Appendix D. Density independence of the negative binomial dispersion parameter.
Some evidence in support of the density independence of the dispersion parameter comes from comparing the dispersion parameter for different time periods, which vary greatly in mean settlement for each species (Fig. D1). The dispersion parameter does vary over time, but there is no indication that it varies with mean settlement density for Ascidia or Watersipora (an analysis of the combined effects of spatial and temporal settlement variation will be the subject of future work). The question of the density dependence of the negative binomial distribution has been important for understanding the consequences of aggregated oviposition among insects whose larvae compete in ephemeral resource patches. In models of these communities, it is necessary to know whether the aggregation of larvae is due to the fact that females lay eggs in clutches, or instead due to the aggregation of multiple egg-laying females in a resource patch. If eggs are laid in clutches, then this form of aggregation is not expected to enhance coexistence, because even at low density conspecifics will still compete within clutches, meaning that the intraspecific competition due to aggregation will not decrease with overall density (Green 1986, Ives 1991). Concomitant with this form of aggregation, the negative binomial dispersion parameter decreases (i.e., clumpiness of larvae increases) as density decreases (Green 1986). We consider this phenomenon to be less important for sessile invertebrates, because in this case larvae are individually dispersing from an adult to many settlement sites, as opposed to adults directly depositing larvae in space.
FIG. D1. Negative binomial dispersion parameter as a function of density. The negative binomial dispersion parameter for each time period was fit separately, and plotted against mean settlement abundance (per 100 cm2) for that time period. (A) Watersipora. (B) Ascidia. These plots include more time periods than Fig. C1 because here we have used additional settlement data recorded after the competition experiment was concluded.
Green, R. F. 1986. Does aggregation prevent competitive exclusion: a response. American Naturalist 128:301–304.
Ives, A. R. 1991. Aggregation and coexistence in a carrion fly community. Ecological Monographs 61:75–94.