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Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle. (English) Zbl 0716.92017

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.

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
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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)
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