- The bacteria Myxococcus xanthus clusters into fruiting bodies in the absence of food. The movement of M. xanthus has three aspects: self-propulsion (run), change of direction due to collision (tumble), and Brownian fluctuations in movement. We propose a minimalist PDE model for the concentrations of left and right moving agents in one-dimension that incorporates these three phenomena as advection, dif- fusion and reaction respectively. We proceed to analyze stationary solutions to this equation, utilizing geometry and dynamical systems theory to show existence of fam- ilies of concentration profiles representing localized peaks or troughs in an otherwise uniform concentration. We verify and extend these existence results by comput- ing cluster and gap type solutions with numerical continuation methods. We also demonstrate temporal stability of the clusters and gaps with numerical experiments. Phenomenologically, we find that these profiles form due to a combination of the left moving population blocking the forward movement of the right moving population and vice-versa, and dampening due to diffusion.