In this course, I will present a few examples of mathematical PDE models for biology, describing the evolution in time of a population of organisms (cells, bacteria, algae…) in interaction and in interaction with its environment. All these models are based on conservation laws and need adapted numerical schemes to be efficiently solved at the numerical level. Here are two examples of the biological systems I will deal with.
In the one dimensional case, I will study different kinds of hyperbolic and parabolic PDE models to describe the phenomenon of chemotaxis, that is to say the motion of organisms due to a chemical substance. I will explain how to discretize efficiently these equations thanks to well-balanced or asymptotic-preserving numerical methods in order to obtain accurate simulations. This example will be extended to the case where chemotaxis systems are set on networks, to describe, for example, the motion of cells on a scaffold or in microfluidic devices.
In the two dimensional case, I will consider models describing the growth of biofilms, such as bacterial or phototrophic biofilms and I will explain how to deal numerically with the momentum conservation equation that contains a pressure term.
Time permitting, other numerical challenges will be presented.