Range expansion by growth and congestion

We introduce here a nonlinear and nonlocal model that describes the range expansion of a population resulting from growth and competition for space. This type of phenomenon underlies the expansion of colonies of immotile cells which motivated this work. These colonies display long-range shuffling induced by congestion, and are also subject to jamming. We start by showing the well-posedness of the general evolution equation. We then derive a singular limit of this model corresponding to a regime where dispersal occurs only from saturated areas. The limiting model, which has the structure of an obstacle free boundary problem in time, provides an effective approach to the description of the range expansion of a population as a result of growth, saturation and dispersion. We establish the main mathematical properties of this singular problem by proving a comparison property and describing the dynamics of the free boundary that delimits the saturated area. We identify traveling wave solutions and characterize the asymptotic speed of spreading of compactly supported solutions.