Today at work we had a journal club about a recent paper in *Nature* that had caused a bit of a stir. It had suggested that the reason we don’t see as many extinctions due to habitat loss as we’d expect from empirical relationships (which I will explain in a moment) is because we’re using the wrong ones.

Unfortunately^{0} I had to prepare a seminar for today, so I didn’t have enough time to read the paper properly, and only got as far as working out that I didn’t understand what the authors were doing. And when, in the journal club, the maths was mentioned they all turned to me for help. Eeek – I wanted someone else to explain it all to me! So after the journal club I decided to really think about the paper, and thanks to our new white board I worked out that it was wrong:

And, thanks to RMV not coordinating their trams and trains during the afternoon commute, I worked out why it was wrong. Now read on…

OK, now we’re below the jump, what’s the problem?

In ecology it’s well know that the number of species you can find in a region is related to the size of the region. This is usually called the Species-Area Relationship, and might look like this:

These predict the number of species we’ll find in an area: if we lose area (for example if Paradise gets paved over), they should predict how many species will go extinct from that area. This is useful for conservation, because we can predict how many species we lose if we remove some land (or sea I suppose, if we’re marine biologists). But, when ecologists have looked at areas where land^{1} has been removed, they see more species than the SAR would predict.

The usual explanation for this is something called the Extinction Debt. This is simply the idea that species hang around for some time before they have all finally shuffled of this mortal coil. So, if we remove half of Wales, it’ll take a bit of time for the species that don’t have enough land to survive in to go extinct: they hang around for a few years declining in number but still managing to hold an Eisteddfod until there’s nobody left.

This new paper that we journal clubbed to day is by He and Hubbell (this is Hubbell of the Unified Neutral Theory of Biogeography fame, and nothing to do with telescopes). They argue that the SAR isn’t a good model for how extinctions will happen, and instead we should use another curve, the Endemics-Area Relationship (EAR). They do this by pointing out that we can calculate extinctions in two ways:

**The SAR way**: if we start with an area*A*, and remove a chunk of size*a*, then we can plug*A*~~i>a into the SAR, and this tell us how many species we have in the remaining area,~~*A**a*.**The EAR way**: When we remove an area*a*, we remove all of the endemics (i.e. the species only living in that area), so these are the species that are lost.

These sound like they should be the same; the SAR way tells us how many species we have after area loss, the EAR way tells us how many species we have lost, and hence the total number of species before the loss minus this is the species we have after loss.. One reason why they might be different is that some species are not quite endemic, so when you remove this area, they don’t have enough space left to live in, so they die off anyway: this is the extinction debt again. But He and Hubbell claim that even without this the EAR and SAR approaches are different. Their argument revolves around this figure:

He and Hubbell use this to calculate two distances: the expected distance from a random point to the first individual of a species (the inner circle, with none of that species in it, which has area *a*^{1}), and the expected distance from a random point to the last individual of a species (the outer circle – of area *a ^{N}* – that contains every member of the species). They then show that if individuals are arranged randomly, these two add up to the total area in the region,

*A*. But, they claim, if individuals are aggregated, the two distances add up to less than the total area

^{2}. The implication of this proof

^{3}is that the number of species remaining plus the number of endemics in that area (i.e. the number lost when

*a*is destroyed) does not equal the total number of species. But if this is true, what has happened to the other species?

Well, nothing. Because it’s not true. Think of it this way: let’s remove a circular area of size

*a*. The number of species lost is the number of endemics in that area, which we can calculate from

*a*

^{N}, the area of the larger circle (there is one circle for each species, and we “sum” over the circles of the different species). What’s left? If the SAR approach is correct, so He & Hubbell suggest, we should be able to plug

*a*into the SAR, to get the species remaining. But because

^{N}*a*

^{1}+

*a*≤

^{N}*A*, we will get the wrong answer. Fortunately, they have forgotten something: this only works if both areas are circles. Now, I want you all to get a sheet of paper and cut a circle out of it. But do it in such a way that you have a circle left.

Done it? Really? I’m impressed.

Actually, you can do it, but you have to be sneaky: the only possible way is to use non-Euclidean geometry (using the surface of a sphere, for example). But the methods for calculating the areas of the circles that He and Hubbell use rely on Euclid. So there’s no help from Mr. Riemann, I’m afraid.

It happens that I can also prove – no numerical simulations necessary – that the EAR and SAR approaches give the same answer; this is what the whiteboard is all about. It’s all a non-problem. I wish, though, that it had been explained more clearly, I think the discussions about the paper on the blogosphere would have been more informed about what the paper was claiming, rather than being about what the paper ignores. Whilst these are reasonable criticisms, they don’t say that the paper is wrong: it might still have been explaining a large part of the disparity between the predictions from SARs and what we have observed.

Overall, good news for ecologists, and particularly for those who have been using SARs: one less thing to worry about. The bad news is that you can really annoy one by asking them why, exactly, this paper is wrong.

## Footnotes

^{0}Yes, yes, I know. For both me and the attendees.

^{1}or sea. I haven’t forgotten you folks.

^{2}Annoyingly, although they claim to prove this result, this is how they do it: “…it is easy to numerically verify for any given

*N*and

*k*that

*a*

^{1}+

*a*≤

^{N}*A*is always true”. Proof by assertion. Now, this looks like it should be true, but that’s certainly no proof any mathematician would recognise.

^{3}or “proof”, rather. See footnote 2.

## Reference

He, F., & Hubbell, S. (2011). Species-area relationships always overestimate extinction rates from habitat loss Nature, 473 (7347), 368-371 DOI: 10.1038/nature09985

Great stuff again, Bob. Are you going to write this up into a paper? The whiteboard is kind of hard to read from here Are the p_i and I_i meant to be the same!?

Is their "proof" actually wrong as stated?

This has to be worth a formal follow-up…

Thanks, Mark. I’m intending to write it up, but I’m hoping the authors will respond first.

p

_{i}and I_{i}weren’t quite the same – I wasn’t sure if I had to go as far as using indicators, hence the I_{i}. If I submit this, it’ll be tidied up.Their proof is for aggregated distributions only, so I think it’s right, but I don’t have a proof either. And I don’t intend looking for one – Irregular Landing Platforms for Pixies are much more interesting to look at.

Ok; so if I’m understanding correctly, the nub of your proof is spotting that

p

_{i}(a) = 1-p_{i}(A-a)and then equality of SAR and EAR is trivial?

Yep, that’s it. I didn’t write it down so consicely, though.

Excellent stuff Bob. There were rumblings in Sheffield that all was not well with that paper, but too many exams to mark just now for anyone actually to read the damn thing!

Editor’s Selections: Physics of The Bends, Modelling Extinctions from Habitat Loss, and Mpemba’s EffectSarah Kendrew selects interesting and notable ResearchBlogging.org posts in the physical sciences, chemistry, engineering, computer science, geosciences and mathematics. She blogs about astronomy at One Small Step.

Not just a great Radiohead album: on…

You write "it has been suggested that the reason we don’t see as many extinctions due to habitat loss as we’d expect from empirical relationships … is because we’re using the wrong ones."

The main reason that this paper has caused a stir is not because it has a technical error. The premise that the relationship makes wrong predictions is simply false. Indeed, He an Hubbell provide no catalogue of predictions and observations to show that the predictions fail. They dismiss studies where, in fact, there is a striking agreement between the number of predicted extinctions and either those observed or species on the way to becoming extinct.

The first in the series is eastern North America. All those Scots, fleeing there after the ’45 cut down half the forest there in a generation or few. (No Eisteddfod, but you can hear the pipes at clan gatherings to this day.) How many species’ ranges did they destroy in their entirety? Well for birds, we know the answer is exactly zero. How many species eventually went extinct: 4 and one is teetering on the edge.

How many would one predict for 30 endemic species with a loss of 50% of the habitat, from the species area curve? The point is which species area curve. There is more than one and the relevant one applies to islands — which is the one you graph. That’s because the forest had become island patches in a sea of farmland. And the answer is 4.5 — which is about as good as one can get.

The series goes on and on; scores of studies from around the world found similar agreements and published them in obvious places, like Nature. (This is why the letter from Tom Brooks in yesterday’s Nature had so many co-signers.)

Bottom line: a flawed piece of mathematics, but an outrageous piece of science that got attention only because it chose to ignore 16 years of perfectly good work by people around the world.

In the supplement of that paper, for random placement model, extinction rates based on EAR or SAR are the same. Why does the sampling area difference of SAR and EAR not play its role in this situation? In addition, random placement SAR and EAR are not mirror images. They are central symetry.