Comparison of linear and nonlinear models for human population dynamics

In standard demographic practice, population projections are commonly based on one-sex linear models of the Lotka-Leslie type. We demonstrate here that such projections based solely on time-invariant, age-specific male fertilities and mortalities are incompatible with those based solely on female fertilities and mortalities. This incompatibility obtains even in the singular case where effective male and female fertility functions are equal, and generate equal ultimate rates of growth. In standard demographic practice, the incompatibility is initially masked, since one-sex fertility functions are generally calculated from the same initial-time data and thereby tautologically forced to initially concur; however, with the passage of any finite time, the incompatibility reasserts itself—the only exception being the uninteresting case where the system is already in the stationary age and sex distribution of balanced exponential growth.An example is adduced of a nonlinear (age-free) system whose true rate of ultimate growth is correctly bracketed by the male and female one-sex rates of ultimate growth. Analysis of more general two-sex models shows that the two one-sex growth rates, calculated for arbitrary male and female initial age distributions, need not bracket the true rate. We show, however, that models exist such that with appropriate choices of initial conditions, this bracketing will occur.