Mathematical models of biological pattern formation
Hours: from 10:30 to 12:10 on every Thursday
From October 7, 2010 to January 27, 2011
Classroom: RM 801 of Godoto Building
This series of lectures is intended as an introduction to the mathematical analysis of
reaction-diffusion systems in developmental biology and a geomeric variational problem concerning
the shape transformation of lipid bilayer membrane.
* To understand quantitatively the mechanism of diffusion-driven instability.
* To learn basics of bifurcation theory and singular perturbation methods.
* To understand basics of geometric variational problems and their gradient flows.
1. Existence of solutions of the initial-boundary value problem for nonlinear parabolic partial
differential equations (three lectures)
2. Stability of stationary solutions and bifurcation (three lectures)
3. Stationary solutions with singular perturbation (three lectures)
4. Very slow motion of interfaces (three lectures)
5. Shape transformation of bilayer membranes: variational problems and gradient flows (three lectures)
$B!!(BThere are many good monographs. To name a few:
 James D. Murray, "Mathematical Biology I. An Introduction", "Mathematical Biology II: Spatial
Models and Biomedical Applications", Third Edition, Springer, 2002, 2003.
 Louis Nirenberg, "Topics in Nonlinear Functional Analysis", Courant Institute, 1973, also
American Mathematical Society, 2001.
 Joel Smoller, "Shock Waves and Reaction-Diffusion Equations", Second Edition, Springer-Verlag, 1994.
 Eberhard Zeidler, "Applied Functional Analysis", Springer 1995.
homework projects or oral examination
- Prerequisits: (i) Basics on the theory of ordinary differential equations such as the existence
and uniqueness of a solution of the initial value problem, presented in the monograph
"Differential Equations and Their Applications" Fourth Edition by Martin Braun, Springer-Verlag, 1992
(ii) Foundamentals on the functional analysis presented, e.g., in .
- Office hours: after the lecture$B!!(B
---application of the theory of nonlinear partial differential equations