  In this paper we develop a theory of general selection systems with discrete
time and explore the evolution of selection systems, in particular,
inhomogeneous populations. We show that the knowledge of the initial
distribution of the selection system allows us to determine explicitly the
system distribution at the entire time interval. All statistical
characteristics of interest, such as mean values of the fitness or any trait
can be predicted effectively for indefinite time and these predictions
dramatically depend on the initial distribution. The Fisher Fundamental theorem
of natural selection (FTNS) and more general the Price equations are the famous
results of the mathematical selection theory. We show that the problem of
dynamic insufficiency for the Price equations and for the FTNS can be resolved
within the framework of selection systems. Effective formulas for solutions of
the Price equations and for the FTNS are derived. Applications of the developed
theory to some other problems of mathematical biology (dynamics of
inhomogeneous logistic and Ricker model, selection in rotifer populations) are
also given. Complex behavior of the total population size, the mean fitness (in
contrast to the plain FTNS) and other traits is possible for inhomogeneous
populations with density-dependent fitness. The temporary dynamics of these
quantities can be investigated with the help of suggested methods.
