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Transport on Networks—A Playground of Continuous and Discrete Mathematics in Population Dynamics

  • Jacek BanasiakEmail author
  • Aleksandra Puchalska
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)

Abstract

We consider structured population models in which the population is subdivided into states according to certain feature of the individuals. We consider various rules allowing individuals to move between the states; it may be physical migration between geographical patches, or the change of the genotype by mutations during mitosis. We shall see that, depending on the type of the migration rule, the models can vary from a system of coupled McKendrick equations to a system of transport equations on a graph. We address the well-posedness of such problems but the main interest is the asymptotic state aggregation that, in the presence of different time scales, allows for a significant simplification of the equations. Interestingly enough, the aggregated equations vary widely, from scalar transport equations to systems of ordinary differential equations.

Keywords

Transport problem on network Asymptotic state lumping Convergence of semigroups Singularly perturbed dynamical systems Structured population dynamics Reducible matrices Long term behaviour. 

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Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of PretoriaPretoriaSouth Africa
  2. 2.Institute of MathematicsŁódź University of TechnologyŁódźPoland
  3. 3.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland

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