1}) + g(u_i;a),$$ in which $i\in\mathbb{Z}$, $d>0$ and $g\colon\mathbb{R}\to\mathbb{R}$ 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 $a,d,k$. In particular, 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. Joint work with 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)