  An implicit Euler finite-volume scheme for an $n$-species population
cross-diffusion system of Shigesada--Kawasaki--Teramoto-type in a bounded
domain with no-flux boundary conditions is proposed and analyzed. The scheme
preserves the formal gradient-flow or entropy structure and preserves the
nonnegativity of the population densities. The key idea is to consider a
suitable mean of the mobilities in such a way that a discrete chain rule is
fulfilled and a discrete analog of the entropy inequality holds. The existence
of finite-volume solutions, the convergence of the scheme, and the large-time
asymptotics to the constant steady state are proven. Furthermore, numerical
experiments in one and two space dimensiona for two and three species are
presented. The results are valid for a more general class of cross-diffusion
systems satisfying some structural conditions.
