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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 Erdös-Rényi, 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 Erdös-Rényi and scale-free networks, the mean-field theory for stationary patterns is constructed.
Highlights
Studies of pattern formation in reaction-diffusion systems far from equilibrium constitute a firmly established research field
As shown in our study, such patterns are possible in networks of diffusively coupled bistable elements, but their properties are significantly different
The behavior of the fronts is highly sensitive to network architecture and degrees of network nodes play an important role here
Summary
Studies of pattern formation in reaction-diffusion systems far from equilibrium constitute a firmly established research field. Starting from the pioneering work by Turing [1] and Prigogine [2], self-organized structures in distributed active media with activator-inhibitor dynamics have been extensively investigated and various non-equilibrium patterns, such as rotating spirals, traveling pulses, propagating fronts or stationary dissipative structures could be observed [3,4]. The attention became turned to network analogs of classical reaction-diffusion systems, where the nodes are occupied by active elements and the links represent diffusive connections between them. Such situations are typical for epidemiology where spreading of diseases over transportation networks takes place [5]. Turing patterns in activator-inhibitor network systems have been considered [11]
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