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: email@example.com
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.