  In this paper we study two models for crowd motion and herding. Each of the
models is of Keller-Segel type and involves two parabolic equations, one for
the evolution of the density and one for the evolution of a mean field
potential. We classify all radial stationary solutions, prove multiplicity
results and establish some qualitative properties of these solutions, which are
characterized as critical points of an energy functional. A notion of
variational stability is associated to such solutions. The dynamical stability
in a neighborhood of a stationary solution is also studied in terms of the
spectral properties of the linearized evolution operator. For one of the two
models, we exhibit a Lyapunov functional which allows to make the link between
the two notions of stability. Even in that case, for certain values of the mass
parameter and all other parameters taken in an appropriate range, we find that
two dynamically stable stationary solutions exist. We further discuss
qualitative properties of the solutions using theoretical methods and numerical
computations.
