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. SwansonEmail author
  • Lucky M. Tedrow
  • Jack Baker
Part of the The Springer Series on Demographic Methods and Population Analysis book series (PSDE, volume 39)


Cohort Change Ratios (CCRs) appear to have been overlooked in regard to a major canon of formal demography, stable population theory. CCRs are explored here 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 as the point when S is equal zero (operationalized as S = 0.000000). 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 (S = .01, S = .05, S = .001, and S = .0005). The regression-based 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. 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. We also find that a regression model works very well in estimating the intrinsic rate of increase from the initial rate of increase, but that 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.


Leslie matrix Stability index Initial conditions 


  1. Alho, J. (2008). Migration, fertility, and aging in stable populations. Demography, 45(3), 641–650.CrossRefGoogle Scholar
  2. Alho, J., & Spencer, B. (2005). Statistical demography and forecasting. New York: Springer.Google Scholar
  3. Arthur, W. B. (1981). Why a population converges to stability. The American Mathematical Monthly, 86(8), 557–563.CrossRefGoogle Scholar
  4. Arthur, W. B., & Vaupel, J. (1984). Some relationships in population dynamics. Population Index, 50(2), 214–226.CrossRefGoogle Scholar
  5. Bacaër, N. (2011). A short history of population dynamics. Dordrecht: Springer.CrossRefGoogle Scholar
  6. Barclay, R. (1958). Techniques of population analysis. New York: Wiley.Google Scholar
  7. Bennett, N., & Horuchi, S. (1984). Mortality estimation from registered deaths in less developed countries. Demography, 21(2), 217–233.CrossRefGoogle Scholar
  8. Caswell, H. (2001). Matrix population models: Construction, analysis, and interpretation (2nd ed.). Sunderland: Sinauer Associates, Inc.Google Scholar
  9. Coale, A. J. (1957). A new method for calculating Lotka’s r – The intrinsic rate of growth in a stable population. Population Studies, 11, 92–94.Google Scholar
  10. Coale, A. J. (1972). The growth and structure of human populations: A mathematical investigation. Princeton: Princeton University Press.Google Scholar
  11. Coale, A. J., & Demeny, P. (1966). Regional model life tables and stable populations. Princeton: Princeton University Press.Google Scholar
  12. Coale, A. J., & Trussell, J. (1974). Model fertility schedules: Variations in the age structure of childbearing in human populations. Population Index, 40(2), 185–258.CrossRefGoogle Scholar
  13. Cohen, J. (1979a). Ergodic theorems in demography. Bulletin of the American Mathematical Society, 1(2), 275–295.CrossRefGoogle Scholar
  14. Cohen, J. (1979b). The cumulative distance from an observed to a stable population age structure. SIAM Journal of Applied Mathematics, 36, 169–175.CrossRefGoogle Scholar
  15. Dublin, L., & Lotka, A. (1925). On the true rate of natural increase: As exemplified by the population of the United States, 1920. Journal of the American Statistical Association, 20, 305–339.Google Scholar
  16. Espenshade, T. (1986). Population dynamics with immigration and low fertility. Population and Development Review, 12, 248–261.CrossRefGoogle Scholar
  17. Espenshade, T., Bouvier, L., & Arthur, W. B. (1982). Immigration and the stable population model. Demography, 19, 125–133.CrossRefGoogle Scholar
  18. Hamilton, C. H., & Perry, J. (1962). A short method for projecting population by age from one decennial census to another. Social Forces, 41, 163–170.CrossRefGoogle Scholar
  19. Hardy, G. F., & Wyatt, F. B. (1911). Report of the actuaries in relation to the scheme of insurance against sickness, disablement, &c., embodied in the national insurance bill. Journal of the Institute of Actuaries XLV, 406–443.Google Scholar
  20. Hobbs, F. (2004). Age and sex composition. In J. Siegel, & D. A. Swanson (Eds.), The methods and materials of demography (2nd ed., pp. 125–173). San Diego: Elsevier Academic Press.Google Scholar
  21. Keyfitz, N. (1968). Introduction to the mathematics of population. Reading: Addison-Wesley.Google Scholar
  22. Keyfitz, N. (1974). A general condition for stability in demographic processes. Canadian Studies in Population, 1, 29–35.Google Scholar
  23. Keyfitz, N. (1977). Introduction to the mathematics of population. New York: Addison-Wesley.Google Scholar
  24. Keyfitz, N. (1980). Multistate demography and its data: A comment. Environment and Planning A, 12, 615–622.CrossRefGoogle Scholar
  25. Keyfitz, N., & Flieger, W. (1968). World population: An analysis of vital data. Chicago: University of Chicago Press.Google Scholar
  26. Kim, Y., & Schoen, R. (1993a). On the intrinsic force of convergence to stability. Mathematical Population Studies, 4(2), 89–102.CrossRefGoogle Scholar
  27. Kim, Y., & Schoen, R. (1993b). Crossovers that link populations with the same vital rates. Mathematical Population Studies, 4(1), 1–19.CrossRefGoogle Scholar
  28. Kim, Y., & Sykes, Z. (1976). An experimental study of weak ergodicity in human populations. Theoretical Population Biology, 10, 150–172.CrossRefGoogle Scholar
  29. Land, K. (1986). Methods for national population forecasts: A review. Journal of the American Statistical Association, 81, 888–901.CrossRefGoogle Scholar
  30. Le Bras, H. (2008). The nature of demography. Princeton: Princeton University Press.Google Scholar
  31. Liaw, K. L. (1980). Multistate dynamics: The convergence of an age-by-region population system. Environment and Planning A, 12, 589–613.CrossRefGoogle Scholar
  32. Lotka, A. J. (1907). Relation between birth rates and death rates. Science (New Series), 26(653), 21–22.Google Scholar
  33. McCann, J. (1973). A more accurate short method of approximating Lotka’s r. Demography, 10(4), 567–570.CrossRefGoogle Scholar
  34. Mitra, S., & Cerone, P. (1986). Migration and stability. Genus, 42(1-2), 1–12.Google Scholar
  35. Nair, S. B., & Nair, P. S. (2010). Momentum of population growth in India. In S. Nangia, N. C. Jana, & R. B. Bhagat, (Eds.), State of natural and human resources in India, Part 2 (pp. 399–424). New Delhi: Concept Publishing Company, Ltd.Google Scholar
  36. Pollak, R. (1986). A re-formulation of the two-sex problem. Demography, 23, 247–259.CrossRefGoogle Scholar
  37. Pollard, A., Yusuf, F., & Pollard, G. (1974). Demographic techniques (3rd ed.). New York: Pergamon Press.Google Scholar
  38. Popoff, C., & Judson, D. (2004). Some methods of estimation for statistically underdeveloped areas. In J. Siegel & D. A. Swanson (Eds.), The methods and materials of demography (2nd ed., pp. 603–641). San Diego: Elsevier Academic Press.Google Scholar
  39. Pressat, R. (2009). Demographic analysis: Projections on natality, fertility and replacement (2nd Paperback Printing). New Brunswick: Aldine Transaction.Google Scholar
  40. Preston, S. (1986). The relation between actual and intrinsic growth rates. Population Studies, 40(3), 343–351.CrossRefGoogle Scholar
  41. Preston, S., & Coale, A. J. (1982). Age structure, growth, attrition, and accession: A new synthesis. Population Index, 48, 217–259.CrossRefGoogle Scholar
  42. Preston, S., Heuveline, P., & Guillot, M. (2001). Demography: Measuring and modeling population processes. Malden: Blackwell Publishing.Google Scholar
  43. Rogers, A. (1985). Regional population projection models (Vol. 4, Scientific Geography Series). Beverly Hills: Sage Publications.Google Scholar
  44. Rogers, A. (1995). Multiregional demography: Principles, methods, and extensions. New York: Wiley.Google Scholar
  45. Rogers, A., Little, J., & Raymer, J. (2010). The indirect estimation of migration. Dordrecht: Springer.CrossRefGoogle Scholar
  46. Rogers-Bennett, L., & Leaf, R. (2006). Elasticity analysis of size-based red and white abalone matrix models: Management and conservation. Ecological Applications, 16(1), 213–224.CrossRefGoogle Scholar
  47. Schoen, R. (1988). Modeling multigroup populations. New York: Plenum Press.CrossRefGoogle Scholar
  48. Schoen, R. (2006). Dynamic population models. Dordrecht: Springer.Google Scholar
  49. Schoen, R., & Kim, Y. (1991). Movement toward stability as a fundamental principle of population dynamics. Demography, 28(3), 455–466.CrossRefGoogle Scholar
  50. Sivamurthy, M. (1982). Growth and structure of human population in the presence of migration. London: Academic.Google Scholar
  51. Smith, S., Tayman, J., & Swanson, D. A. (2013). A practitioner’s guide to state and local population projections. Dordrecht: Springer.CrossRefGoogle Scholar
  52. Stubben, C., & Milligan, B. (2007). Estimating and analyzing demographic models using the popbio package in R. Journal of Statistical Software, 22(11), 1–23.CrossRefGoogle Scholar
  53. Swanson, D. A., & Tayman, J. (2013). Subnational population estimates. Dordrecht: Springer.Google Scholar
  54. Swanson, D. A., & Tayman, J. (2014). Measuring uncertainty in population forecasts: A new approach (pp. 203–215). In M. Marsili & G. Capacci (Eds.), Proceedings of the 6th EUROSTAT/UNECE work session on demographic projections National Institute of Statistics, Rome, Italy.Google Scholar
  55. Swanson, D. A., & Tedrow, L. (2012). Using cohort change ratios to estimate life expectancy in populations with negligible migration: A new approach. Canadian Studies in Population, 39, 83–90.Google Scholar
  56. Swanson, D. A., & Tedrow, L. (2013). Exploring stable population concepts from the perspective of cohort change ratios. The Open Demography Journal, 6, 1–17.CrossRefGoogle Scholar
  57. Swanson, D. A., Schlottmann, A., & Schmidt, R. (2010). Forecasting the population of census tracts by age and sex: An example of the Hamilton-Perry method in action. Population Research and Policy Review, 29(1), 47–63.CrossRefGoogle Scholar
  58. Sykes, Z. M. (1969). Stochastic versions of the matrix model of population dynamics. Journal of the American Statistical Association, 64(325), 111–130.CrossRefGoogle Scholar
  59. Tuljapurkar, S. (1982). Why use population entropy? It determines the rate of convergence. Journal of Mathematical Biology, 13, 325–337.CrossRefGoogle Scholar
  60. United Nations. (1968). The concept of a stable population: Applications to the study of populations with incomplete demographic statistics (Population Studies No. 39). New York: United Nations, Department of Economic and Social Affairs.Google Scholar
  61. United Nations. (2002). Methods for estimating adult mortality. New York: Population Division, United Nations.Google Scholar
  62. United Nations. (2008). 2006 Demographic yearbook. New York: Department of Economic and Social Affairs, United Nations.Google Scholar
  63. Vaupel, J., & Yashin, A. (1985). Heterogeneity’s ruses: Some surprising effects of selection on population dynamics. The American Statistician, 39(August), 176–185.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • David A. Swanson
    • 1
    Email author
  • Lucky M. Tedrow
    • 2
  • Jack Baker
    • 3
  1. 1.Department of SociologyUniversity of California RiversideRiversideUSA
  2. 2.Department of SociologyWestern Washington UniversityBellinghamUSA
  3. 3.Geospatial and population StudiesUniversity of New MexicoAlbuquerqueUSA

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