Peaks, plateaus, canyons, and craters: the complex
geometry of simple mid-domain effect models
Robert K. Colwell1, Nicholas J. Gotelli2, Carsten Rahbek3,
Gary L. Entsminger4, Catherine Farrell5 and Gary R. Graves6
1Department of Ecology and Evolutionary Biology, University of Connecticut, Storrs,
Connecticut, USA, 2Department of Biology, University of Vermont, Burlington, Vermont, USA,
3Center of Macroecology and Evolution, Department of Biology, University of Copenhagen,
Copenhagen, Denmark, 4Acquired Intelligence Inc. and Rocky Mountain Biological Laboratory,
Montrose, Colorado, USA, 5Essex High School, Essex Junction, Vermont, USA and
6Department of Vertebrate Zoology, National Museum of Natural History,
Smithsonian Institution, Washington, DC, USA
ABSTRACT
Background: Geographic ranges, randomly located within a bounded geographical domain,
produce a central hump of species richness (the mid-domain effect, MDE). The hump arises
from geometric constraints on the location of ranges, especially larger ones.
Questions: (1) How do patterns of species richness in one- and two-dimensional MDE
models change as a function of range size? (2) How does dispersal affect these patterns?
Methods: We used a spreading dye algorithm to place assemblages of species of uniform
range size in one-dimensional or two-dimensional bounded domains. In some models, we
allowed dispersal to introduce range discontinuity.
Results: As uniform range size increases from small to medium, a flat pattern of species
richness is replaced by a pair of peripheral peaks, separated by a valley (one-dimensional
models), or by a cratered ring (two-dimensional models) of species richness. With large range
sizes, the peaks or rings fuse to form a central plateau (one-dimensional) or a flat-topped
mound (two-dimensional) of highest species richness. Adding dispersal to the two-dimensional
model weakens the peripheral ring and introduces complex patterns for long-distance dispersal.
Conclusions: Heterogeneous range size distributions (whether theoretical or empirical) used
in most MDE models produce species richness patterns dominated by wide-ranged species,
hiding complex patterns formed by small- to medium-ranged species. These patterns, which are
analogous for one and two dimensions, are complicated further by long-distance dispersal and
discontinuous ranges. Although geometric constraints lead to classic mid-domain effects for
large-ranged species and for mixed range-size frequency distributions, small- and medium-sized
ranges of a uniform size generate more complex patterns, including peaks, plateaus, canyons,
and craters of species richness.
Keywords: biogeography, boundary effects, doughnut, geographical range, geometric
constraints, null models, spatial scale, species richness gradients, stochastic models.
Correspondence: R.K. Colwell, Department of Ecology and Evolutionary Biology, University of Connecticut,
Storrs, CT 06269-3043, USA. e-mail: colwell@uconn.edu
Consult the copyright statement on the inside front cover for non-commercial copying policies.
Evolutionary Ecology Research, 2009, 11: 355?370
? 2009 Robert K. Colwell
INTRODUCTION
Documenting the relative importance of contemporary climate, historical effects, and
geometric constraints on species richness patterns has been a major research focus in
biogeography and macroecology (Willig et al., 2003). Colwell and Hurtt (1994) introduced
one-dimensional stochastic models of species richness in an attempt to explain Rapoport?s
rule, the tendency for range sizes of species to decrease towards the tropics (Stevens, 1989).
While exploring these models, Colwell and Hurtt (1994) stumbled upon a surprising result
that itself had nothing to do with Rapoport?s rule.
Suppose a transect, or domain, is modelled as a one-dimensional line segment, and
geographic ranges are represented as overlapping, randomly sized, smaller line segments,
located at random positions within the domain. In the simplest case, the domain is a line of
length one (the unit line), and geographic ranges are represented by line segments ranging in
length from zero (as a limit) to one. What is the expected pattern of species richness (the
number of overlapping ranges) at any point along the transect? Naively, one might expect
that a random distribution of species along a bounded geographical transect would
produce, on average, a uniform distribution of richness. But this turns out to be wrong.
Under a wide range of model assumptions, the pattern of species richness within the
domain is a hump that declines symmetrically from the centre towards the edges of the
domain. Later, Colwell and Lees (2000) named this phenomenon the ?mid-domain effect?,
or MDE. Whereas previous explanations for species richness gradients could be broadly
classified as either evolutionary (emphasizing historical factors) or ecological (emphasizing
contemporary climate), the mid-domain effect suggested an entirely novel class of causal
explanations: non-uniform species richness gradients might arise because of geometric
constraints imposed by a bounded domain.
Is the mid-domain effect a null model or an explanatory factor?
As of the end of 2008, more than 100 papers had addressed the mid-domain effect,
conceptually, theoretically or as a component of the analysis of empirical data sets.
Conceptually, the issue is whether MDE explains anything we did not already know
about biogeographical patterns of species richness in nature. If so, how much does it explain
and under what conditions? Like any idea that calls for an entirely new way of looking at
an old problem, MDE has, at times, been either too quickly embraced or too quickly
dismissed, but has generally met with appropriate scepticism and gradual acceptance.
Part of the problem from the start has been that MDE models are null models, but not
typical ones (Colwell et al., 2004, 2005). They are null models because they deliberately exclude a
key factor or mechanism believed to be important in nature, with the aim of assessing the
role of the missing factor in shaping natural patterns (Gotelli and Graves, 1996). In the case of
MDE, the factor left out by design is any direct role of environmental gradients in driving
patterns of species richness. No-one doubts that environmental gradients play a strong role
in adaptive evolution of populations and in shaping species distributions and range sizes
(Brown et al., 1996; Parmesan et al., 2005). But the role of such gradients in shaping patterns of overlap
of species ranges (species richness) ? to which MDE models are blind by design ? is less
obvious. This leaves an opening for MDE to serve as a null model of the patterns of range
overlap to be expected in the absence of any direct influence of environment on richness
(e.g. Connolly et al., 2003).
Colwell et al.356
But what if the simple stochastic phenomenon that drives the pattern of richness in
the model contributes, in it own right, to patterns in nature? MDE becomes in this case
not just a null model, but a mechanistic model of the effect of geometric constraints
that arise from the boundedness of domains, which can act in concert or in conflict with
other causes of geographical pattern in species richness, such as primary productivity,
topographic complexity or area. The studies of Jetz and Rahbek (2002) and Brehm et al. (2007)
exemplify many others in that they show MDE acting in concert with environmental
factors, whereas MDE and environmental factors were shown to be spatially discordant
by, for example, McCain (2003) and Lees et al. (1999, with critique and response by Kerr et al., 2006; Currie and
Kerr, 2007; Lees and Colwell, 2007). Interactions between area and MDE emerged from other
studies (e.g. Rahbek, 1997; Sanders, 2002; Bachman et al., 2004; Romdal et al., 2005; Romdal and Grytnes, 2007). Currie
and Kerr (2008) have emphasized the problems of interpretation that arise when MDE
predictions and environmental factors are collinear, although the same issues arise when
environmental variables are collinear among themselves. Hawkins and Diniz-Filho (2002),
Zapata et al. (2003, 2005), and Hawkins et al. (2005) have criticized MDE on several grounds,
but as we have already responded in full to these critiques (Colwell et al., 2004, 2005), we will not
recount these discussions here.
Because any influence of MDE on spatial patterns of richness in nature is expected
to interact in complex ways with other candidate explanatory variables, an appropriate
framework for analysis is multivariate model selection, rather than simple hypothesis testing
(Colwell et al., 2004, 2005). We therefore do not see a basis for the continued treatment of MDE
(most recently, Currie and Kerr, 2008) as a simple null hypothesis, to be rejected outright if it fails to
fully explain patterns in nature.
Range size and geometric constraints in models and in nature
As Colwell and Hurtt (1994) noted, the geometric constraint on range location within a
bounded domain increases linearly with range size. For a unit-line domain, the midpoint
of a range of length r (0 < r ? 1) is geometrically constrained to be located over a distance
of 1 ? r in the middle of the domain. For this reason, in a domain with a gradient of
environmental ?favourability?, a key prediction of MDE theory is that the location
of smaller ranges within a domain is expected to be influenced more by the environment
and less by geometric constraints, compared with larger ranges (Colwell et al., 2004, 2005). Rangel
and Diniz-Filho (2005) demonstrated this effect elegantly for a Hubbell-style neutral
evolutionary model with spatially explicit speciation and extinction. In their model, species
richness is influenced by stochastic range shifts weighted towards the ?favourable? end of
a simple spatial gradient. On shallow gradients, ranges are naturally larger for a given
environmental tolerance (niche breadth) than on steeper gradients (Kirkpatrick and Barton, 1997).
As a result, a strong MDE develops on shallow gradients, whereas species richness
accumulates in a peak near the ?favourable? end of a steep gradient.
The prediction that larger ranges are more likely to show the signature of MDE than
smaller ranges has repeatedly been borne out by empirical studies of richness patterns
(beginning with Pineda and Caswell, 1998; Lees et al., 1999; Jetz and Rahbek, 2001). To assess the role of MDE
for smaller versus larger ranges, these studies deconstructed patterns of species richness by
dividing the empirical distribution of range sizes into quantiles (halves or quartiles) by rank
or by absolute range size. Separate richness predictions are then generated for the ranges
of each quantile. A recent review and meta-analysis of this approach by Dunn et al. (2007)
Geometry of simple mid-domain effect models 357
concludes that, overall, larger ranges fit MDE predictions better than smaller ranges from
the same domain. Figure 1 shows an example for fern species on a tropical elevational
transect (Cardel?s et al., 2006).
Pursuing this approach to richness deconstruction, our aim in this study is to explore
the deep structure of a widely used class of MDE models, ?spreading dye? models (Jetz and
Rahbek, 2001). We do this by simplifying and then varying the data that go into them ? the
distribution of geographical range sizes ? in relation to the size of the geographical domain.
We extend this approach to examine the effect of range discontinuity by relaxing the
spreading dye model assumption of strict range cohesion and introducing long-distance
dispersal. We show that the classic MDE pattern, a mid-domain hump or sloped plateau,
while typical for species with medium to large range sizes, is but one of many patterns that
readily emerge from stochastic models of range location. At smaller range sizes, more
complex landscapes of species richness ? including peaks, plateaus, canyons, and craters ?
are generated by the spreading dye model.
Conceptual and historical context of the spreading dye model
Colwell and Hurtt (1994) explored MDE using a Monte Carlo approach. Lyons and Willig
(1997), likewise pursuing a stochastic approach to modelling Rapoport?s rule, independently
discovered the mid-domain effect (M. Willig, personal communication). Later, they developed a
simple analytical MDE model based on binomial probabilities for range limits on the unit
line (Willig and Lyons, 1998). Eventually, David Lees [Box 2 in Colwell and Lees (2000)] was able to
show that Colwell and Hurtt?s (1994) Model 2, Willig and Lyons?s (1998) binomial model, and
MacArthur?s (1957) Type II (?overlapping niche?) broken stick distribution are mathematic-
ally identical. (Because MacArthur was modelling the relative abundance distribution of
species, he was concerned with the length of the ?pieces? and did not explore the pattern
of their overlap.) While studying the latitudinal distribution of seaweeds along shorelines,
Pielou (1977a, 1977b) was the first to apply this model to the study of biogeographical patterns,
but she considered only pairwise overlap and did not take note of the pattern of multiple-
species overlap (richness) (E.C. Pielou, personal communication).
Fig. 1. Observed and predicted species richness as a function of elevation for ferns, on the Barva
Transect, Costa Rica. The left-hand graph displays results for all fern species, whereas the middle and
right-hand graphs show the 50% of fern species with larger and smaller ranges, respectively. Solid
circles indicate observed richness for each field sampling elevation. Open circles show mean richness
and fine lines give the 95% confidence intervals predicted by a mid-domain effect ?range-shuffling?
model (Connolly, 2005; Colwell and Hurtt, 1994, Model 2, using empirical ranges). The coefficient of determination
for the regression of observed on expected appears on each graph (data from Cardel?s et al., 2006).
Colwell et al.358
In this ?fully stochastic? model, as Colwell and Lees (2000) called it, not only does the
overlap of ranges (species richness) peak in the middle of the domain, but so does the
distribution of range midpoints, because larger ranges are constrained geometrically to
have their midpoints towards the centre of the domain, as first illustrated by Graves (1985).
Reviewing a draft of Colwell and Hurtt (1994), Michael Rosenzweig (personal communication)
questioned whether the midpoint distribution by itself might be driving the humped pattern
of richness. In response, Colwell and Hurtt (1994) created an alternative model (Model 3)
in which range midpoints were constrained to follow a uniform distribution across the
domain. In this doubly constrained ?uniform midpoint? model, range sizes were stochastic
within the midpoint distribution constraint as well as within the constraint posed by the
domain bounds. The shape of the hump changed somewhat, but it was still a hump (Figure 4
in Colwell and Hurtt, 1994). This model treats range size stochastically, conditional upon the
specified range midpoint. Lees et al. (1999) [see also Box 3 in Colwell and Lees (2000)]
developed the opposite model, which treats the range midpoint stochastically, conditional
upon the specified range size. Colwell (2008) provides all three models (as well as others) in the
freeware application RangeModel, for one-dimensional domains.
Rahbek (1997), Pineda and Caswell (1998), Willig and Lyons (1998), and Lees et al. (1999) were
the first to treat MDE as a null model for biogeographical patterns in the natural world.
Pineda and Caswell (1998) introduced the key idea of randomizing the location of empirical
ranges within the domain, while maintaining the empirical range size frequency distribution.
This approach has been applied repeatedly in later studies using both one- and two-
dimensional domains (reviewed by Colwell et al., 2004, 2005 and Dunn et al., 2007; for more recent examples of this
approach, see Cardel?s et al., 2006; Storch et al., 2006; Brehm et al., 2007; Davies et al., 2007; Rahbek et al., 2007; Arita and
V?zquez-Dom?nguez, 2008).
Mid-domain effect models were originally proposed for one-dimensional domains, but
were later extended to two dimensions, initially by Lees et al. (1999), Bokma et al. (2001), and
Jetz and Rahbek (2001, 2002). Bokma and colleagues (2001) developed two approaches. One was
an evolutionary, cellular automaton algorithm that presaged later evolutionary models
by Rangel and Diniz-Filho (2005), Brayard et al. (2005), Rangel et al. (2007), and Arita and
V?zquez-Dom?nguez (2008). In a second approach, Bokma et al. (2001) extended the
binomial model of Willig and Lyons (1998) to two dimensions, in both cases allowing both
range size and range placement to emerge from the model. Arita (2005) further explored
the properties of the two-dimensional binomial model analytically, focusing on range size
distributions in different regions of a hypothetical square domain.
In contrast, Jetz and Rahbek (2001) based their two-dimensional model on the strategy of
Pineda and Caswell (1998) and Lees et al. (1999), both one-dimensional approaches, enforcing a
match of the modelled range size frequency distribution (RSFD) to an a priori RSFD, with
stochastic range location. In this model, each range has a predetermined size ? the range is
drawn either from an empirical RSFD or from a theoretical RSFD, rather than emerging
from colonization and extinction dynamics. The approach of Jetz and Rahbek (2001) has
come to be known as the ?spreading dye model? (or ?spreading dye algorithm?).
The spreading dye algorithm, first suggested by Gotelli and Graves (1996, p. 256), has
become the approach of choice for two-dimensional domains, and has formed the basis
for more complex models of species richness that incorporate both geometric constraints
and other factors (e.g. Jetz and Rahbek, 2002; Rangel and Diniz-Filho, 2003; Storch et al., 2006; Davies et al., 2007;
Rahbek et al., 2007). Connolly (2005) developed an analytical version of a spreading dye model for
one-dimensional domains, and Colwell (2008) implemented a one-dimensional Monte Carlo
Geometry of simple mid-domain effect models 359
version in the freeware application RangeModel. An analytical version of the spreading dye
model has not been developed for two-dimensional domains and may not be possible,
except for geometrically simple domains such as perfect circles, triangles, and squares.
To date, theoretical studies of the mid-domain effect have used heterogeneous
distributions of range sizes (either empirical or simulated) to generate MDE patterns. These
studies have emphasized the interaction between geographic range size and domain size,
usually revealed by partitioning the RSFD into quantiles. We conjectured that it might be
revealing to take the quantile approach to its logical limit in studying the behaviour of
simple MDE models. In this study, we explore the MDE patterns that are expected for
assemblages of species that have identical range sizes. Independently, Connolly (in press) has
explored the spreading dye model with the same strategy.
Objectives
We use Monte Carlo versions of both one- and two-dimensional spreading dye models to
explore the role of range size in driving species richness patterns. We ask two questions: (1)
How do the predictions of MDE models change as a function of geographic range size, in
both one and two dimensions? (2) How do the predictions of MDE models change as the
assumption of strict range cohesion is relaxed and species are allowed to disperse longer
distances within a bounded geographic domain?
MATERIALS AND METHODS
Domains and range sizes
We simulated the random origin and spread of species in a gridded geographic domain.
The one-dimensional domain was 1 ? 100 linear grid cells (line segments) and the two-
dimensional domain was 100 ? 100 grid cells (identical, square cells). We created
assemblages of 100 species, in which all species shared an identical range size. For the
one-dimensional domain, these range sizes were 1, 2, 4, 8, 16, 32, 64, 80, and 100 grid cells.
For the two-dimensional domain, the range sizes were 1, 2, 4, 8, 16, 32, 64, 128, 256, 512,
1024, 2048, 4096, 8192, 9000, and 10,000 grid cells. Because all range sizes are identical
within each assemblage, we can directly assess the contribution of range size to MDE
patterns. In contrast, in most MDE analyses, empirical or simulated range size frequency
distributions are heterogeneous, with a mixture of larger and smaller ranges. When range
sizes are not equal, patterns of species richness are dominated by the effects of widespread
species, making it difficult to assess the spatial patterns and contributions of smaller ranges
(Lees et al., 1999; Jetz and Rahbek, 2001, 2002; Jetz et al., 2004; Lennon et al., 2004).
Simulation models: spreading dye model
We implemented a spreading dye model (Jetz and Rahbek, 2001) by randomly choosing an initial
grid cell to represent the origin of a species? geographic range. Subsequent cells for the range
were also chosen randomly, but range cohesion was enforced by restricting the candidate
cells to those adjacent to a currently occupied cell. In the one-dimensional domain, at each
time step only one or two grid cells were available for range expansion, depending on
whether the species had reached the edge of the domain or not. In the two-dimensional
Colwell et al.360
model, a maximum of eight adjacent cells was available for range expansion, as in a king?s
move in chess. Qualitatively similar results were obtained for two-dimensional models in
which only a maximum of four adjacent, non-diagonal cells were available (a one-square
rook?s move in chess). The simulation continued by randomly choosing and filling grid cells
until the geographic range for a species was complete. The next species in the assemblage
was then seeded in a new random grid cell, without consideration of previously placed
ranges. The procedure was repeated until all 100 species ranges in the assemblage had
been placed. Because no cells outside the boundaries of the domain were permitted to be
occupied, an expanding range that ?hit? a boundary expanded, as necessary, towards the
centre of the domain, in the one-dimensional model, or both inwards and laterally along the
boundary, in the two-dimensional model.
We then determined the total species richness in each grid cell, which could theoretically
range from zero (if a grid cell was never chosen) to 100 (if all species occurred in the grid
cell). As range size is increased, the minimum and average species richness per grid cell
also increases. In the extreme, for models in which the range size equals the domain size,
observed species richness in every grid cell is inevitably 100, with no variation among runs.
We simulated each assemblage 300 times and calculated the average species richness for
each grid cell. For the two-dimensional model, the results were plotted in a bivariate colour
plot, in which the ?hot colours? (red and orange) indicate grid cells with large average species
richness, and the ?cold colours? (blue and green) indicate grid cells with low average species
richness. Colour plots were scaled internally so that variation in richness within the domain
could be visualized most easily.
Stepping-stone model
For the two-dimensional domain, we built a model that relaxed the assumption of strict
range cohesion in the spreading dye model. The initial cell for a range was chosen at
random, just as in the algorithm specified above. To choose the second and subsequent cells,
one of the cells already occupied by the species was chosen at random to be the source of
a colonization event. (For choosing the second cell, this source cell is simply the initial cell.)
A compass orientation (0 to 360
) was selected at random as a dispersal direction for the
subsequent (?destination?) cell. The dispersal distance to the destination cell from the source
cell was chosen as a random variable from a Poisson distribution:
P(d) =
?
de?

d !
,
where P(d) is the probability of dispersing distance d, and ? is the Poisson parameter.
Dispersal distance d was measured in grid cell units. If the destination grid cell selected in
this way was not already occupied by the species in question, it was added to the species?
geographic range. If the destination grid cell was already occupied by the species, or if the
dispersal distance was beyond the map boundary, the same source cell was used and a new
destination grid cell was selected. This process was repeated until the geographic range was
filled for a given species. All species were simulated in the same way (with identical ? values),
and average species richness per grid cell was calculated and plotted as described for
the spreading dye model. All simulations were carried out using BioGeoSim, a dedicated
software package programmed in Delphi for modelling geographic ranges and species
richness patterns in gridded domains (Gotelli et al., 2007).
Geometry of simple mid-domain effect models 361
RESULTS
One-dimensional spreading dye model
Figure 2 depicts the effects of geometric constraints on ranges of different size in a
one-dimensional domain. For an assemblage in which all species have a range size of only
one grid cell, the pattern is purely random and uniform, because no geometric constraints
are in play. As range size increases (range size r = 2 to 4 grid cells), the average species
richness in the centre of the range shows a reduction in the variance, because ranges begin
to overlap more often, so the richness count evens out spatially. Meanwhile, two symmetric
peripheral ?peaks? appear near the edges of the domain, reflecting a mixture of ranges
that landed within the boundaries, but extend into the area of the peaks, and ranges that
originated on or near the edge of the domain and then randomly accumulated back towards
the centre. As range size increases (range size r = 8 to 32 grid cells), the relative height
of these peaks increases, with a progressive narrowing of the canyon floor of richness
near the centre of the domain. This narrowing is caused by the increasing penetration
into the interior of the domain by larger and larger ?reflected? ranges that were forced
inwards.
With the transition from 32 to 64 grid cells per range, range size now exceeds half of the
100-cell domain, so that the canyon floor inverts to a flat-topped plateau. The plateau
lies between 100 ? r on the left and r on the right side domain, with a richness of exactly
100 species, because all ranges must overlap in this region. Thus, as range size r increases
from 64 to 100 grid cells, the width of the richness plateau expands. Connolly?s (2005)
analytical model for the one-dimensional spreading dye model specifies the exact
quantitative expectations for all regions of the richness plots in Fig. 2.
Two-dimensional spreading dye model
Figure 3 shows the colour-graded pattern of average species richness with increasing range
size in a two-dimensional geographic domain. At small range sizes (1 to 8 grid cells), the
pattern is virtually random, although as range size increases, more small-scale aggregations
appear in the domain. (Repeated runs of the model show that these aggregations do not
appear consistently in the same locations.) At larger range sizes (16 to 128 grid cells), an
increasing concentration of species richness emerges in a band running parallel to the edge
of the domain, concentrated even more strongly near the corners at higher range sizes.
These bands correspond to the peripheral peaks in the one-dimensional model, and like
them move towards the centre of the domain as range size increases. The four local richness
peaks near the corners of the domain arise from the intersection of the peripheral bands,
because ranges that are ?reflected? from both adjacent sides overlap in the corners. At still
larger range sizes (256 to 2048 grid cells), the band expands to form a distinct ring or
doughnut, with a crater of species richness in the centre of the domain. The ring contracts
for the same reason the peaks converge in the one-dimensional model: larger ranges are
reflected by the borders and forced further towards the centre. In the transition from 2048 to
4096 grid cells, the doughnut coalesces to a classical mid-domain peak, just as in the
one-dimensional model. As the range size is increased beyond 5000 grid cells, half the size
of the domain, the area of the peak becomes a plateau, and expands towards the edges of
the domain (8192 to 9000 grid cells).
Colwell et al.362
Two-dimensional stepping-stone model
Figure 4 depicts the simultaneous effects of altering dispersal distance (?) and geographic
range size in the stepping-stone model. At very small dispersal distances (? = 0.5 to 1.0),
the model behaves much like the two-dimensional spreading dye model with strict range
cohesion, although the formation of the ring or doughnut of species richness at inter-
mediate range sizes is not so apparent, probably because reflected ranges are less
constrained in their location. As the dispersal distance increases (? = 2 to 16), a broad
Fig. 2. One-dimensional expected species richness plots. The x-axis represents position along the
number line, while the y-axis represents expected species richness. Each plot represents a different
simulation for an artificial assemblage of 100 species with identical range sizes. Each point represents
the average species richness in the one-dimensional domain, based on 300 stochastic runs of the
model. Note that the y-axis scale varies among the plots.
Geometry of simple mid-domain effect models 363
mid-domain peak emerges at intermediate to large geographic range sizes, with only
a narrow band of decreased richness near the edges and corners of the domain.
As ? increases from 16 to 32 grid cell units, the patterns change abruptly, producing a
second wave of peripheral richness peaks that coalesce centrally at high range sizes, but
which fail to become circular. We conjecture that this abrupt shift and pattern are caused by
dispersal distances being so long that many trial destination cells fall outside the domain.
Repeated trials do not find all grid cells in the domain with equal frequency, imitating the
pattern with no dispersal (Fig. 3). As ? increases from 32 to 128 grid cell units, a complex
Fig. 3. Contour plots representing expected numbers of species in a 100 ? 100 cell grid. These plots
were generated using a simple spreading dye model for assemblages with different geographic range
sizes. Blue represents the lowest species density and red represents the highest species density. Each
graph represents the mean of 300 stochastic runs of the model.
Colwell et al.364
Fig. 4. Contour plots of expected species richness in a stepping-stone model. Each row represents
a different value of ?, the Poisson dispersal parameter, and each column represents a different
geographic range size. Domain size and colour scale as in Fig. 3. Each graph represents the mean of
300 stochastic runs of the model.
Fig. 5. Species richness for all 1599 species of birds endemic to Africa (left-hand map) and for 190
small-ranged (each occupying 10 or fewer 1 ? 1 degree quadrats) local endemic bird species (right-
hand map), which tend to occur along the periphery and in mountainous regions (data from Jetz et al.,
2004). The concentration of species along the periphery of the domain resembles some of the patterns
that emerge from spreading dye simulations of small-ranged species (Figs. 2 and 3).
Geometry of simple mid-domain effect models 365
?wavelet? pattern emerges, and geometrically symmetric hot-spots of species richness form
in the domain. With a dispersal distance of 128 grid cells and geographic range sizes of
greater than 4096 cells, the pattern again returns to a simple flat plain of species richness.
These complex patterns cannot be easily interpreted, and they are probably idiosyncratic
to the particular combination of domain size, domain shape, range size, and dispersal
distance.
DISCUSSION
Most previous stochastic models of range location in a bounded domain produce a classic
mid-domain hump or, for smaller ranges, a plateau of richness [Fig. 1 shows an empirical
case; see Figures 2 and 3 in Colwell and Hurtt (1994) for a theoretical example]. But, unlike
our simulations here, previous models ? with the exception of two of the models of
Connolly (in press) ? have consistently used heterogeneous range size frequency distributions
(RSFDs). It has long been appreciated that large-ranged species dominate the overall
pattern of species richness in gradient plots (Pineda and Caswell, 1998; Lees et al., 1999) and maps
(Jetz and Rahbek, 2001), ?voting? in proportion to their range size. Any effects of small and inter-
mediate-sized ranges are masked by the strong mid-domain effect (MDE) of the large-ranged
species.
In a study that developed independently of ours, Connolly (in press) used an analytical
version of the one-dimensional spreading dye model (Equation 2 in Connolly, 2005) and a
two-dimensional simulation model that was identical to our model with full range cohesion
(Fig. 3) to examine richness patterns for assemblages with constant range size. His results
(Connolly, in press, Figure 5B, 5C, and 5D and accompanying text) were identical to ours for the range sizes
he explored (10, 40, and 70% of the domain for the one-dimensional model; 10 and 50% of
the domain for the two-dimensional model). He points out that the abrupt discontinuities
shown by simple, one-dimensional spreading dye models [e.g. Figure 5B in Connolly (in press)
and our Fig. 3] are not evident in alternative models in which ranges limits are determined
stochastically, with no fixed RSFD.
Understanding the distributional pattern of small-ranged species, which often differs
substantially from the pattern for total species richness, requires partitioning ranges into
range-size quantiles (e.g. Jetz and Rahbek, 2001) or separating out small-ranged species for special
study. Major studies of continental-scale species richness in two-dimensional domains
(e.g. Jetz and Rahbek, 2002; Rahbek et al., 2007) and along a variety of more local, one-dimensional
gradients (e.g. Cardel?s et al., 2006; Dunn et al., 2006) reveal that the effects of environment
and geometric constraints on species richness patterns differ strikingly among range-size
quantiles.
The complex two-dimensional patterns of species richness that emerge from the
spreading dye model for small range sizes (Figs. 3 and 4) may provide some insight into
empirical richness maps. The left-hand map in Fig. 5 (based on Jetz et al., 2004) shows the observed
pattern of total species richness for sub-Saharan African birds, on a 1
 latitude ? longitude
grid (1599 species, 1738 map grid cells). Completely obscured by the richness patterns of
larger-ranged species is the right-hand map, which shows the species richness of the subset
of 190 bird species with the very smallest geographic ranges (? 10 grid cells).
The concentrations of these small-ranged species (Fig. 5, right-hand map) in interior
mountainous areas (Cameroon Highlands, Albertine Rift Mountains, Kenya Highlands,
Eastern Zimbabwe mountains, Lesotho Highlands) is obvious, and probably reflects the
Colwell et al.366
effects of topography on speciation rates (Graves, 1985; Rahbek and Graves, 2000, 2001; Fjelds? and
Rahbek, 2006). But a second pattern is less readily explained: small-ranged species also tend
to be concentrated along the coastlines (Fig. 5, right-hand map). This pattern might
be accounted for by a combination of coastal mountains, unusual coastal habitats, and
narrowly distributed coastal climates. But the pattern is also reminiscent of the peripheral
band (doughnut) of richness for small-ranged species that emerges in our two-dimensional
null model (Fig. 3). If the doughnut effect contributes to the empirical pattern of African
birds, the mechanism would likely be the same as in the spreading dye model: a range that
cannot expand past a boundary tends instead to expand laterally, along the edge of the
domain, producing a level of range asymmetry near domain boundaries that cannot be
accounted for by topography, climate or historical factors. In the model, range size is
prescribed; in nature, on the other hand, peripheral ranges that do not expand along the
coast but, instead, occupy a ?truncated? range (Sandel and McKone, 2006) of arbitrarily small size
may have been subject to higher probability of extinction (Colwell and Hurtt, 1994). Only carefully
constructed null models and analyses of environmental and historical factors can uncover
the causes of this and other intriguing patterns of small-ranged species that remain hidden
under the smokescreen of wider-ranged species. Empirical and theoretical studies would
benefit from even further deconstruction of species richness gradients by range size than is
currently practised.
The domains illustrated in Fig. 3 can be viewed as classes of patterns representing
size-based categories of ranges. Any natural range-size frequency distribution can be
approximated by a mixture model composed of a weighted combination of the range sizes
in the figure [or perhaps some other series of discrete range size classes (Lees et al., 1999,
pp. 541?546)]. In this view of the model, the resulting richness pattern for all range sizes
considered together can be envisioned as a weighted sum of the patterns in the figure, or
perhaps a series of layers of varying depth, representing the proportions in the mixture.
In this way, the artificial quality of single-range-size models can be made realistic, while still
preserving the distinct geometric effects that emerge for species of differing ranges.
The two-dimensional stepping-stone model produced an even greater wealth of pattern
(Fig. 4), including the subset of patterns that arose from the strict range cohesion of
the spreading dye model (Fig. 3). As Connolly (2005) noted, introducing dispersal into
a range-based model converts richness from the probability of range overlap to the
probability of cell co-occupancy, with increasingly porous ranges weakening the mid-
domain pattern (bottom rows in Fig. 4). As dispersal distance increases, a phase of constant
species richness throughout the domain reappears, resembling the pattern for small ranges
with cohesive ranges, but of course with much higher richness (as illustrated in Fig. 4 by
the column with range size = 2048). With very long dispersal distances (? ? 32), strange
patterns appear, as most attempts at dispersal point outside the domain, so that trial and
error redirects unsuccessful ?destinations? into the left-hand tail of a widening Poisson
distribution. After many unsuccessful trials, the algorithm eventually finds a small subset of
cells within the domain that meet these constraints and accumulate hot spots of species
richness. These latter parameter combinations are not realistic, but they are included
for the sake of completeness, curiosity, and aesthetics. No-one ever said that constrained
randomness should not be beautiful, as well as instructive.
Geometry of simple mid-domain effect models 367
ACKNOWLEDGEMENTS
We dedicate this paper to Larry Slobodkin, who taught his students to think beyond the constraint of
orthodoxy, while learning from the wisdom of the past. ?Classical?, he liked to say, ?means old, and
mostly wrong, but worth studying?. We are grateful to Hector Arita, Sean Connolly, Walter Jetz, and
Thiago Rangel for their critical comments and useful suggestions. R.K.C. was supported by the
US National Science Foundation, DEB-0072702. N.J.G. and G.L.E. acknowledge NSF grants
DEB-0107403 and DEB 05-41936 for support of modelling and null model research. G.R.G.
was supported by the Alexander Wetmore fund of the Smithsonian Institution and the Danish
National Science Foundation (Grant J. 21-03-0221). C.R. acknowledges the Danish National Science
Foundation (Grant J. 21-03-0221) for support of macroecological research.
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