December  2012, 5(4): 873-900. doi: 10.3934/krm.2012.5.873

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

1. 

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

Received  February 2012 Revised  May 2012 Published  November 2012

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.
Citation: Agnieszka Ulikowska. An age-structured two-sex model in the space of radon measures: Well posedness. Kinetic & Related Models, 2012, 5 (4) : 873-900. doi: 10.3934/krm.2012.5.873
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, (2005).   Google Scholar

[2]

J. A. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics,, Acta Applicandae Mathematicae, (2012), 1.   Google Scholar

[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.   Google Scholar

[4]

C. Castillo-Chávez and W. Huang, The logistic equation revisited,, Math. Biosci., 128 (1995), 199.   Google Scholar

[5]

O. Diekmann and J.A.J. Metz, "The dynamics of Physiologically Structured populations,", Lecture Notes in Biomathematics, (1986).   Google Scholar

[6]

K. Dietz and K. P. Hadeler, Epidemiological models for sexually transmitted diseases,, J. Math. Biol., 26 (1988), 1.   Google Scholar

[7]

L. C. Evans, "Weak Convergence Methods for Nonlinear Partial Differential Equations,", CBMS Regional Conference Series in Mathematics, (1990).   Google Scholar

[8]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (2010).   Google Scholar

[9]

A. Fredrickson, A mathematical theory of age structure in sexual populations: random mating and monogamous models,, Math. Biosci., 10 (1971), 117.  doi: 10.1016/0025-5564(71)90054-X.  Google Scholar

[10]

A. Fredrickson and P. Hsu, Population-changing processes and the dynamics of sexual populations,, Math. Biosci., 26 (1975), 55.  doi: 10.1016/0025-5564(75)90094-2.  Google Scholar

[11]

G. Garnett, An introduction to mathematical models in sexually transmitted disease epidemiology,, Sex Transm. Inf., 78 (2001), 7.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[14]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces,, J. Hyperbolic Differ. Equ., 7 (2010), 733.   Google Scholar

[15]

K. Hadeler and K. Ngoma, Homogeneous models for sexually transmitted diseases,, Rocky Mt. J. Math., 20 (1990), 967.   Google Scholar

[16]

K. P. Hadeler, Pair formation in age-structured populations,, Acta Appl. Math., 14 (1989), 91.   Google Scholar

[17]

K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros, Models for pair formation in bisexual populations,, J. Math. Biol., 26 (1988), 635.  doi: 10.1007/BF00276145.  Google Scholar

[18]

F. Hoppensteadt, "Mathematical Theories of Populations: Demographics, Genetics and Epidemics,", Society for Industrial and Applied Mathematics, (1975).   Google Scholar

[19]

M. Iannelli, M. Martcheva and F. A. Milner, "Gender-structured Population Modeling. Mathematical Methods, Numerics, and Simulations,", Frontiers in Applied Mathematics, (2005).   Google Scholar

[20]

H. Inaba, "An Age-structured Two-sex Model for Human Population Reproduction by First Marriage,", volume \textbf{15} of Working paper series, 15 (1993).   Google Scholar

[21]

H. Inaba, Persistent age distributions for an age-structured two-sex population model,, Math. Population Studies, 7 (2000), 365.   Google Scholar

[22]

D. G. Kendall, Stochastic processes and population growth,, J. Roy. Statist. Soc. Ser. B., 11 (1949), 230.   Google Scholar

[23]

M. Martcheva, Exponential growth in age-structured two-sex populations,, Math. Biosci., 157 (1999), 1.   Google Scholar

[24]

M. Martcheva and F. A. Milner, A two-sex age-structured population model: well posedness,, Math. Population Stud., 7 (1999), 111.   Google Scholar

[25]

M. Martcheva and F. A. Milner, The mathematics of sex and marriage, revisited,, Math. Population Stud., 9 (2001), 123.   Google Scholar

[26]

D. Maxin and F. Milner, The effect of nonreproductive groups on persistent sexually transmitted diseases,, Math. Biosc. and Engin., 4 (2007), 505.  doi: 10.3934/mbe.2007.4.505.  Google Scholar

[27]

F. Milner and K. Yang, The logistic, two-sex, age-structured population model,, J. Biol. Dyn., 3 (2009), 252.   Google Scholar

[28]

J. Prüss and W. Schappacher, Persistent age-distributions for a pair-formation model,, J. Math. Biol., 33 (1994), 17.   Google Scholar

[29]

G. F. Webb, "Structured Population Dynamics,", Banach Center Publications, (2004).   Google Scholar

show all references

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, (2005).   Google Scholar

[2]

J. A. Cañizo, J. A. Carrillo and S. Cuadrado, Measure solutions for some models in population dynamics,, Acta Applicandae Mathematicae, (2012), 1.   Google Scholar

[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.   Google Scholar

[4]

C. Castillo-Chávez and W. Huang, The logistic equation revisited,, Math. Biosci., 128 (1995), 199.   Google Scholar

[5]

O. Diekmann and J.A.J. Metz, "The dynamics of Physiologically Structured populations,", Lecture Notes in Biomathematics, (1986).   Google Scholar

[6]

K. Dietz and K. P. Hadeler, Epidemiological models for sexually transmitted diseases,, J. Math. Biol., 26 (1988), 1.   Google Scholar

[7]

L. C. Evans, "Weak Convergence Methods for Nonlinear Partial Differential Equations,", CBMS Regional Conference Series in Mathematics, (1990).   Google Scholar

[8]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (2010).   Google Scholar

[9]

A. Fredrickson, A mathematical theory of age structure in sexual populations: random mating and monogamous models,, Math. Biosci., 10 (1971), 117.  doi: 10.1016/0025-5564(71)90054-X.  Google Scholar

[10]

A. Fredrickson and P. Hsu, Population-changing processes and the dynamics of sexual populations,, Math. Biosci., 26 (1975), 55.  doi: 10.1016/0025-5564(75)90094-2.  Google Scholar

[11]

G. Garnett, An introduction to mathematical models in sexually transmitted disease epidemiology,, Sex Transm. Inf., 78 (2001), 7.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[14]

P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces,, J. Hyperbolic Differ. Equ., 7 (2010), 733.   Google Scholar

[15]

K. Hadeler and K. Ngoma, Homogeneous models for sexually transmitted diseases,, Rocky Mt. J. Math., 20 (1990), 967.   Google Scholar

[16]

K. P. Hadeler, Pair formation in age-structured populations,, Acta Appl. Math., 14 (1989), 91.   Google Scholar

[17]

K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros, Models for pair formation in bisexual populations,, J. Math. Biol., 26 (1988), 635.  doi: 10.1007/BF00276145.  Google Scholar

[18]

F. Hoppensteadt, "Mathematical Theories of Populations: Demographics, Genetics and Epidemics,", Society for Industrial and Applied Mathematics, (1975).   Google Scholar

[19]

M. Iannelli, M. Martcheva and F. A. Milner, "Gender-structured Population Modeling. Mathematical Methods, Numerics, and Simulations,", Frontiers in Applied Mathematics, (2005).   Google Scholar

[20]

H. Inaba, "An Age-structured Two-sex Model for Human Population Reproduction by First Marriage,", volume \textbf{15} of Working paper series, 15 (1993).   Google Scholar

[21]

H. Inaba, Persistent age distributions for an age-structured two-sex population model,, Math. Population Studies, 7 (2000), 365.   Google Scholar

[22]

D. G. Kendall, Stochastic processes and population growth,, J. Roy. Statist. Soc. Ser. B., 11 (1949), 230.   Google Scholar

[23]

M. Martcheva, Exponential growth in age-structured two-sex populations,, Math. Biosci., 157 (1999), 1.   Google Scholar

[24]

M. Martcheva and F. A. Milner, A two-sex age-structured population model: well posedness,, Math. Population Stud., 7 (1999), 111.   Google Scholar

[25]

M. Martcheva and F. A. Milner, The mathematics of sex and marriage, revisited,, Math. Population Stud., 9 (2001), 123.   Google Scholar

[26]

D. Maxin and F. Milner, The effect of nonreproductive groups on persistent sexually transmitted diseases,, Math. Biosc. and Engin., 4 (2007), 505.  doi: 10.3934/mbe.2007.4.505.  Google Scholar

[27]

F. Milner and K. Yang, The logistic, two-sex, age-structured population model,, J. Biol. Dyn., 3 (2009), 252.   Google Scholar

[28]

J. Prüss and W. Schappacher, Persistent age-distributions for a pair-formation model,, J. Math. Biol., 33 (1994), 17.   Google Scholar

[29]

G. F. Webb, "Structured Population Dynamics,", Banach Center Publications, (2004).   Google Scholar

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