American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography
  • NHM Home
  • This Issue
  • Next Article
    Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity
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

Piotr Gwiazda 1, , Karolina Kropielnicka 2, and  Anna Marciniak-Czochra 3,

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.
Keywords: Escalator Boxcar Train, Particle method, measure-valued solutions., two-sex population models, structured population models.
Mathematics Subject Classification: Primary: 65M75; Secondary: 45K05, 92D2.
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.

[1]

Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233
[2]

Yacouba Simporé, Oumar Traoré. Null controllability of a nonlinear age, space and two-sex structured population dynamics model. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021052
[3]

Anthony Tongen, María Zubillaga, Jorge E. Rabinovich. A two-sex matrix population model to represent harem structure. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1077-1092. doi: 10.3934/mbe.2016031
[4]

L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203
[5]

Rinaldo M. Colombo, Mauro Garavello. Stability and optimization in structured population models on graphs. Mathematical Biosciences & Engineering, 2015, 12 (2) : 311-335. doi: 10.3934/mbe.2015.12.311
[6]

Mustapha Mokhtar-Kharroubi, Quentin Richard. Spectral theory and time asymptotics of size-structured two-phase population models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2969-3004. doi: 10.3934/dcdsb.2020048
[7]

Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063
[8]

Dianmo Li, Zengxiang Gao, Zufei Ma, Baoyu Xie, Zhengjun Wang. Two general models for the simulation of insect population dynamics. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 623-628. doi: 10.3934/dcdsb.2004.4.623
[9]

Bruno Buonomo, Deborah Lacitignola. On the stabilizing effect of cannibalism in stage-structured population models. Mathematical Biosciences & Engineering, 2006, 3 (4) : 717-731. doi: 10.3934/mbe.2006.3.717
[10]

Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049
[11]

Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627
[12]

Agnieszka Ulikowska. An age-structured two-sex model in the space of radon measures: Well posedness. Kinetic and Related Models, 2012, 5 (4) : 873-900. doi: 10.3934/krm.2012.5.873
[13]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032
[14]

Cleopatra Christoforou, Myrto Galanopoulou, Athanasios E. Tzavaras. Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6175-6206. doi: 10.3934/dcds.2019269
[15]

Maria Michaela Porzio, Flavia Smarrazzo, Alberto Tesei. Radon measure-valued solutions of unsteady filtration equations. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022040
[16]

Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Alexandre Thorel. Generalized linear models for population dynamics in two juxtaposed habitats. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2933-2960. doi: 10.3934/dcds.2019122
[17]

Yingli Pan, Ying Su, Junjie Wei. Bistable waves of a recursive system arising from seasonal age-structured population models. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 511-528. doi: 10.3934/dcdsb.2018184
[18]

Thomas Lorenz. Nonlocal hyperbolic population models structured by size and spatial position: Well-posedness. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4547-4628. doi: 10.3934/dcdsb.2019156
[19]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041
[20]

Eduardo Liz, Alfonso Ruiz-Herrera. Delayed population models with Allee effects and exploitation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 83-97. doi: 10.3934/mbe.2015.12.83

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (169)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy