ARTICLE
https://doi.org/10.1038/s41467-019-14267-y OPEN
Dispersion fields reveal the compositional structure
of South American vertebrate assemblages
Michael K. Borregaard1*, Gary R. Graves1,2 & Carsten Rahbek1,3,4
The causes of continental patterns in species richness continue to spur heated discussion.
Hypotheses based on ambient energy have dominated the debate, but are increasingly being
challenged by hypotheses that model richness as the overlap of species ranges, ultimately
controlled by continental range dynamics of individual species. At the heart of this con-
troversy lies the question of whether species richness of individual grid cells is controlled by
local factors, or reflects larger-scale spatial patterns in the turnover of species’ ranges. Here,
we develop a new approach based on assemblage dispersion fields, formed by overlaying the
geographic ranges of all species co-occurring in a grid cell. We created dispersion fields for all
tetrapods of South America, and characterized the orientation and shape of dispersion fields
as a vector field. The resulting maps demonstrate the existence of macro-structures in the
turnover of biotic similarity at continental scale that are congruent among vertebrate classes.
These structures underline the importance of continental-scale processes for species rich-
ness in individual assemblages.
1 Center for Macroecology, Evolution and Climate, GLOBE Institute, University of Copenhagen, Universitetsparken 15, 2100 Copenhagen, Denmark.
2 Department of Vertebrate Zoology, National Museum of Natural History, Smithsonian Institution, Washington, DC 20013, USA. 3 Imperial College London,
Silwood Park, Buckhurst, Road, Ascot, Berkshire SL5 7PY, UK. 4 Danish Institute for Advanced Study, University of Southern Denmark, Odense M 5230,
Denmark. *email: mkborregaard@bio.ku.dk
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The cause of continental patterns of species richness is one make similar predictions for spatial variation in species richness,of the most debated questions in ecology1–3 and was for a resolution to this controversy has not been found, and in recentyears considered the keystone question of macroecology. years relatively little progress has been made on the processes
Though species richness of lowland regions generally correlates underlying patterns of large-scale species richness.
well with contemporary climate4, with mountainous regions Though mechanisms based on (1) energy limiting rates of
correlating less well (refs. 5,6, Fig. 1), the mechanistic processes species origination14, (2) energy limiting local co-occurrence15, or
responsible for this correlation remain unresolved7,8. Hypotheses (3) richness arising as an emergent property of species’ niche
based on a direct local effect of ambient energy on species rich- overlap16 will all lead to the observed correlation between climate
ness have dominated the debate, countered by theories focusing and richness, they represent three different underlying pathways
on species’ niches and range dynamics, positing that species for the generation of continental species richness patterns. We
richness is an emergent property of species range overlap9–13 that argue here that these differences should be reflected in the con-
occurs at scales much greater than the grid cells used in most tinental pattern of species’ turnover among grid cells, making it
analyses. An example of the latter is the idea of tropical niche possible to distinguish between them.
conservatism, which posits that climate–richness correlations Energetic limitation of speciation rates has generally been
arise because many species are evolutionarily adapted to warmer tested by correlating species richness with local energy levels4 or
and more productive environments and thus have ranges by phylogenetically estimating the past diversification rates of all
extending into, or confined to, these areas. Since these hypotheses species whose current ranges overlap a given grid cell. Both of
a b c
Species Species Species
800
200 150
100
400
100
50
0
0 0
d e f
°C mm/yr Species
30 >4000 800
2000
15 400
0 0
0
Fig. 1 Species richness and environmental variables in South America mapped in 1° × 1° grid cells. a Species richness of birds (n= 2869), bmammals (n
= 1146), and c amphibians (n= 2265). dMean annual temperature (°C). e Annual precipitation (mm). f Predicted richness of birds based on a linear model
of temperature and precipitation (R2= 0.63).
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these approaches will lead to biased results in the presence of of compositional similarity, we used the approach of ADFs, which
spatio-temporal range dynamics. Because speciation in terrestrial are created by overlapping the geographic range maps of all
vertebrates is usually allopatric17, newly geminate species will species that occur in a specified grid cell24–27,28. The resulting
generally not co-occur, and thus the species richness of grid cells contour map will peak at the focal cell and decline in all direc-
only increases where the species come into secondary contact. tions from this peak (Fig. 2, Supplementary Figs. 1 and 2). The
This may accumulate species in the area of origination, but if contour slope along any vector illustrates the decay of composi-
range expansion is not spatially random species richness will tional similarity with distance from the focal cell, revealing the
accumulate in the area that ranges expand into, which will greatly biogeographical affiliation of the species assemblage of the focal
weaken the expected spatial link between local speciation rates cell. We generated ADFs for all 1° × 1° grid cells (n= 1689) of the
and species richness. An empirical example of this is the high South American continent for all species of birds (2869 species),
salamander diversity of Amazonia, which is mainly generated by mammals (1146 species), and amphibians (2265 species).
species diversification in the Andes18. Thus speciation rate var- Continental mapping of ADFs for individual grid cells reveals a
iation will only lead to strong energy–richness correlations when clear geographical structure in the species composition of local
range expansion is relatively spatially symmetric around the area assemblages24 in all three vertebrate classes (Fig. 2). The size and
of origination. shape of ADFs exhibited marked geographic variation, reflecting
Energetic limitation of species co-occurrence at large spatial patterns of faunal turnover and the spatial configuration of major
scales19 is an alternative local species–energy mechanism. This biomes. For example, the composition of assemblages of species
hypothesis is based on the theoretical premise that range in 1° × 1° grid cells occurring at opposite ends of the vast
expansion occurs more easily in areas with high energy and many Amazonian ecoregion (~5 million km2) are more similar to one
resources and that local extirpation and range contraction is more another than they are to geographically proximate assemblages in
likely in areas with less available energy. This is predicted to lead adjacent ecoregions that exhibit similar energy levels. ADFs
to range dynamics that are similar across different species, with sampled from the center of the Amazonian ecoregion are
high-energy areas as foci for all species range expansions. relatively symmetrical (Fig. 2a–c), whereas those sampled near
Hypotheses based on range overlap, such as Tropical Niche the periphery of the ecoregion are asymmetrical (Fig. 2d–f). We
Conservatism, in contrast, explicitly posit that range expansion quantified ADF symmetry by projecting a vector from the
history is determined by the spatial configuration of the landscape geographic center of each focal cell to the respective center of
and underlies species richness patterns. Under the mechanism of gravity for each ADF (Fig. 2; see “Methods”). The vector toward
niche conservatism, species with similar adaptations are likely to the center of gravity is a simple representation of the orientation
co-occur deterministically across sites, leading to large-scale and degree of asymmetry of the ADF. The direction of the vector
emergent patterns of range overlap and geographically structured indicates the major axis of asymmetry of the ADF, whereas vector
assemblages whose spatial distribution follow the asymmetric length provides a direct measure of the degree of ADF
configuration of continental structures, such as the turnover of asymmetry. In this context, asymmetry refers to a deviation
ecoregions characterized by different biomes20–22. These pro- from radial symmetry of individual dispersion fields.
cesses are predicted to leave a distinct biogeographical signature
in the continental pattern of species distributions. Because these
patterns result from a deterministic interaction with the physical Symmetry diagrams. We then mapped ADF symmetry diagrams
environment, the pattern of disequilibrium should be predictable for the continental lattice of cells using a color scale to illustrate
and consistent among taxa with shared habitat affinities. the degree of ADF asymmetry and arrows to indicate the direc-
The existence of geographically structured assemblages with a tion of the symmetry vectors (Fig. 3). This continental lattice of
clear signature of biomes would thus support a key prediction of ADFs revealed a hidden biogeographical structure in the species
niche conservatism. Note that the well-established existence of composition of local assemblages that is not readily discerned
biogeographic regions in species distributions23 do not in them- from patterns of species richness (Fig. 3). For all three vertebrate
selves invalidate species–energy theory; however, as outlined groups, ADF symmetry diagrams revealed distinct geographical
above, the energy-based mechanisms will predictably affect spa- patterns in ADF symmetry and assemblage affinity (Fig. 3a–c).
tial patterns of biotic similarity, giving us a previously unexplored ADF vectors for the majority of adjacent grid cells tend to parallel
opportunity to reassess the mechanistic basis of climate–richness one another or nearly so. However, ADFs near vegetation eco-
correlations. tones were highly asymmetrical, and the vectors of ADF sym-
Here we develop a new approach to reveal and visualize geo- metry diagrams straddling ecotones often point in orthogonal
graphic structuring of assemblages. The approach is based on directions away from the ecotones and toward the centers of
assemblage dispersion fields, which are formed by overlaying the abutting ecoregions. In general, the degree of asymmetry
geographic ranges of all species co-occurring in a grid cell. We increased with distance from these centers (Fig. 4). Groups of grid
created dispersion fields for all tetrapods of South America and cells with similar ADFs correspond roughly to the configuration
visualized their geographic structures by expressing their orien- of major vegetation ecoregions (Fig. 3d), revealing, e.g., a clear
tation and shape of dispersion fields as a vector field. These separation of the faunas of Amazonian from the savannah-like
symmetry diagrams support the existence of macrostructures in biomes of the Cerrado and Caatinga, and an extremely rapid
the turnover of biotic similarity at a continental scale. The turnover of assemblages along the eastern versant of the Andes
structures are congruent among vertebrate classes, lending sup- Mountains.
port to the predictions of niche conservatism as a control of Note that, though the arrows on the diagrams might intuitively
continental diversity patterns. be understood as species movement, and indeed should to a
degree reflect general patterns in range expansion, they do not
prove movement and are not hypotheses about historical range
Results and discussion dynamics for the faunas. The ADF symmetry diagrams are a
Assemblage dispersion fields (ADFs). We investigated the geo- representation of the current pattern of biotic similarity, whereas
graphical pattern of compositional similarity for the mammals, patterns of range expansion of individual species and clades can
amphibians, and freshwater and land birds in 1689 cells (1° × 1° only be revealed by focused biogeographical analyses (and in the
latitude–longitude blocks) in South America. To quantify patterns presence of fossils). Instead, the ADF symmetry diagrams reveal
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a b c
63–64° W
2–3° S
Species Species Species
300 150 80
200 100
25
100 50
0 0 0
d e f
53–54° W
11–12° S
Species Species Species
300
50
100
200
50 25
100
0 0 0
Fig. 2 Asymmetry vectors and assemblage dispersion fields (ADFs) for selected 1° × 1° grid cells in Amazonia. The focal cell is marked by a black point.
Colors indicate the number of species shared with the focal cell. The central region (see “Methods”) of the dispersion field is outlined by a black line. The
asymmetry vector extends from the center of the focal cell to the center of gravity (white point) of the central region. ADFs of focal cells are relatively
symmetrical near the center of the Amazonian ecoregion (a–c) and become increasingly asymmetrical near its periphery (d–f), for birds (a, d), mammals
(b, e), and amphibians (c, f).
emergent structures in the current distribution of species that can constrained by the empirical species richness (Fig. 5b) and
be interpreted in the context of understanding the basis of the ecoregion boundaries (Fig. 5c) (see “Methods”; Supplementary
current pattern of species richness. The arrows reveal patterns at a Fig. 7).
larger scale than neighbor-based methods for compositional As a basis for interpreting the empirical ADF symmetry
similarity, with the scale being determined by the species’ diagrams, we also developed simple predictive models under
range sizes. three constraints to operationalize the verbal models presented
above: (1) species originate in grid cells with higher energy levels
and undergo random range expansion (Fig. 5a); (2) species
Null model analysis. We used a null model to correct for two originate randomly and range expansion is limited by a
properties of grid cells that are expected to affect ADF asymmetry maximum level of co-occurring species (Fig. 5b); and (3) species
in the absence of any ecological mechanism29,30: (1) The shape of originate randomly within their native ecoregion and undergo
the continent, which restricts the potential configuration of the range expansion limited by ecoregion boundaries (Fig. 5c). The
ADF; and (2) the range–size frequency distribution of species in emergent pattern in Fig. 5c is broadly similar to the empirical
each grid cell, which will directly affect vector length, and thus pattern. Note that this type of predictive model is necessarily an
asymmetry values. The null model assesses the expected ADF over-simplification and will to a certain extent reflect the choices
symmetry diagrams under random range placement (Fig. 5a) made in implementing the algorithm; we present the results here
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a b c d
Alpine
Cloud forest
Lake
Mediterranean
15 Paramo15 15 Plains — Pampas
Plains — Patagonia
Desert — Monte
10 10 10 Desert —  Peruvo-Chilean
Rainforest — Amazonia
Rainforest — Mata Atlantica
5 5 5 Seasonally dry tropical forestTemperate forest
Woodlands — Cerrado
0 Woodlands — Chaco0 0 Woodlands — Espinal
Fig. 3 ADF symmetry diagrams for South American birds, mammals, and amphibians and major vegetation biomes. Arrows indicate the major axis of
asymmetry of the ADFs for individual cells (n= 1689). The degree of asymmetry is indicated by a color scale (green through red colors represent more
asymmetrical ADFs). This figure summarizes the vector arrows shown in Fig. 2 for all grid cells on the continent. Histograms show the distribution of vector
lengths. The panels show a birds, b mammals, c amphibians, and d vegetation ecoregions. ADF symmetry diagrams using different cutoff values for the
central ADF region are shown in Supplementary Figs. 3 and 5.). The map in d was provided by Tiina Särkinen, originally based on the map of the World’s
ecoregions developed by the WWF (ref. 48; under CC BY 4.0 license: https://creativecommons.org/licenses/by/4.0/).
a b
40
30
20
SES
40
10
20
0
0
–20
–40
0 2 4 6 8
Distance from ecotone
Fig. 4 Null model divergence of ADF vector symmetry. a The deviation of the ADF symmetry vectors of birds from the null model (Supplementary Fig. 7),
calculated as the standardized effect size (SES) deviation of vector lengths after 1000 repetitions of the null model. b The relationship between the
deviation from the null model and the distance from the ecoregion boundary for the subset of grid cells (n= 342) that are completely encompassed within
Amazonia (Supplementary Fig. 9; R2= 0.52). The least square regression explains 52% of the variation.
mainly as a context for discussing the effect of various constraints observed to support greater species richness than predicted purely
on ADF symmetry diagrams. from species–energy dynamics.
Our results support the idea that high species richness at the
base of the Andes is due to high habitat heterogeneity and faunal
Conclusion turnover, rather than a simple response to elevated energy
These results show that the species compositions of local levels31. Our results show that, at the grain sizes of our analyses,
assemblages are consistent with non-random range expansion the cohesive nature of the geographic ranges of species means that
limited by the geographical configuration of habitats, as suggested any process that affects species richness will also affect the species
by niche theory. Though the patterns of ADF asymmetry are composition in grid cells and the large-scale congruence of spe-
complex, they are consistent among three major vertebrate classes cies distributions. The processes behind species richness and
(Fig. 3). Thus range dynamics are not spatially random but species composition can thus not be decoupled at this scale, and
instead result from deterministic processes that will weaken any importantly, employing classic regression methods that deal with
spatial association between rates of speciation and species rich- spatial autocorrelation by imposing a symmetric neighborhood
ness. Patterns of species richness appear instead to be created by variance kernel (e.g., spatial autoregressive and conditional
an interaction between Grinnellian niches of species and geo- autoregressive) is not adequate to evaluate the effect of local
graphical patterns of range overlap among species, which in turn processes on species richness. Other approaches, based on explicit
depends on the configuration of major vegetation types. Fur- mechanistic modeling of assembly processes, are likely to be
thermore, certain areas, such as the eastern Andes Mountains, necessary to resolve this.
exhibited very rapid faunal turnover, reflecting high turnover in The 1° × 1° represents the highest precision we can feasibly
species’ habitats. Grid cells from this area are consistently attain over such a large number of species but entails that many
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Standardized asymmetry
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a Constrained origination b Constrained richness c Constrained range expansion
(weak species-energy) (strong species-energy) (niche conservatism)
Asymmetry
10
5
0
Fig. 5 Predicted assemblage dispersion field (ADF) symmetry diagrams based on predictive models. The arrows reveal the pattern of biotic similarity.
a The predicted ADF diagram from a model where species originate in grid cells according to local energy availability and then spread spatially randomly
from there; b the predicted ADF diagram from a model where species richness is constrained to be identical to the empirical and ranges are constrained to
be spatially cohesive; c the predicted ADF diagram from a model where each species originate in its original vegetation ecoregion and spreads spatially
randomly with the constraint that ecoregion boundaries are crossed with low probability. Vegetation ecoregions are defined by the map shown in Fig. 3d.
areas within each occupied grid cells will not actually be occupied used to build the global dataset from which these distributions are derived, see the
by a given species and that species co-occurring at the grid cell appendix of Holt et al.22
scale may never actually co-occur in local communities. This is The mammal database was based on published range maps from the global
mammal assessment36, as modified by Fritz and Purvis37. Domesticated species
particularly pronounced along ecotone boundaries, where many were excluded from the analysis, as well as portions of the geographic ranges of
species range boundaries are expected to meet, making it hard to non-domesticated species labeled as historical, presence uncertain, introduced
evaluate patterns, e.g., along the Andes. origin, or extinct. Species classified as Data Deficient, Extinct, or Extinct in the
It has long been clear that analyses of species richness must Wild in the IUCN database were excluded. The resulting database was updated to
match the published phylogeny of mammals38. This entailed excluding 130 species
move beyond simple correlative analyses of species richness that were absent from the phylogeny and adding 40 species from PanTHERIA39
numbers and take account of the size, shape, and location of the that were missing from the IUCN database. Finally, 56 species were added by
geographic ranges of species. By doing so here, we are better able splitting IUCN maps to reflect the phylogeny of Bininda-Edmonds et al.40. Range
to evaluate the postulated mechanisms and predictions of com- polygons for the resulting 1146 mammalian species were overlaid on the 1° × 1°
peting hypotheses on the origin of species richness patterns, grid. A species was scored as present in any grid cell that overlapped the geographicrange polygon for that species41.
providing a needed corollary to the template predictions of The amphibian database was based on published range maps from the IUCN42
energy—kinetic theory. The ADF diagrams provide a visually and updated to follow the published taxonomy of amphibians43. Species labeled as
intuitive way of turnover in species composition and the rela- incertae sedis, klepton, and undescribed taxa were excluded, as were species
tionship among different assemblages. The empirical results of classified as introduced or uncertain/introduced. The final dataset consisted ofrange maps for 2265 amphibian species, which were overlaid on the 1° × 1° grid
our study are consistent with recent developments in ecological and scored as present or absent in each grid cell. We note that the amphibian
theory that argue that dynamics at the scale of complete ranges dataset is not quite of the quality of the other two datasets and have been revealed
must be considered in an integrative theory for species richness32. to fit poorly with occupancy records44. We have chosen to include it here for
This entails a revision of species richness theory that expands consistency across the tetrapod groups, but conclusions based on the amphibian
dataset should be interpreted carefully.
geographical grid-cell-based regression analyses of diversity pat-
terns to integrate species niches and geographic range dynamics.
Environmental data. Records of mean annual temperature and annual pre-
cipitation were extracted from the mean monthly climatic database published by
Methods New and co-workers45, which was compiled globally at a 0.5° latitude–longitude
Distribution data. The analyses were based on a distributional database of all resolution for the period 1961–1990 (>3,000,000 data points for each variable).
species of birds, mammals, and amphibians in South America. The continent was
divided into 1° × 1° grid cells corresponding to integer degree lines of latitude and Assemblage dispersion fields. The occurrence of species can be summarized in a
longitude. All continental grid cells that contained land area inside were retained in presence–absence matrix P, where each row of the matrix represents a species, each
the analysis, yielding a total of 1689 1° × 1° grid cells. Land-bridge islands (e.g., column is a grid cell, and a 0 or 1 denotes absence or presence of a species within a
Trinidad) were excluded. grid cell46. The row sums represent the range sizes of species (measured as the total
The bird distributions were extracted from the Copenhagen global bird of occupied grid cells), and the column totals represent species richness values of
database33, which is periodically updated. This database is based on museum grid cells. The ADF matrix D is then defined as
specimens, published sight records, and expert opinion, following the approach
outlined by Rahbek and Graves5,34. The distributions were mapped directly to the D ¼ PTP; ð1Þ
1° × 1° latitude–longitude grid; hence, distributions are not based on a post hoc
fitting of arbitrary-scale polygons to the grid cells (different from the mammal and where T is the transpose operator. D is a symmetrical site-by-site matrix that
amphibian datasets, which are based on IUCN data). The resulting high-quality counts the number of species shared by any two grid cells. Each row (or column)
dataset represents the best current knowledge of bird distributions in South represents the ADF for that cell and describes the number of species that a focal cell
America. To ensure comparability with other sources, the avian species shares with any other cell. Mapping dispersion fields results in the maps presented
delimitation followed the taxonomy published by the SACC35 resulting in a dataset in Fig. 2 and Supplementary Figs. 2 and 3, representing the geographical pattern of
containing 2869 land and freshwater species. For a full list of the >1600 references biotic similarity around the selected focal cells.
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The row (and column) sums of the ADF matrix equal the ADF volume of cells. (yellow to red in the figures), the dispersion fields were highly symmetrical, with
The ADF volume measures the sum of shared species between a specified cell and biotic similarity declining as a basic function of distance in all directions from the
all other cells, or, equivalently, the sum of the range sizes of all species present focal cell. The ADF symmetry diagram exhibited a simple pattern of all vectors
within the focal cell26. Thus, if pi,j refers to the ith row and jth column of P, i.e., the pointing toward the center of the continent (Supplementary Fig. 7c). These
presence of species i in cell j, and ri is the range size of species i, defined as patterns represent the baseline for comparing the results of ecological process.
X To quantify the deviance of empirical ADFs from the null model, we calculatedN
r ¼ p ; ð2Þ standardized effect sizes by repeating the sampling procedure 1000 times andi i;j
j¼1 calculating ADF symmetry values for all grid cells. The deviation between the
empirical and simulated symmetry values was then calculated as (Eq. 7) (ν− μ
then the dispersion field volume is (λ))/σ (λ), where ν is the empirical symmetry value, λ is a (mathematical) vector of
XS symmetry values, and μ and σ are the mean and standard deviation of the
df volume ¼ p r ; ð3Þ simulated symmetry values, respectively.j i;j i
i¼1 We also implemented three different predictive models for the expected ADF
symmetry patterns expected under various constraints on range origination and
where S is the total number of species and N is the total number of grid cells. It also expansion. In model 1, constrained origination, we placed the first grid cell of
quantifies the mean contribution of a cell to all ADFs, since cells with a higher ADF ranges on the continent according to a probability function of available energy and
volume share more species with surrounding cells. Dividing the dispersion field allowed range expansion into each surrounding grid cell to be random. To generate
volume with the species richness of sites yields the mean range of all species present model 3, constrained range expansion, we placed the first grid cell on a random
at the site, what Smith47 labeled the mean average cosmopolitanism of species. grid cell in a vegetation ecoregion chosen with a probability proportional to the
occupancy of that ecoregion by a given species. Random range expansion was
Measurement of ADF asymmetry. The center of gravity of the dispersion field for allowed, but a range expansion event into a grid cell across a boundary was 30× less
a grid cell j was calculated as the point located at the average longitude (x) and likely than an expansion event into a grid cell in the same ecoregion (the number
latitude (y) of the ranges of all species present within cell j. 30 was chosen to ensure that species’ ranges expanded across boundaries
So the mean x and y coordinates of species i is occasionally, but rarely). Sampling from these models followed the protocol
X outlined above. Model 2, constrained species richness, was generated in a differentN
x ¼ 1 p x ð4Þ way, which was designed to maintain the grid cell richness constant, reflecting ai r i;j i;ji j¼1 scenario where the richness of each grid cell is restricted to a maximum value by
local resources. All species ranges were built simultaneously, at each step allowing
XN one species to expand its range by one grid cell adjacent to its range, with a1
y ¼ p y ð5Þ probability equal to the difference between the maximum (i.e., the empirical)i r i;j i;ji j¼1 species richness of the grid cell and the current richness at that point in the
simulation. This allows building random ranges while both maintaining range
and then cohesion and the empirical species richness. At the end of the simulation, absolute
1XS range cohesion would become impossible, and small gaps were allowed (as exists in
df centerXj ¼ pi;jxi ð6Þ the empirical ranges as well). ADF diagrams were built for all three null modelssj i¼1 (Fig. 5).
1XS
df centerY ¼ p y ð7Þ Ecotone analysis for Amazonia. It is a prediction of niche conservatism that thej s i;j ij i¼1 asymmetry of ADF vectors for cells within a major vegetation ecoregion should
Although the contours and symmetry of ADFs exhibit marked geographic decrease with distance from the ecotone boundary. Because the scale of the dis-
variation, the total extents of ADFs (i.e., areas that share at least one species with tribution data (1° × 1° latitude/longitude degrees) was too coarse to adequately
the focal cell) are similar because most assemblages of birds, mammals, and assess this effect for many of the smaller ecoregion, e.g., along the Andes, we tested
amphibians contain a few widespread species that occur over most of the South the conjecture for the largest coherent region, Amazonia. To avoid the confounding
American continent. For example, the Neotropic cormorant (Phalacrocorax influence of disjunct faunas that come into contact in grid cells at ecotones, we
brasilianus) occurs in every cell (n= 1689) in South America, so that all focal cells restricted the analysis to only include cells that are completely within the lowland
shared this species with all other grid cells. The pervasive in uence of widespread (<500 m above sea level) Amazonian ecoregion and also excluded a small numberfl
species forces the center of gravity of ADFs toward the geographic center of South of cells that were isolated behind the highland Tepuis of the Guianan Shield. The
America and obscures regional signals of biotic similarity. This is particularly true cells that were included in the analysis, along with their distance to the outer
in mammals (mean range size= 182.7 grid cells) and birds (mean range size= boundary of Amazonia, are shown in Supplementary Fig. 8. Residuals from the
186.0 grid cells) but less so in amphibians (mean range size= 37.5 grid cells). regression analysis were symmetrical but departed significantly from normality
To visualize the distinct pattern in ADFs, we trimmed away grid cells that according to Kolmogorov–Smirnov test (p < 0.01).
shared only a few species with the focal cell. The cutoff value for trimming
represents a balance between the ability to visualize core ADF asymmetries and the Reporting summary. Further information on research design is available in
need to illustrate the distributional pattern of the entire species assemblage for a the Nature Research Reporting Summary linked to this article.
focal cell. Higher cutoff values reflect increasingly restrictive patterns of biotic
similarity. We arbitrarily set the value at 0.5 for amphibians and increased the value
to 0.6 for birds and 0.7 for mammals, which have larger average range sizes. We Data availability
then calculated the center of mass and the ADF asymmetry vector for the trimmed All data depicted in figures here are available on request from the authors. The
ADF for each focal cell. The resulting spatial patterns of ADFs and asymmetry underlying databases with range maps on amphibians and mammals are available from
diagrams were qualitatively similar for all three vertebrate classes (as seen in the IUCN. The result of computer models are also available upon request.
Supplementary Figs. 4–6).
Code availability
Null and predictive models. We created null models by randomly placing con- All R code necessary to produce the results here will be made available upon acceptance.
tinuous ranges on the South American continent, using a simple spreading dye In addition, R and Julia codes necessary to use the ADF general computational
algorithm30. This algorithm starts by selecting a random starting cell anywhere on framework are available as part of the nodiv package for R (on CRAN) (functions
the continent. The algorithm then selects a random cell adjacent to that occupied dfield_matrix, dfield_explore, dfield_calc, and arrowplot) and the SpatialEcology.jl
and adds it to the range. This process continues until the target range size is package for Julia (can be installed from the official repository, function dispersionfield),
reached. The algorithm is implemented in R. both openly available on GitHub on the MIT free and open license.
To create the null model ADFs, we built a null distribution of random ranges
for each grid cells. Ranges were selected from all possible random ranges in South
America that intersect the focal cell, maintaining the empirical range–size Received: 7 July 2019; Accepted: 19 December 2019;
frequency distribution of that grid cell. This was achieved by first building a
sampling population of random ranges for each possible range size between 2 and
1689 grid cells over the entire domain (the continent), so that all grid cells were
intersected by at least 100 different random ranges of each range size. For each grid
cell, we then picked random ranges from this sampling population according to the
empirical range–size distribution of that grid cell and constructed the ADF.
The resulting ADFs clearly demonstrated an effect of the geometry of the South References
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 “Assemblage dispersion fields reveal the spatial structure of biotic similarity 
among South American vertebrate distributions” 
Borregaard, M.K., Graves, G. & Rahbek, C. 
Submission to Nature Communications 2019 
 
Supplementary Figures 
  
Supplementary Figure 1. Examples of dispersion fields from different grid cells. Each 
row shows the dispersion field for a given grid cell for birds, mammals and amphibians.   
The centroids of the focal cells are: row 1 (a-c): 61.5W 30.5S; row 2 (d-f): 63.5W 3.5N;    
row 3 (g-i): 68.5W11.5S.  
 
 
Supplementary Figure 2. Further examples of dispersion fields from different grid 
cells. Each row shows the dispersion field for a given grid cell for birds, mammals and   
amphibians. The centroids of the focal cells are: row 1 (a-c): 52.5W 3.5N; row 2 (d-f): 7
7.5W 1.5S; row 3 (g-i) 50.5W 24.5S. 
 
2 
 
 
Supplementary Figure 3. An illustration of the dependence of ADF symmetry                
diagrams on the cutoff value used to delimit the central region of the ADF. The number 
at the left of each row indicates the cutoff value. A value of 0 indicates that all grid cells are 
used in calculating the centre of gravity of the ADF, whereas a value of 0.1 indicates that all 
grid cells that contain at least 10% of the species present in the focal cell are considered part 
of the central region. Each row shows the ADF symmetry diagrams for birds (a, d, g),        
mammals (b, e, h) and amphibians (c, f, i). 
 3 
 
 
Supplementary Figure 4. ADF symmetry diagrams for higher cutoff values. The red   
box indicates the diagram that was included in the main text for this taxon. Legend as in    
Supplementary Fig 3. 
 
4 
 
 
Supplementary Figure 5. ADF symmetry diagrams for the highest cutoff values.       
Legend as in Supplementary Fig 3. 
 
 
 
5 
 
 
Supplementary Figure 6. Larger map of South American biomes. The map was  
provided by Tiina Särkinen, originally based on the map of the World’s ecoregions  
developed by the WWF (under CC BY 4.0 license: https://creativecommons.org/licens
es/by/4.0/). 
6 
 
 
 
Supplementary Figure 7. Results from a basic spreading dye null model.  
(a-b) Examples of dispersion fields created by the spreading dye model. Biotic similarity is 
an approximate function of distance, similarly to the predictions of the species-energy  
model. (c) The ADF symmetry diagram expected under a null model of cohesive, non- 
interacting ranges. All symmetry vectors are oriented towards the centre of the continent. 
 
 
7 
 
 
 
Supplementary Figure 8. The cells used for the analysis of the correlation between  
ecotone distance and ADF symmetry. The ecotone around Amazonia is shown as a black 
line. All grid cells intersecting the ecotone are excluded, as well as all cells with a  
maximum altitude over 500meters, which marks a high-altitude fauna that is distinct from  
the Amazonian. 
 
8