We consider a model of the invasion of healthy tissue by cancer cells which is described by a system of nonlinear PDEs consisting of a cross-diffusion-reaction equation and two additional nonlinear ordinary differential equations. We discuss the existence of global classical solutions and formulate a positivity preserving finite element discretization stabilized by the flux-corrected transport approach. Moreover, we present a result on the solvability of this nonlinear discrete problem. The properties of both the model and its discretization are illustrated by numerical results computed using the deal.II library.