  Mathematical theory of selection systems is developed for a wide class of
dynamical models of inhomogeneous populations with discrete time. The Price
equation and its particular case, the Fisher Fundamental theorem of natural
selection (FTNS), are well known general results of the theory. It is known
that the Price equation being a mathematical identity is not dynamically
sufficient, i.e., it does not allow one to predict changes in the mean of a
trait beyond the immediate response if only the value of covariance of the
trait and fitness at this moment is known. We show that the problem of
dynamically insufficiency for the Price equations and for the FTNS can be
correctly overcome if to study these equations subject to given initial
distribution in framework of exact models of the population dynamics. The
knowledge of the entire distribution at any given instant allows making the
exact prediction for indefinite time and this prediction dramatically depends
on the initial distribution. For these models, the current trait distribution
and hence all statistical characteristics of interest, such as mean values of
the fitness or any trait could be computed effectively at any time moment.
