Generalized patterns from local and non local reactions

Giulia Cencetti, Federico Battiston, Timoteo Carletti, Duccio Fanelli

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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.
Original languageEnglish
Article number109707
Pages (from-to)1-8
Number of pages8
Journal Chaos, Solitons & Fractals: the interdisciplinary journal of nonlinear science, and nonequilibrium and complex phenomena
Publication statusPublished - May 2020


  • Pattern formation
  • Network
  • Reaction diffusion systems
  • Linear stability analysis


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  • Dynamical systems on hypergraphs

    Carletti, T. & Fanelli, D., 27 Apr 2022, Higher-Order Systems. Springer, p. 163-180 18 p. (Understanding Complex Systems).

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