RESUMEN
The dynamical
behavior of spike-type solutions to a simplified form of the Gierer--Meinhardt activator-inhibitor model in a
one-dimensional domain is studied asymptotically and numerically in
the limit of small activator diffusivity varepsilon. In the limit varepsilon to, a quasi-equilibrium solution for
the activator concentration that has n localized peaks, or spikes,
is constructed asymptotically using the method of matched asymptotic expansions.
For an initial condition of this form, a differential-algebraic
system of equations describing the evolution of the spike
locations is derived. The equilibrium solutions for this system are
discussed. The spikes are shown to evolve on a slow time scale tau varepsilon
towards a stable equilibrium, provided that the
inhibitor diffusivity is below some threshold and that a certain
stability criterion on the quasi-equilibrium solution is satisfied
throughout the slow dynamics. If this stability condition is not
satisfied initially or else is no longer satisfied at some later
value of the slow time tau,
the quasi-equilibrium profile becomes unstable on a fast O(1) time scale. It is shown numerically that
this O(1) instability
leads to a spike collapse event. The asymptotic theory is compared
with corresponding full numerical results.