% Report from workshop 'How are competitive interactions influenced by traits? A global analysis based on tree radial growth'
% Project leader: Georges Kunstler
% 18/12/2013
This document gives an update on the analyses completed during and after the workshop held in October 2013 at Macquarie University.
**Contact details:** georges.kunstler@gmail.com, Department of Biological Sciences Macquarie University, Sydney, NSW / Irstea EMGR Grenoble France
**Workshop participants:** David A. Coomes, Daniel Falster, Francis Hui, Rob Kooyman, Daniel Laughlin, Lourens Poorter, Mark Vanderwel, Ghislain Vieilledent, Mark Westoby, Joe Wright.
**Other participants and data contributors:** John Caspersen, Hongcheng Zeng, Sylvie Gourlet-Fleury, Bruno Herault, Goran StÃ¥hl, Jill Thompson, Sarah Richardson, Paloma Ruiz, I-Fang Sun, Nathan Swenson, Maria Uriarte, Miguel Zavala, Niklaus E. Zimmermann, Marc Hanewinkel, Jess Zimmerman, Yusuke Onoda, Hiroko Kurokawa, Masahiro Aiba and other.
\newpage
# Background & motivation
It is widely assumed that ecologically dissimilar species compete less
intensely for resources than similar species, and are therefore
more likely to coexist locally than similar species
(the competition-niche similarity hypothesis,
@macarthur_limiting_1967). One way to quantify ecological similarity
between species is via traits, such as leaf, seed and wood
characteristics [@westoby_plant_2002]. Traits influence many aspects
of plant performance, including resource acquisition. Under the *competition-niche similarity hypothesis* higher trait dissimilarity should results in higher resource partitioning at
local scale and less intense competition. This idea underlies numerous ecological analyses
[@kraft_functional_2008; @cornwell_community_2009]. However this
assumption has rarely been tested against field or experimental
outcomes. This is surprising because it is well known that competitive
interactions among vascular plants are more complex. For instance,
most plant species compete for the same limiting resources (water,
light and nutrients), which makes simple local resources partitioning
unlikely. The ranking of competitive ability for these common
limiting resources may be a more important driver of competitive
interaction. If ranking processes are dominant, competitive outcomes
should be more closely related to the hierarchy (or the hierarchical
distance) of relevant functional traits
[@mayfield_opposing_2010; @kunstler_competitive_2012] than trait
dissimilarity. Recent analysis of competitive interactions at local
scale between individual trees (using growth analysis with local
competition index) in mountain forests in the French Alps
[@kunstler_competitive_2012], support this view that competition is
more related to trait hierarchy than trait similarity.
# Objective
Given the importance in the ecological literature of the idea that
trait similarity drives competitive interaction, we will extend the
recent analysis of Kunsteler *et al* [@kunstler_competitive_2012] to other forest ecosystems around the world. For this purpose, we have
assembled several demographic data sets from national forest inventories (NFI)
and from large tropical plots that report individual tree growth. These demographic data sets are combined with data about species functional traits sourced
locally or from global databases (TRY) to test the link between
competition and traits.
# Analysis approach
## General approach
The general approach of the analysis relies on fitting an individual
growth model in which tree growth decreases with increasing abundance
of neighborhood trees. We then consider whether the relative
decrease in growth with increasing neighbor abundance varies with the traits $t_n$ of the neighborhood species in relation to the traits $t_f$ of the focal species[^abiotic-var].
The individual growth model is:
\begin{equation} \label{G1}
G_{f,p,i,t} = G\textrm{max}_{f,p,i} \times s(D_{i,t}) \times g\left(\sum_{n=1}^{N_p} \lambda_{n,f} \times B_{n}\right)
\end{equation}
where:
- $G_{f,p,i,t}$ is the growth (diameter or basal area growth) of
an individual $i$ from species $f$ growing in plot plot $p$ in census $t$,
- $D_i$ is the diameter of the individual $i$,
- $B_{n}$ is the the basal area of neighborhood tree of species $n$,
- $G\textrm{max}_{f,p,i}$ is the maximum growth rate of the focal species $f$ on the plot $p$ for the individual $i$,
- $s$ and $g$ are functions representing the size and the competition effect respectively, and
- $\lambda_{n,f}$ is a parameter representing the growth reduction for a
unit of neighborhood basal area increase of species $n$ on species
$f$.
[^abiotic-var]: Initially I was planning to include abiotic variables to model the variation of the abiotic conditions between plots in the NFI data (climatic variables) or between quadrats in the large tropical plots (soil and topographic variables), but I have decided to not attempt at modeling this directly but only represents this variability through a random plot effect.
## How does $\lambda_{n,f}$ depend on traits of neighborhood and focal species?
The central part of the analysis involves comparing alternative models for $\lambda_{n,f}$ as functions of traits for neighborhood and focal species, $t_n$ and $t_f$ respectively. Initially I was planning to test the two main models explored in
@kunstler_competitive_2012 :
1. $\lambda_{n,f}$ is a function of the absolute distance of traits
($|t{_n} - t{_f}|$) as the classically limiting similarity
hypothesis with
\begin{equation} \label{abs_dist_trait}
\lambda_{n,f} = a + b \times |t_{n} - t_{f}|.
\end{equation}
2. $\lambda_{n,f}$ is a function of a hierarchical distance ($t_{n} - t_{f}$);
\begin{equation} \label{hier_dist_trait}
\lambda_{n,f} = a +b \times (t_{n} - t_{f}).
\end{equation}
The logic behind the hierarchical trait distance model, can be understand through a decomposition of competition in competitive effect and competitive response. The hierarchical trait distance model occurs when the traits conferring a high competitive effect also confer a high competition tolerance[^compreponse]. During the first day of the workshop we discussed the possibility of including a model with separate
links of traits with competitive effect and competitive response. This
model is connected to several papers by Goldberg *et al.*, where
competition is framed in term of effect and response and their links to
traits [@goldberg_competitive_1996]. Two main approaches were
proposed: a multiplicative and an additive
model of competitive effect and response[^inter].
Below I consider the
additive effect-response model because it is simpler. However, I have not ruled out exploring
the multiplicative effect-response model[^equmult].
[^compreponse]: Through out the document I will use competitive response as the inverse of competition tolerance.
[^equmult]: The equations of the multiplicative models are given in the [Appendix 1](#multi).
[^inter]: There was also a detailed discussion of more complex model that would include both effect and response and interactions among both.
## Additive model of competitive effect and response
The general framework for this approach is to consider that
$\lambda_{n,f} = r(t_f) +e(t_n)$ where $r$ and $e$ are
functions for competitive response and effect respectively. A series of models with increasing complexity was identified[^linear-model].
[^linear-model]: Here I present models which are linear functions of the trait but one can easily imagine more complex relations.
1. $\lambda$ is influenced only by the trait of the neighborhood species (competitive effect model):
\begin{equation} \label{effect_trait}
\lambda_{n,f} = a +b \times t_{n}.
\end{equation}
2. $\lambda$ is influenced only by trait of the focal species (competitive response model):
\begin{equation} \label{response_trait}
\lambda_{n,f} = a +b \times t_{f}.
\end{equation}
3. $\lambda$ is influenced by traits of the neighborhood and
focal species (effect-response model):
\begin{equation} \label{response_effect_trait}
\lambda_{n,f} = a +b \times t_{f} +c \times t_{n}.
\end{equation}
The trait hierarchical distance model eq. \ref{hier_dist_trait} is a
sub-case of the effect and response model
eq. \ref{response_effect_trait} where $b=-c$.
During the workshop David Coomes described how to express the model as a function of the community weighted mean of the trait of the neighborhood trees. For the
most complex model eq. \ref{response_effect_trait} this gives:
\begin{equation}
\sum_{n=1}^{N_p} \lambda_{n,f} \times B_n = \sum_{n=1}^{N_p} (a +b \times t_{f}
+c \times t_{n}) \times B_n =B_\textrm{tot} \times (a +b \times t_{f} +c \times \overline{t_{n}})
\end{equation}
where:
- $B_\textrm{tot}$ is the sum of basal area of all neighborhood species,
- $\overline{t_{n}}$ is weighted mean of the trait of the neighborhood
species ($\overline{t_{n}}= \sum_{n=1}^{N_p} P_n \times t_n$ with $P_n$ the
relative basal area abundance of species $n$, $B_n/B_\textrm{tot}$).
Subsequent to the workshop, and in the material I presented at Ecotas13[^ecotas], I decided to
compare the absolute trait distance model eq. \ref{abs_dist_trait} and the
effect-response model eq. \ref{response_effect_trait}.
[^ecotas]: Joint conference of the Ecological Society of Australia and the New Zealand Ecological Society, Nov 2013.
# Data preparation and analysis
The objective was collate data sets spanning
most of the forest biomes of the world, to see if the links
between competition are consistent across these biomes (the
objective was not to have the largest number of data sets). We focused
on five key traits seed mass[^seed-mass], LMA, Leaf N per mass, wood density and maximum
height. Key points for inclusion was good coverage for at least one of the traits of interest. Table
\ref{table-data} presents the different data set used.
[^seed-mass]: Seed mass was not used in subsequent analysis because its link with competitive effect and response is unclear.
## Dividing NFI data by ecoregion
NFI data were split into ecoregions (using local ecoregion classification for each country): these beings regions with broadly similar
ecological conditions. This allowed us to test whether the link between
competitive interactions and traits varies with
abiotic conditions (for
instance in the US there is a large variability between the north and
the south). Division into ecoregions was also necessary for technical reasons (to speed up the
estimation and solve some memory limit issues). Figure
\ref{biomes} presents the position of each ecoregion in
relation to their mean annual temperature and annual precipitation,
overlayed with Whittaker biomes [@whittaker_classification_1962]. Figure \ref{map} presents the positions of the different plots geographically.
## Data formatting
We (mainly Francis Hui, PhD student from UNSW) formatted all tree
data set to match common unit and names (see [Appendix 2,
Variables description and units](#units)). We tested to check whether the range of
variables values (mean and quantile) were within sensible limits and
visually inspected plots of $G$ per $D$ to check for errors.
We extracted SLA ($mm^2/mg$), Leaf N per mass ($mg/g$), wood density
($mg/mm^3$), seed mass ($mg$) and maximum height ($m$), for each species from the TRY data base for
NFI data and from a local
traits database for the large tropical plots[^except]. For most NFI data maximum
height was extracted from the height measurement as the 99% quantile of
the measured height of the species. This need to be update with the method of @king_growth_2006 which is less dependent on sample size. We ran independent test of the extraction of the traits per species to
validate the trait extraction (see Figure \ref{trait} for the range of traits variation)
[^except]: except New Zealand for which a local traits data was available and for the Puerto Rico FIA data which was extracted fromthe Luquillo traits data set - data of N. Swenson.
## Data processing
Next we split each dataset by ecoregion, keeping only
ecoregions where, on average, at least three species contributed more than
5% of the total basal area of each plots. This had the effect of
excluding quasi-monospecific stands.
First we computed the local basal area ($cm^2/m^2$) of neighborhood
competitor per species for each individual tree. For NFI data the
neighborhood was defined as the plot (the size of the plot range from
10 m in radius up to 25 m in radius -for large trees- with some data set having a variable
plot size depending on tree dbh and the New Zealand data the plots are
20x20 m). For the large tropical plots, the neighborhood was defined as
a 15 m radius around the tree.
The community weight mean of the neighboring trees and of the absolute
trait distance between the focal tree and neighborhood trees was calculated using the species level traits data, or filling
missing value with genus level data when it was possible, or
filling the remaining value with the community mean of the trait. All traits were
centered and standardized per data set (a global traits standardization
doesn't seems to provides strikingly different values). We run independent
computation of the community weight means to validate the processing of the data and inspected
histograms of $\overline{t_n}$ to identify errors.
We used only individual tree for which 90% of its neighborhood was
covered with at least genus level traits in subsequent analysis. The
table \ref{table-perc} gives the percentage of the data for which at
least 90% of neighborhood is covered with species or genus level
trait. Paracou and M'Baiki are the only two sites with very low
coverage (this because of missing traits but also because of missing
taxonomic identification).
## Fitting of a mixed linear model
During the workshop we ran estimation using a hierarchical Bayesian model
using [JAGS](http://mcmc-jags.sourceforge.net/). In the subsequent analysis I decided (with the help of Ghislain to test this approach) to start with a linear mixed model approach (function lmer in
package [lme4](http://cran.r-project.org/web/packages/lme4/index.html)
in R cran). The reasons for the change are
1. a log-linear function provides a good
first approximation to the shape of
more complex non-linear functions for the size and competition effect (mainly followingsuch as the one used in the work of C. Canham see
@uriarte_trait_2010), and
2. using lmer was much faster than an estimation with JAGS or [Stan](http://mc-stan.org/).
When the analysis is more advanced I will test whether choice of linear or non-linear functions for functions $s$ and $g$ influences results, by running the same model using Stan.
## Fitted models
The models fitted were based on Equation \label{G1}:
\begin{equation} \label{logG}
\log{G}_{f,p,i,t} = \log{G\textrm{max}}_{f,p,i} + \alpha_f \times D_{i,t} +
\lambda_{n,f} \times (\sum_{n=1}^{N_p} \log{B}_{n})
\end{equation}
where:
- $\log {G}$ is the log basal area growth,
- $\log{G\textrm{max}}$ is the intercept representing log basal area growth with no competition [^Gmax] including a plot $p$ random effect to account for variation of abiotic conditions between plots in NFI or the quadrats in large tropical plots (assuming the same
variance for all species), a random focal species $f$ effect and a random individual $i$ effect when multiple
census are present,
- $\alpha$ represents the dbh slope with a random focal species $f$ effect, and
- $\log{B}_n$ is log basal area of the neighboring species $n$. Here if competitive parameters $\lambda_{n,f}$ is negative this represents competition if positive, facilitation.
[^Gmax]: The term maximum growth is generally used in the non linear models of growth used by C. Canham, I used this term here even if this not strictly identical to a maximum growth.
We compared two alternative models for $\lambda_{n,f}$:
(i) $\lambda$ is a function effect and response traits ($\lambda_{n,f} = a +b \times t_{f} +c
\times t_{n}$) and
(ii) $\lambda$ is a
function the absolute trait distance ($\lambda_{n,f} = a + b \times
|t_{n} - t_{f}|$).
These two models can be expressed in terms of community weighted mean trait value as follows. For the trait effect-response model:
\begin{equation} \label{logG-ER}
\log{G}_{f,p,i,t} = \log{G\textrm{max}}_{f,p,i} + \alpha_f \times D_{i,t} + a \times
\log{B}_\textrm{tot} + b \times \log{B}_\textrm{tot} \times t_f + c \times \log{B}_\textrm{tot} \times \overline{t_{n}}.
\end{equation}
We also fitted version of the model that only included the effect part (not including $b \times \log{B}_\textrm{tot} \times t_f$) or only the response part (not including $c \times \log{B}_\textrm{tot} \times \overline{t_{n}}$).
For the absolute trait distance:
\begin{equation} \label{logGabs}
\log{G}_{f,p,i,t} = \log{G\textrm{max}}_{f,p,i} + \alpha_f \times D_{i,t} + a \times
\log{B}_\textrm{tot} + b \times \log{B}_\textrm{tot} \times \overline{|t_{n} - t_{f}|}.
\end{equation}
where $\overline{|t_{n} - t_{f}|} = \sum_{n=1}^{N_p} P_n \times |t_n -t_f|$.
We then compared these two models to a null model where competition is
constant and independent of focal and neighborhood species trait.
\begin{equation} \label{logG-null}
\log{G}_{f,p,i,t} = \log{G\textrm{max}}_{f,p,t} + \alpha_f \times D_{i,t} + a \times
\log{B}_\textrm{tot}.
\end{equation}
We also fitted a model with no competition ($\log{G\textrm{max}}_{f,p,t} + \alpha_f \times D_{i,t}$).
We compared models using AIC and computed an effect size for the
trait-based model as the difference in $R^2$ to the constant competition model eq. \ref{logG-null} using the approach recently proposed by
@nakagawa_general_2013 (using conditional $R^2$).
# Preliminary results
For several of the NFI data sets (Spain, France, US) the absolute trait distance model was selected as the best model in more ecoregions than any version of the effect-response model (number of best model over all trait and ecoregions: absolute distance=102, Effect=48,Response=5, Effect-response= 26, no competition effect=28, simple competition (no trait)=18). (see Tables \ref{table-aic-SLA} \ref{table-aic-Leaf.N} \ref{table-aic-Wood.density} and \ref{table-aic-Max.height} for full details on model selection by AIC).
The effect size of the models shows a different picture on the figure \ref{boxplot-effectsize} and the figure \ref{boxplot-effectsize-MAP}. The effect size of the effect-response model had often much higher value than the absolute distance models. This was not the case for all ecoregions, with a large proportion still showing low effect sizes[^EffectSize]. Only for maximum height the absolute distance models resulted in effect size similar to the effect-response models.
[^EffectSize]: The effect size represents the increase in $R^2$ of a particular model over the basic diameter growth model (diameter growth variance). It would be better expressed as a percentage of competition explained (species and diameter effect explain more variation than competition so the effect size will always be low) but I need to work more on that point (try to fit a model with a random effect per focal species x neighborhood species in $\lambda$?).
Overall the effect-response models is strongest at low mean annual precipitation (
Figure \ref{boxplot-effectsize-MAP}). This was the case for all traits. This pattern is also visible on the plots of the parameters in function of the MAP of the ecoregion where the maximum value of the parameters is reached for low MAP (see figure \ref{param-trait}). From this figure it is also clear that the model of hierarchical trait distance I used in @kunstler_competitive_2012 is not able to represents adequately the link between traits and competition. In the effect-response models the effect and response parameters are not generally of opposite sign and not of the same magnitude ($b \ne -c$). This means that the competitive effect and response are not necessarily correlated and not related in the same way to the traits. The fact that competitive effect and response are not always correlated was already stressed out by @goldberg_competitive_1996.
Most of the effect-response models fitted show a competitive effect (negative value of the parameters on the figure \ref{param-BATOT}). And overall the average competitive effect of one unit of neighborhood basal area is higher (parameters more negative on figure \ref{param-BATOT}) in ecoregions with lower MAP.
# Future work to do
- Improve the method to estimate the effect-size of the alternative models (ideally what percentage of competition variation is explained by traits).
- Fit a similar model for tree survival
- Explore non-linear model for growth and survival using Stan (probably used the models used by Canham and Uriarte).
- Fit multi-traits models (include multiple traits in effect and response models and either multidimensional distance in the absolute distance model or include all single trait absolute distance). Try to use spike and slab prior for variables selection.
- Try to include traits effect in parameter $Gmax$. This would allows to (1) test if this change the results observed for the traits effect on $\lambda$ (a comment of Maria Uriarte) and (2) test if traits underpin a trade-off between max growth with out competition and competition tolerance.
- Explore if the decrease in the link between trait and competition at high MAP is related in a change in the packing of trait space in this communities.
- Explore the possibility that trait effect may be different for evergreen/deciduous species (leaf traits) or angiosperm/conifer species (wood density). This could be done by fitting different parameters for the trait of evergreen deciduous and conifer in the effect-response model. This is not really possible for the absolute distance model.
- Use an alternative way of dividing the NFI data than the ecoregion (class of MAP and MAT?).
- Try to run a global analysis with all data (memory limit issue to solve).
\newpage
# FIGURES & TABLES
![**Positions of the data sets analysed in the climatic biomes of Whittaker.** The coloured polygons represents the biomes. The points represent the mean position of the data set in the mean annual temperature and annual precipitation space. For the national forest inventory the 95% quantile of the climate within the ecoregion is represented by an error bar. The temperature and precipitation are taken from worldclim [@hijmans_very_2005].\label{biomes}](biome_ecocode_xy.pdf)
\pagebreak
![**Map of the plots of the data sets analysed.** National forest inventory plots are represented by small points and large permanent plot by large points.\label{map}](world_map_all.png)
\pagebreak
\newpage
![**Correlation pairs over all data sets (in log scale).** Each data set is drawn with a different symbols and colors. Traits SLA ($mm^2/mg$), Leaf N per mass ($mg/g$), wood density
($mg/mm^3$), maximum height ($m$). \label{trait}](traits-XY.pdf)
\pagebreak
\pagebreak
\newpage
![**Effect size of the absolute distance models and the effect-response model over all ecoregion for the four traits.** Effect size is computed as the difference of $R_c^2$ between a constant competition model and the tested model. \label{boxplot-effectsize}](R2_boxplot_two.pdf)
\pagebreak
![**Effect size of the absolute distance models and the effect-response model off each ecoregion in function of the mean annual precipitation (MAP) for the four traits.** Effect size is computed as the difference of $R_c^2$ between a constant competition model and the tested model. \label{boxplot-effectsize-MAP}](R2_MAP_two.pdf)
\pagebreak
![**Traits parameters for effect-response model and for the absolute distance model fitted for each ecoregion plotted in function of the mean annual precipitation of the ecoregion.** Results per traits are presented per columns. For the effect-response model the response parameter (in black) and the effect parameter (in red) are respectively the parameter $b$ and $c$ of the equation \ref{logG-ER}. For the absolute distance model the parameter is the parameter $b$ of the equation \ref{logGabs}. A positive value of the trait parameters means that the slope of growth decrease with basal area is either less negative (less competition) or more positive (more facilitation). This is mainly competitive interactions (see Figure \ref{param-BATOT}). \label{param-trait}](parameters_MAP_2models.pdf)
\pagebreak
![**Total basal area parameters for effect-response model fitted for each ecoregion plotted in function of the mean annual precipitation of the ecoregion.** Results per traits are presented per columns. This is the parameter $a$ of the equation \ref{logG-ER}. \label{param-BATOT}](parameters_BATOT_MAP.pdf)
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-------------------------------------------------------------------------------------------------------------------------------
Data name Demographic data Traits data Availability Abiotic variables
-------------------------- ---------------------- ----------------------------- --------------------------- -------------------
BCI Large plot Available with data ok Topography soil
Fushan Large plot Available with data ok Topography soil
Luquillo Large plot Available with data ok Topography soil
La Chonta Large plot Available with data no Topography soil
Paracou Large plots Available with data ok Topography soil
no max height
Mbaiki Large plots Available with data ok Topography soil
no max height
coverage limited
FIA Forest inventory plots TRY ok climate
Canada Forest inventory plots TRY ok climate
France Forest inventory plots TRY ok climate soil
Spain Forest inventory plots TRY ok climate
Sweden Forest inventory plots TRY ok climate
Switzerland Forest inventory plots TRY ok climate
New Zealand Forest inventory plots Landcare ok climate
Australia NSW Medium size plots Available but no LMA ok climate
Japan Large plots Available with ok climate
data but no Leaf N
-------------------------------------------------------------------------------------------------------------------------------
: Description of the data and traits available \label{table-data}
\pagebreak
-----------------------------------------------------------------------------------------------------
set ecoregion N obs total % Leaf N % Seed mass % SLA % Wood density % Max height
-------- ----------- ------------- ---------- ------------- --------- ---------------- --------------
Sweden PA0405 27124 0.965 0.965 0.963 0.9628 0.8972
Sweden PA0436 293342 0.9587 0.9587 0.9586 0.9536 0.9019
Sweden PA0608 187721 0.9501 0.9501 0.9496 0.9445 0.9086
Sweden PA1110 27376 0.9536 0.9536 0.9528 0.9454 0.9084
NVS BeechHumid 19790 0.899 0.9704 0.967 0.9082 0.998
NVS MixedCool 8738 0.8977 0.9698 0.9698 0.9353 1
NVS MixedWarm 4546 0.93 0.9879 0.9573 0.936 0.9879
US DR.DO 25499 0.8165 0.9438 0.8129 0.5829 0.9755
US Ho.Co.Mo 149390 0.7575 0.932 0.7575 0.8457 0.9956
US HU.TE.DO 15146 0.5203 0.991 0.332 0.7428 0.991
US HU.TR.DO 4532 0.2732 0.5812 0.2502 0.2928 0.9762
US Pr.Di 50719 0.5387 0.9428 0.6099 0.7184 0.9861
US Su.Di 517649 0.6648 0.9675 0.6199 0.7621 0.9874
US Su.Mo 11661 0.5784 0.8245 0.2664 0.6485 0.993
US Wa.Co.Di 382200 0.9734 0.9802 0.9763 0.9667 0.9905
US Wa.Co.Mo 106502 0.9834 0.9842 0.9719 0.9713 0.9937
Canada -132 274375 0.9744 0.9747 0.9485 0.946 0.9932
Canada -211 190700 0.9423 0.9492 0.9404 0.9388 0.9921
Canada M211b 7722 0.9128 0.9128 0.8793 0.8723 0.9372
NSW AA 805 0 1 0 0.9553 1
France AB 41900 0.9475 0.9727 0.9497 0.9893 0.997
France C 27261 0.9256 0.9872 0.925 0.9767 0.997
France F 18704 0.9069 0.984 0.9069 0.9841 0.9942
France GDE 47377 0.9808 0.9936 0.9827 0.991 0.9962
France HI 34823 0.9824 0.9938 0.9752 0.9692 0.9961
France JK 14251 0.96 0.9846 0.9537 0.934 0.9872
-----------------------------------------------------------------------------------------------------
: **Number of tree radial growth observation per data sets and ecoregion and percentage of observation with a coverage of the traits of neighborhood tree >90% and observation for the focal species trait.** For the remaining 10% of the neighborhood the missing trait were filled with *genus* mean or with community mean. (continued below) \label{table-perc}
-----------------------------------------------------------------------------------------------------
set ecoregion N obs total % Leaf N % Seed mass % SLA % Wood density % Max height
-------- ----------- ------------- ---------- ------------- --------- ---------------- --------------
Swiss eco.1 25797 0.956 0.9591 0.956 0.9591 0.9734
Swiss eco.2 34404 0.9651 0.9668 0.9634 0.966 0.9847
Swiss eco.3 46682 0.953 0.953 0.9402 0.9371 0.9739
Swiss eco.4 11714 0.9431 0.9431 0.9125 0.8043 0.9727
Swiss eco.5 13674 0.9459 0.9459 0.852 0.7475 0.9556
Swiss eco.6 16230 0.9333 0.9341 0.8834 0.8819 0.9584
Spain PA0406 62985 0.906 0.9066 0.8562 0.8711 0.9972
Spain PA0433 42475 0.961 0.9646 0.9537 0.9584 0.998
Spain PA1208 40609 0.9827 0.9834 0.9285 0.9849 0.9907
Spain PA1209 123531 0.9808 0.9813 0.9225 0.981 0.986
Spain PA1215 48088 0.9754 0.9765 0.9726 0.9711 0.9955
Spain PA1216 58366 0.9764 0.9765 0.9704 0.9772 0.9949
Spain PA1221 13239 0.9579 0.986 0.9447 0.9776 0.9893
Japan ct 5136 0 0.9077 1 1 1
Japan st 1816 0 0.103 0.7605 0.7605 0.8673
Japan wt 10749 0 0.7016 0.9973 0.9973 1
BCI tropical 93838 0.8508 0.8174 0.8725 0.8393 0.9519
Fushan tropical 14701 0.1901 0.0008163 0.9997 0.9465 0.8911
Paracou tropical 92199 0.03612 3.254e-05 0.05546 0.05191 0
Luquillo tropical 14011 0.9374 0.9374 0.9374 0.9374 0.9374
Mbaiki tropical 6377 0.0009409 0 0.0009409 0.007057 0
-----------------------------------------------------------------------------------------------------
\pagebreak
------------------------------------------------
set nocomp simplecomp AD R E ER
-------- -------- ------------ ---- --- --- ----
BCI 0 1 0 0 0 0
Canada 0 1 1 0 1 0
France 0 1 4 0 1 0
Fushan 0 0 0 0 1 0
Japan 2 0 0 0 1 0
Luquillo 0 0 0 0 1 0
Mbaiki 0 0 0 1 0 0
NVS 1 1 0 0 1 0
Paracou 0 1 0 0 0 0
Spain 0 0 6 0 0 1
Sweden 0 0 0 0 2 2
Swiss 0 0 4 0 2 0
US 1 0 4 1 2 1
------------------------------------------------
: **Best models per data set for trait SLA**. nocomp: model with no competitive effect, simplecomp: model with competitive effect constant over all species, AD: model based on trait absolute distance, R: model based only on competitive response on $t_f$, E: model based only on competitive effect on $t_n$, ER: model based on competitive effect and response with $t_n$ and $t_f$. \label{table-aic-SLA}
\pagebreak
------------------------------------------------
set nocomp simplecomp AD R E ER
-------- -------- ------------ ---- --- --- ----
BCI 0 1 0 0 0 0
Canada 0 1 0 0 1 1
France 0 0 3 0 3 0
Fushan 0 1 0 0 0 0
Luquillo 0 0 0 0 1 0
Mbaiki 1 0 0 0 0 0
NVS 2 1 0 0 0 0
Paracou 0 0 0 0 1 0
Spain 0 0 6 0 0 1
Sweden 0 0 0 0 2 2
Swiss 0 1 1 0 4 0
US 1 0 4 1 2 1
------------------------------------------------
: **Best models per data set for trait Leaf.N.** nocomp: model with no competitive effect, simplecomp: model with competitive effect constant over all species, AD: model based on trait absolute distance, R: model based only on competitive response on $t_f$, E: model based only on competitive effect on $t_n$, ER: model based on competitive effect and response with $t_n$ and $t_f$. \label{table-aic-Leaf.N}
\pagebreak
------------------------------------------------
set nocomp simplecomp AD R E ER
-------- -------- ------------ ---- --- --- ----
BCI 0 1 0 0 0 0
Canada 0 0 3 0 0 0
France 0 0 3 0 0 3
Fushan 0 0 0 0 1 0
Japan 2 0 0 1 0 0
Luquillo 0 0 1 0 0 0
Mbaiki 1 0 0 0 0 0
NSW 1 0 0 0 0 0
NVS 2 1 0 0 0 0
Paracou 0 0 0 0 1 0
Spain 0 0 7 0 0 0
Sweden 0 0 0 0 2 2
Swiss 0 0 4 0 2 0
US 1 0 4 1 0 3
------------------------------------------------
: **Best models per data set for trait Wood.density.** nocomp: model with no competitive effect, simplecomp: model with competitive effect constant over all species, AD: model based on trait absolute distance, R: model based only on competitive response on $t_f$, E: model based only on competitive effect on $t_n$, ER: model based on competitive effect and response with $t_n$ and $t_f$. \label{table-aic-Wood.density}
\pagebreak
------------------------------------------------
set nocomp simplecomp AD R E ER
-------- -------- ------------ ---- --- --- ----
BCI 0 1 0 0 0 0
Canada 0 0 3 0 0 0
France 0 0 3 0 3 0
Fushan 0 1 0 0 0 0
Japan 2 0 0 0 1 0
Luquillo 0 0 0 0 1 0
NSW 1 0 0 0 0 0
NVS 2 0 0 0 0 1
Spain 0 0 6 0 0 1
Sweden 0 0 0 0 1 3
Swiss 0 1 1 0 3 1
US 0 1 7 0 1 0
------------------------------------------------
: **Best models per data set for trait Max.height.** nocomp: model with no competitive effect, simplecomp: model with competitive effect constant over all species, AD: model based on trait absolute distance, R: model based only on competitive response on $t_f$, E: model based only on competitive effect on $t_n$, ER: model based on competitive effect and response with $t_n$ and $t_f$. \label{table-aic-Max.height}
\pagebreak
# Appendix 1. Multiplicative model of competitive effect and response {#multi}
The general framework for this approach is to consider that $\lambda_{n,f} = r(t_f) \times e(t_n)$ where $r$ and $e$ are respectively function that relate the competitive response and effect to the trait. We can test a series of model with increasing complexity of trait effect.
1. $\lambda$ can be influence only by the variation in competitive effect through the trait of the neighborhood species:
\begin{equation}
\lambda_{n,f} = a +b \times t_{n}
\end{equation}
2. $\lambda$ can be influence only by the variation in competitive response through the trait of the focal species:
\begin{equation}
\lambda_{n,f} = a +b \times t_{f}
\end{equation}
3. $\lambda$ can be influence by the variation of both competitive effect and response through the trait of the neighborhood and focal species:
\begin{equation}
\lambda_{n,f} = (a +b \times t_{f}) \times (c +d \times t_{n})
\end{equation}
As for the additive model it is then possible to develop the multiplicative model 3 to relate the competition in term of community weighted mean trait of the neighborhood species ($\overline{t_{n}}$).
\begin{equation} \label{multi-er}
\sum_{n=1}^{N_p} \lambda_{n,f} \times B_n = B_\textrm{tot} \times (a +b \times t_{f}) \times (c+ d \times \overline{t_{n}})
\end{equation}
## Comparison of the multiplicative and additive effect and response model
Developing the multiplicative model gives
\begin{equation}
(a +b \times t_{f}) \times (c +d \times t_{n}) = ac+bc \times t_f +ad \times t_n +bd \times t_f \times t_n
\end{equation}
This equation bears some similarity to the additive model plus interaction Equation \label{add-inter} - which is an extension of the effect/response model presented above (equation \label{response_effect_trait}) - which include an interaction between the traits $t_n$ and $t_f$ is:
\begin{equation} \label{add-inter}
\lambda_{n,f} = a' +b' \times t_{f} +c' \times t_{n}+d' \times t_{n} \times t_{f}
\end{equation}
The two models are equal when:
\begin{equation}
a'=ac \mspace{3mu} ;\mspace{3mu} b'=bc\mspace{3mu} ;\mspace{3mu} c'=ad \mspace{5mu} and \mspace{5mu} d'=bd
\end{equation}
The multiplicative model is more constraining than the additive model plus interaction. In other word the additive model with interaction can be fitted to any multiplicative model but the inverse is not true (This would requires adding an interaction in the multiplicative model). For instance, it is not possible to match the hierarchical distance because if $b'$ and $d' \neq 0$ then $d' \neq 0$ as well. More generally, if parameters $a$, $b$ , $c$ and $d$ vary between [-max.r, max.r] then $d'>b'*c'/(max.r^2)$ (or $d'