Predicting the response of forests to climate and land-use change depends on models that can simulate the time-varying distribution of different tree sizes within a forest – so-called

The modelling of the abundances of various tree sizes in tropical forests is important in efforts to improve understanding of land–climate feedbacks and hence anthropogenic climate change. Earth system models (ESMs) are used to model climate but currently have a large range of uncertainty in the prediction of the land carbon sink, with as much as 500 GtC uncertainty by 2100 for a 1 % increase in

These recent DGVMs broadly consist of two different approaches to representing tree size, either based on individual-based models

We follow demographic equilibrium theory (DET)

Amazonia is one of the largest pools of land carbon on the planet

In Sect. 2 below we summarise the theoretical basis for DET and also MSTF, deriving analytical formulae for total forest biomass in each case. Section 3 describes the methods and data, and Sect. 4 describes the results. Finally discussion and conclusions are in Sects. 5 and 6.

The distribution of tree sizes in a forest can be understood in terms of how the growth and mortality of the trees vary with tree size

The governing equation for this process is variously known as the one-dimensional drift or continuity equation

It was shown

This equation can be solved to give an exact solution, if simplifying assumptions of size-independent mortality

This solution is also applicable for other size variables such as tree dry mass

The LTWD distribution has been shown to be a good description of tree trunk diameter distributions in a variety of tropical forests

The total biomass density (kg of dry tree mass per hectare) of the LTWD tree mass distribution can be obtained by integrating Eq. (

As real forests do not satisfy the assumption of infinite maximum tree size, this can lead to errors in the calculated biomass density. A correction to this can be found in terms of

In cases where

Metabolic scaling theory is a theory of scaling of organisms with size, based on theories of metabolism, physics and chemistry

The MST equations also enable the calculation of biomass density (kg of dry tree mass per hectare). In this case only the finite upper bound of

The tree census data used in this study are from the public access permanent sample plots of the RAINFOR

Amazonian allometric regions. Each region, shown by the coloured areas, is defined by geography, rainfall and soil substrate. White circles show location of the forest plots used. The two western regions share common allometry but are split based on rainfall seasonality for analysis purposes.

We selected 124 open-access forest plots (Fig.

The open-access plots of the Amazon RAINFOR dataset consists only of trunk diameter values. To estimate the tree mass, the methodology developed by

The regions were defined by geography and substrate origin

The mass function (kg of tree dry mass), when height was included as one of the parameters used, was

The wood specific gravity

As in our previous study

Maximising the log likelihood

The data were fitted by plot, by allometric region (an aggregated dataset of all plots in that region), and by country (again aggregation of plots), and all the data, from all 124 plots, were grouped together as one large dataset. This allows both the study of the individual plots and the larger-scale patterns across South America.

The probability density function

We fit DET-LTWD twice, once with both parameters

For this situation, where we are only aiming to find the parameter

For the two-parameter case, where both

Substituting Eq. (

Once the parameters

Again, the equations are the same for tree mass, just with the symbols appropriately substituted (e.g.

From the equation for number density

Similarly the MST pdf for mass from Eq. (

To test the biomass density equations, we used the results of the MLE fits to calculate the biomass density predicted by Eqs. (

We chose to measure the biomass density as a function of size in terms of the total mass per unit area from trees with masses equal to or greater than a given size. The main reason for this is that the forest plot data only sampled trees with a trunk diameter equal to or greater than 10 cm. Therefore it makes little sense to measure the biomass density below a given size, as would be the case with a traditional cumulative distribution function. This approach has a second benefit that the mass of a forest above a given size is a much more useful way of easily seeing the contribution of the dominant larger trees to total biomass

A correction term is added to Eqs. (

As the large trees are so rare this correction will be equivalent to adding just one tree of the largest mass

This Eq. (

When the mass data were estimated from the trunk diameter measurements using the methodology of

Figure

The effect of truncating data measured in trunk diameter and then converting to mass using allometry. In

When working with mass data the peak was eliminated from fitting by creating 40 bin edges (39 bins) in log space (base

Fitting the DET-LTWD and MST equations to the trunk diameter size distributions showed a consistent pattern for all the geographical aggregations of plot data. In all cases, except Guyana Shield, the DET-LTWD solutions (both one- and two-parameter versions) more closely captured the curvature of the observed size distribution than the MST solution (Fig.

The two-parameter DET-LTWD fits gave a fitted value of the growth scaling power

Fit to the trunk diameter size distribution for all South American RAINFOR plots as one large dataset. The blue circles show the binned data and the lines show the fitted distribution for each model.

Results of fitting models of the trunk diameter size distributions for the forest plot data aggregated to regions, countries and all plots combined. This table presents the fitted parameters for each model.

In general the one- and two-parameter DET-LTWD solutions were quite similar in terms of the appearance of the fit on the distribution plots. This finding was confirmed using the Akaike information criterion (AIC) and Bayesian information criterion (BIC) (Table

It was only possible to distinguish the quality of the fits for 4 of the 12 geographical aggregations of forest plots. In all four cases (all S. America, Bolivia, Brazilian Shield and N. Western) the two-parameter DET-LTWD fit was favoured, and for the other eight it was not possible to say that the inclusion of the growth scaling power as a fitting parameter improved the fit.

Model comparison for fits to trunk diameter size distributions. This table shows the log likelihood of each model's fit and the corresponding AIC and BIC model comparison criterion. The best model has the lowest AIC or BIC; here the difference is shown compared to the best model, meaning the best model has a score of 0. Models other than the best are strongly rejected if they have a value greater than 10. The best model and those not rejected are shown in bold.

Fitting the models to the individual forest plots (full results in Tables S3 and S4 and Figs. S5 to S13 in the Supplement) again resulted in the DET-LTWD models generally fitting much more closely than MST. Table

Shows the best and acceptable models for the 124 individual forest plots for trunk diameter. Models are labelled as “M” for MST, “1” for the one-parameter DET-LTWD and “2” for the two-parameter DET-LTWD. Columns refer to best-fitting model (lowest BIC score). Rows refer to models that are so good a fit compared to the best that they cannot be rejected, as their BIC score is so close to the best model. For example “1M” means the MST and one-parameter models are not rejected but the two-parameter model is rejected based on BIC. Then the columns in this row show how many forest plots have either the 1 or M model as the best fit.

Plotting just the

Figure

Results of the two-parameter DET-LTWD MLE fits for trunk diameter data from all 124 individual forest plots. The fitted mortality-to-growth ratio

If it is assumed that for any fixed value of

Equation (

This trade-off would take place in each forest plot with the dominant strategy in each plot depending on local conditions that are affecting growth and mortality. To test if the trade-off could explain the results, fitting parameters

All fitting was performed on mass data after trees smaller than

Fit to the mass size distribution for all South American RAINFOR plots as one large dataset. The blue circles show the binned data and the lines show the fitted distribution for each model. The peak in the distribution is clearly shown. The fitting is only performed on trees with mass greater than the mass of the peak.

The two-parameter fits gave a fitted value of the growth scaling power

Results of fitting the models of the mass size distributions for the forest plot data aggregated to regions, countries and all plots combined. Shown are the fitted parameters for each model.

As with the trunk diameter, fits for the two DET-LTWD solutions were, in general, quite similar in terms of the appearance on the mass distribution plots. Again the AIC and BIC fitting metrics were barely able to distinguish which DET-LTWD model best fit the data (Table

Model comparison for fits to mass size distributions. This table shows the log likelihood of each model's fit and the corresponding AIC and BIC model comparison criterion. The best model has the lowest AIC or BIC; here the difference is shown compared to the best model, meaning the best model has a score of 0. Models other than the best are strongly rejected if they have a value greater than 10. The best model and those not rejected are shown in bold.

Fitting the models to the individual forest plots (full results in Tables S5 and S6 and Figs. S14 to S22 in the Supplement) again resulted in the DET-LTWD models often fitting much more closely than MST. All fitting was performed on mass data after trees smaller than

Shows the best and acceptable models for the 124 individual forest plots for mass. Models are labelled as “M” for MST, “1” for the one-parameter DET-LTWD and “2” for the two-parameter DET-LTWD. Columns refer to the best-fitting model (lowest BIC score). Rows refer to models that are so good a fit compared to the best that they cannot be rejected, as their BIC score is so close to the best model. For example “1M” means the MST and one-parameter models are not rejected but the two-parameter model is rejected based on BIC. Then the columns in this row show how many forest plots have either the 1 or M model as the best fit.

Figure

Results of the two-parameter DET-LTWD MLE fits for mass data from all 124 individual forest plots. The fitted mortality-to-growth ratio

Plotting just the

The biomass density Eqs. (

The value of

It is apparent that the MST biomass density equation is inferior to the DET-LTWD-derived biomass density equation from the DET theory. For all aggregations the biomass density was overestimated by MST, and in many cases by a considerable margin. The comparison of the different DET-LTWD biomass density equations was found to favour the two-parameter fit using the finite upper bound (6 regions out of 12). Four areas had better estimates with the two-parameter fit using the infinite upper bound (all S. America, Bolivia, Peru and Guyana Shield).

Interestingly, two regions (S. Western and Ecuador) had a worse fit for the two-parameter DET-LTWD. The S. Western region, though, fits the biomass within 2 % regardless of the choice of upper bound or DET model, so the very slight difference in the biomass density prediction is almost certainly not significant for this region. When the reverse cumulative biomass density, defined as biomass density of all trees above a given tree mass, is plotted for Ecuador (see Figs. S27 and S28) the error comes from the shape of the tail of the distribution, which is much flatter than theory. This flat tail could be due to it being a region with a smaller number of trees (4159) or could be due to higher mortality for large trees in this region.

Model biomass comparison. Table shows the percentage difference between each model of the biomass density predicted by the parameters obtained from fitting the mass distribution using MLE and the allometric mass in the dataset. This comparison is only for data where the tree mass is greater than the peak in the mass distribution

To look deeper at the relationship between model choice and predicted biomass density, the analysis was repeated for the individual forest plots. In Fig.

Comparison of the biomass density prediction based on the size-distribution fits to the mass data and to the allometric biomass density in each of the 124 forest plots. Results are plotted for both the one- and two-parameter fits and for both the assumption of infinite and finite maximum tree size. The finite tree size case is limited to the largest tree mass

The relative root mean squared error (RMSE) of the biomass density prediction of the 124 forest plots using the parameters fitted via MLE to the mass size distribution. The table compares the results from the different DET-LTWD models and the MST model. The range column indicates the integration limits of the biomass density calculation. The DET-LTWD model assumes no maximum size and by default integrates out to infinity. This can be corrected in terms of the largest tree mass

For the small individual forest plots, finite maximum tree size has a larger effect on accuracy than using the two-parameter DET-LTWD over the one-parameter version.

In this paper we show that the left-truncated Weibull distribution (LWTD), which is consistent with the demographic equilibrium theory (DET) when the mortality is size independent and the growth is a power law of tree size, fits the observed tree-size distributions for 124 forest plots across Amazonia. Our fitting was undertaken with either two free parameters or with one free parameter and the growth scaling power

We found that this conversion of trunk diameter to mass introduces a peak in the mass distribution that is purely an artefact of the conversion. The peak is due to the variation in mass of trees of a given trunk diameter, due to height and wood density variation leading to some small mass trees being in effect “missing” from the mass distribution. If the diameter-to-mass relationship were purely one to one, then the artefact peak would not occur. This peak has implications for anyone using mass size distributions converted from trunk diameter data. Our solution was to fit only to trees with mass greater than the mass distribution peak.

The model fitting shows that Amazon size distributions are generally better fit by the DET-LTWD-based models than MSTF. The two- and one-parameter DET-LTWD fits were often not significantly different enough from each other for comparison by AIC or BIC (which balance the quality of the fit against the number of unknown parameters) to choose which is the best description of the size distributions. The few plots and regions (including all plots combined) where one model was found to have a significantly better AIC or BIC score all favoured the two-parameter model.

The best-fit growth scaling exponent

The clustering of

It was suggested

We find the fitted two-parameter DET-LTWD

MSTF was rarely a good fit at the plot, regional or all-plots level for either trunk diameter or mass distributions, and it significantly overestimated total biomass density, so we reject the MSTF model as a good model of forest size distributions. This rejection is consistent with the recent study by

There was a strong correlation between sample size and how likely MSTF was to be considered either the best model or an acceptable model, with small sample sizes favouring MSTF. This correlation suggests that small sample sizes may lead to difficulty in identifying the best model or even wrongly choosing the best model, most likely as rarer large trees are more likely to be absent from a small sample. Meaning, where practical, larger forest plots of at least 1000 stems are desirable when analysing size distributions.

All three models of size distribution were used to predict total biomass density using the integration of the analytical form of their respective mass distributions. One interesting implication of the resulting equations for DET is that mortality and growth only ever appear in the form of the ratio

When considering how well the models predicted total biomass density from the fitted size distribution, the biggest source of error at the plot scale is the model assumption of infinite maximum tree size. However, this can be corrected for and allows the one-parameter DET-LTWD to estimate biomass density with a relative root mean square error of 10 % over the 124 forest plots and the two-parameter DET-LTWD within 6 %. Conversely, the MST model consistently overestimated the biomass density, often by a considerable margin. The regional scale, which has larger sample size, showed much better prediction of the biomass density, and the two-parameter DET-LTWD with finite upper bound had the smallest error in biomass density. This suggests the DET-LTWD model is a useful model of biomass for large-scale applications such as being used to initialise a DGVM based on the continuity equation

One of our priorities for further work is to investigate whether the commonality found in the values of

This study demonstrates that demographic equilibrium theory (DET) is able to fit measured tree-size distributions in Amazonian forests. The fitted growth scaling parameter

Code is available on reasonable request to the corresponding author.

The supplement related to this article is available online at:

JRM and PMC conceived the project. JRM carried out the data analysis, wrote the paper and prepared the figures. KZ, APKA and CH gave much invaluable advice on analysis, mathematics, and the general direction of the project, as well as commented on the paper.

The authors declare that they have no conflict of interest.

This work and its contributors (Jonathan R. Moore, Arthur P. K. Argles, Kai Zhu, Chris Huntingford and Peter M. Cox) were supported by the European Research Council (ERC) ECCLES project and by the Newton Fund through the Met Office Climate Science for Service Partnership Brazil (CSSP Brazil), as well as by a Faculty Research Grant awarded by the Committee on Research from the University of California, Santa Cruz (Kai Zhu), and the UK Centre of Ecology and Hydrology (CEH) National Capability Fund (Chris Huntingford).

We also wish to thank Ted Feldpausch for his many helpful comments and advice regarding Amazon forests, their allometry and analysis.

We particularly wish to thank the hard-working teams of researchers working to gather the RAINFOR data and share them through the ForestPlots network. The principal investigators (PIs) who worked on each of the forest plots (see Table S2 for details) used that we wish to thank are Samuel Almeida, Esteban Álvarez Dávila, Luiz Aragão, Alejandro Araujo-Murakami, Luzmila Arroyo, Timothy Baker, Jorcely Barroso, Roel Brienen, Fernando Cornejo Valverde, Maria Cristina Peñuela-Mora, William Farfan-Rios, Ted Feldpausch, Eurídice Honorio Coronado, Ben Hur Marimon Junior, Eliana Jimenez-Rojas Jon Lloyd, Yadvinder Malhi, Alexander Parada Gutierrez, Guido Pardo, Beatriz Marimon, Casimiro Mendoza, Irina Mendoza Polo, Abel Monteagudo-Mendoza, David Neill, Nadir Pallqui Camacho, Oliver Phillips, Nigel Pitman, Hirma Ramírez-Angulo, Freddy Ramirez Arevalo, Zorayda Restrepo Correa, Miles Silman, Javier Silva Espejo, Marcos Silveira, John Terborgh, Geertje van der Heijden, Rodolfo Vasquez Martinez, Emilio Vilanova Torre, Luis Valenzuela Gamarra and Vincent Vos.

This research has been supported by the European Research Council (ERC) ECCLES project (grant no. 742472).

This paper was edited by Akihiko Ito and reviewed by two anonymous referees.