Applications of stochastic semigroups to cell cycle models

Katarzyna Pichór; Ryszard Rudnicki

doi:10.3934/dcdsb.2019099

Article Contents

2019, Volume 24, Issue 5: 2365-2381. Doi: 10.3934/dcdsb.2019099 open access

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Applications of stochastic semigroups to cell cycle models

Katarzyna Pichór^1, and
Ryszard Rudnicki^2, ,

1.
Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland
2.
Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland

^* Corresponding author: Ryszard Rudnicki
^* Corresponding author: Ryszard Rudnicki

Received: December 2017

Revised: January 2019

Early access: March 2019

Published: March 2019

This research was partially supported by the National Science Centre (Poland) Grant No. 2017/27/B/ST1/00100

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Abstract

We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.

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Mathematics Subject Classification: Primary: 47D06; Secondary: 60J75 92C37.

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Figure 1. Evolution of maturity of a mother cell: (1) – resting phase; (2) – proliferating phase and a daughter cell: (3) – resting phase; (4) – proliferating phase

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Figure 2. The set $ X $

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