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March  2016, 11(1): 123-143. doi: 10.3934/nhm.2016.11.123

## The Escalator Boxcar Train method for a system of age-structured equations

 1 Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa 2 Institute of Mathematics, University of Gdańsk, Poland 3 Institute of Applied Mathematics, Interdisciplinary Center of Scienti c Computing and BIOQUANT, University of Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg

Received  April 2015 Revised  August 2015 Published  January 2016

The Escalator Boxcar Train method (EBT) is a numerical method for structured population models of McKendrick -- von Foerster type. Those models consist of a certain class of hyperbolic partial differential equations and describe time evolution of the distribution density of the structure variable describing a feature of individuals in the population. The method was introduced in late eighties and widely used in theoretical biology, but its convergence was proven only in recent years using the framework of measure-valued solutions. Till now the EBT method was developed only for scalar equation models. In this paper we derive a full numerical EBT scheme for age-structured, two-sex population model (Fredrickson-Hoppensteadt model), which consists of three coupled hyperbolic partial differential equations with nonlocal boundary conditions. It is the first step towards extending the EBT method to systems of structured population equations.
Citation: Piotr Gwiazda, Karolina Kropielnicka, Anna Marciniak-Czochra. The Escalator Boxcar Train method for a system of age-structured equations. Networks and Heterogeneous Media, 2016, 11 (1) : 123-143. doi: 10.3934/nhm.2016.11.123
##### References:
 [1] Å. Brännström, L. Carlsson and D. Simpson, On the convergence of the escalator boxcar train, SIAM J. Numer. Anal., 51 (2013), 3213-3231. doi: 10.1137/120893215. [2] A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000, The one-dimensional Cauchy problem. [3] R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst., 23 (2009), 733-753. doi: 10.3934/dcds.2009.23.733. [4] A. M. de Roos, Errata: "Numerical methods for structured population models: The escalator boxcar train'', Numer. Methods Partial Differential Equations, 5 (1989), p169. [5] A. G. Fredrickson, A mathematical theory of age structure in sexual populations: Random mating and monogamous marriage models, Math. Biosci., 10 (1971), 117-143. doi: 10.1016/0025-5564(71)90054-X. [6] P. Gwiazda, J. Jabłoński, A. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded lipschitz distance, Numer Meth Part Differ Equat, 30 (2014), 1797-1820. doi: 10.1002/num.21879. [7] P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differential Equations, 248 (2010), 2703-2735. doi: 10.1016/j.jde.2010.02.010. [8] P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 (2010), 733-773. doi: 10.1142/S021989161000227X. [9] K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102, Evolution and control in biological systems (Laxenburg, 1987). doi: 10.1007/BF00046676. [10] 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. [11] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975, Regional Conference Series in Applied Mathematics. [12] H. Inaba, An age-structured two-sex model for human population reproduction by first marriage, Working Paper Series, 15. [13] M. Martcheva and F. A. Milner, A two-sex age-structured population model: Well posedness, Math. Population Stud., 7 (1999), 111-129. doi: 10.1080/08898489909525450. [14] A. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1926), 98-130. doi: 10.1017/S0013091500034428. [15] S. Müller and M. Ortiz, On the $\Gamma$-convergence of discrete dynamics and variational integrators, J. Nonlinear Sci., 14 (2004), 279-296. doi: 10.1007/BF02666023. [16] J. Prüss and W. Schappacher, Persistent age-distributions for a pair-formation model, J. Math. Biol., 33 (1994), 17-33. doi: 10.1007/BF00160172. [17] A. Ulikowska, An age-structured, two-sex model in the space of radon measures: Well posedness, Kinet Relat Mod, 5 (2012), 873-900. doi: 10.3934/krm.2012.5.873. [18] P. E. Zhidkov, On a problem with two-time data for the Vlasov equation, Nonlinear Anal., 31 (1998), 537-547. doi: 10.1016/S0362-546X(97)00420-3.

show all references

##### References:
 [1] Å. Brännström, L. Carlsson and D. Simpson, On the convergence of the escalator boxcar train, SIAM J. Numer. Anal., 51 (2013), 3213-3231. doi: 10.1137/120893215. [2] A. Bressan, Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000, The one-dimensional Cauchy problem. [3] R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst., 23 (2009), 733-753. doi: 10.3934/dcds.2009.23.733. [4] A. M. de Roos, Errata: "Numerical methods for structured population models: The escalator boxcar train'', Numer. Methods Partial Differential Equations, 5 (1989), p169. [5] A. G. Fredrickson, A mathematical theory of age structure in sexual populations: Random mating and monogamous marriage models, Math. Biosci., 10 (1971), 117-143. doi: 10.1016/0025-5564(71)90054-X. [6] P. Gwiazda, J. Jabłoński, A. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded lipschitz distance, Numer Meth Part Differ Equat, 30 (2014), 1797-1820. doi: 10.1002/num.21879. [7] P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differential Equations, 248 (2010), 2703-2735. doi: 10.1016/j.jde.2010.02.010. [8] P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 (2010), 733-773. doi: 10.1142/S021989161000227X. [9] K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102, Evolution and control in biological systems (Laxenburg, 1987). doi: 10.1007/BF00046676. [10] 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. [11] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975, Regional Conference Series in Applied Mathematics. [12] H. Inaba, An age-structured two-sex model for human population reproduction by first marriage, Working Paper Series, 15. [13] M. Martcheva and F. A. Milner, A two-sex age-structured population model: Well posedness, Math. Population Stud., 7 (1999), 111-129. doi: 10.1080/08898489909525450. [14] A. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1926), 98-130. doi: 10.1017/S0013091500034428. [15] S. Müller and M. Ortiz, On the $\Gamma$-convergence of discrete dynamics and variational integrators, J. Nonlinear Sci., 14 (2004), 279-296. doi: 10.1007/BF02666023. [16] J. Prüss and W. Schappacher, Persistent age-distributions for a pair-formation model, J. Math. Biol., 33 (1994), 17-33. doi: 10.1007/BF00160172. [17] A. Ulikowska, An age-structured, two-sex model in the space of radon measures: Well posedness, Kinet Relat Mod, 5 (2012), 873-900. doi: 10.3934/krm.2012.5.873. [18] P. E. Zhidkov, On a problem with two-time data for the Vlasov equation, Nonlinear Anal., 31 (1998), 537-547. doi: 10.1016/S0362-546X(97)00420-3.
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