Stability of Least Energy Patterns of the Shadow System for an Activator-Inhibitor Model

Wei Ming Ni; Izumi Takagi; Eiji Yanagida

doi:10.1007/BF03168574

Stability of Least Energy Patterns of the Shadow System for an Activator-Inhibitor Model

Wei Ming Ni, Izumi Takagi, Eiji Yanagida

Institute of Liberal Arts and Sciences

Research output: Contribution to journal › Article › peer-review

28 Citations (Scopus)

Abstract

Stability of stationary solutions to the shadow system for the activator-inhibitor system proposed by Gierer and Meinhardt is considered in higher dimensional domains. It is shown that a stationary solution with minimal "energy" is stable in a weak sense if the inhibitor reacts sufficiently fast, while it is unstable whenever the reaction of the inhibitor is slow. Moreover, the loss of stability results in a Hopf bifurcation.

Original language	English
Pages (from-to)	259-272
Number of pages	14
Journal	Japan Journal of Industrial and Applied Mathematics
Volume	18
Issue number	2
DOIs	https://doi.org/10.1007/BF03168574
Publication status	Published - 2001 Jun

Keywords

Hopf bifurcation
Reaction-diffusion system
Shadow system
Spike-layer
Stability

ASJC Scopus subject areas

Engineering(all)
Applied Mathematics

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10.1007/BF03168574

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abstract = "Stability of stationary solutions to the shadow system for the activator-inhibitor system proposed by Gierer and Meinhardt is considered in higher dimensional domains. It is shown that a stationary solution with minimal {"}energy{"} is stable in a weak sense if the inhibitor reacts sufficiently fast, while it is unstable whenever the reaction of the inhibitor is slow. Moreover, the loss of stability results in a Hopf bifurcation.",

keywords = "Hopf bifurcation, Reaction-diffusion system, Shadow system, Spike-layer, Stability",

author = "Ni, {Wei Ming} and Izumi Takagi and Eiji Yanagida",

year = "2001",

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AU - Ni, Wei Ming

AU - Takagi, Izumi

AU - Yanagida, Eiji

PY - 2001/6

Y1 - 2001/6

N2 - Stability of stationary solutions to the shadow system for the activator-inhibitor system proposed by Gierer and Meinhardt is considered in higher dimensional domains. It is shown that a stationary solution with minimal "energy" is stable in a weak sense if the inhibitor reacts sufficiently fast, while it is unstable whenever the reaction of the inhibitor is slow. Moreover, the loss of stability results in a Hopf bifurcation.

AB - Stability of stationary solutions to the shadow system for the activator-inhibitor system proposed by Gierer and Meinhardt is considered in higher dimensional domains. It is shown that a stationary solution with minimal "energy" is stable in a weak sense if the inhibitor reacts sufficiently fast, while it is unstable whenever the reaction of the inhibitor is slow. Moreover, the loss of stability results in a Hopf bifurcation.

KW - Hopf bifurcation

KW - Reaction-diffusion system

KW - Shadow system

KW - Spike-layer

KW - Stability

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JF - Japan Journal of Industrial and Applied Mathematics

IS - 2

ER -

Stability of Least Energy Patterns of the Shadow System for an Activator-Inhibitor Model

Abstract

Keywords

ASJC Scopus subject areas

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