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Published in:
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2014 | OriginalPaper | Chapter

1. Introduction

Authors : Juncheng Wei, Matthias Winter

Published in: Mathematical Aspects of Pattern Formation in Biological Systems

Publisher: Springer London

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Abstract

We introduce general two-component reaction-diffusion systems and Turing instability. Then we specialise on the Gierer-Meinhardt system for hydra. We discuss amplitude equations, order parameters and analytical methods for spiky patterns.

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next chapter Existence of Spikes for the Gierer-Meinhardt System in One Dimension
Literature
13.
Berding, C., Haken, H.: Pattern formation in morphogenesis. Analytical treatment of the Gierer-Meinhardt system on a sphere. J. Math. Biol. 14, 133–151 (1982) MathSciNetCrossRefMATH
17.
Blair, S.S.: Limb development: marginal fringe benefits. Curr. Biol. 7, R686–R690 (1997) CrossRef
30.
31.
Crandall, M.G., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52, 161–180 (1973) MathSciNetCrossRefMATH
34.
Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993) CrossRef
47.
Doelman, A., Gardner, R.A., Kaper, T.J.: Stability analysis of singular patterns in the 1-D Gray-Scott model: a matched asymptotic approach. Physica D 122, 1–36 (1998) MathSciNetCrossRefMATH
48.
Doelman, A., Eckhaus, W., Kaper, T.J.: Slowly modulated two-pulse solutions in the Gray-Scott model I: asymptotic construction and stability. SIAM J. Appl. Math. 61, 1080–1102 (2000) MathSciNetCrossRefMATH
49.
Doelman, A., Gardner, R.A., Kaper, T.J.: Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 50, 443–507 (2001) MathSciNetCrossRefMATH
51.
Doelman, A., Gardner, R.A., Kaper, T.J.: A stability index analysis of 1-D patterns of the Gray-Scott model. Mem. Am. Math. Soc. 155(737), xii + 64 pp. (2002)
73.
Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik (Berlin) 12, 30–39 (1972) CrossRef
75.
Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38, 29–43 (1983) CrossRef
76.
Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilites to the system A+2B→3B, B→C. Chem. Eng. Sci. 39, 1087–1097 (1984) CrossRef
84.
Haken, H., Olbrich, H.: Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis. J. Math. Biol. 6, 317–331 (1978) MathSciNetCrossRefMATH
122.
Kuramoto, Y., Tsuzuki, T.: On the formation of dissipative structures in reaction-diffusion systems—reductive perturbation approach. Prog. Theor. Phys. 54, 687–699 (1975) CrossRef
145.
Meinhardt, H.: Models of Biological Pattern Formation. Academic Press, London (1982)
152.
Meinhardt, H.: The Algorithmic Beauty of Sea Shells, 4th edn. Springer, Berlin (2009) CrossRef
155.
Meinhardt, H., Klingler, M.: A model for pattern formation on the shells of molluscs. J. Theor. Biol. 126, 63–69 (1987) MathSciNetCrossRef
157.
Mielke, A.: The Ginzburg-Landau equation in its role as a modulation equation. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, vol. 2, pp. 759–834. Elsevier, Amsterdam (2002)
158.
Mielke, A., Schneider, G.: Attractors for modulation equations on unbounded domains: existence and comparison. Nonlinearity 8, 743–768 (1995) MathSciNetCrossRefMATH
159.
Mimura, M., Nishiura, Y.: Spatial patterns for an interaction-diffusion equation in morphogenesis. J. Math. Biol. 7, 243–263 (1979) MathSciNetCrossRefMATH
166.
Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edn. Interdisciplinary Applied Mathematics, vol. 18. Springer, Berlin (2003)
168.
Ni, W.-M.: Diffusion, cross-diffusion, and their spike-layer steady states. Not. Am. Math. Soc. 45, 9–18 (1998) MATH
183.
Nishiura, Y., Fujii, H.: Stability of singularly perturbed solutions to systems of reaction-diffusion equations. SIAM J. Math. Anal. 18, 1726–1770 (1987) MathSciNetCrossRefMATH
214.
Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81, 389–400 (1979) MathSciNetCrossRef
225.
228.
Taniguchi, M.: A uniform convergence theorem for singular limit eigenvalue problems. Adv. Differ. Equ. 8, 29–54 (2003) MATH
232.
Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 237, 37–72 (1952) CrossRef
Metadata
Title
Introduction
Authors
Juncheng Wei
Matthias Winter
Copyright Year
2014
Publisher
Springer London
Book
Mathematical Aspects of Pattern Formation in Biological Systems

Print ISBN: 978-1-4471-5525-6

Electronic ISBN: 978-1-4471-5526-3

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-1-4471-5526-3_1

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