Speaker
Description
Motivated by the modeling of bacteria microcolony morphogenesis across multiple scales, we explore in this talk models for a spatial population of interacting, growing and dividing particles. Starting from a microscopic stochastic model, we first write the corresponding stochastic differential equation satisfied by the empirical measure, and rigorously derive its mesoscopic (mean-field) limit. We then take an interest in the so-called localization limit, to reach a macroscopic (large-scale) model. The scaling consists in assuming that the range of interaction between individuals is very small compared to the size of the domain. In proving the localization limit using compactness arguments, the difficulties are twofold: first, growth and division render the system non-conservative, preventing the use of energy estimates. Second, the size of the particles, being a continuous trait, leads to new difficulties in obtaining compactness estimates. We first show rigorously the localization limit in the case without growth and fragmentation, under smoothness and symmetry assumptions for the interaction kernel. We then perform a thorough numerical study in order to compare the three modeling scales and study the different limits in situations not covered by the theory yet. These works provide a better understanding of the link between the micro- meso- and macro- scales for interacting particle systems.
