  In this paper we combine the three universalisms: pairwise interactions
concept, dynamical systems theory and relative entropy analysis to develop a
theory of entropy issues. We introduce two hypotheses concerning the structure
and types properties of the system's entities interactions and derive
generalized replicator dynamic equations. Then we construct energy-like and
entropy-like Lyapunov-Meyer functions (LMF) for these equations. We show that
energy-like LMF contains no information about the equilibrium of the system and
is a substantial generalization of the Fisher's fundamental theorem of natural
selection. If there are exists nontrivial equilibrium point for generalized
replicator system then we construct entropy-like LMF for this system and prove
that it is a relative entropy function or the function of information
divergence. 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. As result we establish the set of next important links: nonlinear
pairwise interaction - generalized Fisher (replicator) equations -
Lyapunov-Meyer functions - relative entropy - distance measure -
LDF-transformation - duality. In particular it follows from these links that
nonlinear pairwise interaction is the origin of all known entropy functions.
