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A model for the growth of a size-structured population reproducing by fission and living under changing (nutrient) conditions is formulated and analysed. It is shown that the dynamics is asymptotically described by an o.d.e. total population model, while the size distribution becomes stationary.

For a wide class of host-parasitoid models, a reduction to Arnold's normal form can be carried out in an explicit way. In the case of Hopf bifurcation, the shape and size of the elliptic limit curve can be derived in terms of the parameters of the model. Some models have a rich bifurcation behaviour with both supercritical and subcritical Hopf bifurcation, and with a transition zone in the parameter plane for which there exists a pair of limit curves, one stable and one unstable. The theory is confirmed and illustrated by numerical experiments.

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