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 differetial equation for  the resource concentration). As many results for such systems are available, 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 and a model  introduced by Gurney-Nisbet and Jones et al., and next obtain various ecological insights by analytical or numerical studies of special cases.