  In this work we present the convergence of a positivity preserving
semi-discrete finite volume scheme for a coupled system of two non-local
partial differential equations with cross-diffusion. The key to proving the
convergence result is to establish positivity in order to obtain a discrete
energy estimate to obtain compactness. We numerically observe the convergence
to reference solutions with a first order accuracy in space. Moreover we
recover segregated stationary states in spite of the regularising effect of the
self-diffusion. However, if the self-diffusion or the cross-diffusion is strong
enough, mixing occurs while both densities remain continuous.
