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Existence, stability, and dynamics of ring and near-ring solutions to the saturated Gierer--Meinhardt model in the semistrong regime

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dc.contributor.author Ward, Michael J.
dc.contributor.author Moyles, Iain R.
dc.date.accessioned 2018-02-09T16:19:44Z
dc.date.available 2018-02-09T16:19:44Z
dc.date.issued 2017
dc.identifier.uri http://hdl.handle.net/10344/6547
dc.description peer-reviewed en_US
dc.description.abstract We analyze a singularly perturbed reaction-diffusion system in the semi-strong diffusion regime in two spatial dimensions where an activator species is localized to a closed curve, while the inhibitor species exhibits long range behavior over the domain. In the limit of small activator diffusivity we derive a new moving boundary problem characterizing the slow time evolution of the curve, which is defined in terms of a quasi steady-state inhibitor diffusion field and its properties on the curve. Numerical results from this curve evolution problem are illustrated for the Gierer-Meinhardt model (GMS) with saturation in the activator kinetics. A detailed analysis of the existence, stability, and dynamics of ring and near-ring solutions for the GMS model is given, whereby the activator concentrates on a thin ring concentric within a circular domain. A key new result for this ring geometry is that by including activator saturation there is a qualitative change in the phase portrait of ring equilibria, in that there is an S-shaped bifurcation diagram for ring equilibria, which allows for hysteresis behavior. In contrast, without saturation, it is well-known that there is a saddle-node bifurcation for the ring equilibria. For a near-circular ring, we develop an asymptotic expansion up to quadratic order to fully characterize the normal velocity perturbations from our curve-evolution problem. In addition, we also analyze the linear stability of the ring solution to both breakup instabilities, leading to the disintegration of a ring into localized spots, and zig-zag instabilities, leading to the slow shape deformation of the ring. We show from a nonlocal eigenvalue problem that activator saturation can stabilize breakup patterns that otherwise would be unstable. Through a detailed matched asymptotic analysis, we derive a new explicit formula for the small eigenvalues associated with zig-zag instabilities, and we show that they are equivalent to the velocity perturbations induced by the near-circular ring geometry. Finally, we present full numerical simulations from the GMS PDE system that confirm the predictions of the analysis. en_US
dc.language.iso eng en_US
dc.publisher Society for Industrial and Applied Mathematics en_US
dc.relation.ispartofseries Siam Journal on Applied Dynamical Systems;16 (1), pp. 597-639
dc.relation.uri https://doi.org/10.1137/16M1060327
dc.subject homoclinic orbits en_US
dc.subject zig-zag instability en_US
dc.subject nonlocal eigenvalue problem en_US
dc.subject WKB stability en_US
dc.subject hysteresis en_US
dc.title Existence, stability, and dynamics of ring and near-ring solutions to the saturated Gierer--Meinhardt model in the semistrong regime en_US
dc.type info:eu-repo/semantics/article en_US
dc.type.supercollection all_ul_research en_US
dc.type.supercollection ul_published_reviewed en_US
dc.date.updated 2018-02-09T15:59:37Z
dc.identifier.doi 10.1137/16M1060327
dc.contributor.sponsor SFI en_US
dc.relation.projectid SFI/13/IA/1923 en_US
dc.rights.accessrights info:eu-repo/semantics/openAccess en_US
dc.internal.rssid 2705575
dc.internal.copyrightchecked Yes
dc.identifier.journaltitle Siam Journal On Applied Dynamical Systems
dc.description.status peer-reviewed


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