A simulation and time series analysis of reaction-diffusion equations in biological pattern formation

A computer program was modified to model the dynamics of morphogen concentrations in a developing eye of a Xenopus laevis frog. The dynamics were modelled because it is believed that the behavior of the morphogen concentrations determine how the developing eye maps to the brain. The eye in the xenophus grows as a series of rings, and thus this is the model used. The basis for the simulation are experiments done by Sullivan et al. Following the experiment, aIl eye ring is "split" in half, inverted, and then "pasted" onto a donor half. The purpose of the program is to replicate and analyze the results that were found experimentally: a graft made on a north to south axis (dorsal to ventral) produces a change in vision along the east to west axis (anterior to posterior). Four modified Gierer-Meinhardt reaction-diffusion equations are used to simulate the operation. In the second part of the research, the program was further modified and a time series analysis was done on the results. It was found that the modified Gierer-Meinhardt equations demonstrated chaotic behavior under certain conditions. The dynamics included fixed points, limit cycles, transient chaos, intermittent chaos, and strange attractors. The creation and destruction of fractal torii was found.