# American Institute of Mathematical Sciences

doi: 10.3934/krm.2020046

## Kinetic modelling of colonies of myxobacteria

 1 University of Vienna, Faculty for Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria 2 University College London, Dept. of Mathematics, 25 Gordon Street, WC1H 0AY London, UK

* Corresponding author: Christian Schmeiser

Received  January 2020 Revised  August 2020 Published  September 2020

A new kinetic model for the dynamics of myxobacteria colonies on flat surfaces is derived formally, and first analytical and numerical results are presented. The model is based on the assumption of hard binary collisions of two different types: alignment and reversal. We investigate two different versions: a) realistic rod-shaped bacteria and b) artificial circular shaped bacteria called Maxwellian myxos in reference to the similar simplification of the gas dynamics Boltzmann equation for Maxwellian molecules. The sum of the corresponding collision operators produces relaxation towards nematically aligned equilibria, i.e. two groups of bacteria polarized in opposite directions.

For the spatially homogeneous model a global existence and uniqueness result is proved as well as exponential decay to equilibrium for special initial conditions and for Maxwellian myxos. Only partial results are available for the rod-shaped case. These results are illustrated by numerical simulations, and a formal discussion of the macroscopic limit is presented.

Citation: Sabine Hittmeir, Laura Kanzler, Angelika Manhart, Christian Schmeiser. Kinetic modelling of colonies of myxobacteria. Kinetic & Related Models, doi: 10.3934/krm.2020046
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Graphic illustration of the collision rules. (a): Alignment collisions with two-step geometric algorithm to regularize it. (b): Already invertible reversal collisions
Support of two group data (solid lines, purple)
Two group initial conditions with the same mass in $\mathbb{T}^1_{+}$ and $\mathbb{T}^1_{-}$; rod shaped bacteria. Left: uniform distributions within $\mathbb{T}^1_{+}$ and $\mathbb{T}^1_{-}$. Right: vacuum around $\pm\pi/2$
Left: initial condition with uniform distribution in $\mathbb{T}^1_{+}$ and vacuum everywhere else. Right: initially two concentrated patches at a distance somewhat bigger than $\pi/2$ (yellow at the left end). Outer stripes created by reversal, then fill-in by alignment, followed by concentration towards opposite directions. The mean angles $\bar \varphi_+$ (red line) and $\bar \varphi_-$ (dotted red line) in the two groups change significantly
Instability of constant positive steady states. Left: random initial perturbation, leading to an unpredictable equilibrium direction. Right: initial perturbation at one direction, which eventually becomes the equilibrium direction. Note that this differs from the simulations in Figure 4, left, by the fact that a positive state is perturbed, and therefore reversal collisions are active
Left: The evolution of the inverse square root of the variance $V[f]$ from the simulation depicted on the left side of Figure 4, supporting the validity of Haff's law for rod shaped myxos. Right: Semi-log plot of $V[f]$ for a simulation with the same initial data, but for Maxwellian myxos, demonstrating exponential decay to equilibrium as shown in Lemma 4.2 a)
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