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Turing pattern formation in the Brusselator system with nonlinear diffusion.

G Gambino1, M C Lombardo, M Sammartino

  • 1University of Palermo, Department of Mathematics, Via Archirafi, 34, 90123 Palermo, Italy.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 16, 2013
PubMed
Summary
This summary is machine-generated.

Density-dependent nonlinear diffusion promotes Turing pattern formation in the Brusselator system. This nonlinear diffusion also leads to phenomena like hysteresis and traveling waves, unlike classical linear diffusion.

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Area of Science:

  • Chemical kinetics
  • Nonlinear dynamics
  • Pattern formation

Background:

  • The Brusselator system is a classic model for studying pattern formation in chemical reactions.
  • Nonlinear diffusion can significantly alter pattern formation dynamics compared to linear diffusion.
  • Understanding pattern formation is crucial in fields ranging from chemistry to biology.

Purpose of the Study:

  • To investigate the impact of density-dependent nonlinear diffusion on pattern formation within the Brusselator system.
  • To compare the effects of nonlinear diffusion with classical linear diffusion on pattern emergence.
  • To analyze the conditions favoring Turing patterns and oscillatory instabilities.

Main Methods:

  • Linear stability analysis to determine instability boundaries (Turing and oscillatory).
  • Weakly nonlinear multiple scales analysis to derive amplitude equations for stationary patterns.
  • Ginzburg-Landau equation derivation for traveling patterning waves.
  • Analysis of pattern formation in 1D and 2D spatial domains.

Main Results:

  • Nonlinear diffusion enhances the occurrence of Turing patterns compared to linear diffusion.
  • Observed phenomena include stable supercritical and subcritical Turing patterns, hysteresis, and multiple stable solution branches.
  • Traveling wave fronts precede pattern formation in large domains.
  • Radially symmetric target patterns emerge, with derived outer amplitude equations and inner core solutions.

Conclusions:

  • Density-dependent nonlinear diffusion plays a crucial role in pattern formation, favoring Turing patterns.
  • The study reveals complex behaviors such as hysteresis and traveling waves driven by nonlinear diffusion.
  • Nonlinear diffusion offers a richer landscape for pattern formation dynamics in reaction-diffusion systems.