The latitudinal gradient in species richness,
perhaps the most conspicuous
feature of global biogeography, has
intrigued ecologists and biogeographers
continuously since the times of de Candolle
(Ref. 1, pp. 1270­1276) and Wallace
(Ref. 2, pp. 65­68). In spite of this long
history, consensus on the causes of tropical
peaks in species richness remains elusive3­5.
The overwhelming range of
hypotheses proposed to account for
changes of species richness (Palmer6
lists 120 named hypotheses for variation
in species richness or coexistence, and
Rohde7 identifies 28 specifically applied
to the latitudinal gradient) makes the
task of sorting out which factors are
causal and which are incidental a daunting
one. Increasingly, using computationally
intensive methods, ecologists and
biogeographers are looking for answers
at a regional or a global scale, putting
models and hypotheses7­15 to the test
against increasingly comprehensive distributional
data encompassing the still
poorly known tropics15­19.
This search has taken an unexpected
turn. In spite of the plethora of climatic,
ecological, evolutionary and historical
explanations proposed to explain biogeographic
diversity patterns, something
fundamental has, until recently, been
almost completely ignored: the geometry
of species ranges in relation to geographical
boundaries. Based both on simulations11,12,14
and on analytical null models13,15,
it is now clear that a mid-domain
peak or plateau in species richness is
inevitable for virtually any set of ranges,
theoretical or empirical, when these
ranges are randomly placed within a
bounded geographical domain, in the
complete absence of any supposition of
environmental gradients within the
domain. Qualitatively, this result survives
a wide range of changes in model
details and assumptions. Quantitatively,
it explains a surprisingly large proportion
of geographic variation in species
richness for diverse groups of organisms,
for the empirical cases so far examined.
In spite of conceptual antecedents in the
ecological literature dating back more
than 40 years20, and a substantial literature
on formally analogous problems in
niche overlap and phenology (Box 1),
this important result seems to have gone
unnoticed until 1994 (Ref. 11).
What do the geometric
models predict?
There is a growing consensus that the
regional pool of species is the key determinant
of the species composition of
more or less unsaturated local communities21,22.
Based on regional biotas, geometric
models11,13­15 demonstrate that
the stochastic placement of species
ranges between shared geographic
boundaries can generate precise predictions
of species richness at points
between the boundaries. For example, in
the case of random, one-dimensional
placement of ranges between two boundaries,
the null models predict a convex,
symmetrical pattern of species richness:
this pattern is either parabolic11,13 or
quasi-parabolic11,15 depending upon
alternative distributions of range sizes
and of midpoints (Fig. 1; Boxes 2 and 3).
An instructive way to grasp this idea
quickly is to experiment with a null
model that simulates range size and randomizes
range placement within differently
defined boundaries11 (Fig. 1). (A
graphical simulation program, that not
only implements the principal stochastic
range model variants, but also allows
input of empirical range-size frequency
The mid-domain effect: geometric
constraints on the geography of
species richness
Robert. K. Colwell and David C. Lees
Geographic patterns of species richness are influenced by many factors, but the role
of shared physiographical and physiological boundaries in relation to range-size
distributions has been surprisingly neglected, in spite of the fact that such geometric
constraints lead to mid-domain richness peaks even without environmental
gradients (the mid-domain effect). Relying on null models, several recent studies
have begun to quantify this problem using simulated and empirical data. This
approach promises to transform how we perceive geographic variation in diversity,
including the long unresolved latitudinal gradient in species richness. The question is
not whether geometry affects such patterns, but by how much.
Robert Colwell is at the Dept of Ecology and Evolutionary Biology, U-43, University of Connecticut,
Storrs, CT 06269-3043, USA (colwell@uconnvm.uconn.edu); David Lees is at the
Dept of Palaeontology and the Biogeography and Conservation Laboratory (Dept of Entomology),
Natural History Museum, Cromwell Road, South Kensington,
London, UK SW7 5BD (dcl@nhm.ac.uk).
70 0169-5347/00/$ ­ see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S0169-5347(99)01767-X TREE vol. 15, no. 2 February 2000
40 Rausher, M.D. (1992) The measurement of
selection on quantitative traits: biases due to
environmental covariances between traits and
fitness. Evolution 46, 616­626
41 Crespi, B.J. and Bookstein, F.L. (1989) A
path-analytic model for the measurement of
selection on morphology. Evolution 43,
18­28
42 Schluter, D. and Nychka, D. (1994)
Exploring fitness surfaces. Am. Nat. 143,
597­616
43 Gould, S.J. and Vrba, E.S. (1982) Exaptation ­ a
missing term in the science of form.
Paleobiology 8, 4­15
44 Clausen, J. et al. (1940) Experimental Studies on
the Nature of Plant Species. I. Effect of Varied
Environment on Western North American
Plants, Carnegie Institution
45 Travisano, M. (1997) Long-term experimental
evolution in Escherichia coli. VI. Environmental
constraints on adaptation and divergence.
Genetics 146, 471­479
46 Stratton, D.A. and Bennington, C.C. (1996)
Measuring spatial variation in natural selection
using randomly-sown seeds of Arabidopsis
thaliana. J. Evol. Biol. 9, 215­228
47 Pigliucci, M. and Schlichting, C.D. (1997) On the
limits of quantitative genetics for the study of
phenotypic evolution. Acta Biotheoretica 45,
143­160
48 Houle, D. (1991) Genetic covariance of fitness
correlates: what genetic correlations are made
of and why it matters. Evolution 45, 630­648
49 Gromko, M.H. (1995) Unpredictability of
correlated response to selection: pleiotropy
and sampling interact. Evolution 49,
685­693
50 Steppan, S.J. (1997) Phylogenetic analysis of
phenotypic covariance structure. I. Contrasting
results from matrix correlation and common
principal component analyses. Evolution 51,
571­586
PERSPECTIVES
TREE vol. 15, no. 2 February 2000 71
distributions and exports the results, is
freely downloadable from the web http://
viceroy.eeb.uconn.edu/rangemodel).
To date, two analytical versions of the
one-dimensional geometric model have
appeared (Boxes 2 and 3)13,15. The first of
these13 generates (with confidence intervals)
the predicted species richness
for the fully stochastic interaction of
range size versus range midpoint within
predefined boundaries; the second15 supports
analytical estimation of expected
species richness, based on empirical
range-size frequency distributions, and
extends the approach, tentatively, to two
dimensions.
The richness gradient predicted by
the geometric models certainly matches,
in its bilateral symmetry and its unimodality,
the latitudinal patterns of
species richness documented for most
higher taxa11,13. The models explain a
remarkably high proportion of empirical
variation in latitudinal richness for some
taxa13 (Fig. 2). A striking prediction of
these models is that richness patterns on
elevational gradients and on bathymetric
gradients should also be unimodal, with
a mid-gradient richness peak11,14,15; this
prediction of a mid-gradient richness
peak defies the longstanding dogma23
that species richness decreases monotonically
with increasing elevation and
depth24,25. In fact, most data and recent
meta-analysis suggest that richness patterns
with a medial peak might be more
the rule than the exception for both
the elevational gradient15,16,23,26 and the
bathymetric gradient14,27, although the
PERSPECTIVES
Fig. 1. A fully stochastic null model for species richness gradients within a bounded domain11. For particular sets of species, the unit domain might represent
elevation from sea level to mountaintop16, ocean depth from the surface to the abyss14, distance from one end of a large island to another (Fig.
3), latitude from the northern to the southern end of the continental New World (Fig. 2a), or latitude from the northern to the southern limit of the distribution
of a clade (Fig. 2b). In (a), the range size for each species is plotted against its range midpoint (500 species shown). In this model (Box 2), midpoints
and range values are generated as a uniform random coverage of feasible values. In (b), the ranges for a subset (50 species) of the points in (a)
are shown as horizontal lines centred on their midpoints. Because the domain is bounded at 0 and 1, all midpoint-range coordinate pairs ­ the
points in (a) and (b) ­ must lie within the isosceles triangle. For any point x in the domain, richness is computed as the number of horizontal range lines
that a vertical line at x (the broken line) would intersect. In (c), the closed circles show the pattern of species richness across the domain for the points
in (a) and (b). The open circles plot species richness when maximum range size is limited to half the domain (0.5) and the crosses show richness for a
maximum range size of 0.25. The ordinate in (c) scales richness as a proportion of all species in the simulation. In all cases, the richness peaks at the
domain midpoint. Only the top curve is parabolic and peaks at a proportional richness of 0.5. Note the more pronounced mid-domain effect when larger
ranges are permitted. Modified, with permission, from Ref. 11.
0
(a)
Rangesize
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0.5
1.0
(c)
Speciesrichness
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1.00
(b)
Rangesize
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1.0
1.0x
Trends in Ecology & Evolution
Location on domainLocation on domain
Box 1. Antecedents and analogues
The species range has long been studied as the fundamental unit of species richness gradients29­33. Although some modeling approaches have incorporated
aspects of range size evolution, distribution or overlap29,34­37, species richness has traditionally been viewed as a function of area, history and
climatic or biological gradients, rather than geometric constraints. Colwell and Hurtt11 were apparently the first to show that mid-domain species richness
peaks can arise from geometric constraints alone, although analogous problems in ecology have been studied for more than 40 years.
In his classic paper on the relative abundance of species, MacArthur20 proposed two `broken stick' models. The first of these (n independent, uniform
random breaks in a single unit `stick', producing n 1 pieces) became the focus of much further attention (e.g. May38) ­ notwithstanding MacArthur's
attempt to bury it39. MacArthur's second model was a random, `overlapping niche' model: two independent, uniform random hits to the unit `stick'
defined the endpoints of each species (n pairs of points for n species). However, in spite of the niche overlap metaphor, MacArthur was concerned only
with the distribution of the length of the pieces, not the patterns of overlap among them along the unit line. Quickly dismissed for its poor fit to relative
abundance distributions, the two-hit broken stick model was resurrected by Pielou40,41 (and Dale42,43) to assess the randomness of distributional overlap
among species along gradients. Later, Pielou extended this approach, theoretically and empirically, to an analysis of latitudinal overlap among congeneric
seaweed species31,36,37. But neither MacArthur nor Pielou examined the expected distribution of range overlaps as a function of location on the
domain; thus, they failed to discover the mid-domain peak of richness.
The overlap of phenological events within a temporal domain is precisely analogous to (one-dimensional) geographical range overlap within a spatial
domain. To test for non-random spacing of flowering periods, Cole44 and others (reviewed by Gotelli and Graves45) computed expected pairwise overlaps
and randomized empirical flowering periods within the flowering season. Cole44 and Sugihara46 extended Pielou's41 analytical approach to phenology
and to utilization spectra in general. Nonetheless, no one in the phenology debate examined the expected number of simultaneously flowering
species as a function of time of year for randomized flowering periods, for which the mid-domain effect predicts a mid-season flowering peak.
Perhaps the most obvious manifestation of geometric constraints on patterns of geographic range on a bounded domain is the limitation placed on
the relation between range size and range midpoint (Fig. 1b). Species with midpoints near a boundary necessarily have relatively small ranges or, conversely,
species with increasingly larger ranges must have their midpoints increasingly nearer the centre of the domain. Pielou31,37 was apparently first
to recognize (and plot) this constraint, followed by Graves47 (elevation), Rhode et al.48 (latitude), and Pineda27 (depth). Although their contributions
pointed out the boundary constraint on range placement ­ ultimately the cause of richness peaks in stochastic range models ­ none of these authors
noted the expected mid-domain peak.
72 TREE vol. 15, no. 2 February 2000
richness peak is often not precisely in the
middle of the gradient.
Another pattern predicted by the
models (but in this case predicted by no
other hypothesis) is that, regardless of
latitude, the richness of terrestrial
groups should peak in the middle of
large, isolated, continuous biomes to
which they are endemic, as a result of the
constraints imposed by biome boundaries.
Lees et al.15 documented a striking
confirmation of this prediction for Madagascan
rainforest (Fig. 3).
Naturally, boundaries vary in their
potential to limit species distributions.
For example, mountaintops set absolute
elevational limits for non-airborne organisms,
and land­sea limits fortified by
deeper ocean trenches offer strong
resistance to range expansion within
most terrestrial clades. Other limits
might be more yielding. Lyons and
Willig12 included physiological frontiers
along with topographic features, such as
orographic barriers, in the category of
`soft' boundaries; whereas Colwell and
Hurtt11 used `hard' to refer to any naturally
definable biogeographic barrier that
presents some degree of resistance to
dispersal. Thus, defining, quantifying or
even ranking barrier resistance is likely
to remain a challenge, because the effectiveness
of boundaries depends greatly
on the temporal scale on which they are
considered and on the phylogenetic constraints.
At least, physiological limits to
the geographic distribution of a clade
might be just as effective as sharp physiographical
boundaries on the shorter
time scales most relevant to macroecological
patterns28.
The ongoing debate over Rapoport's
rule8 and the idea that the evolution of
broadened physiological tolerance
favours larger range sizes towards the
harsher extremes of environmental gradients
have motivated much of the
recent work discussed here. The relationship
between Rapoport's rule and
the geometric models is complex (Box 4),
but the models clarify the constraints on
range size variation within domains and
highlight the limitations of Rapoport's
rule as an explanation for richness
patterns.
The mid-domain effect: a geometry
of ranges and richness
We call the geometric theory of species
richness gradients the `mid-domain
effect'. We define the mid-domain effect
as `the increasing overlap of species
ranges towards the centre of a shared
geographic domain due to geometric
boundary constraints in relation to the
distribution of species' range sizes and
midpoints'. Here, we place the limits of a
PERSPECTIVES
Box 2. A fully stochastic null model for species richness gradients
Model 2 of Colwell and Hurtt11 is, in essence, a fully neutral model. No assumptions are made as
regards range size and range placement, except for the geometric constraints of boundaries. This
model is fundamentally equivalent to the two-hit broken stick model of MacArthur20 described in
Box 1, and the binomial model of Willig and Lyons13.
To demonstrate this equivalence and to relate this model to the others in Box 3, the probability
of intersecting a geometrically constrained line segment generated by chance alone was
sought13. This is the unconditional probability p(x,r) that a range of size r overlaps a sampling point
at x, where x is any point within the domain (Fig. 1b). According to Bayes' Theorem, p(x,r) p(r)
p(x|r). The distribution of range sizes determines
the probability p(r). For a random uniform
distribution of midpoint densities within the
midpoint constraint triangle, as shown in the figure
here, p(r) is a linear decreasing function of
range size, p(r) (2 2r) (Fig. 1b)11,20. We know
the conditional probability p(x|r) from the Lees
model15 (Box 3) for the triangular areas in the
plot of r against x figured here. (The bars represent
two different ranges at their leftmost positions.)
The two diagonals x r and x (1 r)
bound an hourglass-shaped region where p(x|r)
is maximal for the two triangles {A, B} as
{1, r/(1 r)}, whereas for {C, C'} p(x|r) increases
with x up to these diagonals as {x/(1 r),
(1 x)/(1 r)}15. We next transform the r axis to
the range size distribution for this model by
applying the inverse of its ranked distribution
function (2r r2) (Refs 11, 20), which is
[1 (1 r)0.5] (Ref. 11). We then integrate, for
any value of x, the range intersection probability
densities resulting from the product of p(r) and
p(x|r) for {A, B, C, C'} in the figure, namely {2 2r, 2r, 2x, 2 2x}. These values increase towards one
at the intersection of x r and x (1 r). The integral for this distribution of r evaluates to 2x 2x2,
a true parabolic curve describing proportional species richness.
This is precisely the same result that Willig and Lyons13 elegantly derived using a binomial
model. Suppose two points placed at random on a unit domain define the endpoints of a species'
range. (This is exactly the 2-hit broken stick model of MacArthur20, discussed in Box 1.) Now consider
a sampling point x on the domain. Then the proportion of ranges for which both endpoints
of the range fall to the left of x is x2 and the proportion falling to the right of x is (1 x)2. Therefore,
the proportion that actually intersect the sampling point is 1 x2 (1 x)2 2x 2x2. From this
result, the expected peak proportional richness at mid-domain for this model is 0.5, as shown
mathematically by Willig and Lyons13 and graphically by Colwell and Hurtt11.
1
0
0 1Location on domain (x)
Rangesize(r)
B
A
C C
r =
x
r=
1
­
x
Trends in Ecology & Evolution
(Online Fig. I)
Box 3. Constrained null models for species richness gradients
In addition to the fully stochastic model discussed in Box 2, we can distinguish two other fundamental
geometric null models for one dimension, constrained according to the dependence
structure of either range size or range placement. These correspond to the model of Lees et al.15
and Model 3 of Colwell and Hurtt11. As for the fully stochastic case (Box 2), each of these two
models generates a convex, symmetrical pattern of species richness owing to the mid-domain
effect, differing only quantitatively. (The simulation program of Colwellimplements all three models:
http://viceroy.eeb.uconn.edu/rangemodel)
The model of Lees et al.15 effectively randomizes range placements given range sizes. This
probabilistic, analytical null model assumes a uniform distribution of range sizes, but can be partitioned
to approximate empirical range size distributions (see Box 5). Here, we seek p(x|r), probability
densities derived15 for the triangles in the figure in Box 2, whose integration for this distribution
of r evaluates to the proportional species richness function:
­(1 x)ln(1 x)­xln(x)
This describes a quasi-parabolic curve whose peak at x 0.5 takes the value ln(2), or about
0.69 of the total species pool endemic to the domain. Pineda and Caswell14 introduced a Monte
Carlo version of the same model.
The second constrained geometric null model (Model 3 of Colwell and Hurtt11) is based on
ranges drawn at random from geometrically feasible values for a given range midpoint (although
left or right endpoint could just as well be used). In this case, we seek p(x|m). With a uniform random
distribution of midpoints (as in Colwell and Hurtt's Model 3), this model generates a hollowcurve
range size frequency distribution that seems biologically quite realistic35, but produces a
build-up of particularly small ranges towards the domain boundary, where ranges are allowed
only a small amplitude11. As indicated graphically by Colwell and Hurtt11, the expected peak
proportional richness for this model (for a proportional maximum range size value of unity) is
approximately 0.3, with a medially rather flattened, quasi-parabolic, species richness curve.
Pineda and Caswell14 also used this model in its Monte Carlo form. No analytical version of this
model has yet appeared.
TREE vol. 15, no. 2 February 2000 73
domain (in its mathematical sense of a
union of a connected set and its boundary
points) in a geographic context,
either physiographically (e.g. the landsea
interface14), biologically (e.g. biome
interfaces, such as tree-line or the limits
of rainforest in Madagascar15), as the
limits of a region of endemism (e.g. the
northern and southern range limits of
New World marsupials12) or arbitrarily
(e.g. 20°N to 20°S latitude). Although we
focus on one-dimensional views of geographical
domains and their geometric
representation on a line, the concept
applies equally to geographical areas and
to volumes. Although clearly related to
earlier models of geometric constraints
(Box 1), the mid-domain effect focuses on
the emergent, macroecological pattern
that these constraints predict, rather
than on the effects of constraints on individual
ranges. (The effect has also been
referred to, without explicit definition, as
the `geometric constraint model'14 or `the
Périnet effect'26).
In assessing the role of the middomain
effect in specific cases, an important
element of this definition is that the
species considered must, collectively,
share the same boundaries. In other
words, species ranges that are not fully
contained in the geographical domain
under consideration are excluded
(although such taxa can still be analysed
using different domain limits). The resulting
shape, intensity and, thus, detectability
of the mid-domain effect depends
on the taxa involved; the distribution of
their range sizes and range midpoints;
the existence, location and limiting potential
of the boundary constraints; the
sampling position or region; and on countervailing
or intensifying biological, climatic
or historical determinants of
species richness. Thus, confirmation or
rejection of the mid-domain effect in any
particular case requires statistical evaluation
of an appropriate null model in
relation to observed patterns13­15.
PERSPECTIVES
Fig. 2. Comparison of actual latitudinal patterns of species richness for New World marsupials (points) with patterns predicted by the fully stochastic
null model10 (solid curves) (Fig. 1; Box 2). In each graph, the central curve represents the predicted mean richness for each latitude; the outer lines
define the 95% confidence bands. The same empirical data points are shown, with model predictions, for three different latitudinal domains: (a) the
continental New World, (b) the distributional limits of all New World marsupials and (c) the smallest domain within which 95% of all New World marsupial
species occur. Reproduced, with permission, from Ref. 13.
-60 -40 -20 0
Speciesrichness
0
10
20
30
40
20 40 60
(a)
-60 -40 -20 0
Speciesrichness
0
10
20
30
40
20 40 60
(b)
-60 -40 -20 0
Latitude (degrees)
Speciesrichness
0
10
20
30
40
20 40 60
(c)
TrendsinEcology&Evolution
Latitude (degrees)Latitude (degrees)
Fig. 3. Latitudinal proportional species richness, based on interpolated ranges, for species strictly
endemic to the Madagascan rainforest biome15. In (a) observed richness for ten faunal groups
(enariine beetles, tiger beetles, butterflies, ctenuchiine moths, chameleons, frogs, birds, tenrecs,
rodents and lemurs) is compared with an analytical null model (Boxes 3 and 5). The model computes
expected richness based on either two or four equal partitions of range size15. The middle
curve represents all 637 species (closed squares, observed; solid curve, 4-partition analytical
model). The top curve represents all 182 wide-ranging species ­ species with ranges that cover
at least half the domain (open squares, observed; solid curve, 2-partition analytical model). The
lower curve represents all 455 narrow-ranging species ­ species with ranges that cover at most
half the domain (open circles, observed; solid curve, 2-partition analytical model). Modified, with
permission, from Ref. 15. In (b) the same data are plotted as for the middle curve in (a) for all 637
Madagascan rainforest species, showing the fit to 4-partition analytical geometric null model,
which explains 85% of the variance (uncompensated for spatial autocorrelation).
(a)
26
0
0.2
0.4
0.6
0.8
1.0
24 22 20 18
Latitude (°S)
Speciesrichness(proportional)
16 14 12
(b)
0
0
100
R 2
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y = 1.06 x
200
300
400
100 200
Actual species richness (interpolated)
Predictedspeciesrichness
300 400
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74 TREE vol. 15, no. 2 February 2000
Are the predictions of the
geometric models confirmed?
Once the effects of geometry are
removed, we can look afresh at environmental
and historical influences on richness
gradients. The surprise will be if the
mid-domain effect turns out, after more
than a hundred years of biological and
historical hypotheses, to be the principal
factor determining the general shape of
empirical richness curves along gradients.
To date, four major studies have
assessed this heretical conjecture.
In their study of the distribution of
bats and marsupials throughout the continental
New World, Willig and Lyons13
found that, when latitudinal distributional
limits were used to approximate
geographic limits for these groups as a
whole, a geometric null model (Box 2)
can explain 69­94% of variation in
species richness patterns over the entire
transect (Fig. 2). Richness for marsupials
reaches its peak well south of the equator
(at about 20°S), out of phase with
climatic gradients, thus weakening
explanations based on climate or energy.
In an analysis of the distribution of
nearly 1200 species of insects and vertebrates,
including 637 species endemic to
the Madagascan rainforest (a biome
encompassing about 13 degrees of latitude
and 2100 m of elevation entirely
south of the equator), Lees et al.15 identified
geometry as the most important
cause of species richness gradients. A
geometric null model explained far more
of the variation (85% for latitude alone,
75% for latitude and longitude as orthogonal
dimensions) than either area or
energy (Fig. 3; Boxes 3 and 5). Latitudinally
and longitudinally, species richness
peaked at approximately the (middomain)
position and the level expected
from the model, given the location
of biome boundaries, the empirical
species range size distributions and the
size of the regional species pool. However,
for at least one large adaptive radiation
examined15,26, richness peaked
somewhat below the middle of the elevational
domain (as is common for
elevational richness gradients23), suggesting
a significant modifying influence
of other factors.
In a comprehensive study of the
elevational distribution of 2800 species
of birds in all tropical biomes of South
America (including both western and
eastern slopes of the Andes), Rahbek16
found that, once effects of area are factored
out, a mid-elevation peak in species
richness emerges ­ as predicted by geometric
models, although at an elevation
below the true elevational midpoint. He
concluded that the mid-elevation peak is
best explained by boundary constraints
PERSPECTIVES
Box 4. The geometric models and Rapoport's rule
Following Rapoport32, Stevens8 reported greater average latitudinal range among species at
higher latitudes, for certain taxa, a pattern he called Rapoport's rule. Extending the idea to both
elevational24 and bathymetric25 richness gradients, based on an analogy between low latitudes,
low elevations and shallow waters (and their opposites), Stevens supposed a monotonic
decrease in richness (now called into question for both elevation and depth14­16,23,26,27) and
increasing average range size at higher elevations and greater depths49. Stevens8,24,25 also proposed
that narrow ranges in tropical, low elevation and shallow-water communities might actually
cause high richness by the spill-over or mass effect of Schmida and Whittaker50: a hypothesis
that Stevens termed the Rapoport rescue effect.
Geometric constraints complicate any consideration of Rapoport's rule11,12,17,27. Colwell and
Hurtt11 showed that incorporating the boundary constraint in two basic null models of range size
versus range midpoint (Boxes 2 and 3) produces a pattern of average range size across a domain
that is either flat or opposite to that predicted by Rapoport's rule. Lyons and Willig12 went further,
arguing that a valid test of Rapoport's rule ought first to account for an expected negative correlation
between latitudinal range midpoint and range size forced by the boundary constraint27,44,47.
Using bounded null models to assess this correlation for New World bats and marsupials, they
showed that average range size for these groups does indeed decline with latitude (opposite to
Rapoport's rule) but declines less than predicted by the null models. Although Lyons and Willig12
suggested that this result is `in the spirit of Rapoport's rule', such a pattern certainly does not support
Stevens'8 mechanism for high tropical species richness, which invokes absolutely smaller
ranges among tropical species than among species at higher latitudes.
By contrast with the two Colwell models, the Lees model15 (Box 3) produces a pattern consistent
with Rapoport's rule. Lyons and Willig's12 observation that boundary constraints force a
steady decrease in range size as range midpoints become closer to a boundary (e.g. at high latitudes),
in opposition to Rapoport's rule, is complicated by the interaction of differing range size
distributions with boundaries11. Whether the final Rapoport curve is concave (standard), flat or
convex (reversed) depends upon the range size frequency distribution. Random placement of
ranges drawn from theoretical or empirical size frequency distributions with a higher proportion
of large ranges (such as the Lees model, which assumes a uniform distribution of range sizes)
can produce a standard Rapoport effect, whereas range size distributions with a preponderance
of smaller ranges tend to yield a reverse effect.
Ironically, high richness at domain centres is disproportionately the result of any wide-ranging
species, regardless of whether they predominate in a particular model11,14,15,49. Likewise, enforcing
a gradient of increasing average range size towards high latitudes, in accordance with
Rapoport's rule, generates a reverse latitudinal richness gradient, even without boundary constraints,
because of increased overlap of wider-ranging species at high latitudes11,49. Finally,
Rapoport's rule and the geometric models make entirely different predictions about richness gradients
and their causes on elevational and bathymetric transects11,14, because geometric constraints
predict mid-domain richness peaks, which Rapoport's rule alone cannot do.
Box 5. Evaluating the mid-domain effect for empirical range size distributions
The constrained models of Box 3 have been extended to empirical distributions of range sizes
and of midpoints. For the empirical version of the constrained range-size model, two approaches
have been taken to shuffle empirical range size distributions. Both assume a uniform distribution
of midpoints, within boundary constraints. Lees' approach, an analytical method based on partitioning
of range sizes, integrates the probability densities shown for the figure in Box 2 between
the relevant limits of r for each range size class, then weights the result by the proportion of
species in each class, before summing these terms15 (Fig. 3). In the second, Monte Carlo
approach, the empirical distribution of range sizes is sampled and each sampled range is placed
on the domain at random, given the midpoint constraints imposed by the size of the range. The
simulation program of Colwell (http://viceroy.eeb.uconn.edu/rangemodel) implements this
approach for empirical range sizes; the simulations of Pineda and Caswell14 (their Figs 5 and 6)
use a similar method for range placements. In an empirical version of the constrained midpoint
model, Lyons and Willig12 (in their `pseudorandom' simulations) assigned a range at random,
within boundary constraints, to each empirical midpoint. We suspect that incorporating both
empirical range size and range placement distributions into a geometric null model would produce
a model so constrained that statistical comparison with empirical richness patterns would
have little power to reject a false null model45. Evaluation of the fit of these null models to corresponding
empirical data should include not only comparison of predicted and observed richness
patterns on the domain13­15, but also comparison of patterns of midpoint placement (when
ranges are constrained) or range size distributions (when midpoints are constrained).
The methodology for empirical tests needs much further work, especially in devising ways to
separate the mid-domain effect (inevitably present) from biological and historical causes of richness
peaks. The challenge of applying fully analytical two- and three-dimensional geometric null
models to quantitative biogeographical patterns still remains, although Taylor and Gaines51 simulated
two-dimensional ranges on a sphere. As an approximation, Lees et al.15 produced a twodimensional,
geometric species richness map by iterating the one-dimensional, analytical null
model of Lees (see above) along the smaller (longitudinal) axis, given regional latitudinal range
size frequencies. The history of null models in other areas of ecology45 suggests that no single
null model for the mid-domain effect will be, or should be, declared the `correct' one. Rather,
as with other clusters of imperfect models, we are likely to find that `truth is the intersection of
independent lies'52.
TREE vol. 15, no. 2 February 2000 75
(the ocean and the mountaintops), with
other factors affecting the details of the
gradient.
In an analysis of bathymetric richness
gradients for northwest Atlantic gastropods
and polychaetes, Pineda and
Caswell14 compared the parabolic richness
gradients for these groups with patterns
formed by randomizing the placement
of empirical depth ranges. As
expected, the null model also produced
mid-domain richness peaks, but, for
both groups, model peaks and empirical
peaks were only partially matched in
location, magnitude and curvature. They
concluded that ` ... while geometric constraints
can contribute to parabolic patterns,
[the] observed patterns ... result
from a non-random distribution of
species along the depth gradient'.
In summary, work completed so far
indicates a strong influence of the middomain
effect in shaping geographic
patterns of richness, when evaluated
against competing biological and climatic
hypotheses. The components of
these patterns that cannot be predicted
by geometric null models (e.g. off-centre
elevational or bathymetric richness
peaks or large residuals for individual
sampling sites) stand out as appropriate
targets of research for non-geometric
explanations.
Stochastic versus deterministic
perspectives
Although comparison with an appropriately
constructed and parameterized
geometric null model is necessary to
detect the probable influence of the middomain
effect, the demonstration of
such influence neither assumes nor
proves that species' ranges, or the
boundaries that constrain them, are individually
random in shape and in placement,
with respect to ecological, evolutionary
and historical causes. Of course,
they are not, but the seeming paradox is
a familiar one. No biologist would consider
it disturbingly paradoxical that
genes and environment largely determine
the individual heights of a sample
of 1000 adult women, whereas the distribution
of their heights neatly fits a normal
curve based on mean and variance
in height. Analogously, the mid-domain
effect is a macroecological outcome of
geometric constraints that, for large
numbers of taxa with their own history
and adaptations, shape richness patterns
along geographic gradients.
These constraints arise from boundaries
(both geographic limits and biological
thresholds) and from an emergent statistical
property of taxonomically defined
biotas: their range size and midpoint
distributions.
How, then, are we to view the real
world relevance of a geometric null
model? There seem to be two, non-exclusive
approaches to interpreting empirical
species richness patterns. On the one
hand, we can assume that range sizes
and range placements are governed by a
strong element of chance. On the other
hand, we could view a good fit between
null models and observed species richness
patterns from a deterministic perspective.
Thus, ranges largely determined
by the diverse adaptations and the
histories of species in a regional assemblage
might nonetheless fit a geometric
null model, rather than yielding an emergent
richness response to an environmental
gradient. Lyons and Willig19
express this view: `Within a larger taxon
such as bats or marsupials, no single factor
may attain hegemony in limiting the
distribution of constituent species, and
as a consequence, range boundaries may
have a geographic distribution that is
quite similar to those produced by
chance, even though different deterministic
factors account for the limitations of
each species.'
A specific way to reconcile the middomain
effect with determinism is to
view range overlap from a phylogenetic
perspective at the species level or above.
Sister populations that have diverged
allopatrically or parapatrically might
emerge over evolutionary time with gene
frequency distributions tuned to different
adaptive peaks along a cline. If speciation
reaches completion, their ranges,
given the opportunity, should expand or
shift independently once freed from the
evolutionary yoke of gene flow from
source populations at the centre of the
ancestral species' range29. Eventually,
the two ranges might overlap, thus
enriching a local biota. Averaged over
many such cladistic events, the most
probable area of overlap will be at middomain.
Indeed, if each of the two (continuous)
ranges expands to at least half
the available domain, both must occur
over mid-domain regions, the essence of
the mid-domain effect. Such wide-ranging
species tend to mask any environmental
influence on local species richness14,15.
However, in the case of two species
whose ranges each span at most half the
domain (narrow-ranging species), the
contribution to richness at mid-domain
will average less than two. The probability
of two narrow-ranging species both
occurring at mid-domain therefore
decreases from one towards zero as the
range size becomes smaller15 (Boxes 2
and 3). Thus, the geometric theory of
species richness generates a specific and
highly testable prediction: that wideranging
species or higher taxa within a
regional assemblage are considerably
more likely to show patterns in accordance
with geometric theory than
narrow-ranging taxa14,15,26 (Fig 3a). Geographic
patterns of species richness in
narrow-ranging taxa, less constrained by
geometry, are more likely to reflect environmental
and historical factors. Therefore,
the most discerning way to test for
geometric effects is to categorize data by
range size class14,15,26.
Prospects
Incorporating the geometry of ranges
and of richness into local, regional and
global perspectives on biodiversity
promises far-reaching consequences.
Geometric null models are capable of
precisely predicting the expected shape
and local magnitude of the interaction
between geographic domains and species
ranges for classes of different range size.
Thus, by considering deviations from
null model expectations, the prospects
are good for disentangling biological, climatic
and historical factors that affect
species richness at different spatial and
temporal scales (Box 5). The mid-domain
effect emerges as a compellingly parsimonious
foundation for a general theory
of patterns of species richness along different
gradients at a range of scales.
In effect, biogeographers and ecologists
have simply had the wrong null
model at the back of their minds for the
past 150 years, by assuming that, were
there no climatic, physical or biological
gradients, species richness would be the
same at all latitudes, elevations and
depths. The mid-domain effect shows
that mid-domain richness peaks are to be
expected in the absence of such gradients.
Departure from the expected richness
peak, under an appropriate null
model, but not the peak itself, requires
biological or historical explanation at
geographic scales.
Acknowledgements
We thank Hal Caswell, Blaine Cole,
Ron Etter, Steve Gaines, Nick Gotelli,
Gary Graves, Chris Humphries,
George Hurtt, Kate Lyons, Clive Moncrieff,
Mike Palmer, Chris Pielou, Jesús Pineda,
Carsten Rahbek, Brian Rosen,
Sacha Spector, Dick Vane-Wright,
Joseph Veech, Paul Williams,
Michael Willig and anonymous
referees for consultation, for comments
on this article and/or for providing
articles in press. Support was provided
by the US National Science
Foundation (DEB-9706976), the University
of Connecticut (Colwell), the
Biotechnology and Biological Science
Research Council, the Leverhulme
Foundation and the National
Environmental Research Council (Lees).
PERSPECTIVES
76 TREE vol. 15, no. 2 February 2000
PERSPECTIVES
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