Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability

Equilibrium solutions to the one-dimensional Gierer-Meinhardt model in the form of sequences of spikes of different heights are constructed asymptotically in the limit of small activator diffusivity e. For a pattern with k spikes, the construction yields k(1) spikes that have a common small amplitude and k(2) = k-k(1) spikes that have a common large amplitude. A k-spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. It is shown that such solutions exist when the inhibitor diffusivity D is less than some critical value D-m that depends upon k(1), on k(2), and on other parameters associated with the Gierer-Meinhardt model. It is also shown that these asymmetric k-spike solutions bifurcate from the symmetric solution branch s(k), for which k spikes have equal height. These asymmetric solutions provide connections between the branch s(k) and the other symmetric branches s(j), for j=1,...,k-1. The stability of the asymmetric k-spike patterns with respect to the large O(1) eigenvalues and the small O(epsilon(2)) eigenvalues is also analyzed. It is found that the asymmetric patterns are stable with respect to the large O(1) eigenvalues when D > D-e, where D-e depends on k(1) and k(2), on certain parameters in the model, and on the specific ordering of the small and large spikes within a given k-spike sequence. Numerical values for D-e are obtained from numerical solutions of a matrix eigenvalue problem. Another matrix eigenvalue problem that determines the small eigenvalues is derived. For the examples considered, it is shown that the bifurcating asymmetric branches are all unstable with respect to these small eigenvalues.