2001

The Stability and Dynamics of Spike-Type Solutions to the Gierer-Meinhardt Model

A well-known system of partial differential equations, known as the Gierer-Meinhardt system, has been used to model cellular differentiation and morphogenesis. The system is of reaction-diffusion type and involves the determination of an activator and an inhibitor concentration field. Long-lived isolated spike solutions for the activator model the localized concentration profile that is responsible for cellular differentiation. In a biological context, the Gierer-Meinhardt system has been used to model such events as head determination in the hydra and heart formation in axolotl.

This thesis involves a careful numerical and asymptotic analysis of the Gierer-Meinhardt system in one dimension and a limited analysis of this system in a multi-dimensional setting. We begin by studying a reduced model, referred to as the shadow system, which results from simplifying the Gierer-Meinhardt model in the limit of inhibitor diffusivity tending to infinity. This reduced model is studied in both one and in several spatial dimensions. In Section 2 we study the stability and dynamics of interior spike profiles for this reduced model. We find that any n-spike profile, with n > 1, is unstable on a fast time scale. Profiles with a single interior spike are also unstable but on an exponentially slow time scale. In this case the spike tends towards the closest point on the boundary. In Section 3 we examine the behaviour of a spike profile in which the spike is confined to the boundary. This scenario is studied in the case of a two and a three dimensional domain. It is found that the spike moves in the direction of increasing boundary curvature and increasing boundary mean curvature in two and three dimensions, respectively. Stable spike equilibria correspond to local maxima of these curvatures. We then study the case of a spike confined to a flat portion of the boundary in two dimensions. In this case it is found that the spike moves on an exponentially slow time scale.

The remainder of this thesis examines the full Gierer-Meinhardt system in a one-dimensional spatial domains. In Section 4 we study the stability properties of n-spike equilibrium solutions to the full system. A necessary and sufficient condition is found for the linear stability of an n-spike solution. In Section 5 we study the dynamics of spike profiles. We derive a system of ordinary differential equations which govern the motions of the spikes in one spatial dimension. Numerical computations of this asymptotic system is compared with numerical computations of the full system. In Section 6 we study the effects of precursor gradients. The mathematical result of these spatial inhomogeneities in the chemical reaction is that some of the coefficients in the equations are no longer constant in space. We study the effects of spatially varying activator and inhibitor decay rates as well as a spatially inhomogeneous activator diffusivity. It is found that these spatial inhomogeneities can effect both the dynamics and equilibrium position of the spikes.