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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
Blair, S.S.: Limb development: marginal fringe benefits. Curr. Biol. 7, R686–R690 (1997)
Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)
Crandall, M.G., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52, 161–180 (1973)
Cross, M., Hohenberg, P.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)
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)
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)
Doelman, A., Gardner, R.A., Kaper, T.J.: Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 50, 443–507 (2001)
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)
Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik (Berlin) 12, 30–39 (1972)
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)
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)
Haken, H., Olbrich, H.: Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis. J. Math. Biol. 6, 317–331 (1978)
Kuramoto, Y., Tsuzuki, T.: On the formation of dissipative structures in reaction-diffusion systems—reductive perturbation approach. Prog. Theor. Phys. 54, 687–699 (1975)
Meinhardt, H.: Models of Biological Pattern Formation. Academic Press, London (1982)
Meinhardt, H.: The Algorithmic Beauty of Sea Shells, 4th edn. Springer, Berlin (2009)
Meinhardt, H., Klingler, M.: A model for pattern formation on the shells of molluscs. J. Theor. Biol. 126, 63–69 (1987)
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)
Mielke, A., Schneider, G.: Attractors for modulation equations on unbounded domains: existence and comparison. Nonlinearity 8, 743–768 (1995)
Mimura, M., Nishiura, Y.: Spatial patterns for an interaction-diffusion equation in morphogenesis. J. Math. Biol. 7, 243–263 (1979)
Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edn. Interdisciplinary Applied Mathematics, vol. 18. Springer, Berlin (2003)
Ni, W.-M.: Diffusion, cross-diffusion, and their spike-layer steady states. Not. Am. Math. Soc. 45, 9–18 (1998)
Nishiura, Y., Fujii, H.: Stability of singularly perturbed solutions to systems of reaction-diffusion equations. SIAM J. Math. Anal. 18, 1726–1770 (1987)
Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81, 389–400 (1979)
Takagi, I.: Stability of bifurcating solutions of the Gierer-Meinhardt system. Tohoku Math. J. 31, 221–246 (1979)
Taniguchi, M.: A uniform convergence theorem for singular limit eigenvalue problems. Adv. Differ. Equ. 8, 29–54 (2003)
Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B, Biol. Sci. 237, 37–72 (1952)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag London
About this chapter
Cite this chapter
Wei, J., Winter, M. (2014). Introduction. In: Mathematical Aspects of Pattern Formation in Biological Systems. Applied Mathematical Sciences, vol 189. Springer, London. https://doi.org/10.1007/978-1-4471-5526-3_1
Download citation
DOI: https://doi.org/10.1007/978-1-4471-5526-3_1
Publisher Name: Springer, London
Print ISBN: 978-1-4471-5525-6
Online ISBN: 978-1-4471-5526-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)