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An Explicitly Solvable Nonlocal Eigenvalue Problem and theStability of a Spike for a Sub-Diffusive Reaction-Diffusion System

Published online by Cambridge University Press:  24 April 2013

Y. Nec  and
M. J. Ward
Y. Nec
Affiliation:
Department of Mathematics, University of British Columbia 1984 Mathematics Road, Vancouver, V6T1Z2, BC, Canada
M. J. Ward*
Affiliation:
Department of Mathematics, University of British Columbia 1984 Mathematics Road, Vancouver, V6T1Z2, BC, Canada
*
⋆Corresponding author. E-mail: ward@math.ubc.ca
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Abstract

The stability of a one-spike solution to a general class of reaction-diffusion (RD)system with both regular and anomalous diffusion is analyzed. The method of matchedasymptotic expansions is used to construct a one-spike equilibrium solution and to derivea nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O(1) time-scale. For a particular sub-class of the reactionkinetics, it is shown that the discrete spectrum of this NLEP is determined in terms ofthe roots of certain simple transcendental equations that involve two key parametersrelated to the choice of the nonlinear kinetics. From a rigorous analysis of thesetranscendental equations by using a winding number approach and explicit calculations,sufficient conditions are given to predict the occurrence of Hopf bifurcations of theone-spike solution. Our analysis determines explicitly the number of possible Hopfbifurcation points as well as providing analytical formulae for them. The analysis isimplemented for the shadow limit of the RD system defined on a finite domain and for aone-spike solution of the RD system on the infinite line. The theory is illustrated fortwo specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation isunique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopfbifurcation.

Keywords

matched asymptotic expansionsnonlocal eigenvalue problemwinding numberHopf bifurcationsub–diffusion

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Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 2: Anomalous diffusion , 2013 , pp. 55 - 87
Copyright
© EDP Sciences, 2013

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