The entities we study in science fall into two categories: those which can be classified in a way a human can understand, and those which are unclassifiably wild. Numbers are in the first class—you would agree that although you cannot list all the whole numbers, you have a good sense of what numbers are out there….
In the second class are things like networks (in mathematical lingo, graphs) and beetles. There doesn't appear to be any nice, orderly structure on the set of all beetles, and we've got no way to predict what kinds of novel species will turn up. All we can do is observe some features that most beetles seem to share, most of the time. But there's no periodic table of beetles, and there probably couldn't be.
Mathematicians are much happier when a mathematical subject turns out to be of the first, more structured, type. We are much sadder when a subject turns out to be a variegated mass of beetles.
Nicely put. But beetles — and biodiversity — are not such a tangled, unknowable mass as you might think. For example, for trees there is a power-law relationship between number of species found in a place and the number of higher groups — genera and families (which are often seen as fairly arbitrary groups constructed for human convenience). Generally, there are also patterns in the relative diversity of different groups — crudely put, most higher groups contain only a few lower taxa (i.e. species), but a few are fantastically diverse. Beetles are one of these — they are often called a ‘hyperdiverse’ group (nematode worms have also been labeled such).
Hyperdiverse groups take up a large chunk of the diversity within their taxonomic level, and are found at all levels: so insects are a hyperdiverse group of animals, beetles are a hyperdiverse group of insects, and the chrysomelidae are a hyperdiverse group within the beetles. (There are also similar patterns in relative abundance — crudely put, most species are rare, a few are very common.)
Of course, describing a pattern isn’t the same as explaining it, and what’s still missing are theories (or at least, theories that most people agree on) that show why these patterns in nature are as they are, and whether they have to be like that. That’s one of the things that my book is about.