Selection of the ground state on a compact metric graph

We show that the ground state in the Fisher--KPP model on a compact metric graph with Dirichlet conditions on boundary vertices is either trivial (zero) or nontrivial and strictly positive. For positive initial data, we prove that the trivial ground state is globally asymptotically stable if the edges of the metric graph are uniformly small and the nontrivial ground state is globally asymptotically stable if the edges are uniformly large. For the intermediate case, we find a sharp criterion for the existence, uniqueness and global asymptotic stability of the trivial versus nontrivial ground state. Besides standard methods based on the comparison theory, energy minimizers, and the lowest eigenvalue of the graph Laplacian, we develop a novel method based on the period function for differential equations to characterize the nontrivial ground state in the particular case of flower graphs.