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Uniqueness of limit cycles for a limiting case of the chemostat: does it justify the use of logistic growth rates?
Linnaeus University, Faculty of Technology, Department of Mathematics.ORCID iD: 0000-0002-7261-0399
Malmö University.
2015 (English)In: Electronic journal on the qualitative theory of differential equations, ISSN 1417-3875, E-ISSN 1417-3875, no 47, 1-14 p.Article in journal (Refereed) Published
Abstract [en]

On infinitesimally short time intervals various processes contributing to population change tend to operate independently so that we can simply add their contributions (Metz and Diekmann, The dynamics of physiologically structured populations, 1986, p. 3). This is one of the cornerstones for differential equations modeling in general. Complicated models for processes interacting in a complex manner may be built up, and not only in population dynamics. The principle holds as long as the various contributions are taken into account exactly. In this paper we discuss commonly used approximations that may lead to non-removable dependency terms potentially affecting the long run qualitative behavior of the involved equations. We prove that these terms do not produce such effects in the simplest and most interesting biological case, but the general case is left open. Our main result is a rather complete analysis of an important limiting case. Once complete knowledge of the qualitative properties of simple models is obtained, it greatly facilitates further studies of more complex models. A consequence of our analysis is that standard methods can be applied. However, the application of those methods is far from straightforward and require non-trivial estimates in order to make them valid for all values of the parameters. We focus on making these proofs as elementary as possible.

Place, publisher, year, edition, pages
2015. no 47, 1-14 p.
Keyword [en]
global stability, uniqueness of limit cycles, chemostat model
National Category
Mathematical Analysis
Research subject
Mathematics, Applied Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-45826DOI: 10.14232/ejqtde.2015.1.47ISI: 000358722000001OAI: oai:DiVA.org:lnu-45826DiVA: diva2:847934
Note

I ArXiv med titeln "Logistic approximations and their consequences to bifurcations patterns and long-run dynamics"

Available from: 2015-08-21 Created: 2015-08-21 Last updated: 2017-01-24Bibliographically approved

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CiteExportLink to record
Permanent link

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Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
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