A stochastic Keller-Segel model of chemotaxis

TL;DR

This paper develops stochastic extensions of the Keller-Segel chemotaxis model, incorporating fluctuations and inertia effects, and derives related kinetic and hydrodynamic equations, including connections to the Cattaneo and telegraph models.

Contribution

It introduces stochastic chemotaxis models that include fluctuations and inertia, extending the classical Keller-Segel framework and deriving exact kinetic equations.

Findings

Derivation of the exact kinetic equation for cell density.

Recovery of the Keller-Segel model in the mean field limit.

Establishment of connections with the Cattaneo and telegraph equations.

Abstract

We introduce stochastic models of chemotaxis generalizing the deterministic Keller-Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. Following Dean's approach, we derive the exact kinetic equation satisfied by the density distribution of cells. In the mean field limit where statistical correlations between cells are neglected, we recover the Keller-Segel model governing the smooth density field. We also consider hydrodynamic and kinetic models of chemotaxis that take into account the inertia of the particles and lead to a delay in the adjustment of the velocity of cells with the chemotactic gradient. We make the connection with the Cattaneo model of chemotaxis and the telegraph equation.