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

Piotr Gwiazda; Karolina  Kropielnicka; Anna Marciniak-Czochra; Piotr Gwiazda; Karolina  Kropielnicka; Anna Marciniak-Czochra

doi:10.3934/nhm.2016.11.123

Networks and Heterogeneous Media

2016, Volume 11, Issue 1: 123-143. doi: 10.3934/nhm.2016.11.123

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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
3.
Institute of Applied Mathematics, Interdisciplinary Center of Scientic Computing and BIOQUANT, University of Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg

Received: 01 April 2015 Revised: 01 August 2015
Primary: 65M75; Secondary: 45K05, 92D25.

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.
- Particle method,
- Escalator Boxcar Train,
- structured population models,
- two-sex population models,
- measure-valued solutions
Citation: Piotr Gwiazda, Karolina Kropielnicka, Anna Marciniak-Czochra. The Escalator Boxcar Train method for a system of age-structured equations[J]. Networks and Heterogeneous Media, 2016, 11(1): 123-143. doi: 10.3934/nhm.2016.11.123

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Abstract

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.

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