Kinetic model for myxobacteria with directional diffusion

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 [S. Hittmeir, L. Kanzler, A. Manhart, C. Schmeiser, Kinet. Relat. Models, 14(1):1–24, 2021], 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 whether 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.

Keywords

myxobacteria, inelastic Boltzmann equation, hypocoercivity, entropy, bifurcation, small diffusion parameter, fixed-point, decay to equilibrium

2010 Mathematics Subject Classification

35B32, 35B40, 35Q20

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This work has been supported by the Austrian Science Fund, grants no. W1245 and F65.

Received 27 September 2021

Received revised 3 March 2022

Accepted 8 April 2022

Published 27 December 2022