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.
Highlights
The modeling of the cell cycle has a long history [27]
The second group is formed by continuous-time models characterizing the time evolution of distribution of cell maturity [6, 19, 29] or cell size [8, 11]
The mathematical model is given by a stochastic operator P which describes the relation between densities of maturity of new born cells in consecutive generations
Summary
The modeling of the cell cycle has a long history [27]. The core of the theory was formulated in the late sixties [16, 30, 38]. The mathematical model is given by a stochastic operator P which describes the relation between densities of maturity of new born cells in consecutive generations. The novelty of our model is that it consists a system of three differential equations which describes age, maturity, and phase of a cell. Since we include into the model age, maturity and phase of a cell, our process satisfies the Markov property. The evolution of densities of the PDMP corresponding to our model leads directly to a continuous-time stochastic semigroup {P (t)}t≥0.
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