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Periodic reversals of the direction of motion in systems of self-propelledrod shaped bacteria enable them to effectively resolve traffic jams formedduring swarming and maximize their swarming rate. In this paper, a connectionis found between a microscopic one dimensional cell-based stochastic model ofreversing non-overlapping bacteria and a macroscopic non-linear diffusionequation describing dynamics of the cellular density. Boltzmann-Matano analysisis used to determine the nonlinear diffusion equation corresponding to thespecific reversal frequency. Macroscopically (ensemble-vise) averagedstochastic dynamics is shown to be in a very good agreement with the numericalsolutions of the nonlinear diffusion equation. Critical density $p_0$ isobtained such that nonlinear diffusion is dominated by the collisions betweencells for the densities $p>p_0$. An analytical approximation of the pairwisecollision time and semi-analytical fit for the total jam time per reversalperiod are also obtained. It is shown that cell populations with high reversalfrequencies are able to spread out effectively at high densities. If the cellsrarely reverse then they are able to spread out at lower densities but are lessefficient at spreading out at at higher densities.

Macroscopic model of self-propelled bacteria swarming with regular reversals