TUDMATH SS2024 Modul Math-MA-34 (MMAM)

In recent decades, the field of Mathematical Biology has gained a lot of popularity for several reasons. Not only do new biological insights push the frontiers of our knowledge, but the need to understand complex biological processes requires refined descriptions thereof --- a chance for mathematics to make a contribution.

We shall revisit some models introduced in MOSIM-1 & MOSIM-2. The migratory part of these models is obtained using a coarse-graining procedure of a system of coupled stochastic differential equations (SDEs), which leads to Fokker-Planck-type equations (MOSIM-2). Coupled with reaction phenomena (MOSIM-1), the resulting models belong to the family of reaction-drift-diffusion equations.

The purpose of this lecture series is to introduce analytical tools to address the well-posedness of these models and to study their behaviour. Particularly, we introduce classical linear parabolic theory, consider vanishing viscosity limits for hyperbolic equations, and, as 'grand finale' we consider the localisation limit of nonlocal interaction equations and the resulting nonlinear degenerate parabolic equations -- a timely hot topic in applied analysis.

Course dates:

Prerequisits:

- MOSIM-1, MOSIM-2 will help understand the models but are not necessary, strictly speaking.

- Analysis 1,2, and an introductory course on Functional Analysis

- acquaintance with Ordinary Differential Equations is beneficial

Literature:

Pavliotis: Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations, 2014

Perthame: Transport equations in biology. 2006

Perthame: Parabolic equations inbiology.. 2015

Brezis: Functional analysis, Sobolev spaces and partial differential equations. 2011

Evans. Partial Differential Equations. 1998