IntroductionTop

Competition among individual trees is a fundamental ecological process that plays a major role in population dynamics, survival, growth and species replacement (Peet & Christensen, 1987). By definition, competition is ‘‘an interaction between the individuals, leading to a reduction in the survival, growth and reproduction of the competing individuals” (Begon et al., 1996). Several case studies have been conducted in ecology and forestry to develop, improve or modify different competition indices (CIs). Such indices quantify the competition level for an individual tree and are classified into two major groups of non-spatially explicit indices (e.g. Biging & Dobbertin (1995) and Schröder & Gadow (1999)) and spatially explicit indices (e.g. Hegyi (1974) and Alemdag (1978)). Non-spatial indices are functions of stand level variables, or of the initial dimensions of the trees, and therefore do not require the trees coordinates. Whenever spatial indices are used to measure the influence of local neighbours on a central tree (the subject tree), the dimensions and the relative location of neighbour trees are required for the computation (Tomé & Burkhart, 1989; Corral Rivas et al., 2005).

For several species and forest conditions, the effectiveness of different CIs on tree diameter or basal area growth has been examined (Munro, 1974; Martin & Ek, 1984; Pukkala & Kolström, 1987; Holmes & Reed, 1991; Contreras et al., 2011). Since several aspects of stand density and neighbour sizes influence the tree growth, non-spatial CIs with simple structures are parsimonious to quantify the competitive status of trees in each stand. On the other hand, as ecology is spatial (Berger & Hildenbrandt, 2000), therefore by increasing the interval distance, the negative interaction of neighbours will decrease and spatial CIs take the explicit description of tree spacing into account. Additionally the identity of neighbouring species is an important factor in the characterization of their competitive effect (Bella, 1971; Zhao et al., 2006). Competition can occur among conspecific individuals, plants of same species, and hetero-specific individuals, plants of different species, termed intraspecific and interspecific competition, respectively. The competition behaviour of different species can be differentiated by using Ellenberg et al.’s system (1991). It is the most widely used indicator species system, which compares the response of different species to edaphic and climatic parameters, such as light, temperature, moisture and nitrogen at a 9-point scale for each.

To investigate the effect of competition on the diameter growth of trees, we focused our study on silver birch (Betula pendula Roth). Silver birch occurs naturallyin northern temperate and boreal forests and it is an essential ecological and commercial broadleaved tree species (Hynynen et al., 2010). As a pioneer tree species (Fischer et al., 2002), birch is light demanding, and if it grows as a dominant tree with low competitive effects of neighbours, in a stand with relatively wide spacing, birch maintains its vitality and vigorous growth (Hynynen et al., 2010). In Estonia, birch is the second most abundant tree species in terms of forest cover (31.2%) and the coverage is expanding (Yearbook of Forest, 2013). A few attempts have been made to study birch growth related to the negative interaction of tree competitive status in stands (Jõgiste, 1998; Prévosto et al., 1999; Andreassen & Tomter, 2003; Damgaard & Weiner, 2008; Kaitaniemi & Lintunen, 2010).

The main objective of this study was to investigate the adequacy of different spatial and non-spatial CIs to explain single-tree silver birch diameter growth in Estonia. Further objectives were to find the best competitor selection method for Estonian birch stands and evaluate the differences in competitive ability of different species by employing Ellenberg’s species-specific light indicator values. Specifically, we hypothesized that (i) the CIs contribute to explain diameter increments in silver birch; (ii) spatial indices perform better than non-spatial indices; (iii) selecting the potential competitors based on the concept of the influence zone is superior to variable competition zone radii; and (iv) considering the species-specific competition improves the ability of spatial indices to account for growth variability.

Materials and methodsTop

Competition indices

The competition for each subject tree was quantified using 18 different CIs, consisting of 7 non-spatial and 11 spatial indices (Table 2). The indices described below were selected from the literature, taking into consideration the available tree variables for this study, and their simplicity to describe the competition situation of a tree, as it is difficult to understand the statistical qualities of an index with the combination of several primary variables (Weiglet & Jolliffe, 2003).

Table 2. The list of competition indices tested to use in the tree diameter growth model*

The first seven indices in Table 2 are non-spatial indices. In a plot, BA-g_jCI proposed by Steneker & Jarvis (1963), is the sum of the basal area (g) of the neighbouring trees j for a subject tree i (m² ha^-1); BAL presented by Wykoff et al. (1982) is the sum of the basal area of trees larger than the subject tree (m² ha^-1). Sdr sums up the d of neighbours divided by the subject tree d in the plot (ha^-1). The index dr_g calculates the ratio of the diameter of the subject tree to the quadratic mean diameter of the plot (Hamilton, 1986) and BAr is another form of Sdr that considers g instead d. The index BALr is the ratio of BAL to the cumulative basal area of the plot (Vanclay, 1991) and finally BALMOD (Schröder & Gadow, 1999) modifies BALr by dividing it into the relative spacing index as following:

where S is plot area (m²), N is the number of trees on plot, and H_Dom is the stand dominant height (m) (mean height of hundred thickest trees per hectare (Assmann, 1970)).

The next three competition indices Sl, SOr and SOdr in Table 2 are so-called influence-zone overlap indices, which assume that a horizontal circle surrounding the subject tree can represent the active competition area, and that competition occurs where neighbouring trees overlap their influence zone with the subject tree’s influence zone. The radius of these circles is thought to be equal to the expected growing space of open-grown trees, and usually is a function of tree size (Corral Rivas et al., 2005).

Finally, the last eight indices in Table 2 are size-ratio spatial CIs. The idea of this type of indices was derived from the hypothesis that competition effect has positive relationship with the size of neighbouring trees and negative relationship with their distance from the subject tree (Tomé & Burkhart, 1989). For spatial indices of Heg (Hegyi, 1974), Almdg (Alemdag, 1978), Sdrl1(Lorimer, 1983), Sdrl2 (Martin & Ek, 1984), and SBAr (Daniels et al., 1986) the diameter at breast height performs as a tree size indicator. SAng1 (Lin, 1974)is the sum of horizontal angles. Since the average elevation angle of the brightest region of the sky over the growth season can be approximated by angle of 45° (Stadt & Lieffers, 2000), the 45° gauge was employed for this index. The index SAng2 sums up the horizontal angles originating from the subject tree centre and spanning the diameter of each competitor (Rouvinen & Kuuluvainen, 1977), and SdrAng calculates the sum of the horizontal angles multiplied by the ratios of the diameter of the competitors and the subject trees (Fig. 2).

Figure 2. Schematic of the horizontal angles originating from the subject tree centre and spanning the diameter (at breast height) of each competitor tree within the competition zone used to calculate indices Sang2 & SdrAng; *dx is the diameter at breast height (cm), lx is the distance between the subject tree and its competitors (m).

Methods of competitor identification (SM)Top

As well as the mathematical formulation, the value of a competition index depends on the method used to define competitors for the subject tree (Bigging & Dobbertin, 1992). Among different proposed methods to choose the potential competitors, we tested four approaches. The first two approaches, approaches 1 and 2, were based on the concept of an influence-zone that assumes an imaginary circle whose centre is constituted by the subject tree (Staebler, 1951) and trees inside this circle are competitors. The last two approaches, approaches 3 and 4, identified competitors based on variable competition zone radii, often weighted by dimensions of the subject tree and its neighbours (Daniels, 1976; Ford & Diggle, 1981):

1. The radius of influence zone was defined as a fraction of the stand’s average height for each plot; CZR_0.4h was set equal to 0.4 average height of plot (Sims et al., 2009).
2. Based on Lee & Gadow (1997) the influence zone radius was calculated using the following equations:
   
   
   
   where CZR_k is dynamic radius, N is the number of trees per hectare, and k is a constant number.
   The function calculates average distance between the neighbours. The values of k equal to two (CZR_k2) and three (CZR_k3) multiply this distance by two and three, respectively to define CZR. Within the influence zone, trees were considered to be active competitors if d_{j ≥ 0.3}d_i (where d_j is d of competitor and d_i is the d of subject tree) and they were beyond the crown projection of other competing trees, considering a competition elimination angle of 30˚ (CEA=30˚).
3. The Bitterlich method (1952) was used to identify the competitors in variable plot radii samplings. BAF₁, BAF₂ and BAF₄tested three basal area factors (BAF) equal to 1, 2 and 4 m² ha^-1, respectively. A tree was considered a competitor if its distance to the subject tree was:
   
   
   where l_ij is the distance between the subject tree i and the neighbouring tree j and d_iis the diameter of the subject tree. The values of BAF equal to 1, 2 and 4 correspond to the opening angles of β =1.15˚, 1.62˚ and 2.30˚, respectively. Therefore, when the BAF values and boundary angles increase, fewer trees meet the criteria for being considered as competitors (Lorimer, 1983; Tomé & Burkhart, 1989).
4. Finally, the reserved search-conemethod (Pretzsch, 2009) or angular height method (Richards et al., 2008) applied height angle from the base of the subject tree to identify the competing neighbours. For a search-cone opening angle β,set up at the stem base of the subject tree, competitors are neighbouring trees whose heights are greater than a critical distance, determined as the following:
   
   
   
   where l_ij is the distance between the subject tree and the competitor tree, and h_i is the subject tree height. If the apex of the reversed search-cone is at the crown base height of the subject tree (cbh_j) then a neighbouring tree with height h_j is a competitor when:
   
   
   
   We tested the opening angle β equal to 100˚, 80˚, and 60˚, respectively where the apex was set up either at the stem base (SCH₁₀₀, SCH₈₀ and SCH₆₀) or at thecrown base height (SCHCr₁₀₀, SCHCr₈₀ and SCHCr₆₀).

In all the above-mentioned methods, in order to avoid the interference from the competitive effects of non-measured trees beyond the plot borders, we computed CIs only for interior trees on each plot where the neighbours’ information was available for them. After determining the competitors, we calculated the spatially explicit CIs for each subject tree. Four spatially explicit CIs (Sl, SOr, SOdr, and Sdrl1) were based on the influence zone concept and only the first two approaches of competitor selection (CZR_0.4h and CZR_k) were applicable to quantify the mentioned indices. Moreover, the allometric crown radius model, (developed by Lang et al., 2007) was used to calculate the crown radius that was required for quantifying indices SOr and SOdr as well as fitting the Eqs. (7) and (8):

where R_cr is the crown radius (m), d and h are the diameter at breast height (cm) and the total height of the tree (m) respectively, and a₁ and a₂are estimation parameters (Table 3).

Table 3. The values of parameters for the used allometric crown model for main tree species

Statistical and comparative analyses of competition indices

Preliminary analysis was carried out to pre-select adequate CI candidates to include in our growth model. As suggested by Pedersen et al. (2013) we applied the Spearman rank correlation (Spearman’s rho) to characterize the relationship between the 5-year tree diameter increment (i_d5) and the competition indices. The Spearman correlation is able to consider potential nonlinear trends frequently seen in growth and competition studies, besides it is valid for the data size larger than 10 (Siegel, 1956) which was applicable to our data. The existence of a pairwise relationship between i_d5 and CIs was proved using the t-test. Based on the Spearman rank correlation results, the four best CIs (two non-spatial and two spatial CIs) were selected for further analyses.

Then, we constructed a linear multiple regression model (Wimberly & Bare, 1996; Jõgiste, 2010) between i_d5 (cm) and some predictor variables that influence diameter growth. In a preliminary assessment, non-linear extra sum of square method (Bates & Watts, 1988) was applied to evaluate the effect of plots on growth. For this purpose, we considered the simple model of diameter growth as a function of tree diameter. In order to differentiate the study plots, we introduced dummy variables to the defined simple model. Then, we compared the two mentioned models using F-test and a significant effect of plots was detected (F=8.56; P<0.0001). Therefore, predictor variables presenting the initial stand status were also included in the growth models (Eqs. (7) and (8)). Additionally, for each combination of selected variables, the variance inflation factors (VIF) were calculated to certify that our multiple models were not influenced by multicollinearity amongst explanatory variables. We only implemented the combination of variables with VIFs<10 (Soares & Tomé, 2001; Corral Rivas et al., 2005). Eventually, the growth model was fitted by improvising some initial stand variables along with the tree variables.

In order to evaluate the efficiency of the chosen CIs to improve the prediction ability of growth function, the numerical value of each of those indices, two non-spatial CIs and two spatial CIs, was independently added to the previous growth function:

where b_k are coefficients to be estimated, i_d5is the 5-year tree diameter increment (cm), d is the subject tree diameter at breast height (cm) that integrates the past competitive interactions (Soares & Tomé, 1999), cr is the ratio between the crown width and the tree height that depicts the vigour of trees of similar size (Schröder et al., 2002). The relative diameter dr is the ratio between the subject tree diameter and the quadratic mean diameter of the stand that represents the dominance of the subject tree in relation to other trees in the stand, RS is the stand relative spacing,and SI₁₀₀ is the stand site index. Nilson (2005) model was used to estimate the average height of the stands at reference age 100 years (m) and CI is the competition measure for the subject tree. R statistical software version 3.1.2 (R Development Core Team, 2014) was employed to carry out all the required analyses for this research.

Before proceeding with the subsequent analyses, the existence of any correlation among residuals was explored. For this purpose, the growth model was fit using the lme function from the nlme package in R as following:

For the recent linear mixed effect model, i_d5 is the dependant variable; b is a vector of fixed effects consisting of the same explanatory variables of Eq. (7); u is a vector of random effects including tree, plot, and growth interval (measurements); e is a vector of random errors; X and Z are design matrices relating the 5-year diameter growth to fixed and effect random effects, respectively. The previous and recent models were compared in terms of AIC (Akaike’s Information Criterion) where ΔAIC = AIC _{multiple model}-AIC _{mixed model}. In addition, to ensure that there was not any remaining within-group correlation, the recent model was checked with an auto-regressive structure (AR1). The mixed effect models, those with and without auto-regressive structures, were compared using ANOVA (analysis of variances).

The relative quality of growth functions, with and without CIs, were estimated using R² (Adjusted-R²), the root mean square error (RMSE, calculated using the rmse function for the model residuals in R), AIC and Akaike weights (AIC_w). The probability that model is the best with the lowest expected information loss is illustrated by the smallest value of AIC and the biggest AIC_w (Wagenmakers & Farrell, 2004). Additionally, the performance and the contribution of each CI to the growth model were assessed with the mean square error reduction (MSER).

where MSE₇ and MSE₈ are the mean square errors of models 7 and 8, respectively.

Finally, the efficiency of CIs in different stand stages and the contribution of different species in the competition load of a subject tree were evaluated. Stand development stages were defined by the age of the silver birch, as the dominant tree species. First, to differentiate the effect of different neighbouring species on competition, tree diameters were weighted differently. For that purpose, tree diameters were multiplied by their corresponding Ellenberg’s species-specific light transmission coefficients (Ellenberg et al., 1991) from one (plants in deep shade) to nine (plants in full light); then, the selected spatial CIswere recalculated using the new weighted diameters. Table 4 provides the Ellenberg’s light values for more frequent tree species in Estonia. Finally, subject trees were divided into three subdivisions of young (<35 years), middle-aged (35-69 years) and old stands (≥70 years). For each age group, the regression analyses for the selected CIs and tree diameter growth were repeated separately.

Table 4. The Ellenberg’s species-specific light coefficients for more frequent tree species in Estonia

ResultsTop

In Table 5 the Spearman’s rank correlation coefficients between the tree diameter growth and non-spatial CIs and also the combination of spatially explicit CIs and the competitor selection methods are presented. Table 5 shows that the competitor selecting approaches significantly affect the growth prediction ability of spatial indices and the competition selection methods of CZR_k3, CZR_0.4h and SCH₆₀ demonstrated greater values of Spearman’s rho, respectively. Among different spatial indices in these three neighbours selecting methods, SdrAng and Heg were well correlated with diameter increment. However, the values of rho for Heg were slightly lower than SdrAng. None of the alternatives of Bitterlich method (BAF₁, BAF₂ and BAF₄) showed to be an appropriate selection method of competitors. Furthermore, BAL and BALMOD as the best non-spatial CIsdid notperform better than the superior spatial indices SdrAngand Heg.The results presented in Table 5 are based on the analyses of 2,742 subject trees that were presented in different neighbours’ selection methods and 18 different non-spatial and spatial CIs are quantified and available for them.

Table 5. The Spearman’s rank correlation coefficients between the 5-year tree diameter increment and competition indices for a sample of 2,742 subject trees presenting in different neighbours’ selection methods

The comparison of the linear mixed effect models and the linear multiple models detected the improvement in linear mixed effect regressions in terms of AIC, but the ANOVA comparison between the mixed effect models, with and without an auto-regressive structure, did not show significant remaining within-group correlation (P-value>0.05). Subsequently, the growth model was fit into linear mixed effect regression (Eq. (9)) with no auto-regressive structure for further analyses in this study. All explanatory variables used for Eq. (7) were considered as mixed effects and proved significant (P-value<0.05), also VIF indicated no problem with multiclollinearity, all values being less than eight. In order to test the contribution of selected CIs, they were devised into the recent growth model.

Table 6 illustrates the statistical measures of mixed effect models, including R², RMSE, MSER, AIC, AIC_w, and also DAIC. The indices comparisons were done for different sample size of subject trees based on each neighbour selection method. Generally, the contributions of CIs were significant but not very large in magnitude, and among the CIs added to the model, SdrAng presented the most significant contribution, no matter which competitor selection method was used. After that, Heg was found to be important in efficiency to improve the growth model. Non-spatial CIs of BAL and BALMOD showed less contributions to the growth model than the spatial indices, except for SCH₆₀ where BAL appeared slightly better than Heg CI.

Table 6. Contribution of the competition indices to tree diameter growth model

The results of analyses for different age groups (Table 7) demonstrated that competition had stronger prediction ability in younger stands, and spatially CIs proved to be better than non-spatial ones. As shown in Tables (6) and (7), the Ellenberg’s light values performed a slight improvement for some models in order to describe the species-specific effect. The profiles of the R² and AIC did not show considerable variation between the two methods of calculating selected spatial CIs, with and without Ellenberg’s values (Table 6.) However, statistical measures for young stands viewed a slight improvement for including species-specific values in competition quantifications (Table 7).

Table 7. Contribution of the competition indices to tree diameter growth in different age groups

DiscussionTop

Computing the correlation coefficient of tree growth, and determining the efficiency of CIs when added to a tree growth model, have been widely used (Burkhart & Tomé, 2012). In the current study, adding the CIs to the growth model slightly improved the model, which can be partially due to the inclusion of relative dimensions of the trees in model. Relative dimensions measure the hierarchical position of the subject tree within the stand, and indirectly indicate the competitive status of the trees (Burkhart & Tomé, 2012).

Results from comparing different CIs proposed that spatial CIs of SdrAng and Heg were the best indices suitable to quantify the competition status of birch trees, respectively. Several studies (Castagneri et al., 2008; Contreras et al., 2011) have reported that SdrAng can describe a greater proportion of the investigated variation in growth models. Also, Heg demonstrated superior performance to non-spatial CIs in many studies (Alemdag, 1978; Pukkala & Kolström, 1987; Holmes & Reed, 1991; Mailly et al., 2003). The indices of SdrAng and Heg assign greater weight to the closer and bigger competitor trees (Wimberly & Bare, 1996) and it was following along the Cole & Lorimer (1994) hypothesis that noticeable competitive stress occurs by immediate competitors surrounding the subject tree crown.

The results we obtained for non-spatial CIs showed that BAL and BALMOD improved the predictive ability of Eq. (9), although in a smaller amount than when using the CIs of SdrAng or Heg. Some studies including BAL or BALMOD found an improvement (large or modest) in model performance (e.g. Biging & Dobbertin, 1995; Corral Rivas et al., 2005). However, similar to our study, several other studies suggested that spatial measures provided more precise growth prediction (Boivin et al., 2010; Contreras et al., 2011), and to the contrary, many studies did not report any superiority of spatial indices to non-spatial ones (Soares & Tomé, 1999; Stadt et al., 2007; Roberts & Harrington, 2008). The superiority of size-ratio CIs of SdrAng and Heg that used the d as indicator of size was probably because of the actual correlation between the subject tree’s diameter increment and its d (Holmes & Reed, 1991); however, the strength of competitive stress explained by such correlations might be unclear (Brand & Magnussen, 1988; Larocque, 2002).

While non-spatial CIs are simple functions of the stand or a tree’s dimensions, the selection of the neighbours that affect the growth of a subject tree is of crucial importance when calculating spatial indices. Concerning the competitor selection methods, the best results were acquired with those based on the influence zone and competition elimination angle concepts. Several studies showed that the competition status of a tree could potentially vary depending on the radius of influence zone (Pukkala & Kolström, 1987; He & Duncan, 2000; Nanami et al., 2005). The CZR_k was a multiple of average distance between the trees in the plot and highly affected by stand density. In our study plots, considering k equal to three and CEA equal to 30° proved to be a good fit to select the adequate number of active competitors. Although some studies (e.g. Alvarez et al. 2003) found better results using a different angle gauge, the angle gauge of 30° provided satisfactory results in some other studies (e.g. Lee & Gadow, 1997; Corral Rivas et al., 2005; Zhang et al., 2009).

The next superior competition selection approach, CZR_0.4h, was simple in practice and in accordance with studies showing that, the optimal influence zone radius strongly depended on the tree’s initial dimensions (D’Amato & Puettmann, 2004; Sims et al., 2009). Considering the third suitable competitor selection method, similar to several other studies, the opening angle of 50°-60° performed well (Biging & Dobbertin, 1995; Pretzsch, 2009; Oheimb et al., 2011) where bigger angles (80° and 100° in this study) mainly decreased the merit of the search-cone method used to detect the competitors (Richards et al., 2008). In contrast to the CZR_k3, the two methods of CZR_0.4h and SCH₆₀ gave more weight to tree height than distance, and since in our study, there was a lack of height measures for all trees, the selecting system of CZR_k3was preferable to identify competitors for central trees.

Despite the fact that the identity of neighbouring species is an important factor in the characterization of their competitive effects (Bella, 1971; Zhao et al., 2006; Kaitaniemi & Lintunen, 2010; Bošelá et al., 2013), no significant improvement appeared in recalculating the selected indices using Ellenberg’s light values except for young trees. One possible explanation is that in our study plots about two-thirds of the trees analysed were birch with the same light factors. Consequently, giving weight to different species did not significantly change the values of measured competition. In addition, interspecific competition mainly caused by Norway spruce appeared inferior due to differences in temporal growth patterns and shade-tolerance (Tahvanainen & Forss, 2008; Hynynen et al., 2011). Furthermore, the influence of competition on diameter growth was not strongly impacted by the number of species in the local neighbourhood as suggested by Oheimb et al. (2011). The slight improvement in young stands might be due to the nature of Ellenberg’s light values that refer to the preferences of the early stage of the tree life cycle. During the early stage, when light-demanding birch trees rapidly occupy regeneration areas, Norway spruce tends to appear more shade-tolerant; consequently, weighting them differently in competition measures is justified. Moreover the performance of CIs changed slightly by stand development. In young stands spatial indices performed better while in older stands, non-spatial indices showed superior results. In the early stage, pioneer birches grow quite fast and vigorously (Hynynen et al., 2011), and spatial CIs explain competition effects better in dense young stands, since they account for the short distances between the neighbouring trees, that are competing for resources. In older stands, due to mortality induced by different factors, including competition (Sims et al., 2009), the number of trees decline and non-spatial CIs are adequate for competition studies.

The overall results of this study provided a better understanding of competition in birch stands. Although spatial CIs performed better than non-spatial CIs, the reported differences between the spatial and non-spatial indices are relatively small. The spatial indices require tree attributes and locations, and the recording of such information is expensive and time consuming. Therefore, we suggest applying the spatial indices only when studying the competition in the natural development of young stands, where the stands are usually dense, because these types of indices give more weight to trees that are closer to the subject tree. In the middle-aged and old managed stands, an efficient measure of competition is possible by employing the non-spatial indices that do not require as many field measurements.