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Saturation Effects in Population Dynamics: Use Branching Processes or Dynamical Systems?

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Mathematical Modeling of Biological Systems, Volume II

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Summary

This chapter deals with the behavior of a branching population undergoing saturation effects when it becomes too large. We study in particular the limits of the prediction given in the setting of the deterministic dynamical system related to the stochastic branching process modeling the evolution of the population. We also generalize the usual Markovian branching processes of order one to size-dependent branching processes that may have a longer memory and give conditions leading to an almost sure extinction of the process while the dynamical system is persistent. The notion of reproductive rate is explained and generalized. We give some examples, in particular the amplification process in the polymerase chain reaction (PCR).

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Author information

Authors and Affiliations

  1. Applied Mathematics and Informatics unity, INRA, 78352 Jouy-en-Josas Cedex, France

    Christine Jacob

Authors
  1. Christine Jacob
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Editor information

Editors and Affiliations

  1. Center for Information Services and High Performance Computing, Technische Universität Dresden, 01062 Dresden, Germany

    Andreas Deutsch

  2. Departamento de Matemáticas, Universidad de Alcalá, 28871 Alcalá de Henares, Spain

    Rafael Bravo de la Parra

  3. Department of Theoretical Biology, Utrecht University, Padualaan 8, 3584 CD Utrecht, The Netherlands

    Rob J. de Boer

  4. Mathematisch Instituut, Utrecht University, Budapestlaan 6, 3584 CH Utrecht, The Netherlands

    Odo Diekmann

  5. Mathematical Sciences, Chalmers University of Technology, Göteborg University, 412 96 Göteborg, Sweden

    Peter Jagers

  6. Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin katu 2b, 00014 Helsinki, Finland

    Eva Kisdi

  7. Fakultät für Gesundheitswissenschaften, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany

    Mirjam Kretzschmar

  8. Institute of Physiology, Academy of Sciences of the Czech Republic, Videnska 1083, 142 20 Prague 4, Czech Republic

    Petr Lansky

  9. Section Theoretical Evolutionary Biology, Institute of Evolutionary and Ecological Sciences, Rijksuniversiteit Leiden, Kaiserstraat 63, 2311 GP Leiden, The Netherlands

    Hans Metz

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Jacob, C. (2008). Saturation Effects in Population Dynamics: Use Branching Processes or Dynamical Systems?. In: Deutsch, A., et al. Mathematical Modeling of Biological Systems, Volume II. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4556-4_30

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