Geodesic
The Stability and Slow Dynamics of Two-Spike Patterns for a Class of Reaction-Diffusion System
Y. Nec1 ; M.J. Ward1
1 Department of Mathematics, University of British Columbia 1984 Mathematics Road, Vancouver, V6T1Z2, BC, Canada
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 206-232.

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The slow dynamics and linearized stability of a two-spike quasi-equilibrium solution to a general class of reaction-diffusion (RD) system with and without sub-diffusion is analyzed. For both the case of regular and sub-diffusion, the method of matched asymptotic expansions is used to derive an ODE characterizing the spike locations in the absence of any 𝒪(1) time-scale instabilities of the two-spike quasi-equilibrium profile. These fast instabilities result from unstable eigenvalues of a certain nonlocal eigenvalue problem (NLEP) that is derived by linearizing the RD system around the two-spike quasi-equilibrium solution. For a particular sub-class of the reaction kinetics, it is shown that the discrete spectrum of this NLEP is determined by the roots of some simple transcendental equations. From a rigorous analysis of these transcendental equations, explicit sufficient conditions are given to predict the occurrence of either Hopf bifurcations or competition instabilities of the two-spike quasi-equilibrium solution. The theory is illustrated for several specific choices of the reaction kinetics.
DOI : 10.1051/mmnp/20138513

Y. Nec 1 ; M.J. Ward 1

1 Department of Mathematics, University of British Columbia 1984 Mathematics Road, Vancouver, V6T1Z2, BC, Canada 
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Y. Nec; M.J. Ward. The Stability and Slow Dynamics of Two-Spike Patterns for a Class of Reaction-Diffusion System. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 206-232. doi : 10.1051/mmnp/20138513. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138513/

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