  We prove the existence of solutions of a cross-diffusion parabolic population
problem. The system of partial differential equations is deduced as the limit
equations satisfied by the densities corresponding to an interacting particles
system modeled by stochastic differential equations. According to the values of
the diffusion parameters related to the intra and inter-population repulsion
intensities, the system may be classified in terms of an associated matrix. For
proving the existence of solutions when the matrix is positive definite, we use
a fully discrete finite element approximation in a general functional setting.
If the matrix is only positive semi-definite, we use a regularization technique
based on a related cross-diffusion model under more restrictive functional
assumptions. We provide some numerical experiments demonstrating the weak and
strong segregation effects corresponding to both types of matrices.
