Exploring Stable Population Concepts from the Perspective of Cohort Change Ratios: Estimating the Time to Stability and Intrinsic R from Initial Information and Components of Change

David A. Swanson, University of California, Riverside
Lucky M. Tedrow, Western Washington University
Jack Baker, University of New Mexico

Cohort Change Ratios (CCRs) have a long history of use in demography. Except in their restrictive form as survival rates, CCRs, appear, however, to have been overlooked in regard to a major canon of formal demography, stable population theory. We believe that it is worthwhile to move beyond this restrictive form and examine full CCRs because they contain information about both migration and mortality. As a means of exploiting this information, CCRs are explored as a tool for examining the transient dynamics of a population as it moves toward the stable equivalent that is captured in most formal demographic models based on asymptotic population dynamics. We employ simulation and a regression-based approach to model trajectories toward this stability. This examination is done in conjunction with the Leslie Matrix and data for 62 countries selected from the US Census Bureau’s International Data Base. We use an Index of Stability (S), which defines stability at the point when it is equal to 0.0000. The Index also is used to define initial stability for a given population and four subsequent “quasi-stable” points on the temporal path to stability. The Stability Index can be readily calculated and provides an easy-to-interpret measure of the distance to stability. We use the ergodicity theorem as a guide in that it directly states that initial conditions are “forgotten” when a population reaches stability and only vital rates play a role. Given this, we not only explore the effect of the initial Stability Index and vital rates, but also the effect of migration on the temporal path to stability. We use a series of regression models for this purpose. The analysis reveals that the initial conditions as defined by the initial Stability Index along with fertility and migration play a role in determining time to stability up until the quasi-stable point of S =.0005 is reached. After this point, the initial conditions are no longer a factor and mortality joins the fertility and migration components in determining the remaining time to stability. These findings are consistent with ergodicity. Overall all, we find that fertility and mortality have an inverse relationship with time to stability while migration has a positive relationship. The initial Stability Index has an inverse relationship with time to quasi-stability at S=.01, S=.005, S=.001, and S=.0005. Continuing the use of regression analysis, we also find that a regression model works very well in estimating the intrinsic rate of increase from the initial rate of increase, but this model can be improved by adding the components of change. We also compare time to stability and intrinsic r as estimated using the CCR Leslie Matrix approach to, respectively, estimates of time to stability and intrinsic r found using analytic methods and find that the former are consistent with the latter. We discuss our findings and provide suggestions for future work.