Scaling limits for a population model with growth, division and cross-diffusion

TL;DR

This paper investigates a multiscale modeling approach for bacterial microcolonies, deriving from microscopic stochastic models to macroscopic PDEs, with entropy estimates and numerical comparisons across scales.

Contribution

It rigorously derives mesoscopic and macroscopic limits from a microscopic stochastic model of bacterial populations with growth and division.

Findings

Successful derivation of mean-field limit from stochastic model

Entropy estimates enable localization at the macroscopic scale

Numerical comparisons validate multiscale modeling approach

Abstract

Originally motivated by the morphogenesis of bacterial microcolonies, the aim of this article is to explore models through different scales for a spatial population of interacting, growing and dividing particles. We start from a microscopic stochastic model, write the corresponding stochastic differential equation satisfied by the empirical measure, and rigorously derive its mesoscopic (mean-field) limit. Under smoothness and symmetry assumptions for the interaction kernel, we then obtain entropy estimates, which provide us with a localization limit at the macroscopic level. Finally, we perform a thorough numerical study in order to compare the three modeling scales.