  This paper is devoted to the study of systems of reaction-cross diffusion
equations arising in population dynamics. New results of existence of weak
solutions are presented, allowing to treat systems of two equations in which
one of the cross diffusions is convex, while the other one is concave. The
treatment of such cases involves a general study of the structure of Lyapunov
functionals for cross diffusion systems, and the introduction of a new scheme
of approximation, which provides simplified proofs of existence.
