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We study a one-dimensional three-state run-and-tumble model motivated by thebacterium Caulobacter crescentus which displays a cell cycle between twonon-proliferation mobile phase and a proliferating sedentary phase. Our modelimplements kinetic transitions between the two mobile and one sedentary statesdescribed in terms of their number densities, where mobility is allowed withdifferent running speeds in forward and backward direction. We start byanalyzing the stationary states of the system and compute the mean andsquared-displacements for the distribution of all cells, as well as for thenumber density of settled cells. The latter displays a surprisingsuper-ballistic scaling ~t^3 at early times. Including repulsive and attractiveinteractions between the mobile cell populations and the settled cells, weexplore the stability of the system and then employ numerical methods to studystructure formation in the fully nonlinear system. We find traveling waves ofbacteria, whose occurrence is quantified in a nonequilibrium state diagram.

A one-dimensional three-state run-and-tumble model with a `cell cycle'