Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T11:18:56.442Z Has data issue: false hasContentIssue false
  • English
  • Français

Hopf bifurcation from spike solutions for the weak coupling Gierer–Meinhardt system

Part of: Physiological, cellular and medical topics Partial differential equations

Published online by Cambridge University Press:  17 March 2020

DANIEL GOMEZ ,
LINFENG MEI Open the ORCID record for LINFENG MEI [Opens in a new window]  and
JUNCHENG WEI
DANIEL GOMEZ
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, BCCanadaV6T 1Z2 emails: dagubc@math.ubc.ca; jcwei@math.ubc.ca
LINFENG MEI
Affiliation:
College of Mathematics and Computer Sciences, Zhejiang Normal University, Jinhua321004, Zhejiang, P.R. China email: lfmei@outlook.com
JUNCHENG WEI
Affiliation:
Department of Mathematics, The University of British Columbia, Vancouver, BCCanadaV6T 1Z2 emails: dagubc@math.ubc.ca; jcwei@math.ubc.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

The Hopf bifurcation from spike solutions for the classical Gierer–Meinhardt system in a onedimensional interval is considered. The existence of time-periodic solution near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of the limit cycle are determined, and it is shown that the limit cycle is unstable.

Keywords

Gierer-Meinhard systemHopf bifurcationunstable limit cyclespike solutionnonlinear eigenvalue problem

MSC classification

Primary: 35B10: Periodic solutions 92C40: Biochemistry, molecular biology
Secondary: 35B32: Bifurcation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 1 , February 2021 , pp. 113 - 145
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

L. Mei is supported by the National Natural Science Foundation of China (11771125, 11371117). J. Wei is partially supported by NSERC of Canada. D. Gomez is supported by NSERC of Canada.

References

Crandall, M. G. & Rabinowitz, P. H. (1977) The Hopf bifurcation theorem in infinite dimensions. Arch. Rational Mech. Anal. 67(1), 53–72.CrossRefGoogle Scholar
Dancer, E. N. (2001) On stability and Hopf bifurcations for chemotaxis systems. Methods Appl. Anal. 8(2), 245–256.Google Scholar
Doelman, A., Gardner, R. A. & Kaper, T. J. (2001) Large stable pulse solutions in reaction-diffusion equations. Indiana Univ. Math. J. 50(1), 443–507.CrossRefGoogle Scholar
Doelman, A., Kaper, T. J. & Promislow, K. (2007) Nonlinear asymptotic stability of the semistrong pulse dynamics in a regularized Gierer-Meinhardt model. SIAM J. Math. Anal. 38(6), 1760–1787.CrossRefGoogle Scholar
Doelman, A., Kaper, T. J. & van der Ploeg, H. (2001) Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer-Meinhardt equation. Methods Appl. Anal. 8(3), 387–414.Google Scholar
Gierer, A. & Meinhardt, H. (1972) A theory of biological pattern formation. Kybernetik, Continued Biol. Cybern. 12(1), 30–39.Google ScholarPubMed
Gomez, D., Mei, L. & Wei, J. (2020) Stable and unstable periodic spiky solutions for the Gray-Scott system and the Schnakenberg system. J. Dyn. Diff. Equ. 32(1), 442–481.Google Scholar
Henry, D. (1981) Geometric Theory of Semilinear Parabolic Equations, Vol. 840, Lecture Notes in Mathematics. Springer-Verlag, Berlin, New York.CrossRefGoogle Scholar
Iron, D., Ward, M. J. & Wei, J. (2001) The stability of spike solutions to the one-dimensional Gierer-Meinhardt model. Phys. D 150(1–2), 25–62.CrossRefGoogle Scholar
Kielhöfer, H. (2004) Bifurcation Theory: An Introduction with Applications to PDEs, Vol. 156. Applied Mathematical Sciences. Springer, New York, NY.CrossRefGoogle Scholar
Lin, C.-S. & Ni, W.-M. (1988) On the diffusion coefficient of a semilinear Neumann problem. In: Calculus of Variations and Partial Differential Equations (Trento, 1986), Vol. 1340. Search Results Featured snippet from the web Image result for Lecture Notes in Math link.springer.com Lecture Notes in Mathematics. Springer, Berlin, pp. 160–174.CrossRefGoogle Scholar
Maini, P. K., Woolley, T. E., Gaffney, E. A. & Baker, R. E. (2016). Turing's theory of developmental pattern formation. In: The Once and Future Turing. Cambridge University Press, Cambridge, pp. 131–143.CrossRefGoogle Scholar
Meinhardt, H. (1982). Models of Biological Pattern Formation. The Virtual Laboratory. Academic Press, London. With Contributions and Images by Prusinkiewicz, P. & Fowler, D. R. With 1 IBMPC floppy disk (3.5 inch; HD).Google Scholar
Ni, W.-M. & Takagi, I. (1993) Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70(2), 247–281.CrossRefGoogle Scholar
Ni, W.-M., Takagi, I. & Yanagida, E. (2001). Stability of least energy patterns of the shadow system for an activator-inhibitor model. Japan J. Indust. Appl. Math. 18(2), 259–272. Recent Topics in Mathematics Moving Toward Science and Engineering.CrossRefGoogle Scholar
Ruuth, S. J. (1995) Implicit-explicit methods for reaction-diffusion problems in pattern formation. J. Math. Biol. 34(2), 148–176.CrossRefGoogle Scholar
Sun, W., Ward, M. J. & Russell, R. (2005) 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(4), 904–953.CrossRefGoogle Scholar
Takagi, I. (1986) Point-condensation for a reaction-diffusion system. J. Diff. Equ. 61(2), 208–249.CrossRefGoogle Scholar
van der Ploeg, H. & Doelman, A. (2005) Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations. Indiana Univ. Math. J. 54(5), 1219–1301.Google Scholar
Veerman, F. Breathing pulses in singularly perturbed reaction-diffusion systems. Nonlinearity 28(7), 2211–2246.CrossRefGoogle Scholar
Ward, M. J. & Wei, J. (2003a) Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model. European J. Appl. Math. 14(6), 677–711.CrossRefGoogle Scholar
Ward, M. J. & Wei, J. (2003b) Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Sci. 13(2), 209–264.CrossRefGoogle Scholar
Wei, J. (1999) On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates. European J. Appl. Math. 10(4), 353–378.CrossRefGoogle Scholar
Wei, J. & Winter, M. (2001) Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case. J. Nonlinear Sci. 11(6), 415–458.CrossRefGoogle Scholar
Wei, J. & Winter, M. (2007) Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in R1. Methods Appl. Anal. 14(2), 119–163.Google Scholar
Wei, J. & Winter, M. (2014) Mathematical Aspects of Pattern Formation in Biological Systems, Vol. 189. Applied Mathematical Sciences. Springer, London.CrossRefGoogle Scholar