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Instability of turing patterns in reaction-diffusion-ODE systems.

Anna Marciniak-Czochra1, Grzegorz Karch2, Kanako Suzuki3

  • 1Institute of Applied Mathematics, Interdisciplinary Center for Scientific Computing (IWR) and BIOQUANT, University of Heidelberg, Heidelberg, 69120, Germany.

Journal of Mathematical Biology
|June 17, 2016
PubMed
Summary
This summary is machine-generated.

This study reveals that reaction-diffusion equations coupled with ordinary differential equations, crucial for modeling cell processes, do not support stable Turing patterns due to diffusion-driven instability affecting all solutions.

Keywords:
AutocatalysisPattern formationReaction-diffusion equationsTuring instabilityUnstable stationary solutions

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Area of Science:

  • Mathematical Biology
  • Chemical Kinetics
  • Systems Biology

Background:

  • Reaction-diffusion systems model pattern formation in biological and chemical systems.
  • Coupling with ordinary differential equations (ODEs) introduces complexity, particularly in modeling cell signaling interactions.
  • Understanding pattern formation in these hybrid systems is key to deciphering cellular processes like growth and differentiation.

Purpose of the Study:

  • To analyze the pattern formation phenomenon in reaction-diffusion equations coupled with ODEs.
  • To investigate the stability of solutions in a prototype model combining a reaction-diffusion equation and an ODE.
  • To determine the conditions under which stable patterns can emerge in such coupled systems.

Main Methods:

  • Focus on stability analysis of solutions for a prototype reaction-diffusion-ODE model.
  • Investigate diffusion-driven instability (Turing instability) under autocatalysis conditions.
  • Employ spectral analysis of linear operators to rigorously prove nonlinear instability.

Main Results:

  • Reaction-diffusion-ODE systems exhibit distinct behavior compared to classical reaction-diffusion models.
  • Diffusion-driven instability arises from autocatalysis of the non-diffusing component.
  • This instability destabilizes both constant and continuous spatially heterogeneous stationary solutions, precluding stable Turing patterns.
  • Nonlinear instability is rigorously demonstrated through continuous spectrum analysis.

Conclusions:

  • Stable Turing patterns are unattainable in the studied reaction-diffusion-ODE systems.
  • The mechanism causing instability affects all types of stationary solutions, including continuous ones.
  • Results extend to discontinuous patterns for specific nonlinearities, highlighting fundamental differences from classical models.