Hydrodynamic limit of a stochastic model of proliferating cells with chemotaxis

TL;DR

This paper analyzes a hybrid stochastic model of proliferating cells with chemotaxis, demonstrating that as the cell number increases, the model converges to a Patlak-Keller-Segel-type system, linking individual behavior to macroscopic dynamics.

Contribution

It introduces a new hybrid stochastic model coupling branching diffusion with PDEs and proves its convergence to a well-known chemotaxis system in the hydrodynamic limit.

Findings

Model converges to Patlak-Keller-Segel system as cell number increases

Movement of descendants converges to a mean-field stochastic process

Provides a rigorous link between individual-based and continuum models

Abstract

A hybrid stochastic individual-based model of proliferating cells with chemotaxis is presented. The model is expressed by a branching diffusion process coupled to a partial differential equation describing concentration of a chemotactic factor. It is shown that in the hydrodynamic limit when number of cells goes to infinity the model converges to the solution of nonconservative Patlak-Keller-Segel-type system. A nonlinear mean-field stochastic model is defined and it is proven that the movement of descendants of a single cell in the individual model converges to this mean-field process.