An operatortheoretic formulation of asynchronous exponential growth
HTML articles powered by AMS MathViewer
 by G. F. Webb PDF
 Trans. Amer. Math. Soc. 303 (1987), 751763 Request permission
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 maturitytime model of cell population growth and a transition probability model of cell population growth.References

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), 329351.
 Felix E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1960/61), 22–130. MR 209909, DOI 10.1007/BF01343363
 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, DOI 10.1007/BF00277748 O. Diekmann, H. Lauwerier, T. Aldenberg and J. Metz, Growth, fission, and the stable size distribution, J. Math. Biol. 18 (1983), 135148.
 Martin Eisen, Mathematical models in cell biology and cancer chemotherapy, Lecture Notes in Biomathematics, vol. 30, SpringerVerlag, BerlinNew York, 1979. MR 635622
 Willy Feller, On the integral equation of renewal theory, Ann. Math. Statistics 12 (1941), 243–267. MR 5419, DOI 10.1214/aoms/1177731708
 G. Greiner, A typical PerronFrobenius theorem with applications to an agedependent population equation, Infinitedimensional systems (Retzhof, 1983) Lecture Notes in Math., vol. 1076, Springer, Berlin, 1984, pp. 86–100. MR 763356, DOI 10.1007/BFb0072769
 G. Greiner and R. Nagel, On the stability of strongly continuous semigroups of positive operators on $L^{2}(\mu )$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 2, 257–262. MR 728436
 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
 Jack K. Hale, Ordinary differential equations, Pure and Applied Mathematics, Vol. XXI, WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1969. MR 0419901
 Kenneth B. Hannsgen, John J. Tyson, and Layne T. Watson, Steadystate size distributions in probabilistic models of the cell division cycle, SIAM J. Appl. Math. 45 (1985), no. 4, 523–540. MR 796094, DOI 10.1137/0145031
 Kenneth B. Hannsgen and John J. Tyson, Stability of the steadystate size distribution in a model of cell growth and division, J. Math. Biol. 22 (1985), no. 3, 293–301. MR 813400, DOI 10.1007/BF00276487
 H. J. A. M. Heijmans, Structured populations, linear semigroups and positivity, Math. Z. 191 (1986), no. 4, 599–617. MR 832818, DOI 10.1007/BF01162350
 Peter Jagers, Branching processes with biological applications, Wiley Series in Probability and Mathematical Statistics—Applied Probability and Statistics, WileyInterscience [John Wiley & Sons], LondonNew YorkSydney, 1975. MR 0488341 —, 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/NorthHolland, New York and Amsterdam, 1978, 2129.
 Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, SpringerVerlag New York, Inc., New York, 1966. MR 0203473
 W. Kerscher and R. Nagel, Asymptotic behavior of oneparameter semigroups of positive operators, Acta Appl. Math. 2 (1984), no. 34, 297–309. MR 753697, DOI 10.1007/BF02280856
 A. Lasota and M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), no. 1, 43–62. MR 737168, DOI 10.1007/BF00275930
 Sol I. Rubinow, Mathematical problems in the biological sciences, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 10, 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. MR 0462660 —, Agestructured 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. 389410.
 Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, SpringerVerlag, New YorkHeidelberg, 1974. MR 0423039 F. R. Sharpe and A. J. Lotka, A problem in age distributions, Philos. Mag. 21 (1911), 435438.
 C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395–418. MR 382808, DOI 10.1090/S00029947197403828083
 G. F. Webb, Theory of nonlinear agedependent population dynamics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 89, Marcel Dekker, Inc., New York, 1985. MR 772205
 G. F. Webb, A model of proliferating cell populations with inherited cycle length, J. Math. Biol. 23 (1986), no. 2, 269–282. MR 829138, DOI 10.1007/BF00276962
 Kôsaku Yosida, Functional analysis, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 123, SpringerVerlag New York, Inc., New York, 1968. MR 0239384
Additional Information
 © Copyright 1987 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 303 (1987), 751763
 MSC: Primary 47D05; Secondary 47B55, 92A15
 DOI: https://doi.org/10.1090/S00029947198709027967
 MathSciNet review: 902796