% Report from workshop 'How are competitive interactions influenced by traits? A global analysis based on tree radial growth'

% Project leader: Georges Kunstler

% 16/12/2013

% 18/12/2013

This document gives an update on the analyses completed during and after the workshop held in October 2013 at Macquarie University.

**Contact details:** georges.kunstler@gmail.com, Department of Biological Sciences Macquarie University, Sydney, NSW / Irstea EMGR Grenoble France

**Workshop participants:** David A. Coomes, Daniel Falster, Rob Kooyman, Daniel Laughlin, Lourens Poorter, Mark Vanderwel, Ghislain Vieilledent, Mark Westoby, Joe Wright

**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.

**Other participants and Data contributors:** J. Caspersen, H. Zeng, S. Gourlet-Fleury, B. Herault, G. Ståhl, J. Thompson, S. Richardson, P. Ruiz, I-F. Sun, N. Swenson, M Uriarte, M. Zavala, N. E. Zimmermann, M. Hanewinkel, J. Zimmerman, Yusuke Onoda, Hiroko Kurokawa, Masahiro Aiba, MRN Québec and other

\newpage

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@@ -21,7 +22,7 @@ more likely to coexist locally than similar species

@macarthur_limiting_1967). One way to quantify ecological similarity

between species is via traits, such as leaf, seed and wood

characteristics [@westoby_plant_2002]. Traits influence many aspects

of plant performance, including resource acquisition. Under the *competition-niche similarity hypothesis* higher dissimilarity should results in higher resource partitioning at

of plant performance, including resource acquisition. Under the *competition-niche similarity hypothesis* higher trait dissimilarity should results in higher resource partitioning at

local scale and less intense competition. This idea underlies numerous ecological analyses

[@kraft_functional_2008; @cornwell_community_2009]. However this

assumption has rarely been tested against field or experimental

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@@ -105,7 +106,7 @@ proposed: a multiplicative and an additive

model of competitive effect and response[^inter].

Below I consider the

additive effect-response model because it is simpler. However, I have not ruled out to exploring

additive effect-response model because it is simpler. However, I have not ruled out exploring

the multiplicative effect-response model[^equmult].

[^compreponse]:Through out the document I will use competitive response as the inverse of competition tolerance.

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@@ -209,8 +210,9 @@ validate the trait extraction (see Figure \ref{trait} for the range of traits va

## Data processing

Next we split each dataset by ecoregion, keeping only

ecoregions where in average at least three species contributed more than

5% of the average basal area of the plots, thereby excluding quasi-monospecific stands.

ecoregions where, on average, at least three species contributed more than

5% of the total basal area of each plots. This had the effect of

excluding quasi-monospecific stands.

First we computed the local basal area ($cm^2/m^2$) of neighborhood

competitor per species for each individual tree. For NFI data the

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@@ -226,8 +228,8 @@ missing value with genus level data when it was possible, or

filling the remaining value with the community mean of the trait. All traits were

centered and standardized per data set (a global traits standardization

doesn't seems to provides strikingly different values). We run independent

computation of the $t_c$ to validate the processing of the data and inspected

histograms of $t_c$ to identify errors.

computation of the community weight means to validate the processing of the data and inspected

histograms of $\overline{t_n}$ to identify errors.

We used only individual tree for which 90% of its neighborhood was

covered with at least genus level traits in subsequent analysis. The

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@@ -240,13 +242,13 @@ taxonomic identification).

## Fitting of a mixed linear model

During the workshop we ran estimation using a hierarchical Bayesian model

using [JAGS](http://mcmc-jags.sourceforge.net/). In the subsequent analysis I decided to start with a linear mixed model approach (function lmer in

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

the functions for the size and competition effect (mainly following the work of C. Canham for instance see

more complex non-linear functions for the size and competition effect (mainly followingsuch as the one used in the work of C. Canham see

@uriarte_trait_2010), and

2. using lmer was much faster than an estimation with JAGS or [Stan](http://mc-stan.org/).

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@@ -278,7 +280,7 @@ We compared two alternative models for $\lambda_{n,f}$:

\times t_{n}$) and

(ii) $\lambda$ is a

function the absolute trait distance ($\lambda_{n,f} = a + b \times

$).

|t_{n} - t_{f}|$).

These two models can be expressed in terms of community weighted mean trait value as follows. For the trait effect-response model:

\begin{equation} \label{logG-ER}

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@@ -331,6 +333,7 @@ Most of the effect-response models fitted show a competitive effect (negative va

- Explore non-linear model for growth and survival using Stan (probably used the models used by Canham and Uriarte).

- Fit multi-traits models (include multiple traits in effect and response models and either multidimensional distance in the absolute distance model or include all single trait absolute distance). Try to use spike and slab prior for variables selection.

- Try to include traits effect in parameter $Gmax$. This would allows to (1) test if this change the results observed for the traits effect on $\lambda$ (a comment of Maria Uriarte) and (2) test if traits underpin a trade-off between max growth with out competition and competition tolerance.

- Explore if the decrease in the link between trait and competition at high MAP is related in a change in the packing of trait space in this communities.

- Explore the possibility that trait effect may be different for evergreen/deciduous species (leaf traits) or angiosperm/conifer species (wood density). This could be done by fitting different parameters for the trait of evergreen deciduous and conifer in the effect-response model. This is not really possible for the absolute distance model.

- Use an alternative way of dividing the NFI data than the ecoregion (class of MAP and MAT?).

- Try to run a global analysis with all data (memory limit issue to solve).

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@@ -341,7 +344,7 @@ Most of the effect-response models fitted show a competitive effect (negative va

\newpage

# FIGURES

# FIGURES & TABLES

![**Positions of the data sets analysed in the climatic biomes of Whittaker.** The coloured polygons represents the biomes. The points represent the mean position of the data set in the mean annual temperature and annual precipitation space. For the national forest inventory the 95% quantile of the climate within the ecoregion is represented by an error bar. The temperature and precipitation are taken from worldclim [@hijmans_very_2005].\label{biomes}](biome_ecocode_xy.pdf)

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@@ -354,11 +357,13 @@ Most of the effect-response models fitted show a competitive effect (negative va

\newpage

![**Correlation pairs over all data sets 9in 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

![**Correlation pairs over all data sets (in log scale).** Each data set is drawn with a different symbols and colors. Traits SLA ($mm^2/mg$), Leaf N per mass ($mg/g$), wood density

($mg/mm^3$), maximum height ($m$). \label{trait}](traits-XY.pdf)

\pagebreak

\pagebreak

\newpage

![**Effect size of the absolute distance models and the effect-response model over all ecoregion for the four traits.** Effect size is computed as the difference of $R_c^2$ between a constant competition model and the tested model. \label{boxplot-effectsize}](R2_boxplot_two.pdf)

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@@ -378,7 +383,6 @@ Most of the effect-response models fitted show a competitive effect (negative va

Data name Demographic data Traits data Availability Abiotic variables

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@@ -660,7 +664,7 @@ Luquillo 0 0 0 0 1 0

\pagebreak

## Multiplicative model of competitive effect and response

# Appendix 1. Multiplicative model of competitive effect and response {#multi}

The general framework for this approach is to consider that $\lambda_{n,f} = r(t_f) \times e(t_n)$ where $r$ and $e$ are respectively function that relate the competitive response and effect to the trait. We can test a series of model with increasing complexity of trait effect.

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@@ -682,14 +686,14 @@ The general framework for this approach is to consider that $\lambda_{n,f} = r(t

\end{equation}

As for the additive model it is then possible to develop the multiplicative model 3 to relate the competition in term of community weighted mean trait of the neighborhood species.

As for the additive model it is then possible to develop the multiplicative model 3 to relate the competition in term of community weighted mean trait of the neighborhood species ($\overline{t_{n}}$).

\begin{equation}

\sum_{n=1}^{N_p} \lambda_{n,f} \times B_n = B_\textrm{tot} \times (a +b \times t_{f}) \times (c+ d \times \bar{t_{n}})

\begin{equation}\label{multi-er}

\sum_{n=1}^{N_p} \lambda_{n,f} \times B_n = B_\textrm{tot} \times (a +b \times t_{f}) \times (c+ d \times \overline{t_{n}})

\end{equation}

## Appendix 1. Comparison of the multiplicative and additive effect and response model {#multi}

## Comparison of the multiplicative and additive effect and response model

Developing the multiplicative model gives

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@@ -697,19 +701,19 @@ Developing the multiplicative model gives

In comparison the additive model plus interaction i which is an extension of the model presented above which include an interaction between the traits $t_n$ and $t_f$ is:

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:

\lambda_{n,f} = a' +b'\times t_{f} +c'\times t_{n}+d'\times t_{n} \times t_{f}

\end{equation}

Thus the two models are equal when:

The two models are equal when:

\begin{equation}

a_{add}=ac \mspace{3mu} ;\mspace{3mu} b_{add}=bc\mspace{3mu} ;\mspace{3mu} c_{add}=ad \mspace{5mu} and \mspace{5mu} d_{add}=bd

a'=ac \mspace{3mu} ;\mspace{3mu} b'=bc\mspace{3mu} ;\mspace{3mu} c'=ad \mspace{5mu} and \mspace{5mu} d'=bd

\end{equation}

The multiplicative model is more constraining than the additive model plus interaction. In other word the additive model with interaction can be fitted to any multiplicative model but the inverse is not true (This would requires adding an interaction in the multiplicative model). For instance, it is not possible to match the hierarchical distance because if $b_{add}$ and $d_{add}\neq 0$ then $d_{add}\neq = 0$ as well. More generally, if parameters $a$, $b$ , $c$ and $d$ vary between [-max.r, max.r] then $d_{add}>b_{add}*c_{add}/(ma.r^2)$ (or $d_{add}<b_{add}*c_{add}/(-ma.r^2)$). Thus it is not possible to have a strong traits effect on response and effect and no interaction.

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 $d'<b'*c'/(-max.r^2)$). Thus it is not possible to have a strong traits effect on response and effect and no interaction. From first principle I think it is difficult to decide which model (equation \label{response_effect_trait} or equation \label{multi-er}) is the most likely.