Collective motion driven by nutrient consumption

TL;DR

This paper analyzes a mathematical model of collective cell motion driven by nutrient consumption, revealing how cell density concentrates and nutrients exhibit discontinuities, with detailed analysis of the system's limiting behavior.

Contribution

It provides a rigorous analysis of a coupled PDE/ODE system modeling nutrient-driven cell movement, including the limit behavior with and without diffusion.

Findings

Cell density concentrates as a moving Dirac mass in the limit.

Nutrient distribution develops a discontinuity in the limit.

The system can be interpreted as a heterogeneous monostable equation.

Abstract

A classical problem describing the collective motion of cells, is the movement driven by consumption/depletion of a nutrient. Here we analyze one of the simplest such model written as a coupled Partial Differential Equation/Ordinary Differential Equation system which we scale so as to get a limit describing the usually observed pattern. In this limit the cell density is concentrated as a moving Dirac mass and the nutrient undergoes a discontinuity. We first carry out the analysis without diffusion, getting a complete description of the unique limit. When diffusion is included, we prove several specific a priori estimates and interpret the system as a heterogeneous monostable equation. This allow us to obtain a limiting problem which shows the concentration effect of the limiting dynamics.