Skip to main content
Article thumbnail
Location of Repository

Biological pattern formation on two-dimensional domains: A nonlinear bifurcation analysis

By G. C. Cruywagen, P. K. Maini and J. D. Murray

Abstract

A tissue interaction model for skin organ pattern formation is presented. Possible spatially patterned solutions on rectangular domains are investigated. Linear stability analysis suggests that the model can exhibit pattern formation. A weakly nonlinear two-dimensional perturbation analysis is then carried out. This demonstrates that when bifurcation occurs via a simple eigenvalue, patterns such as rolls, squares, and rhombi can be supported by the model equations. Our nonlinear analysis shows that more complex patterns are also possible if bifurcation occurs via a double eigenvalue. Surprisingly, hexagonal patterns could not develop from a primary bifurcation

Topics: Biology and other natural sciences
Year: 1997
DOI identifier: 10.1137/S0036139996297900
OAI identifier: oai:generic.eprints.org:457/core69

Suggested articles

Citations

  1. (1988). A nonlinear analysis of a mechanochemical model for biological pattern formation,
  2. (1994). Bifurcating spatial patterns arising from travelling waves in a tissue interaction model,
  3. (1986). Cell adhesion molecules in the regulation of animal form and tissue pattern,
  4. (1984). Cell traction models for generating pattern and form in morphogenesis,
  5. (1987). Complex spatial patterns from tissue interactions|an illustrative model,
  6. (1990). Model for complex skin patterns, doi
  7. (1965). Morphology and proliferation during early feather development, doi
  8. (1988). Neuron-glia cell adhesion molecule interacts with neurons and astroglia via dierent binding mechanisms,
  9. (1970). Nonlinear dynamic stability: A formal theory,
  10. (1992). On a tissue interaction model for skin pattern formation,
  11. (1989). Pattern formation models and developmental constraints,
  12. (1974). Primary and Secondary Waves in Developmental Biology,
  13. (1988). Principles of Applied Mathematics, Transformation and Approximation,
  14. (1992). Sequential pattern formation in a model for skin morphogenesis,
  15. (1990). Size dependent pigmentation pattern formation in embryos of Alligator Mississippiensis: Time of initiation of pattern generation mechanism,
  16. (1985). Some aspects of the weakly non-linear theory of the morphological instability, doi
  17. (1997). Spatial pattern formation in chemical and biological systems, Faraday Transactions,
  18. (1995). Spatio-temporal patterns in a mechanical model for mesenchymal morphogenesis,
  19. (1952). The chemical basis of morphogenesis, doi
  20. (1994). Threshold bifurcation in tissue interaction models for spatial pattern generation,
  21. (1994). Travelling waves in a tissue interaction model for skin pattern formation,
  22. (1997). Unravelling the Turing bifurcation using spatially varying diusion coecients,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.