  We consider an aggregation model for two interacting species. The coupling
between the species is via their velocities, that incorporate self- and
cross-interactions. Our main interest is categorizing the possible steady
states of the considered model. Notably, we identify their regions of existence
and stability in the parameter space. For assessing the stability we use a
combination of variational tools (based on the gradient flow formulation of the
model and the associated energy), and linear stability analysis (perturbing the
boundaries of the species' supports). We rely on numerical investigations for
those steady states that are not analytically tractable. Finally we perform a
two-scale expansion to characterize the steady state in the limit of
asymptotically weak cross-interactions.
