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Topics in the stability of localized patterns for some reaction-diffusion systems

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dc.contributor.author Rozada, Ignacio
dc.date.accessioned 2012-08-22T17:40:03Z
dc.date.available 2012-08-22T17:40:03Z
dc.date.copyright 2012 en
dc.date.issued 2012-08-22
dc.identifier.uri http://hdl.handle.net/2429/43012
dc.description.abstract In the first part of this thesis, we study the existence and stability of multi-spot patterns on the surface of a sphere for a singularly perturbed Brusselator and Schnakenburg reaction-diffusion model. The method of matched asymptotic expansions, tailored to problems with logarithmic gauge functions, is used to construct both symmetric and asymmetric spot patterns. There are three distinct types of instabilities of these patterns that are analyzed: self-replication instabilities, amplitude oscillations of the spots, and competition instabilities. By using a combination of spectral theory for nonlocal eigenvalue problems together with numerical computations, parameter thresholds for these three different classes of instabilities are obtained. For the Brusselator model, our results point towards the existence of cycles of creation and destruction of spots, and possibly to chaotic dynamics. For the Schnakenburg model, a differential-algebraic ODE system for the motion of the spots on the surface of the sphere is derived. In the second part of the thesis, we study the existence and stability of mesa solutions in one spatial dimension and the corresponding planar mesa stripe patterns in two spatial dimensions. An asymptotic analysis is used in the limit of a large diffusivity ratio to construct mesa patterns in one spatial dimension for a general class of two-component reaction-diffusion systems that includes the well-known Gierer Meinhardt activator-inhibitor model with saturation (GMS model), and a predator-prey model. For such one-dimensional patterns, we study oscillatory instabilities of the pattern by way of a Hopf bifurcation and from a reduction to a limiting ODE-PDE system. In addition, explicit thresholds are derived characterizing transverse instabilities of planar mesa-stripe patterns in two spatial dimensions. The results of our asymptotic theory as applied to the GMS and predator-prey systems are confirmed with full numerical results. en
dc.language.iso eng en
dc.publisher University of British Columbia en
dc.rights Attribution-NonCommercial 2.5 Canada
dc.rights.uri http://creativecommons.org/licenses/by-sa/3.0/
dc.title Topics in the stability of localized patterns for some reaction-diffusion systems en
dc.type Electronic Thesis or Dissertation en
dc.degree.name Doctor of Philosophy - PhD en
dc.degree.discipline Mathematics en
dc.degree.grantor University of British Columbia en
dc.date.graduation 2012-11 en
dc.degree.campus UBCV en
dc.description.scholarlevel Graduate en

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