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

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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 $\varepsilon \to 0$, 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 $\tau=\varepsilon^2 t$ 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 $\tau$, 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.