Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example.
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.
Diekmann
Odo
O
Department of Mathematics, University of Utrecht, P. O. Box 80010, 3508 TA, Utrecht, The Netherlands.
Gyllenberg
Mats
M
Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, 00014, Helsinki, Finland.
Metz
J A J
JAJ
Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, 00014, Helsinki, Finland.|Institute of Biology and Mathematical Institute, Leiden University, P. O. Box 9505, 2300 RA, Leiden, The Netherlands.|Evolution and Ecology Program, International Institute of Applied Systems Analysis, 2361, Laxenburg, Austria.
Nakaoka
Shinji
S
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan. snakaoka@ms.u-tokyo.ac.jp.
de Roos
Andre M
AM
Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, P. O. Box 94248, 1090 GE, Amsterdam, The Netherlands.
D000465:Algorithms; D000818:Animals; D049628:Body Size / Q000502:physiology; D003198:Computer Simulation; D003621:Daphnia / Q000502:physiology*; D017753:Ecosystem; D020387:Food Chain; D006128:Growth / Q000502:physiology; D008954:Models, Biological; D011156:Population Density; D011157:Population Dynamics; D012098:Reproduction / Q000502:physiology