progress.md 41.5 KB
 Daniel Falster committed Dec 17, 2013 1 2 % Report from workshop 'How are competitive interactions influenced by traits? A global analysis based on tree radial growth' % Project leader: Georges Kunstler  Georges Kunstler committed Dec 19, 2013 3 % 18/12/2013  Georges Kunstler committed Dec 17, 2013 4   Daniel Falster committed Dec 18, 2013 5 This document gives an update on the analyses completed during and after the workshop held in October 2013 at Macquarie University.  Georges Kunstler committed Dec 17, 2013 6   Daniel Falster committed Dec 17, 2013 7   Daniel Falster committed Dec 18, 2013 8 **Contact details:** georges.kunstler@gmail.com, Department of Biological Sciences Macquarie University, Sydney, NSW / Irstea EMGR Grenoble France  Georges Kunstler committed Dec 17, 2013 9   Georges Kunstler committed Dec 19, 2013 10 11 12 **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 Ståhl, 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.  Georges Kunstler committed Dec 17, 2013 13 14   Daniel Falster committed Dec 17, 2013 15 \newpage  Georges Kunstler committed Dec 17, 2013 16   Daniel Falster committed Dec 17, 2013 17 # Background & motivation  Georges Kunstler committed Dec 17, 2013 18 It is widely assumed that ecologically dissimilar species compete less  Daniel Falster committed Dec 17, 2013 19 20 21 intensely for resources than similar species, and are therefore more likely to coexist locally than similar species (the competition-niche similarity hypothesis,  Georges Kunstler committed Dec 17, 2013 22 23 24 @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  Georges Kunstler committed Dec 19, 2013 25 of plant performance, including resource acquisition. Under the *competition-niche similarity hypothesis* higher trait dissimilarity should results in higher resource partitioning at  Georges Kunstler committed Dec 17, 2013 26 local scale and less intense competition. This idea underlies numerous ecological analyses  Georges Kunstler committed Dec 17, 2013 27 28 29 30 31 32 [@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  Georges Kunstler committed Dec 17, 2013 33 unlikely. The ranking of competitive ability for these common  Georges Kunstler committed Dec 17, 2013 34 35 36 37 38 39 40 41 42 43 44 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.  Daniel Falster committed Dec 17, 2013 45 # Objective  Georges Kunstler committed Dec 17, 2013 46 Given the importance in the ecological literature of the idea that  Daniel Falster committed Dec 17, 2013 47 48 49 50 51 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  Georges Kunstler committed Dec 17, 2013 52 53 competition and traits.  Daniel Falster committed Dec 18, 2013 54 # Analysis approach  Daniel Falster committed Dec 17, 2013 55   Daniel Falster committed Dec 18, 2013 56 ## General approach  Daniel Falster committed Dec 17, 2013 57 58 59 60 61  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].  Georges Kunstler committed Dec 17, 2013 62   Daniel Falster committed Dec 17, 2013 63 The individual growth model is:  Georges Kunstler committed Dec 17, 2013 64 \begin{equation} \label{G1}  Daniel Falster committed Dec 18, 2013 65 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)  Georges Kunstler committed Dec 17, 2013 66 67 \end{equation}  Daniel Falster committed Dec 17, 2013 68 69 70 where: - $G_{f,p,i,t}$ is the growth (diameter or basal area growth) of  Daniel Falster committed Dec 18, 2013 71 72 73 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 $n$,  Georges Kunstler committed Dec 18, 2013 74 - $G\textrm{max}_{f,p,i}$ is the maximum growth rate of the focal species $f$ on the plot $p$ for the individual $i$,  Daniel Falster committed Dec 18, 2013 75 - $s$ and $g$ are functions representing the size and the competition effect respectively, and  Daniel Falster committed Dec 17, 2013 76 - $\lambda_{n,f}$ is a parameter representing the growth reduction for a  Georges Kunstler committed Dec 17, 2013 77 unit of neighborhood basal area increase of species $n$ on species  Daniel Falster committed Dec 17, 2013 78 79 $f$.  Daniel Falster committed Dec 18, 2013 80   Georges Kunstler committed Dec 17, 2013 81   Georges Kunstler committed Dec 17, 2013 82 [^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.  Georges Kunstler committed Dec 17, 2013 83   Daniel Falster committed Dec 18, 2013 84 ## How does $\lambda_{n,f}$ depend on traits of neighborhood and focal species?  Georges Kunstler committed Dec 17, 2013 85   Daniel Falster committed Dec 18, 2013 86 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  Georges Kunstler committed Dec 17, 2013 87 88 89 @kunstler_competitive_2012 : 1. $\lambda_{n,f}$ is a function of the absolute distance of traits  Georges Kunstler committed Dec 17, 2013 90 91  ($|t{_n} - t{_f}|$) as the classically limiting similarity hypothesis with  Georges Kunstler committed Dec 17, 2013 92 \begin{equation} \label{abs_dist_trait}  Daniel Falster committed Dec 18, 2013 93 \lambda_{n,f} = a + b \times |t_{n} - t_{f}|.  Georges Kunstler committed Dec 17, 2013 94 95 96 \end{equation} 2. $\lambda_{n,f}$ is a function of a hierarchical distance ($t_{n} - t_{f}$); \begin{equation} \label{hier_dist_trait}  Daniel Falster committed Dec 18, 2013 97 \lambda_{n,f} = a +b \times (t_{n} - t_{f}).  Georges Kunstler committed Dec 17, 2013 98 99 \end{equation}  Georges Kunstler committed Dec 18, 2013 100 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  Georges Kunstler committed Dec 17, 2013 101 links of traits with competitive effect and competitive response. This  Daniel Falster committed Dec 18, 2013 102 103 model is connected to several papers by Goldberg *et al.*, where competition is framed in term of effect and response and their links to  Daniel Falster committed Dec 17, 2013 104 105 traits [@goldberg_competitive_1996]. Two main approaches were proposed: a multiplicative and an additive  Daniel Falster committed Dec 18, 2013 106 107 108 model of competitive effect and response[^inter]. Below I consider the  Georges Kunstler committed Dec 19, 2013 109 additive effect-response model because it is simpler. However, I have not ruled out exploring  Daniel Falster committed Dec 17, 2013 110 the multiplicative effect-response model[^equmult].  Georges Kunstler committed Dec 17, 2013 111   Daniel Falster committed Dec 17, 2013 112 [^compreponse]: Through out the document I will use competitive response as the inverse of competition tolerance.  Georges Kunstler committed Dec 17, 2013 113   Georges Kunstler committed Dec 18, 2013 114 [^equmult]: The equations of the multiplicative models are given in the [Appendix 1](#multi).  Georges Kunstler committed Dec 17, 2013 115 116 117  [^inter]: There was also a detailed discussion of more complex model that would include both effect and response and interactions among both.  Daniel Falster committed Dec 17, 2013 118 ## Additive model of competitive effect and response  Georges Kunstler committed Dec 17, 2013 119 120  The general framework for this approach is to consider that  Daniel Falster committed Dec 18, 2013 121 122 $\lambda_{n,f} = r(t_f) +e(t_n)$ where $r$ and $e$ are functions for competitive response and effect respectively. A series of models with increasing complexity was identified[^linear-model].  Georges Kunstler committed Dec 17, 2013 123   Daniel Falster committed Dec 18, 2013 124 [^linear-model]: Here I present models which are linear functions of the trait but one can easily imagine more complex relations.  Georges Kunstler committed Dec 17, 2013 125   Daniel Falster committed Dec 18, 2013 126 1. $\lambda$ is influenced only by the trait of the neighborhood species (competitive effect model):  Georges Kunstler committed Dec 17, 2013 127 \begin{equation} \label{effect_trait}  Daniel Falster committed Dec 18, 2013 128 \lambda_{n,f} = a +b \times t_{n}.  Georges Kunstler committed Dec 17, 2013 129 130 \end{equation}  Daniel Falster committed Dec 18, 2013 131 2. $\lambda$ is influenced only by trait of the focal species (competitive response model):  Georges Kunstler committed Dec 17, 2013 132 \begin{equation} \label{response_trait}  Daniel Falster committed Dec 18, 2013 133 \lambda_{n,f} = a +b \times t_{f}.  Georges Kunstler committed Dec 17, 2013 134 135 \end{equation}  Daniel Falster committed Dec 18, 2013 136 137 3. $\lambda$ is influenced by traits of the neighborhood and focal species (effect-response model):  Georges Kunstler committed Dec 17, 2013 138 \begin{equation} \label{response_effect_trait}  Daniel Falster committed Dec 18, 2013 139 \lambda_{n,f} = a +b \times t_{f} +c \times t_{n}.  Georges Kunstler committed Dec 17, 2013 140 141 142 143 144 145 \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 $b=-c$.  Daniel Falster committed Dec 18, 2013 146 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  Georges Kunstler committed Dec 17, 2013 147 148 149 most complex model eq. \ref{response_effect_trait} this gives: \begin{equation}  Georges Kunstler committed Dec 17, 2013 150 \sum_{n=1}^{N_p} \lambda_{n,f} \times B_n = \sum_{n=1}^{N_p} (a +b \times t_{f}  Georges Kunstler committed Dec 18, 2013 151 +c \times t_{n}) \times B_n =B_\textrm{tot} \times (a +b \times t_{f} +c \times \overline{t_{n}})  Georges Kunstler committed Dec 17, 2013 152 153 \end{equation}  Daniel Falster committed Dec 18, 2013 154 155 where:  Georges Kunstler committed Dec 18, 2013 156 157 158 - $B_\textrm{tot}$ 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  Daniel Falster committed Dec 17, 2013 159 relative basal area abundance of species $n$, $B_n/B_\textrm{tot}$).  Georges Kunstler committed Dec 17, 2013 160   Daniel Falster committed Dec 18, 2013 161 162 163 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}.  Georges Kunstler committed Dec 17, 2013 164   Georges Kunstler committed Dec 18, 2013 165 [^ecotas]: Joint conference of the Ecological Society of Australia and the New Zealand Ecological Society, Nov 2013.  Georges Kunstler committed Dec 17, 2013 166   Daniel Falster committed Dec 18, 2013 167 # Data preparation and analysis  Georges Kunstler committed Dec 17, 2013 168   Daniel Falster committed Dec 18, 2013 169 170 The objective was collate data sets spanning most of the forest biomes of the world, to see if the links  Georges Kunstler committed Dec 17, 2013 171 between competition are consistent across these biomes (the  Georges Kunstler committed Dec 17, 2013 172 objective was not to have the largest number of data sets). We focused  Georges Kunstler committed Dec 17, 2013 173 on five key traits seed mass[^seed-mass], LMA, Leaf N per mass, wood density and maximum  Daniel Falster committed Dec 18, 2013 174 height. Key points for inclusion was good coverage for at least one of the traits of interest. Table  Georges Kunstler committed Dec 17, 2013 175 176 \ref{table-data} presents the different data set used.  Georges Kunstler committed Dec 17, 2013 177 [^seed-mass]: Seed mass was not used in subsequent analysis because its link with competitive effect and response is unclear.  Georges Kunstler committed Dec 17, 2013 178   Daniel Falster committed Dec 18, 2013 179 ## Dividing NFI data by ecoregion  Georges Kunstler committed Dec 17, 2013 180   Daniel Falster committed Dec 18, 2013 181 182 183 184 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  Georges Kunstler committed Dec 17, 2013 185 instance in the US there is a large variability between the north and  Daniel Falster committed Dec 18, 2013 186 187 the south). Division into ecoregions was also necessary for technical reasons (to speed up the estimation and solve some memory limit issues). Figure  Georges Kunstler committed Dec 17, 2013 188  \ref{biomes} presents the position of each ecoregion in  Daniel Falster committed Dec 18, 2013 189 190 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.  Georges Kunstler committed Dec 17, 2013 191   Daniel Falster committed Dec 18, 2013 192 ## Data formatting  Georges Kunstler committed Dec 17, 2013 193   Daniel Falster committed Dec 18, 2013 194 We (mainly Francis Hui, PhD student from UNSW) formatted all tree  Georges Kunstler committed Dec 18, 2013 195 196 data set to match common unit and names (see [Appendix 2, Variables description and units](#units)). We tested to check whether the range of  Daniel Falster committed Dec 18, 2013 197 variables values (mean and quantile) were within sensible limits and  Georges Kunstler committed Dec 17, 2013 198 visually inspected plots of $G$ per $D$ to check for errors.  Georges Kunstler committed Dec 17, 2013 199 200  We extracted SLA ($mm^2/mg$), Leaf N per mass ($mg/g$), wood density  Georges Kunstler committed Dec 17, 2013 201 202 ($mg/mm^3$), seed mass ($mg$) and maximum height ($m$), for each species from the TRY data base for NFI data and from a local  Daniel Falster committed Dec 18, 2013 203 traits database for the large tropical plots[^except]. For most NFI data maximum  Georges Kunstler committed Dec 17, 2013 204 height was extracted from the height measurement as the 99% quantile of  Daniel Falster committed Dec 18, 2013 205 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  Georges Kunstler committed Dec 17, 2013 206 207 208 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.  Georges Kunstler committed Dec 17, 2013 209   Daniel Falster committed Dec 18, 2013 210 ## Data processing  Georges Kunstler committed Dec 17, 2013 211   Daniel Falster committed Dec 18, 2013 212 Next we split each dataset by ecoregion, keeping only  Georges Kunstler committed Dec 19, 2013 213 214 215 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.  Georges Kunstler committed Dec 17, 2013 216   Georges Kunstler committed Dec 17, 2013 217 First we computed the local basal area ($cm^2/m^2$) of neighborhood  Georges Kunstler committed Dec 17, 2013 218 competitor per species for each individual tree. For NFI data the  Georges Kunstler committed Dec 17, 2013 219 220 221 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  Daniel Falster committed Dec 18, 2013 222 20x20 m). For the large tropical plots, the neighborhood was defined as  Georges Kunstler committed Dec 17, 2013 223 224 a 15 m radius around the tree.  Georges Kunstler committed Dec 18, 2013 225 The community weight mean of the neighboring trees and of the absolute  Daniel Falster committed Dec 18, 2013 226 trait distance between the focal tree and neighborhood trees was calculated using the species level traits data, or filling  Georges Kunstler committed Dec 17, 2013 227 228 229 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  Daniel Falster committed Dec 17, 2013 230 doesn't seems to provides strikingly different values). We run independent  Georges Kunstler committed Dec 19, 2013 231 232 computation of the community weight means to validate the processing of the data and inspected histograms of $\overline{t_n}$ to identify errors.  Georges Kunstler committed Dec 17, 2013 233 234 235 236 237 238 239  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  Daniel Falster committed Dec 17, 2013 240 taxonomic identification).  Georges Kunstler committed Dec 17, 2013 241   Daniel Falster committed Dec 18, 2013 242 ## Fitting of a mixed linear model  Georges Kunstler committed Dec 17, 2013 243   Georges Kunstler committed Dec 17, 2013 244 During the workshop we ran estimation using a hierarchical Bayesian model  Georges Kunstler committed Dec 19, 2013 245 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  Georges Kunstler committed Dec 17, 2013 246 package [lme4](http://cran.r-project.org/web/packages/lme4/index.html)  Daniel Falster committed Dec 18, 2013 247 248 249 250 in R cran). The reasons for the change are 1. a log-linear function provides a good first approximation to the shape of  Georges Kunstler committed Dec 19, 2013 251 more complex non-linear functions for the size and competition effect (mainly followingsuch as the one used in the work of C. Canham see  Daniel Falster committed Dec 18, 2013 252 253 254 @uriarte_trait_2010), and 2. using lmer was much faster than an estimation with JAGS or [Stan](http://mc-stan.org/).  Georges Kunstler committed Dec 18, 2013 255 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.  Daniel Falster committed Dec 18, 2013 256 257 258 259 260  ## Fitted models The models fitted were based on Equation \label{G1}:  Georges Kunstler committed Dec 17, 2013 261 \begin{equation} \label{logG}  Daniel Falster committed Dec 18, 2013 262 263 \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})  Georges Kunstler committed Dec 17, 2013 264 265 \end{equation}  Daniel Falster committed Dec 18, 2013 266 267 where:  Georges Kunstler committed Dec 18, 2013 268 - $\log {G}$ is the log basal area growth,  Daniel Falster committed Dec 18, 2013 269 - $\log{G\textrm{max}}$ 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  Georges Kunstler committed Dec 17, 2013 270 variance for all species), a random focal species $f$ effect and a random individual $i$ effect when multiple  Daniel Falster committed Dec 18, 2013 271 272 273 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.  Georges Kunstler committed Dec 17, 2013 274 275  [^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.  Georges Kunstler committed Dec 17, 2013 276   Daniel Falster committed Dec 18, 2013 277 We compared two alternative models for $\lambda_{n,f}$:  Georges Kunstler committed Dec 17, 2013 278   Georges Kunstler committed Dec 18, 2013 279 280 281 (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  Georges Kunstler committed Dec 17, 2013 282 function the absolute trait distance ($\lambda_{n,f} = a + b \times  Georges Kunstler committed Dec 19, 2013 283 |t_{n} - t_{f}|$).  Georges Kunstler committed Dec 17, 2013 284   Georges Kunstler committed Dec 18, 2013 285 286 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}  Daniel Falster committed Dec 18, 2013 287 \log{G}_{f,p,i,t} = \log{G\textrm{max}}_{f,p,i} + \alpha_f \times D_{i,t} + a \times  Georges Kunstler committed Dec 18, 2013 288 \log{B}_\textrm{tot} + b \times \log{B}_\textrm{tot} \times t_f + c \times \log{B}_\textrm{tot} \times \overline{t_{n}}.  Georges Kunstler committed Dec 17, 2013 289 290 \end{equation}  Georges Kunstler committed Dec 18, 2013 291 292 293 294 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}  Daniel Falster committed Dec 18, 2013 295 \log{G}_{f,p,i,t} = \log{G\textrm{max}}_{f,p,i} + \alpha_f \times D_{i,t} + a \times  Georges Kunstler committed Dec 18, 2013 296 \log{B}_\textrm{tot} + b \times \log{B}_\textrm{tot} \times \overline{|t_{n} - t_{f}|}.  Georges Kunstler committed Dec 17, 2013 297 298 \end{equation}  Georges Kunstler committed Dec 18, 2013 299 300 where $\overline{|t_{n} - t_{f}|} = \sum_{n=1}^{N_p} P_n \times |t_n -t_f|$.  Georges Kunstler committed Dec 17, 2013 301   Daniel Falster committed Dec 18, 2013 302 We then compared these two models to a null model where competition is  Georges Kunstler committed Dec 17, 2013 303 constant and independent of focal and neighborhood species trait.  Georges Kunstler committed Dec 17, 2013 304 \begin{equation} \label{logG-null}  Daniel Falster committed Dec 18, 2013 305 306 \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}.  Georges Kunstler committed Dec 17, 2013 307 308 \end{equation}  Daniel Falster committed Dec 18, 2013 309 310 311 312 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  Georges Kunstler committed Dec 17, 2013 313 314 @nakagawa_general_2013 (using conditional $R^2$).  Georges Kunstler committed Dec 17, 2013 315 # Preliminary results  Georges Kunstler committed Dec 17, 2013 316   Daniel Falster committed Dec 18, 2013 317 318 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).  Georges Kunstler committed Dec 18, 2013 319 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.  Georges Kunstler committed Dec 17, 2013 320   Daniel Falster committed Dec 18, 2013 321 [^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 $\lambda$?).  Georges Kunstler committed Dec 17, 2013 322   Georges Kunstler committed Dec 18, 2013 323 Overall the effect-response models is strongest at low mean annual precipitation (  Daniel Falster committed Dec 18, 2013 324 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.  Georges Kunstler committed Dec 17, 2013 325   Daniel Falster committed Dec 18, 2013 326 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.  Georges Kunstler committed Dec 17, 2013 327   Georges Kunstler committed Dec 17, 2013 328 # Future work to do  Georges Kunstler committed Dec 17, 2013 329 330   Daniel Falster committed Dec 18, 2013 331 - Improve the method to estimate the effect-size of the alternative models (ideally what percentage of competition variation is explained by traits).  Georges Kunstler committed Dec 17, 2013 332 333 - 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).  Daniel Falster committed Dec 18, 2013 334 - 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.  Georges Kunstler committed Dec 17, 2013 335 - 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.  Georges Kunstler committed Dec 19, 2013 336 - 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.  Daniel Falster committed Dec 17, 2013 337 - 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.  Georges Kunstler committed Dec 17, 2013 338 339 340 341 - 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).  Georges Kunstler committed Dec 17, 2013 342 343 344 345 346  \newpage  Georges Kunstler committed Dec 19, 2013 347 # FIGURES & TABLES  Georges Kunstler committed Dec 17, 2013 348   Daniel Falster committed Dec 17, 2013 349 ![**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)  Georges Kunstler committed Dec 17, 2013 350   Georges Kunstler committed Dec 17, 2013 351 \pagebreak  Georges Kunstler committed Dec 17, 2013 352   Georges Kunstler committed Dec 17, 2013 353 354 355 356 357  ![**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  Georges Kunstler committed Dec 18, 2013 358 359 \newpage  Georges Kunstler committed Dec 19, 2013 360 ![**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  Georges Kunstler committed Dec 17, 2013 361 362 363 ($mg/mm^3$), maximum height ($m$). \label{trait}](traits-XY.pdf) \pagebreak  Georges Kunstler committed Dec 17, 2013 364   Georges Kunstler committed Dec 19, 2013 365 366 \pagebreak  Georges Kunstler committed Dec 17, 2013 367 368 \newpage  Daniel Falster committed Dec 17, 2013 369 ![**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)  Georges Kunstler committed Dec 17, 2013 370 371 372  \pagebreak  Daniel Falster committed Dec 17, 2013 373 ![**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)  Georges Kunstler committed Dec 17, 2013 374 375 376  \pagebreak  Daniel Falster committed Dec 17, 2013 377 ![**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)  Georges Kunstler committed Dec 17, 2013 378 379 380  \pagebreak  Daniel Falster committed Dec 17, 2013 381 ![**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)  Georges Kunstler committed Dec 17, 2013 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430  \pagebreak ------------------------------------------------------------------------------------------------------------------------------- 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 -----------------------------------------------------------------------------------------------------  Daniel Falster committed Dec 17, 2013 431  set ecoregion N obs total % Leaf N % Seed mass % SLA % Wood density % Max height  Georges Kunstler committed Dec 17, 2013 432 -------- ----------- ------------- ---------- ------------- --------- ---------------- --------------  Daniel Falster committed Dec 17, 2013 433  Sweden PA0405 27124 0.965 0.965 0.963 0.9628 0.8972  Georges Kunstler committed Dec 17, 2013 434   Daniel Falster committed Dec 17, 2013 435  Sweden PA0436 293342 0.9587 0.9587 0.9586 0.9536 0.9019  Georges Kunstler committed Dec 17, 2013 436   Daniel Falster committed Dec 17, 2013 437  Sweden PA0608 187721 0.9501 0.9501 0.9496 0.9445 0.9086  Georges Kunstler committed Dec 17, 2013 438   Daniel Falster committed Dec 17, 2013 439  Sweden PA1110 27376 0.9536 0.9536 0.9528 0.9454 0.9084  Georges Kunstler committed Dec 17, 2013 440   Daniel Falster committed Dec 17, 2013 441  NVS BeechHumid 19790 0.899 0.9704 0.967 0.9082 0.998  Georges Kunstler committed Dec 17, 2013 442   Daniel Falster committed Dec 17, 2013 443  NVS MixedCool 8738 0.8977 0.9698 0.9698 0.9353 1  Georges Kunstler committed Dec 17, 2013 444   Daniel Falster committed Dec 17, 2013 445  NVS MixedWarm 4546 0.93 0.9879 0.9573 0.936 0.9879  Georges Kunstler committed Dec 17, 2013 446   Daniel Falster committed Dec 17, 2013 447  US DR.DO 25499 0.8165 0.9438 0.8129 0.5829 0.9755  Georges Kunstler committed Dec 17, 2013 448   Daniel Falster committed Dec 17, 2013 449  US Ho.Co.Mo 149390 0.7575 0.932 0.7575 0.8457 0.9956  Georges Kunstler committed Dec 17, 2013 450   Daniel Falster committed Dec 17, 2013 451  US HU.TE.DO 15146 0.5203 0.991 0.332 0.7428 0.991  Georges Kunstler committed Dec 17, 2013 452   Daniel Falster committed Dec 17, 2013 453  US HU.TR.DO 4532 0.2732 0.5812 0.2502 0.2928 0.9762  Georges Kunstler committed Dec 17, 2013 454   Daniel Falster committed Dec 17, 2013 455  US Pr.Di 50719 0.5387 0.9428 0.6099 0.7184 0.9861  Georges Kunstler committed Dec 17, 2013 456   Daniel Falster committed Dec 17, 2013 457  US Su.Di 517649 0.6648 0.9675 0.6199 0.7621 0.9874  Georges Kunstler committed Dec 17, 2013 458   Daniel Falster committed Dec 17, 2013 459  US Su.Mo 11661 0.5784 0.8245 0.2664 0.6485 0.993  Georges Kunstler committed Dec 17, 2013 460   Daniel Falster committed Dec 17, 2013 461  US Wa.Co.Di 382200 0.9734 0.9802 0.9763 0.9667 0.9905  Georges Kunstler committed Dec 17, 2013 462   Daniel Falster committed Dec 17, 2013 463  US Wa.Co.Mo 106502 0.9834 0.9842 0.9719 0.9713 0.9937  Georges Kunstler committed Dec 17, 2013 464   Daniel Falster committed Dec 17, 2013 465  Canada -132 274375 0.9744 0.9747 0.9485 0.946 0.9932  Georges Kunstler committed Dec 17, 2013 466   Daniel Falster committed Dec 17, 2013 467  Canada -211 190700 0.9423 0.9492 0.9404 0.9388 0.9921  Georges Kunstler committed Dec 17, 2013 468   Daniel Falster committed Dec 17, 2013 469  Canada M211b 7722 0.9128 0.9128 0.8793 0.8723 0.9372  Georges Kunstler committed Dec 17, 2013 470   Daniel Falster committed Dec 17, 2013 471  NSW AA 805 0 1 0 0.9553 1  Georges Kunstler committed Dec 17, 2013 472   Daniel Falster committed Dec 17, 2013 473  France AB 41900 0.9475 0.9727 0.9497 0.9893 0.997  Georges Kunstler committed Dec 17, 2013 474   Daniel Falster committed Dec 17, 2013 475  France C 27261 0.9256 0.9872 0.925 0.9767 0.997  Georges Kunstler committed Dec 17, 2013 476   Daniel Falster committed Dec 17, 2013 477  France F 18704 0.9069 0.984 0.9069 0.9841 0.9942  Georges Kunstler committed Dec 17, 2013 478   Daniel Falster committed Dec 17, 2013 479  France GDE 47377 0.9808 0.9936 0.9827 0.991 0.9962  Georges Kunstler committed Dec 17, 2013 480   Daniel Falster committed Dec 17, 2013 481  France HI 34823 0.9824 0.9938 0.9752 0.9692 0.9961  Georges Kunstler committed Dec 17, 2013 482   Daniel Falster committed Dec 17, 2013 483  France JK 14251 0.96 0.9846 0.9537 0.934 0.9872  Georges Kunstler committed Dec 17, 2013 484 485 486 487 ----------------------------------------------------------------------------------------------------- : **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} -----------------------------------------------------------------------------------------------------  Daniel Falster committed Dec 17, 2013 488  set ecoregion N obs total % Leaf N % Seed mass % SLA % Wood density % Max height  Georges Kunstler committed Dec 17, 2013 489 -------- ----------- ------------- ---------- ------------- --------- ---------------- --------------  Daniel Falster committed Dec 17, 2013 490  Swiss eco.1 25797 0.956 0.9591 0.956 0.9591 0.9734  Georges Kunstler committed Dec 17, 2013 491   Daniel Falster committed Dec 17, 2013 492  Swiss eco.2 34404 0.9651 0.9668 0.9634 0.966 0.9847  Georges Kunstler committed Dec 17, 2013 493   Daniel Falster committed Dec 17, 2013 494  Swiss eco.3 46682 0.953 0.953 0.9402 0.9371 0.9739  Georges Kunstler committed Dec 17, 2013 495   Daniel Falster committed Dec 17, 2013 496  Swiss eco.4 11714 0.9431 0.9431 0.9125 0.8043 0.9727  Georges Kunstler committed Dec 17, 2013 497   Daniel Falster committed Dec 17, 2013 498  Swiss eco.5 13674 0.9459 0.9459 0.852 0.7475 0.9556  Georges Kunstler committed Dec 17, 2013 499   Daniel Falster committed Dec 17, 2013 500  Swiss eco.6 16230 0.9333 0.9341 0.8834 0.8819 0.9584  Georges Kunstler committed Dec 17, 2013 501   Daniel Falster committed Dec 17, 2013 502  Spain PA0406 62985 0.906 0.9066 0.8562 0.8711 0.9972  Georges Kunstler committed Dec 17, 2013 503   Daniel Falster committed Dec 17, 2013 504  Spain PA0433 42475 0.961 0.9646 0.9537 0.9584 0.998  Georges Kunstler committed Dec 17, 2013 505   Daniel Falster committed Dec 17, 2013 506  Spain PA1208 40609 0.9827 0.9834 0.9285 0.9849 0.9907  Georges Kunstler committed Dec 17, 2013 507   Daniel Falster committed Dec 17, 2013 508  Spain PA1209 123531 0.9808 0.9813 0.9225 0.981 0.986  Georges Kunstler committed Dec 17, 2013 509   Daniel Falster committed Dec 17, 2013 510  Spain PA1215 48088 0.9754 0.9765 0.9726 0.9711 0.9955  Georges Kunstler committed Dec 17, 2013 511   Daniel Falster committed Dec 17, 2013 512  Spain PA1216 58366 0.9764 0.9765 0.9704 0.9772 0.9949  Georges Kunstler committed Dec 17, 2013 513   Daniel Falster committed Dec 17, 2013 514  Spain PA1221 13239 0.9579 0.986 0.9447 0.9776 0.9893  Georges Kunstler committed Dec 17, 2013 515   Daniel Falster committed Dec 17, 2013 516  Japan ct 5136 0 0.9077 1 1 1  Georges Kunstler committed Dec 17, 2013 517   Daniel Falster committed Dec 17, 2013 518  Japan st 1816 0 0.103 0.7605 0.7605 0.8673  Georges Kunstler committed Dec 17, 2013 519   Daniel Falster committed Dec 17, 2013 520  Japan wt 10749 0 0.7016 0.9973 0.9973 1  Georges Kunstler committed Dec 17, 2013 521   Daniel Falster committed Dec 17, 2013 522  BCI tropical 93838 0.8508 0.8174 0.8725 0.8393 0.9519  Georges Kunstler committed Dec 17, 2013 523   Daniel Falster committed Dec 17, 2013 524  Fushan tropical 14701 0.1901 0.0008163 0.9997 0.9465 0.8911  Georges Kunstler committed Dec 17, 2013 525   Daniel Falster committed Dec 17, 2013 526 Paracou tropical 92199 0.03612 3.254e-05 0.05546 0.05191 0  Georges Kunstler committed Dec 17, 2013 527   Daniel Falster committed Dec 17, 2013 528 Luquillo tropical 14011 0.9374 0.9374 0.9374 0.9374 0.9374  Georges Kunstler committed Dec 17, 2013 529   Daniel Falster committed Dec 17, 2013 530  Mbaiki tropical 6377 0.0009409 0 0.0009409 0.007057 0  Georges Kunstler committed Dec 17, 2013 531 532 533 534 535 ----------------------------------------------------------------------------------------------------- \pagebreak ------------------------------------------------  Daniel Falster committed Dec 17, 2013 536  set nocomp simplecomp AD R E ER  Georges Kunstler committed Dec 17, 2013 537 -------- -------- ------------ ---- --- --- ----  Daniel Falster committed Dec 17, 2013 538  BCI 0 1 0 0 0 0  Georges Kunstler committed Dec 17, 2013 539   Daniel Falster committed Dec 17, 2013 540  Canada 0 1 1 0 1 0  Georges Kunstler committed Dec 17, 2013 541   Daniel Falster committed Dec 17, 2013 542  France 0 1 4 0 1 0  Georges Kunstler committed Dec 17, 2013 543   Daniel Falster committed Dec 17, 2013 544  Fushan 0 0 0 0 1 0  Georges Kunstler committed Dec 17, 2013 545   Daniel Falster committed Dec 17, 2013 546  Japan 2 0 0 0 1 0  Georges Kunstler committed Dec 17, 2013 547   Daniel Falster committed Dec 17, 2013 548 Luquillo 0 0 0 0 1 0  Georges Kunstler committed Dec 17, 2013 549   Daniel Falster committed Dec 17, 2013 550  Mbaiki 0 0 0 1 0 0  Georges Kunstler committed Dec 17, 2013 551   Daniel Falster committed Dec 17, 2013 552  NVS 1 1 0 0 1 0  Georges Kunstler committed Dec 17, 2013 553   Daniel Falster committed Dec 17, 2013 554 Paracou 0 1 0 0 0 0  Georges Kunstler committed Dec 17, 2013 555   Daniel Falster committed Dec 17, 2013 556  Spain 0 0 6 0 0 1  Georges Kunstler committed Dec 17, 2013 557   Daniel Falster committed Dec 17, 2013 558  Sweden 0 0 0 0 2 2  Georges Kunstler committed Dec 17, 2013 559   Daniel Falster committed Dec 17, 2013 560  Swiss 0 0 4 0 2 0  Georges Kunstler committed Dec 17, 2013 561   Daniel Falster committed Dec 17, 2013 562  US 1 0 4 1 2 1  Georges Kunstler committed Dec 17, 2013 563 564 565 566 567 568 ------------------------------------------------ : **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 ------------------------------------------------  Daniel Falster committed Dec 17, 2013 569  set nocomp simplecomp AD R E ER  Georges Kunstler committed Dec 17, 2013 570 -------- -------- ------------ ---- --- --- ----  Daniel Falster committed Dec 17, 2013 571  BCI 0 1 0 0 0 0  Georges Kunstler committed Dec 17, 2013 572   Daniel Falster committed Dec 17, 2013 573  Canada 0 1 0 0 1 1  Georges Kunstler committed Dec 17, 2013 574   Daniel Falster committed Dec 17, 2013 575  France 0 0 3 0 3 0  Georges Kunstler committed Dec 17, 2013 576   Daniel Falster committed Dec 17, 2013 577  Fushan 0 1 0 0 0 0  Georges Kunstler committed Dec 17, 2013 578   Daniel Falster committed Dec 17, 2013 579 Luquillo 0 0 0 0 1 0  Georges Kunstler committed Dec 17, 2013 580   Daniel Falster committed Dec 17, 2013 581  Mbaiki 1 0 0 0 0 0  Georges Kunstler committed Dec 17, 2013 582   Daniel Falster committed Dec 17, 2013 583  NVS 2 1 0 0 0 0  Georges Kunstler committed Dec 17, 2013 584   Daniel Falster committed Dec 17, 2013 585 Paracou 0 0 0 0 1 0  Georges Kunstler committed Dec 17, 2013 586   Daniel Falster committed Dec 17, 2013 587  Spain 0 0 6 0 0 1  Georges Kunstler committed Dec 17, 2013 588   Daniel Falster committed Dec 17, 2013 589  Sweden 0 0 0 0 2 2  Georges Kunstler committed Dec 17, 2013 590   Daniel Falster committed Dec 17, 2013 591  Swiss 0 1 1 0 4 0  Georges Kunstler committed Dec 17, 2013 592   Daniel Falster committed Dec 17, 2013 593  US 1 0 4 1 2 1  Georges Kunstler committed Dec 17, 2013 594 595 596 597 598 599 600 ------------------------------------------------ : **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 ------------------------------------------------  Daniel Falster committed Dec 17, 2013 601  set nocomp simplecomp AD R E ER  Georges Kunstler committed Dec 17, 2013 602 -------- -------- ------------ ---- --- --- ----  Daniel Falster committed Dec 17, 2013 603  BCI 0 1 0 0 0 0  Georges Kunstler committed Dec 17, 2013 604   Daniel Falster committed Dec 17, 2013 605  Canada 0 0 3 0 0 0  Georges Kunstler committed Dec 17, 2013 606   Daniel Falster committed Dec 17, 2013 607  France 0 0 3 0 0 3  Georges Kunstler committed Dec 17, 2013 608   Daniel Falster committed Dec 17, 2013 609  Fushan 0 0 0 0 1 0  Georges Kunstler committed Dec 17, 2013 610   Daniel Falster committed Dec 17, 2013 611  Japan 2 0 0 1 0 0  Georges Kunstler committed Dec 17, 2013 612   Daniel Falster committed Dec 17, 2013 613 Luquillo 0 0 1 0 0 0  Georges Kunstler committed Dec 17, 2013 614   Daniel Falster committed Dec 17, 2013 615  Mbaiki 1 0 0 0 0 0  Georges Kunstler committed Dec 17, 2013 616   Daniel Falster committed Dec 17, 2013 617  NSW 1 0 0 0 0 0  Georges Kunstler committed Dec 17, 2013 618   Daniel Falster committed Dec 17, 2013 619  NVS 2 1 0 0 0 0  Georges Kunstler committed Dec 17, 2013 620   Daniel Falster committed Dec 17, 2013 621 Paracou 0 0 0 0 1 0  Georges Kunstler committed Dec 17, 2013 622   Daniel Falster committed Dec 17, 2013 623  Spain 0 0 7 0 0 0  Georges Kunstler committed Dec 17, 2013 624   Daniel Falster committed Dec 17, 2013 625  Sweden 0 0 0 0 2 2  Georges Kunstler committed Dec 17, 2013 626   Daniel Falster committed Dec 17, 2013 627  Swiss 0 0 4 0 2 0  Georges Kunstler committed Dec 17, 2013 628   Daniel Falster committed Dec 17, 2013 629  US 1 0 4 1 0 3  Georges Kunstler committed Dec 17, 2013 630 631 632 633 634 635 ------------------------------------------------ : **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 ------------------------------------------------  Daniel Falster committed Dec 17, 2013 636  set nocomp simplecomp AD R E ER  Georges Kunstler committed Dec 17, 2013 637 -------- -------- ------------ ---- --- --- ----  Daniel Falster committed Dec 17, 2013 638  BCI 0 1 0 0 0 0  Georges Kunstler committed Dec 17, 2013 639   Daniel Falster committed Dec 17, 2013 640  Canada 0 0 3 0 0 0  Georges Kunstler committed Dec 17, 2013 641   Daniel Falster committed Dec 17, 2013 642  France 0 0 3 0 3 0  Georges Kunstler committed Dec 17, 2013 643   Daniel Falster committed Dec 17, 2013 644  Fushan 0 1 0 0 0 0  Georges Kunstler committed Dec 17, 2013 645   Daniel Falster committed Dec 17, 2013 646  Japan 2 0 0 0 1 0  Georges Kunstler committed Dec 17, 2013 647   Daniel Falster committed Dec 17, 2013 648 Luquillo 0 0 0 0 1 0  Georges Kunstler committed Dec 17, 2013 649   Daniel Falster committed Dec 17, 2013 650  NSW 1 0 0 0 0 0  Georges Kunstler committed Dec 17, 2013 651   Daniel Falster committed Dec 17, 2013 652  NVS 2 0 0 0 0 1  Georges Kunstler committed Dec 17, 2013 653   Daniel Falster committed Dec 17, 2013 654  Spain 0 0 6 0 0 1  Georges Kunstler committed Dec 17, 2013 655   Daniel Falster committed Dec 17, 2013 656  Sweden 0 0 0 0 1 3  Georges Kunstler committed Dec 17, 2013 657   Daniel Falster committed Dec 17, 2013 658  Swiss 0 1 1 0 3 1  Georges Kunstler committed Dec 17, 2013 659   Daniel Falster committed Dec 17, 2013 660  US 0 1 7 0 1 0  Georges Kunstler committed Dec 17, 2013 661 662 663 664 665 666 ------------------------------------------------ : **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  Georges Kunstler committed Dec 19, 2013 667 # Appendix 1. Multiplicative model of competitive effect and response {#multi}  Georges Kunstler committed Dec 17, 2013 668   Georges Kunstler committed Dec 17, 2013 669 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.  Georges Kunstler committed Dec 17, 2013 670   Georges Kunstler committed Dec 17, 2013 671 1. $\lambda$ can be influence only by the variation in competitive effect through the trait of the neighborhood species:  Georges Kunstler committed Dec 17, 2013 672 673 674 675 676 677 678 679 680 681 \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}  Georges Kunstler committed Dec 17, 2013 682 3. $\lambda$ can be influence by the variation of both competitive effect and response through the trait of the neighborhood and focal species:  Georges Kunstler committed Dec 17, 2013 683 684 685 686 687 688  \begin{equation} \lambda_{n,f} = (a +b \times t_{f}) \times (c +d \times t_{n}) \end{equation}  Georges Kunstler committed Dec 19, 2013 689 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 ($\overline{t_{n}}$).  Georges Kunstler committed Dec 17, 2013 690   Georges Kunstler committed Dec 19, 2013 691 692 \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}})  Georges Kunstler committed Dec 17, 2013 693 694 695 \end{equation}  Georges Kunstler committed Dec 19, 2013 696 ## Comparison of the multiplicative and additive effect and response model  Georges Kunstler committed Dec 17, 2013 697 698 699 700 701 702 703  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}  Georges Kunstler committed Dec 19, 2013 704 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:  Georges Kunstler committed Dec 17, 2013 705   Georges Kunstler committed Dec 19, 2013 706 707 \begin{equation} \label{add-inter} \lambda_{n,f} = a' +b' \times t_{f} +c' \times t_{n}+d' \times t_{n} \times t_{f}  Georges Kunstler committed Dec 17, 2013 708 709 \end{equation}  Georges Kunstler committed Dec 19, 2013 710 The two models are equal when:  Georges Kunstler committed Dec 17, 2013 711 712  \begin{equation}  Georges Kunstler committed Dec 19, 2013 713 a'=ac \mspace{3mu} ;\mspace{3mu} b'=bc\mspace{3mu} ;\mspace{3mu} c'=ad \mspace{5mu} and \mspace{5mu} d'=bd  Georges Kunstler committed Dec 17, 2013 714 715 \end{equation}  Georges Kunstler committed Dec 19, 2013 716 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'