All ETDs from UAB

Advisory Committee Chair

Marius Nkashama

Advisory Committee Members

Shangbing Ai

Carmeliza Navasca

Hemant Tiwari

James Wang

Document Type

Dissertation

Date of Award

2016

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

This thesis is devoted to study one of the famous models in biological spatial pattern formation, the Gierer-Meinhardt system. Gierer-Meinhardt system was used to model head formation in the hydra. Typical experiments on hydra involve removing part of the head region and transplanting it to other parts of the body column. Then, a new head will form if and only if the transplanted area is sufficiently far from the (old) head. We will prove that solutions of this system exist locally and globally in time under some suitable conditions on the coefficients. Both the full and shadow Gierer-Meinhardt system will be considered. Shadow systems are mostly used when one of the diffusion coefficients are large to approximate reaction-diffusion systems. We consider a Robin boundary condition for the activator and Neumann boundary condition for the inhibitor for both full and shadow Gierer-Meinhardt system. We use analytic semigroup theory in function spaces to prove that there exist a local nonnegative solution to the Gierer-Meinhardt system. The proofs of the global in time existence are based on a priori estimates of the solutions. Numerical simulations using MATLAB pdepe based on finite difference method are used to graphically show the solution profiles of the Gierer-Meinhardt system.

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