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An operator-theoretic formulation of asynchronous exponential growth

Author: G. F. Webb
Journal: Trans. Amer. Math. Soc. 303 (1987), 751-763
MSC: Primary 47D05; Secondary 47B55, 92A15
MathSciNet review: 902796
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Abstract: A strongly continuous semigroup of bounded linear operators $ T(t)$, $ t \geqslant 0$, in the Banach space $ X$ has asynchronous exponential growth with intrinsic growth constant $ {\lambda _0}$ provided that there is a nonzero finite rank operator $ {P_0}$ in $ X$ such that $ {\lim _{t \to \infty }}{e^{ - {\lambda _0}t}}T(t) = {P_0}$. Necessary and sufficient conditions are established for $ T(t)$, $ t \geqslant 0$, to have asynchronous exponential growth. Applications are made to a maturity-time model of cell population growth and a transition probability model of cell population growth.

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  • [1] G. I. Bell and E. C. Anderson, Cell growth and division. I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J. 7 (1967), 329-351.
  • [2] Felix E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1960/1961), 22–130. MR 0209909
  • [3] O. Diekmann, H. J. A. M. Heijmans, and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol. 19 (1984), no. 2, 227–248. MR 745853, 10.1007/BF00277748
  • [4] O. Diekmann, H. Lauwerier, T. Aldenberg and J. Metz, Growth, fission, and the stable size distribution, J. Math. Biol. 18 (1983), 135-148.
  • [5] Martin Eisen, Mathematical models in cell biology and cancer chemotherapy, Lecture Notes in Biomathematics, vol. 30, Springer-Verlag, Berlin-New York, 1979. MR 635622
  • [6] Willy Feller, On the integral equation of renewal theory, Ann. Math. Statistics 12 (1941), 243–267. MR 0005419
  • [7] G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Infinite-dimensional systems (Retzhof, 1983) Lecture Notes in Math., vol. 1076, Springer, Berlin, 1984, pp. 86–100. MR 763356, 10.1007/BFb0072769
  • [8] G. Greiner and R. Nagel, On the stability of strongly continuous semigroups of positive operators on 𝐿²(𝜇), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 2, 257–262. MR 728436
  • [9] Günther Greiner, Jürgen Voigt, and Manfred Wolff, On the spectral bound of the generator of semigroups of positive operators, J. Operator Theory 5 (1981), no. 2, 245–256. MR 617977
  • [10] Jack K. Hale, Ordinary differential equations, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. MR 0419901
  • [11] Kenneth B. Hannsgen, John J. Tyson, and Layne T. Watson, Steady-state size distributions in probabilistic models of the cell division cycle, SIAM J. Appl. Math. 45 (1985), no. 4, 523–540. MR 796094, 10.1137/0145031
  • [12] Kenneth B. Hannsgen and John J. Tyson, Stability of the steady-state size distribution in a model of cell growth and division, J. Math. Biol. 22 (1985), no. 3, 293–301. MR 813400, 10.1007/BF00276487
  • [13] H. J. A. M. Heijmans, Structured populations, linear semigroups and positivity, Math. Z. 191 (1986), no. 4, 599–617. MR 832818, 10.1007/BF01162350
  • [14] Peter Jagers, Branching processes with biological applications, Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1975. Wiley Series in Probability and Mathematical Statistics—Applied Probability and Statistics. MR 0488341
  • [15] -, Balanced exponential growth: What does it mean and when is it there?, Biomathematics and Cell Kinetics, Development in Cell Biology, Vol. 2, A. Valleron and P. Macdonald, Eds., Elsevier/North-Holland, New York and Amsterdam, 1978, 21-29.
  • [16] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • [17] W. Kerscher and R. Nagel, Asymptotic behavior of one-parameter semigroups of positive operators, Acta Appl. Math. 2 (1984), no. 3-4, 297–309. MR 753697, 10.1007/BF02280856
  • [18] A. Lasota and M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), no. 1, 43–62. MR 737168, 10.1007/BF00275930
  • [19] Sol I. Rubinow, Mathematical problems in the biological sciences, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Ten lectures delivered at the Regional Conference on Mathematical Problems in the Biological Sciences of 5–9 June 1972 at Michigan State University; Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 10. MR 0462660
  • [20] -, Age-structured equations in the theory of cell populations, Studies in Mathematical Biology, Vol. 16, Part II, Populations and Communities, S. Levin, Ed., The Mathematical Association of America, Washington, D. C., 1978, pp. 389-410.
  • [21] Helmut H. Schaefer, Banach lattices and positive operators, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 215. MR 0423039
  • [22] F. R. Sharpe and A. J. Lotka, A problem in age distributions, Philos. Mag. 21 (1911), 435-438.
  • [23] C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395–418. MR 0382808, 10.1090/S0002-9947-1974-0382808-3
  • [24] G. F. Webb, Theory of nonlinear age-dependent population dynamics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 89, Marcel Dekker, Inc., New York, 1985. MR 772205
  • [25] G. F. Webb, A model of proliferating cell populations with inherited cycle length, J. Math. Biol. 23 (1986), no. 2, 269–282. MR 829138, 10.1007/BF00276962
  • [26] Kôsaku Yosida, Functional analysis, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York Inc., New York, 1968. MR 0239384

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Article copyright: © Copyright 1987 American Mathematical Society