\chapter[Regeneration height growth]{Competition between two species with contrasting shade tolerances} % Chapter title
\label{ch:rege} % For referencing the chapter elsewhere, use \autoref{ch:mathtest}

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\color{gray} {\slshape You will find something more in woods than in books. Trees and stones will teach you that which you can never learn from masters.}
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Saint Bernard
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The management of mixed stands with continuous-cover forestry systems has been increasingly promoted to improve forest biodiversity, resiliency, sustainability, and ecosystem services \citep{Schutz1997}. However, this approach relies upon the natural regeneration \citep{Pommerening2004} of species that can have contrasting shade tolerances.
 
Light availability in the understory determines competitive outcomes between species with different shade tolerances; light availability can be controlled by forest managers through adequate control of canopy closure. Maintaining a closed canopy promotes regeneration of shade-tolerant species, whereas opening canopy gaps promotes the regeneration of less shade-tolerant species. According to \citet{Kobe1995}, light-demanding species are characterized by fast growth, allowing them to outcompete neighboring trees in high-light environments, whereas shade-tolerant species are characterized by slow growth and low mortality, which allows their persistence in low-light environments.

However, these assumptions can be questioned as they hardly explain why forest managers rarely succeed in promoting sessile oak (\textit{Quercus petraea} (Matt.) Liebl.) regeneration beneath canopies that also contain European beech (\textit{Fagus sylvatica} L.) in Europe \citep{VonLupke1999}.
 
The regeneration of oak and beech has been intensively studied in European forests. In particular, many studies have detailed the shade tolerance of both species (\autoref{tab:31shadetolerance}). Beech juveniles have greater abilities to survive and grow in shade than oak juveniles. They have greater morphological plasticity, which can presumably enable them to maintain a positive carbon balance in low-light environments. Oak juveniles have greater light requirements than beech juveniles \citep{Collet1997, Emborg1998, Collet2001, Collet2006, Stancioiu2006b, Balandier2007, Petritan2007, Petritan2009, Wagner2010}. According to the definition of shade tolerance that was proposed by \citet{Kobe1995} and from greenhouse experiments \citep{Dreyer2005}, in high-light understories, the most light-demanding species, oak, is expected to outgrow the shade-tolerant beech.

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\begin{table}[htb]
\caption[Shade tolerance of oak and beech.]{Literature review of the shade tolerance of sessile oak and European beech. The columns titled beech and oak provide indications regarding the variable of interest and development stage (if specified) for beech and oak, respectively.}
\footnotesize
\begin{tabularx}{\linewidth}{p{5cm}lXXp{9cm}}
\toprule
\textsc{variable of interest} & \textsc{development stage} & \textsc{beech} & \textsc{oak} & \textsc{references} \\ 
\midrule
 Transmittance at light compensation point & 1\textsuperscript{st}-year seedling & very low & very low & \citet{Turbang1954}, \citet{Welander1998}, \citet{Chaar1999}, \citet{Nicolini2000} \\ 
 & Young seedling & \SIrange{2}{5}{\percent} & \SI{10}{\percent} & \citet{Madsen1997}, \citet{LeDuc1998}, \citet{VonLupke1999}, \citet{Emborg2000}, \citet{Collet2001},\citet{Collet2006}, \citet{Petritan2007}\\
 Transmittance at saturating growth & young seedling & \SI{10}{\percent} & \SI{20}{\percent} & \citet{Dineur1951}, \citet{Emborg1998}, \citet{VonLupke1998}, \citet{Stancioiu2006} \\ 
 & old seedling ($>$10 years) & \SI{20}{\percent} & \SI{30}{\percent} &  \citet{Jarret2004}, \citet{Petritan2009}\\ 
Morphological plasticity to different light conditions &  & high & low & \citet{Farque2001}, \citet{Collet1996}, \citet{Stancioiu2006}, \citet{Wagner2010} \\ 
Sensitivity to ground vegetation competition & & medium & low & \citet{Newbold1981}, \citet{Collet1996}, \citet{Coll2004}, \citet{Lof2004}, \citet{Wagner2010} \\
\bottomrule
\end{tabularx}
\label{tab:31shadetolerance}
\end{table}

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Little information has been published concerning the \textit{in situ} dynamics of advanced and well-established natural regeneration of the two species in mixed stands. Indeed, both species have mostly been studied separately, at the seedling stage, and under controlled conditions. Our understanding of interspecific competition is therefore largely inferred by extrapolating information on the autoecology of the two species (\autoref{tab:31shadetolerance}). Consequently, scientific investigation is required to better explain the difficulties that are faced by forest managers who want to maintain the coexistence of oak and beech.
 
Additionally, modeling the growth of natural regeneration has raised two methodological issues. First, it requires assessing a relevant indicator of understory light conditions, which depend upon the interception of diffuse and direct radiation by trees. The percentage of transmitted diffuse radiation has often been considered to represent accurately the percentage of total transmitted radiation \citep{Petritan2007, Petritan2009}. Nevertheless, direct radiation carries more energy than does diffuse radiation, but direct radiation that is transmitted under the canopy is limited in time and space (sunflecks) due to canopy irregularities and constant change of sunlight direction \citep{Bonhomme1993}. Second, \textit{in situ} experiments cannot control all the different factors that influence sapling growth. The growth of beech and oak regeneration is expected to vary from site to site due to both biotic and abiotic factors. In particular, different herbaceous species are known to compete with the saplings for supplies of nutrients, water and light \citep{Collet1996, Coll2003, Coll2004, Wagner2010}, or to produce allelochemicals \citep{Timbal1990, Dolling1996, Jaderlund1996}. Local variations in climate and soil \citep{Turbang1954} are additional factors that could influence regeneration development.
 
Within this context, the growth of natural oak and beech regeneration was monitored and modeled to test that:
\begin{itemize}
\item The less shade-tolerant oak grows faster in high-light environments than the shade-tolerant beech;
\item Regeneration growth is reduced by interspecific competition between saplings; 
\item Regeneration growth responds better to the availability of diffuse radiation than to the availability of direct radiation;
\item Regeneration growth varies from site to site and this ``site effect'' is correlated with the local climate and soil conditions.
\end{itemize}

\section{Methods}
\subsection{Study sites}
We selected 23 sites within the study area (\autoref{fig:21map} and \autoref{fig:A11}). The sites had well-established regeneration (\SIrange{10}{300}{\centi\metre} high) that spanned a wide range of stand structures and compositions in mixed stands of the Belgian Ardennes forests. In spring 2007, regeneration areas (\SIrange{100}{6500}{\square\metre}) were fenced off to protect them from browsing by wildlife. Inside each fence, we sampled the saplings within \SIrange{5}{31} square plots of \SI{4}{\square\metre} (total of 242 plots). The plots were laid out every \SI{4}{\metre} following a square grid (\autoref{fig:31plot}). Inside the fences and within \SI{20}{\metre} outside the fences, we measured and mapped every tree with a diameter at breast height (dbh) greater than \SI{13}{\centi\metre}.

\begin{figure}[htb]
\includegraphics[width=\textwidth]{fig31plot.pdf}
\caption[Plot layout in a site.]{Plot layout in a site. Experimental plots were laid out following North-South and West-East transects inside a 2-m tall fence.}
\label{fig:31plot}
\end{figure}

We performed a soil and humus description \citep{Jabiol1995} as well as floristic surveys (Braun-Blanquet method), in order to compute levels of the supply of nutrients and water at every site \citep{Bartoli2000}. Similarly to Ellenberg’s method, every species indicates a range of nutrient and water conditions, and site values correspond to a weighted average of the indexes of found species. The variation of soil richness between the sites appeared limited with index values ranging between 0.8 and 2 on a scale of 0 (very acid soils) to 6 (calcareous soil). The local variations of climate between sites were also assessed using a regional climate model and computing approximately 30 indexes of temperature, precipitation, and evapotranspiration \citep{Tychon2000}. They corresponded to averages, standard deviations, minimums, and maximums computed either per season or per year.

Studied stands had complex vertical and horizontal structures and included a wide range of dbh and height classes. Stand composition was also diverse, varying from monospecific beech stands to stands almost completely dominated by oak (\autoref{tab:32rege}). Other species (mainly \textit{Carpinus betulus} L., \textit{Betula pendula} Roth, \textit{Betula pubescens} Ehrh., \textit{Acer pseudoplatanus} L., and \textit{Acer pseudoplatanus} L.) accounted for up to \SI{20}{\percent} of stand composition. 
The percentage of above canopy light measured in the understory (PACL) ranged between \SI{1}{\percent} and \SI{61}{\percent} for plots dominated by oak, and between \SI{2}{\percent} and \SI{43}{\percent} for plots dominated by beech. This encompassed a wide gradient ranging from a close canopy to a canopy with a gap size varying up to approximately \SI{1200}{\square\metre} (area without overtopping crown).

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\begin{table}[p]
\caption[Main characteristics of the study sites.]{Main characteristics of the study sites. The number of sites and the number of plots per site ($n$) are given in the two first columns. The structure and composition of the study stands is shown by the minimum and maximum tree diameter (dbh), the average basal area and the average proportion of oak. Regeneration is characterized by the minimum and maximum height, age and total sapling density. The last column contains the minimum and maximum of the percentage of above canopy light (PACL). The table is sorted by average PACL.}
\footnotesize
\centering
\begin{tabularx}{\linewidth}{rrXrrrXrrXrrXrXr}
\toprule
  &   & & \multicolumn{ 3}{c}{\textsc{overstory}} & \multicolumn{1}{l}{} & \multicolumn{ 2}{c}{\textsc{oak saplings}} & \multicolumn{1}{l}{} & \multicolumn{ 2}{c}{\textsc{beech saplings}} & \multicolumn{1}{l}{} & \textsc{all saplings} & \multicolumn{1}{l}{} & \textsc{light} \\ 
\cmidrule{4-6} \cmidrule{8-9} \cmidrule{11-12} \cmidrule{14-14} \cmidrule{16-16} 
Site & $n$ & & dbh & basal area & oak basal area &  & height & age &  & height & age &  & density &  & PACL \\
& & & \si{\centi\metre} & \si{\square\metre\per\hectare} & \si{\percent} &   & \si{\centi\metre} & year &   & \si{\centi\metre} & year &   & \si{\per\square\metre} &   & \si{\percent} \\ 
\midrule
17 & 6 & & 9 - 74 & 25 & 80 &  & 26 - 39 & 4 - 8 &  & 49 - 127 & 4 - 12 &  & 10 – 36 &  & 6 - 10 \\ 
22 & 18 & & 12 - 78 & 16 & 74 &  & 105 - 209 & 7 - 12 &  &  &  &  & 6 – 13 &  & 3 - 14 \\ 
25 & 20 & & 13 - 81 & 22 & 0 &  &  &  &  & 28 - 248 & 5 - 16 &  & 3 – 110 &  & 2 - 25 \\ 
12 & 10 & & 13 - 81 & 22 & 68 &  & 16 - 171 & 5 - 12 &  & 39 - 220 & 5 - 10 &  & 12 – 39 &  & 5 - 23 \\ 
15 & 31 & & 6 - 72 & 23 & 80 &  & 18 - 58 & 3 - 6 &  & 29 - 145 & 5 - 13 &  & 3 – 25 &  & 1 - 26 \\ 
19 & 12 & & 8 - 84 & 15 & 0 &  &  &  &  & 176 - 252 & 9 - 14 &  & 4 – 52 &  & 2 - 13 \\ 
26 & 6 & & 7 - 80 & 14 & 6 &  &  &  &  & 133 - 265 & 7 - 12 &  & 4 – 14 &  & 2 - 43 \\ 
3 & 7 & & 7 - 66 & 18 & 17 &  & 23 - 80 & 3 - 6 &  & 42 - 210 & 4 - 21 &  & 21 – 94 &  & 10 - 19 \\ 
2 & 6 & & 6 - 74 & 17 & 25 &  & 73 - 186 & 6 - 17 &  & 155 - 245 & 10 - 15 &  & 5 – 19 &  & 8 -23 \\ 
10 & 9 & & 6 - 60 & 23 & 85 &  & 123 - 249 & 12 - 13 &  &  &  &  & 10 – 29 &  & 10 - 22 \\
29 & 15 & & 7 - 67 & 20 & 51 &  & 25 - 204 & 4 - 12 &  & 105 - 264 & 6 - 17 &  & 10 – 43 &  & 10 - 21 \\ 
8 & 11 & & 6 - 63 & 19 & 89 &  & 76 - 243 & 1 - 13 &  &  &  &  & 6 – 69 &  & 9 - 27 \\ 
20 & 5 & & 7 - 92 & 19 & 37 &  &  &  &  & 159 - 231 & 11 - 18 &  & 14 – 19 &  & 10 - 27 \\
4 & 7 & & 6 - 74 & 25 & 73 &  & 168 - 255 & 13 - 18 &  &  &  &  & 4 – 95 &  & 13 - 20 \\ 
14 & 7 & & 7 - 73 & 20 & 41 &  & 74 - 134 & 7 - 17 &  & 174 - 274 & 10 - 19 &  & 5 – 30 &  & 16 - 22 \\ 
24 & 11 & & 6 - 55 & 11 & 93 &  & 166 - 273 & 10 - 15 &  &  &  &  & 17 - 75 &  & 10 - 34 \\ 
28 & 18 & & 14 - 65 & 10 & 6 &  &  &  &  & 122 - 246 & 9 - 12 &  & 6 – 15 &  & 14 - 32 \\
23 & 15 & & 6 - 77 & 21 & 58 &  & 80 - 245 & 7 - 13 &  & 196 - 277 & 7 - 11 &  & 4 – 28 &  & 7 - 29 \\ 
13 & 2 & & 7 - 73 & 14 & 51 &  & 21 - 111 & 7 - 14 &  & 203 - 248 & 12 - 17 &  & 8 – 45 &  & 12 - 13 \\ 
1 & 3 & & 6 - 67 & 11 & 28 &  & 13 - 43 & 3 - 5 &  & 99 - 152 & 4 - 11 &  & 25 – 67 &  & 20 - 26 \\ 
18 & 5 & & 6 - 80 & 7 & 19 &  & 47 - 234 & 8 - 14 &  & 208 - 272 & 9 - 14 &  & 3 – 12 &  & 4 - 35 \\ 
11 & 9 & & 7 - 68 & 14 & 91 &  & 165 - 236 & 11 - 17 &  &  &  &  & 7 – 13 &  & 42 -45 \\ 
9 & 8 & & 7 - 50 & 11 & 95 &   & 175 - 240.3 & 12 - 13 &   &   &   &   & 3 – 53 &   & 14 - 61 \\ 
\bottomrule
\end{tabularx}
\label{tab:32rege}
\end{table}

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\subsection{Regeneration height growth}
In this study, we monitored sapling height growth between 2009 and 2011. In 2009 and 2011, we measured sapling height using a telescopic meter stick to the nearest centimeter of the 3 tallest saplings of oak and beech in each plot. Saplings measured in 2009 were not tagged and therefore the saplings measured in 2011 could have been different. The mean annual height increment in site $i$ and plot $j$ ($iH_{ijs}$) was computed for each species $s$ (\autoref{eq:31ih}). 

\begin{equation}
iH_{ijs} = \frac{\overline{H11}_{ijs} - \overline{H09}_{ijs}}{2}
\label{eq:31ih}
\end{equation}

where $\overline{H09}_{ijs}$ and $\overline{H11}_{ijs}$ are, respectively, the average height of the three tallest saplings of species $s$ in site $i$ and plot $j$ measured during year 2009 and 2011. $iH_{ijs}$ therefore corresponds to the annual height growth of oak or beech saplings two years after the installation of fences.
 
We counted the sapling in every \SI{4}{\square\metre} plot for every species in 2007 and 2012. In 2009, the sapling density was computed as the average of these two counts.
 
Up to five different species (beech, oak with mainly \textit{Carpinus betulus} L., \textit{Betula} sp., \textit{Acer pseudoplatanus} L. and \textit{Coryllus avellana} L.) were found to coexist in the plots. In order to take into account the effects of the potential competition between species, we identified the species dominance, i.e., whether $\overline{H11}_{ijs}$ was the greatest for the species $s$ in plot $ij$.

In 2011, we selected five representative saplings of oak and beech within each site, inside the fences but outside the \SI{4}{\square\metre} plots, to determine sapling age. Sapling age was determined by counting the number of bud scars and growth rings \citep{Collet1997}. The rings were counted on stem sections at a height of \SI{5}{\centi\metre} using a binocular microscope. The sections were carefully sanded beforehand with sandpaper with a grit designation of up to 2000.

\subsection{Understory light conditions}
To estimate light availability just above the saplings, hemispherical photographs were taken before sunrise during mid-summer 2010 or 2011 above the regeneration at the center of the plot. The photographs were used to compute three indexes of light availability for the whole growing season (from 1\textsuperscript{st} April to 31\textsuperscript{st} October): the percentage of total above canopy light (PACL), the percentage of diffuse above canopy light (DIFF) and the percentage of direct above canopy light (DIR). Photographs were thresholded with PiafPhotem software \citep{Adam2006} and light was calculated with GLA software \citep{Frazer1999}. Additional details are provided in \autoref{sec:Aphoto}.

We validated our PACL estimates by comparing them with measures of the Photosynthetically Active Radiation (PAR) carried out with sensors (Solem S.A., Palaiseau, France) in five sites during one day in July 2010. The relationship between the estimates and the measures was highly significant ($r=0.91$, $P<0.001$, $n=70$) with a slope not significantly different from 1.

\subsection{Statistical analyses}
We modeled the height growth of saplings ($iH$) in relation to the initial height ($H$), understory light (PACL) and species dominance ($SDOM$). Similarly to other studies \citep{Pacala1994, Kunstler2005, Stancioiu2006b, Petritan2007, Wagner2010}, height growth of saplings ($iH$) was modeled with a nonlinear saturated relationship with PACL, which meant that growth increased at an increasing rate with PACL up to a certain point (inflection point) and then saturated progressively (asymptote). We attempted to add additional explanatory variables such as regeneration age and sapling density. Nevertheless, adding these latter variables did not improve significantly the models (\autoref{sec:Amodeling}). We used mixed models to take into account that the observed growth depended upon a random site factor and a set of explanatory variables ($H$, PACL and $SDOM$). Site characteristics were not included in the model as fixed factors because we did not sample the sites across a gradient of environmental conditions. The sites were thus considered as random repetitions of the experiment within the Belgian Ardennes. The mixed modeling approach enabled to quantify and explain the between-site variation. Based on Akaike’s Information Criterion (AIC) and residual dispersal (\autoref{sec:Amodeling}), we selected the logistic model among others. For beech (\autoref{eq:3beechmodel}), it was a function of both the initial height ($H_{ij}$) and the percentage of above canopy light (PACL). For oak, the selected model (\autoref{eq:3oakmodel}) took additionally into account the species dominance ($SDOM$), i.e., whether or not oak saplings dominated saplings of other species.

\begin{equation}
iH_{ij} = (\alpha_j + b \cdot \sqrt{H_{ij}}) \cdot \frac{1}{1+\exp{(1 - \frac{\textnormal{PACL}_{ij}}{c})}} + \epsilon_{ij}   
\label{eq:3beechmodel}
\end{equation}

and

\begin{equation}
iH_{ij} = (\alpha_j + b \cdot \sqrt{H_{ij}}) \cdot \frac{1}{1+\exp{(1 - \frac{\textnormal{PACL}_{ij}}{c_{SDOM}})}} + \epsilon_{ij} 
\label{eq:3oakmodel}
\end{equation}

with 

\begin{align*}
\alpha_j & \sim dN (0, \theta_\alpha) \nonumber \\
\epsilon_{ij} & \sim dN (0, \theta_\epsilon) \nonumber 
\end{align*}

We further tested whether adding the percentage of transmitted direct radiation improved significantly models containing the percentage of transmitted diffuse radiation. We replaced PACL in the model with a linear combination of the percentage of above direct (DIR) and diffuse light (DIFF): $e \cdot \textnormal{DIR} + (1 – e) \cdot \textnormal{DIFF}$. The null hypothesis was that direct light would not influence height growth if the diffuse light was already taken into account, i.e., $e=0$. According to the ratio between diffuse and direct radiation measured by the meteorological institute of Belgium, the mean annual proportion of direct radiation, $e$, equals 0.46 above canopy.

Finally, we computed the best linear unbiased estimates of the random factor $\alpha$ (BLUP$_\alpha$) and tested the Pearson’s correlations between these estimates and indexes of nutrient and water supply as well as microclimatic indexes.

All of the statistical analyses were performed within the R environment \citep{R2013} with a significance level of 0.05. Nonlinear mixed models were adjusted with the nlme package \citep{Pinheiro2011} using the restricted maximum likelihood approach.

\section{Results}
\subsection{Regeneration characteristics}
Details of the studied regeneration for every site are given in \autoref{tab:32rege} and average characteristics are given in \autoref{tab:33regemean}. Regeneration density strongly varied across sites, from 0 to 110 saplings \si{\per\square\metre}. Species composition also varied across sites. The proportion of other admixed species was large in some plots (max. \SI{98}{\percent}), but exceeded \SI{50}{\percent} in only 28 plots.
 
On average, beech saplings had higher initial height and height increment than oak saplings. Harvested saplings of the two species were 11 years old in average, but the studied regenerations were clearly uneven-aged. Moreover, in plots with a mixture of beech and oak, oak saplings were on average 1.5 years younger (two-way mixed ANOVA, $F=5.8$, $P<0.001$).
  
Negative height increment occurred ($n=18$) because the three tallest measured saplings were not necessary the same for the two measurements. Saplings measured the first time were not tagged. Before the second measurement, some measured saplings could have died and be no longer among the three tallest saplings.

The Pearson’s correlations between PACL and the initial height of oak and beech saplings were respectively 0.387 ($P<0.001$) and 0.042 ($P=0.608$). At the beginning of the study, high oak regenerations were thus mostly found in plots with higher light levels.
 
Overtopped and overtopping oak regenerations occurred in 104 and 70 plots, respectively. Overtopped and overtopping beech regenerations occurred in 120 and 29 plots, respectively. In half of the plots with overtopped oak regeneration, oak saplings represented the major proportion of the total sapling count. By contrast, overtopped regeneration of beech involved only a few individuals.

\begin{table}[htb]
\caption[Main characteristics of the measured regenerations.]{Main characteristics of the measured regenerations. Average (minimum and maximum) sapling height, height increment and age for oak, beech and other species encountered in the plots. The last column indicates the number of subplots for which the species dominates the regeneration.}
\footnotesize
\begin{tabularx}{\textwidth}{lrRRRR}
\toprule
\textsc{species} & $n$ & \textsc{height in 2009} & \textsc{height increment} & \textsc{age in 2009} & \textsc{dominant rege.} \\ 
 &  & \si{\centi\metre} & \si{\centi\metre} & year & \\ 
\midrule
Beech & 149 & 153 (23;277) & 26 (-9;62) & 10 (4;21) & 120 \\ 
Oak & 174 & 125 (13;273) & 16 (-10;52) & 10 (1;19) & 70 \\ 
Other & 93 & 118 (7;285) & 20 (-66;84) & / & 51 \\ 
\bottomrule
\end{tabularx}
\label{tab:33regemean}
\end{table}

\subsection{Height growth models}
The modeling of height growth for the two species showed that beech saplings grew on average faster than oak saplings, whatever the light conditions (\autoref{fig:32model}). For the two species, as PACL increased, growth increased following a sigmoid curve and then reached a horizontal asymptote. This asymptote increased with sapling initial height. For instance, in high-light conditions, the height increment of small beech saplings ($H_{ij} = $ \SI{50}{\centi\metre}) was found to be approximately \SI{20}{\centi\metre}, whereas the height increment of taller beech saplings ($H_{ij} = $ \SI{300}{\centi\metre}) was up to \SI{50}{\centi\metre}. Between saplings of oak and beech of similar height, the beech saplings had a greater height increment than the oak saplings.

Moreover, beech saplings reached their asymptotic growth at lower light levels than oak saplings. Overtopping beeches, overtopping oaks, and overtopped oaks reached \SI{90}{\percent} of their asymptotic growth at, respectively, \SI{12}{\percent}, \SI{20}{\percent} and \SI{29}{\percent} of above canopy light. Indeed, the inflection point of the models, denoted by parameter $c$ in \autoref{eq:3beechmodel} and \autoref{eq:3oakmodel}, varied significantly between models (\autoref{tab:3model}).
 
The good dispersal of residuals indicated no evidence of model bias. However, residual scatterplots indicated a substantial residual variation (\autoref{sec:Amodeling}) that was greater in the model for beech ($\theta_\epsilon = $ \SI{10.5}{\centi\metre}) than for oak ($\theta_\epsilon = $ \SI{6.8}{\centi\metre}). The between-site variation, $\theta_\alpha$, was about \SI{7}{\centi\metre} in both models.

\begin{table}[htb]
\caption[Parameters of the model of regeneration growth.]{Parameter estimates (confidence intervals with $\alpha$ level of 0.05) of the selected models presented in \autoref{eq:3beechmodel} and \autoref{eq:3oakmodel}. $b$ is the parameters of the asymptotic height growth and $c$ is the inflection point. In the model for oak, there were two estimates for $c$ (denoted by $c_{SDOM}$ in \autoref{eq:3oakmodel}): one for overtopping regeneration and one for overtopped regeneration. $\theta_\alpha$ and $\theta_\epsilon$ are, respectively, the standard deviation associated to the random factor (between-site variation) and the residual error (within-site variation).}
\footnotesize
\begin{tabularx}{\textwidth}{lXXXrr}
\toprule
\textsc{species} & $b$ & $c$ & $c$ & $\theta_\alpha$ & $\theta_\epsilon$ \\ 
	 &  & (overtopping) & (overtopped) &  & \\ 
\midrule
Oak & 2.059 (1.642;2.476) & 6.058 (3.468;8.648) & 10.257 (6.469;14.046) & 7.255 & 6.759 \\ 
\addlinespace[2mm]
Beech & 2.431 (2.035;2.828) & 3.964 (2.378;5.550) & & 7.261 & 10.491 \\ 
\bottomrule
\end{tabularx}
\label{tab:3model}
\end{table}

\begin{figure}[htb]
\includegraphics[width=\textwidth]{fig32model.pdf}
\caption[Model of the height growth of oak and beech.]{Model of the height growth of oak and beech saplings in mixed forests in the Belgium Ardennes. Beech showed a higher growth rate under all light conditions. The presence of taller saplings of beech reduced the growth of the neighboring oak saplings (overtopped oak saplings).}
\label{fig:32model}
\end{figure}

\subsection{Diffuse and direct radiation}
We tested whether adding the percentage of transmitted direct radiation (DIR) into a model already including the percentage of transmitted diffuse radiation (DIFF) improved the prediction of height increment by replacing PACL in \autoref{eq:3beechmodel} and \autoref{eq:3oakmodel} with $e \cdot \textnormal{DIR} + (1 - e) \cdot \textnormal{DIFF}$. The null hypothesis was accepted for beech ($e=0.300$, $P=0.640$) which means that DIR did not influence significantly the height growth of beech regeneration. At the opposite, the null hypothesis was rejected for oak ($e=1.001$, $P=0.005$). This indicated that the height growth of oak responded mainly to DIR. The same conclusion would have been obtained by adjusting models including only the overtopping saplings. 

\subsection{Analyzing site effect} 
For the oak saplings, the ``site effect'', estimated using BLUP$\alpha$, was positively correlated with soil richness ($r=0.562$, $P=0.015$) and mean annual temperature ($r=0.638$, $P=0.004$). The site effect was negatively correlated with altitude ($r=-0.631$, $P=0.005$) and mean annual precipitation ($r=-0.689$, $P=0.002$). In addition, altitude was inversely correlated with mean temperature ($r=-0.522$, $P=0.026$) but not significantly with mean precipitation.

For beech saplings, the ``site effect'' was only positively correlated with soil richness ($r=0.553$, $P=0.001$). 

\section{Discussion} 
Our large \textit{in situ} sampling of beech and oak regenerations enabled us to adjust mixed nonlinear models of height growth according to initial height, light availability and species dominance (\autoref{eq:3beechmodel} and \autoref{eq:3oakmodel}). We further used the adjusted models to ascertain (\textit{i}) the height growth of the two studied species under different light conditions, (\textit{ii}) the effects of species dominance, (\textit{iii}) the influence of direct radiation, (\textit{iv}) the between-site variations, (\textit{v}) and silvicultural implications. 

\subsection{Height growth ranking}
The ecological theory of forest succession postulates that under high-light levels, light-demanding species outcompete shade-tolerant species. According to \citet{Kobe1995}, a high capacity to survive under low-light levels is offset by lower growth rate under high-light levels. Light demanding species would thus allocate preferentially resources to height growth and hence risk dying from light starvation \citep{Messier1999}.
 
Our findings indicated well that oak saplings have greater light requirements than beech saplings. We found that the beech and oak saplings needed more than \SI{10}{\percent} and \SI{20}{\percent} of above canopy light (PACL), respectively, to exhibit more than \SI{90}{\percent} of the maximum height growth (\autoref{fig:32model}). In accordance with previous studies \citep{Farque2001, Stancioiu2006b, Petritan2007, Petritan2009}, we found that under higher levels of PACL, any increase in PACL induced little variation in height growth (asymptote). Nevertheless, our findings indicated that oak saplings, the less shade-tolerant species, generally exhibited a lower height increment than beech saplings even under high-light conditions (PACL $>$ \SI{20}{\percent}, \autoref{fig:32model}). 

This observation is partly in contradiction with the definition of shade tolerance proposed by \citet{Kobe1995}, but similar observations have already been reported with other species \citep{Walter1996, Kunstler2005}. It highlights the strong ability of beech to survive in deep shade \citep{Wagner2010}, and lead us to infer that oak saplings have an insignificant chance of survival under beech saplings. The lower height growth of oak means, that in mixed clumps of oak and beech, oak saplings are naturally suppressed. Nevertheless, the height growth might be traded off with adaptations to low nutrient and water supply \citep{Kobe1995, Walter1996, Beaudet1998, Collet2001, Wagner2010} or herbaceous competition \citep{Walter1996, Coll2003, Coll2004}.

In this study, we focused on the variations of saplings height growth with light availability and we admit that the response of diameter growth and mortality rate could have led to different conclusions. Nevertheless, height growth has been shown to be a good proxy for sapling mortality \citep{Kobe1995, Walter1996, Kunstler2005, Petritan2007}.

Beech saplings were on average taller and older at the beginning of the experiment, which gave them a competitive advantage over oak saplings. Due to their shade-tolerant nature and the higher frequency of seed production by adult trees, beech saplings often pre-exist in the understory before the canopy opening \citep{Wagner2010}. They maintain large seedling populations in the understory while waiting for more suitable light conditions. Consequently, beech saplings are often established before oak saplings. However, the small age differences between the oak and beech saplings in this study are unlikely to be responsible for the differences in height growth. Indeed, we found that sapling age did not appear as a significant explanatory variable in height growth models. This result is in agreement with the findings of \citet{Collet2001} and partly in agreement with those of \citet{Emborg1998}.

Growth ranking might depend upon tree ontogeny and size \citep{Delagrange2004, Niinemet2006, Balandier2007}, and availability of nutrient and water \citep{Kobe1995, Walter1996}. Our results might therefore be limited to the studied ontogenetic stage characterized by saplings with a height of less than \SI{300}{\centi\metre} and within the conditions of the Belgian Ardennes.

Moreover, we ensured strict protection of the studied saplings from herbivorous browsing. As deer prefer browsing oak over beech \citep{Gill1992}, oak regeneration is more severely damaged than beech regeneration and the dominance of beech is intensified by ungulate pressure. 

\subsection{Competition}
Overtopping saplings of beech (and hornbeam) significantly reduced the height growth of overtopped oak saplings. This shifted leftward the inflection point from \SI{10}{\percent} to \SI{6}{\percent} of above canopy light. Such a reduction was expected because PACL was measured above the regeneration and overtopped species receive only the PACL that was not intercepted by overtopping regenerations. Overtopped beeches were not significantly affected. The levels of transmitted radiation in plots with overtopped beeches were probably too high (average PACL of \SI{15}{\percent}) to model the effect of interspecific competition on the growth of beech saplings. In such conditions (PACL $>$ \SI{10}{\percent}), a small decrease in transmitted radiation does not really affect beech growth (\autoref{tab:3model} and \autoref{fig:32model}). Moreover, diameter growth may be more affected by competition than height growth \citep{Collet2006, Prevosto2007}. 

\subsection{Direct radiation}
We tested whether the height growth of oak and beech saplings was sensitive to direct radiation. Our results suggested that stem elongation of oak saplings was promoted by direct radiation in contrast with the height growth of beech saplings. This result confirmed the higher light requirement of oak saplings. In addition, beech growth was sensitive to light change when PACL was less than \SI{10}{\percent}. In these conditions, gaps are generally very small and the periods with transmitted direct radiation are very short \citep{Chazdon1991}. This stresses the importance of using total PAR radiation to model the growth of less shade-tolerant species. \citet{Diaci2002} and \citet{Diaci2007} demonstrated the relative importance of diffuse and direct radiation on regeneration success of Norway spruce, beech and pedunculate oak in Slovenia. In particular, they reported the successful development of beech under diffuse canopy openings, something that is in accordance with our results. On the other hand, sites with higher levels of direct radiation might also be drier \citep{Diaci2007}. We studied solely the effect of light supply, while, in another study, changes in light conditions were shown to induce changes in microclimatic and soil variables \citep{Aussenac2000}.
 
\subsection{Between-site variation}
In this study, we showed that between-site variations was substantial ($\theta_\alpha \approx$ \SI{7.2}{\centi\metre}) and in the same order of magnitude that the within-site error ($\theta_\epsilon =$ \SI{6.7}{\centi\metre} and \SI{10.5}{\centi\metre} for oak and beech, respectively). We attempted to explain this variation with both biotic and abiotic characteristics of the sites. 

Soil richness was the only significantly correlated variable for both species. Both species are known to require similar levels of nutrient availability \citep{Piedallu2009} and the investigated range of soil richness was small between the sites and it was computed from a floristic survey. These variations might hence denote complex interactions between canopy composition, canopy closure and ground vegetation.
 
In contrast with beech saplings, the height growth of oak saplings responded positively to mean annual temperature and negatively to mean annual precipitation. Oak has higher mean annual temperature requirements and its optimum temperature is about \SI{11}{\degreeCelsius} in Western Europe \citep{Piedallu2009}. Rainfall is abundant throughout the study area and should not affect negatively sapling growth in the absence of soil waterlogging. However, abundant rainfall implies greater cloud cover and precipitation that are negatively correlated with temperature. Sites with abundant rainfall could then be characterized by higher cloud cover and lower temperatures, which might negatively affect oak growth within the study area. Taken together, these climatic considerations, underlined the sub-montane trend of the climate of the Ardennes that is less convenient for oak than for beech regeneration, even in high-light environments. Possible additional explanations for within- and between-site variations may lie in genetic variation, incidence of disease, canopy history, unmeasured soil and microclimatic conditions.

\section{Conclusion}
We sampled \textit{in situ} advanced regenerations within the whole range of conditions encountered in beech and oak forests managed with a continuous-cover forestry system in the Belgian Ardennes. Transmittance ranged from \SI{1}{\percent} to \SI{60}{\percent} under heterogeneous canopies. Transmittance below \SI{3}{\percent} is typical under closed canopy \citep{Emborg1998}, whereas, after canopy release, the level of transmittance might increase up to \SI{15}{\percent} \citep{Collet2001}, \SI{30}{\percent} \citep{Pacala1994} or even beyond \SI{60}{\percent} according to our data. We observed that beech saplings naturally outgrow oak saplings. Beech saplings reached an optimum growth at \SI{10}{\percent} of above-canopy light whereas oak saplings needed twice as much light. In addition, our sapling age analysis highlighted that beech saplings were usually established before oak saplings. After canopy opening, these pre-existing beeches derived greater benefit from the increase of light availability than younger and smaller oaks. In these conditions, oaks are rapidly suppressed and mixed stands of oak and beech evolve naturally toward pure beech stands. The two species have mainly coexisted because beech naturally regenerates under well-established oak stands where oak has previously been favored by selective thinning, coppicing and plantations \citep{Claessens2010}. The reverse situation is unlikely to occur naturally without frequent disturbances. 




