  In the present work, we investigate 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 show that when the convective part of the system,
the chemotactic term, is dominant, then straightforward numerical methods for
the studied system may be unstable. We present an implicit finite element
method using conforming $P_1$ or $Q_1$ finite elements to discretize the model
in space and the $\theta$-method for discretization in time. The discrete
problem is stabilized using a nonlinear flux-corrected transport approach. It
is proved that both the nonlinear scheme and the linearized problems used in
fixed-point iterations are solvable and positivity preserving. Several
numerical experiments are presented in 2D using the deal.II library to
demonstrate the performance of the proposed method.
