Analytical models for β-diversity and the power-law scaling of β-deviation

β-diversity is a primary biodiversity pattern for inferring community assembly. A randomized null model that generates a standardized β-deviation has been widely used for this purpose. However, the null model has been much debated and its application is limited to abundance data. Here we derive analytical models for β-diversity to address the debate, clarify the interpretation, and extend the application to occurrence data. The analytical analyses show unambiguously that the standardized β-deviation is a quantification of the effect size of non-random spatial distribution of species on β-diversity for a given species abundance distribution. It robustly scales with sampling effort following a power law with exponent of 0.5. This scaling relationship offers a simple method for comparing β-diversity of communities of different sizes. Assuming logseries distribution for the metacommunity species abundance distribution, our model allows for calculation of the standardized β-deviation using occurrence data plus a datum on the total abundance. Our theoretical model justifies and generalizes the use of the β null model for inferring community assembly rules.

expected for real ecosystems (Harte 2011, p150). This is because in real applications the size 1 8 0 of a local community is usually much smaller than the area of the metacommunity (M = 1 8 1 metacommunity area / local-community size). However, we would recommend using the 1 8 2 exact formulas in real applications, particularly when the sample size is small, since they 1 8 3 impose no computing challenge. However, the approximations offer analytical simplicity and 1 8 4 we will use them to derive the scaling relationship between β -deviation and the sampling  parameter p is fully determined by S and N allows β null to be parameterized using only the substituting equation (2) into The variance in equation (2b) is the expected conditional variance due to species spatial No variation due to SAD arises because in 1 9 7 Kraft et al.'s randomization approach the empirical SAD is used. However, in scientific 1 9 8 inference, the empirical SAD ought to be considered as a sample of the underlying theoretical be written as (Appendix S1): , where the subscript Φ indicates that the variation is due 2 0 3 to SAD. This leads naturally to a quantification of the variance of β -deviation due to the 2 0 4 metacommunity SAD: In reality, species almost always show aggregated instead of random spatial distribution 2 0 8 due to a variety of ecological processes such as dispersal limitation, habitat filtering, and Reed 2006; see Appendix S1 for the derivation): values of k and λ M; see Appendix S1). Like equation (2), the approximation here is valid Until now all the derivations in the above assume that the metacommunity is fully sampled.

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That is, all the M local communities (that comprise the metacommunity) are sampled. In real 2 3 5 applications we can only sample a fraction of a metacommunity and sampling intensity also 2 3 6 changes from study to study. This sampling incompleteness can have consequences. As shown in Bennet & Gilbert (2016), β -deviation is subject to the effect of sampling effort.

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Considering equation (5a), when only m (out of M) local communities are sampled, the 2 3 9 parameter k will not change but both S(m) and λ (m) will change with m. We used several well-studied data sets from the literature to evaluate the analytical results confirm the performance of our analytical models (2) and (4). This data set has been this assumption and to evaluate potential consequences due to violations of this assumption, a Kolmogorov-Smirnov goodness-of-fit test was performed for each of the metacommunities. With respect to the NBD model (equation 4), we tested its performance by plotting the following discussion, we will address and clarify the controversies surrounding the use of the 3 3 4 randomized β -deviation. Our discussion will make it clear that β -deviation is an important random spatial distribution of species clearly indicates that the randomization procedure of for a given SAD.

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The misinterpretation of β -deviation as "a standard effect size of β -diversity deviations concern about the null model and led to attempts to develop alternative null models or  in Appendix S1). It is also known that the parameter λ is a function of the ratio between total (2013) and Xu et al. (2015) analyzed the raw (i.e., β obsβ null ), instead of the standardized, 3 6 0 β -deviation. From our models (2) and (4), it is clear that the raw β -deviation is also dependent 3 6 1 on γ , thus not serving as a correction for the γ -dependence. A more fundamental problem with 3 6 2 the raw β -deviation is that it is not a measure of effect size and thus is of little use for "fixed-fixed" null model that preserves both row and column sums of the community matrix to correct for the γ -dependence. However, the "fixed-fixed" null model does not have an analytical solution and its ecological interpretation is not clear.

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Another major criticism on β -deviation is that it is sampling-effort dependent, meaning the application in estimating β -deviation from occurrence data could potentially be affected