Partial differential equations are often used in biology to model spatially structured systems: forest propagation, inflammation dynamics, cell polarization... These models can be used, for example, to accurately simulate the behavior of an organ. In other cases, particularly when dealing with less well understood biological systems, partial differential equations can offer a qualitative description of complex phenomena.
In this course, we will focus on two ecological issues, through the study of recent work. First, we will look at propagation phenomena described by non-linear parabolic equations. We will then see that a good understanding of linear equations (elliptical or parabolic) enables us to study the behavior of non-linear models. In the second part of the course, we will study the dynamics of collective movements described by kinetic equations. To understand the dynamics of this second type of partial differential equation, we will identify a fast time scale (which will be local in space) and a slow time scale (which will govern the spatial dynamics of the system).
We will then be able to describe the solutions using macroscopic models.
The two ecological issues discussed in this course will enable us to understand a diversity of questions that can arise around partial differential equation models in biology: modeling, links with other models (in particular stochastic models), numerical simulations. We will also discuss the possible roles of mathematical analysis in the study of biological problems. The mathematical methods covered (linear/non-linear models, slow/fast dynamics) are relevant to many mathematical problems in biology.
Moreover, these methods have links with arguments used in probability and the study of dynamic systems.
- Profesor: Raoul Gael