Competition Indices and Their Relationship With Basal Area Increment of Araucaria

Models that report the effect of competition are important for forest management since forests with higher levels of competition have lower increment rates, and their use is necessary to plan forest interventions. Thus, this study aimed to assess the effect of competition in the basal area increment of individual trees of Araucaria angustifolia (Bertol.) Kuntze in a natural forest. A total of 397 subject trees were measured, covering the diametric range. The dendrometric and morphometric characteristics of subject trees and their competitors were obtained, and 22 distance-dependent and distance-independent competition indices were calculated, in addition to increment cores extracted radially from the trunk at diameter at breast height. The relationship between models of periodic annual increment in basal area based on competition indices has allowed to obtain R values of 0.425 and Syx% ≥ 50.2. The multivariate technique of principal component analysis has shown that three principal components explain 78.43% of total variation. The first component was responsible for explaining 52.95%, with similar eigenvector for 11 competition indices, evidencing that these models can be used to describe especies competition, although they show different variables and mathematical equations in calculations. Results show the importance of competition to predict increment of Araucaria in individual trees.


Introduction
One of the most important facts to simulate tree growth is to assess the effects of competition, i.e., the influence of characteristics of the population where the tree is found.These characteristics can be measured through competition indices, which are algebraic relationships used to quantify the effect of higher or lower resource availability for a subject tree in relation to other competitor trees.The use of these competition indices has become an important tool for forest management worldwide (Pedersen et al., 2013).
It is possible to find in the literature competition indices with various complex ways of calculation, mainly the ones described by Gerrard (1969), Bella (1971), Arney (1973), Hegyi (1974), Ek and Monserud (1974), Glover and Hool (1979), Lorimer (1983), Corona and Ferrara (1989), Mugasha (1989), Rouvinen and Kuuluvainen (1997), Castagneri et al. (2008), among others.These competition indices can be divided into two classes: (a) distance-independent indices that use non spatial measurements, based on tree size distribution in a given area; (b) distance-dependent indices, in which competitors are identified based on their size and the distance in relation to the subject tree (Wimberly & Bare, 1996).
analysis reduces the data to be analysed, especially when they consist of a great number of inter-related variables (Jobson, 1992).With this analysis, the first components responsible for most variation of original data are determined, and they can be used to summarize data with little loss of information (Everitt & Dunn, 2010).In summary, it is necessary to calculate the variance-covariance matrix, or correlation matrix, finding eigenvalues and eigenvectors, in order to determine the principal components, and, finally, write the linear combinations, which will be the new variables, or principal components, being each principal component a linear combination of all original variables, mutually independent, and they will retain, in order of estimation, most variation from the initial data (Gotelli & Ellison, 2011).
In view of the above, the present study aimed to assess the effect of competition on the periodic annual increment in basal area of individual trees of Araucaria angustifolia (Bertol.)Kuntze in a natural forest, as a way to help in activities of silvicultural interventions of this species.The specific objectives were: (a) characterize tree growth and its relationship to distance-dependent and distance-independent competition indices; (b) adjust statistical models to describe the periodic annual increment in basal area according to competition indices; and (c) apply the multivariate technique of principal component analysis (PCA) in competition indices, aiming to interpret the influence of each index on the composition of data variation.

Material and Methods
Trees growing under competition in a natural forest were sampled in the municipalities of Lages (SC) and São Francisco de Paula (RS).The climate in these municipalities is humid subtropical according to Köppen classification, without a dry season and with temperate summer (Cfb) (Alvares et al., 2013).Location and climate characteristics of the studied places are indicated in Table 1.Note.MAT = Mean Annual Temperature, in °C; MAP = Mean Annual Precipitation, in mm.
In Lages, the study was carried out in a private rural property with 83.5 hectares of araucaria forest, 30km from the municipality seat, in the Central Plateau of Santa Catarina.In the forest, 28 species were identified according to the survey for the Forest Management Plan, in 1999 (Table 2), in which is presented that 10 species express 82.9% of the total importance value index (IV), whereas other 18 species represent 17.1%.In São Francisco de Paula the study was performed in São Francisco de Paula National Forest (FLONA-SFP), 27 km from the municipality seat, in the Campos de Cima da Serra region, northeast Rio Grande do Sul.In the araucaria forest, trees belonging to the plots (1546; 1537; 1539; 1543 and 1545) of the Brazilian Long-Term Ecological Research (LTER) were sampled.A total of 74 species were identified in the studied fields, 10 species had higher importance value index (IV), 71.4% from the total, and the other 64 species represented 28.6% (Table 2).Note.N = number of species; RD = relative density, in %; RDo = relative dominance, in %; RF = relative frequency, in %; IV = importance value index, in %; NI = no identify.
A total of 397 subject trees of Araucaria angustifolia (Bertol.)Kuntze distributed in diameter classes were intentionally selected within the natural forest, aiming to obtain sample trees in the whole diametric range, taking into account the smallest class core of 10cm, and intervals of 10cm between diameter classes.The intentional selection of trees is justified by the need to assess the inherent characteristics of each individual.Araucaria trees in competition were named 'subject tree', and the neighbouring trees were named 'competitor trees'.From the 397 subject trees, 308 (77.6%) were measured in Lages, SC; the other 89 trees (22.4%) were measured at FLONA in São Francisco de Paula, RS.
The criteria to select competitor trees in relation to the subject tree was based on crown dimensions, and consequently tree height assessed regarding capacity to compete for light and growth space.The neighbouring trees with contact between crowns in a 360° turn from the subject tree were considered competitors.When the competitor tree was taller or shorter than the subject tree, with no contact between crowns, it was not selected, since the aim was to select trees with effective competition.Due to the high dominance of Araucaria angustifolia (Bertol.)Kuntze in the studied sites, subject trees were influenced by intraspecific competition (Table 2).Thus, when considering contact between crowns, the selection criteria of competitor trees may be represented by the Equation (1): Where, dist ij = horizontal distance between subject tree (i) and competitor tree (j); rci = radius of crown of subject tree (i) in meters; rcj = radius of crown of competitor tree (j) in meters.The crown radius of trees (cr) was calculated dividing by half the value obtained for the crown diameter model (cd) based on the DBH for Lages, SC: cd = 1.3149 + 0.2112•DBH; R 2 = 0.838; Syx % = 13.7%, and São Francisco de Paula, RS: cd = 0.8947 + 0.2032•DBH; R 2 = 0.853; Syx % = 15.4%.
Diameter at breast height (DBH), total height (h), crown insertion height (cih), and 8 crown radii in the direction of the cardinal points: north (N), northeast (NE), east (E), southeast (SE), south (S), southwest (SW), west (W), and northwest (NW), were measured for each subject tree and their respective competitors.Crown insertion height was determined as the height from ground level to insertion of live crown.Diameter was measured with diametric tape, in centimeters, and horizontal distances (dist ij ) and height were measured with Vertex IV hypsometer, in meters, both with declivity corrected, as well as the crown radii, with accuracy down to centimeters.Crown diameter (cd) was determined by doubling the root mean square of the eight crown radii (mcr).The crown horizontal projection area (CHPA) of araucaria trees was calculated considering the circular shape as CHPA = π•mcr 2 .
The calculated competition indices considered the dimensional variables between subject trees (i) and their respective competitors (j): diameter at breast height (DBH), horizontal distances (dist), basal area of individual tree (g), height (h), crown diameter (cd), crown horizontal projection area (CHPA), potential crown horizontal projection area for araucaria (CHPA pot ), crown overlap area (COA), and mean crown radius (mcr) (Table 3).Note.DBH = diameter at breast height, in cm; dist = horizontal distances, in m; g = basal area of individual tree, in m 2 ; h = total height, in m; cd = crown diameter, in m; CHPA = crown horizontal projection area, in m 2 ; CHPA pot = potential crown horizontal projection area for araucaria, we used the equation: cd pot = 4.8601 + 0.2038·DBH (Costa, 2015), therefore: CHPA pot = π (cd pot /2) 2 , in m 2 ; COA = crown overlap area, in m 2 ; mcr = mean crown radius, in m; subject tree (i); competitor tree (j).
The calculation of crown overlap areas (COA) between subject tree (i) and each competitor (j) was performed through an algorithm developed in Visual Basic language for Microsoft Excel™ 2013 spreadsheet editor.For this purpose, the equation of crown overlap areas presented by several authors (Lee & Gadow, 1997;Gadow, 2003;Álvarez Taboada et al., 2003) was programmed, the algorithm was adapted to situations in which CHPA were secants to one another, internally and externally disjoined; and crowns that could be externally and internally tangent and concentrically disjoined were verified, however not found in any of the 1560 analyzed trees.
The periodical increment was established annually in the last five years through growth rings measured in increment cores sampled radially from the trunk with Pressler borer.Two increment cores were removed from each tree, but due to the difficulty to quantify increments of individuals in high competition, fieldwork had to be repeated when necessary to collect new increment cores from the same tree.The periodic annual increment in basal area of each tree was calculated by the Equation ( 2): Where, PAIg = periodic annual increment in basal area, in cm 2 /year; g = basal area of individual tree at the end of the period, in cm 2 ; g 5 = basal area of individual tree in the beginning of the period, in cm 2 ; t = number of assessed years: for the present study, the period of the last five years was established, according to what is indicated for this type of model (Hasenauer, 2006;Pretzsch, 2009).
To assess the influence of competition (CI) in the periodic annual increment in basal area (PAIg) of the subject tree, the nonlinear alometric model was adjusted by Equation ( 3): Where, PAIg = periodic annual increment in basal area, in cm 2 year -1 ; CI = competition indices; β 0 and β 1 = estimated regression coefficients; ε = residual error.
All the statistics were processed in the Statistical Analysis System (Sas, 2004).Data were characterized through descriptive statistics and adjusted models were assessed regarding coefficient of determination (R 2 ), standard error of estimate percentage (Syx%), F value, and its probability in the analysis of variance.The principal components analysis was used to reduce the number of variables, verify the influence of the 22 competition indices in the principal components, and their importance in relation to total data variation.

Results and Discussion
The calculation of competition indices (Table 3) for the 397 subject trees allowed to charachterize each one with the analysis of descriptive statistics (Table 4).Values of coefficient of variation for the 22 competition indices oscillated within a range of 17.0% and 650.2% indicating high dimensional variability (DBH, horizontal distances, heights, basal areas, crown diameters, crown horizontal projection areas, crown intersection areas) of subject araucaria trees and each of their competitors.Negative values of CI13 occurred when the height of the subject tree was smaller than that of their competitor.The minimum value of 100.0 for CI19 and zero for CI20, CI21, and CI22 resulted from nine subject trees whose crowns were assymmetric, leading to the decrease of mean crown radius in the calculation, and reducing this value until there was no crown intersection, contradicting the results found in fieldwork.Note.PAIg = periodic annual increment in basal area, in cm 2 year -1 ; [CI1 … CI22] = competition indices; Q1 = first quartile; Q3 = third quartile; CV% = coefficient of variation.
By adjusting the regression model (3), a relationship was found between the models of periodic annual increment in basal area and the 22 competition indices calculated for araucaria trees (Table 5).Results evidenced that all equations showed regression, and increment in basal area at individual tree level reached values of up to 0.425 for R 2 and Syx% ≥ 50.2.Note.[CI1 … CI22] =competition indices; Std.Er. = standard error; β 0 , β 1 = estimated regression coefficients; R 2 = coefficient of determination; Syx% = standard error of the estimate in percentage; Prob.>|t| = probability of significance for the t value; Prob.>F = probability of significance to F value.
Regardless of the high variability in the relationship between basal area increment and competition indices (Table 5), the results found are in accordance with studies performed with other species (Shi & Zhang, 2003;Castagneri et al., 2008).The relationship can be improved when aggregated to independent variables such as DBH (Contreras et al., 2011), crown attributes, basal area and density of stand (Corral Rivas et al., 2005).For araucaria trees, variance of basal area increment was studied based in dendrometric and morphometric variables, showing improvement in the description of species growth (Costa et al., 2015).However, the inclusion of other independent variables that characterize the present dimensions, site quality, crown dimension variables, and the effect of competition measured by an index has also improved the predictive performance of the model of basal area increment for araucaria in south Brazil (Costa, 2015).
According to the statistical criteria for adjustment (R 2 ) and accuracy (Syx%) used to assess the performance of models of basal area increment of araucaria (Table 5), the four best equations were selected through observed and estimated values based on competition indices CI4, CI7, CI11 and CI18 (Figure 1).The competition indices (Figures 1a,1b and 1d) showed inversely proportional relationships to growth, i.e. as competition increased, there was a decrease of basal area increment of araucaria trees.On the other hand, CI11 was directly proportional to tree growth, showing that value increase reflects on lower influence of competitor trees on the subject tree.

(d)
The principal components analysis was efficient, reducing the number of variables, and allowing to identify that from the 22 indices calculated, the eigenvalue of the first component (Figure 2a) was 11.65, contributing with 52.95% of the total explained variance (Figure 2b).In the same relationship, the eigenvalue of the second principal component was 3.67, 16.68% of the total explained variance, and the third component showed cumulative total explained variance of 78.43% (Figure 2b).Thus, these three components were taken as examples since they explain most variability of the data observed.The first component (PC1) explained 52.95% of the total data variance, whose eigenvectors were similar in 11 competition indices, varying between 0.23 and 0.27 (Table 6), approximately, confirming that although they showed different variables and mathematical equations used for the calculation, competition indices can be used to describe species competitiveness.When compared to the model adjustments (Table 5), it was found that these competition indices (included in PC1) show intermediary adjustments (R 2 ) and moderate accuracy (Syx%).In the second component (PC2), four indices stood out, with eigenvectors varying between 0.32 and 0.48, explaining only 16.68% of total variance of data.Based on the values of Table 6, competition indices in PC2 presented the lowest adjustments (R 2 ) and the highest errors (Syx%), as seen in Table 5.The third principal component (PC3) was responsible for 8.81% of total explained variance, being CI1, CI10 and CI11 the indices with the highest eigenvectors.Competition indices that were directly proportional to basal area increment of araucaria trees (Table 5) were grouped in this component.Note.[CI1 … CI22] = competition indices; PC1, PC2, PC3 = principal components 1, 2 and 3.
The biological process that involves competition between trees is more complex than it may be described by only one mathematical competition index (Daniels et al., 1986).However, several competition indices are used to model tree growth and production (Burkhart & Tomé, 2012).In general, it is recommended to use competition indices that express the biological behavior of Araucaria trees, associated with the ease and convenience of field measurement and application in a natural forest.

Conclusion
Competition indices showed high variability, and when adjusted through regression techniques, they allowed to explain the variation of periodic annual increment in basal area in a moderate way, with high estimation errors.
The principal components analysis was efficient and allowed to find eleven potential competition indices to describe increment in basal area of araucaria trees.
The models assessed help to model growth of Araucaria trees together with independent variables that characterize tree size, crown vitality, site, among other factors.

Figure 2 .
Figure 2. (a) Eigenvalue value in function of the principal component; (b) % of the proportion of explained and cumulative variance according to the principal component for Araucaria angustifolia (Bertol.)Kuntze sampled in a natural forest

Table 1 .
Location and climatic characteristics of the studied areas

Table 2 .
Horizontal phytosociological analysis: density, dominance, frequency and importance value index of the sites studied in the Araucaria forest

Table 4 .
Statistical summary of competition indices calculated for Araucaria angustifolia (Bertol.)Kuntze sampled in natural forest

Table 5 .
Estimated coefficients and adjustment and precision statistics of periodic annual increment models in basal area (PAIg) in function of competition indices (CI) for Araucaria angustifolia (Bertol.)Kuntze in natural forest

Table 6 .
Eigenvectors of the main components (PC) for competition indices of Araucaria angustifolia (Bertol.)Kuntze sampled in natural forest