  The existence of global-in-time weak solutions to reaction-cross-diffusion
systems for an arbitrary number of competing population species is proved. The
equations can be derived from an on-lattice random-walk model with general
transition rates. In the case of linear transition rates, it extends the
two-species population model of Shigesada, Kawasaki, and Teramoto. The
equations are considered in a bounded domain with homogeneous Neumann boundary
conditions. The existence proof is based on a refined entropy method and a new
approximation scheme. Global existence follows under a detailed balance or weak
cross-diffusion condition. The detailed balance condition is related to the
symmetry of the mobility matrix, which mirrors Onsager's principle in
thermodynamics. Under detailed balance (and without reaction), the entropy is
nonincreasing in time, but counter-examples show that the entropy may increase
initially if detailed balance does not hold.
