An operator-theoretic formulation of asynchronous exponential growth. (English) Zbl 0654.47021

Webb, G. F.

Trans. Am. Math. Soc. 303, 751-763 (1987).

A strongly continuous semigroup of bounded operators \(\{\) T(t),t\(\geq 0\}\) in a Banach space X is said to have asynchronous exponential growth with intrinsic growth constant \(\lambda_ 0\in {\mathbb{R}}\) if there is a finite range operator \(P_ 0\) in X such that \[ \lim_{t\to \infty}e^{- \lambda_ 0t}T(t)=P_ 0. \] The author aims to establish necessary and sufficient conditions for \(\{\) T(t),t\(\geq 0\}\) to have asynchronous exponential growth. He first proves a necessary and sufficient condition in terms of the spectral properties of the generator of the semigroup. Then he studies how to guarantee that such properties are fulfilled. To this end he considers the case in which X is a Banach lattice and makes use of results of the Tübingen school. Finally, he applies his result to models describing cell population growth.

Reviewer: P.de Mottoni

MSC:

47D03	Groups and semigroups of linear operators
92D25	Population dynamics (general)

Keywords:

strongly continuous semigroup of bounded operators; asynchronous exponential growth; spectral proprties of the generator of the semigroup; cell population growth

References:

[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. · Zbl 0104.07502
[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. · Zbl 0543.92021
[4]	O. Diekmann, H. Lauwerier, T. Aldenberg and J. Metz, Growth, fission, and the stable size distribution, J. Math. Biol. 18 (1983), 135-148. · Zbl 0533.92016
[5]	Martin Eisen, Mathematical models in cell biology and cancer chemotherapy, Lecture Notes in Biomathematics, vol. 30, Springer-Verlag, Berlin-New York, 1979. · Zbl 0414.92005
[6]	Willy Feller, On the integral equation of renewal theory, Ann. Math. Statistics 12 (1941), 243 – 267. · Zbl 0026.23001
[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. · Zbl 0568.47031
[8]	G. Greiner and R. Nagel, On the stability of strongly continuous semigroups of positive operators on \?²(\?), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 2, 257 – 262. · Zbl 0527.47027
[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. · Zbl 0469.47032
[10]	Jack K. Hale, Ordinary differential equations, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. · Zbl 0433.34003
[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. · Zbl 0577.92019
[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. · Zbl 0571.92023
[13]	H. J. A. M. Heijmans, Structured populations, linear semigroups and positivity, Math. Z. 191 (1986), no. 4, 599 – 617. · Zbl 0595.92009
[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. · Zbl 0356.60039
[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. · Zbl 0148.12601
[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. · Zbl 0562.47031
[18]	A. Lasota and M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), no. 1, 43 – 62. · Zbl 0529.92011
[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. · Zbl 0269.92001
[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. · Zbl 0296.47023
[22]	F. R. Sharpe and A. J. Lotka, A problem in age distributions, Philos. Mag. 21 (1911), 435-438. · JFM 42.1030.02
[23]	C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc. 200 (1974), 395 – 418. · Zbl 0299.35085
[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. · Zbl 0555.92014
[25]	G. F. Webb, A model of proliferating cell populations with inherited cycle length, J. Math. Biol. 23 (1986), no. 2, 269 – 282. · Zbl 0596.92017
[26]	Kôsaku Yosida, Functional analysis, Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York Inc., New York, 1968.