  We investigate systems of self-propelled particles with alignment
interaction. Compared to previous work, the force acting on the particles is
not normalized and this modification gives rise to phase transitions from
disordered states at low density to aligned states at high densities. This
model is the space inhomogeneous extension of a previous work by Frouvelle and
Liu in which the existence and stability of the equilibrium states were
investigated. When the density is lower than a threshold value, the dynamics is
described by a non-linear diffusion equation. By contrast, when the density is
larger than this threshold value, the dynamics is described by a hydrodynamic
model for self-alignment interactions previously derived in Degond and Motsch.
However, the modified normalization of the force gives rise to different
convection speeds and the resulting model may lose its hyperbolicity in some
regions of the state space.
