An age-structured two-sex model in the space of radon measures: Well posedness

Agnieszka Ulikowska

doi:10.3934/krm.2012.5.873

Article Contents

2012, Volume 5, Issue 4: 873-900. Doi: 10.3934/krm.2012.5.873

This issue Previous Article Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities Next Article

An age-structured two-sex model in the space of radon measures: Well posedness

Agnieszka Ulikowska^1,

1.
University of Warsaw, ul. Banacha 2, 02-097 Warsaw

Received: January 31, 2012

Revised: April 30, 2012

Published: November 2012

Abstract Related Papers Cited by

Abstract

In the following paper a well-posedness of an age-structured two-sex population model in a space of Radon measures equipped with a flat metric is presented. Existence and uniqueness of measure valued solutions is proved by a regularization technique. This approach allows to obtain Lipschitz continuity of solutions with respect to time and stability estimates. Moreover, a brief discussion on a marriage function, which is the main source of a nonlinearity, is carried out and an example of the marriage function fitting into this framework is given.

Keywords:

Population dynamics,
age-structured two-sex population model,
Radon measures,
measure valued solutions,
flat metric.

Mathematics Subject Classification: 28A33, 35F10, 35L50, 35B30.

Citation:

Related Papers

Cited by

References

[1]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005.
[2]	J. A. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics, Acta Applicandae Mathematicae, (2012), 1-16.
[3]	J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, Journal of Differential Equations, 252 (2012), 3245-3277.
[4]	C. Castillo-Chávez and W. Huang, The logistic equation revisited, Math. Biosci., 128 (1995), 199-316.
[5]	O. Diekmann and J.A.J. Metz, "The dynamics of Physiologically Structured populations," Lecture Notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986.
[6]	K. Dietz and K. P. Hadeler, Epidemiological models for sexually transmitted diseases, J. Math. Biol., 26 (1988), 1-25.
[7]	L. C. Evans, "Weak Convergence Methods for Nonlinear Partial Differential Equations," CBMS Regional Conference Series in Mathematics, 74, published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990.
[8]	L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.
[9]	A. Fredrickson, A mathematical theory of age structure in sexual populations: random mating and monogamous models, Math. Biosci., 10 (1971), 117-143.doi: 10.1016/0025-5564(71)90054-X.
[10]	A. Fredrickson and P. Hsu, Population-changing processes and the dynamics of sexual populations, Math. Biosci., 26 (1975), 55-78.doi: 10.1016/0025-5564(75)90094-2.
[11]	G. Garnett, An introduction to mathematical models in sexually transmitted disease epidemiology, Sex Transm. Inf., 78 (2001), 7-12.
[12]	P. Gwiazda, G. Jamróz and A. Marciniak-Czochra, Models of discrete and continuous cell differentiation in the framework of transport equation, SIAM J. Math. Anal., 44 (2012), 1103-1133.
[13]	P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, Journal of Differential Equations, 248 (2010), 2703-2735.
[14]	P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 2010, 733-773.
[15]	K. Hadeler and K. Ngoma, Homogeneous models for sexually transmitted diseases, Rocky Mt. J. Math., 20 (1990), 967-986.
[16]	K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102.
[17]	K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros, Models for pair formation in bisexual populations, J. Math. Biol., 26 (1988), 635-649.doi: 10.1007/BF00276145.
[18]	F. Hoppensteadt, "Mathematical Theories of Populations: Demographics, Genetics and Epidemics," Society for Industrial and Applied Mathematics, Philadelphia, PA, 1975.
[19]	M. Iannelli, M. Martcheva and F. A. Milner, "Gender-structured Population Modeling. Mathematical Methods, Numerics, and Simulations," Frontiers in Applied Mathematics, 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2005.
[20]	H. Inaba, "An Age-structured Two-sex Model for Human Population Reproduction by First Marriage," volume 15 of Working paper series, Institute of Population Problems, Tokyo, 1993.
[21]	H. Inaba, Persistent age distributions for an age-structured two-sex population model, Math. Population Studies, 7 (2000), 365-398.
[22]	D. G. Kendall, Stochastic processes and population growth, J. Roy. Statist. Soc. Ser. B., 11 (1949), 230-264.
[23]	M. Martcheva, Exponential growth in age-structured two-sex populations, Math. Biosci., 157 (1999), 1-22.
[24]	M. Martcheva and F. A. Milner, A two-sex age-structured population model: well posedness, Math. Population Stud., 7 (1999), 111-129.
[25]	M. Martcheva and F. A. Milner, The mathematics of sex and marriage, revisited, Math. Population Stud., 9 (2001), 123-141.
[26]	D. Maxin and F. Milner, The effect of nonreproductive groups on persistent sexually transmitted diseases, Math. Biosc. and Engin., 4 (2007), 505-522.doi: 10.3934/mbe.2007.4.505.
[27]	F. Milner and K. Yang, The logistic, two-sex, age-structured population model, J. Biol. Dyn., 3 (2009), 252-270.
[28]	J. Prüss and W. Schappacher, Persistent age-distributions for a pair-formation model, J. Math. Biol., 33 (1994), 17-33.
[29]	G. F. Webb, "Structured Population Dynamics," Banach Center Publications, 63, Polish Acad. Sci., Warsaw, 2004.