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. Hu
ell3,4
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
eversing 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’, refe
ing to species ‘committed
to extinction’ owing to habitat loss and reduced population size but
not yet extinct during a non-equili
ium period6,7. Here we show
that the extinction debt as cu
ently 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, i
espective 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 greate
than 160%. Although we conclude that extinctions caused by habitat
loss require greater loss of habitat than previously thought, ou
esults 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
ates (see also ref. 7). Most predictions of species extinction rates,
including those in the Millennium Ecosystem Assessment, are infe
ed
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 occu
ed3,6,17. 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
a
ive 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 XXXXXXXXXX, Republic of Panama.
Area of first
encounte
Area of last
encounte
Figure 1 | Sampling differences for SAR and EAR. Range distribution of a
species (blue area), and an a
itrary starting sample point, indicated by 1.
Regardless of the starting location, a sampling frame of a
itrary 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.
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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-
er 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, fo
the special case of species distributed randomly in space, extinction
ates estimated from the backward random placement SAR and from
the forward random placement EAR are the same, and the SAR and
EAR are mi
or 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 mi
or-image relationship only holds for randomly distri-
uted species, however. Almost all species in nature are clumped, not
andomly 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 co
esponding EAR
model (Supplementary Information A),
l 5 Sloss/SA 5 1 – (1 – a/A)
z (3)
are not mi
or-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
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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