Technion - Israel Institute of Technology Department of Mathematics ===================================== PDE AND APPLIED MATHEMATICS SEMINAR ===================================== DATE: Tuesday, April 24, 2012 SPEAKER: Yana Nec, University of British Columbia, Vancouver TITLE: Title: Dynamics of spike-type solutions to a one dimensional Gierer-Meinhardt model with anomalous diffusion PLACE: Room 814, Amado Mathematics Building, Technion TIME: 14:30 ABSTRACT: The Gierer-Meinhardt model was modified to include first sub-diffusion and then Levy flights. For the former the dynamics and stability of spike-type patterns was studied in a one dimensional spatial domain. A differential algebraic system was derived to characterise the dynamics of an n-spike quasi-equilibrium pattern in the presence of sub-diffusion. With sub-diffusive effects it is shown that quasi-equilibrium spike patterns exist for diffusivity ratios asymptotically smaller than for the case of regular diffusion, and that the spikes approach their equilibrium locations at an algebraic, rather than exponential, rate in time. A new non-local eigenvalue problem was derived to examine the stability of an n-spike pattern. For a two spike pattern sub-diffusion has little effect on the competition instability threshold, whereas the threshold associated with an oscillatory instability of the spike profile increases significantly. Furthermore, for a multi-spike pattern it is shown that an asynchronous oscillatory instability of the spike profile, rather than a synchronous oscillatory instability characteristic of the case of regular diffusion, is the dominant instability when the anomaly index $\g$ is below a certain threshold. With Levy flights the model is shown to give rise to patterns of spikes with algebraically decaying tails. The spike shape is given by a solution to a fractional differential equation. Near an equilibrium formation the spikes drift according to the differential equations of the form known for Fickian diffusion, but with a new homoclinic. A non-local eigenvalue problem of a new type is formulated and studied. The system is less stable due to the Levy flights, though the behaviour of eigenvalues is changed mainly quantitatively. For further info: Yehuda Pinchover <pincho@techunix.technion.ac.il> For past and future Applied Math/PDE seminars see: <http://www.math.technion.ac.il/pde/seminar.html> --------------------------------------------------------- Technion Math. Net (TECHMATH) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Yehuda Pinchover <pincho@tx.technion.ac.il>