×

Strong ergodicity for perturbed dual semigroups and application to age- dependent population dynamics. (English) Zbl 0761.92028

An age-dependent population model described by a Lotka-McKendric-Von Foerster system is studied using the perturbation theory of dual semigroups in a sun-reflexive Banach space. A brief summary of the results for dual semigroups theory and perturbation theory of dual semigroups in a sun-reflexive Banach space is given. Also asymptotic properties of the evolutionary system generated by a Lipschitz continuous perturbation of a \(C_ 0\)-semigroup are given. The concept of strong ergodicity for the evolutionary system is introduced and this concept is applied to Lotka’s renewal equation to prove strong ergodicity of the age structured population model with time dependent vital rates. The controllability of a demographic model is also studied by using the total fertility rate as a control.
Reviewer: S.Totaro (Firenze)

MSC:

92D25 Population dynamics (general)
47D06 One-parameter semigroups and linear evolution equations
93B05 Controllability
45K05 Integro-partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Artzrouni, M., Generalized stable population theory, J. Math. Biol., 21, 363-381 (1985) · Zbl 0567.92013
[2] Artzrouni, M., On the convergence of infinite products of matrices, Linear Algebra Appl., 74, 11-21 (1986) · Zbl 0592.15010
[3] Artzrouni, M., The rate of convergence of a generalized stable population, J. Math. Biol., 24, 405-422 (1986) · Zbl 0609.92031
[4] Birkhoff, G., Uniformly semi-primitive multiplicative processes, II, J. Math. Mech., 14, No. 3, 507-512 (1965) · Zbl 0131.33401
[5] Birkhoff, G., Lattice Theory (1967), Amer. Math. Soc.,: Amer. Math. Soc., Providence, RI · Zbl 0126.03801
[6] Bushell, P. J., On the projective contraction ratio for positive linear mappings, J. London. Math. Soc., 6, No. 2, 256-258 (1973) · Zbl 0255.47048
[7] Butzer, P. L.; Berens, H., Semi-Groups of Operators and Approximation (1967), Springer-Verlag: Springer-Verlag Berlin · Zbl 0164.43702
[8] Clément, Ph; Diekmann, O.; Gyllenberg, M.; Heijmans, H. J.A. M.; Thieme, H. R., Perturbation theory for dual semigroups. I. The sun-reflexive case, Math. Ann., 277, 709-725 (1987) · Zbl 0634.47039
[9] Clément, Ph; Heijmans, H. J.A. M., One-Parameter Semigroups, (CWI Monographs (1987), North-Holland: North-Holland Amsterdam) · Zbl 0636.47051
[10] Clément, Ph; Diekmann, O.; Gyllenberg, M.; Heijmans, H. J.A. M.; Thieme, H. R., Perturbation theory for dual semigroups. II. Time-dependent perturbations in the sun-reflexive case, (Proc. Roy. Soc. Edinburgh, 109A (1988)), 145-172 · Zbl 0661.47015
[11] Clément, Ph; Diekmann, O.; Gyllenberg, M.; Heijmans, H. J.A. M.; Thieme, H. R., Perturbation theory for dual semigroups. III. Nonlinear Lipschitz continuous perturbation in the sun-reflexive case, (da Prato, G.; Iannelli, M., Volterra Integrodifferential Equations in Banach Spaces and Applications (1989), Longman: Longman New York), 67-89 · Zbl 0634.47039
[12] Diekmann, O.; Heijmans, H. J.A. M.; Thieme, H. R., On the stability of the cell-size distribution II: Time-periodic developmental rates, Comput. Math. Appl., 12A, Nos. 4/5, 491-512 (1986) · Zbl 0667.92015
[13] Golubitsky, M.; Keeler, E. B.; Rothschild, M., Convergence of the age structure: Applications of the projective metric, Theoret. Population Biol., 7, 81-93 (1975) · Zbl 0297.92014
[14] Heijmans, H. J.A. M., Semigroup theory for control on sun-reflexive Banach space, CWI Report, AM-R8607 (1986), Amsterdam · Zbl 0627.93031
[15] Hille, E.; Phillips, R. S., Functional Analysis and Semigroups (1957), Amer. Math. Soc.,: Amer. Math. Soc., Providence, RI · Zbl 0078.10004
[16] Inaba, H., A semigroup approach to the strong ergodic theorem of the multistate stable population process, Math. Population Stud., 1, No. 1, 49-77 (1988) · Zbl 0900.92122
[17] Inaba, H., Weak ergodicity of population evolution processes, Math. Biosci., 96, 195-219 (1989) · Zbl 0698.92020
[18] Kim, Y. J., Dynamics of populations with changing rates: Generalization of the stable population theory, Theoret. Population Biol., 31, 306-322 (1987) · Zbl 0619.92007
[19] Langhaar, H. L., General population theory in the age-time continuum, J. Franklin Inst., 293, No. 3, 199-214 (1972) · Zbl 0268.92011
[20] Lopez, A., Problems in Stable Population Theory (1961), Office of Population Research: Office of Population Research Princeton, NJ
[21] Madsen, R. W.; Conn, P. S., Ergodic behavior for nonnegative kernels, Ann. Probab., 1, No. 6, 995-1013 (1973) · Zbl 0272.60052
[22] (Metz, J. A.J; Diekmann, O., The Dynamics of Physiologically Structured Populations. The Dynamics of Physiologically Structured Populations, Lect. Notes Biomath, Vol. 68 (1986), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0614.92014
[23] Rudin, W., Real and Complex Analysis (1974), McGraw-Hill: McGraw-Hill New York
[24] Seneta, E., Non-negative Matrices and Markov Chains (1981), Springer-Verlag: Springer-Verlag Berlin · Zbl 0471.60001
[25] Song, J.; Tuan, C. H.; Yu, J. Y., Population Control in China: Theory and Applications (1985), Praeger: Praeger New York
[26] Song, J.; Yu, J. Y., Population System Control (1988), Springer-Verlag: Springer-Verlag Berlin
[27] Thieme, H. R., Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7, 253-277 (1984) · Zbl 0566.45016
[28] Thompson, M., Asymptotic growth and stability in population with time dependent vital rates, Math. Biosci., 42, 267-278 (1978) · Zbl 0402.92025
[29] Webb, G. F., Theory of Nonlinear Age-Dependent Population Dynamics (1985), Dekker: Dekker New York/Basel · Zbl 0555.92014
[30] Webb, G. F., An operator theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303, No. 2, 751-776 (1987) · Zbl 0654.47021
[31] Yosida, K., Functional Analysis (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0217.16001
[32] Yu, J. Y.; Guo, B. Z.; Zhu, G. T., Asymptotic expansion in \(L[0, r_m]\) for the population evolution and controllability of the population system, J. Systems Sci. Math. Sci., 7, No. 2, 97-104 (1987) · Zbl 0633.92012
[33] Yu, J. Y.; Guo, B. Z.; Zhu, G. T., The control of the semi-discrete population evolution system, J. Systems Sci. Math. Sci., 7, No. 3, 214-219 (1987), [Chinese] · Zbl 0637.92014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.