Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell Population in the Light of Suns and Stars

  • Y. Elalaoui
  • L. Alaoui


A model describing the interactions between cells of several levels of differentiation and malignancy by tracking their lineages while taking into consideration the proliferation process is analyzed. The model is structured by age and is formulated by a system of partial differential equations. The analysis of the model is based on the perturbation theory for dual semigroups of operators applied to a renewal equation of the type u(t) = ϕ(ut). The operator ϕ associated with the model is determined and compactness and spectral properties are established to conclude the asynchronous exponential growth property for the model and the characterization of associated Malthusian coefficient by only using the properties of ϕ.


  1. 1.
    L. Alaoui, Population dynamics and translation semigroups. Dissertation, University of Tübingen, 1995zbMATHGoogle Scholar
  2. 2.
    L. Alaoui, Generators of translation semigroups and asymptotic behavior of the Sharpe-Lotka model. Diff. Int. Equ. 9, 343–362 (1996)MathSciNetzbMATHGoogle Scholar
  3. 3.
    L. Alaoui, A cell cycle model and translation semigroups. Semigroup Forum 54(1), 135–153 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    L. Alaoui, Age-dependent population dynamics and translation semigroups. Semigroup Forum 57, 186–207 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. Alaoui, Nonlinear homogeneous retarded differential equations and population dynamics via translation semigroups. Semigroup Forum 63, 330–356 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Alaoui, O. Arino, Compactness and spectral properties for positive translation semigroups associated with models of population dynamics. Diff. Int. Equ. 6, 459–480 (1993)MathSciNetzbMATHGoogle Scholar
  7. 7.
    L. Alaoui, Y. El Alaoui, AEG property of a cell cycle model with quiescence in the light of translation semigroups. Int. J. Math. Anal. 9(51), 2513–2528 (2015)CrossRefGoogle Scholar
  8. 8.
    O. Arino, M. Kimmel, G.F. Webb, Mathematical modeling of the loss of telomere sequences. J. Theor. Biol. 177, 45–57 (1995)CrossRefGoogle Scholar
  9. 9.
    O. Arino, E. sánchez, G.F. Webb, Polynomial growth dynamics of telomere loss in a heterogeneous cell population. Dynam. Control Discrete Impuls. Syst. 3, 263–282 (1997)Google Scholar
  10. 10.
    T.H. Brummendorf, J. Mak, K.M. Sabo, G.M. Baerlocker, K. Dietz, J.L. Abowitz, P.M. Lansdorp, Longitudinal studies of telomere length in feline blood cells: implications for hematopoeitic stem cell turnover in vivo. Exp. Hematol. 30, 1147–1152 (2002)CrossRefGoogle Scholar
  11. 11.
    Ph. Clement, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, H.R. Thieme, Perturbation theory for dual semigroups. I. The sun-reflexive case. Math. Ann. 277, 709–725 (1987)zbMATHGoogle Scholar
  12. 12.
    Ph. Clement, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, H.R. Thieme, Perturbation theory for dual semigroups. II. Time-dependent perturbations in the sunreflexive case. Proc. R. Soc. Edinburgh Sect. A 109, 145–172 (1988)CrossRefGoogle Scholar
  13. 13.
    Ph. Clement, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, H.R. Thieme, Perturbation theory for dual semigroups. III. Nonlinear Lipschitz continuous perturbations in the sun-reflexive case, in Volterra Integrodifferential Equations in Banach Spaces and Applications, ed. by G. da Prato, M. Iannelli. Pitman Research Notes in Mathematics Series, vol. 190 (Longman Scientific and Technical, Harlow, 1989), pp. 67–89Google Scholar
  14. 14.
    Ph. Clement, O. Diekmann, M. Gyllenberg, H.J.A.M. Heijmans, H.R. Thieme, Perturbation theory for dual semigroups. IV. The intertwining formula and the canonical pairing, in Trends in Semigroup Theory and Applications, ed. by Ph. Clèment, S. Invernizzi, E. Mitidieri, I.I. Vrabie (Dekker, New York, 1989), pp. 95–116Google Scholar
  15. 15.
    O. Diekmann, M. Gyllenberg, Equations with infinite delay: blending the abstract and the concrete. J. Differ. Equ. 252(2), 819–851 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    O. Diekmann, S.A. Van Gils, S.M. Verduyn Lunel, H.-O. Walther, Delay Equations: Functional, Complex and Nonlinear Analysis (Springer, New York, 1995)CrossRefGoogle Scholar
  17. 17.
    O. Diekmann, P. Getto, M. Gyllenberg, Stability and bifurcation analysis of volterra functional equations in the light of suns and stars. SIAM J. Math. Anal. 39(4), 1023–1069 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    J. Dyson, R. Villella-Bressan, G.F. Webb, Asymptotic behaviour of solutions to abstract logistic equations. Math. Biosci. 206, 216–232 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    J. Dyson, E. Sánchez, R. Villella-Bressan, G.F. Webb, Stabilization of telomeres in nonlinear models of proliferating cell lines. J. Theor. Biol. 244, 400–408 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Y. El Alaoui, L. Alaoui, Asymptotic behavior in a cell proliferation model with unequal division and random transition using translation semigroup. Indian J. Sci. Technol. 10(28), 1–8 (2017)CrossRefGoogle Scholar
  21. 21.
    K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer, New York, 2000)zbMATHGoogle Scholar
  22. 22.
    L. Hayflick, P.S. Moorhead, The serial cultivationn of human diploid strains. Expt. Cell. Res. 25, 585–621 (1961)CrossRefGoogle Scholar
  23. 23.
    G. Kapitanov, A mathematical model of cancer stem cell lineage population dynamics with mutation accumulation and telomere length hierarchies. Math. Model. Nat. Phenom. 7(1), 136–165 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    P. Olofsson, M. Kimmel, Stochastic models of telomere shortening. Math. Biosci. 158, 75–92 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    A.M. Olovnikov, Principle of marginotomy in template synthesis of polynucleotides. Dokl. Akad. Nark. S.S.S.R 201, 1496–1499 (1971)Google Scholar
  26. 26.
    A.M. Olovnikov, A theory of marginotomy. J. Theor. Biol. 41, 181–190 (1973)CrossRefGoogle Scholar
  27. 27.
    I. Sidorov, D. Gee, D.S. Dimitrov, A Kinetic model of telomer shorteningin infants and adults. J. Theor. Biol. 226, 169–175 (2002)CrossRefGoogle Scholar
  28. 28.
    I. Sidorov, K.S. Hirsch, C.B. Harley, D.S. Dimitrov, Cancer cell dynamics in presence of telomerase inhibitors: analysis of in vitro data. J. Theor. Biol. 219, 225–233 (2004)MathSciNetCrossRefGoogle Scholar
  29. 29.
    J. Van Neerven, The Adjoint of a Semigroup of Linear Operators. Lecture Notes in Mathematics, vol. 1529 (Springer, Berlin, 1992)Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Y. Elalaoui
    • 1
  • L. Alaoui
    • 2
  1. 1.Faculty of Sciences, Department of MathematicsMohammed V UniversityAgdalMorocco
  2. 2.International University of RabatSala Al-JadidaMorocco

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