Finite difference approximations for measure-valued solutions of a hierarchicallysize-structured population model

Azmy S. Ackleh; Vinodh K. Chellamuthu; Kazufumi Ito

doi:10.3934/mbe.2015.12.233

Article Contents

2015, Volume 12, Issue 2: 233-258. Doi: 10.3934/mbe.2015.12.233 open access

This issue Previous Article Preface to ``Modeling with Measures" Next Article Riemann problems with non--local point constraints and capacity drop

Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model

1.
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010
2.
Department of Mathematics, Box 8205, North Carolina State University, Raleigh, NC 27695-8205

Received: March 31, 2014

Revised: August 31, 2014

Published: November 2014

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Abstract

We study a quasilinear hierarchically size-structured population model presented in [4]. In this model the growth, mortality and reproduction rates are assumed to depend on a function of the population density. In [4] we showed that solutions to this model can become singular (measure-valued) in finite time even if all the individual parameters are smooth. Therefore, in this paper we develop a first order finite difference scheme to compute these measure-valued solutions. Convergence analysis for this method is provided. We also develop a high resolution second order scheme to compute the measure-valued solution of the model and perform a comparative study between the two schemes.

Keywords:

Hierarchically size-structured population model,
measure-valued solutions,
finite-difference approximations,
convergence analysis.

Mathematics Subject Classification: Primary: 35A05, 35D05, 35F25, 35L67, 92D25.

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