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Kolmogorov’s Differential Equations and Positive Semigroups on First Moment Sequence Spaces

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An Erratum to this article was published on 08 January 2008

Abstract

Spatially implicit metapopulation models with discrete patch-size structure and host-macroparasite models which distinguish hosts by their parasite loads lead to infinite systems of ordinary differential equations. In several papers, a this-related theory will be developed in sufficient generality to cover these applications. In this paper the linear foundations are laid. They are of own interest as they apply to continuous-time population growth processes (Markov chains). Conditions are derived that the solutions of an infinite linear system of differential equations, known as Kolmogorov’s differential equations, induce a C 0-semigroup on an appropriate sequence space allowing for first moments. We derive estimates for the growth bound and the essential growth bound and study the asymptotic behavior. Our results will be illustrated for birth and death processes with immigration and catastrophes.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL, 32611–8105, USA

    Maia Martcheva

  2. Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287-1804, USA

    Horst R. Thieme & Thanate Dhirasakdanon

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  1. Maia Martcheva
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  2. Horst R. Thieme
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  3. Thanate Dhirasakdanon
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Correspondence to Maia Martcheva.

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An erratum to this article can be found at http://dx.doi.org/10.1007/s00285-007-0154-y

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Martcheva, M., Thieme, H.R. & Dhirasakdanon, T. Kolmogorov’s Differential Equations and Positive Semigroups on First Moment Sequence Spaces. J. Math. Biol. 53, 642–671 (2006). https://doi.org/10.1007/s00285-006-0002-5

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  • DOI: https://doi.org/10.1007/s00285-006-0002-5

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