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Mathematical Biosciences and Engineering

2010, Volume 7, Issue 1: 37-49. doi: 10.3934/mbe.2010.7.37
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Structured populations with diffusion in state space

  • 1.
    School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287
  • Received: 01 May 2009 Accepted: 29 June 2018 Published: 01 January 2010

  • MSC : Primary: 92D25; Secondary: 35K57, 49L25.

  • The classical models for populations structured by size have two features which may cause problems in biologically realistic modeling approaches: the structure variable always increases, and individuals in an age cohort that are identical initially stay identical throughout their lives. Here a diffusion term is introduced in the partial differential equation which mathematically amounts to adding viscosity. This approach solves both problems but it requires to identify appropriate boundary (recruitment) conditions. The method is applied to size-structured populations, metapopulations, infectious diseases, and vector-transmitted diseases.

    Citation: Karl Peter Hadeler. Structured populations with diffusion in state space[J]. Mathematical Biosciences and Engineering, 2010, 7(1): 37-49. doi: 10.3934/mbe.2010.7.37

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  • The classical models for populations structured by size have two features which may cause problems in biologically realistic modeling approaches: the structure variable always increases, and individuals in an age cohort that are identical initially stay identical throughout their lives. Here a diffusion term is introduced in the partial differential equation which mathematically amounts to adding viscosity. This approach solves both problems but it requires to identify appropriate boundary (recruitment) conditions. The method is applied to size-structured populations, metapopulations, infectious diseases, and vector-transmitted diseases.


  • This article has been cited by:

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    3. Carles Barril, Àngel Calsina, Sílvia Cuadrado, Jordi Ripoll, On the basic reproduction number in continuously structured populations, 2021, 44, 0170-4214, 799, 10.1002/mma.6787
    4. Physiologically structured populations with diffusion and dynamic boundary conditions, 2011, 8, 1551-0018, 503, 10.3934/mbe.2011.8.503
    5. J. Z. Farkas, A. Y. Morozov, A. Morozov, Modelling Effects of Rapid Evolution on Persistence and Stability in Structured Predator-Prey Systems, 2014, 9, 0973-5348, 26, 10.1051/mmnp/20149303
    6. Jacek Banasiak, Aleksandra Falkiewicz, Proscovia Namayanja, Asymptotic state lumping in transport and diffusion problems on networks with applications to population problems, 2016, 26, 0218-2025, 215, 10.1142/S0218202516400017
    7. Karl-Peter Hadeler, 2017, Chapter 2, 978-3-319-65620-5, 79, 10.1007/978-3-319-65621-2_2
    8. Qiang-Jun Xie, Ze-Rong He, Chun-Guo Zhang, Harvesting Renewable Resources of Population with Size Structure and Diffusion, 2014, 2014, 1085-3375, 1, 10.1155/2014/396420
    9. Agnieszka Bartłomiejczyk, Henryk Leszczński, Agnieszka Marciniak, Rothe’s method for physiologically structured models with diffusion, 2018, 68, 0139-9918, 211, 10.1515/ms-2017-0094
    10. Andrea Pugliese, Fabio Milner, A structured population model with diffusion in structure space, 2018, 77, 0303-6812, 2079, 10.1007/s00285-018-1246-6
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  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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