Modeling Morphogenesis with Reaction-Diffusion Equations using Galerkin Spectral Methods

摘要

This project studies the nonhomogeneous steady-state solutions of the Gray-Scott model, a system of nonlinear partial differential equations that has received attention in the past decade in the context of pattern formation and morphogenesis. Morphogenesis, or birth of shape', is the biological term for the initial formation of patterns that occur in development as cells begin to differentiate. The model is a two morphogen reaction-diffusion system in which individual molecules display complex self-organization in aggregate. The project is divided into two main parts. The first part develops the Galerkin Spectral method for application to the two species reaction-diffusion system. Limitations and capabilities of the Galerkin Spectral method are discussed in the context of the heat equation, the Burgers equation, and the Allen-Cahn equation. The second part analyzes the stability of equilibria in the Gray-Scott model in terms of reaction and diffusion parameters. A region of Hopf bifurcation is identified for the diffusionless system, and conditions for diffusion driven instability are developed. We show in particular that diffusion driven instability will occur only when the diffusion constants of each morphogen are different in any two species reaction-diffusion equation. We then show some numerical simulations of pattern formation in the Gray-Scott model using MATLAB programs to implement the Galerkin Spectral method.