The authors consider the strongly coupled cross-diffusion system
\[ \partial_t u - \text{div}\bigg[ \left( \begin{matrix} D_1 & 0 \\ 0 & D_2 \end{matrix} \right) \nabla {\mathbf u}\bigg] \text{div}[A({\mathbf u}) \nabla {\mathbf u}] = \left( \begin{matrix} F({\mathbf u}) \\ G({\mathbf u}) \end{matrix} \right) \quad \text{in } \Omega \times [0,T), \]
\(\Omega \subset \mathbb{R}^2 \text{or}\, \mathbb{R}^3\) a bounded domain, subject to initial and Neumann boundary conditions, where \({\mathbf u}:=(u,v)\) are two population densities and the reaction terms are given by
\[ F(u,v):=u(a_1-b_1u-c_1v), \quad G(u,v):= v(a_2-b_2u-c_2v). \]
If \(A=0\), the classical Lotka-Volterra two species competition model with self-diffusion is obtained. It is assumed that the diffusion matrix \(A=((A_{ij}))\) satisfies for all \({\mathbf w} \in \mathbb{R}^2\) and \(u,v \geq 0\)
\[ A_{12}(0,v)=0, \,\, A_{21}(u,0)=0, \,\, (A(u,v){\mathbf w} ,{\mathbf w} ) \geq \tfrac{1}{C}\, \| A(u,v) \| {\mathbf w}\|^2, \]
together with a polynomial growth condition with exponent depending on the dimension of \(\Omega\).
The existence of a non-negative unique weak solution \({\mathbf u}\) is shown based on a priori estimates and \(L^1\)-compactness. The authors discretize the given problem in space using a finite volume scheme and in time using backward differentiation. With analoguous methods the discrete problem is analyzed giving the existence and uniquness of a discrete non-negative solution and the convergence of a subsequence to \({\mathbf u}\). Nicely chosen numerical examples with \(d=1\) and \(d=2\) complete the paper.