A Monte Carlo simulation for kinetic chemotaxis models: an application to the traveling population wave

TL;DR

This paper develops a Monte Carlo simulation combining bacterial run-and-tumble dynamics with chemical transport to model and analyze traveling bacterial waves, validating the approach against experimental data and asymptotic solutions.

Contribution

It introduces a coupled Monte Carlo and finite volume simulation framework for kinetic chemotaxis models, capturing microscopic bacterial behavior and macroscopic chemical transport.

Findings

Successfully reproduces experimentally observed bacterial traveling waves.

Validates Monte Carlo method against asymptotic solutions in the small Knudsen number limit.

Demonstrates the method's ability to reveal microscopic and macroscopic dynamics.

Abstract

A Monte Carlo simulation of chemotactic bacteria is developed on the basis of the kinetic model and is applied to a one-dimensional traveling population wave in a microchannel. In this simulation, the Monte Carlo method, which calculates the run-and-tumble motions of bacteria, is coupled with a finite volume method to calculate the macroscopic transport of the chemical cues in the environment. The simulation method can successfully reproduce the traveling population wave of bacteria that was observed experimentally and reveal the microscopic dynamics of bacterium coupled with the macroscopic transports of the chemical cues and bacteria population density. The results obtained by the Monte Carlo method are also compared with the asymptotic solution derived from the kinetic chemotaxis equation in the continuum limit, where the Knudsen number, which is defined by the ratio of the mean free path of bacterium to the characteristic length of the system, vanishes. The validity of the Monte Carlo method in the asymptotic behaviors for small Knudsen numbers is numerically verified.