Intrinsically dynamic population models

Intrinsically dynamic models (IDMs) depict populations whose cumulative growth rate over a number of intervals equals the product of the long term growth rates (that is the dominant roots or dominant eigenvalues) associated with each of those intervals. Here the focus is on the birth trajectory produced by a sequence of population projection (Leslie) matrices. The elements of a Leslie matrix are represented as straightforward functions of the roots of the matrix, and new relationships are presented linking the roots of a matrix to its Net Reproduction Rate and stable mean age of childbearing. Incorporating mortality changes in the rates of reproduction yields an IDM when the subordinate roots are held constant over time. In IDMs, the birth trajectory generated by any specified sequence of Leslie matrices can be found analytically. In the Leslie model with 15 year age groups, the constant subordinate root assumption leads to reasonable changes in the age pattern of fertility, and equations (27) and (30) provide the population size and structure that result from changing levels of net reproduction. IDMs generalize the fixed rate stable population model. They can characterize any observed population, and can provide new insights into dynamic demographic behavior, including the momentum associated with gradual or irregular paths to zero growth.dynamic models, dynamic population models, eigenvalues, Leslie matrices, population momentum

The metastable birth trajectory

The metastable model generalizes the stable population model by allowing net maternity to change exponentially over age and time. As a result, the metastable model generates an exponentially quadratic birth trajectory, which is characterized by a constant proportion of births by age of mother. The metastable model is well suited to analyzing steady fertility declines and transitions between two regimes of fixed vital rates.dynamic population models, metastable population, quadratic exponential trajectory, stable population

Toward a General Model for Populations with Changing Rates

Formal demography has yet to move beyond assuming that demographic rates are constant over time, an assumption that is both unrealistic and constraining. To generalize the fixed rate stable model to the changing rate dynamic model, this paper explores the mathematical regularities that underlie the behavior of all populations. At any time, the composition of a population can be expressed in terms of current circumstances, using the rates of a "latent" stable model. Closed form solutions for the equations governing dynamic multistate models are not always possible, but are presented for certain special cases. Those solutions provide opportunities for specifying dynamic models of potentially great value, especially for analyses of cyclical and hierarchical populations.dynamic, mathematical demography, multistate, population models, stable population

Age-specific growth, reproductive values, and intrinsic r

The age-specific growth function of an observed population and the reproductive value function based on the population´s current vital rates determine the intrinsic rate of growth implied by those vital rates through the simple relationship given in equation (1). That equation establishes the analytical significance of age-specific growth, and leads to relationships that quantify a population´s approach to stability and that specify the extraordinarily close connection between reproductive values and population momentum.age-specific growth, intrinsic r, population momentum, reproductive value, stable equivalent, stable population

On the Impact of Spatial Momentum

Momentum, the population growth that occurs after a fall in fertility to replacement level, has long been appreciated as a factor in the future population growth of many countries. This paper argues that another aspect of growing populations - their high proportion rural - is also a source of significant growth, and refers to the additional growth attributable to geographical redistribution as spatial momentum. Using simplifying assumptions, a model for analyzing spatial momentum is developed based on population composition, rates of growth, and levels of interregional migration. Calculations are then done using (i) hypothetical populations exhibiting a range of plausible demographic behavior, and (ii) the population of Mexico, 1970. The results show that spatial momentum can have a substantial impact on ultimate population size under commonly encountered circumstances.population growth, population momentum, spatial momentum, urbanization

A behaviorally-based approach to measuring inequality

The measurement of inequality is often made using observed population-based distributions, such as the distribution of income or the distribution of members of different groups across neighborhoods. Unfortunately, such distributions confound the behavior of a given year with earlier events that influence the composition of the population. Here, we advocate measuring inequality using current behavioral measures and their compositional implications, and show how such measures may be obtained from frequently available data. The approach is then applied to trends in inequality between men and women in the distribution of ages at death. Observed death distributions indicate that, since 1970, mortality in 4 Western countries experienced increases in inequality that recently leveled off. In contrast, life table death distributions, which solely reflect the implications of a given year’s mortality rates, reveal a peak in inequality followed (in 3 of the 4 countries) by appreciable declines. The results are insensitive to whether inequality is measured by entropy, the Gini Index, or the Index of Dissimilarity. However, the type of distribution analyzed---whether observed or behaviorally derived---can make a significant difference in the results obtained. Because behaviorally derived distributions reflect the inequality implications of actual behavior, they are recommended for greater use in analyses of inequality.behaviorally-based, entropy, Gini Index, index of dissimilarity, inequality, measurement

Age-specific contributions to changes in the period and cohort life expectancy

Period life expectancy has increased more slowly than its cohort counterpart. This paper explores the differences between life expectancies at a given time (the gap) and the time required for period life expectancy to reach the current level of cohort life expectancy (the lag). Additionally, to understand the disparity between the two life expectancies we identify and compare age-specific contributions to change in life expectancy. Using mortality models and historical data for Sweden, we examine the effect of mortality changes over time. Our results indicate that the widening of the gap between the two life expectancies is primarily a consequence of the dramatic mortality decline at older ages that occurred during the twentieth century. These results imply that the divergence between the two measures is likely to become even greater in the future as reductions in deaths are concentrated at older ages.age-specific decomposition, cohort life expectancy, gap and lag, life expectancy, mortality models, period life expectancy

Changing mortality and average cohort life expectancy

Period life expectancy varies with changes in mortality, and should not be confused with the life expectancy of those alive during that period. Given past and likely future mortality changes, a recent debate has arisen on the usefulness of the period life expectancy as the leading measure of survivorship. An alternative aggregate measure of period mortality which has been seen as less sensitive to period changes, the cross-sectional average length of life (CAL) has been proposed as an alternative, but has received only limited empirical or analytical examination. Here, we introduce a new measure, the average cohort life expectancy (ACLE), to provide a precise measure of the average length of life of cohorts alive at a given time. To compare the performance of ACLE with CAL and with period and cohort life expectancy, we first use population models with changing mortality. Then the four aggregate measures of mortality are calculated for England and Wales, Norway, and Switzerland for the years 1880 to 2000. CAL is found to be sensitive to past and present changes in death rates. ACLE requires the most data, but gives the best representation of the survivorship of cohorts present at a given time.cohort life expectancy, cross-sectional average length of life (CAL), life expectancy, mortality, mortality tempo, period life expectancy

A diminishing population whose every cohort more than replaces itself

We observe that a dynamic population model can have period fertility that is always below replacement and cohort fertility that is always above replacement. We ask whether such a paradoxical population will get larger or smaller, and show that it must become smaller. Cohort replacement does not imply population replacement, and emphasizing fertility timing and cohort fertility ignores the issue of relative cohort size. The resolution of this apparent paradox reinforces the importance of the level of period fertility in demographic analysis.Bongaarts-Feeney adjusted fertility, cohort, cohort fertility, dynamic population model, fertility, fertility timing, period fertility

Estimating multistate transition rates from population distributions

The ability to estimate transition rates (or probabilities) from population distributions has many potential applications in demography. Iterative Proportional Fitting (IPF) has been used for such estimation, but lacks a meaningful behavioral, or demographic, foundation. Here a new approach, Relative State Attractiveness (RSA), is advanced. It assumes that states become more (or less) attractive, and that rates respond accordingly. The RSA estimation procedure is developed and applied to model and actual data where the underlying rates are known. Results show that RSA provides accurate estimates under a wide range of conditions, usually yielding values similar to those produced by IPF. Both methods are then applied to U.S. data to provide new estimates of interregional migration between the years 1980 and 1990.entropy, estimation techniques, iterative proportional fitting, multistate models