I couldn't understand at all this fucking paper (actually I think it's wrong). So I decided to derive the equations myself using common sense and my math skills, and I realized a simplification that I hadn't seen it before. Using that, I can derive a single equation (yes!) that describes the evolution of the density I care about. I have to double check to see if it's right, but if it's ok it will be hard to solve since it's highly nonlinear. More on that later...
