Qualitative Properties of a Cell Proliferating Model with Multi-phase Transition and Age Structure

Qualitative Properties of a Cell Proliferating Model with Multi-phase Transition and Age Structure

Volume 5, Issue 6, Page No 01-08, 2020

Author’s Name: Youssef El Alaoui1,a), Larbi Alaoui2

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1Faculty of Sciences, University Mohammed V – Agdal, Rabat, 10010, Morocco
2International University of Rabat, Sala Al-Jadida, 11100, Morocco

a)Author to whom correspondence should be addressed. E-mail: is.youssefelalaoui@gmail.com

Adv. Sci. Technol. Eng. Syst. J. 5(6), 01-08 (2020); a  DOI: 10.25046/aj050601

Keywords: Semigroup, Dual semigroup, Asynchronous exponential, growth

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Abstract

In this paper we study a cell division cycle modeled by a system of partial differential equations with an age structure. This model translates the many regulatory mechanisms within the cell cycle where it introduces the notion of phases. The individual cell can be either in I phases where the transition between theses phases are ordered and unidirectional. The model is related to the suns and stars caluculus via the dual semigroups of operators that are considered as solution of an abstract integral equation equivalent to a Volterra type equation of the form w(t) = Φ(wt). We will determine the core operator Φ and prove that the semigroup solution of the model possesses the asynchronous exponential property. The model permits different types of controls where the provided framework allows better control on the model parameters and yields the characterization of the intrinsic rate of natural increase through properties of the core operator Φ. Finally, we demonstrate that the asymptotic behavior of the model is governed by the simple dominant eigenvalue and its associated eigenvector, that leads to the dispersion of the cell structure through the future generations.

Received: 30 August 2020, Accepted: 18 October 2020, Published Online: 08 November 2020

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