Theor Ecol (2011) 4:195–200
DOI 10.1007/s12080-011-0113-5
ORIGINAL PAPER
Theory predicts a rapid transition from niche-structured
to neutral biodiversity patterns across a speciation-rate
gradient
Ryan A. Chisholm & Stephen W. Pacala
Received: 1 October 2010 / Accepted: 13 January 2011 / Published online: 3 February 2011
# Springer Science+Business Media B.V. 2011
Abstract A central challenge in community ecology is to
predict patterns of biodiversity with mechanistic models.
The neutral model of biodiversity is a simple model that
appears to provide parsimonious and accurate predictions
of biodiversity patterns in some ecosystems, even though
it ignores processes such as species interactions and niche
structure. In a recent paper, we used analytical techniques
to reveal why the mean predictions of the neutral model
are robust to niche structure in high diversity but not lowdiversity ecosystems. In the present paper, we explore this
phenomenon further by generating stochastic simulated
data from a spatially implicit hybrid niche-neutral model
across different speciation rates. We compare the resulting
patterns of species richness and abundance with the
patterns expected from a pure neutral and a pure niche
model. As the speciation rate in the hybrid model
increases, we observe a surprisingly rapid transition from
an ecosystem in which diversity is almost entirely
governed by niche structure to one in which diversity is
statistically indistinguishable from that of the neutral
model. Because the transition is rapid, one prediction of
our abstract model is that high-diversity ecosystems such
as tropical forests can be approximated by one simple
Electronic supplementary material The online version of this article
(doi:10.1007/s12080-011-0113-5) contains supplementary material,
which is available to authorized users.
R. A. Chisholm : S. W. Pacala
Department of Ecology and Evolutionary Biology,
Princeton University,
Princeton, NJ 08544, USA
R. A. Chisholm (*)
Smithsonian Tropical Research Institute,
MRC0580-12, Unit 9100, Box 0948,
DPO, AA 34002-9998, USA
e-mail: [email protected]
model—the neutral model—whereas low-diversity ecosystems such as temperate forests can be approximated by
another simple model—the niche model. Ecosystems that
require the hybrid model are predicted to be rare,
occurring only over a narrow range of speciation rates.
Keywords Biodiversity . Community ecology . Neutral
theory . Niche model . Emergent pattern . Speciation rate
Introduction
In a seminal paper, Simon Levin (1992) argued that the
mechanisms that generate patterns in ecological systems
often operate on different temporal and spatial scales from
those on which the patterns are observed. He cautioned
against perceptual biases that can arise from observing
ecosystems on particular scales, and he proposed that, in
some cases, macroscopic patterns could best be understood
as emergent properties of the collective behaviour of many
units at smaller scales. This perspective seems particularly
germane to the neutral theory of biodiversity, which
purports to explain macroscopic patterns of biodiversity
but has aroused controversy largely because it ignores
many processes that ecologists have observed and measured in the field.
The origins of the neutral theory of biodiversity can be
traced back to Caswell (1976) and Hubbell (1979, 1997)
and beyond that to the theory of island biogeography
(MacArthur and Wilson 1967; Losos and Ricklefs 2010).
Caswell (1976) was the first to apply neutral techniques
from population genetics (Kimura 1968) to ecological
communities. Hubbell (1979) did not draw directly on
population genetics but demonstrated that species abundance distributions (SADs) in tropical forests could be
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reasonably well described by a simple random-walk model.
The neutral theory of biodiversity was subsequently
popularised in a 2001 book (Hubbell 2001) and has since
become a major focus of ecological research (Chisholm and
Lichstein 2009; Chisholm and Pacala 2010; Volkov et al.
2003, 2007; O'Dwyer and Green 2010; Rosindell et al.
2008, 2010; Etienne 2005; Alonso and McKane 2004).
Neutral theory assumes that all species are equivalent on
a per-capita basis, thereby ignoring a multitude of processes
whose existence has been empirically demonstrated in real
ecosystems. As such, the neutral theory of biodiversity
plays a similar role to the neutral theory of population
genetics in that it generates a null hypothesis that can be
used to infer when and where non-neutral processes are
operating (Leigh 2007). According to the null hypothesis,
we expect neutral patterns to be observed only when nonneutral processes are absent or weak. However, as alluded
to above, neutral theory generates a second hypothesis,
which is that neutral biodiversity patterns are emergent
macroscopic phenomena (Holt 2006) and may in fact be
robust to non-neutral processes at some spatial and
temporal scales (Chave 2004).
The notion that simple patterns can emerge from more
complex ones is now familiar to biologists (Levin 1992;
Couzin et al. 2005; Follows et al. 2007; Scheffer and van
Nes 2006; Lawton 1999; Harte 2003), but, in the context of
neutral theory, this idea has been greeted with skepticism
by some ecologists (e.g. Leigh 2007; Clark 2009). Of
course, it may not be true that neutral theory emerges from
more complex ecosystem processes, but the possibility
certainly seems worth investigating given the good fit of
neutral predictions to some biodiversity patterns in some
ecosystems (Volkov et al. 2007; O’Dwyer and Green 2010;
Bell 2001). The neutral model has been criticised for its
failure to include biological detail (Clark 2009), but the
sheer complexity of ecological systems compels us to
ignore most biological detail in any model that we build
(Lawton 1999). In the present case, our question is: when is
the limited complexity of the neutral model sufficient to
provide a useful approximation to a model of biodiversity
with simple niche structure? By studying these abstract
models, we hope also to gain qualitative insights into when
neutral models might provide accurate and parsimonious
predictions of empirical patterns of biodiversity.
In a recent paper (Chisholm and Pacala 2010), we
showed analytically that the neutral SAD emerges from a
more complex niche-structured model in the limit of high
diversity, thus providing some justification for why the
neutral model provides an apparently good fit to SADs
from tropical forests and coral reefs (Volkov et al. 2007),
even though niche processes are known to operate in these
systems (Wright 2002; Connolly et al. 2005). In the present
paper, we seek to establish where and how rapidly along a
Theor Ecol (2011) 4:195–200
gradient of diversity the transition from niche to neutral
pattern occurs. We do not seek to explain the existence of
the diversity gradient in the first place, but observe simply
that there are many empirical examples of such gradients,
such as latitudinal species richness gradients (Wiens and
Donoghue 2004; Allen and Gillooly 2006). We model our
diversity gradient by adjusting a single speciation rate
parameter, but we acknowledge that this abstract parameter
stands in for an interplay of speciation, dispersal and
extinction processes and that the mechanisms that determine speciation rates in practice are complex and incompletely understood (Rosindell et al. 2010; Allen and
Gillooly 2006).
In principle, the transition from niche to neutral pattern
along the speciation-rate gradient could occur very slowly,
with the implication that hybrid niche-neutral models would
be required to model patterns of species abundances in
most ecosystems. Alternatively, the transition could occur
rapidly, in which case ecosystems with low speciation rates
may be well approximated by simple niche models with
only one species per niche and ecosystems with high
speciation rates may be well approximated by neutral
models. We investigated this problem using the framework
of our previous study (Chisholm and Pacala 2010) but with
numerical (Etienne 2005) instead of analytical techniques.
Methods
We based our analysis on a spatially implicit hybrid nicheneutral model (henceforth referred to as the “hybrid
model”) in which each of several non-interacting niches
operates according to its own neutral zero-sum dynamics
(Chisholm and Pacala 2010). In the model, a semi-isolated
local community is connected to a much larger metacommunity by immigration. This model includes only the
fundamental processes of birth, death, speciation, immigration and niche segregation. The model is based on the
spatially implicit neutral model of biodiversity (Hubbell
2001).
The model’s dynamics operate as follows. The landscape
is divided into K niches, with each niche covering a
proportion βi of the landscape. In general, the βi are
arbitrary (with the constraint that Σβi =1), but, in the
simulations, we generate the βi from a broken-stick model
(see below). Each species can survive in only one niche, but
multiple species may exist in a single niche (i.e. niches are
non-overlapping). In the metacommunity, there are JM
individuals. At each timestep, a randomly selected individual in the metacommunity dies and is replaced by the
offspring of another randomly selected individual (without
replacement) from the same niche with probability 1–ν.
With probability ν<<1, a speciation event occurs, in which
Theor Ecol (2011) 4:195–200
case the dying individual is replaced by an individual of a
new species. We assume that the value of ν is independent
of the community’s niche structure (K and βi). The
metacommunity is assumed to be at equilibrium relative
to the much smaller local community, which has J<<JM
individuals. The local community has the same niche
distribution (same distribution of βi) as the metacommunity.
At each timestep in the local community, a randomly
selected individual dies and is replaced by the offspring of
another randomly selected individual (again without replacement) from the same niche within the local community with probability 1–m. With probability m, the dying
individual is replaced by an individual from the metacommunity (but still within the same niche). The parameter m is
neutral theory’s immigration parameter (Hubbell 2001;
Chisholm and Lichstein 2009). For the special case of only
one niche (K=1), the hybrid model described here reduces
to the spatially implicit neutral model (Hubbell 2001).
We generated stochastic realisations of a local community at equilibrium under the hybrid model using sampling
techniques and software implementations of these techniques developed by Etienne (2005). The software is written
in PARI/GP (The PARI~Group 2008; for the code, see
supplementary information to Etienne 2005). For the
special case of K=1 (the neutral model), we were able to
apply the code directly. For the general case, we first broke
the community into K niches according to a broken-stick
algorithm (MacArthur 1957), ran the code independently
for each niche and then combined the data for each niche
back into a single community. To implement the brokenstick algorithm, we generated K–1 numbers from the
uniform random distribution on the interval [0,1] and used
these to partition the interval into K fragments with lengths
βi .
From the output of each simulation, we generated the
following statistics: the species richness (S), the loglikelihood of the neutral model given the observed SAD
and the log-likelihood of a pure broken-stick niche model
(i.e. with one species per niche and no neutral drift) given
the observed SAD. The log-likelihood of the neutral model
given the observed SAD was computed using code from
Etienne (2005). The log-likelihood of the pure broken-stick
niche model given the observed SAD was computed from
the following formula for the likelihood:
f ðy1 ; :::; y2 Þ ¼
S!
1
y1 !:::y2 ! J 1
S1
where yi is the observed number of species with abundance
i, and where we assume that the number of broken-stick
niches is equal to the number of observed species S (this is
the only sensible value for the number of niches if we are
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assuming a pure broken-stick niche model). In the formula
above, the combinatorial expresses the number of possible
ways in which the “stick” can be broken, and the other
factor represents the number of combinations of breaks that
will lead to the same SAD (i.e. it accounts for the fact that
the breaks are ordered but the species abundances are
unordered; we have verified this formula numerically). We
also compute the posterior probability that the data came
from a neutral model given a dichotomous uninformative
prior [P(neutral) = P(broken-stick) = 0.5]. This posterior
probability is an indicator of the strength of the evidence
in favour of the neutral model versus a pure broken-stick
niche model, once the SAD has been observed.
For each speciation rate (ν), we repeated the simulation
20 times. Stochasticity arises in the model from the
sampling procedure and, in the case of hybrid model, from
the broken-stick algorithm. In our main simulations, we set
J=20,000 and m=0.1 for the local community, which
corresponds roughly to the values for trees ≥10 cm
diameter-at-breast-height in the 50-ha permanent plot on
Barro Colorado Island, Panama (Hubbell et al. 1999, 2005;
Condit 1998). For the metacommunity, we set JM =1012,
although the magnitude of this value is only meaningful
relative to the speciation rate and cannot be interpreted
literally because of the spatially implicit assumption of the
model. We set the number of niches in the hybrid model to
K=16. We repeated the entire analysis for a range of
speciation rates from ν=10−14 to ν=4×10−10 (we used 15
equally spaced values of ν on a logarithmic scale). In
subsequent simulations, we tested the effects of different
values of K (4 and 64) and m (0.01 and 0.5) on the results.
Results
At low speciation rates (ν<10−12), the species richness is
determined solely by the number of niches (Fig. 1), and the
distribution of abundances is statistically indistinguishable
from those of a pure broken-stick niche model (Fig. 2). At
high speciation rates (ν>10−11), the patterns of diversity in
the hybrid model are statistically indistinguishable from
those of the neutral model (Figs. 1 and 3). The transition
point between niche and neutral pattern occurs approximately where the speciation rate in the hybrid model
produces species richness equal to twice the number of
niches (see Fig. 1 for K=16; see Electronic Supplementary
Material for other values of K). This transition is rapid, as
evidenced by the fact that the posterior probability of the
neutral model is very close to P=0 for ν<10−12 and very
close to P=1 for ν>10−11 (Fig. 4). There is only a narrow
range of speciation rates in which statistical evidence does
not clearly discriminate between the neutral versus the pure
broken-stick niche model (Fig. 4).
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Fig. 1 Species richness (S) in a local community at equilibrium as a
function of speciation rate for the neutral and hybrid models (m=0.1,
J=20,000, JM =1012, K=16). Dashed lines show 95% confidence
intervals. The dotted line corresponds to S=K=16, which is the
species richness in a system completely dominated by niche structure
Qualitatively similar results were observed for different
values of K and m (see Electronic Supplementary Material),
although the transition from niche-dominated to neutraldominated pattern was less rapid for small K (occurring
over roughly two orders of magnitude of ν for K=4) and
more rapid for large K (occurring over roughly half an order
of magnitude of ν for K=64). The transition from nichedominated to neutral-dominated pattern occurred earlier (at
smaller values of ν) for larger values of m and substantially
earlier for smaller values of K.
Fig. 2 Log-likelihood of the pure (broken-stick) niche model given
species abundance data generated from the hybrid model (thick lines)
and from the broken-stick model itself (thin lines), for different values
of the speciation rate. Other parameters are m=0.1, J=20,000, JM =
1012, K=16. See text for model details. Higher values of the loglikelihood for a particular speciation rate indicate a greater likelihood
of the broken-stick model given the data, but absolute values of the
likelihood are not meaningful in themselves. Dashed lines show 95%
confidence intervals (estimated from 20 simulations)
Theor Ecol (2011) 4:195–200
Fig. 3 Log-likelihood of the neutral model given species abundance
data generated from the hybrid model (thick lines) and from the
neutral itself (thin lines), for different values of the speciation rate.
Other parameters are m=0.1, J=20,000, JM =1012, K=16. See text for
model details. Higher values of the log-likelihood for a particular
speciation rate indicate a greater likelihood of the neutral model given
the data, but absolute values of the likelihood are not meaningful in
themselves. Dashed lines show 95% confidence intervals (estimated
from 20 simulations)
Discussion
Our observation of a rapid transition from a model
ecosystem in which species richness and relative abundance
are almost entirely governed by niche structure to one in
which they are statistically indistinguishable from neutral
patterns was not anticipated a priori (Figs. 1, 2, 3 and 4).
The transition occurs near a critical speciation rate that
produces, on average, two species per niche in the hybrid
model. The intuitive interpretation of this result is that, at
low speciation rates, there is ample time between speciation
Fig. 4 Posterior probability that an observed species abundance
distribution from the hybrid model comes from a neutral model, given
an uninformative dichotomous prior that specifies equal probabilities
for the neutral model and the pure (broken-stick) niche model (P
(neutral) = P(broken stick)=0.5), for different values of the speciation
rate. Other parameters are m=0.1, J=20,000, JM =1012, K=16. See
text for model details. Dashed lines show 95% confidence intervals
(estimated from 20 simulations)
Theor Ecol (2011) 4:195–200
events for neutral drift to eliminate ecologically equivalent
species that occupy the same niche in the hybrid model, and
so each niche almost always contains a single species.
Thus, providing that the number of niches is larger than the
species richness predicted by the neutral model (which has
one niche), the species richness of the hybrid model is
approximately equal to the number of niches. As the
speciation rate increases beyond the critical speciation rate,
more and more niches contain at least two ecologically
equivalent species, whose dynamics drift relative to one
another and hence drift relative to the species in the other
niches also. Of course, the constraint that the sum of all
species’ abundances within a niche must equal the size of
that niche produces higher order correlations and some
signature of non-neutral dynamics. But, the constraints
imposed by niches quickly become weak as increasing
speciation rates produce ever more species per niche. Our
results imply that only a small amount of neutral drift is
required to overwhelm any signal of niche structure in
biodiversity patterns.
We emphasise that the focus of our study here has been
on the rate of transition from niche-dominated to neutraldominated pattern with increasing diversity. The existence
of the transition in the first place was established
analytically in our previous paper (Chisholm and Pacala
2010), and, intuitively, results from the fact that as diversity
increases, the zero-sum assumption (Etienne et al. 2007)
within each niche becomes less important from the
perspective of an individual species and thus species’
random walks become indistinguishable from those in a
neutral world.
Although the discrete-niche structure in our model may be
a reasonable approximation for real ecosystems (Scheffer and
van Nes 2006; MacArthur 1957), continuous-niche models
(e.g. Tilman 2004; Purves and Pacala 2005) are a priority for
future research. We suspect that our discrete-niche results
will generalise to continuous-niche models because the
mechanism for the rapid transition from niche to neutral
pattern is simply that within-niche drift produced by withinniche diversity mimics pure drift, and, in a continuous-niche
model, one can visualise species occupying the same point in
niche space drifting relative to one another and species very
close to one another in niche space having nearly neutral
dynamics because they are nearly ecologically equivalent
(see Purves and Pacala 2005). As diversity increases in a
continuous-niche model, more and more species get packed
into the same niche space, nearest-neighbour distances in
niche space decrease, and the deterministic forces that
regulate the relative abundances of near neighbours become
weaker and weaker relative to drift. Once species are
sufficiently close together in niche space that two near
neighbours’ abundances approximately drift relative to each
other, we suspect that the macroscopic pattern will be one in
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which every species’ abundance drifts relative to that of
every other. We note that our results for gradients of
speciation rates should also generalise to gradients of
extinction rates, because the patterns observed depend
fundamentally only on gradients of diversity rather than
speciation rates specifically.
The transition from left to right in Figs. 1 and 2 can be
thought of as a caricature of the transition we might observe
in walking along a transect from a boreal forest to a tropical
forest. In the boreal forest, arrival rates of new species are
low enough, or loss rates are high enough, that very few
species coexist (Hubbell 2001), and so their abundance is
largely regulated by niche structure. As we move towards
the equator, we enter realms that have experienced a greater
arrival of new tree species (again through speciation or
immigration), or lower loss rates, and at some point, we
observe biodiversity patterns that exhibit some characteristics of niche and neutral models (i.e. around ν=10−12 in
Figs. 1, 2, 3 and 4). However, this niche-neutral hybrid
zone is rather narrow (as predicted by our model), and, as
we continue to move still further towards the equator, we
rapidly encounter forests whose tree diversity patterns are
well approximated by pure neutral models. One caveat to
these results is that, if species richness is very low (which
occurs when ν and K are both low), it becomes statistically
more difficult to distinguish between niche and neutral
patterns, making the niche-neutral hybrid zone appear
larger (e.g. Figs. S1 to S6). In practical terms, this means
that, in systems with very few species, it may very difficult
to infer much about biodiversity mechanisms from static
snapshots of SADs.
The reader will notice that we deliberately refer to “the
arrival of new species” in the previous paragraph rather
than “the speciation rate”. This is because the speciation
rate in the spatially implicit neutral model, on which our
analysis is based, can only be interpreted qualitatively as
some measure of the rate of origin of new species
(Rosindell et al. 2010). The spatially explicit neutral model,
by contrast, lends itself to more direct interpretations of the
speciation process. A more quantitative interpretation of our
results depends, therefore, on continued progress in
analysing the spatially explicit model (O’Dwyer and Green
2010; Vanpeteghem and Haegeman 2010; Chave and Leigh
2002). In particular, there remains the core challenge of
developing a full spatially explicit theory that predicts
species abundance distributions and addresses dynamic as
well as static patterns of biodiversity (Leigh 2007).
In summary, our theoretical study predicts that most
ecosystems will exhibit patterns of diversity that are either
strongly niche-structured or indistinguishable from neutral.
These predictions are a direct consequence of the rapid
emergence of the neutral model from the hybrid model in
our simulations. If our predictions are correct, this should
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be a boon for macroscopic biodiversity modelling because
it obviates the need for such relatively complex hybrid
models.
Acknowledgements We thank Rampal Etienne for advice on
running his code. We thank James Rosindell for helpful discussions.
R.A.C. acknowledges STRI and SIGEO/CTFS for financial support.
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