title: Discrete limit and monotonicity properties of the Floquet eigenvalue in an age structured cell division cycle model.
abstract: We consider a cell population described by an age-structured partial differential equation with time periodic coefficients. We assume that division only occurs within certain time intervals at a rate [Formula: see text] for cells who have reached minimal positive age (maturation). We study the asymptotic behavior of the dominant Floquet eigenvalue, or Perron-Frobenius eigenvalue, representing the growth rate, as a function of the maturation age, when the division rate [Formula: see text] tends to infinity (divisions become instantaneous). We show that the dominant Floquet eigenvalue converges to a staircase function with an infinite number of steps, determined by a discrete dynamical system. This indicates that, in the limit, the growth rate is governed by synchronization phenomena between the maturation age and the length of the time intervals in which division may occur. As an intermediate result, we give a sufficient condition which guarantees that the dominant Floquet eigenvalue is a nondecreasing function of the division rate. We also give a counter example showing that the latter monotonicity property does not hold in general.
authors: INRIA and CMAP, École Polytechnique, CNRS UMR 7641, CMAP, École Polytechnique, 91128, Palaiseau cedex, France. stephane.gaubert@inria.fr.
authors: Stéphane
authors: S
authors: Gaubert
authors: INRIA and Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, 69622, Villeurbanne cedex, France. thomas.lepoutre@inria.fr.
authors: Thomas
authors: T
authors: Lepoutre
mesh_terms: D000818:Animals; D002453:Cell Cycle; D002455:Cell Division; D002940:Circadian Rhythm; D003198:Computer Simulation; D006801:Humans; D055641:Mathematical Concepts; D008954:Models, Biological; D013997:Time Factors
keywords: Cell cycle; Circadian rhythms; Delay differential equations; Floquet eigenvalue; Perron-Frobenius theory; Structured PDEs