Pattern formation in a two-component reaction–diffusion system with delayed processes on a network

Julien Petit, Malbor Asllani, Duccio Fanelli, Ben Lauwens, Timoteo Carletti

Research output: Contribution to journalArticle

Abstract

Reaction–diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W-function allowed us to get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts

whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator–inhibitor variant without delay. Numerical results gained from the Mimura–Murray model support the theoretical approach.

Original languageEnglish
Pages (from-to)230-249
Number of pages20
JournalPhysica A: Statistical Mechanics and its Applications
Volume462
DOIs
Publication statusPublished - 17 Jun 2016

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Pattern Formation
Reaction-diffusion System
Time Delay
time lag
Turing Instability
Turing Patterns
Complex Networks
Hopf Bifurcation
Diffusion Coefficient
diffusion coefficient
Numerical Results
Model
Framework

Keywords

  • nonlinear dynamics
  • spatio-temporal patterns
  • Complex Networks
  • delay differential equations
  • Turing waves

Cite this

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title = "Pattern formation in a two-component reaction–diffusion system with delayed processes on a network",
abstract = "Reaction–diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W-function allowed us to get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator–inhibitor variant without delay. Numerical results gained from the Mimura–Murray model support the theoretical approach.",
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T1 - Pattern formation in a two-component reaction–diffusion system with delayed processes on a network

AU - Petit, Julien

AU - Asllani, Malbor

AU - Fanelli, Duccio

AU - Lauwens, Ben

AU - Carletti, Timoteo

PY - 2016/6/17

Y1 - 2016/6/17

N2 - Reaction–diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W-function allowed us to get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator–inhibitor variant without delay. Numerical results gained from the Mimura–Murray model support the theoretical approach.

AB - Reaction–diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W-function allowed us to get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator–inhibitor variant without delay. Numerical results gained from the Mimura–Murray model support the theoretical approach.

KW - nonlinear dynamics

KW - spatio-temporal patterns

KW - Complex Networks

KW - delay differential equations

KW - Turing waves

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JO - Physica A: Statistical Mechanics and its Applications

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