Online Supplementary Information
Metabolic zero-sum dynamics predict long-term changes in a tropical forest
S.K. Morgan Ernest, Ethan P. White, and James H. Brown
Supplemental Methods:
Temperature trend analysis:
Supplemental Figure 1. Temperature trend at BCI based on annual mean temperature data from
the clearing weather station.
While the changes in temperature over the course of the study are clearly more complicated than
a simple linear regression, the predominant feature – that temperatures over the last decade are
warmer than temperatures over the previous decade are robust to a variety of different statistical
models (power-law, exponential, moving averages, etc.). However, the statistical form of the
trend is irrelevant to the conclusions of the paper because we used the actual data for calculating
metabolic rates, not a statistical model of the trend.
Accounting for Jensen’s inequality in self-thinning rule analyses:
Traditional self-thinning analyses are based on even-age stands of trees. Because trees that were
planted/recruited at the same time tend to have similar sizes these stands will typically exhibit
low variance in individual size. Therefore analyses of these data simply plot average mass or
diameter as a function of abundance and fit the related exponent. In contrast, in most natural
forests there is large variation in the size of individuals and thus this approach fails to accurately
capture the underlying hypothesis that there is a constant amount of something (e.g., space in
3/2’s thinning) that is being divided up into fewer, larger, pieces when the trees are larger. To
capture this hypothesis in mixed age stands it is proper to take the average of the non-linear
function relating diameter and abundance, not the function evaluated at the average value of
diameter (or mass), for analysis. This is important because these two values are not equal to one
another when the variance of diameter is not equal to 0. For a more detailed discussion of
Jensen’s inequality see Savage (2004) and White et al. (2004).
Spatial Scale Analysis:
To determine the dependence of our results on the spatial scale of analysis, we subdivided the 50
ha plot into non-overlapping subplots that were 10, 20, 25, 50, 100, 125, 250, and 500 m on a
side. We fit the exponent in equation 2 using reduced major axis regression on log-transformed
data for each subplot and determined whether the mean value of the exponent across subplots
differed significantly from -1.
Supplemental Results:
Constancy of different ecosystem properties
We compared the dynamics of all readily available system properties for the BCI tree community
to confirm that the zero-sum dynamic for total metabolic flux was at least as well supported as
that for other properties/currencies (Supplemental Figure 2). In the context of this study we are
primarily interested in the presence of directional changes in the different currencies. This is
difficult to establish with only four data points because the power of regressions with n = 4 is
generally low. To help deal with this issue we performed post-hoc power analyses on the nonsignificant regressions to determine the number of additional censuses that would be necessary to
achieve a significant result if the observed trend continued. We estimated the observed trend and
variability using linear regression and then simulated time-series from 4 to 10 censuses long
using these values. We then determined the proportion of the regressions that was significant for
each of the different numbers of censuses (Supplemental Figure 3). Biomass data is from (Chave
et al. 2008).
It should be noted that measures of variability, such as the coefficient of variation, cannot be
used to reliably compare the stability of these currencies, because each currency is estimated
with a different degree of error. Abundance and basal area are presumably estimated with only
very small amounts of measurement error. Biomass estimates will contain greater error due to
their reliance on statistical models based on tree diameter, height, and wood density. However
these models have been extensively studied (Chave et al. 2004; Chave et al. 2005; Chave et al.
2008) and contain relatively large numbers of parameters and so should estimate biomass with
less error than the models used to estimate total metabolic flux. Total metabolic flux models
should have the greatest error in the estimates because of their reliance on simple statistical
models, with relatively few parameters, that have been relatively poorly studied. Higher error in
the estimates will lead to higher observed variability regardless of actual differences in the
stability of the true value of the property. Therefore we have relied upon directional trends for
this comparison. However, when biomass and total metabolic flux are estimated using
equivalently simple models (i.e., we estimated biomass using the equation from (Enquist et al.
1999)) their coefficients of variation are similar, with the variability in biomass (0.0236) being
somewhat larger than the variability in total metabolic flux (0.0194).
Supplemental Figure 2. Comparison of observed trends in abundance (a), total metabolic flux
(b), biomass (c), and total basal area (d). Solid lines are OLS regression results. Regressions
were clearly significant for abundance (p=0.018; r2=96.4%), clearly non-significant for total
metabolic flux (p=0.974; r2=0.7%), and marginal for biomass (p=0.260; r2=54.7%) and total
basal area (p=0.107; r2=79.7%).
Supplemental Figure 3. Power analyses for the three regressions in Supplemental Figure 2 that
were not significant at p = 0.05. These analyses indicate that the sample size was not sufficient to
have confidence in the observed lack of significance for the biomass (yellow) and basal area
(green) trends and that only 1-2 additional censuses would be necessary for relationships to
become significant if the observed trends hold. In contrast even after an additional 6 censuses
there is no change in the probability that total metabolic flux (blue) will exhibit a significant
trend and that probability is fixed at the chosen Type 1 error rate, suggesting that the lack of
trend in total metabolic flux is real.
Supplemental Discussion:
Effects of size and temperature on biomass dynamics: Studies of tropical forests often focus
on biomass because of its relevance to questions related to forestry harvests and carbon storage.
Studies examining biomass trends in tropical forests around the world found an overall trend
toward increasing biomass, but there was considerable variation among forests (Phillips et al.
1998; Chave et al. 2008). Application of an explicitly metabolic framework, and quantification
of carbon flux and storage, may help to explain why biomass is increasing in some forests while
declining in others. When the masses of individual organisms are measured directly, the
metabolic zero-sum framework presented here can be extended to predict how total biomass
(Mtot) should change in zero-sum communities with shifts in average body size (
and hence
metabolic rate (White et al. 2004). This can be done by taking a basic equation relating biomass
and abundance (
and substituting in Eqn 1 (
),
which yields
M tot 
M
M
,

B
M34
(Supp. 1)
demonstrating that the total biomass of a community in a metabolic zero-sum steady state will be
determined by the form of the individual size distribution. While Supplemental Eq 1 has been
shown to predict biomass trends in small mammal communities (White et al. 2004), it is more
complex to make predictions for forests because mass is not measured directly; it is estimated
from statistical correlations using height, diameter, and wood density. Using equation 3 and one
of the simpler models (Enquist et al. 1999) for converting from diameter to mass, M=ρD8/3,
Supplemental Eq 1 can be modified as:
M tot 
D 8 / 3
D  e  E / kT
(Supp. 2)
where ρ is wood density. Because wood density within a species may be a function of
environmental temperature (Kilpelainen et al. 2005) and trunk diameter (Woodcock & Shier
2002), it is necessary to understand the interrelationships among wood density, diameter, and
temperature; relationships that are not yet well understood. Without this understanding, it is
unclear whether a metabolic zero-sum theory would predict increasing, decreasing, or stable
biomass in the forest at BCI given observed changes in diameter and temperature. Standing
biomass appears to have decreased slightly or remained stable at BCI over the last few decades
(Chave et al. 2008) (see Supplemental Figures 2 & 3). Clearly, continued long-term monitoring
of tropical forests and a quantitative understanding of covariations among diameter, wood
density, and temperature are needed to assess whether a metabolic zero-sum framework can
accurately predict biomass dynamics in forests.
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