  This paper deals with a class of macroscopic models for cell migration in a
saturated medium for two-species mixtures. Those species tend to achieve some
motion according to a desired velocity, and congestion forces them to adapt
their velocity. This adaptation is modelled by a correction velocity which is
chosen minimal in a least-square sense. We are especially interested in two
situations: a single active species moves in a passive matrix (cell migration)
with a given desired velocity, and a closed-loop Keller-Segel type model, where
the desired velocity is the gradient of a self-emitted chemoattractant. We
propose a theoretical framework for the open-loop model (desired velocities are
defined as gradients of given functions) based on a formulation in the form of
a gradient flow in the Wasserstein space. We propose a numerical strategy to
discretize the model, and illustrate its behaviour in the case of a prescribed
velocity, and for the saturated Keller-Segel model.
