negative binomial encounters...
Most progress has been made in untangling these effects in host-parasitoid systems. A good starting point is the model constructed by May (1978), in which he ignored precise details and argued simply that the distribution of host-parasitoid encounters was not random but aggregated. In particular, he assumed that this distribution could be described by a particular statistical model, the negative binomial. In this case
(in contrast to Section 10.2.3), the proportion of hosts not encountered at all is given by:
where k is a measure of the degree of aggregation; maximal aggregation at k = 0, but a random distribution (recovery of the Nicholson-Bailey model) at k = If this is incorporated into the Nicholson-Bailey model (Equations 10.14 and 10.15), then we have:
... that stabilize dynamics
The behavior of a version of this model, which also includes a density-dependent host rate of increase, is illustrated in Figure 10.13, from which it is clear that the system is given a marked boost in stability by the incorporation of significant levels of aggregation (k < 1). Of particular importance is the existence of stable systems with low values of H*/K; i.e. aggregation appears capable of generating stable host abundances well below the host's normal carrying capacity. This coincides with the conclusion drawn from Figure 10.11.
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