An operatortheoretic formulation of asynchronous exponential growth
Author:
G. F. Webb
Journal:
Trans. Amer. Math. Soc. 303 (1987), 751763
MSC:
Primary 47D05; Secondary 47B55, 92A15
MathSciNet review:
902796
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Abstract 
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Similar Articles 
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Abstract: A strongly continuous semigroup of bounded linear operators , , in the Banach space has asynchronous exponential growth with intrinsic growth constant provided that there is a nonzero finite rank operator in such that . Necessary and sufficient conditions are established for , , 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.
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O. Diekmann, H. Heijmans and H. Thieme, On the stability of the cell size distributions, J. Math. Biol. 19 (1984), 227248. MR 745853 (86a:92025)
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O. Diekmann, H. Lauwerier, T. Aldenberg and J. Metz, Growth, fission, and the stable size distribution, J. Math. Biol. 18 (1983), 135148.
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M. Eisen, Mathematical models in cell biology, Lecture Notes in Biomathematics, Vol. 30, SpringerVerlag, Berlin, Heidelberg and New York, 1979. MR 635622 (83c:92017)
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W. Feller, On the integral equation of renewal theory, Ann. Math. Statist. 12 (1941), 243267. MR 0005419 (3:151c)
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G. Greiner, A typical PerrronFrobenius theorem with application to an agedependent population equation, InfiniteDimensional Systems, Proceedings, Retzhof 1983, F. Kappel and W. Schappacher, Eds., Lecture Notes in Math., Vol. 1076, SpringerVerlag, Berlin, Heidelberg and New York, 1984. MR 763356 (86b:47073)
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G. Greiner and R. Nagel, On the stability of strongly continuous semigroups of positive operators on , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 257262. MR 728436 (85b:47044)
 [9]
G. Greiner, J. Voigt and M. Wolff, On the spectral bound of the generator of semigroups of positive operators, J. Operator Theory 5 (1981), 245256. MR 617977 (82h:47039)
 [10]
J. K. Hale, Ordinary differential equations, Interscience Series on Pure and Appl. Math., Vol. 21, WileyInterscience, New York, 1969. MR 0419901 (54:7918)
 [11]
K. Hannsgen, J. Tyson and L. Watson, Steadystate distributions in probabilistic models of the cell division cycle, SIAM J. Appl. Math. 45 (1985), 523540. MR 796094 (87e:92030)
 [12]
K. Hannsgen and J. Tyson, Stability of the steadystate size distribution in a model of cell growth and divison, J. Math. Biol. 22 (1985), 293301. MR 813400 (87d:92041)
 [13]
H. Heijmans, Structured populations, linear semigroups, and positivity (to appear). MR 832818 (87m:35211)
 [14]
P. Jagers, Branching processes with biological applications, Wiley, New York, 1975. MR 0488341 (58:7890)
 [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/NorthHolland, New York and Amsterdam, 1978, 2129.
 [16]
T. Kato, Perturbation theory for linear operators, SpringerVerlag, Berlin, Heidelberg and New York, 1966. MR 0203473 (34:3324)
 [17]
W. Kerscher and R. Nagel, Asymptotic behavior of oneparameter semigroups of positive operators, Acta Appl. Math. 2 (1984), 297310. MR 753697 (86e:47047)
 [18]
A. Lasota and M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), 4362. MR 737168 (85h:92010)
 [19]
S. Rubinow, Mathematical problems in the biological sciences, CBMS Regional Conf. Ser. Appl. Math., no. 10, SIAM, Philadelphia, Pa., 1973, pp. 5373. MR 0462660 (57:2633)
 [20]
, 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.
 [21]
H. Schaefer, Banach lattices and positive operators, SpringerVerlag, Berlin, Heidelberg and New York, 1974. MR 0423039 (54:11023)
 [22]
F. R. Sharpe and A. J. Lotka, A problem in age distributions, Philos. Mag. 21 (1911), 435438.
 [23]
C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equation, Trans. Amer. Math. Soc. 200 (1974), 395418. MR 0382808 (52:3690)
 [24]
G. F. Webb, Theory of nonlinear agedependent population dynamics, Monographs and Textbooks in Pure and Appl. Math. Series, Vol. 89, Dekker, New York, 1985. MR 772205 (86e:92032)
 [25]
, A model of proliferating cell populations with inherited cycle length, J. Math. Biol. 23 (1986), 269282. MR 829138 (87h:92074)
 [26]
K. Yosida, Functional analysis, 2nd ed., SpringerVerlag, Berlin, Heidelberg and New York, 1968. MR 0239384 (39:741)
 [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), 329351.
 [2]
 F. E. Browder, On the spectral theory of elliptic differential operators, Math. Ann. 142 (1961), 22130. MR 0209909 (35:804)
 [3]
 O. Diekmann, H. Heijmans and H. Thieme, On the stability of the cell size distributions, J. Math. Biol. 19 (1984), 227248. MR 745853 (86a:92025)
 [4]
 O. Diekmann, H. Lauwerier, T. Aldenberg and J. Metz, Growth, fission, and the stable size distribution, J. Math. Biol. 18 (1983), 135148.
 [5]
 M. Eisen, Mathematical models in cell biology, Lecture Notes in Biomathematics, Vol. 30, SpringerVerlag, Berlin, Heidelberg and New York, 1979. MR 635622 (83c:92017)
 [6]
 W. Feller, On the integral equation of renewal theory, Ann. Math. Statist. 12 (1941), 243267. MR 0005419 (3:151c)
 [7]
 G. Greiner, A typical PerrronFrobenius theorem with application to an agedependent population equation, InfiniteDimensional Systems, Proceedings, Retzhof 1983, F. Kappel and W. Schappacher, Eds., Lecture Notes in Math., Vol. 1076, SpringerVerlag, Berlin, Heidelberg and New York, 1984. MR 763356 (86b:47073)
 [8]
 G. Greiner and R. Nagel, On the stability of strongly continuous semigroups of positive operators on , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 257262. MR 728436 (85b:47044)
 [9]
 G. Greiner, J. Voigt and M. Wolff, On the spectral bound of the generator of semigroups of positive operators, J. Operator Theory 5 (1981), 245256. MR 617977 (82h:47039)
 [10]
 J. K. Hale, Ordinary differential equations, Interscience Series on Pure and Appl. Math., Vol. 21, WileyInterscience, New York, 1969. MR 0419901 (54:7918)
 [11]
 K. Hannsgen, J. Tyson and L. Watson, Steadystate distributions in probabilistic models of the cell division cycle, SIAM J. Appl. Math. 45 (1985), 523540. MR 796094 (87e:92030)
 [12]
 K. Hannsgen and J. Tyson, Stability of the steadystate size distribution in a model of cell growth and divison, J. Math. Biol. 22 (1985), 293301. MR 813400 (87d:92041)
 [13]
 H. Heijmans, Structured populations, linear semigroups, and positivity (to appear). MR 832818 (87m:35211)
 [14]
 P. Jagers, Branching processes with biological applications, Wiley, New York, 1975. MR 0488341 (58:7890)
 [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/NorthHolland, New York and Amsterdam, 1978, 2129.
 [16]
 T. Kato, Perturbation theory for linear operators, SpringerVerlag, Berlin, Heidelberg and New York, 1966. MR 0203473 (34:3324)
 [17]
 W. Kerscher and R. Nagel, Asymptotic behavior of oneparameter semigroups of positive operators, Acta Appl. Math. 2 (1984), 297310. MR 753697 (86e:47047)
 [18]
 A. Lasota and M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), 4362. MR 737168 (85h:92010)
 [19]
 S. Rubinow, Mathematical problems in the biological sciences, CBMS Regional Conf. Ser. Appl. Math., no. 10, SIAM, Philadelphia, Pa., 1973, pp. 5373. MR 0462660 (57:2633)
 [20]
 , 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.
 [21]
 H. Schaefer, Banach lattices and positive operators, SpringerVerlag, Berlin, Heidelberg and New York, 1974. MR 0423039 (54:11023)
 [22]
 F. R. Sharpe and A. J. Lotka, A problem in age distributions, Philos. Mag. 21 (1911), 435438.
 [23]
 C. C. Travis and G. F. Webb, Existence and stability for partial functional differential equation, Trans. Amer. Math. Soc. 200 (1974), 395418. MR 0382808 (52:3690)
 [24]
 G. F. Webb, Theory of nonlinear agedependent population dynamics, Monographs and Textbooks in Pure and Appl. Math. Series, Vol. 89, Dekker, New York, 1985. MR 772205 (86e:92032)
 [25]
 , A model of proliferating cell populations with inherited cycle length, J. Math. Biol. 23 (1986), 269282. MR 829138 (87h:92074)
 [26]
 K. Yosida, Functional analysis, 2nd ed., SpringerVerlag, Berlin, Heidelberg and New York, 1968. MR 0239384 (39:741)
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DOI:
http://dx.doi.org/10.1090/S00029947198709027967
PII:
S 00029947(1987)09027967
Article copyright:
© Copyright 1987
American Mathematical Society
