Finger revival and reconnections near breakthrough in unstable growth processes
Résumé
Transport networks, such as vasculature or river networks, provide key functions in organisms and the environment. Their performance and robustness to damage strongly depend on the topology, which can either be branched or contain loops. They often emerge as a result of unstable growth processes, in which growing branches compete for the available flux. This leads to effective repulsion between the branches and screening of the shorter ones. Although this classical picture easily produces branched structures, it does not capture attraction between the branches leading to the emergence of reticulated networks. We show that a striking transition from repulsive to attractive interactions between the branches takes place as the leading branch reaches the boundary of the system. The shorter branches revive then and grow toward the leading one forming loops, which changes the topology of the growing network. This results in a characteristic hierarchical pattern which can be observed in a large variety of systems: dissolving fractures, viscous fingering, discharge patterns, and even gastrovascular canals on jellyfish.