zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $C\sb 0$-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.

MSC:
 92D25 Population dynamics (general) 47D06 One-parameter semigroups and linear evolution equations 93B05 Controllability 45K05 Integro-partial differential equations
Full Text:
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. No. 3, 507-512 (1965) · Zbl 0131.33401 [5] Birkhoff, G.: Lattice theory. (1967) · Zbl 0153.02501 [6] Bushell, P. J.: On the projective contraction ratio for positive linear mappings. J. London math. Soc. 6, No. No. 2, 256-258 (1973) · Zbl 0255.47048 [7] Butzer, P. L.; Berens, H.: Semi-groups of operators and approximation. (1967) · 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) · 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, 145-172 (1988) · 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. Volterra integrodifferential equations in Banach spaces and applications, 67-89 (1989) · Zbl 0675.47036 [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, No. 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) [15] Hille, E.; Phillips, R. S.: Functional analysis and semigroups. (1957) · 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. 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. No. 3, 199-214 (1972) · Zbl 0268.92011 [20] Lopez, A.: Problems in stable population theory. (1961) [21] Madsen, R. W.; Conn, P. S.: Ergodic behavior for nonnegative kernels. Ann. probab. 1, No. No. 6, 995-1013 (1973) · Zbl 0272.60052 [22] Metz, J. A. J; Diekmann, O.: Lect. notes biomath. 68 (1986) [23] Rudin, W.: Real and complex analysis. (1974) · Zbl 0278.26001 [24] Seneta, E.: Non-negative matrices and Markov chains. (1981) · Zbl 0471.60001 [25] Song, J.; Tuan, C. H.; Yu, J. Y.: Population control in China: theory and applications. (1985) [26] Song, J.; Yu, J. Y.: Population system control. (1988) · Zbl 0671.92019 [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) · Zbl 0555.92014 [30] Webb, G. F.: An operator theoretic formulation of asynchronous exponential growth. Trans. amer. Math. soc. 303, No. No. 2, 751-776 (1987) · Zbl 0654.47021 [31] Yosida, K.: Functional analysis. (1980) · Zbl 0435.46002 [32] Yu, J. Y.; Guo, B. Z.; Zhu, G. T.: Asymptotic expansion in L[0, rm] for the population evolution and controllability of the population system. J. systems sci. Math. sci. 7, No. 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. No. 3, 214-219 (1987) · Zbl 0637.92014