Abstract
We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.
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Acknowledgments
This paper had a very long maturation period, but the finishing touch was much catalysed by visits of Odo Diekmann and Hans Metz to the Department of Mathematics and Statistics of the University of Helsinki. They are grateful to this Department for excellent working conditions as well as financial support. Moreover, Odo Diekmann, Mats Gyllenberg and Hans Metz thank the Oberwolfach Research in Pairs (in our case Triples) programme for facilitation in conceiving this paper and during delivery. Finally, all authors are grateful to Roger Nisbet for very constructive criticism.
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Dedicated to Horst Thieme on the occasion of his 60th birthday.
M. Gyllenberg was partially supported by the Academy of Finland and S. Nakaoka was partly supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
An erratum to this article is available at http://dx.doi.org/10.1007/s00285-017-1148-z.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Diekmann, O., Gyllenberg, M., Metz, J.A.J. et al. Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example. J. Math. Biol. 61, 277–318 (2010). https://doi.org/10.1007/s00285-009-0299-y
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DOI: https://doi.org/10.1007/s00285-009-0299-y
Keywords
- Physiologically structured population models
- Size-structure
- Delay equations
- Linearised stability
- Characteristic equation