2:00 PM

CAB 657

Dr. Michael Ward

Department of Mathematics
University of British Columbia

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

Pulse-Splitting and Oscillatory Instabilities of Localized Solutions to the Gierer-Meinhardt Model


Since the pioneering work of Turing in 1952, there have been many studies determining the conditions for the onset of instabilities of spatially homogeneous patterns in reaction-diffusion systems. Various types of weakly nonlinear theories, many of them associated with the complex Ginzburg-Landau equation, have been used to study the weakly nonlinear development of these Turing patterns. However, in the singularly perturbed limit of small diffusivity, many reaction-diffusion systems allow for the existence of equilibrium solutions that have a high degree of spatial heterogeneity. A very common pattern of this type is a spike pattern, whereby one of the components of the system becomes spatially localized. In contrast to the study of the stability of spatially homogeneous solutions, the instabilities and dynamics of localized patterns are not nearly as well understood. In this talk, we highlight different types of dynamics and instabilities that occur for spike-type solutions to the singularly perturbed Gierer-Meinhardt reaction-diffusion system. Depending on the range of parameters in the system, there can either be a slow evolution of a spike towards the midpoint of the domain, a sudden oscillatory instability triggered by a Hopf bifurcation leading to an intricate temporal oscillation in the height of the spike, or a pulse-splitting instability leading to the creation of new spikes in the domain. Criteria for the onset of these oscillatory and pulse-splitting instabilities are obtained through rigorous, asymptotic, and numerical, techniques. A central feature in the study of the Hopf bifurcation of multi-spike solutions is the analysis of a nonlocal eigenvalue problem.