TY - JOUR

T1 - Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

AU - Diekmann, O.

AU - Gyllenberg, M.

AU - Metz, J.A.J.

AU - Nakaoka, S.

AU - de Roos, A.M.

PY - 2010

Y1 - 2010

N2 - 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

AB - 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

KW - Mathematics

KW - Wiskunde en computerwetenschappen

KW - Landbouwwetenschappen

KW - Wiskunde: algemeen

U2 - 10.1007/s00285-009-0299-y

DO - 10.1007/s00285-009-0299-y

M3 - Article

SN - 0303-6812

VL - 61

SP - 277

EP - 318

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

ER -