Lyapunov-Meyer functions and distance measure from generalized Fisher's equations

The main goal of this report is to do the next step in the investigation of generalized Fisher's (replicator) equations. Recently Lyapunov-Meyer function was constructed by the author for above equation as relative entropy. In this paper we prove that negative relative entropy is a convex function for a probability space and receive new distance measure between two probability distributions. Also we use Legendre-Donkin-Fenchel transformation for dual coordinates. In particular it follows from these cross-disciplinary issue that nonlinear pairwise interactions is the origin of all known entropy functions.