Julijana Gjorgjieva

Harvey Mudd College Mathematics 2006

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Thesis Proposal: Turing Pattern Dynamics for Spatiotemporal Models with Growth and Curvature
Thesis Advisor: Professor Jon Jacobsen
Second Reader: Professor Pablo Padilla
E-Mail: jgjorgjieva@hmc.edu
Final Report: Turing Pattern Dynamics for Spatiotemporal Models with Growth and Curvature

Turing Pattern Dynamics for Spatiotemporal Models with Growth and Curvature

Turing theory plays an important role in real biological pattern formation problems, such as solid tumor growth and animal coat patterns. To understand how patterns form and develop over time due to growth, we consider spatiotemporal patterns, in particular Turing patterns, for reaction-diffusion systems on growing surfaces with curvature. Of particular interest is isotropic growth of the sphere, where growth of the domain occurs in the same proportion in all directions. Applying a modified linear stability analysis and a separation of timescales argument, we derive the necessary and sufficient conditions for a diffusion-driven instability of the steady state and for the emergence of spatial patterns. Finally, we explore these results using numerical simulations.