Online Appendix S3. Further details on model parameterization
Online Appendix S3 contains further details on parameterization of our models for two tropical
forest tree communities.
The strategy for parameterization of a model for a particular combination of 50 ha plot site (BCI
or Pasoh) and DBH threshold (10 or 1 cm) has two main parts. Firstly, where possible,
parameters were fixed at particular values according to observations. In total, this was done for
three of the eight model parameters. The time interval for our model was taken to be DT = 5 yr,
commensurate with the typical time between censuses for the BCI and Pasoh plots. Condit et al.
(2006) found that the mortality rates of tree species at these two plots followed a lognormal
distribution. Chisholm et al. (2014) subsequently estimated values of the two lognormal
parameters for all pairs of consecutive censuses for the two plots. In our model, the expected
mortality rate of species was assumed to take a constant value m because we assumed that
environmental variance only acts on recruitment rates (see main text and Online Appendix S1).
For a particular forest plot, m was set equal to the mean of a lognormal distribution with
parameters taking the average values over all corresponding census pairs (Table S1). The
carrying capacity for each species in a plot, K s , was fixed according to the upper bound of the
maximum log2 abundance class for species observed in the censuses for that plot. A log2
abundance class was used following the classic method of constructing species-abundance
distribution (SADs) introduced by Preston (1948). The observed upper bound could have arisen
because density dependent mechanisms were strong enough to prevent species moving into the
next class. However, because of uncertainty in this implication and the potential high sensitivity
of SADs to K s , we also tested K s values that were two and four times higher than the observed
upper bound and found that they gave quantitatively similar results (Appendix S5: Tables S3 and
S7). If a community carrying capacity, K c , was modeled instead of K s , then K c was set equal to
the mean of the observed community abundances. We fixed different values of K s and K c for
each DBH threshold (Table S1).
Secondly, plausible ranges were derived for the remaining five parameters, using observations
and other studies as a guide. g , F1 and F 2 are the parameters of an asymmetric Laplace
distribution (ALD) that describe the probability distribution of possible instantaneous per-capita
net recruitment rates for species, denoted by r . Its probability density function (pdf) is
ì f exp -(g - r )f ,
(
ï
2)
f (r ) = í
ïî f exp (-(r - g )f1 ),
r <g
r ³g
,
(S.1)
where f = f1f2 (f1 + f2 ) (Kotz et al. 2001). Our use of an ALD rather than a normal distribution,
as in the model by Kalyuzhny et al. (2015), can be justified by statistically comparing fits of both
distributions to empirical data. For each census interval of the BCI and Pasoh, considering a
DBH threshold of 1 cm, Chisholm et al. (2014) fitted an ALD to empirical values of r and then
derived corresponding maximum likelihood and AIC values (their Table S1). The likelihood for
each species was calculated as L (g , f1, f2 | D, mi,o ) = ò L ( r | D, mi,o ) L (g, f1, f2 | r )dr , where D
refers to the tree census data and mi,o refers to the instantaneous mortality rate for species i that
was estimated from the data. In this equation, L ( r | D, mi,o ) was specified as in Appendix S5 of
Chisholm et al. (2014) and L (g , f1, f2 | r ) was specified by equation (S.1). The total likelihood
was then calculated as the sum of the likelihoods for each species, and its maximum estimated
using Gibbs sampling with Metropolis updating (described in more detail in Appendix S5 of
Chisholm et al. 2014). In this study, we used this method to fit corresponding ALDs for a DBH
threshold of 10 cm. In addition, we fitted a normal distribution to the empirical values of r for
each census interval at each site and for each DBH threshold, using the same method but with the
pdf of a normal distribution replacing that of an ALD. We found that for a DBH threshold of 10
cm and either BCI or Pasoh, the AIC value for an ALD was lower than that for a normal
distribution for all but one census interval (Table S2). With a DBH threshold of 1 cm and for
either site, the AIC value for an ALD was lower for all census intervals (Table S2).
At a particular site and for a particular DBH threshold, the ranges for g , F1 and F 2 were
constructed using minimum and maximum values from the ALDs fitted to data from the
corresponding census intervals (described above). In the corresponding neutral model, the
instantaneous net recruitment rates were fixed over all species and time-steps at a value equal to
the mean of an ALD, which is M r = g + (1 f1 ) - (1 f2 ) . This reduces the number of recruitment
parameters from three to one. The minimum and maximum values of M r , which define the
range of M r , were calculated as the combinations of g , F1 and F 2 giving the lowest and
highest value of g + (1 f1 ) - (1 f2 ) , respectively.
In each time step, the modeled local community receives I immigrants from a large surrounding
metacommunity. For BCI and considering all individuals with DBH ³10 cm, the average
number of new individuals was calculated over all seven censuses, and the range of I was taken
to range from zero to 0.3 of this average number, i.e. [1, 700] (Table S1). This represents a range
for the migration rate (proportion of new individuals that are immigrants) of zero to 0.3,
encompassing previous migration rate estimates for trees in 50 ha tropical forest communities
(Hubbell 2001, Etienne 2005, Volkov et al. 2007, Chisholm and Lichstein 2009). The range of I
for the 1 cm DBH threshold and the ranges of I for Pasoh and both DBH thresholds were derived
analogously (Table S1).
The expected SAD for the metacommunity was taken to be a log-series distribution with
fundamental diversity parameter q (Hubbell 2001). For the purposes of sampling immigrants
during a simulation, a set of 1,000 metacommunities was constructed using the log-series with an
arbitrarily large expected size of JM =10 9 . Different values of JM can be used without affecting
local community dynamics, since the shape of the expected metacommunity SAD was
determined only by q . The number of species in each of the 1,000 metacommunities, SM , was
determined using
æ J ö
SM (q ) = q ln ç1+ M ÷ , (S.2)
è
q ø
which was derived from the definition of the log-series. It is noted that the metacommunity SAD
obtained from neutral theory is not exactly a log-series, but does tend towards it in the limit of
large SM and JM (Alonso and McKane 2004) – such large values of SM and JM values are
found in tropical forest tree communities at the 50 ha scale.
The species abundances in each metacommunity were ranked in ascending order from 1 to SM .
Then, the metacommunity proportional abundance of species i, pi,M , was set equal to the average
ith-ranked species abundance over the 1,000 metacommunities divided by the average total
number of individuals. The range of q for a metacommunity surrounding the BCI or Pasoh plot
was determined by assuming that SM was between two and four times the number of species in
the local plot and inverting equation (S.2) to solve for q (Table S1). The ranges encompass
previous estimates of q for the BCI and Pasoh plots (Volkov et al. 2003, 2007).
parameter
definition
value(s)
BCI
DT
m
Time interval
Instantaneous percapita mortality rate
Ks
Carrying capacity
for each species
population
Kc
Carrying capacity
for community
g
F1
Parameters for
asymmetric Laplace
distribution of
instantaneous percapita net
recruitment rates
F2
I
Total number of
immigrants per time
interval
data source(s)
Pasoh
5 yr
0.0378 yr-1
(DBH ³10 cm)
0.0567 yr-1
(DBH ³1 cm)
CTFS census
data
0.0217 yr-1
(DBH ³10 cm)
0.0260 yr-1
(DBH ³1 cm)
211 (DBH ³10 cm), 210 (DBH ³10 cm),
216 (DBH ³1 cm)
214 (DBH ³1 cm)
21,000
(DBH ³10 cm),
226,000
(DBH ³1 cm)
27,900
(DBH ³10 cm),
312,000
(DBH ³1 cm)
[-0.0142, 0.0013]
(DBH ³10 cm)
[-0.0185, 0.0072]
(DBH ³1 cm)
[-0.0086, 0.0046]
(DBH ³10 cm)
[-0.0147, -0.0073]
(DBH ³1 cm)
[43.3, 72.2]
(DBH ³10 cm)
[28.4, 73.2]
(DBH ³1 cm)
[57.2, 96.4]
(DBH ³10 cm)
[47.2, 196.7]
(DBH ³1 cm)
[73.9, 221.4]
(DBH ³10 cm)
[27.4, 95.5]
(DBH ³1 cm)
[131.4, 244.4]
(DBH ³10 cm)
[80.0, 176.0]
(DBH ³1 cm)
[1, 700]
(DBH ³10 cm)
[1, 8,600]
(DBH ³1 cm)
[1, 800]
(DBH ³10 cm)
[1, 6,700]
(DBH ³1 cm)
Chisholm et al.
(2014); CTFS
census data
CTFS census
data
CTFS census
data
Chisholm et al.
(2014); CTFS
census data
CTFS census
data
q
Fundamental
biodiversity number
[34.9, 73.0]
[99.2, 208.0]
Table S1. Plausible range of values for each parameter of the dynamic community model used in
this study, which incorporates both demographic and environmental variance. Definitions and
values for each parameter are presented, together with data sources used to derive values.
site
census interval DBH threshold
(cm)
AIC for ALD
AIC for normal
distribution
BCI
1–2
10
1704.7
1704.5
BCI
2–3
10
1844.6
1872.8
BCI
3–4
10
1694.3
1707.5
BCI
4–5
10
1666.3
1688.3
BCI
5–6
10
1706.1
1753.6
BCI
6–7
10
1763.4
1870.7
BCI
1–2
1
3176.5
3248.7
BCI
2–3
1
3371.3
3479.5
BCI
3–4
1
3090.9
3211.9
BCI
4–5
1
3240.5
3401.2
BCI
5–6
1
3238.0
3455.9
BCI
6–7
1
3198.0
3345.8
Pasoh
1–2
10
3948.1
3972.2
Pasoh
2–3
10
4537.3
4619.1
Pasoh
3–4
10
4303.1
4326.9
Pasoh
4–5
10
4240.2
4237.7
Pasoh
1–2
1
6357.4
6427.2
Pasoh
2–3
1
8044.2
8386.8
Pasoh
3–4
1
7483.4
7706.7
Pasoh
4–5
1
8245.0
8715.2
Table S2. For each combination of site, census interval and DBH threshold, AIC values for
ALDs and normal distributions fitted to instantaneous per-capita net recruitment rates for species
populations, as calculated from census data.
Literature Cited for Online Appendix S3
Alonso, D., and A. J. McKane. 2004. Sampling Hubbell’s neutral theory of biodiversity. Ecology
Letters 7:901–910.
Condit, R. et al. 2006. The importance of demographic niches to tree diversity. Science 313:98–
101.
Chisholm, R. A., and J. W. Lichstein. 2009. Linking dispersal, immigration and scale in the
neutral theory of biodiversity. Ecology Letters 12:1385–1393.
Chisholm, R. A. et al. 2014. Temporal variability of forest communities: empirical estimates of
population change in 4000 tree species. Ecology Letters 17:855–865.
Etienne, R. S. 2005. A new sampling formula for neutral biodiversity. Ecology Letters 8:253–
260.
Hubbell, S. P. 2001. The Unified Neutral Theory of Biodiversity and Biogeography. Princeton
University Press, Princeton, USA.
Kalyuzhny, M., R. Kadmon, and N. M. Shnerb. 2015. A neutral theory with environmental
stochasticity explains static and dynamic properties of ecological communities. Ecology Letters
18:572–580.
Kotz, S., T. J. Kozubowski, and K. Podgórski. 2001. Asymmetric Laplace distributions. Pages
133–178 in S. Kotz, T. J. Kozubowski, and K. Podgórski, editors. The Laplace distribution and
generalizations. Birkhäuser Boston, Boston, USA.
Preston, F. W. 1948. The commonness, and rarity, of species. Ecology 29:254–283.
Volkov, I., J. R. Banavar, S. P. Hubbell, and A. Maritan. 2003. Neutral theory and relative
species abundance in ecology. Nature 424:1035–1037.
Volkov, I., J. R. Banavar, S. P. Hubbell, and A. Maritan. 2007. Patterns of relative species
abundance in rainforests and coral reefs. Nature 450:45–49.