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Meta-stability of the Gierer Meinhardt equations

University of British Columbia

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 this system in one dimension for a specific parameter set and a limited analysis of this system in a multidimensional setting. Numerical analysis has revealed that once the spikes form they continue to move on an extremely slow time scale. This type of phenomenon is a general indicator of meta-stable behaviour. By perturbing off of an isolated spike solution an exponentially small eigenvalue of the linearized operator was found. This small eigenvalue accounted for the extremely slow motion found numerically and thus was used to obtain an equation of motion for the location of the spike. The Gierer Meinhardt system is analyzed in the limit of small activator diffusivity for both a finite inhibitor diffusivity and for an asymptotically large inhibitor diffusivity. In this thesis, the mathematical techniques used include the method of matched asymptotic expansions, spectral theory and numerical computations.

Thesis/Dissertation

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2009-03-24

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http://hdl.handle.net/2429/6402

Mathematics

University of British Columbia

UBCV

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