# Multiple asymptotics of Kinetic Equations with Internal States

## TL;DR

This paper investigates the derivation of macroscopic models from microscopic assumptions in bacterial movement, revealing how different asymptotics lead to classical or flux-limited Keller-Segel equations and introducing new equilibrium equations with singular solutions.

## Contribution

It provides a mathematical framework connecting microscopic internal states to macroscopic chemotaxis models, highlighting the impact of response stiffness and adaptation time on resulting equations.

## Key findings

- Derivation of macroscopic equations from microscopic internal states.
- Identification of conditions leading to Keller-Segel and flux-limited Keller-Segel models.
- Discovery of new equilibrium equations with singular solutions.

## Abstract

The run and tumble process is well established in order to describe the movement of bacteria in response to a chemical stimulus. However the relation between the tumbling rate and the internal state of bacteria is poorly understood. The present study aims at deriving models at the macroscopic scale from assumptions on the microscopic scales. In particular we are interested in comparisons between the stiffness of the response and the adaptation time. Depending on the asymptotics chosen both the standard Keller-Segel equation and the flux-limited Keller-Segel (FLKS) equation can appear. An interesting mathematical issue arises with a new type of equilibrium equation leading to solution with singularities.