Beyond Turing: The Stability and Dynamics of Localized Patterns in Reaction-Diffusion Systems

Since the pioneering work of Turing in 1952, there have been many studies of instabilities of spatially homogeneous patterns in reaction-diffusion systems. These previous results provide general criteria for the onset of different types of instabilities as well as providing a normal form analysis valid near the onset of the instability. However, in the singularly perturbed limit, many reaction-diffusion systems can give rise to solutions that have a high degree of spatial heterogeneity. Examples of this class of solutions include grain boundaries in materials science, vortices in superconductivity, hot-spot solutions in the microwave heating of ceramics, and spike-type solutions related to biological morphogenesis. In contrast to spatially homogeneous solutions, the instabilities and the dynamics of these localized patterns are not nearly as well understood. In this talk, we begin by giving a brief survey of problems and general results in pattern formation theory. Then, we highlight novel types of dynamical behaviors that occur for spike-type solutions to the Gierer-Meinhardt system modeling morphogenesis. These behaviors include both self-replication instabilities, whereby new spikes are created across the domain, and intricate temporal instabilities initiated through a subcritical Hopf bifurcation. These problems are studied using a combination of rigorous, asymptotic, and numerical methods. A nonlocal eigenvalue problem is found to be central to the stability analysis. Our results are compared with recent results of Ni, Takagi, and Yanagida.

Joint work with Juncheng Wei (Chinese U. of Hong Kong).