Selection of the ground state on a compact metric graph

TL;DR

This paper investigates the stability and existence of trivial and nontrivial ground states in the Fisher--KPP model on compact metric graphs, providing criteria based on edge size and introducing a novel method for flower graphs.

Contribution

It introduces a new approach using the period function to analyze ground states on flower graphs and establishes sharp criteria for their stability and existence.

Findings

Trivial ground state is stable on small graphs.

Nontrivial ground state is stable on large graphs.

A new method based on the period function characterizes ground states on flower graphs.

Abstract

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.