Persistent age distributions for an age‐structured two‐sex population modelFootnote*

An earlier version of this paper was presented at the Workshop on Nonlinear Demography, Max Planck Institute for Demographic Research, Rostock, Germany, May 26–28, 1998. The author was partially supported by a Grant‐in Aid for Scientific Research from the Japan Ministry of Education, Science and Culture. I thank Evert van Imhoff and anonymous referees for their kind suggestions.

References

• Arbogast , T. and Milner , F.A. 1989 . A finite difference method for a two‐sex model of population dynamics . SIAM J. Numer. Anal. , 26 (6) : 1474 – 1486 .
   Web of Science ®Google Scholar
• Castillo‐Chavez , C. , Busenberg , S. and Gerow , K. 1991a . “ Pair formation in structured populations ” . In Differential Equations with Applications in Biology, Physics, and Engineering , Edited by: Goldstein , J.A. , Kappel , F. and Schappacher , W. 47 – 65 . New York : Marcel Dekker .
   Google Scholar
• Castillo‐Chavez , C. and Busenberg , S. 1991b . “ On the solution of the two‐sex mixing problem ” . In Differential Equations Models in Biology, Epidemiology and Ecology , Lecture Notes in Biomathematics 92 Edited by: Busenberg , S. and Martelli , M. 80 – 98 . Berlin : Springer‐Verlag .
   Google Scholar
• Castillo‐Chavez , C. and Huang , W. 1995 . The logistic equation revisited: The two‐sex case . Math. Biosci. , 128 : 299 – 316 .
   PubMed Web of Science ®Google Scholar
• Da Prato , G. and Sinestrari , E. 1987 . Differential operators with non dense domain . Annali della Scuola Normale Superiore di Pisa , 14 (2) : 285 – 344 .
   Google Scholar
• Dietz , K. and Hadeler , K.P. 1988 . Epidemiological models for sexually transmitted diseases . J. Math. Biol. , 26 : 1 – 25 .
   PubMed Web of Science ®Google Scholar
• Fredrickson , A.G. 1971 . A mathematical theory of age structure in sexual populations: random mating and monogamous marriage models . Math. Biosci. , 10 : 117 – 143 .
   Google Scholar
• Grabosch , A. 1989 . Translation semigroups and their linearizations on spaces of integrable functions . Transactions of the American Mathematical Society , 311 ( 1 ) : 357 – 390 .
   Web of Science ®Google Scholar
• Hadeler , K.P. , Waldstäter , R. and Wörz‐Busekros , A. 1988 . Models for pair formation in bisexual populations . J. Math. Biol. , 26 : 635 – 649 .
   PubMed Web of Science ®Google Scholar
• Hadeler , K.P. 1989a . Pair formation in age‐structured populations . Acta Applicandae Mathematicae , 14 : 91 – 102 .
   PubMed Web of Science ®Google Scholar
• Hadeler , K.P. 1989b . “ Modeling AIDS in structured populations ” . In Bulletin of the International Statistical Institute, Proceedings of the 47th Session Volume LIII , 83 – 99 . Paris Book 1
   Google Scholar
• Hadeler , K.P. and Ngoma , K. 1990 . Homogeneous models for sexually transmitted diseases . Rocky Mountain Journal of Mathematics , 20 (4) : 967 – 986 .
   Web of Science ®Google Scholar
• Hadeler , K.P. 1992a . Periodic solutions of homogeneous equations . Journal of Differential Equations , 95 : 183 – 202 .
   Web of Science ®Google Scholar
• Hadeler , K.P. 1992b . “ Structured population models for HIV infection pair formation and non‐constant infectivity ” . In AIDS Epidemiology Methodological Issues , Edited by: Jewell , N.P. , Dietz , K. and Farewell , V.T. 156 – 173 . Boston : Birkhäuse .
   Google Scholar
• Hoppensteadt , F. 1975 . Mathematical Theories of Populations: Demographics, Genetics and Epidemics. , Philadelphia : Society for Industrial and Applied Mathematics .
   Google Scholar
• Hsu , P.H. and Fredrickson , A.G. 1975 . Population‐changing processes and the dynamics of sexual populations . Math. Biosci. , 26 : 55 – 78 .
   Google Scholar
• Iannelli , M. and Martcheva , M. 1997 . A semigroup approach to the well posedness of an age‐structured two‐sex population model . Dynamic Systems and Applications , 6 : 353 – 370 .
   Google Scholar
• Inaba , H. 1993 . An age‐structured two‐sex model for human population reproduction by first marriage , Working Paper Series No. 15 Tokyo : Institute of Population Problems .
   Google Scholar
• Inaba , H. 1995 . Human population reproduction via first marriage . Math. Popul. Studies , 5 (2) : 123 – 144 .
   Google Scholar
• Inaba , H. 1997 . “ Calculating Ro for HIV infection via pair formation ” . In Advances in Mathematical Population Dynamics —Molecules, Cells and Man , Edited by: Arino , O. , Axelrod , D. and Kimmel , M. 355 – 382 . Singapore : World Scientific .
   Google Scholar
• Keilman , N. 1985 . Nuptiality models and the two‐sex problem in national population forecasts . European Journal of Population , 1 : 207 – 235 .
   PubMed Web of Science ®Google Scholar
• Keyfitz , N. The mathematics of sex and marriage . Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Biology and Health . vol. 4 , pp. 89 – 108 . Berkeley : Univ. of California Press .
   Google Scholar
• Keyfitz , N. 1977 . Introduction to the Mathematics of Population with revisions. , Reading, Massachusetts : Addison‐Wesley .
   Google Scholar
• Martcheva , M. and Milner , F.A. 1996 . The mathematics of sex and marriage, revisited . preprint
   Google Scholar
• Martcheva , M. and Milner , F.A. 1999 . A two‐sex age‐structured population model: Well posedness . Math. Popul. Studies , 7 (2) : 111 – 129 .
   PubMedGoogle Scholar
• Martcheva , M. 1999 . Exponential growth in age‐structured two‐sex populations . Math. Biosci. , 157 : 1 – 22 .
   PubMed Web of Science ®Google Scholar
• Matsumoto , T. , Oharu , S. and Thieme , H.R. 1996 . Nonlinear perturbations of a class of integral semigroups . Hiroshima Mathematical Journal , 26 (3) : 433 – 473 .
   Google Scholar
• McFarland , D.D. 1972 . “ Comparison of alternative marriage models ” . In Population Dynamics , Edited by: Greville , T.N.E. 89 – 106 . New York and London : Academic Press .
   Google Scholar
• Metz , J.A.J. and Diekmann , O. , eds. 1986 . Dynamics of Physiologically Structured Populations. , Lecture Notes in Biomathematics 86 Berlin : Springer‐Verlag .
   Google Scholar
• Milner , F.A. 1988 . A finite element method for a two‐sex model of population dynamics . Numerical Methods for Partial Differential Equations , 4 : 329 – 345 .
   Google Scholar
• Milner , F.A. and Rabbiolo , G. 1992 . Rapidly converging numerical algorithms for models of population dynamics . J. Math. Biol. , 30 (7) : 733 – 753 .
   PubMed Web of Science ®Google Scholar
• Pazy , A. 1983 . Semigroups of Linear Operators and Applications to Partial Differential Equations. , Berlin : Springer‐Verlag .
   Google Scholar
• Pollak , R.A. 1986 . A reformulation of the two‐sex problem . Demography , 23 (2) : 247 – 259 .
   PubMed Web of Science ®Google Scholar
• Pollak , R.A. 1987 . The two‐sex problem with persistent unions: A generalization of the birth matrix‐mating rule model . Theor. Popul. Biol. , 32 : 176 – 187 .
   PubMed Web of Science ®Google Scholar
• Pollak , R.A. 1990a . Two‐sex demographic models . Journal of Political Economy , 98 (2) : 399 – 420 .
   Web of Science ®Google Scholar
• Pollak , R.A. 1990b . “ Two‐sex population models and classical stable population theory ” . In Convergent Issues in Genetics and Demography , Edited by: Adams , J. , Hermalin , A. , Lam , D. and Smouse , P. 317 – 333 . Oxford : Oxford University Press .
   Google Scholar
• Pollard , J.H. 1973 . Mathematical Models for the Growth of Human Populations. , Cambridge : Cambridge Univ. Press .
   Google Scholar
• Pollard , J.H. 1975 . Modelling human populations for projection purposes ‐ some of the problems and challenges . Austral. J. Statist. , 17 (2) : 63 – 76 .
   Google Scholar
• Pollard , J.H. The continuing attempt to incorporate both sexes into marriage analysis . International Population Conference . Mexico. vol. 1 , pp. 291 – 309 . Liege : IUSSP .
   Google Scholar
• Prüss , J. and Schappacher , W. 1994a . Persistent age‐distributions for a pair formation model . J. Math. Biol , 33 : 17 – 33 .
   PubMed Web of Science ®Google Scholar
• Prüss , J. and Schappacher , W. 1994b . “ Semigroup methods for age‐structured population dynamics ” . In Jahrbuch Überblicke Mathematik 1994 , Edited by: Chatterji , S.D. , Fuchssteiner , B. , Kulisch , U. and Liedl , R. 74 – 90 . Wiesbaden : Vieweg .
   Google Scholar
• Schoen , R. 1988 . Modeling Multigroup Populations. , New York and London : Plenum Press .
   Google Scholar
• Staroverov , O.V. 1977 . Reproduction of the structure of the population and marriage. (Russian) . Ekonomika i matematiceskije metody , 13 : 72 – 82 .
   Google Scholar
• Thieme , H.R. 1990 . Semiflows generated by Lipschitz perturbations of non‐densely defined operators . Differential and Integral Equations , 3 (6) : 1035 – 1066 .
   Google Scholar
• Thieme , H.R. 1991 . “ Analysis of age‐structured population models with additional structure ” . In Mathematical Population Dynamics , Edited by: Arino , O. , Axelrod , D. E. and Kimmel , M. 115 – 126 . New York : Marcel Dekker .
   Google Scholar
• Tucker , S.L. and Zimmerman , S.O. 1988 . A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables . SIAM J. Appl. Math. , 48 (3) : 549 – 591 .
   Web of Science ®Google Scholar
• Waldstätter , R. 1990 . Models for Pair Formation with Applications to Demography and Epidemiology. , University of Tübingen . Ph. D. Dissertation
   Google Scholar
• Webb , G.F. 1985 . Theory of Nonlinear Age‐Dependent Population Dynamics. , New York and Basel : Dekker .
   Google Scholar
• Webb , G.F. 1993 . “ Asynchronous exponential growth in differential equations with homogeneous nonlinearities ” . In Differential Equations in Banach Spaces , Lecture Notes in Pure and Applied Mathematics 148 Edited by: Dore , G. , Favini , A. , Obrecht , E. and Venni , A. 225 – 233 . New York : Dekker .
   Google Scholar
• Yellin , J. and Samuelson , P.A. 1974 . A dynamical model for human population . Proc. Nat. Acad. Sci. USA , 71 (4) : 2813 – 2817 .
   PubMed Web of Science ®Google Scholar
• An earlier version of this paper was presented at the Workshop on Nonlinear Demography, Max Planck Institute for Demographic Research, Rostock, Germany, May 26–28, 1998. The author was partially supported by a Grant‐in Aid for Scientific Research from the Japan Ministry of Education, Science and Culture. I thank Evert van Imhoff and anonymous referees for their kind suggestions.
   Google Scholar