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Persistent Discrete-Time Dynamics on Measures

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Progress on Difference Equations and Discrete Dynamical Systems (ICDEA 2019)

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Abstract

A discrete-time structured population model is formulated by a population turnover map F on the cone of finite nonnegative Borel measures that maps the structural population distribution of a given year to the one of the next year. F has a first order approximation at the zero measure (the extinction fixed point), which is a positive linear operator on the ordered vector space of real measures and can be interpreted as a basic population turnover operator. A spectral radius can be defined by the usual Gelfand formula and can be interpreted as basic population turnover number. We continue our investigation (Thieme, H.R.: Discrete-time population dynamics on the state space of measures, Math. Biosci. Engin. 17:1168–1217 (2020). doi: 10.3934/mbe.2020061) in how far the spectral radius serves as a threshold parameter between population extinction and population persistence. Emphasis is on conditions for various forms of uniform population persistence if the basic population turnover number exceeds 1.

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Acknowledgements

The author thanks Azmy Ackleh for useful hints and Odo Diekmann and Eugenia Franco for helpful suggestions.

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Authors and Affiliations

  1. School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287-1804, USA

    Horst R. Thieme

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  1. Horst R. Thieme
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Correspondence to Horst R. Thieme .

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Editors and Affiliations

  1. Department of Mathematics, University College London, London, UK

    Steve Baigent

  2. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

    Martin Bohner

  3. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

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Thieme, H.R. (2020). Persistent Discrete-Time Dynamics on Measures. In: Baigent, S., Bohner, M., Elaydi, S. (eds) Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-030-60107-2_4

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