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Asymptotic Behavior in a Cell Proliferation Model with Unequal Division and Random Transition using Translation Semigroups


  • Faculty of Science, University Mohamed V, Agdal, Rabat, Morocco
  • International University of Rabat, Sala Al-Jadida, Rabat – 11100, Morocco


Objectives: We provide a theoretical framework for the mathematical analysis of a cell cycle model described by a delay integral equation to get properties on the asymptotic behavior of the solutions. Methods: We relate the model to the class of translation semigroups of operators that are associated with a core operator φ and are solutions of equations of the type m (t) = Φ(mt). Then by using the theory developed for such class of semigroups we establish results on the existence, uniqueness, positivity, compactness and spectral properties of the solution semigroup in order to conclude the asynchronous exponential growth (AEG) property for the model. Findings: The framework yields an innovative analysis method for the model where only conditions on the parameters of its associated core operator are considered. It allows better control of the parameters for getting the AEG property and the derivation in an automatic way of characterizations of associated generator, spectral properties and AEG property only in terms of the core operator φ Indeed the Malthusian parameter λ0 is characterized as the only solution of the equation )r(Φ ˜λ)= 1 where(Φ ˜λ) :=Φ( eλ⊗), coincides with the spectral bound of the generator of the solution semigroup and is a dominant eigen value of this generator. Application/ Improvements: The provided framework will pave the way for the study of other aspects such as oscillations, bifurcation and chaos to get better insights of the dynamics of the model solutions.


Asymptotic Behavior, Cell Proliferation, Translation Semigroups

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  • Arino O, Kimmel M. Comparison of approaches to modeling of cell population dynamics. SIAM Journal on Applied Mathematics. 1993; 53(5):1480–504. Crossref
  • Arino O, Kimmel M, Zerner M. Analysis of a cell population model with unequal division and random transition. In: Arino A, Axelrod D, Kimmel M. eds. Proceeding of the 2nd International Conference M.P.D, New York: 1991. p.3–12.
  • Kimmel M, Axelrod DE. Branching Processes in Biology. Springer; 2015. Crossref
  • Guo X, Bernard A, Orlandob DA, Haase SB, Harteminka AJ. Branching process deconvolution algorithm reveals a detailed cell-cycle transcription program. Proceedings of the National Academy of Sciences of the United States of America. 2013; 110 (10):968–77.Crossref PMCid:PMC3593847
  • Alaoui L, Alaoui YE. AEG property of a cell cycle model with quiescence in the light of translation semigroups. International Journal of Mathematical Analysis. 2015; 9(51):2513–28. Crossref
  • Alaoui L, Khaladi M. Rate of convergence to equilibria of the Lotka-Sharpe-McKendrick model. International Journal of Mathematical Analysis. 2015; 9(1):39–60. Crossref
  • Alaoui L. Nonlinear homogeneous retarded differential equations and population dynamics via translation semigroups. Semigroup Forum. 2001; 63:330–56. Crossref
  • Alaoui L. Age-dependent population dynamics and translation semigroups. Semigroup Forum. 1998; 57:186–207. Crossref
  • Alaoui L. A cell cycle model and translation semigroups. Semigroup Forum. 1997; 54:135–53. Crossref
  • Alaoui L. Generators of translation semigroups and asymptotic behavior of the Sharpe-Lotka model. Differential and Integral Equations. 1996; 9(2):343–62.
  • Alaoui L, Arino O. Compactness and spectral properties for positive translation semigroups associated with models of population dynamics. Differential and Integral Equations. 1993; 6(2):459–80.
  • Billy F, Clairambault J, Delaunay F, Feillet C, Robert N. Age structured cell population model to study the influence of growth factors on cell cycle dynamics. Mathematical Biosciences and Engineering. 2013; 10(1):1–17. Crossref, PMid:23311359
  • Kapitanov G, A Mathematical Model of Cancer Stem Cell Lineage Population Dynamics with Mutation Accumulation and Telomere Length Hierarchies. Mathematical Modeling of Natural Phenomena. 2012; 7(1):136–65. Crossref
  • Alexanderian A, Gobbert MK, Fister KR, Gaff H, Lenhart S, Schaefer E. An age structured model for the spread of epidemic cholera: Analysis and simulation. Nonlinear Analysis: Real World Applications. 2011; 12(6):3483–98. Crossref
  • Inaba G, Nishiurab G. The Basic Reproduction Number of an Infectious Disease in a Stable Population: The Impact of Population Growth Rate on the Eradication Threshold. Mathematical Modelling of Natural Phenomena. 2008; 3(7):194–228. Crossref
  • Yang J, Wang X. Existence of a Nonautonomous SIR Epidemic Model with Age Structure. Advances in Difference Equations. 2010. Crossref
  • Grabosch A. Translation semigroups and their linearizations on spaces of integrable functions. Transactions of the American Mathematical Society. 1989; 311(1):351–91. Crossref
  • Arino O, Kimmel M. Asymptotic analysis of a cell cycle model based on unequal division. SIAM Journal on Applied Mathematics. 1987; 47(1):128–45. Crossref
  • Kimmel M, Darzynkiewicz Z, Arino O, Traganos F. Analysis of a model of cell cycle based on unequal division of mitotic constituents to daughter cells during cytokinesis. Journal of Theoretical Biology. 1984; 110(4):637–64. Crossref
  • Zerner M. Quelques proprietes spectrales des operateurs positifs. Journal of Functional Analysis. 1987; 72(2):381– 417. Crossref


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