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Pattern formation (II): The Turing Instability

Authors: Yan Guo and Hyung Ju Hwang
Journal: Proc. Amer. Math. Soc. 135 (2007), 2855-2866
MSC (2000): Primary 35K57, 35Pxx, 92Bxx
Published electronically: May 14, 2007
MathSciNet review: 2317962
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the classical Turing instability in a reaction-diffusion system as the secend part of our study on pattern formation. We prove that nonlinear dynamics of a general perturbation of the Turing instability is determined by the finite number of linear growing modes over a time scale of $ \ln\frac{1}{\delta},$ where $ \delta$ is the strength of the initial perturbation.

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  • 1. C. Bardos, Y. Guo, and W. Strauss, Stable and unstable ideal plane flows, Chinese Ann. Math. Ser. B 23 (2002), no. 2, 149–164. Dedicated to the memory of Jacques-Louis Lions. MR 1924132, 10.1142/S0252959902000158
  • 2. V. Castets, E. J. Boissonade, P. De Kepper, Experimental evidence for a sustained Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64 (1990), 2953-2956.
  • 3. R. Dillon, P. K. Maini, and H. G. Othmer, Pattern formation in generalized Turing systems. I. Steady-state patterns in systems with mixed boundary conditions, J. Math. Biol. 32 (1994), no. 4, 345–393. MR 1279745, 10.1007/BF00160165
  • 4. Peter Grindrod, Patterns and waves, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1991. The theory and applications of reaction-diffusion equations. MR 1136256
  • 5. Yan Guo, Instability of symmetric vortices with large charge and coupling constant, Comm. Pure Appl. Math. 49 (1996), no. 10, 1051–1080. MR 1404325, 10.1002/(SICI)1097-0312(199610)49:10<1051::AID-CPA2>3.3.CO;2-U
  • 6. Y. Guo, H.J. Hwang, Pattern formation (I): The Keller-Segel Model, preprint.
  • 7. Yan Guo, Chris Hallstrom, and Daniel Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys. 245 (2004), no. 2, 297–354. MR 2039699, 10.1007/s00220-003-1017-z
  • 8. A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30-39.
  • 9. Yan Guo and Walter A. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math. 48 (1995), no. 8, 861–894. MR 1361017, 10.1002/cpa.3160480803
  • 10. Hyung Ju Hwang and Yan Guo, On the dynamical Rayleigh-Taylor instability, Arch. Ration. Mech. Anal. 167 (2003), no. 3, 235–253. MR 1978583, 10.1007/s00205-003-0243-z
  • 11. K.J. Lee, W.D. McCormick, J.E. Pearson, H.L. Swinney, Experimental observation of self-replication spots in a reaction-diffusion system. Nature 369 (1994), 215-218.
  • 12. H. Meinhardt, Models of biological pattern formation, Academic Press, London (1982).
  • 13. J. D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer-Verlag, Berlin, 1989. MR 1007836
  • 14. Wei-Ming Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), no. 1, 9–18. MR 1490535
  • 15. Q. Quyang, H. Swinney, Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352 (1991), 610-612.
  • 16. J.J. Tyson, Classification of instabilities in chemical reaction systems, J. Chem. Phys. 62 (1975), 1010.
  • 17. A. Turing, The chemical basis of morphogenesis, Phil. Trans. Roc. Soc. B 237 (1952), 37-72.
  • 18. Evelyn Sander and Thomas Wanner, Pattern formation in a nonlinear model for animal coats, J. Differential Equations 191 (2003), no. 1, 143–174. MR 1973286, 10.1016/S0022-0396(02)00156-0

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Additional Information

Yan Guo
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912

Hyung Ju Hwang
Affiliation: School of Mathematics, Trinitiy College Dublin, Dublin 2, Ireland & Department of Mathematics, Postech, Pohang 790-784, Korea

Received by editor(s): October 19, 2005
Received by editor(s) in revised form: May 26, 2006
Published electronically: May 14, 2007
Communicated by: Walter Craig
Article copyright: © Copyright 2007 American Mathematical Society