The growth of world population appears to be a leading global problem and expresses the cumulative result of all economic, social and cultural activities that make up the history of mankind. All of humanity is viewed as a dynamic system. Using synergetic methods, a phenomenological mathematical model of the world demographic process has been developed, which is now culminating in the demographic transition. The development of humanity is determined by global interaction of an informational nature. The quantitative statistical nonlinear theory of the growth of the Earth's population is of interest for anthropology and demography, history and sociology, population genetics and epidemiology, psychology, the origin and evolution of man, and also provides grounds for drawing some qualitative conclusions about periodization and stability. The proposed methods are applied to discuss modern stabilization and aging of the world and Russian populations.