The Dynamics of Multispike Solutions to the One-Dimensional Gierer--Meinhardt Model

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 ε. In the limit ε → 0, 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-algebraicsystem 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 τ=ε² t towards a stable equilibrium, provided that the inhibitor diffusivity D 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 τ, 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.