The existence and stability of asymmetric spike patterns for the Schnakenberg model

Asymmetric spike patterns are constructed for the two-component Schnakenburg reaction-diffusion system in the singularly perturbed limit of a small diffusivity of one of the components. 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. Explicit conditions for the existence and linear stability of these asymmetric spike patterns are determined using a combination of asymptotic techniques and spectral properties associated with a certain nonlocal eigenvalue problem. These asymmetric solutions are found to bifurcate from symmetric spike patterns at certain critical values of the parameters. Two Interesting conclusions are that asymmetric patterns can exist for a reaction-diffusion system with spatially homogeneous coefficients under Neumann boundary conditions and that these solutions can be linearly stable on an O(1) time scale.