  We study a one-dimensional three-state run-and-tumble model motivated by the
bacterium Caulobacter crescentus which displays a cell cycle between two
non-proliferation mobile phase and a proliferating sedentary phase. Our model
implements kinetic transitions between the two mobile and one sedentary states
described in terms of their number densities, where mobility is allowed with
different running speeds in forward and backward direction. We start by
analyzing the stationary states of the system and compute the mean and
squared-displacements for the distribution of all cells, as well as for the
number density of settled cells. The latter displays a surprising
super-ballistic scaling ~t^3 at early times. Including repulsive and attractive
interactions between the mobile cell populations and the settled cells, we
explore the stability of the system and then employ numerical methods to study
structure formation in the fully nonlinear system. We find traveling waves of
bacteria, whose occurrence is quantified in a nonequilibrium state diagram.
