Traveling and pinned fronts in bistable reaction-diffusion systems on network

TL;DR

This paper investigates traveling and pinned fronts in bistable reaction-diffusion systems on various networks, revealing how network architecture influences pattern formation and pinning phenomena.

Contribution

It extends the study of bistable reaction-diffusion patterns to complex networks, providing analytical and mean-field insights into front pinning and propagation.

Findings

Traveling fronts can propagate across different network types.

Pinning of fronts depends on node degree and network structure.

Analytical conditions for front pinning are derived for regular trees.

Abstract

Traveling fronts and stationary localized patterns in bistable reaction-diffusion systems have been broadly studied for classical continuous media and regular lattices. Analogs of such non-equilibrium patterns are also possible in networks. Here, we consider traveling and stationary patterns in bistable one-component systems on random Erd\"os-R\'enyi, scale-free and hierarchical tree networks. As revealed through numerical simulations, traveling fronts exist in network-organized systems. They represent waves of transition from one stable state into another, spreading over the entire network. The fronts can furthermore be pinned, thus forming stationary structures. While pinning of fronts has previously been considered for chains of diffusively coupled bistable elements, the network architecture brings about significant differences. An important role is played by the degree (the number of connections) of a node. For regular trees with a fixed branching factor, the pinning conditions are analytically determined. For large Erd\"os-R\'enyi and scale-free networks, the mean-field theory for stationary patterns is constructed.