Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction Read chapter Read first chapter

2014 | OriginalPaper | Chapter

10. Spikes for Other Two-Component Reaction-Diffusion Systems

Authors : Juncheng Wei, Matthias Winter

Published in: Mathematical Aspects of Pattern Formation in Biological Systems

Publisher: Springer London

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We present results on the existence and stability of multiple spikes for reaction-diffusion systems of the activator-substrate type: The Schnakenberg and Gray-Scott models. We will consider the Schnakenberg model in one space dimension and the Gray-Scott model in two space dimensions. We will conclude by considering flow-distributed spikes, namely the influence of convection on the existence and stability of spikes in the case of the Schnakenberg model. In this chapter we focus on the main results and their biological relevance but skip most of the proofs.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter The Gierer-Meinhardt System with Saturation
next chapter Reaction-Diffusion Systems with Many Components
Literature
12.
Benson, D.L., Maini, P.K., Sherratt, J.A.: Unravelling the Turing bifurcation using spatially varying diffusion coefficients. J. Math. Biol. 37, 381–417 (1998) MathSciNetCrossRefMATH
25.
Chen, X., Kowalczyk, M.: Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system. Adv. Differ. Equ. 6, 847–872 (2001) MathSciNetMATH
26.
46.
Doelman, A., Kaper, T.J., Zegeling, P.A.: Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity 10, 523–563 (1997) MathSciNetCrossRefMATH
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
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)
52.
Doelman, A., Kaper, T.J., Peletier, L.A.: Homoclinic bifurcations at the onset of pulse self-replication. J. Differ. Equ. 231, 359–423 (2006) MathSciNetCrossRefMATH
56.
57.
Ei, S., Nishiura, Y., Ueda, K.: 2 n splitting or edge splitting: a manner of splitting in dissipative systems. Jpn. J. Ind. Appl. Math. 18, 181–205 (2001) MathSciNetCrossRefMATH
58.
59.
Ei, S., Mimura, M., Nagayama, M.: Interacting spots in reaction-diffusion systems. Discrete Contin. Dyn. Syst. 14, 31–62 (2006) MathSciNetMATH
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
77.
Guckenheimer, J.: On a codimension two bifurcation. In: Dynamical Systems and Turbulence, Warwick 1980, Coventry, 1979/1980. Lecture Notes in Mathematics, vol. 898, pp. 99–142. Springer, Berlin (1981) CrossRef
85.
Hale, J.K., Peletier, L.A., Troy, W.C.: Stability and instability of the Gray-Scott model: the case of equal diffusion constants. Appl. Math. Lett. 12, 59–65 (1999) MathSciNetCrossRefMATH
86.
Hale, J.K., Peletier, L.A., Troy, W.C.: Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis. SIAM J. Appl. Math. 61, 102–130 (2000) MathSciNetCrossRefMATH
98.
Iron, D., Ward, M.J.: The dynamics of multi-spike solutions to the one-dimensional Gierer-Meinhardt system. SIAM J. Appl. Math. 62, 1924–1951 (2002) MathSciNetCrossRefMATH
100.
Iron, D., Wei, J., Winter, M.: Stability analysis of Turing patterns generated by the Schnakenberg model. J. Math. Biol. 49, 358–390 (2004) MathSciNetCrossRefMATH
106.
Keener, J.P.: Secondary bifurcation in nonlinear diffusion reaction equations. Stud. Appl. Math. 55, 187–211 (1976) MathSciNetMATH
107.
112.
Kolokolnikov, T., Ward, M.J., Wei, J.: The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the low-feed regime. Stud. Appl. Math. 115, 21–71 (2005) MathSciNetCrossRefMATH
113.
Kolokolnikov, T., Ward, M.J., Wei, J.: The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime. Physica D 202, 258–293 (2005) MathSciNetCrossRefMATH
115.
Kolokolnikov, T., Ward, M.J., Wei, J.: Spot self-replication and dynamics for the Schnakenberg model in a two-dimensional domain. J. Nonlinear Sci. 19, 1–56 (2009) MathSciNetCrossRefMATH
163.
Muratov, C.B., Osipov, V.V.: Static spike autosolitons in the Gray-Scott model. J. Phys. A, Math. Gen. 33, 8893–8916 (2000) MathSciNetCrossRef
164.
Muratov, C.B., Osipov, V.V.: Stability of the static spike autosolitons in the Gray-Scott model. SIAM J. Appl. Math. 62, 1463–1487 (2002) MathSciNetCrossRefMATH
181.
Nishiura, Y.: Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal. 13, 555–593 (1982) MathSciNetCrossRefMATH
182.
Nishiura, Y.: Far-from-Equilibrium-Dynamics. Translations of Mathematical Monographs, vol. 209. Am. Math. Soc., Providence (2002) 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
184.
Nishiura, Y., Ueyama, D.: A skeleton structure of self-replicating dynamics. Physica D 130, 73–104 (1999) CrossRefMATH
185.
Nishiura, Y., Ueyama, D.: Spatio-temporal chaos for the Gray-Scott model. Physica D 150, 137–162 (2001) CrossRefMATH
186.
Nishiura, Y., Teramoto, T., Ueda, K.: Scattering and separators in dissipative systems. Phys. Rev. E 67, 056210 (2003) CrossRef
203.
Pearson, J.E.: Complex patterns in a simple system. Science 261, 189–192 (1993) CrossRef
212.
213.
Satnoianu, R.A., Maini, P.K., Menzinger, M.: Parameter domains for Turing and stationary flow-distributed waves: I. The influence of nonlinearity. Physica D 160, 79–102 (2001) MathSciNetCrossRefMATH
214.
Schnakenberg, J.: Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol. 81, 389–400 (1979) MathSciNetCrossRef
223.
Sun, W., Ward, M.J., Russell, R.: The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: competition and oscillatory instabilities. SIAM J. Appl. Dyn. Syst. 4, 904–953 (2005) MathSciNetCrossRefMATH
241.
Ward, M.J., Wei, J.: The existence and stability of asymmetric spike patterns for the Schnakenberg model. Stud. Appl. Math. 109, 229–264 (2002) MathSciNetCrossRefMATH
250.
Wei, J.: Existence, stability and metastability of point condensation patterns generated by the Gray-Scott system. Nonlinearity 12, 593–616 (1999) MathSciNetCrossRefMATH
254.
Wei, J.: Pattern formations in two-dimensional Gray-Scott model: existence of single-spot solutions and their stability. Physica D 148, 20–48 (2001) MathSciNetCrossRefMATH
267.
Wei, J., Winter, M.: Existence and stability of multiple-spot solutions for the Gray-Scott model in \({\mathbb{R}}^{2}\). Physica D 176, 147–180 (2003) MathSciNetCrossRefMATH
268.
274.
279.
286.
Wittenberg, R.W., Holmes, P.: The limited effectiveness of normal forms: a critical review and extension of local bifurcation studies of the Brusselator PDE. Physica D 100, 1–40 (1997) MathSciNetCrossRefMATH
Metadata
Title
Spikes for Other Two-Component Reaction-Diffusion Systems
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_10

Premium Partner

    Image Credits