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Explicitly solvable eigenvalue problem and bifurcation delay in sub-diffusive Gierer–Meinhardt model

Published online by Cambridge University Press:  29 January 2016

YANA NEC
YANA NEC*
Affiliation:
Thompson Rivers University†, 900 McGill road, Kamloops, British Columbia, Canada email: cranberryana@gmail.com
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Abstract

A spike solution is constructed on the infinite line for a sub-diffusive version of the Gierer–Meinhardt reaction – diffusion model. A non-local eigenvalue problem governs the spike's stability and is explicitly solvable for a certain choice of the kinetic parameters. Its solution generalises former results for the Gierer–Meinhardt model with regular diffusion, and the normal and anomalous systems' properties are juxtaposed. It is shown that a Hopf bifurcation occurs in the sub-diffusive system for larger values of the time parameter τo as compared to the normal counterpart, rendering the anomalous system more stable. Asymptotic solutions are obtained near important values of the diffusion anomaly index γ and collectively shown to be valid over most of the applicable range 0 < γ < 1. A bifurcation delay scenario is described for the sub-diffusive system, and the WKB exponent is computed analytically.

Keywords

Sub-diffusionreaction – diffusionspike solutionnon-local eigenvalue problembifurcation delay
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 5 , October 2016 , pp. 699 - 725
Copyright
Copyright © Cambridge University Press 2016 

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