Asymptotic behaviour for solutions to reaction–diffusion equations on a root-like metric graph *

In this paper we consider symmetric solutions of reaction–diffusion equations on an unbounded root-like metric graph. We first prove a version of the zero number diminishing properties on the graph, and then use them to show a convergence result for bounded and symmetric solution of general equations. As application, we provide a complete spreading-transition-vanishing trichotomy result on the asymptotic behaviour for solutions of the bistable reaction–diffusion equation on the graph.