Qualitative behavior of solutions to cross-diffusion systems from population dynamics

A general class of cross-diffusion systems for two population species in a bounded domain with no-flux boundary conditions and Lotka–Volterra-type source terms is analyzed. Although the diffusion coefficients are assumed to depend linearly on the population densities, the equations are strongly coupled. Generally, the diffusion matrix is neither symmetric nor positive definite. Three main results are proved: the existence of global uniformly bounded weak solutions, their convergence to the constant steady state in the weak competition case, and the uniqueness of weak solutions. The results hold under appropriate conditions on the diffusion parameters which are made explicit and which contain simplified Shigesada–Kawasaki–Teramoto population models as a special case. The proofs are based on entropy methods, which rely on convexity properties of suitable Lyapunov functionals.