The stability of spike solutions to the one-dimensional Gierer–Meinhardt model

The stability properties of an N-spike equilibrium solution to a simplified form of the Gierer–Meinhardt activator–inhibitor model in a one-dimensional domain is studied asymptotically in the limit of small activator diffusivity ε. The equilibrium solution consists of a sequence of spikes of equal height. The two classes of eigenvalues that must be considered are the O(1) eigenvalues and the O(ε2) eigenvalues, which are referred to as the large and small eigenvalues, respectively. The spike pattern is stable when the parameters in the Gierer–Meinhardt model are such that both sets of eigenvalues lie in the left half-plane. For a certain range of these parameters and for N≥2 and ε→0, it is shown the O(1) eigenvalues are in the left half-plane only when D<DN, where DN is some explicit critical value of the inhibitor diffusivity D. For N≥2 and ε→0, it is also shown that the small eigenvalues are real and that they are negative only when D<DN∗, where DN∗ is another critical value of D, which satisfies DN∗<DN. Thus, when N≥2 and ε≪1, the spike pattern is stable only when D<DN∗. An explicit formula for DN∗ is given. For the special case N=1, it is shown that a one-spike equilibrium solution is stable when D<D1(ε), where D1(ε) is exponentially large as ε→0, and is unstable when D>D1(ε). An asymptotic formula for D1(ε) is given. Finally, the dynamics of a one-spike solution is studied by deriving a differential equation for the trajectory of the center of the spike.