1 TITLE: Dispersal ability determines the scaling properties of species abundance 2 distributions: a case study using arthropods from the Azores 3 4 Luís Borda-de-Água1,2*, Robert J. Whittaker3,4, Pedro Cardoso5,6, François Rigal6,7, 5 Ana M. C. Santos6,8,9, Isabel R. Amorim6, Aristeidis Parmakelis6,10, Kostas A. 6 Triantis6,10, Henrique M. Pereira1,2,11,12 & Paulo A. V. Borges6 7 8 1 Theoretical Ecology and Biodiversity, and Infraestruturas de Portugal Biodiversity 9 Chair, CIBIO/InBio, Centro de Investigação em Biodiversidade e Recursos 10 Genéticos, Laboratório Associado, Universidade do Porto. Campus Agrário de 11 Vairão, 4485-661 Vairão, Portugal. 12 2 CEABN/InBio, Centro de Ecologia Aplicada “Professor Baeta Neves”, Instituto 13 Superior de Agronomia, Universidade de Lisboa, Tapada da Ajuda, 1349-017 Lisboa, 14 Portugal. 15 3School of Geography and the Environment, University of Oxford, South Parks Rd, 16 Oxford OX1 3QY, UK. 17 4Center for Macroecology, Evolution and Climate, Natural History Museum of 18 Denmark, University of Copenhagen, Copenhagen, Denmark. 19 5Finnish Museum of Natural History, University of Helsinki, Helsinki, Finland. 20 6cE3c – Centre for Ecology, Evolution and Environmental Changes / Azorean 21 Biodiversity Group and Universidade dos Açores - Departamento de Ciências 22 Agrárias, Rua Capitão João d’Ávila, São Pedro, 9700-042 Angra do Heroísmo, 23 Terceira, Azores, Portugal. 24 7Environment and Microbiology Team, Université de Pau et des Pays de l’Adour, 25 IPREM UMR CNRS 5254, Pau Cedex, France. 1 26 8Departament of Biogeography and Global Change, Museo Nacional de Ciencias 27 Naturales (MNCN-CSIC), 28006 Madrid, Spain. 28 9Forest Ecology & Restoration Group, Department of Life Sciences, Universidad de 29 Alcalá. 28805 Alcalá de Henares, Madrid, Spain. 30 10Department of Ecology and Taxonomy, Faculty of Biology, National and 31 Kapodistrian, University of Athens, Athens GR-15784, Greece. 32 11German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig 33 Deutscher Platz 5e, 04103 Leipzig, Germany 34 12Institute of Biology, Martin Luther University Halle-Wittenberg, Am Kirchtor 1, 35 06108 Halle (Saale), Germany. 36 37 Emails: 38 Luís Borda-de-Água: [email protected] 39 Robert J. Whittaker: [email protected] 40 Pedro Cardoso: pedro.cardoso@ helsinki.fi 41 François Rigal: [email protected] 42 Ana M. C. Santos: [email protected] 43 Isabel R. Amorim: [email protected] 44 Aris Parmakelis: [email protected] 45 Kostas A. Triantis: [email protected] 46 Henrique M. Pereira: [email protected] 47 Paulo A. V. Borges: [email protected] 48 49 50 * Corresponding author: Luís Borda-de-Água (Email: [email protected]) 2 51 ABSTRACT 52 53 Species abundance distributions (SAD) are central to the description of diversity and 54 have played a major role in the development of theories of biodiversity and 55 biogeography. However, most work on species abundance distributions has focused 56 on one single spatial scale. Here we used data on arthropods to test predictions 57 obtained with computer simulations on whether dispersal ability influences the rate of 58 change of SADs as a function of sample size. To characterize the change of the shape 59 of the SADs we use the moments of the distributions: the skewness and the raw 60 moments. In agreement with computer simulations, low dispersal ability species 61 generate a hump for intermediate abundance classes earlier than the distributions of 62 high dispersal ability species. Importantly, when plotted as function of sample size, 63 the raw moments of the SADs of arthropods have a power law pattern similar to that 64 observed for the SAD of tropical tree species, thus we conjecture that this might be a 65 general pattern in ecology. The existence of this pattern allows us to extrapolate the 66 moments and thus reconstruct the SAD for larger sample sizes using a procedure 67 borrowed from the field of image analysis based on scaled discrete Tchebichef 68 moments and polynomials. 69 70 Keywords 71 arthropods, Azores, Laurisilva forest, Macaronesia, macroecology, moments, 72 scaling, skewness, species abundance distribution, Tchebichef moments and 73 polynomials 74 75 3 76 77 INTRODUCTION 78 79 The number of species and their relative abundance are important components of 80 species diversity and community structure1-3. A key ecological question concerning 81 species richness patterns is how the number of species scales as a function of area, i.e. 82 the form taken by the species–area relationship (SAR). This is a well studied pattern4. 83 However, and in clear contrast, there has been much less attention to how the 84 abundance of species scales with area5, although this is a fundamental question in 85 understanding how patterns of community assembly emerge and their consequences 86 for ecosystem functioning. Here we examine the role of dispersal in determining how 87 the relative abundance of species changes with increasing spatial scale, with a view to 88 enabling the projection of abundance distributions to larger scales. 89 90 The relative abundance of species has played a major role in describing and 91 understanding ecological communities1,3. There are several graphical ways to depict 92 the relative abundance of species3,6,7. One such approach is the species abundance 93 distribution (SAD), which consists of plotting the size class of the number of 94 individuals (log-transformed) on the x-axis and the number of species in each class on 95 the y-axis (for instance, Fig. 1a,b). Some of the seminal studies on SADs concerned 96 the scaling properties of these distributions. For example, Preston1 emphasized that 97 the shape of the observed distributions was a function of sample size and introduced 98 the concept of the “veil line”, by which he was referring to a line shifting to the left of 99 the distribution revealing additional rarer species as sample size increased. However, 100 more recently, studies on SADs have focused on the statistical distribution that best 4 101 fits the data at one specific scale8-11. Nevertheless, in addition to studying the SAD at 102 one specific scale, there is also merit in describing how the SAD changes as a 103 function of a scaling variable, such as area5,12,13. Identifying the patterns associated 104 with the scaling properties of the SADs is important because, first, it provides a 105 mathematical framework where experiments can be devised in order to understand the 106 processes generating the observed diversity patterns, and, second, it suggests ways to 107 forecast the distributions for larger areas, with obvious implications for conservation 108 studies. Other authors have recently addressed how to forecast the SAD at large 109 spatial scales14-18, but here we use a method5 which combines the patterns observed 110 for the moments of the SADs with methods developed in the field of image analysis19. 111 112 A priori knowledge of the importance of natural processes for some species 113 groups may justify the deconstruction of a SAD into the distributions of those 114 groups20,21. Indeed, computer simulations developed in some of our previous work22 115 showed that the distributions of high and low dispersal ability species develop at 116 different paces when sample size increases (Fig. 1, adapted from ref. 22). These 117 results motivated us to explore the deconstruction of the SAD into the distributions of 118 these two groups6,23. These computer simulations were spatially explicit and 119 implemented the basic tenets of neutral theory24 (namely, that in a community, all 120 individuals, independently of the species, have the same probability of dying, 121 reproducing, speciating and dispersing), but we assumed two communities: one with 122 high (Fig. 1a) and one with low (Fig. 1b) dispersal ability species (for more details on 123 the simulations, see ref. 22). The important feature revealed by Fig. 1 is that the SADs 124 of both communities are monotonically decreasing functions for small sample sizes, 125 but when sample size increases the maximum of the SADs for low dispersal ability 5 126 species moves towards intermediate abundance classes faster than that of the SADs of 127 high dispersal ability species, which retain a larger proportion of singletons even for 128 relatively large sample sizes. 129 The shape of the SADs of low dispersal ability species changes more rapidly 130 with increasing sample size because these species are more aggregated in space. 131 Clustering implies that when sample size increases we tend to find more individuals 132 of the same species, but relatively fewer new species, hence species quickly shift 133 towards intermediate abundance classes as sample size increases. On the other hand, 134 species with high dispersal ability tend to be more spatially mixed, thus when sample 135 size increases we find a large number of new species, but with many more species 136 remaining at low abundances. For very large sample sizes the two distributions are 137 again similar, because for both communities of high and low dispersal ability species, 138 new species are mainly rare ones; new intermediate or large abundance species have 139 already been identified (for the role of spatial aggregation in the context of SADs, see 140 also ref. 25). However, as we will see, our empirical data only describe the transitions 141 in the distributions for small sample sizes obtained with the simulations, that is, the 142 change from monotonically decreasing functions to functions with a maximum 143 developing for intermediate abundance classes. 144 145 Our main purpose is to quantify how SADs change with scale and for that we 146 use the moments of the empirical data. This means that we do not attempt to fit SADs 147 at a particular scale with a given probability distribution, such as the logseries or the 148 lognormal.