Technion - Israel Institute of Technology
          Department of Mathematics
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
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
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.
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