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Species-area relationships always overestimate extinction rates from habitat loss LETTER doi:10.1038/nature09985 Species–area relationships always overestimate extinction rates from habitat loss Fangliang He1,2 & Stephen P. Hubbell3,4 Extinction from habitat loss is the signature conservation problem of the twenty-first century1. Despite its importance, estimating extinction rates is still highly uncertain because no proven direct methods or reliable data exist for verifying extinctions. The most widely used indirect method is to estimate extinction rates by reversing the species–area accumulation curve, extrapolating back- wards to smaller areas to calculate expected species loss. Estimates of extinction rates based on this method are almost always much higher than those actually observed2–5. This discrepancy gave rise to the concept of an ‘extinction debt’, referring to species ‘committed to extinction’ owing to habitat loss and reduced population size but not yet extinct during a non-equilibrium period6,7. Here we show that the extinction debt as currently defined is largely a sampling artefact due to an unrecognized difference between the underlying sampling problems when constructing a species–area relationship (SAR) and when extrapolating species extinction from habitat loss. The key mathematical result is that the area required to remove the last individual of a species (extinction) is larger, almost always much larger, than the sample area needed to encounter the first individual of a species, irrespective of species distribution and spatial scale. We illustrate these results with data from a global network of large, mapped forest plots and ranges of passerine bird species in the continental USA; and we show that overestimation can be greater than 160%. Although we conclude that extinctions caused by habitat loss require greater loss of habitat than previously thought, our results must not lead to complacency about extinction due to habitat loss, which is a real and growing threat. The Millennium Ecosystem Assessment1 predicts that near-term extinction rates could be as high as 1,000 to 10,000 times background rates (see also ref. 7). Most predictions of species extinction rates, including those in the Millennium Ecosystem Assessment, are inferred from applying the SAR to rates of habitat loss8–14. The wide discrep- ancy between the rates of species extinction predicted by this method and the extinction rates actually recorded, has fuelled a continuing debate about how to explain the discrepancy2,4,15–20. The main issue is that, almost always, more species are left after a given loss of habitat than the number of species predicted to remain, based on the SAR. The most frequent interpretation is that the excess species are ‘committed to extinction’. The term ‘extinction debt’ was coined to refer to species’ populations that were no longer viable but were facing certain extinc- tion due to habitat destruction that had already occurred3,6,17. The consensus on the most likely reason for the extinction debt is that there is a time lag for populations to go extinct after severe losses in population size6,21. Here we show that extinction rates estimated from the SAR are all overestimates. We define extinction rate as the fractional loss of species over a defined period accompanied by a given loss of habitat. These overestimates are due to the false assumption that the sampling problem for extinction is simply the reverse of the sampling problem for the SAR. The area that must be added to find the first individual of a species is in general much smaller than the area that must be removed to eliminate the last individual of a species (Fig. 1). Therefore, on average, it takes a much greater loss of area to cause the extinction of a species than it takes to add the species on first encounter, except in the degenerate case of a species having a single individual. We show mathematically that this is a necessary result of fundamental sampling differences between the SAR and the endemics–area relationship (EAR). Only in a very special and biologically unrealistic case, when all species are randomly and independently distributed in space, is it possible to derive the EAR from the SAR. Although this special case almost never occurs in nature, we examine this simple case first to clarify the nature of the problem. Then we relax these assumptions and consider the general case of aggregated species distributions. The problem has gone unnoticed for so long because the traditional method for estimating extinction uses the power-law SAR, S 5 cAz, which has no sampling theory relating it to species distributions (Supplementary Information A). To develop a sampling theory, we must consider the spatial distribution of species explicitly (Supplemen- tary Information B and C). We derive the SAR and EAR from nearest- neighbour distances under two situations, random dispersion and clumped dispersion. We construct an SAR from the probability of encountering the first nearest neighbour of a species (a new species is added every time the sampling frame a encounters the first indi- vidual of the given species). In contrast, we construct the EAR from the probability of encountering the last neighbour of a species (a species is added only after all individuals are contained within frame a). We arrive at the species–area curve for randomly and independently dis- tributed species as (Supplementary Information B): 1State Key Laboratory of Biocontrol and School of Life Sciences, Sun Yat-sen University, Guangzhou 510275, China. 2Department of Renewable Resources, University of Alberta, Edmonton, Alberta, T6G 2H1, Canada. 3Department of Ecology and Evolutionary Biology, University of California, Los Angeles, California 90095, USA. 4Center for Tropical Forest Science, Smithsonian Tropical Research Institute, Unit 0948, APO AA 34002-0948, Republic of Panama. Area of first encounter Area of last encounter Figure 1 | Sampling differences for SAR and EAR. Range distribution of a species (blue area), and an arbitrary starting sample point, indicated by 1. Regardless of the starting location, a sampling frame of arbitrary shape (here circular) with an area of a size sufficient to contact the species for the first time is always less than the sample area needed to encompass the entire range of the species. The SAR (species accumulation) is constructed from sample areas of first contact, and the EAR (species extinction) is constructed from areas of last contact. 3 6 8 | N A T U R E | V O L 4 7 3 | 1 9 M A Y 2 0 1 1 Macmillan Publishers Limited. All rights reserved©2011 www.nature.com/doifinder/10.1038/nature09985 S1a~S{ XS i~1 1{ a A � �Ni ð1Þ and the endemics–area curve as: SNa ~ XS i~1 a A � �Ni ð2Þ where Ni is the total abundance of species i and S is the total number of species in the region A. Equations (1) and (2), derived from nearest- neighbour distances, are identical to the classical random placement models22–25. Let the total area be A and let a sub-area a be lost. For randomly and independently distributed species, we can calculate the expected num- ber of species lost with a loss of area a from the SAR (equation (1)) as Sloss 5 S 2 SA 2 a. This is identical to the EAR calculated directly from equation (2): Sloss~S{S 1 A{a~ XS i~1 a A � �Ni ~SNa . This proves that, for the special case of species distributed randomly in space, extinction rates estimated from the backward random placement SAR and from the forward random placement EAR are the same, and the SAR and EAR are mirror images (Fig. 2 and Supplementary Fig. 1). This case is true because, under random placement, the total area A is equal to the sum of the areas of encountering the first individual and the last individual of a species. From the probability models of the nearest- neighbour distance, the expected area needed to sample the first indi- vidual is a1 5 A/(N 1 1), and the expected area for the last individual is aN 5 NA/(N 1 1) (Supplementary Information B). Thus a1 1aN 5 A. Note that aN . a1 is always true except when N 5 1. This mirror-image relationship only holds for randomly distri- buted species, however. Almost all species in nature are clumped, not randomly distributed26. For aggregated species, one can show that a1 1aN , A with aN $ a1 remaining true (Supplementary Information C and Supplementary Fig. 2). This leads to S{S1A{a=S N a . The more spatially aggregated species distributions are, the stronger the inequality aN $ a1 becomes. These results are completely general and explain the discrepancy between the backward SAR and forward EAR methods as well as why the backward SAR method systematically overestimates extinction rates. These results apply to sample areas on any spatial scale. We can assess the magnitude of overestimation by the backward SAR method precisely in cases where we know the species composition and spatial location of each individual of each species or spatial range of each species. To illustrate this, we use spatially explicit data from eight large stem-mapped plots from a global forest dynamics network. We also perform the analysis on biogeographical spatial scales for passerine species in the continental USA (see Methods). The results show that the classic power-law SAR model, S 5 cAz, and its corresponding EAR model (Supplementary Information A), l 5 Sloss/SA 5 1 – (1 – a/A) z (3) are not mirror-image curves. In equation (3), Sloss is the number of species lost (endemic) to destroyed sub-area a. Because of the differ- ence in sampling procedure of encountering species and losing species, the slopes z of the power-law model S 5 cAz and EAR (3) are not the same. The fit of the power-law SAR and EAR to species–area and endemics–area data respectively lead to two very different slopes 0 10 2 0 30 40 5 0 Yasuni Pasoh Changbai N um b er o f s p ec ie s N um b er o f s p ec ie s Korup Fushan Area AreaArea (ha) Area (ha) N um b er o f s p ec ie s 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 5 10 15 20 25 0.1 0.1 0.5 2.0 10 0.5 2.0 10 10 1,000 3,000 5 50 500 5,000 50 0.1 0.5 2.0 10 50 0.1 0.5 2.0 100 5 10 15 20 25 5×10–3 10–1 5 1,000 600 200 0 800 600 400 200 0 500 400 300 200 100 0 100 80 60 40 20 0 250 200 150 100 50 0 200 50 10 5 2 1 103 100 10–2 5×103 101 5×10–1 5×10–2 5×100 10–1 5×10–3 103 101 10–1 101 10–1 10–2 Figure 2 | Species– and endemics–area curves for six of the nine data sets in Table 1. The second and fourth columns are the plots on a log–log scale. The upper and lower