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
DOI:
https://doi.org/10.1090/S0002-9939-07-08850-8
Published electronically:
May 14, 2007
MathSciNet review:
2317962
Full-text PDF
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 where
is the strength of the initial perturbation.
- 1. Bardos, C; Y. Guo; W. Strauss: Stable and unstable ideal plane flows. Dedicated to the memory of Jacques-Lious Lions, Chinese Ann. Math. Ser B. 23 (2002), no 2, 149-164.MR 1924132 (2003k:35192)
- 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, 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 (95e:92004)
- 4. P. Grinrod, Patterns and Waves: The Theory and Applications of Reaction-Diffusion equations, Oxford: Clarendon, 1991. MR 1136256 (92k:35145)
- 5. Y. Guo: Instability of symmetric vortices with large charge and coupling constant. Comm. Pure Appl. Math. 49 (1996) no. 8, 1051-1080. MR 1404325 (97e:35175)
- 6. Y. Guo, H.J. Hwang, Pattern formation (I): The Keller-Segel Model, preprint.
- 7. Y. Guo, C. Hallstrom, and D. Spirn, Dynamics near an unstable Kirchhoff ellipse, Comm. Math. Phys. 245, (2004) 297-354. MR 2039699 (2005m:76039)
- 8. A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972), 30-39.
- 9. Y. Guo, W. Strauss: Instability of periodic BGK equilibria. Comm. Pure Appl. Math. 48 (1995) no. 3, 861-894.MR 1361017 (96j:35252)
- 10. H-J, Hwang; Y. Guo: On the dynamical Rayleigh-Taylor instability. Arch. Ration. Mech. Anal. 167 (2003), no. 3, 235-253. MR 1978583 (2004f:76064)
- 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. Murray, Mathematical Biology, Springer-Verlag. MR 1007836 (90g:92001)
- 14. W.M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states. Notices Amer. Math. Soc. 45 (1998), no. 1, 9-18. MR 1490535 (99a:35132)
- 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. E. Sander, T. Wanner, Pattern formation in a nonlinear model for animal coats. J. Diff. Eqs. 191 (2003) 143-174.MR 1973286 (2004b:35148)
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35K57, 35Pxx, 92Bxx
Retrieve articles in all journals with MSC (2000): 35K57, 35Pxx, 92Bxx
Additional Information
Yan Guo
Affiliation:
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email:
guoy@dam.brown.edu
Hyung Ju Hwang
Affiliation:
School of Mathematics, Trinitiy College Dublin, Dublin 2, Ireland & Department of Mathematics, Postech, Pohang 790-784, Korea
Email:
hjhwang@postech.edu
DOI:
https://doi.org/10.1090/S0002-9939-07-08850-8
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