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Turing patterns in network-organized activator–inhibitor systems

Abstract

Turing instability in activator–inhibitor systems provides a paradigm of non-equilibrium self-organization; it has been extensively investigated for biological and chemical processes. Turing instability should also be possible in networks, and general mathematical methods for its treatment have been formulated previously. However, only examples of regular lattices and small networks were explicitly considered. Here we study Turing patterns in large random networks, which reveal striking differences from the classical behaviour. The initial linear instability leads to spontaneous differentiation of the network nodes into activator-rich and activator-poor groups. The emerging Turing patterns become furthermore strongly reshaped at the subsequent nonlinear stage. Multiple coexisting stationary states and hysteresis effects are observed. This peculiar behaviour can be understood in the framework of a mean-field theory. Our results offer a new perspective on self-organization phenomena in systems organized as complex networks. Potential applications include ecological metapopulations, synthetic ecosystems, cellular networks of early biological morphogenesis, and networks of coupled chemical nanoreactors.

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Figure 1: Linear stability analysis.
Figure 2: Critical Turing modes of a scale-free network.
Figure 3: Localization of Laplacian eigenvectors in scale-free networks.
Figure 4: Nonlinear evolution and a stationary Turing pattern.
Figure 5: Hysteresis and multistability.
Figure 6: Stationary Turing patterns compared with the mean-field bifurcation diagrams.

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Acknowledgements

This work was supported by the Volkswagen Foundation, Germany, and by the MEXT, Japan (Global COE Program ‘The Next Generation of Physics, Spun from Universality and Emergence’ and Kakenhi Grant No. 19760253).

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Both authors designed the study, carried out the analysis, and contributed to writing the paper. H.N. performed numerical simulations.

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Correspondence to Hiroya Nakao or Alexander S. Mikhailov.

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Nakao, H., Mikhailov, A. Turing patterns in network-organized activator–inhibitor systems. Nature Phys 6, 544–550 (2010). https://doi.org/10.1038/nphys1651

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