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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
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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
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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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.

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