Journal of Statistical Physics

, Volume 148, Issue 3, pp 565–578 | Cite as

A Markovian Growth Dynamics on Rooted Binary Trees Evolving According to the Gompertz Curve

  • C. Landim
  • R. D. Portugal
  • B. F. Svaiter


Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields (Probab. Theory Relat. Fields 79(4):509–542, 1988) model. Fix n≥1 and β>0. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate β(nk)/n, where k is the distance from the node to the root. Denote by Z n (t) the number of nodes with no descendants at time t and let T n =β −1 nln(n/ln4)+(ln2)/(2β). We prove that 2n Z n (T n +), τ∈ℝ, converges to the Gompertz curve exp(−(ln2) e βτ ). We also prove a central limit theorem for the martingale associated to Z n (t).


Aging Random binary trees Gompertz curve Growth processes 


  1. 1.
    Aldous, D., Shields, P.: A diffusion limit for a class of randomly-growing binary trees. Probab. Theory Relat. Fields 79(4), 509–542 (1988) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Baxter, M.A., Wynn, R.F., Jowitt, S.N., Wraith, J.E., Fairbairn, L.J., Ellington, I.: Study of telomere length reveals rapid aging of human marrow stromal cells following in vitro expansion. Stem Cells 22(5), 675–682 (2004) CrossRefGoogle Scholar
  3. 3.
    Best, K., Pfaffelhuber, P.: The Aldous-Shields model revisited with applications to cellular ageing. Electron. Commun. Probab. 15, 475–488 (2010) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1999) MATHCrossRefGoogle Scholar
  5. 5.
    Coe, J.B., Mao, Y.: Gompertz mortality law and scaling behavior of the Penna model. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 72(5 Pt 1), 051925 (2005). ADSCrossRefGoogle Scholar
  6. 6.
    Dean, D.S., Majumdar, S.N.: Phase transition in a generalized Eden growth model on a tree. J. Stat. Phys. 124, 1351–1376 (2006) MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Gompertz, B.: On the nature of the function expressive of the law of human mortality, and on a new method of determining the value of life contingencies. Philos. Trans. R. Soc. Lond. 115, 513–585 (1825) CrossRefGoogle Scholar
  8. 8.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288. Springer, Berlin (1987) MATHGoogle Scholar
  9. 9.
    Harley, C.B., Vaziri, H., Counter, C.M., Allsopp, R.C.: The telomere hypothesis of cellular aging. Exp. Gerontol. 27(4), 375–382 (1992) CrossRefGoogle Scholar
  10. 10.
    Hayflick, L.: The limited in vitro lifetime of human diploid cell strains. Exp. Cell Res. 37, 614–636 (1965) CrossRefGoogle Scholar
  11. 11.
    Laird, A.K.: Dynamics of relative growth. Growth 29(3), 249–263 (1965) MathSciNetGoogle Scholar
  12. 12.
    Laird, A.K.: Dynamics of growth in tumors and in normal organisms. In: National Cancer Institute Monograph, vol. 30, pp. 15–28 (1969) Google Scholar
  13. 13.
    Portugal, R.D., Land, M.G.P., Svaiter, B.F.: A computational model for telomere-dependent cell-replicative aging. Biosystems 91(1), 262–267 (2008) CrossRefGoogle Scholar
  14. 14.
    Wallenstein, S., Brem, H.: Statistical analysis of wound-healing rates for pressure ulcers. Am. J. Surg. 188(1A Suppl), 73–78 (2004) CrossRefGoogle Scholar
  15. 15.
    Winsor, C.P.: The Gompertz curve as a growth curve. Proc. Natl. Acad. Sci. USA 18(1), 1–8 (1932) ADSMATHCrossRefGoogle Scholar
  16. 16.
    Wright, S.: Book review. J. Am. Stat. Assoc. 21, 494 (1926) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil
  2. 2.CNRS UMR 6085Université de RouenSaint-Étienne-du-RouvrayFrance
  3. 3.Faculty of MedicineFederal University of Rio de JaneiroRio de JaneiroBrazil

Personalised recommendations