Aggregation by Parasitoids and Predators: Effects on Equilibrium and Stability

Abstract

Predators and insect parasites (parasitoids) sometimes aggregate in patches containing high prey (host) densities because they search longer there, causing higher death (parasitism) rates in patches with more prey (hosts). This behavior can stabilize the otherwise unstable, discrete-time, Nicholson-Bailey model, typically at the cost of increasing the host equilibrium density. The key feature that produces both effects in this model appears to be the absence of within-generation dynamics; in particular, parasitoids are not able to reaggregate in response to changes in local host density within the generation. We examine the effects of aggregation on the otherwise neutrally stable Lotka-Volterra model. We allow the parasitoid to reaggregate continually as local host density changes. Aggregation appears as positive covariance between the parasitoid and host density across patches. It affects the model by altering the initially linear functional response. Aggregation always increases parasitoid efficiency and hence reduces host equilibrium density. We examine the effects of aggregation on model stability in several circumstances: (1) the parasitoid responds to a signal ranging from absolute to relative difference in local host density; (2) aggregation is linear or accelerating in response to this signal; (3) the variance of the host distribution is related in various ways to the mean host density (the host may have a Poisson, negative-binomial, or unspecified distribution). When account is taken of the probable limits set on parameter values by theoretical considerations and field conditions, it appears that aggregation is typically destabilizing. Although aggregation reduces host equilibrium density, h*, stability conditions usually require h* to be large. Stability is not affected by the magnitude of the spatial variance of host density, per se, but rather by the rate of change of the variance with respect to the mean. The chances of stability are increased if the parasitoid or predator has an accelerating aggregative response and possibly if it responds to absolute differences in local host density over a wide range of host densities, though the stabilizing effects in the latter case are likely to be weak or limited to small perturbations. We also examine aggregation that is independent of local prey (host) density. The Nicholson-Bailey model can be stabilized if, for example, the distribution of parasitism among patches is highly heterogeneous but unrelated to host density in the patch. Again, strong aggregation increases the Nicholson-Bailey host equilibrium density. Such aggregation has no effect on stability or host equilibrium in our continuous-time model. We discuss critical assumptions in the models. The results point to the need for more information about the signal to which aggregating predators actually respond and about the form of the response in the field.