Asymptotics of the Lebowitz-Rubinow-Rotenberg model of population development

Adam Gregosiewicz

doi:10.3934/dcdsb.2018260

Article Contents

2019, Volume 24, Issue 6: 2443-2472. Doi: 10.3934/dcdsb.2018260

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Asymptotics of the Lebowitz-Rubinow-Rotenberg model of population development

Adam Gregosiewicz^1,2,

1.
Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland
2.
Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-656 Warsaw, Poland

Received: July 31, 2017

Early access: October 2018

Published: June 2019

This research was partly supported by Polish National Science Centre grant 2014/15/N/ST1/03110

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Abstract

We study a mathematical model of cell populations dynamics proposed by J. Lebowitz and S. Rubinow, and analysed by M. Rotenberg. Here, a cell is characterized by her maturity and speed of maturation. The growth of cell populations is described by a partial differential equation with a boundary condition. In the first part of the paper we exploit semigroup theory approach and apply Lord Kelvin's method of images in order to give a new proof that the model is well posed. A semi-explicit formula for the semigroup related to the model obtained by the method of images allows two types of new results. First of all, we give growth order estimates for the semigroup, applicable also in the case of decaying populations. Secondly, we study asymptotic behavior of the semigroup in the case of approximately constant population size. More specifically, we formulate conditions for the asymptotic stability of the semigroup in the case in which the average number of viable daughters per mitosis equals one. To this end we use methods developed by K. Pichór and R. Rudnicki.

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Mathematics Subject Classification: Primary: 35Q92, 47D06; Secondary: 92D25.

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Figure 1. The set Ω

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Figure 2. ${\tilde{\Omega }}$ is the union of $\Omega_i$'s

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Figure 3. The set $\bigcup_{i = 1}^4 (1+\Omega_i)$ is shaded

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