  We analyze a macroscopic model with a maximal density constraint which
describes short range repulsion in biological systems. This system aims at
modeling finite-size particles which cannot overlap and repel each other when
they are too close. The parts of the fluid where the maximal density is reached
behave like incompressible fluids while lower density regions are compressible.
This paper investigates the transition between the compressible and
incompressible regions. To capture this transition, we study a one-dimensional
Riemann problem and introduce a perturbation problem which regularizes the
compressible-incompressible transition. Specific difficulties related to the
non-conservativity of the problem are discussed.
