On Mathematical Theory of Selection: Discrete-Time Models
  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. 
Georgy P. Karev