Tyrcha, Joanna Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle. (English) Zbl 0716.92017 J. Math. Biol. 26, No. 4, 465-475 (1988). Summary: A new mathematical model of the cell cycle is presented which generalizes the probabilistic/deterministic model of A. Lasota and M. C. Mackey [ibid. 19, 43-62 (1984; Zbl 0529.92011)] and the tandem model of J. J. Tyson and K. B. Hannsgen [ibid. 23, 231-246 (1986; Zbl 0582.92020)]. By the use of a multiplicative (exponential) Lyapunov function a stability theorem is proved, parallel to the results of Lasota-Mackey. Some open problems related to the tandem model are also solved. Cited in 1 ReviewCited in 11 Documents MSC: 92D25 Population dynamics (general) 92C99 Physiological, cellular and medical topics 47N30 Applications of operator theory in probability theory and statistics 47G10 Integral operators 60J99 Markov processes Keywords:Markov operator; asymptotic stability; cell cycle; probabilistic/deterministic model; tandem model; stability theorem Citations:Zbl 0529.92011; Zbl 0582.92020 PDFBibTeX XMLCite \textit{J. Tyrcha}, J. Math. Biol. 26, No. 4, 465--475 (1988; Zbl 0716.92017) Full Text: DOI References: [1] Lasota, A., Mackey, M. C.: Globally asymptotic properties of proliferating cell populations. J. Math. Biol. 19, 43–62 (1984) · Zbl 0529.92011 · doi:10.1007/BF00275930 [2] Lasota, A., Mackey, M. C.: Probabilistic properties of deterministic systems. Cambridge: Cambridge University Press 1985 · Zbl 0606.58002 [3] Lasota, A., Yorke, J. A.: Exact dynamical systems and the Frobenius-Perron operator. Trans. Am. Math. Soc. 273, 375–384 (1982) · Zbl 0524.28021 · doi:10.1090/S0002-9947-1982-0664049-X [4] Painter, P. R., Marr, A. G.: Mathematics of microbial populations. Ann. Rev. Microbiol. 22, 519–549 (1968) · doi:10.1146/annurev.mi.22.100168.002511 [5] Tyson, J. J., Diekmann, O.: Sloppy size control of the cell division cycle. J. Theor. Biol. 118, 405–426 (1986) · doi:10.1016/S0022-5193(86)80162-X [6] Tyson, J. J., Hannsgen, K. B.: The distributions of the cell size and generation time in a model of the cell cycle incorporating size control and random transitions. J. Theor. Biol. 113, 29–62. (1985) · doi:10.1016/S0022-5193(85)80074-6 [7] Tyson, J. J., Hannsgen, K. B.: Cell growth and division: a deterministic/probabilistic model of the cell cycle. J. Math. Biol. 23, 231–246 (1986) · Zbl 0582.92020 · doi:10.1007/BF00276959 [8] Tyson, J. J.: The coordination of cell growth and division – intentional or incidental? BioEssays 2 (1972) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.