Modelling Pattern Formation in Reaction-Diffusion Systems

An investigation of Turing's Theory of Morphogenesis with special reference to highly non-linear and bistable systems

by Robin Engelhardt

Department of Chemistry

Laboratory III

H.C.Oersted Institute

University of Copenhagen

Denmark

e-mail: kel3re@unidhp.uni-c.dk, robin@santafe.edu

June 1994

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Contents

• Preface
• Introduction
• Reaction Kinetics
  • General Terminology
  • Enzyme Kinetics
  • Cooperative Behaviour
  • Autocatalysis
  • Multiple Steady State Systems
• Mathematical Tools
  • Linear Differential Equations
  • Nonlinear Differential Equations
  • Bifurcation Theory
    • Saddle-node Bifurcation
    • Hopf Bifurcation
  • Partial Differential Equations
    • Theory of Catastrophes
• The Turing Mechanism
  • Linear Stability Analysis for Two-Component Systems
    • The Homogeneous System
    • The Reaction-Diffusion System
  • The Dispersion Relation and the Diffusion Ratio
  • Activation-Inhibition and Activation Substrate-Depletion
  • The Gel-Reactor
• Initial Growth and Damping
  • The Lengyel-Epstein Model
  • Couplings and Amplifications
  • The Selkov Model
  • Concluding Remarks
• Cooperative Systems
  • Turings Mechanism with High Hill Numbers
    • The Selkov Model
    • The Brusselator
    • The Schnakenberg Model
    • The Gierer-Meinhardt Model
    • The Lengyel-Epstein Model
• Bistable Systems
  • A New Pattern Forming Mechanism in Bistable Systems
  • General Mechanism
    • The Diffusion Ratio in Bistable Systems
  • Three Component Systems and Turing-Saddle node Interaction
    • The Homogeneous Case
    • The Reaction-Diffusion System
    • Turing-Saddle node Interaction
  • Labyrinthic Patterns
• The Edblom-Orban-Epstein Reaction
  • The Ten- and Four Variable Model
  • Reduction to a Two-variable Model
  • The Reduced Non-Oscillatory EOE-Model
    • The Homogeneous System without Diffusion
    • The Saddle-node Bifurcation
    • Patterns in the Bistable Region
    • Front Dynamics and Morphological Instabilities
    • Localized Structures
• Appendix and References
  • Ginzburg-Landau Parameters for the Selkov Model
  • Scaling of the Reduced EOE-Model
  • The Routh-Hurwitz Condition
  • Descartes' Rule of Signs
• Publications
• References

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