6 - Species Similarity and Magnitude

Summary

We define diversity measures that take account of the varying similarities between species, and show how they can be used. We state an unexpected theorem on maximizing diversity: there is a single abundance distribution that maximizes diversity from all viewpoints simultaneously. There follows a broad-brush survey of magnitude, which is closely related to maximum diversity and is defined in the very wide generality of enriched categories. In the case of metric spaces, magnitude encodes fundamental geometric invariants of size (such as volume, surface area and dimension) and is related to the concept of capacity in potential theory.