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March 29, 2017

Functional Equations VIII: Measuring Biodiversity

Posted by Tom Leinster

Posts entitled “Such-and-Such Part 8” can be intimidating…! But in this week’s instalment of the functional equations course, we just began a new topic. So if you’ve been interested in this course at a distance, without having actually dived in, now is a decent entry point.

This week, I introduced the difficult conceptual question of how to quantify biological diversity. A partial answer is given by what ecologists call the Hill numbers, which are the exponentials of what others call the Rényi entropies. I explained what these are, how they behave, and how they’re linked to other mathematical concepts.

There were hardly any actual functional equations this week, but they’ll come back soon! This week was mainly just background and intuition.

You can find the notes from this week’s session on pages 30–34 of the notes so far.

Posted at March 29, 2017 2:32 AM UTC

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Re: Functional Equations VIII: Measuring Biodiversity

What do you think of the work of Tomas Veloz and others to use Chemical Organization Theory (COT) to measure and quantify biodiversity?

https://www.pims.math.ca/scientific-event/170111-mbstv

http://users.minet.uni-jena.de/csb/prj/ot/

Posted by: Jeffery Winkler on March 31, 2017 9:56 PM | Permalink | Reply to this

Re: Functional Equations VIII: Measuring Biodiversity

I wasn’t aware of it. At first glance, it looks as if it’s less about the quantification of diversity than the question of how ecosystems respond to changes in diversity. Of course, that’s very important too! But it seems to be a different theme than what I’ve mostly been concerned with, which is essentially “what exactly should the word ‘diversity’ mean?”

Posted by: Tom Leinster on March 31, 2017 10:09 PM | Permalink | Reply to this

Re: Functional Equations VIII: Measuring Biodiversity

On page 31 of your notes, you show diagrams of two different communities of birds, where it’s more species versus better balance.

Well, let’s say you have two communities of birds, with the same number of species, and also the same number of individuals per species. However, in the first one, all members of the same species are clones, and in the second one, there is genetic diversity within each species. I would say that the second is more diverse.

Now, let’s say you have two communities, each with the same number of species, and the same number of individuals per species. However, in the first one, you have five species of birds. In the second one, the five species are a bird, elephant, oak tree, amoeba, and human. I would say the second is more diverse.

So it would seem that biological diversity includes not just the number of species, and number of individuals per species, but the genetic diversity within species, and the genetic diversity between species.

Posted by: Jeffery Winkler on April 4, 2017 4:26 AM | Permalink | Reply to this

Re: Functional Equations VIII: Measuring Biodiversity

That’s absolutely right. In fact, your questions could act as an advertisement for the remaining classes.

Last week, I talked about diversity in terms of a very crude model of a biological community, wherein the community is modelled in terms of the relative abundances (proportions) of the species.

Today, I’ll talk about diversity in terms of a refined model that takes into account the similarity between species. That similarity can be genetic, phylogenetic, morphological, etc. It’s based on a paper that Christina Cobbold and I wrote a few years ago.

As you say, treating each species as a monolithic block may also be too crude. (And in any case, boundaries between species can be debateable.) In that case, you can subdivide further until you get the resolution you need. In the extreme, you can treat each individual as a species. Christina and I showed in our paper that our measures behave sensibly under this kind of refinement process.

A particularly testing situation is that of bacteria or viruses, where there may be no “species” classification at all. (In the virus world, the analogue of “species” is “serotype”; it’s a somewhat arbitrary division.) Practitioners can measure genetic similarity between two DNA samples in various ways, and that has to be the starting point for any calculation of diversity. (You won’t see them peering down a microscope trying to identify the species of each bacterial cell.) Again, the measures we proposed have good, sensible properties in this kind of situation.

Notes should be coming soon!

Posted by: Tom Leinster on April 4, 2017 8:33 AM | Permalink | Reply to this

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