Skip to main content

Pattern Formation

  • Chapter
  • First Online:
  • 2015 Accesses

Part of the book series: Springer Monographs in Mathematics ((SMM))

Abstract

Surface patterns are common in biological objects. The stripes of zebras, the spots of the leopard are ubiquitous in the popular scientific literature [137], but also invertebrates and plants show various patterns. In the present context the patterns on the shells of molluscs (especially snails and mussels) are of special interest. These patterns are not produced all at once. Since the shell gradually grows by adding material to the outer edge and since the pattern is not changed later, the pattern is essentially located in a space-time continuum, with the space axis parallel to the edge and the time axis perpendicular to the edge.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   69.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Y. Bar-Yam, Dynamics of Complex Systems (Perseus Books, Cambridge, 1997)

    MATH  Google Scholar 

  2. A. Budd, C. McDougall, K. Green, B.M. Degnan, Control of shell pigmentation by secretory tubules in the abalone mantle. Front. Zool. 11, 62 (2014)

    Article  Google Scholar 

  3. M. Cross, H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems (Cambridge University Press, Cambridge, 2009)

    Book  MATH  Google Scholar 

  4. A. Deutsch, S. Doormann, Cellular Automaton Modeling of Biological Pattern Formation (Birkhäuser, Basel, 2004)

    Google Scholar 

  5. R.L. Devaney, Chaotic Dynamical Systems (Westview Press, New York, 2003)

    MATH  Google Scholar 

  6. B. Ermentrout, Neural networks as spatio-temporal pattern-forming systems. Rep. Prog. Phys. 61, 353–430 (1998)

    Article  Google Scholar 

  7. J. Greenberg, S. Hastings, Spatial patterns for discrete models of diffusion in excitable media. SIAM J. Appl. Math. 34, 515–523 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Markus, Modelling morphogenetic processes in excitable media using cellular automata. Biomed. Biochem. Acta 49, 681–696 (1990)

    Google Scholar 

  9. H. Meinhardt, The Algorithmic Beauty of Shells (Springer, Berlin, 1995)

    Book  MATH  Google Scholar 

  10. H. Meinhardt, M. Klinger, A model of pattern formation on shells of molluscs. J. Theor. Biol. 126, 63–89 (1987)

    Article  MathSciNet  Google Scholar 

  11. J. Müller, C. Kuttler, Methods and Models in Mathematical Biology (Springer, Berlin, 2015)

    Book  MATH  Google Scholar 

  12. J. Murray, Mathematical Biology (Springer, Berlin, 1989)

    Book  MATH  Google Scholar 

  13. B. Schönfisch, A. de Roos, Synchronous and asynchronous update in cellular automata. BioSystems 51, 123–143 (1999)

    Article  Google Scholar 

  14. A. Turing, The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37–72 (1952)

    Article  MathSciNet  Google Scholar 

  15. C. Waddington, R. Cowe, Computer simulation of a molluscan. J. Theor. Biol. 25, 219–225 (1969)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Hadeler, KP., Müller, J. (2017). Pattern Formation. In: Cellular Automata: Analysis and Applications. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-53043-7_12

Download citation

Publish with us

Policies and ethics