| Abstract: |
| We consider a morphogenetic mechanism generating filamentary lines by the
trail of a traveling spike. The underlying model, suggested by
Meinhardt, consists of three components: activator, substrate and excitator.
We analyze this system in one space dimension for the singular limit of small
activator diffusivity. Using Liapunov-Schmidt reduction, we rigorously prove
the existence and (orbital) stability of a traveling spike solution with a
trail, and most importantly, derive a formula for the pulse speed. The result
is in contrast to two-component system where the spikes are often stationary in
space. In the current case, the asymmetric far-field information of the
excitator provides the key driving force for the motion of the spike. This is
joint work with Matthias Winter. |
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