Species-area relationships don’t overestimate extinction rates from habitat loss

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.
Unfortunately⁰ 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 land¹ 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 Ai>a into the SAR, and this tell us how many species we have in the remaining area, Aa.
• 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 a¹), 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 area². The implication of this proof³ 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^N into the SAR, to get the species remaining. But because a¹ + 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

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