Skip to main content

Generalization of Carey’s equality and a theorem on stationary population

Journal of Mathematical Biology volume 71, pages 583–594 (2015)Cite this article

Abstract

Carey’s Equality pertaining to stationary models is well known. In this paper, we have stated and proved a fundamental theorem related to the formation of this Equality. This theorem will provide an in-depth understanding of the role of each captive subject, and their corresponding follow-up duration in a stationary population. We have demonstrated a numerical example of a captive cohort and the survival pattern of medfly populations. These results can be adopted to understand age-structure and aging process in stationary and non-stationary population models.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

43,34 €

Price includes VAT (Finland)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2

References

  • Brookmeyer R, Gail MH (1988) A method for obtaining short-term projections and lower bounds on the size of the AIDS epidemic. J Am Stat Assoc 83(402):301–308

    Article  MATH  Google Scholar 

  • Carey JR, Papadopoulos N, Müller H-G, Katsoyannos B, Kouloussis N, Wang J-L, Wachter K, Yu W, Liedo P (2008) Age structure changes and extraordinary life span in wild medfly populations. Aging Cell 7:426–437

    Article  Google Scholar 

  • Carey JR, Müller H-G, Wang J-L, Papadopoulos NT, Diamantidis A, Kouloussis NA (2012) Graphical and demographic synopsis of the captive cohort method for estimating population age structure in the wild. Exp Gerontol 47:787–791

    Article  Google Scholar 

  • Cox DR (1962) Renewal theory. Methuen and Co, London

    MATH  Google Scholar 

  • Feller W (1941) On the integral equation of renewal theory. Ann Math Stat 12(3):243–267

    MathSciNet  Article  Google Scholar 

  • Feller W (1971) An introduction to the probability theory and its applications, vol II, 2nd edn. Wiley, New York

    Google Scholar 

  • Goldstein JR (2009) Life lived equals life left in stationary populations. Demogr Res 20:3–6

    Article  Google Scholar 

  • Goswami A, Rao BV (2006) A course in applied stochastic processes. Hindustan Book Agency (India), New Delhi

    MATH  Google Scholar 

  • Karlin S (1969) A first course in stochastic process. Academic Press Inc, New York

    Google Scholar 

  • Kim YJ, Aron JL (1989) On the equality of average age and average expectation of remaining life in a stationary population. SIAM Rev 31(1):110–113. doi:10.1137/1031005

    MathSciNet  Article  MATH  Google Scholar 

  • Lawler GF (2006) Introduction to stochastic process (2/e). Chapman & Hall, London

    Google Scholar 

  • Lotka AJ (1907) Relation between birth rates and death rates. Science 26:21–22

    Article  Google Scholar 

  • Lotka AJ (1939a) A contribution to the theory of self-renewing aggregates with special reference to industrial replacement. Ann Math Stat 10:1–25

    Article  Google Scholar 

  • Lotka AJ (1939b) On an integral equation in population analysis. Ann Math Stat 10:144–161

    Article  Google Scholar 

  • Lotka AJ (1956) Elements of mathematical biology. Dover Publications, New York Inc

    MATH  Google Scholar 

  • Müller HG, Wang J-L, Carey JR, Caswell-Chen EP, Chen C, Papadopoulos N, Yao F (2004) Demographic window to aging in the wild: constructing life tables and estimating survival functions from marked individuals of unknown age. Aging Cell 3:125–131

    Article  Google Scholar 

  • Preston SH, Heuveline P, Guillot M (2001) Demography: measuring and modeling population processes. Blackwell Publishers, Malden

    Google Scholar 

  • Ryder NB (1975) Notes on stationary populations. Popul Index 41(1):3–28

    Article  Google Scholar 

  • Rao ASRS, Carey JR (2014) Behavior of Carey’s Equality in two-dimensions: age and proportion of population (manuscript in-preparation)

  • Rao ASRS (2015) Incubation periods under various anti-retroviral therapies in homogeneous mixing and age-structured dynamical models: a theoretical approach. Rocky Mountain J Math (to appear in 2015). http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rmjm/1379596735&page=record

  • Rao ASRS (2014) Population stability and momentum. Not Am Math Soc 61(9):1162–1165

  • Vaupel JW (2009) Life lived and left: Carey’s Equality. Demogr Res 20(2009):7–10

    Article  Google Scholar 

Download references

Acknowledgments

We thank the organizers of the Keyfitz Centennial Symposium on Mathematical Demography sponsored by the Mathematical Biosciences Institute, Ohio State University, June 2013. Research by JRC supported by NIA/NIH Grants P01 AG022500-01 and P01 AG08761-10.

Author information

Affiliations

  1. Georgia Regents University, 1120 15th Street, Augusta, GA, 30912, USA

    Arni S. R. Srinivasa Rao

  2. Department of Entomology, University of California, Davis, CA, 95616, USA

    James R. Carey

  3. Center for the Economics and Demography of Aging, University of California, Berkeley, CA, 94720, USA

    James R. Carey

Authors
  1. Arni S. R. Srinivasa Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James R. Carey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arni S. R. Srinivasa Rao.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Srinivasa Rao, A.S.R., Carey, J.R. Generalization of Carey’s equality and a theorem on stationary population. J. Math. Biol. 71, 583–594 (2015). https://doi.org/10.1007/s00285-014-0831-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-014-0831-6

Keywords

  • Captive cohort
  • Life expectancy
  • Symmetric patterns

Mathematics Subject Classification

  • 92A17
  • 05.99
  • 60K05