Kinetic modelling of colonies of myxobacteria

Figure 1.  Graphic illustration of the collision rules. (a): Alignment collisions with two-step geometric algorithm to regularize it. (b): Already invertible reversal collisions

Figure 2.  Support of two group data (solid lines, purple)

Figure 3.  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 $

Figure 4.  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

Figure 6.  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

Figure 5.  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)