Estimating the global number of tropical tree species, and Fisher's paradox.

How many tropical tree species are there in the world? Slik et al. (1) address this question in an elegant way, but this question is far harder than one might initially suppose. Significant uncertainties still remain, which I attribute in large part to an unsolved conundrum that I call “Fisher’s paradox.” Except for a few charismatic groups of organisms such as birds and butterflies, ecologists still do not know the numbers of species in most taxonomically or functionally defined groups, including tropical trees. Surveying herbaria and published species descriptions cannot tell us how many tropical tree species remain undiscovered and undescribed. Experienced tropical botanists and plant biogeographers may provide expert opinions on how many tropical tree species there are, but Slik et al. note that it is difficult to assess the accuracy of such opinions. Is there some way to come up with a more reliable estimate? Slik et al. think so and use a method first applied to estimating tree species richness in Amazonia (2, 3). The method involves extrapolating estimates of species richness to an entire biogeographic region from tree species inventories in many sample plots scattered across the region. The accuracy of this extrapolation relies on a curious stability property of Fisher’s logseries (4), a statistical distribution widely used to describe patterns of relative species abundance in ecological communities, including tropical tree communities. The stability property has to do with the relative stability of Fisher’s α, the diversity parameter of the logseries, in the face of changes in sample size. The fit of the logseries to relative tree species abundance data over large numbers of plots is often remarkably precise. Consider Fig. 1, which is a fit of the logseries to rank species abundance data for 4,962 tree species in 1,170 plots scattered across Amazonia (figure S6 in 3). On the y axis is the logarithm of species abundance, plotted against the rank in species abundance on the x axis, from the most to least abundant species, left to right. Slik et al. estimated species richness for trees having a trunk diameter of 10 cm or larger at breast height (DBH) in the three main tropical forest-containing biogeographic regions of the world: Central and South America, the tropical Indo-Pacific, and continental tropical Africa. Slik et al. computed their species richness estimates in three steps. First, they estimated the asymptotic value of Fisher’s α in each biogeographic region. Second, they estimated N, the total number of individual trees >10-cm DBH in each region. Finally, they calculated the number of species S in the region or world from the equation S= α lnð1+N=αÞ, the formula for the logseries that relates S to total number of individuals N and Fisher’s α (4). Slik et al. obtain minimum and maximum estimates for total tropical tree species richness in the world of 40,517 and 53,345, respectively, depending on uncertainties in estimating Fisher’s α and other estimation problems, such as unidentified species. However, there are other issues with the estimation, associated with Fisher’s paradox. To explain this paradox, it is useful to unpack the steps in the estimation procedure by explaining Fisher’s logseries in a bit more detail. Ecologists commonly write the logseries as φðnÞ= αðxn=nÞ, where φðnÞ is the number of species having n individuals in a sample, α is Fisher’s alpha, the diversity parameter, and x is a parameter very close to, but slightly less than, unity. Because x≈ 1, the logseries is almost a perfect hyperbolic function of n, φðnÞ≈ α=n. From this, we draw two conclusions. First, when n = 1, we see that φð1Þ≈ α, so Fisher’s α is the number of singleton species having a single individual in the sample. Second, we see that the singleton category is always the category with the most species, no matter how large a sample one takes because all abundance categories for n > 1 have fewer than α species by a factor 1/n. In big samples when n can be large, the frequency of species at any given abundance n is typically small because S << N, so instead of plotting φðnÞ versus n, ecologists plot the logarithm of the abundance of individual species on the y axis against species rank in abundance on the x axis. Fig. 1 is an example of such a rank abundance curve. At higher ranks after the most abundant species have been plotted, log species abundance becomes very nearly linearly and negatively related to species rank in species-rich samples. As sample sizes increase, the logseries retains the same linearity and it also retains very nearly the same position on the y axis, provided that the y axis is plotted as log of fractional or percentage relative abundance instead of the log of absolute abundance. As sample sizes increase, more and more rare species are added to the sample, and the rank abundance curve extends further down and to the right to ever rarer species. It is this invariance of the slope and y-axis position of the logseries rank abundance curve as one collects larger and larger samples, that allows ecologists to extrapolate 10000