Kinetic Model for Myxobacteria with Directional Diffusion

TL;DR

This paper develops a kinetic model for myxobacteria colonies that incorporates directional diffusion, analyzing its effects on collective behavior, stability, and bifurcations through mathematical proofs and numerical simulations.

Contribution

It extends existing kinetic models by including Brownian forcing, proving global existence and decay, and analyzing bifurcations in low diffusion regimes.

Findings

Large diffusion ensures global stability and exponential decay.

Existence of nonuniform stable equilibria in small diffusion regimes.

Numerical simulations support theoretical results.

Abstract

In this article a kinetic model for the dynamics of myxobacteria colonies on flat surfaces is investigated. The model is based on the kinetic equation for collective bacteria dynamics introduced in arXiv:2001.02711, which is based on the assumption of hard binary collisions of two different types: alignment and reversal, but extended by additional Brownian forcing in the free flight phase of single bacteria. This results in a diffusion term in velocity direction at the level of the kinetic equation, which opposes the concentrating effect of the alignment operator. A global existence and uniqueness result as well as exponential decay to uniform equilibrium is proved in the case where the diffusion is large enough compared to the total bacteria mass. Further, the question wether in a small diffusion regime nonuniform stable equilibria exist is positively answered by performing a formal bifurcation analysis, which revealed the occurrence of a pitchfork bifurcation. These results are illustrated by numerical simulations.