Commit a45b528a authored by kunstler's avatar kunstler
Browse files

last minor edit to the text before reminder

parent 436ef78d
################# FUNCTION TO EXTRACT DECTED OUTLIER AND FORMAT TRY DATA
## Georges Kunstler
############################################ 14/06/2013
### just testing this out! ##############
### install all unstallled packages
......@@ -189,6 +188,7 @@ fun.turn.list.in.DF <- function(sp, res.list) {
fun.extract.format.sp.traits.TRY <- function(sp, sp.syno.table, data) {
### test data sp and sp.syno.table match
require(gdata)
browser()
sp.syno.table[["Latin_name_syn"]] <-
trim(as.character(sp.syno.table[["Latin_name_syn"]]) )
data[["Latin_name"]] <- trim(as.character(data[["Latin_name"]]))
......
......@@ -8,13 +8,13 @@ We developed the equation of $\alpha_{c,f} = \alpha_{0,f} - \alpha_t \, t_f + \a
Where:
$B_{i,tot} = \sum_{c=1}^{C_p} {B_{i,c,p,s}}$,
$B_{i,tot} = \sum_{c=1}^{N_p} {B_{i,c,p,s}}$,
$B_{i,t_c} = \sum_{c=1}^{C_p} {t_c \times B_{i,c,p,s}}$,
$B_{i,t_c} = \sum_{c=1}^{N_p} {t_c \times B_{i,c,p,s}}$,
$B_{i,\vert t_c - t_f \vert} = \sum_{c=1}^{C_p} {\vert t_c - t_f \vert \times B_{i,c,p,s}}$,
$B_{i,\vert t_c - t_f \vert} = \sum_{c=1}^{N_p} {\vert t_c - t_f \vert \times B_{i,c,p,s}}$,
and $C_p$ is the number of species on the plot $p$.
and $N_p$ is the number of species on the plot $p$.
## Details on sites
......@@ -241,7 +241,7 @@ and $C_p$ is the number of species on the plot $p$.
## Trait effects and potential mechanisms
The most important driver of individual growth was individual tree size with a positive effect on basal area growth (see Extended data Table 3). This is unsurprising as tree size is known to be a key drivers of tree growth[@Stephenson-2014; @Enquist-1999]. Then there was a consistent negative effect of the total basal area of neighbouring competitors across all biomes. The dominance of a competitive effect for the growth of adult trees (diameter at breast height > 10cm), agree well with the idea that facilitation processes are generally limited to the regeneration phase rather than at the adult stage [@Callaway-1997].
The most important driver of individual growth was individual tree size with a positive effect on basal area growth (see Extended data Table 3). This is unsurprising as tree size is known to be a key driver of tree growth[@Stephenson-2014; @Enquist-1999]. Then there was a consistent negative effect of the total basal area of neighbouring competitors across all biomes. The dominance of a competitive effect for the growth of adult trees (diameter at breast height >= 10cm dbh), agrees well with the idea that facilitation processes are generally limited to the regeneration phase rather than at the adult stage [@Callaway-1997]. The variation of $\alpha_0$ between biomes is limited with large overlap of their confidences intervals.
In term of traits effects, Wood density (WD) was strongly negatively associated with maximum growth, in agreement with the idea that shade-intolerant species with low wood density have faster growth in absence of competition (in full light conditions) than shade tolerant species[@Nock-2009; @Wright-2010]. One advantage of low wood density is clearly that it is cheaper to build light than dense wood, thus for the same biomass growth a low wood density species will have a higher basal area increment than a high wood density species[@Enquist-1999]. Other advantages of light wood may include higher xylem conductivity[@Chave-2009], though for angiosperms this is a correlated trait rather than an automatic consequence. A countervailing advantage for high wood density species was their better tolerance to competition (less growth reduction per unit of basal area of competitors), which is in line with the idea that these species are more shade tolerant[@Chave-2009; @Nock-2009; @Wright-2010]. This has generally been related to the higher survival associated with high wood density[@Kraft-2010], via resistance to mechanical damage, herbivores and pathogens[@Chave-2009; @Kraft-2010], but may also be connected to a lower maintenance respiration[@Larjavaara-2010]. For growth, the lower respiration may lead to a direct advantage in deep shade, but the correlation might also arise through correlated selection for high survival rate and for high growth in shade. Finally, high wood density was also weakly correlated with stronger competitive effect, especially in tropical forest where the confidence interval did not span zero. This might possibly have been mediated by larger crowns (both in depth and radius)[@Poorter-2006a; @Aiba-2009], casting a deeper shade.
......@@ -249,7 +249,7 @@ SLA was positively correlated with maximum basal area growth (growth without com
Maximum height was weakly positively correlated with maximum growth rate (confidence intervals spanned zero except for temperate rain forest). Previous studies[@Poorter-2006a; @Poorter-2008; @Wright-2010] found mixed support for this relationship. Possible mechanisms are contradictory: maximum height may be associated with greater access to light and thus faster growth, but at the same time life history strategies might be expected to select for slower growth in long-lived plants[@Poorter-2008]. Maximum height was negatively correlated with tolerance to competition (confidence intervals spanned zero except for temperate rain forest and taiga), in line with the idea that sub-canopy trees are more shade-tolerant[@Poorter-2006a]. There was however a tendency for species with tall maximum height to have stronger competitive effect (though with wider confidence intervals intercepting zero), that might be explained by greater light interception from taller trees.
Our results raised the question whether there is a coordination between trait values conferring strong competitive effect and trait values conferring high competitive tolerance. Competitive effect and tolerance are the two central elements of the species competitive ability[@Goldberg-1991]. One may expect that because of intra-specific competition, species with strong competitive effect should have evolved a high tolerance to competition. However, in agreement with previous studies[@Goldberg-1990; @Goldberg-1991; @Wang-2010], we found little evidence for such coordination. It was present only for wood density, where high density conferred better competitive tolerance and also stronger competitive effect (but with wide confidence intervals). For SLA there was no clear coordinations. For maximum height as explained above there was a tendency for short maximum height to lead to high tolerance to competition but to low competitive effect. This interesting because a trade-off between competitive tolerance and maximum height has been proposed as fundamental mechanisms of coexistence of species in size-structured population in the stratification theory of species coexistence[@Kohyama-2009]. Finally the lack of support for coordination between tolerance and effect is important because it means that competitive interaction is not well described as a trait hierarchy relating a focal species to its competitors (measured as $t_c -t_f$ and thus assuming $\alpha_e = \alpha_t$ as in @Kunstler-2012; @Kraft-2014; @Lasky-2014). Traits of competitors alone or of focal plants alone may convey more information. If traits are strongly linked to either competitive effect or competitive tolerance, this still means that some trait values will have an advantage in competitive interactions.
Our results raised the question whether there is a coordination between trait values conferring strong competitive effect and trait values conferring high competitive tolerance. Competitive effect and tolerance are the two central elements of the species competitive ability[@Goldberg-1991]. One may expect that because of intra-specific competition, species with strong competitive effect should have evolved a high tolerance to competition. However, in agreement with previous studies[@Goldberg-1990; @Goldberg-1991; @Wang-2010], we found little evidence for such coordination. There was only such a tendency for wood density and SLA. High wood density conferred better competitive tolerance and also stronger competitive effect, but with wide confidence intervals intercepting zero for the later. High SLA conferred stronger competitive effect and higher tolerance of competition, but with wide confidence intervals intercepting zero for the later. For maximum height as explained above there was a tendency for short maximum height to lead to high tolerance to competition but to low competitive effect. This is interesting because a trade-off between competitive tolerance and maximum height has been proposed as a fundamental mechanisms of coexistence of species in size-structured population in the stratification theory of species coexistence[@Kohyama-2009]. Finally the lack of support for coordination between tolerance and effect is important because it means that competitive interaction is not well described as a trait hierarchy relating a focal species to its competitors (measured as $t_c -t_f$ and thus assuming $\alpha_e = \alpha_t$ as in @Kunstler-2012; @Kraft-2014; @Lasky-2014). Traits of competitors alone or of focal plants alone may convey more information than the trait hierarchy. These processes of traits linked to either competitive effect or competitive tolerance, nevertheless, still leads to some trait values having an advantage in competitive interactions.
Given that the effect sizes we report for effects of traits on competitive interaction are modest, the question arises whether the three traits available to us (wood density, SLA, and maximum height) were the best candidates. It is possible that traits more directly related to mechanisms of competition -- for instance for competition for light, the leaf area index of the competitors or the compensation point at leaf or whole-plant level -- may be more powerful. It is also possible that if traits measured at the individual level were available, rather than species averages, this might strengthen predictive power[@Kraft-2014].
......
......@@ -8,13 +8,13 @@ We developed the equation of $\alpha_{c,f} = \alpha_{0,f} - \alpha_t \, t_f + \a
Where:
$B_{i,tot} = \sum_{c=1}^{C_p} {B_{i,c,p,s}}$,
$B_{i,tot} = \sum_{c=1}^{N_p} {B_{i,c,p,s}}$,
$B_{i,t_c} = \sum_{c=1}^{C_p} {t_c \times B_{i,c,p,s}}$,
$B_{i,t_c} = \sum_{c=1}^{N_p} {t_c \times B_{i,c,p,s}}$,
$B_{i,\vert t_c - t_f \vert} = \sum_{c=1}^{C_p} {\vert t_c - t_f \vert \times B_{i,c,p,s}}$,
$B_{i,\vert t_c - t_f \vert} = \sum_{c=1}^{N_p} {\vert t_c - t_f \vert \times B_{i,c,p,s}}$,
and $C_p$ is the number of species on the plot $p$.
and $N_p$ is the number of species on the plot $p$.
## Details on sites
......@@ -54,7 +54,7 @@ writeLines(unlist(list.t[dat[["Country"]]]))
## Trait effects and potential mechanisms
The most important driver of individual growth was individual tree size with a positive effect on basal area growth (see Extended data Table 3). This is unsurprising as tree size is known to be a key drivers of tree growth[@Stephenson-2014; @Enquist-1999]. Then there was a consistent negative effect of the total basal area of neighbouring competitors across all biomes. The dominance of a competitive effect for the growth of adult trees (diameter at breast height > 10cm), agree well with the idea that facilitation processes are generally limited to the regeneration phase rather than at the adult stage [@Callaway-1997].
The most important driver of individual growth was individual tree size with a positive effect on basal area growth (see Extended data Table 3). This is unsurprising as tree size is known to be a key driver of tree growth[@Stephenson-2014; @Enquist-1999]. Then there was a consistent negative effect of the total basal area of neighbouring competitors across all biomes. The dominance of a competitive effect for the growth of adult trees (diameter at breast height >= 10cm dbh), agrees well with the idea that facilitation processes are generally limited to the regeneration phase rather than at the adult stage [@Callaway-1997]. The variation of $\alpha_0$ between biomes is limited with large overlap of their confidences intervals.
In term of traits effects, Wood density (WD) was strongly negatively associated with maximum growth, in agreement with the idea that shade-intolerant species with low wood density have faster growth in absence of competition (in full light conditions) than shade tolerant species[@Nock-2009; @Wright-2010]. One advantage of low wood density is clearly that it is cheaper to build light than dense wood, thus for the same biomass growth a low wood density species will have a higher basal area increment than a high wood density species[@Enquist-1999]. Other advantages of light wood may include higher xylem conductivity[@Chave-2009], though for angiosperms this is a correlated trait rather than an automatic consequence. A countervailing advantage for high wood density species was their better tolerance to competition (less growth reduction per unit of basal area of competitors), which is in line with the idea that these species are more shade tolerant[@Chave-2009; @Nock-2009; @Wright-2010]. This has generally been related to the higher survival associated with high wood density[@Kraft-2010], via resistance to mechanical damage, herbivores and pathogens[@Chave-2009; @Kraft-2010], but may also be connected to a lower maintenance respiration[@Larjavaara-2010]. For growth, the lower respiration may lead to a direct advantage in deep shade, but the correlation might also arise through correlated selection for high survival rate and for high growth in shade. Finally, high wood density was also weakly correlated with stronger competitive effect, especially in tropical forest where the confidence interval did not span zero. This might possibly have been mediated by larger crowns (both in depth and radius)[@Poorter-2006a; @Aiba-2009], casting a deeper shade.
......@@ -62,7 +62,7 @@ SLA was positively correlated with maximum basal area growth (growth without com
Maximum height was weakly positively correlated with maximum growth rate (confidence intervals spanned zero except for temperate rain forest). Previous studies[@Poorter-2006a; @Poorter-2008; @Wright-2010] found mixed support for this relationship. Possible mechanisms are contradictory: maximum height may be associated with greater access to light and thus faster growth, but at the same time life history strategies might be expected to select for slower growth in long-lived plants[@Poorter-2008]. Maximum height was negatively correlated with tolerance to competition (confidence intervals spanned zero except for temperate rain forest and taiga), in line with the idea that sub-canopy trees are more shade-tolerant[@Poorter-2006a]. There was however a tendency for species with tall maximum height to have stronger competitive effect (though with wider confidence intervals intercepting zero), that might be explained by greater light interception from taller trees.
Our results raised the question whether there is a coordination between trait values conferring strong competitive effect and trait values conferring high competitive tolerance. Competitive effect and tolerance are the two central elements of the species competitive ability[@Goldberg-1991]. One may expect that because of intra-specific competition, species with strong competitive effect should have evolved a high tolerance to competition. However, in agreement with previous studies[@Goldberg-1990; @Goldberg-1991; @Wang-2010], we found little evidence for such coordination. It was present only for wood density, where high density conferred better competitive tolerance and also stronger competitive effect (but with wide confidence intervals). For SLA there was no clear coordinations. For maximum height as explained above there was a tendency for short maximum height to lead to high tolerance to competition but to low competitive effect. This interesting because a trade-off between competitive tolerance and maximum height has been proposed as fundamental mechanisms of coexistence of species in size-structured population in the stratification theory of species coexistence[@Kohyama-2009]. Finally the lack of support for coordination between tolerance and effect is important because it means that competitive interaction is not well described as a trait hierarchy relating a focal species to its competitors (measured as $t_c -t_f$ and thus assuming $\alpha_e = \alpha_t$ as in @Kunstler-2012; @Kraft-2014; @Lasky-2014). Traits of competitors alone or of focal plants alone may convey more information. If traits are strongly linked to either competitive effect or competitive tolerance, this still means that some trait values will have an advantage in competitive interactions.
Our results raised the question whether there is a coordination between trait values conferring strong competitive effect and trait values conferring high competitive tolerance. Competitive effect and tolerance are the two central elements of the species competitive ability[@Goldberg-1991]. One may expect that because of intra-specific competition, species with strong competitive effect should have evolved a high tolerance to competition. However, in agreement with previous studies[@Goldberg-1990; @Goldberg-1991; @Wang-2010], we found little evidence for such coordination. There was only such a tendency for wood density and SLA. High wood density conferred better competitive tolerance and also stronger competitive effect, but with wide confidence intervals intercepting zero for the later. High SLA conferred stronger competitive effect and higher tolerance of competition, but with wide confidence intervals intercepting zero for the later. For maximum height as explained above there was a tendency for short maximum height to lead to high tolerance to competition but to low competitive effect. This is interesting because a trade-off between competitive tolerance and maximum height has been proposed as a fundamental mechanisms of coexistence of species in size-structured population in the stratification theory of species coexistence[@Kohyama-2009]. Finally the lack of support for coordination between tolerance and effect is important because it means that competitive interaction is not well described as a trait hierarchy relating a focal species to its competitors (measured as $t_c -t_f$ and thus assuming $\alpha_e = \alpha_t$ as in @Kunstler-2012; @Kraft-2014; @Lasky-2014). Traits of competitors alone or of focal plants alone may convey more information than the trait hierarchy. These processes of traits linked to either competitive effect or competitive tolerance, nevertheless, still leads to some trait values having an advantage in competitive interactions.
Given that the effect sizes we report for effects of traits on competitive interaction are modest, the question arises whether the three traits available to us (wood density, SLA, and maximum height) were the best candidates. It is possible that traits more directly related to mechanisms of competition -- for instance for competition for light, the leaf area index of the competitors or the compensation point at leaf or whole-plant level -- may be more powerful. It is also possible that if traits measured at the individual level were available, rather than species averages, this might strengthen predictive power[@Kraft-2014].
......
......@@ -5,9 +5,9 @@
## ```
## ![Map of the plot locations of all data sets analysed. LPP plots are represented with a large points and NFI plots with small points (The data set of Panama comprise both a 50ha plot and a network of 1ha plots).](image/worldmapB.png)
## ![Map of the plot locations of all data sets analysed. LPP plots are represented with a large points and NFI plots with small points (The data set of Panama comprises both a 50ha plot and a network of 1ha plots).](image/worldmapB.png)
## \newpage
## \newpage
## ![Variation of the four parameters linking the three studied traits with maximum growth and competition - maximum growth ($t_f \, m_1$), tolerance to competition ($t_f \, \alpha_t$), competitive effect ($t_c \, \alpha_e$) and limiting similarity ($|t_f - t_c| \, \alpha_l$ ($t_c$ was fixed at the lowest value and $t_f$ varying from quantile 5 to 95\%). The shaded area represents the 95% confidence interval of the prediction (including uncertainty associated with $\alpha_0$ or $m_0$).](../../figs/figres4b.pdf)
......@@ -43,6 +43,7 @@ dat.2[dat.2$set == 'BCI',1] <- 'Panama'
dat.2[dat.2$set == 'Fushan',1] <- 'Taiwan'
dat.2[dat.2$set == 'Luquillo',1] <- 'Puerto Rico'
dat.2[dat.2$set == 'Mbaiki',1] <- 'Central African Republic'
dat.2[dat.2$set == 'Paracou',1] <- 'French Guiana'
var.names <- colnames(dat.2)
var.names[2] <- '# of trees'
......@@ -62,12 +63,12 @@ dat.2 <- dat.2[, 1:11]
dat.2[,5:11] <- dat.2[,5:11]*100
pandoc.table(dat.2[, 1:6],
caption = "Data description, with number of individual tree, species and plot in NFI data and quadrat in LPP data, and percentage of angiosperm and evergreen species.",
caption = "Data description, with number of individual trees, species and plots in NFI data and quadrats in LPP data, and percentage of angiosperm and evergreen species.",
digits = c(3,3,3,0,0), split.tables = 200, split.cells = 35,
justify = c('left', rep('right', 5)), keep.trailing.zeros = TRUE)
pandoc.table(dat.2[, c(1,9:11)],
caption = "Traits coverage in each sites. Percentage of species with species level trait data.",
caption = "Traits coverage in each site. Percentage of species with species level trait data.",
digits = 1, split.tables = 200, split.cells = 25,
justify = c('left', rep('right', 3)),
keep.trailing.zeros = TRUE)
......@@ -121,19 +122,19 @@ mat.param.mean.sd <- matrix(paste0(round(mat.param, 3),
mat.param <- rbind(mat.param.mean.sd,
round(mat.R2m, 4),
round(mat.R2c, 4),
format(round(mat.AIC, 0), scientific = TRUE, digits = 4),
format(round(mat.AIC.0, 0), scientific = TRUE, digits = 4))
round(mat.AIC- mat.AIC, 0),
round(mat.AIC.0 - mat.AIC, 0))
colnames(mat.param) <- c('Wood density', 'SLA', 'Maximum height')
row.names(mat.param) <- c('$m_0$', '$\\gamma$', '$m_1$', '$\\alpha_0$',
'$\\alpha_i$', '$\\alpha_r$',
'$\\alpha_s$', '$R^2_m$*', '$R^2_c$*', 'AIC', 'AIC no trait')
'$\\alpha_e$', '$\\alpha_t$',
'$\\alpha_s$', '$R^2_m$*', '$R^2_c$*', '$\\Delta$ AIC', '$\\Delta$ AIC no trait')
##+ Table2_Effectsize, echo = FALSE, results='asis', message=FALSE
pandoc.table(mat.param[c(1,3,2,4:11), ], caption = "Standaridized parameters estimates and standard error (in bracket) estimated for each traits, $R^2$* of models and AIC of the model and AIC of a model with no trait effect. See section Method for explanation of parameters",
pandoc.table(mat.param[c(1,3,2,4:11), ], caption = "Standardized parameters estimates and standard error (in bracket) estimated for each trait, $R^2$* of models and $\\Delta$ AIC of the model and of a model with no trait effect. Best model have a $\\Delta$ AIC of zero. See section Method for explanation of parameters",
digits = 3, justify = c('left', rep('right', 3)),
emphasize.strong.cells = bold.index, split.tables = 200)
## \* We report the conditional and marginal $R^2$ of the models using the methods of reference[^1], modified by reference[^2]. AIC is the Akaike's Information Criterion (as defined by reference[^3]), and the best-fitting model was identified as the one with the smallest AIC.
## \* We report the conditional and marginal $R^2$ of the models using the methods of reference[^1], modified by reference[^2]. $\Delta$ AIC is the difference in AIC between the model and the best model (lowest AIC). AIC is the Akaike's Information Criterion (as defined by reference[^3]), and the best-fitting model was identified as the one with a $\Delta$ AIC of zero. $\Delta$ AIC greater than 10 shows strong support for the best model^3^.
## [^1]: Nakagawa, S. & Schielzeth, H. A general and simple method for obtaining R2 from generalized linear mixed-effects models. Methods in Ecology and Evolution 4, 133–142 (2013).
## [^2]: Johnson, P. C. D. Extension of Nakagawa and Schielzeth’s R2GLMM to random slopes models. Methods in Ecology and Evolution 5, 944–946 (2014).
......
% Methods (1942/max 3000 words)
# Model and analysis
To examine the link between competition and traits we used a neighbourhood modelling framework[@canham_neighborhood_2006; @uriarte_trait_2010; @ruger_functional_2012; @kunstler_competitive_2012; @lasky_trait-mediated_2014] to model the growth of a focal tree of species $f$ as a product of its maximum growth rate (determined by its traits and size) together with reductions due to competition from individuals growing in the local neighbourhood. Specifically, we assumed a relationship of the form
\begin{equation} \label{G1}
G_{i,f,p,s} = G_{\textrm{max} \, f,p,s} \, D_{i}^{\gamma_f} \, \exp\left(\sum_{c=1}^{N_p} {-\alpha_{c,f} B_{i,c,p,s}}\right),
\end{equation}
where:
- $G_{i,f,p,s}$ and $D_{i,f,p,s}$ are the the annual basal area growth and diameter at breast height of individual $i$ from species $f$, plot $p$ and data set $s$,
- $G_{\textrm{max} \, f,p,s}$ is the potential growth rate in basal area growth for species $f$ on plot $p$ in data set $s$, i.e. in absence of competition,
- $\gamma_f$ determines the rate at which growth changes with size for species $f$, modelled with a normally distributed random effect of species $\epsilon_{\gamma, f}$ (as $\gamma_f = \gamma_0 + \epsilon_{\gamma, f}$ where $\epsilon_{\gamma, f} \sim N(0,\sigma_{\gamma})$)
- $N_p$ is the number of competitor species on plot $p$ ,
- $\alpha_{c,f}$ is the per unit basal area effect of individuals from species $c$ on growth of an individual in species $f$, and
- $B_{i,c,p,s}= 0.25\, \pi \, \sum_{j \neq i} w_j \, D_{j,c,p,s}^2$ is the sum of basal area of all individuals trees $j$ of the species $c$ competiting with the tree $i$ within the plot $p$ and data set $s$, where $w_j$ is a constant based on subplot size where tree $j$ was measured. Note that $B_{i,c,p,s}$ include all trees in the plot excepted the tree $i$.
Values of $\alpha_{c,f}> 0$ indicate competition, whereas $\alpha_{c,f}$ < 0 indicates facilitation.
Log-transformation of eq. \ref{G1} leads to a linearised model of the form
\begin{equation} \label{logG1}
\log{G_{i,f,p,s}} = \log{G_{\textrm{max} \, f,p,s}} + \gamma_f \, \log{D_{i,f,p,s}} + \sum_{c=1}^{N_p} {-\alpha_{c,f} B_{i,c,p,s}}.
\end{equation}
To include the effect of a focal trees' traits, $t_f$, on its growth, we let:
\begin{equation} \label{Gmax}
\log{G_{\textrm{max} \, f,p,s}} = m_{0} + m_1 \, t_f + \epsilon_{G_{\textrm{max}}, f} + \epsilon_{G_{\textrm{max}}, p} + \epsilon_{G_{\textrm{max}}, s}.
\end{equation}
Here $m_0$ is the average maximum growth, $m_1$ gives the effect of the focal trees trait, and $\epsilon_{G_{\textrm{max}}, f}$, $\epsilon_{G_{\textrm{max}}, p}$, $\epsilon_{G_{\textrm{max}}, s}$ are normally distributed random effect for species $f$, plot or quadrat $p$ (see below), and data set $s$ \[where $\epsilon_{G_{\textrm{max}, f}} \sim N(0,\sigma_{G_{\textrm{max}, f}})$; $\epsilon_{G_{\textrm{max}, p}} \sim N(0,\sigma_{G_{\textrm{max}, p}})$ and $\epsilon_{G_{\textrm{max}, s}} \sim N(0,\sigma_{G_{\textrm{max}, s}})$].
To include trait effects on competition presented in Fig. 1, competitive interactions were modelled using an equation of the form[^note]:
\begin{equation} \label{alpha}
\alpha_{c,f}= \alpha_{0,f} - \alpha_t \, t_f + \alpha_e \, t_c + \alpha_s \, \vert t_c-t_f \vert
\end{equation}
where:
- $\alpha_{0,f}$ is the trait independent competition for the focal species $f$, modelled with a normally distributed random effect of species $f$ and a normally distributed random effect of data set $s$ \[as $\alpha_{0,f} = \alpha_0 + \epsilon_{\alpha_0, f}+ \epsilon_{\alpha_0, s}$, where $\epsilon_{\alpha_0, f} \sim N(0,\sigma_{\alpha_0, f})$ and $\epsilon_{\alpha_0, s} \sim N(0,\sigma_{\alpha_0, s})$],
- $\alpha_t$ is the **tolerance to competition** of the focal species, i.e. change in competition tolerance due to traits $t_f$ of the focal tree with a normally distributed random effect of data set $s$ included \[$\epsilon_{\alpha_t,s} \sim N(0,\sigma_{\alpha_t})$],
- $\alpha_{e}$ is the **competitive effect**, i.e. change in competition effect due to traits $t_c$ of the competitor tree with a normally distributed random effect of data set $s$ included \[$\epsilon_{\alpha_i,s} \sim N(0,\sigma_{\alpha_i})$], and
- $\alpha_s$ is the effect of **trait similarity**, i.e. change in competition due to absolute distance between traits $\vert{t_c-t_f}\vert$ with a normally distributed random effect of data set $s$ included \[$\epsilon_{\alpha_s,s} \sim N(0,\sigma_{\alpha_s})$].
[^note]:For fitting the model the equation of $\alpha_{c,f}$ was developped with species basal area in term of community weighted mean of the trait, see Supplementary methods for more details.
Eqs. \ref{logG1}-\ref{alpha} were then fitted to empirical estimates of growth, given by
\begin{equation} \label{logGobs} G_{i,f,p,s} = 0.25 \pi \left(D_{i,f,p,s,t+1}^2 - D_{i,f,p,s,t}^2\right).
\end{equation}
To estimate standardised coefficients (one type of standardised effect
size)[@schielzeth_simple_2010], response and explanatory variables
were standardized (divided by their standard deviations) prior to
analysis. Trait and diameter were also centred to facilitate
convergence. The models were fitted using $lmer$ in
lme4[@lme4] with R[@RCRAN]. We fitted two versions of this
model. In the first version parameters $m_{0}, m_1,
\alpha_0,\alpha_t,\alpha_i,\alpha_s$ were estimated as constant across
all biomes. In the second version, we repeated the same analysis as
the first version but provided for different fixed estimates of these parameters for each biome. This enabled us to explore variation between biomes. Because some biomes had few observations, we merged some biomes with similar climate. Tundra was merged with taiga, tropical rainforest and tropical seasonal forest were merged into tropical forest, and deserts were not included in this final analysis as too few data were available.
# Data
## Growth data
Our main objective was to collate data sets spanning the dominant forest biomes of
the world. Data sets were included if they (i) allowed both growth rate of individual trees and the local abundance of competitors to be estimated, and (ii) had good (>40%) coverage for at least one
of the traits of interest (SLA, wood density, and maximum height).
The data sets collated fell into two broad categories: (1) national forest inventories (NFI), in which trees above a given diameter were sampled in a network of small plots (often on a regular grid) covering the country; (2) large permanent plots (LPP) ranging in size from 0.5-50ha, in which the x-y coordinates of all trees above a given diameter were recorded. These LPP were mostly located in tropical regions. The minimum diameter of recorded trees varied among sites from 1-12cm. To allow comparison between data sets, we restricted our analysis to trees greater than 10cm. Moreover, we excluded from the analysis any plots with harvesting during the growth measurement period, that were identified as a plantations, or overlapping a forest edge. Finally, we selected only two consecutive census dates for each tree to avoid having to account for repeated measurements, as less than a third of the data had repeated measurements. See the Supplementary Methods and Extended Data Table 1 for more details on the individual data sets.
Basal area growth was estimated from diameter measurements recorded
across successive time points. For the French NFI, these data were
obtained from short tree cores. For all other data sets, diameter at
breast height ($D$) of each individual was recorded at multiple census
dates. We excluded trees (i) with extreme positive or negative
diameter growth rates, following criteria developed at the BCI site
[@condit_mortality_1993] (see the R package
[CTFS R](http://ctfs.arnarb.harvard.edu/Public/CTFSRPackage/)),
(ii) that were a palm or a tree fern species, or (iii) that were
measured at different height in two consecutive censuses.
For each individual tree, we estimated the local abundance of competitor species as the sum of basal area for all individuals > 10cm diameter within a specified neighbourhood. For LPPs, we defined the neighbourhood as being a circle with 15m radius. This value was selected based on previous studies showing the maximum radius of interaction to lie in the range 10-20m[@uriarte_neighborhood_2004; @uriarte_trait_2010]. To avoid edge effects, we also excluded trees less than 15m from the edge of a plot. To account for variation of abiotic conditions within the LPPs, we divided plots into regularly spaced 20x20m quadrats.
For NFI data coordinates of individual trees within plots were generally not available, thus neighbourhoods were defined based on plot size. In the NFI from the United States, four sub-plots of 7.35m located within 20m of one another were measured. We grouped these sub-plots to give a single estimate of the local competitor abundance. Thus, the neighbourhoods used in the competition analysis ranged in size from 10-25 m radius, with most plots 10-15 m radius.
We extracted mean annual temperature (MAT) and mean annual sum of precipitation (MAP) from the [worldclim](http://www.worldclim.org/) data base [@hijmans_very_2005], using the plot latitude and longitude. MAT and MAP data were then used to classify plots into biomes, using the diagram provided by @ricklefs_economy_2001 (after Whittaker).
## Traits
Data on species functional traits were extracted from existing sources. We focused on wood density, species specific leaf area (SLA) and maximum height, because these traits have previously been related to competitive interactions and are available for large numbers of species [@wright_functional_2010; @uriarte_trait_2010; @ruger_functional_2012; @kunstler_competitive_2012; @lasky_trait-mediated_2014] (see Extended data Table 2 for traits coverage). Where available we used data collected locally; otherwise we sourced data from the [TRY](http://www.try-db.org/) trait data base [@kattge_try_2011]. Local data were available for most tropical sites and species (see Supplementary methods). Several of the NFI data sets also provided height measurements, from which we computed a species' maximum height as the 99% quantile of observed values (for France, US, Spain, Switzerland). For Sweden we used the estimate from the French data set and for Canada we used the estimate from the US data set. Otherwise, we extracted measurement from the TRY database. We were not able to account for trait variability within species between sites.
For each focal tree, our approach required us to also account for the traits of all competitors present in the neighbourhood. Most of our plots had good coverage of competitors, but inevitably there were some trees where trait data were lacking. In these cases we estimated trait data as follows. If possible, we used the genus mean, and if no genus data was available, we used the mean of the species present in the country. However, we restricted our analysis to plots where (i) the percentage of basal area of trees with no species level trait data was less than 10%, and (ii) no genus level data was less than 5%.
\newpage
# References
\documentclass[a4paper,11pt]{article}
\usepackage{lmodern}
\usepackage{amssymb,amsmath}
\usepackage{amsmath}
\usepackage{ifxetex,ifluatex}
\usepackage{fixltx2e} % provides \textsubscript
\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
......@@ -10,7 +10,7 @@
\ifxetex
\usepackage{mathspec}
\usepackage{xltxtra,xunicode}
\else
\else
\usepackage{fontspec}
\fi
\defaultfontfeatures{Mapping=tex-text,Scale=MatchLowercase}
......@@ -19,11 +19,11 @@
\usepackage{ms}
% use upquote if available, for straight quotes in verbatim environments
\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
% use microtype if available
\IfFileExists{microtype.sty}{%
\usepackage{microtype}
\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
}{}
% % use microtype if available
% \IfFileExists{microtype.sty}{%
% \usepackage{microtype}
% \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
% }{}
\usepackage[numbers,super,sort&compress]{natbib}
\ifxetex
......@@ -68,7 +68,7 @@ reductions due to competition from individuals growing in the local
neighbourhood. Specifically, we assumed a relationship of the form
\begin{equation} \label{G1}
G_{i,f,p,s} = G_{\textrm{max} \, f,p,s} \, D_{i}^{\gamma_f} \, \exp\left(\sum_{c=1}^{N_p} {-\alpha_{c,f} B_{i,c,p,s}}\right),
G_{i,f,p,s} = G_{\textrm{max} \, f,p,s} \, D_{i,f,p,s}^{\gamma_f} \, \exp\left(\sum_{c=1}^{N_p} {-\alpha_{c,f} B_{i,c,p,s}}\right),
\end{equation}
where:
......@@ -86,9 +86,10 @@ where:
\item
\(\gamma_f\) determines the rate at which growth changes with size for
species \(f\), modelled with a normally distributed random effect of
species \(\epsilon_{\gamma, f}\) {[}as
\(\gamma_f = \gamma_0 + \epsilon_{\gamma, f}\) where
\(\epsilon_{\gamma, f} \sim N(0,\sigma_{\gamma})\){]}
species \(\varepsilon_{\gamma, f}\) {[}as
\(\gamma_f = \gamma_0 + \varepsilon_{\gamma, f}\) where
\(\varepsilon_{\gamma, f} \sim \mathcal{N} (0,\sigma_{\gamma})\) -- a
normal distribution of mean 0 and standard deviation $\sigma_{\gamma}${]}
\item
\(N_p\) is the number of competitor species on plot \(p\) ,
\item
......@@ -99,7 +100,7 @@ where:
the sum of basal area of all individuals trees \(j\) of the species
\(c\) competiting with the tree \(i\) within the plot \(p\) and data
set \(s\), where \(w_j\) is a constant based on subplot size where
tree \(j\) was measured. Note that \(B_{i,c,p,s}\) include all trees
tree \(j\) was measured. Note that \(B_{i,c,p,s}\) include all trees of species $c$
in the plot excepted the tree \(i\).
\end{itemize}
......@@ -117,18 +118,18 @@ To include the effect of a focal trees' traits, \(t_f\), on its growth,
we let:
\begin{equation} \label{Gmax}
\log{G_{\textrm{max} \, f,p,s}} = m_{0} + m_1 \, t_f + \epsilon_{G_{\textrm{max}}, f} + \epsilon_{G_{\textrm{max}}, p} + \epsilon_{G_{\textrm{max}}, s}.
\log{G_{\textrm{max} \, f,p,s}} = m_{0} + m_1 \, t_f + \varepsilon_{G_{\textrm{max}}, f} + \varepsilon_{G_{\textrm{max}}, p} + \varepsilon_{G_{\textrm{max}}, s}.
\end{equation}
Here \(m_0\) is the average maximum growth, \(m_1\) gives the effect of
the focal trees trait, and \(\epsilon_{G_{\textrm{max}}, f}\),
\(\epsilon_{G_{\textrm{max}}, p}\), \(\epsilon_{G_{\textrm{max}}, s}\)
the focal trees trait, and \(\varepsilon_{G_{\textrm{max}}, f}\),
\(\varepsilon_{G_{\textrm{max}}, p}\), \(\varepsilon_{G_{\textrm{max}}, s}\)
are normally distributed random effect for species \(f\), plot or
quadrat \(p\) (see below), and data set \(s\) {[}where
\(\epsilon_{G_{\textrm{max}, f}} \sim N(0,\sigma_{G_{\textrm{max}, f}})\);
\(\epsilon_{G_{\textrm{max}, p}} \sim N(0,\sigma_{G_{\textrm{max}, p}})\)
\(\varepsilon_{G_{\textrm{max}, f}} \sim \mathcal{N} (0,\sigma_{G_{\textrm{max}, f}})\);
\(\varepsilon_{G_{\textrm{max}, p}} \sim \mathcal{N} (0,\sigma_{G_{\textrm{max}, p}})\)
and
\(\epsilon_{G_{\textrm{max}, s}} \sim N(0,\sigma_{G_{\textrm{max}, s}})\){]}.
\(\varepsilon_{G_{\textrm{max}, s}} \sim \mathcal{N} (0,\sigma_{G_{\textrm{max}, s}})\){]}.
To include trait effects on competition presented in Fig. 1, competitive
interactions were modelled using an equation of the form\footnote{For
......@@ -149,30 +150,30 @@ where:
species \(f\), modelled with a normally distributed random effect of
species \(f\) and a normally distributed random effect of data set
\(s\) {[}as
\(\alpha_{0,f} = \alpha_0 + \epsilon_{\alpha_0, f}+ \epsilon_{\alpha_0, s}\),
where \(\epsilon_{\alpha_0, f} \sim N(0,\sigma_{\alpha_0, f})\) and
\(\epsilon_{\alpha_0, s} \sim N(0,\sigma_{\alpha_0, s})\){]},
\(\alpha_{0,f} = \alpha_0 + \varepsilon_{\alpha_0, f}+ \varepsilon_{\alpha_0, s}\),
where \(\varepsilon_{\alpha_0, f} \sim \mathcal{N} (0,\sigma_{\alpha_0, f})\) and
\(\varepsilon_{\alpha_0, s} \sim \mathcal{N} (0,\sigma_{\alpha_0, s})\){]},
\item
\(\alpha_t\) is the \textbf{tolerance of competition} of the focal
species, i.e.~change in competition tolerance due to traits \(t_f\) of
the focal tree with a normally distributed random effect of data set
\(s\) included
{[}\(\epsilon_{\alpha_t,s} \sim N(0,\sigma_{\alpha_t})\){]},
{[}\(\varepsilon_{\alpha_t,s} \sim \mathcal{N} (0,\sigma_{\alpha_t})\){]},
\item
\(\alpha_{e}\) is the \textbf{competitive effect}, i.e.~change in
competition effect due to traits \(t_c\) of the competitor tree with a
normally distributed random effect of data set \(s\) included
{[}\(\epsilon_{\alpha_i,s} \sim N(0,\sigma_{\alpha_i})\){]}, and
{[}\(\varepsilon_{\alpha_i,s} \sim \mathcal{N} (0,\sigma_{\alpha_i})\){]}, and
\item
\(\alpha_s\) is the effect of \textbf{trait similarity}, i.e.~change
in competition due to absolute distance between traits
\(\vert{t_c-t_f}\vert\) with a normally distributed random effect of
data set \(s\) included
{[}\(\epsilon_{\alpha_s,s} \sim N(0,\sigma_{\alpha_s})\){]}.
{[}$\varepsilon_{\alpha_s,s} \sim \mathcal{N} (0,\sigma_{\alpha_s})${]}.
\end{itemize}
Eqs. \ref{logG1}-\ref{alpha} were then fitted to empirical estimates of
growth, given by
growth based on change in diameter between time $t$ and $t+1$, given by
\begin{equation} \label{logGobs} G_{i,f,p,s} = 0.25 \pi \left(D_{i,f,p,s,t+1}^2 - D_{i,f,p,s,t}^2\right).
\end{equation}
......@@ -289,7 +290,7 @@ less than 10\%, and (ii) no genus level data was less than 5\%.
\newpage
\clearpage
\section{References}\label{references}
\bibliographystyle{naturemag}
......
......@@ -121,7 +121,7 @@ address:
- code: 23
address: Plant Sciences (IBG-2), Forschungszentrum Jülich GmbH, Jülich, Germany.
- code: 24
address: Department of Ecology, Evolution and Environmental Biology, Columbia University, New York, NY 10027.
address: Department of Ecology, Evolution and Environmental Biology, Columbia University, New York, NY 10027, United States of America.
- code: 25
address: Landcare Research, PO Box 40, Lincoln 7640, New Zealand.
- code: 26
......@@ -133,7 +133,7 @@ address:
- code: 29
address: Department of Forest Resource Management, Swedish University of Agricultural Sciences (SLU), Skogsmarksgränd, Umeå, Sweden.
- code: 30
address: Department of Plant Biology, Michigan State University, East Lansing, Michigan, United States of America.
address: Department of Biology, University of Maryland, College Park, Maryland, United States of America.
- code: 31
address: Centre for Ecology and Hydrology−Edinburgh, Bush Estate, Penicuik, Midlothian EH26 0QB United Kingdom.
- code: 32
......
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