Clumps and patches in self-aggregation of finite size particles

New model equations are derived for dynamics of self-aggregation of finite-size particles. Differences from standard Debye-Hückel [1] and Keller-Segel [2] models are: a) the mobility μ of particles depends on the locally-averaged particle density and b) linear diffusion acts on that locally-averaged particle density. The cases both with and without diffusion are considered here. Surprisingly, these simple modifications of standard models allow progress in the analytical description of evolution as well as the complete analysis of stationary states. When