Species Traits Shape the Relationship Between Local and Regional Species

Species traits shape the relationship between local and regional species abundance distributions 1,2 3,4 MURILO DANTAS DE MIRANDA, LUIS BORDA-DE-AGUA, 1,2,3,4, 5,6 HENRIQUE MIGUEL PEREIRA, AND THOMAS MERCKX

1Institute of Biology, Martin Luther University Halle-Wittenberg, Am Kirchtor 1, 06108 Halle (Saale), Germany 2German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Deutscher Platz 5e, 04103 Leipzig, Germany 3CIBIO/InBIO, Centro de Investigacß~ao em Biodiversidade e Recursos Geneticos, Universidade do Porto, Campus Agrario de Vair~ao, Rua Padre Armando Quintas, 4485-601 Vair~ao, Portugal 4CEABN/InBIO, Centro de Ecologia Aplicada “Professor Baeta Neves”, Instituto Superior de Agronomia, Universidade de Lisboa, Tapada da Ajuda, 1349-017 Lisboa, Portugal 5Behavioural Ecology and Conservation Group, Biodiversity Research Centre, Earth and Life Institute, UCLouvain, Croix du Sud 4-5, bte L7.07.04, BE-1348 Louvain-la-Neuve, Belgium 6Department of Ecology and Genetics, University of Oulu, 3000, FI-90014 Oulu, Finland

Citation: Dantas de Miranda, M., L. Borda-de-Agua, H. M. Pereira, and T. Merckx. 2019. Species traits shape the relationship between local and regional species abundance distributions. Ecosphere 10(5):e02750. 10.1002/ecs2.2750

Abstract. The species abundance distribution (SAD) depicts the relative abundance of species within a community, which is a key concept in ecology. Here, we test whether SADs are more likely to either follow a lognormal-like or follow a logseries-like distribution and how that may change with spatial scale. Our results show that the shape of SADs changes from logseries-like at small, plot scales to lognormal-like at large, land- scapescales.However,therateatwhichtheSAD’s shape changes also depends on species traits linked to the spatial distribution of individuals. Speciﬁcally, we show for oligophagous and small macro-moth species that a logseries distribution is more likely at small scales and a lognormal distribution is more likely at large scales, whereas the logseries distribution ﬁts well at both small and large scales for polyphagous and large species. We also show that SAD moments scale as power laws as a function of spatial scale, and we assess the performance of Tchebichef moments and polynomials to reconstruct SADs at the landscape scale from information at local scales. Overall, the method performed well and reproduced the shapes of the empirical distributions.

Key words: body size; countryside biogeography; host-plant specialization; Lepidoptera; moths; multi-habitat landscapes; raw moments; spatial scale; species abundance distribution; species traits; Tchebichef moments; Tchebichef polynomials.

Received 22 October 2018; revised 2 April 2019; accepted 3 April 2019. Corresponding Editor: Robert R. Parmenter. Copyright: © 2019 The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. E-mail: [email protected]

INTRODUCTION species can also be useful in applied ecology and biodiversity management (Matthews and Whit- The species abundance distribution (SAD) taker 2015). As such, both theoretical and empir- describes the relative abundance of species ical studies have examined the inﬂuence on within a community, which is a central concept SADs of several environmental and biological in ecology and essential for theories on biodiver- variables, such as elevational and latitudinal sity and biogeography (McGill et al. 2007, Mat- gradients, niche differentiation, dispersal, and thews and Whittaker 2014, Arellano et al. 2017). ecological disturbance (Whittaker 1975, Hub- Analyses of SADs that enable the identiﬁcation bell 1979, 2001, Magurran 2004, Arellano et al. of patterns in both commonness and rarity of 2017).

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Species abundance distributions—the distribu- others have focused on upscaling, trying to pre- tion of species into abundance classes—can take dict the regional SAD from SADs obtained at a variety of shapes. Fisher et al. (1943) first smaller scales using various methods, such as derived the logseries distribution to fit the SAD maximum-entropy and Bayesian methods of an abundance data set of Lepidoptera, thus (Magurran 2004, Harte et al. 2009, Zillio and He implicitly stating that it was a monotonically 2010). The method that we apply here, devel- decreasing distribution, with the maximum fre- oped by Borda-de-Agua et al. (2012), uses the quency for the singleton class. This was chal- raw moments of SADs (see Eq. 1 below) to lenged by Preston (1948, 1962) who pointed out describe the scaling properties of the SAD fol- that his data on bird species abundance led to a lowed by the application of the discrete Tchebi- SAD that had a bell-shaped curve with the maxi- chef moment and polynomial method to mum no longer occurring for the singleton class, reconstruct it at large scales. the rarest species. Furthermore, he pointed out The description of a SAD based on its that this maximum for intermediate classes was moments was first applied to data on tree and progressively revealed as more data were being shrub species in a 50-ha plot of tropical rainforest collected, that is, the concept of the veil line, in (Borda-de-Agua et al. 2012). By testing the the process revealing the tail of rare species. method within the 50-ha plot and then extrapo- Therefore, Preston clearly acknowledged that the lating the SAD for areas up to 5 km2, the authors shape of SADs was a function of the sample size, demonstrated the advantages of the Tchebichef and he suggested that SADs converge on a log- moment method over other upscaling methods. normal distribution as sample size increases. The The basic idea of this method is to describe the veiling line can arise for several reasons. One is SAD shape at different scales using the (raw) that some species may first go undetected and an moments of the SAD (the more the better): The increase in sampling effort may reveal more spe- first raw moment is the mean, the second raw cies; Chao et al. (2015) have suggested methods moment is related to the variance, and the third to deal with this issue. Another reason is that and fourth raw moments are related to the skew- species are not present everywhere, often show- ness and the kurtosis, respectively. When using ing some degree of aggregation, and hence, an this method, we are not concerned with the SAD increase in the size of the sampled area leads to at one spatial scale but, instead, with how the an increase in the number of species and, obvi- moments change as a function of the size of the ously, of individuals. In this paper, we deal solely area sampled and, especially, at the patterns with the latter issue. exhibited by the moments as a function of the The unveiling process results from increasing size of the area sampled. Previous work has sampling. Since collecting data to construct shown that the moments have a power law large-scale SADs is a time-consuming and eco- behavior as a function of area size (Borda-de-A nomically expensive endeavor, it remains elusive gua et al. 2012, 2017). If the moments exhibit a what the true shape of the SAD is for very large pattern, then their values can be extrapolated for samples, such as entire biogeographical (meta)- larger areas and the reconstruction of the SAD communities (but see ter Steege et al. 2006 and for hitherto unsampled area sizes. Different alter- Hubbell et al. 2008). It is a major challenge in natives exist for the reconstruction of a distribu- ecology to describe the regional SAD using sam- tion using (some) of its moments. Here, we use a ples characterized by a relatively small number method based on discrete Tchebichef moments of individuals or with data that cover only small and polynomials (Mukundan et al. 2001, Borda- areas relatively to the extent of the entire com- de-Agua et al. 2012). In summary, our approach munity. A key point that will move the study of consists of calculating the raw moments of the regional SADs forward is understanding how SADs at different scales, determining which ten- SADs in general are affected by spatial scale dencies the moments exhibit when plotted as a (McGill et al. 2007). Some studies have focused function of the sampled area, extrapolating the on downscaling (i.e., how species aggregation moments’ values for larger areas, and then using affects the shape of local SADs within a region; the discrete Tchebichef moments and polynomi- Dewdney 1998, Green and Plotkin 2007), while als to reconstruct the SAD.

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Actually, the above approach is closely We do so using a nested multi-site design in related to the one which is often adopted by three multi-habitat landscapes. Our focus is on ecologists studying the species–area relationship the species-rich group of macro-moths, which (SAR). These studies typically focus on how the allows us to test the effect of two traits linked to number of species changes as a function of spatial distribution: body size and host-plant area, a relationship that is well characterized by specialization. a power law. While the SAR relates to one single variable—the number of species—and is METHODS described by a single curve, for the description of the evolution of the SAD across scales, we Study area and sampling need to describe the evolution of a distribution We gathered species abundance data on function. Because a distribution is characterized macro-moths in the Castro Laboreiro area (ap- by its moments, such as the mean and variance, proximately 42°20 N, 8°100 W) within the we need a curve to describe the evolution of Peneda-Gerês National Park, NW Portugal each moment: one to describe the evolution of (Fig. 1). The landscape within our study area its mean, another to describe the evolution of consists mainly of scrub (78.4%), forest (10.5%), its variance, and so on. Thus, we use several and meadows (9.8%). Sampling was conducted raw moments, and in analogy with SAR stud- in three 1.6-km2 multi-habitat landscapes that ies, we plot the raw moments of the SAD as a differed in dominant habitat type: a meadow- function of area. Our emphasis is on how the dominated, scrub-dominated, and forest-domi- SAD changes as a function of area and not on nated landscape. For each landscape, 28 fixed which distribution gives the best fit of the SAD light-trap sites were selected using a semi- at all scales. Because moments describe any dis- nested design with four levels, each corre- tribution, our approach provides a continuous sponding to a different spatial scale (Proencßa description of the progression of any empirical and Pereira 2013; Fig. 1). This sampling design SAD across different area sizes. follows an almost fractal arrangement: There is Here, we test whether SADs are more likely to a trap-level scale of ~20 9 20 m (i.e., based on follow lognormal-like or logseries-like distribu- an attraction-to-light radius of ~10 m; Merckx tions, and whether this is affected by scale. By and Slade 2014; then there are four of these logseries-like, we mean distributions that are samples forming a 80 9 80 m2, i.e., 4 9 20, monotonically decreasing functions, and by log- which is nested in a 320 9 320 m2, i.e., 4 9 80, normal-like, we mean distributions that when and this arrangement is then embedded in a plotted on a logarithmic scale exhibit a maxi- 1280 9 1280 m2, i.e., 4 9 320; Fig. 1). Hence- mum for intermediate classes and are approxi- forth, expressions such as “we sampled (extrap- mately bell-shaped; hereafter, we refer to these olated to) an area of size X” are shorthand for distributions as logseries and lognormal, respec- “we sampled (extrapolated to) a set of plots tively. The logseries SAD implies that most indi- with total area corresponding to the total area viduals belong to a few species and that most of the sampling traps, following a design as species are represented by a few individuals, shown in Fig. 1, and whose corners form a whereas the lognormal SAD implies a higher square of area X.” number of species with intermediate abundance During both 2011 and 2012, macro-moths were and smaller numbers of rare and abundant spe- light-trapped between May and September, cies (Magurran 2004). Also, since some species using identical equipment (i.e., heath pattern 6W traits (e.g., mobility, niche preference) are linked actinic light traps). Each of the 84 sites was sam- to the spatial distribution of individuals, we test pled three times per year, resulting in a total of whether and how such traits affect the scale six samples per site over both years. Per sam- dependency of SADs. Moreover, we analyze how pling night seven sites were simultaneously sam- SAD moments relate to spatial scale, and we pled so that they were sampled under identical assess the performance of the Tchebichef weather conditions, covering three spatial scales: moments and polynomials to reconstruct SADs (1) trap-level scale, (2) 80 9 80 m, and (3) from the local plot scale to the landscape scale. 320 9 320 m (Fig. 1). Moreover, all 84 sites (i.e.,

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NP Peneda-Gerês

Meadow-dominated

Scrub-dominated

Castro *Laboreiro

Forest-dominated 1280 m 1280 m

042 Km

Habitat types Sampling site Forest Scrub Meadow

Fig. 1. Location of the study area in Peneda-Gerês National Park, Northern Portugal. Eighty-four fixed light- trap sampling sites (~400 m2) were part of a semi-nested (80 9 80 m, 320 9 320 m) sampling design in three multi-habitat study landscapes (1280 9 1280 m). twelve sampling nights) were sampled in as weather conditions favorable for moth flight short a period as possible so as to avoid seasonal activity (i.e., minimum night temperature >10°C; differences in species composition, while the maximum wind speed <20 km/h; no persistent overnight sampling only took place during rain; see Merckx et al. 2012a, b).

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Species abundance distribution models with average wingspan smaller than the overall For each sampling site, which corresponds to median (32.5 mm) were classified as small, the the trap-level scale, macro-moth abundance data others as large. Host-plant specialization was were lumped and then fitted to histogram-type classified into two classes: oligophagous spe- data with logseries and truncated lognormal dis- cies, whose larvae only feed on plant species tributions using maximum likelihood. For the from the same family, and polyphagous species, next spatial scale (i.e., 80 9 80 m), we lumped whose larvae are able to feed on several plant all sampling sites within these areas and fitted families. Data on host-plant specialization were logseries and lognormal distributions for each obtained from the same sources as above. An area. This procedure was repeated three more overview of all species classifications can be times for the other spatial scales (i.e., found in Appendix S1: Table S1. All statistical 320 9 320 m, 1280 9 1280 m, and the sum of analyses were run in the statistical software the three 1280 9 1280 m landscapes). Because environment R version 3.1.1 with packages we were merely interested in assessing the rela- mass and sads (R Development Core Team tive quality of fit among the logseries and lognor- 2014, Prado et al. 2018). mal distribution across scales, we used corrected Akaike information criteria (AICc) in order to SAD raw moments, Tchebichef moments, and select the best-fitting model. Specifically, we cal- Tchebichef polynomials culated the difference between the AICc values In principle, a probability distribution can be Δ ( AICc) corresponding to the lognormal and the reconstructed directly from its moments (Borda- logseries distributions, using the threshold of de-Agua et al. 2012). However, in practice this is Δ >| | AICc 2 to establish whether both distribu- not viable because of the large number of tions are significantly different (Burnham and moments required and the numerical instabilities Anderson 2002, Slik et al. 2015). Accordingly, the associated with the high order moments. Thus, lognormal SAD model is considered to provide a other methods have to be sought (Mukundan fi Δ ≤ Δ ≥ better t when AICc 2 while AICc 2 et al. 2001). Following Borda-de-Agua et al. indicates a better fit for the logseries model. (2012), we reconstructed the probability density < Δ < Models characterized by 2 AICc 2 were function using the estimated discrete Tchebichef classified as intermediate ones, with both the log- moments and polynomials. Tchebichef moments normal and logseries having a similar level of are a general non-parametric tool to describe the support. We then created ordinal logistic regres- shape of any distribution. The idea is to fitahis- sion models (Guenette and Villard 2004, Ruther- togram, f(x)withasumofTchebichefpolynomials, ~ ford et al. 2007) to test the relation between tnðxÞ, weighed by the corresponding Tchebichef spatial scale (ln (area)) and the probability for moments, Tn, according to the formula these lognormal, intermediate, and logseries NX1 SADs. Next, we evaluated the goodness of fit ~ f ðxÞ¼ TntnðxÞ and deviance. The strength of the association n¼0 ’ 2 (McFadden s R ) was calculated as 1 (Lmod/ In principle, we need as many Tchebichef Lnull), where Lmod is the log-likelihood value for fi moments as the number of bins of the histogram. the tted model and Lnull is the log-likelihood for the null model which includes only an intercept. Further information about discrete Tchebichef moments and polynomials can be found in Ordinal logistic regression models too were used to test how species traits (body size and Mukundan et al. (2001) and Borda-de-Agua host-plant specialization) affect the SAD shape et al. (2012). probability (three classes: lognormal, intermedi- The Tchebichef moments are related to the raw ate, and logseries). Species-specific average moments (Appendix S3). The raw moment of wingspan (mm) was used as a proxy for body order n of the SAD from a given community is size (Hamback€ et al. 2007). Mean values were calculated as XS obtained from www.lepidoptera.eu, topped up 1 M ¼ xn (1) with values from Manley (2009) for a few spe- n S j cies where information was missing. Species j¼1

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9 where S is the number of species, and xj is the scales (i.e., 1280 1280 m or the individual land- log2-transformed number of individuals of species scapes, and the sum of the three landscapes) using j. For each spatial scale, we calculated the average the moments obtained at the trap level, of the moments obtained from each set of lumped 80 9 80 m and 320 9 320 m scales; that is, we sampling sites and then plotted this average as a used the smallest three scales of the sampling function of the corresponding spatial scale. As design to extrapolate the SADs (1) to the full data mentioned previously, the number of Tchebichef for each landscape and (2) to the entire data set. moments is limited to the number of bins of the We used chi-square tests to compare the extrapo- histogram. Specifically, if the histogram has N lated and observed SADs on histograms and did bins, then the maximum number of Tchebichef this separately for each landscape as well as for moments is N and it requires the same number of the sum of the three landscapes. raw moments (Mukundan et al. 2001, Borda-de- Agua et al. 2012). Although in our analysis the RESULTS number of moments varied between 0 and 11— because there were 12 bins in the histogram of all In total, we collected 22,825 individuals landscapes—we limited the number of moments belonging to 378 species. Most species belonged to 9, both separately for each landscape and for all to two families: Noctuidae (39.4%) and landscapes together, as the extrapolations Geometridae (38.9%). The two most abundant appeared sensitive for the highest two moments. species were the noctuid Xestia agathina and the In practice, we observe that the calculation of geometrid Pachycnemia hippocastanaria, compris- higher moments has numerical instabilities, ing 8.2% and 6.9% of all individuals collected, because of the very large numbers that result from respectively, 15.3% of the species were singletons, the large exponents involved in the computation and 38.3% had five individuals or less. of higher order moments (Mukundan 2004). Thus, For all macro-moth species, ordinal logistic the number of moments used has to be chosen regression showed that the probabilities of moving judiciously, often by trial and error, and currently, from a logseries to intermediate/lognormal distri- there are no rules to determine the number of bution (or from logseries/intermediate to lognor- moments to be used. However, these numerical mal) increased as ln-area increases (Fig. 2). This instabilities are very noticeable, leading to distri- means that the probability of being logseries dis- butions with large oscillations with negative val- tributed is highest at small areas and that the ues.Inordertobetterquantifythebestnumberof probability of being lognormal distributed is high- moments to use, we determined the number of est at large areas (ln-area = 0.395; t = 4.436; P < moments that minimizes the sum of the difference 0.0001; Fig. 2; Appendix S2: Fig. S1; Appendix S1: in the histogram of the real data and that obtained Table S3). with the Tchebichef method. Our results also show the importance of host- 2 ¼ : 2 ¼ For each site, data were lumped. We calculated plant specialization (ROligophagy 0 14; RPolyphagy : 2 ¼ : 2 ¼ : the moments for each sampling site (Eq. 1), and 0 00) and body size (RSmall 0 27; RLarge 0 01), then, for each moment, we calculated the average with the difference in R2 values indicating a fitof from all sampling sites. For the next spatial scale the model only for oligophagous and small spe- (i.e., 80 9 80 m), we lumped all sampling sites cies. This shows that for oligophagous and small within such areas, calculated the moments for species, a logseries distribution is more likely at each area, and for each moment we then averaged small areas, and a lognormal distribution is more = the values across all these areas. This procedure likely at large areas (AreaOligophagy 0.412; = < = = was repeated one more time for the next spatial t 4.463; P 0.0001; AreaSmall 0.717; t 5.786; scale (i.e., 320 9 320 m). Next, we fitted a linear P < 0.0001). For polyphagous and large body size regression in order to assess the relationship species groups, the logseries distribution gener- between the ln-transformed area and the ln- ally provided the best relative fitatbothsmall transformed moment, lnðMnðAÞÞ ¼ an þ bn lnðAÞ, and large spatial scales (Fig. 2; Appendix S1: where an and bn are parameters estimated from Table S3). the regression of that particular moment. Then, Plotting the ln-transformed moments of order we extrapolated the moments for the largest two 1–9 as a function of the ln-transformed area

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All species (R2 = 0.16*) Oligophagous species (R2 = 0.14*) Polyphagous species (R2 = 0.00) 1.0 0.8 0.6 0.4

Probability 0.2 0.0 5e2 5e3 5e4 5e5 5e6 5e2 5e3 5e4 5e5 5e6 5e2 5e3 5e4 5e5 5e6 ln(Area (m2)) Small species (R2 = 0.27*) Large species (R2 = 0.01) 1.0 0.8 0.6 lognormal probability 0.4

intermediate probability Probability 0.2 log-series probability 0.0 5e2 5e3 5e4 5e5 5e6 5e2 5e3 5e4 5e5 5e6 ln(Area (m2)) ln(Area (m2))

Fig. 2. Probabilities for three types of species abundance distribution (SAD)—lognormal, logseries, and an intermediate combination of both (2 < ΔAIC < 2)—as a function of area (ln m2). Panels depict relationships for all macro-moth species as well as for separate species groups, contrasting oligophagous versus polyphagous species, and small versus large species. Solid, dotted, and dashed lines represent lognormal, intermediate, and logseries SADs, respectively, based on ordinal logistic regression model output. R2 values and significance levels of the probabilities when changing distribution as function of ln-area (P < 0.05) are given for each panel. shows that there is an almost linear relationship from smaller spatial scales and whether spatial (Fig. 3; Appendix S1: Table S2), similar to previ- scale and species traits affect the shape of SADs. ous findings (Borda-de-Agua et al. 2012, 2017). We showed that the shape of the SADs changes We used this relationship to extrapolate the SAD across spatial scales, but we also showed that this to larger scales. For the meadow-, forest-, and is less noticeable for polyphagous and large spe- scrub-dominated landscapes, the SAD was cies. extrapolated to an area of 1280 9 1280 m A simple qualitative model helps interpret our (Fig. 1). Next, the SAD was also extrapolated to results for the different groups of species. Imag- the sum of all three landscapes combined. Con- ine two communities with exactly the same num- sistently, the predicted extrapolated curves do ber of species and relative abundance, or in other not statistically differ from the empirical distribu- words with the total number of individuals tions (extrapolated SADs: gray lines in Fig. 4, occurring at the same density. In one of the com- v2 ¼ = Appendix S2: Figs. S2, S3; Meadow 120, df munities, individuals of the same species tend to = v2 ¼ = = 110, P 0.24; Scrub 108, df 99, P 0.25; be strongly aggregated, while in the other com- v2 ¼ = = v2 ¼ Forest 132, df 121, P 0.23; Three landscapes 108, munity individuals of different species are spa- df = 99, P = 0.25). Also, note that the distribu- tially well mixed. We do not invoke any tions at smaller spatial scales, which contain the particular reason for the level of clustering of information used to forecast, are very different both communities, but it could be due to disper- from the extrapolated distributions (Appendix S2: sal (Borda-de-Agua et al. 2017), which is likely Fig. S1). the case for small versus large-bodied macro- moth species, or it could be due to niche prefer- DISCUSSION ences, which is likely the case for oligophagous moths. For small sample sizes, if one increases The main goals of this study were to assess the area sampled in the aggregated community, how well Tchebichef moments and polynomials we find a small number of species, but each rep- are able to predict the regional SAD using SADs resented by several individuals. Furthermore,

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Fig. 3. Plots of ln-transformed moments (1–9) of the species abundance distribution as a function of spatial scale (ln m2). Panels depict relationships for all macro-moths as well as for separate species groups, contrasting oligophagous versus polyphagous species, and small versus large species. Dashed lines represent the best fitof linear regression models. when we increase the sample size one tends to sample size for the SAD to finally start develop- find more individuals of the same species. Thus, ing a maximum for intermediate abundance as we increase the sampling scale, the SAD classes; Borda-de-Agua et al. (2007) performed quickly develops a maximum for intermediate simulations that illustrate these behaviors. abundance classes, although we should obvi- In line with our previous explanations, com- ously not exclude the presence of some rare spe- plexity arises as different species traits interact to cies in the sample. On the other hand, if one determine SADs (Gaston et al. 2000). For increases sample size in the mixed community instance, although polyphagous moth species from small to intermediate sample sizes, we tend typically have wider distribution ranges than oli- to find more species, but all with small abun- gophagous species (Spitzer et al. 1984, Spitzer dances because species are well mixed, the SAD and Leps 1988, Quinn et al. 1997), the latter are classes corresponding to rare species keep also typically smaller than polyphagous species. increasing, and the maximum for intermediate Also, large moth species tend to occur at lower classes only occurs for larger sample sizes than it densities at the local scale than small species does for the case of the aggregated community. (Peters 1986, Nieminen et al. 1999). Moreover, Hence, most species tend to remain rare and the large macro-moth species (e.g., noctuids) are typ- € SAD classes corresponding to rare species keep ically relatively mobile (Ockinger et al. 2010, increasing. This trend will eventually stop Sekar 2012, Slade et al. 2013). This agrees with because the rarest species will accumulate more our previous simple model and it may explain individuals and not many more rare species will why the SADs of small species followed the log- be found. Thus, overall, most species will have series distribution at the local scale, and changed intermediate abundances, but it requires a larger to the lognormal distribution at larger scales,

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Fig. 4. Species abundance distributions (SAD) for all macro-moth species, both separated according to landscape type (forest-, scrub-, and meadow-dominated), and for all three landscapes combined. Histograms represent the observed data at these spatial scales while gray lines represent the number of species as predicted by the moments (n = 9). For the meadow-, forest-, and scrub-dominated landscapes, the SAD (gray line) was extrapolated to an area of 1280 9 1280 m and to an area that summed these three landscapes (Fig. 1). while the SADs for the large moth species fol- showed that evenly spaced species are rare in lowed the logseries distribution at all spatial abundance and dominate the rare mode of the scales. As we mentioned before, the higher dis- SAD, while clustered species dominate the persal ability of large species tends to homoge- intermediate mode. nize them in the community, making the SAD Some studies have shown that plant species steeper and placing the mode at the rare species. distributions are important predictors of moth On the other hand, small species tend to be abundance, diversity, and distribution (Kitching more aggregated, and hence more common at a et al. 2000, Hilt and Fiedler 2006, Novotny et al. given spatial scale, due to their low to interme- 2006). If host plants are generally characterized diate mobility levels (Nieminen et al. 1999, by a patchy distribution, we would expect that Borda-de-Agua et al. 2017). As a consequence, oligophagous species are more aggregated than theSADsofthisspeciesgroupshowalognor- polyphagous ones (Lindstrom€ et al. 1994). This mal shape at larger spatial scales. Likewise, interpretation would explain why the shape of Mouquet and Loreau (2003) observed that spe- the SADs for oligophagous species also changed cies rank-abundance distributions were strongly from logseries at a local scale to lognormal at lar- affected by the level of dispersal between spatial ger scales, while no change was observed for scales. Dornelas and Connolly (2008) also polyphagous species.

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The capability of forecasting the SAD for larger CONCLUSIONS areas than those sampled is one of the most inter- esting features of our method. There are, how- We show that the shape of SADs generally ever, two important assumptions that need to be changes across spatial scales and that species met (1) the correct identification of the pattern traits are able to affect the rate of this change; associated with the moments within the sampled small species are characterized by SADs that spatial scales and (2) this pattern needs to remain change more rapidly than those of large species, valid for the extrapolated areas. Here, and con- while SADs of polyphagous species do not exhi- trary to previous applications (Borda-de-Agua bit any change contrary to those of oligophagous et al. 2012, 2017), we only have a small number species. Although studying species richness of area sizes as a base to fit the pattern. However, across spatial scales—SARs—is common practice because the sampling design follows a fractal (Rosenzweig 1995), our findings demonstrated pattern, it is ideal to assess the patterns on a loga- the importance of considering relative species rithmic scale and, in particular, to identify possi- abundance via the SAD across spatial scales. ble power laws. Note nevertheless that the Indeed, our approach is similar to that used by method also works when the sampling scheme is ecologists when analyzing species richness not fractal, such as the scheme used in the paper across spatial scales, but we added the analysis where the method was originally developed of species abundance across scales. We predict (Borda-de-Agua et al. 2012) or in Borda-de-Agua this analysis to be important both from a theoret- et al. (2017). Although we recognize that we ical perspective, thanks to the patterns it reveals, used only three points based on which we fitted and from a practical perspective, thanks to possi- the moments, they nevertheless follow approxi- ble applications to conservation and manage- mately straight lines for several orders of magni- ment of species. tude. More work, both empirical and theoretical, needs to be carried out in order to assess the gen- ACKNOWLEDGMENTS erality of this pattern across taxa and across space. With regard to the second assumption, the We thank Gabriel Arellano and Tom Matthews for pattern remains valid for the extrapolated areas. their constructive comments. The Brazilian National Counsel of Technological and Scientific Development However, extrapolating for even larger areas fi should be done with care given the uncertainty CNPq provided nancial support to Murilo Dantas de Miranda (process 237206/2012-9) while the Portuguese on the extrapolation of the coefficients of the Science Foundation FCT provided financial support to Tchebichef moments and the increasing habitat Thomas Merckx (postdoc grant SFRH/BPD/74393/ heterogeneity at larger scales. Indeed, although 2010). Luıs Borda-de-Agua was funded by the FCT our method partially reveals the inherent project MOMENTOS (PTDC/BIA-BIC/5558/2014). Per- heterogeneity in the landscape, at very large mits were granted to carry out light trapping within scales, we risk combining very different commu- the Peneda-Gerês National Park. nities which may translate into sudden changes in the scaling patterns of the moments or of the LITERATURE CITED SAR. Overall, we believe that our results are in Arellano, G., M. N. Umana,~ M. J. Macıa, M. I. Loza, A. agreement with Preston’s veil line concept, and Fuentes, V. Cala, and P. M. Jørgensen. 2017. The for those groups studied here that did not exhibit role of niche overlap, environmental heterogeneity, landscape roughness and productivity in shaping it, the reason is likely to be due to paucity of species abundance distributions along the Ama- data. However, we do not assume that the distri- zon-Andes gradient. Global Ecology and Biogeog- butions are converging to a lognormal distribu- raphy 26:191–202. tion. The reason is that empirical (ter Steege et al. Borda-de-Agua, L., P. A. Borges, S. P. Hubbell, and H. 2006), simulation (Borda-de-Agua et al. 2007), M. Pereira. 2012. Spatial scaling of species abun- and theoretical (Hubbell 2001) results have dance distributions. Ecography 35:549–556. hinted at the possibility of the distributions Borda-de-Agua, L., S. P. Hubbell, and F. He. 2007. Scal- reverting to a logseries type of curve for very ing biodiversity under neutrality. Pages 347–375 in large areas. D. Storch, P. A. Marquet and J. H. Brown, editors.

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