Rarefaction and extrapolation of species richness using an area‐based Fisher's logseries

Abstract

Fisher's logseries is widely used to characterize species abundance pattern, and some previous studies used it to predict species richness. However, this model, derived from the negative binomial model, degenerates at the zero‐abundance point (i.e., its probability mass fully concentrates at zero abundance, leading to an odd situation that no species can occur in the studied sample). Moreover, it is not directly related to the sampling area size. In this sense, the original Fisher's alpha (correspondingly, species richness) is incomparable among ecological communities with varying area sizes. To overcome these limitations, we developed a novel area‐based logseries model that can account for the compounding effect of the sampling area. The new model can be used to conduct area‐based rarefaction and extrapolation of species richness, with the advantage of accurately predicting species richness in a large region that has an area size being hundreds or thousands of times larger than that of a locally observed sample, provided that data follow the proposed model. The power of our proposed model has been validated by extensive numerical simulations and empirically tested through tree species richness extrapolation and interpolation in Brazilian Atlantic forests. Our parametric model is data parsimonious as it is still applicable when only the information on species number, community size, or the numbers of singleton and doubleton species in the local sample is available. Notably, in comparison with the original Fisher's method, our area‐based model can provide asymptotically unbiased variance estimation (therefore correct 95% confidence interval) for species richness. In conclusion, the proposed area‐based Fisher's logseries model can be of broad applications with clear and proper statistical background. Particularly, it is very suitable for being applied to hyperdiverse ecological assemblages in which nonparametric richness estimators were found to greatly underestimate species richness.

The Full Hierarchical Structuration of Species Abundances Reliably Inferred from the Numerical Extrapolation of Still Partial Samplings: A Case Study with Marine Snail Communities in Mannar Gulf (India)

“…The following expression stands for the estimated abundance, a i , of the unrecorded species of rank i (thus for i > R 0 ): Besides, it is easy to verify that another consequence of these preferred ranges is that the selected estimator will always provide the highest estimate, as compared to the other estimators. Interestingly, this mathematical consequence, of general relevance, is in line with the already admitted opinion that all non-parametric estimators provide under-estimates of the true number of missing species [8,9,[54][55][56]. Also, this shows that the approach initially proposed in [57] -which has regrettably suffered from its somewhat difficult implementation in practice -might be advantageously reconsidered, now, in light of the very simple selection key above, of far much easier practical use.…”

“…More generally, sampling incompleteness is doomed to become the unavoidable consequence of the generalization of "rapid assessments" and "quick surveys", at least when having to deal with species-rich communities, as is usually the case with terrestrial or marine invertebrates faunas, especially under tropical climate. In turn, such partial inventories prevent a reliable appreciation of the key descriptive and functional aspects of the internal organization within species communities [7][8][9]. Hence, the recently acknowledged necessity to implement relevant numerical extrapolations applied to both: (i) the as-recorded species accumulation process, beyond the already reached sampling-size and (ii) the as-recorded distribution of species abundances in the studied community.…”

Analysing the Role of Environmental Stresses on Species Richness and the Process of Hierarchical Structuring of Species Abundances in Marine Gastropods communities at Suva (Fiji Islands)

“…Now, the usually high diversity of marine gastropod communities -which precisely makes them especially attractive for such studiesinevitably leads to the need to address a methodological issue which cannot be ruled out: the difficulty and often the virtual impossibility to complete samplings so as to reach (or at least to closely approach) exhaustive inventories. And this, in turn, can result in severely unreliable inferences, since sampling incompleteness not only delivers undetermined underestimates of the true species richness but also hampers any comprehensive and unbiased approach to the hierarchical structuration of species abundances within communities [6][7][8][9]. Hence, the necessity to implement a reliable procedure of numerical extrapolation of partial sampling [10] able to provide estimates with minimised bias of (i) the number of the still unrecorded species and (ii) the distribution of abundance of these unrecorded species.…”

Extinction debt in local habitats: quantifying the roles of random drift, immigration and emigration

“…Suppose in the initial stage, there is a local area of size a that is part of the region (a , A), and the local species abundance distribution (SAD) is given by S j (0|a), which represents the number of species with abundance j at the initial time in local area a. Accordingly, the expected total biodiversity loss at time t at the local level, under random drift, immigration and emigration processes, can be estimated as To better incorporate instant habitat destruction on the transition behaviour of the species richness with Fisher's explicit statistical background, and accordingly, estimate the expected species loss over time, we assume there are Sð0 j AÞ ¼ 1000 species at the initial time point t = 0 at the regional scale; and the SAD at either the regional or local scale follows an area-based logseries abundance model [22]. To be specific, the area-based model is given as follows.…”

“…Detailed derivation of equation (2.15) can be found in our previous paper [22] and is thus omitted here. Therefore, the expected species richness with abundance j in the local ecological community a in the entire region A, given the parameter value α A , is given by S j ð0 j a,a a Þ ¼ Sð0 j AÞ Â f( j j a a ,a A ):…”