Discrete limit and monotonicity properties of the Floquet eigenvalue in an age structured cell division cycle model

TL;DR

This paper analyzes the asymptotic behavior of the growth rate in an age-structured cell division model with periodic coefficients, revealing a staircase limit and conditions for monotonicity of the Floquet eigenvalue.

Contribution

It introduces a detailed study of the Floquet eigenvalue's limit behavior and provides conditions for its monotonicity, including a counterexample showing when monotonicity fails.

Findings

Floquet eigenvalue converges to a staircase function as division rate increases

Structural condition guarantees nondecreasing Floquet eigenvalue

Counterexample shows monotonicity does not always hold

Abstract

We consider a cell population described by an age-structured partial differential equation with time periodic coefficients. We assume that division only occurs after a minimal age (majority) and within certain time intervals. We study the asymptotic behavior of the dominant Floquet eigenvalue, or Perron-Frobenius eigenvalue, representing the growth rate, as a function of the majority age, when the division rate 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. As an intermediate result, we give a structural 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.