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Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems.

Thomas E Woolley1

  • 1Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK. woolleyt1@cardiff.ac.uk.

Bulletin of Mathematical Biology
|August 11, 2022
PubMed
Summary

Turing

Area of Science:

  • Mathematical Biology
  • Developmental Biology
  • Pattern Formation

Background:

  • Turing's theory explains spatial pattern generation in biological development.
  • The Turing bifurcation is commonly believed to occur via a pitchfork bifurcation.

Purpose of the Study:

  • To investigate the nature of the Turing bifurcation under different boundary conditions.
  • To challenge the dogma of pitchfork bifurcations in Turing models.
  • To explore the impact of boundary conditions on pattern formation.

Main Methods:

  • Weakly nonlinear analysis to derive algebraic results.
  • Application to Schnakenberg kinetics.
  • Numerical exploration of boundary condition complexities.
Keywords:
Transcritical bifurcationTuring instability

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Main Results:

  • The Turing bifurcation is transcritical under fixed boundary conditions, not always pitchfork.
  • Boundary conditions significantly influence the bifurcation type.
  • Combined kinetics and boundary conditions introduce unique complexities.

Conclusions:

  • The type of Turing bifurcation depends on boundary conditions.
  • Sensitivity analyses must include boundary condition variations.
  • Fixed boundary conditions necessitate considering transcritical bifurcations in Turing models.