The Stability and Dynamics of Spikes for a Reaction-Diffusion System

Many classes of singularly perturbed reaction-diffusion systems possess localized solutions where the gradient of the solution is very large only in the vicinity of certain points in the domain. An example of a problem where such spikes occur is the Geirer-Meinhardt (GM) activator-inhibitor system modeling biological morphogenesis. In the limit of a small activator diffusivity, this system has been used to model many situations including spot-type patterns on sea-shells and head formation in the Hydra. Most of the previous work on this system over the past twenty years has been based either on full numerical simulations or on a linearized Turing-type stability analysis around spatially homogeneous steady-state solutions. However, this type of linearized analysis is not appropriate for determining the stability of spike-type patterns. In this talk we will survey some recent results on the existence and stability of symmetric and asymmetric equilibrium spike patterns for the GM model. The inhibitor diffusivity is found to be a critical parameter. A key result that is obtained is that there exists a sequence of critical values D_n of the inhibitor diffusivity D for which an n-spike symmetric equilibrium solution is stable if D < D_n and unstable if D > D_n. An explicit formula for D_n is given. The dynamics of spike patterns is also characterized in a one-dimensional domain and partial results are obtained in a multi-dimensional context. The mathematical tools used include asymptotic analysis, spectral analysis of nonlocal eigenvalue operators, dynamical systems, and numerical and matrix analysis.

Joint work with David Iron (graduate student at UBC), and Prof. Juncheng Wei (Chinese University of Hong Kong).