Aggregation of Risk: Relationships Among Host-Parasitoid Models

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Abstract
We explore the relationships among host-parasitoid models involving aggregation, with particular emphasis on May''s negative binomial model. Models in which parasitoid density in a patch is strictly a function of host density in the patch, with no "error" about this function, are pure-regression models. Those in which there is random variation in the number of parasitoids per patch, with no relationship between local parasitoid density and local host density, are pure-error models. The key factor in these models is not the distribution of parasitoids per se, but the distribution of the relative risk of parasitism, p, which in the present formulation can result from variation in the number of parasitoids in a patch, Pj, or from variation in host vulnerability, a. We show that May''s model, and the Bailey et al. (1962) model of which it is a special case, arises naturally from pure-error models. By contrast, it is very difficult to obtain May''s model from biologically plausible pure-regression models. There are severe contraints to obtaining May''s model under pure regression: (1) the parasitoids must aggregate in response to relative host density only, and must be unresponsive to the absolute number of hosts in a patch: (2) they must be unresponsive to parasitoid density; (3) the distribution of hosts must be independent of average host density; (4) once the parasitoid aggregation function is established, only one corresponding host distribution yields May''s model; (5) p must be gamma-distributed with constant parameters. It is not possible to obtain May''s model if parasitoid aggregation to local host density involves a transit time between patches. Only the first three constraints apply to obtaining the model of Bailey et al. When parasitoids aggregate to local host density and the above constraints are not met, modifications of May''s model may arise in which the constants a and k are replaced by functions of average host density. There is then no necessary relationship between aggregation and stability. By contrast, random aggregation in space, independent of local host density, leads to May''s model (or, more generally, to that of Bailey et al.) and serves as a powerful stabilizing force in the Nicholson-Bailey context. However, stability is obtained at the cost of lowered parasitoid efficiency. Where parasitoid aggregation in relation to local host density occurs under the constraints listed above, we find that aggregation to patches with few prey is as effective as aggregation to patches with many prey in yielding both May''s model and local stability. Under the identified constraints, any process (e.g., egg depletion or handling time) that strongly concentrates relative risk in sparser patches, thus producing inverse spatial density dependence, will promote stability as effectively as does spatial density dependence.