Analysis of biological pattern formation models
In this thesis we examine mathematical models which have been suggested as possibile mechanisms for forming certain biological patterns. We analyse them in detail attempting to produce the requisite patterns both analytically and numerically. A reaction diffusion system in two spatial dimensions with anisotropic diffusion is examined in detail and the results compared with certain snakeskin patterns. We examine two other variants to the standard reaction diffusion system: a system where the reaction kinetics and the diffusion coefficients depend upon the cell density suggested as a possible model for the segmentation sequence in Drosophila and a system where the model parameters have one dimensional spatial gradients. We also analyse a model derived from known cellular processes used to model the branching behaviour in bryozoans and show that, in one dimension, such a model can, in theory, give all the required solution behaviour. A genetic switch model for pattern elements on butterfly wings is also briefly examined to obtain expressions for the solution behaviour under coldshock.