Appendix: Measures of diverfsity

How do these results compare with standard indices of diversity? Patil and Taillie¹³ note that in a diverse community, the typical species is rare. They suggest defining diversity as the average rarity of species within a community.¹⁴

How do we formalize this?

$$\hbox{Community} = C = \{s; \pi_1, \pi_2, \ldots, \pi_s\} \quad ,$$

where $\pi_i$ is the proportion of all individuals that are of species $i$, and $s$ is the total number of species.

$$\pi_i = {N_i \over \sum_{k=1}^s N_k} \quad .$$

Let $R(\pi_i)$ be our measure of rarity for a species with a particular frequency of occurrence. Then the average rarity of species in the community is:

$$\Delta(C) = \sum_{k=1}^s\pi_kR(\pi_k) \quad .$$

A common species has a large $\pi_k$ (close to 1) and a small $R(\pi_k)$ (because we are not surprised to find it). Conversely, a rare species has a small $\pi_k$, and a large $R(\pi_k)$.