Dynamics of inhomogeneous populations and global demography models
  The dynamic theory of inhomogeneous populations developed during the last decade predicts several essential new dynamic regimes applicable even to the well-known, simple population models. We show that, in an inhomogeneous population with a distributed reproduction coefficient, the entire initial distribution of the coefficient should be used to investigate real population dynamics. In the general case, neither the average rate of growth nor the variance or any finite number of moments of the initial distribution is sufficient to predict the overall population growth. We developed methods for solving the heterogeneous models and explored the dynamics of the total population size together with the reproduction coefficient distribution. We show that, typically, there exists a phase of hyper-exponential growth that precedes the well-known exponential phase of population growth in a free regime. The developed formalism is applied to models of global demography and the problem of population explosion predicted by the known hyperbolic formula of world population growth. We prove here that the hyperbolic formula presents an exact solution to the Malthus model with an exponentially distributed reproduction coefficient and that population explosion is a corollary of certain implicit unrealistic assumptions. Alternative models of world population growth are derived; they show a notable phenomenon, a transition from protracted hyperbolical growth (the phase of hyper-exponential development) to the brief transitional phase of exponential growth and, subsequently, to stabilization. The model solutions are consistent with real data and produce relatively accurate forecasts. 
Georgy P. Karev