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

Abstract

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(m t ). 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 λ 0 is characterized as the only solution of the equation )r(Φ ˜ λ )= 1 where(Φ ˜ λ ) :=Φ( e λ ⊗), 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.