Appendix D: all forest stands generated by the forest factory. Bohn et al. 2016, RSOS

This R-Workspace includes all analysed forest stands. "forests" contains the forest properties (BA,Theta, Omega, meanHeight), the productivity of each year (AWP2000 – AWP2004), the species richness and a column called mixture. The number in this column refers to the line in "mixtureSpecies" wherein the mixtures are described. The numbers in these strings refers to the rows of "species", which also contains the parameters of the species

Appendix A: Additional information regarding method and validation from The importance of forest structure to biodiversity–productivity relationships

While various relationships between productivity and biodiversity are found in forests, the processes underlying these relationships remain unclear and theory struggles to coherently explain them. In this work, we analyse diversity–productivity relationships through an examination of forest structure (described by basal area and tree height heterogeneity). We use a new modelling approach, called ‘Forest Factory’, which generates various forest stands and calculates their annual productivity (above wood increment). Analysing approximately 300 000 forest stands, we find that mean forest productivity does not increase with species diversity. Instead forest structure emerges as the key variable. Similar patterns can be observed by analysing 5054 forest plots of the German National Forest Inventory. Furthermore, we group the forest stands into nine forest structure classes, in which we find increasing, decreasing, invariant and even bell-shaped relationships between productivity and diversity. In addition, we introduce a new index, called optimality, which describes the ratio of realized to the maximal possible productivity (by shuffling species identities). The optimality and forest structure indices explain the obtained productivity values quite well (<i>R</i>² between 0.7 and 0.95), whereby the influence of these attributes varies within the nine forest structure classes

Appendix C: detailed Model describtion from The importance of forest structure to biodiversity–productivity relationships

While various relationships between productivity and biodiversity are found in forests, the processes underlying these relationships remain unclear and theory struggles to coherently explain them. In this work, we analyse diversity–productivity relationships through an examination of forest structure (described by basal area and tree height heterogeneity). We use a new modelling approach, called ‘Forest Factory’, which generates various forest stands and calculates their annual productivity (above wood increment). Analysing approximately 300 000 forest stands, we find that mean forest productivity does not increase with species diversity. Instead forest structure emerges as the key variable. Similar patterns can be observed by analysing 5054 forest plots of the German National Forest Inventory. Furthermore, we group the forest stands into nine forest structure classes, in which we find increasing, decreasing, invariant and even bell-shaped relationships between productivity and diversity. In addition, we introduce a new index, called optimality, which describes the ratio of realized to the maximal possible productivity (by shuffling species identities). The optimality and forest structure indices explain the obtained productivity values quite well (<i>R</i>² between 0.7 and 0.95), whereby the influence of these attributes varies within the nine forest structure classes

On the Challenge of Fitting Tree Size Distributions in Ecology

<div><p>Patterns that resemble strongly skewed size distributions are frequently observed in ecology. A typical example represents tree size distributions of stem diameters. Empirical tests of ecological theories predicting their parameters have been conducted, but the results are difficult to interpret because the statistical methods that are applied to fit such decaying size distributions vary. In addition, binning of field data as well as measurement errors might potentially bias parameter estimates. Here, we compare three different methods for parameter estimation – the common maximum likelihood estimation (MLE) and two modified types of MLE correcting for binning of observations or random measurement errors. We test whether three typical frequency distributions, namely the power-law, negative exponential and Weibull distribution can be precisely identified, and how parameter estimates are biased when observations are additionally either binned or contain measurement error. We show that uncorrected MLE already loses the ability to discern functional form and parameters at relatively small levels of uncertainties. The modified MLE methods that consider such uncertainties (either binning or measurement error) are comparatively much more robust. We conclude that it is important to reduce binning of observations, if possible, and to quantify observation accuracy in empirical studies for fitting strongly skewed size distributions. In general, modified MLE methods that correct binning or measurement errors can be applied to ensure reliable results.</p> </div

Effects of binning and random measurement errors on parameter estimation using different MLE methods.

<p>(a) MLE including binning (<i>multinomial MLE</i>) and (b) MLE accounting for measurement errors (<i>Gaussian MLE</i>). We evaluate virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. Solid lines represent the mean estimates and shaded areas show the standard deviation (of (a) 1000 values and (b) 250 values). (a) Effect of binning on parameter estimates. Virtual data are classified into classes of certain bin width (x-axis in cm). (b) Effect of random measurement errors on parameter estimates. An error value generated from a Gaussian distribution with mean cm and an assumed standard deviation (x-axis in cm) is added to each virtual data value.</p

Supplementary material from What drives the spatial distribution and dynamics of local species richness in tropical forest?

Detailed description of the simulation model, analytical model and additional figure

Presentation of the three assumed truncated frequency distributions used in our investigations.

<p>Presentation of the three assumed truncated frequency distributions used in our investigations.</p

Scheme of the evaluation procedure of virtual data sets.

<p>Scheme of the evaluation procedure of virtual data sets.</p

Effect of errors on <i>Akaike weights</i> for the correct determination of the underlying distribution.

<p>In each row virtual data sets of sample size  = 500 which originate from the three truncated distributions (power-law, negative exponential and Weibull distribution) are evaluated. Weights are calculated supposing these distributions (power-law, negative exponential and Weibull distribution) with (a)–(c) <i>multinomial MLE</i> and (d)–(f) <i>Gaussian MLE</i>. The highest <i>Akaike weight</i> determines the best fit of a frequency distribution to the data. (a)–(c) Effect of binning of virtual data sets with used bin width (x-axis in cm) on <i>Akaike weights</i>. (d)–(f) Effect of random measurement errors added to the virtual data sets on <i>Akaike weights</i>, whereby errors are Gaussian distributed with mean cm and assumed standard deviation (x-axis in cm). Solid lines represent the mean of <i>Akaike weights</i> and shaded areas show the standard deviation (of (a)–(c) 1000 values and (d)–(f) 250 values).</p

Analyses of virtual data including different levels of measurement errors.

<p>We evaluate 1,000 virtual data sets of sample size  = 500 from a truncated power-law, a truncated negative exponential and a truncated Weibull distribution. An error value generated from a Gaussian distribution with mean cm and an assumed standard deviation (x-axis in cm) is added to each virtual data point before applying <i>standard MLE</i>. (a) Effect of random measurement errors on parameter estimates of the three investigated distributions. (b)–(d) Effect of random measurement errors on <i>Akaike weights</i> supposing three distributions (power-law, negative exponential and Weibull distribution) for (b) power-law distributed virtual data, (c) negative exponentially distributed virtual data and (d) Weibull distributed virtual data. The highest <i>Akaike weight</i> determines the best fit of a frequency distribution to the data. Solid lines represent the mean values and shaded areas show the standard deviation (of 1,000 calculated values).</p