Colloquium
3:30 p.m., Friday
Math 100
Michael Ward
UBC
The Stability and Dynamics of Spikes for a ReactionDiffusion System
Many classes of singularly perturbed reactiondiffusion systems possess
localized solutions where the gradient of the solution is very large
only in the vicinity of certain points in the domain. An example of
a problem where such spikes occur is the GeirerMeinhardt (GM)
activatorinhibitor system modeling biological morphogenesis. In the
limit of a small activator diffusivity, this system has been used to
model many situations including spottype patterns on seashells and
head formation in the Hydra. Most of the previous work on this
system over the past twenty years has been based either on full
numerical simulations or on a linearized Turingtype stability analysis
around spatially homogeneous steadystate solutions. However, this type of
linearized analysis is not appropriate for determining the stability
of spiketype patterns. In this talk we will survey some recent results
on the existence and stability of symmetric and asymmetric equilibrium
spike patterns for the GM model. The inhibitor diffusivity is
found to be a critical parameter. A key result that is obtained is that
there exists a sequence of critical values D_n of the inhibitor
diffusivity D for which an nspike symmetric equilibrium solution
is stable if D < D_n and unstable if D > D_n. An explicit formula for
D_n is given. The dynamics of spike patterns is also characterized in
a onedimensional domain and partial results are obtained in a
multidimensional context. The mathematical tools used include asymptotic
analysis, spectral analysis of nonlocal eigenvalue operators, dynamical
systems, and numerical and matrix analysis.
Joint work with David Iron (graduate student at UBC), and Prof.
Juncheng Wei (Chinese University of Hong Kong).
