Math. Model. Nat. Phenom.
Volume 8, Number 5, 2013Bifurcations
|Page(s)||206 - 232|
|Published online||18 September 2013|
The Stability and Slow Dynamics of Two-Spike Patterns for a Class of Reaction-Diffusion System
Department of Mathematics, University of British Columbia 1984
Mathematics Road, Vancouver, V6T1Z2, BC,
⋆ Corresponding author. E-mail: firstname.lastname@example.org
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.
Mathematics Subject Classification: 35Q53 / 34B20 / 35G31
Key words: matched asymptotic expansions / bifurcation / spikes / nonlocal eigenvalue problem / Hopf bifurcation / sub-diffusion
© EDP Sciences, 2013
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