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Generalized patterns from local and non local reactions

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  • Cencetti, Giulia
  • Battiston, Federico
  • Carletti, Timoteo
  • Fanelli, Duccio

Abstract

A class of systems is considered, where immobile species associated to distinct patches, the nodes of a network, interact both locally and at a long-range, as specified by an (interaction) adjacency matrix. Non local interactions are treated in a mean-field setting which enables the system to reach a homogeneous consensus state, either constant or time dependent. We provide analytical evidence that such homogeneous solution can turn unstable under externally imposed disturbances, following a symmetry breaking mechanism which anticipates the subsequent outbreak of the patterns. The onset of the instability can be traced back, via a linear stability analysis, to a dispersion relation that is shaped by the spectrum of an unconventional reactive Laplacian. The proposed mechanism prescinds from the classical Local Activation and Lateral Inhibition scheme, which sits at the core of the Turing recipe for diffusion driven instabilities. Examples of systems displaying a fixed-point or a limit cycle, in their uncoupled versions, are discussed. Taken together, our results pave the way for alternative mechanisms of pattern formation, opening new possibilities for modeling ecological, chemical and physical interacting systems.

Suggested Citation

  • Cencetti, Giulia & Battiston, Federico & Carletti, Timoteo & Fanelli, Duccio, 2020. "Generalized patterns from local and non local reactions," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    • Handle: RePEc:eee:chsofr:v:134:y:2020:i:c:s0960077920301090
    DOI: 10.1016/j.chaos.2020.109707
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    References listed on IDEAS

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    2. D. Bullara & Y. De Decker, 2015. "Pigment cell movement is not required for generation of Turing patterns in zebrafish skin," Nature Communications, Nature, vol. 6(1), pages 1-7, November.
    3. Malbor Asllani & Joseph D. Challenger & Francesco Saverio Pavone & Leonardo Sacconi & Duccio Fanelli, 2014. "The theory of pattern formation on directed networks," Nature Communications, Nature, vol. 5(1), pages 1-9, December.
    4. Steven H. Strogatz, 2001. "Exploring complex networks," Nature, Nature, vol. 410(6825), pages 268-276, March.
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    1. Zheng, Qianqian & Shen, Jianwei & Xu, Yong & Pandey, Vikas & Guan, Linan, 2022. "Pattern mechanism in stochastic SIR networks with ER connectivity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).

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