Stochastic aggregative responses and spatial patterns of parasitism in patchy host-parasitoid interactions.

Assuming random search by parasitoids within host-containing patches, and a constant search rate, current host-parasitoid models suggest that positive "searching time aggregation" by parasitoids in patches of high host density should tend to produce spatially density dependent parasitism at the patch level. However, these models view the aggregative response as a deterministic process, ignoring variability in searching time (T ) allocation among patches of equal host density, and it is not clear that stochastic analogues of these deterministic models would predict the same result.This question is examined by adding a stochastic aggregative response to the well-known "random parasitoid equation," the deterministic equation on which most existing models have been based. Simulations, based on data collected in an earlier laboratory study, indicate that this stochastic model generates very different relationships between parasitoid searching behavior and spatial patterns of parasitism than are predicted using the deterministic approach. The stochastic model suggests that "positive" aggregative responses, in which patches of high host density receive larger allocations of searching time (on the average) than patches containing lower densities, may fail to produce spatially density dependent parasitism at the patch level if searching time allocation is also more variable at the higher densities. Similarly, a "flat" response, in which mean searching times do not vary among patches of different host density, may lead to density dependent, density independent, or inversely density dependent parasitism, depending on the variance of the searching time values among patches at different density levels. The different predictions generated by the deterministic and stochastic models can be explained on purely mathematical grounds.When models are written in units of total foraging time (T ), different equations are usually required to describe the spatial features of host-parasitoid and predator-prey interactions. Because the model considered here is written in units of active searching time (T ) it should, in cases in which the underlying assumptions hold, be capable of describing these different interactions in the framework of a single ("unified") equation. This equation may also apply to some plant-herbivore systems and, to indicate its potential generality, might be referred to as a "random forager" equation.