The Slow Dynamics of Two-Spike Solutions for the Gray--Scott and Gierer--Meinhardt Systems: Competition and Oscillatory Instabilities

Abstract

The dynamics and instability mechanisms of both one- and two-spike solutions to the Gierer--Meinhardt (GM) and Gray--Scott (GS) models are analyzed on a bounded one-dimensional spatial domain. For each of these nonvariational two-component systems, the semistrong spike-interaction limit where the ratio $O(\varepsilon^{-2})$ of the two diffusion coefficients is asymptotically large is studied. In this limit, differential equations for the spike locations, with speed $O(\varepsilon^2) \ll 1$, are derived. To determine the stability of the spike patterns, nonlocal eigenvalue problems, which depend on the instantaneous spike locations, are derived and analyzed. For these nonlocal eigenvalue problems, it is shown that eigenvalues can enter into the unstable right half-plane either along the real axis or through a Hopf bifurcation, leading to either a competition instability or an oscillatory instability of the spike pattern, respectively. Competition instabilities occur only for two-spike patterns and numerically are shown to lead to the annihilation of a spike. Oscillatory instabilities occur for both one- and two-spike solutions. For two-spike solutions this instability typically synchronizes the oscillations of the spike amplitudes. Since the nonlocal eigenvalue problems depend on the instantaneous spike locations, the key feature of these instabilities is that they can be dynamic in nature and can be triggered at some point during the slow evolution of a spike pattern that is initially stable at t=0. The asymptotic theory is compared with full numerical simulations and previous theoretical results.