  A cross-diffusion system with Lotka-Volterra reaction terms in a bounded
domain with no-flux boundary conditions is analyzed. The system is a nonlocal
regularization of a generalized Busenberg-Travis model, which describes
segregating population species with local averaging. The partial velocities are
the solutions of an elliptic regularization of Darcy's law, which can be
interpreted as a Brinkman's law. The following results are proved: the
existence of global weak solutions; localization limit; boundedness and
uniqueness of weak solutions (in one space dimension); exponential decay of the
solutions. Moreover, the weak-strong uniqueness property for the limiting
system is shown.
