Maxime Breden, Technical University Munich

A study of the equilibria of a cross-diffusion system in population dynamics

vendredi 14 décembre 2018, 11h00 - 12h00

Salle du conseil, espace Turing

Cross-diffusion is a mechanism used in population dynamics to model a repulsive effect between individuals. Mathematically, this corresponds to adding a nonlinear diffusion term to classical reaction-diffusion systems. Cross-diffusion allows to obtain a richer variety of solutions, whose qualitative behavior seems to better fit observations (spatial segregation phenomenom), but it also complicates the mathematical study of these solutions.

In this talk, I will explain how this problem can be tackled by combining numerical simulations with a posteriori estimates, to obtain computerassisted proofs. First, I will present the general strategy behind this kind of computer-assisted techniques, namely to apply a fixed point theorem in a neighborhood of a numerical solution, which then yields the existence of a true solution. Then, I will illustrate how this techniques can be applied to study inhomogeneous steady states of the SKT triangular system:

\(\left\{\begin{eqnarray*} \frac{\partial u}{\partial t} &=& \Delta\left((d_1+d_{12}v)u\right) + \left( r_1-a_1u-b_1v\right)u \\\frac{\partial v}{\partial t} &=& \delta (d_2v) +\left(r_2-b_2v-a_2v\right) v\end{eqnarray*} \right.\)

This is the result of a joint work with R. Castelli (VU Amsterdam).