Eur J Forest Res (2005) 124: 133–142 DOI 10.1007/s10342-005-0061-y O R I GI N A L P A P E R J. J. Corral Rivas Æ J. G. Álvarez. González Oscar Aguirre Æ F. J. Hernández The effect of competition on individual tree basal area growth in mature stands of Pinus cooperi Blanco in Durango (Mexico) Received: 22 July 2004 / Accepted: 28 February 2005 / Published online: 13 May 2005
Springer-Verlag 2005 Abstract In this paper, we evaluated how well-selected distance-dependent and distance-independent competition indices explain individual tree basal area growth of trees, growing in mature and even-aged stands of Pinus cooperi Blanco. A total of 18 competition measures were analyzed of which six do not need tree location (distance-independent) and 12 that utilize tree location (distance-dependent). The competition situation of a stand and/or an individual tree was studied using 11 different competitor selection methods. The mean square error reduction relative to no-competition was used to judge the performance of each competition index. It was found that the best distance-independent competition indices performed as well as the best distance-dependent competition indices. It was concluded that the BALMOD-index would be a good competition index to be incorporated into further individual tree basal area growth models for the study area. Keywords Pinus cooperi Æ Competition indices Æ Individual basal area growth J. J. C. Rivas (&) Institut für Waldinventur und Waldwachstum, Georg-August-Universität Göttingen, Büsgenweg 5, 37077 Göttingen, Germany E-mail: jcorral@gwdg.de Tel.: +49-551-393554 Fax: +49-551-399787 J. G. Á. González Departamento de Enxeñerı́a Agroforestal, Escola Politécnica Superior de Lugo, Universidade de Santiago de Compostela, Campus Universitario s/n, 27002 Lugo, Spain O. Aguirre Facultad de Ciencias Forestales, Universidad Autónoma de Nuevo León, 67700 Linares, NL, Mexico F. J. Hernández Instituto Tecnológico Forestal, Mesa del Tecnológico s/n, 34950 El Salto, P.N., Durango, Mexico Introduction Competition among trees implies that resource supplies are below levels capable of supporting optimal growth of two or more trees (Brand and Magnussen 1988; Holmes and Reed 1991; Gadow and Hui 1999; Pretzsch 2002). Tomé and Burkhart (1989) and Preztsch (2001) have shown that individual tree growth models can be used to analyze how individual tree basal area growth varies with different levels of competition. The effects of competitor trees on the growth of individual trees have long been studied by numerous authors in an attempt to predict growth of different species as accurately as possible (e.g. Clark and Evans 1954; Alemdag 1978; Lorimer 1983; Tomé and Burkhart 1989; Biging and Dobbertin 1995; Bachmann 1998; Alvarez et al. 2003). Competition indices may be distance-independent or distance-dependent. The distance-independent ones do not require tree coordinates since they are simple functions of stand-level variables or of the initial dimensions of the subject tree, while distance-dependent models require the dimensions and the relative locations of several neighbors for the computation. Distance-independent indices are easy to calculate and less demanding in data field measurements, which is an advantage. In Mexico, only few investigations on competition among trees for resources have been reported (Rodrı́guez 1987; Romero 1993; Valles et al. 1998; Torres 2000). Research carried out in other countries has shown improvements in diameter growth prediction for different species when a competition index was included (e.g. Bella 1971; Pukkala and Kolström 1987; Tomé and Burkhart 1989; Biging and Dobbertin 1992, 1995; Pretzsch et al. 2002; Alvarez et al. 2003). In these studies, a single tree competition index could not be shown to be superior, but some indices seemed to perform better with certain species depending on specific situations. Thus, it seems reasonable to examine the possibilities of using competition indices in growth models under Mexican forest conditions. 134 The objectives of this work are: (a) to compare the predictive capability of selected distance-independent competition indices against distance-dependent ones and (b) to identify the best competition index and/or best combination of a competition index with a method for selecting competitors. The selection method should not require too many field measurements and should be easily incorporated into an individual tree basal area growth model for Pinus cooperi in the forest region of El Salto, Durango, Mexico. Materials and methods Data In the Mexican forests, P. cooperi Blanco is one of the most important commercial timber species, due to its good wood properties, wide distribution range and the harvested timber volume. P. cooperi occurs mainly in the Sierra Madre Occidental in southern Chihuahua, eastern Sinaloa, and Durango, at elevations of 1,500–2,800 m. The species may form pure stands, but usually is mixed with other pines, junipers, or various species of Quercus (Garcia and Gonzalez 1998). The stands are moderately managed with an average cutting cycle of 10–15 years. Data from eight permanent plots located in mature and even-aged stands of P. cooperi Blanco, each covering an area of 0.5 ha, were used in this study. The stands originated from natural regeneration and are approximately 60-years-old. The plots were established in January 1998 and have been remeasured each year as part of a long-term experiment designed to evaluate growth and survival in the forest Ejido1 known as La Victoria, in the province of El Salto, Durango (Mexico). The study area is located in the Sierra Madre Occidental, between 2340¢–2447¢ latitude and 10521¢–10529¢ longitude, 100 km to the southwest of Durango City (Fig. 1). The following measurements were taken for each tree within the plots: X and Y coordinates (m), total height (m), diameter at breast height (cm), average crown diameter at the base of the live crown (m) and crown length (m). In addition, the variables basal area, stem number per hectare, site index at a reference age of 50 years using the equation proposed by Corral et al. (2004), and relative spacing index were calculated for each plot. The relative spacing index is a stand density measure, which is expressed in this work by the ratio of the average distance between the trees growing in ploti (m) and the dominant ploti-height (m). For computing the effect of competition on individual tree growth, a 10m buffer zone was used around the plot perimeter, the effective plot size was thus reduced from 100·50 m to 90·40 m. A growth period of 5 years was used in this study because the annual diameter increment was not Fig. 1 Map showing the location of Ejido La Victoria near El Salto City in the State of Durango in Mexico distinguishable in many of the sample trees. The plot characteristics are presented in Table 1. Competition indices In this work the competition situation of a tree was described with 18 different competition indices (Table 2). The first six indices (CI1–CI6) are distance-independent whereas the others (CI7 to CI18) are distancedependent ones. The indices described below were selected for evaluation in this paper due to their good performance in previous studies (e.g. Pukkala and Kolström 1987; Tomé and Burkhart 1989; Biging and Dobbertin 1992, 1995; Valles et al. 1998; Bachmann 1998; Alvarez et al. 2003). Selected distance-independent competition indices In this study, some of the most widely used stand density measures and indices which incorporate the vertical and horizontal distribution of foliage were examined. CI1 represents Reineke’s (1933) stand density index, which is based on the number of trees per hectare of plots and on its quadratic mean diameter. In order to compute this index the slope of the regression line (b1=1.6918) from the stand density guide for P. cooperi developed by Cruz and Castañeda (1999) was used. CI2 is the canopy closure that expresses the area of the crown, projected on a horizontal plane, as a fraction of the total ground area of the stand. CI3 represents the crown competition factor, defined as the percent of the growing area that is occupied by the crown projection of trees assuming that each individual is open-grown. For this study, we used Cruz and Castañeda’s (1999) equation to predict opengrown crown width in all indices: mcwi ¼ 0:1503di þ 2:0241 ð1Þ 1 Ejido is a land that belongs to communal groups who live and manage their natural resources with some level of governmental control where mcwi is the maximum crown width (m) that could be reached by the tree i if it was open-grown and di is its 135 Table 1 Values of the main stand variables at the first inventory and mean, maximum and minimum values of the individual tree basal area increments for each sample plot Site index was defined as the dominant height at a reference age of 50 years calculated by using the equations proposed by Corral et al. (2004) Plot 1 2 3 4 5 6 7 8 Trees/ha 248 126 116 122 134 150 168 250 Basal area (m2 ha
1) 25.5 14.4 15.6 10.8 14.8 17.1 13.8 16.5 Mean dbh (cm) 34.6 37.3 40.5 32.8 36.2 37.2 30.8 27.8 Dominant height (m) 24.8 23.8 25.9 20.6 24.4 21.8 20.3 17.9 Site index (m) 22.3 21.2 23.2 18.1 21.7 19.2 17.7 15.4 Individual tree increment (cm2) basal area Mean Maximum Minimum 115.1 111.3 87.7 123.0 104.9 96.9 77.6 61.5 292.1 230.9 209.1 267.0 201.3 191.4 179.0 168.0 6.6 34.6 21.8 37.7 26.7 21.4 8.4 8.5 Table 2 Competition indices and corresponding formulas, tested for use in the individual tree basal area growth models Index Source Equation Distance-independent competition indices: CI1 Reineke (1933) CI2 Canopy closure CI3 Krajicek et al. (1961) CI4 Wykoff et al. (1982) CI5 Schröder and Gadow (1999) CI6 Biging and Dobbertin (1995) Distance-dependent competition indices: CI7 Staebler (1951) CI8 Gerrard (1969) CI9 Bella (1971) CI10 Hegyi (1974) CI11 Alemdag (1978) CI12 Martin and Ek (1984) CI13 Daniels et al. (1986) CI14 Braathe (1980, cited in Pukkala and Kolström 1987) CI15 Biging and Dobbertin (1992) CI16 Biging and Dobbertin (1995) CI17 Valles et al. (1998) CI18 Valles et al. (1998) 10ðlog N þ1:691 log dg
1:691Þ P ( ni=1(pÆcw2i )/4)/S P ( ni=1(pÆmcw2i)/4)/S Pn 2 i=1(pÆdmax i )/4 = BALi (1
[1
(BALi/G)])/RS Pn i=1ccj/S P i „ jLij P i „ jOij/Zi P i „ j(OijÆdj)/(ZiÆdi) P i „ jdj/(di Æ distij) P i „ j{p[(distijÆdi)/(di+dj)] P 2 P (dj/distij)/ (dj/distij)} i „ j(dj/di)Æ exp ((16Ædistij)/(di+dj)) P (d2i Æn)/ i „ jd2j P i „ jhi/(hjÆ distij) P i „ jcsaj/(csai Æ distij) P ccj/(cci Æ (distij+1)) 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 P 
csaj =csai þ hj =hi =n d i þqh i =distffiij þ i6¼j ffiffiffiffiffiffiffiffiffiffiffiffiffi .j =d  P nc  csai csaj p i6¼j i „ j N Trees per ha in the plot, dg quadratic mean diameter (cm), n trees per plot, cwi crown width of the subject tree (m), S plot size (m2), mcwi maximum crown width of the subject tree estimated using Eq. 1 (m), dmaxi dbh of the trees larger than subject tree (m), BALi basal area of trees larger than the subject tree (m2 ha
1), G basal area of the plot (m2 ha
1), RS relative spacing index of plot, Lij distance between competitor tree and subject tree that overlap their influence zones (m), Oij are of the influence zone overlap between competitor and subject tree (m2), Zi area of subject tree influence zone (m2), di subject tree dbh (cm), dj competitor tree dbh (cm), distij distance between competitor and subject tree (m), nc number of competitors, hi subject tree height (m), hj competitor tree height (m), csai crown surface area of the subject tree (m2), csaj crown surface area of the competitor (m2), exp base of the natural logarithm, ccj crown cross-sectional-area of competitor evaluated at 66% of the subject tree’s crown height from the top (m2), cci crown cross-sectional-area of the subject tree evaluated at 66% of its crown height from the top (m2) diameter at breast height (cm). CI4 is the sum of the basal areas of the trees larger than the subject tree (BAL) proposed by Wykoff et al. (1982). C5 is a modification of BAL, the BALMOD proposed by Schröder and Gadow (1999), which includes the relative spacing index. CI6 is a crown cross-sectional area index and was calculated for each tree as the percent of the growing area that is occupied by the crown projection of trees estimated at 66% of the subject tree’s crown length measured from the top. Each competitor and subject tree’s contribution to CI6 is found by taking a horizontal slice through the canopy at this height. Competitors’ contribution to CI6 may differ with associated point of evaluation. If live crown of a competitor tree was higher than the point of evaluation, the contribution is then defined as the competitor’s total live crown area, otherwise, the area at the point of evaluation was taken into account. Afterward individual contributions are summed up and ex- 136 Fig. 2 a Crown cross-sectional area evaluated at 66% of the subject tree’s crown height from top (CI6). The subject tree and competitor’s contributions to CI6 are displayed with bold lines. b Competitors are chosen with a height angle gauge of 60 from the subject tree’s base panded to a per hectare basis, which is then divided by the plot area to produce an individual tree competition index (see Fig. 2a). We selected this height because some studies have shown that the greatest reduction in residual sum of squares of diameter and height growth models occurs when the evaluation point is equal or close to the one used in this study (e.g. Krumland 1982; Biging and Dobbertin 1995; Nagel et al. 2003). As a model for estimating the tree’s crown parameters was not available, a paraboloid form for the tree crown of conifer mature stands was assumed in order to compute the crown cross-sectional surface of the trees (see Rondeux 1993, p 59). Selected distance-dependent competition indices Two types of distance-dependent indices were evaluated: influence-zone overlap indices and size-ratio indices. The first three indices CI7 (Staebler 1951), CI8 (Gerrard 1969), and CI9 (Bella 1971) are influence-zone overlap indices which assume that the area, in a horizontal projection, over which a tree competes for the resources of the site, can be represented by a circle. The radius of the circle is a function of tree size and is thought to be equal to the expected growing space of open-grown trees estimated using Eq. 1. Those neighboring trees that overlap their influence zone to subject tree’s influence zone are potential competitors. Nine size-ratio competition indices were computed and evaluated. These indices are based on the hypothesis that the competition effect of a neighboring tree increases with increasing size and decreasing distance. The first four indices use the diameter at breast height as an indicator of size: CI10 (Hegyi 1974), CI11 (Alemdag 1978), CI12 (Martin and Ek 1984) and CI13 (Daniels et al. 1986). CI14 is a size-ratio index proposed by Braathe (1980; cited in Pukkala and Kolström 1987) that uses the total tree height as an indicator of size. CI15 is a size-ratio index using the crown surface area and it is based on the assumption that within species growth is correlated to a tree’s crown foliage area (Kramer 1988; Biging and Dobbertin 1992, 1995; Nagel et al. 2002). CI16 is a size-ratio index using crown cross-sectional area at 66% of the subject tree’s crown length with competitors chosen with a vertical angle gauge of 60 from the base of the subject tree (see Fig. 2b) as was proposed by Biging and Dobbertin (1992). The last two size-ratio indices, C17 and C18 were developed by Valles et al. (1998) to analyze the competition effect for P. cooperi in Mexico. From a total of nine distance-dependent competition indices analyzed, these two indices had the highest contribution to the tested diameter growth model (P<0.05). The CI18 was first proposed by Glover and Hool (1979) as a distance independent-competition index for a basal area ratio predictor of loblolly pine mortality. This index was modified by Valles et al. (1998) as a distance-dependent competition index and such modification is currently used in the forest growth simulator SICREMAS developed for the forest region of San Dimas, Durango forest region (Valles and Gutiérrez 2000). Competitor selection methods The value of a competition index for each individual tree depends on the mathematical formulation of the relationship between the chosen variables as well as on the method used to define neighboring trees as competitors (Bigging and Dobbertin 1992). Various methods to choose the trees that compete with a subject tree have been proposed. Some of them consider as competitors all the trees that are included within a circle with a fixed radius whose centre is constituted by the subject tree (Hegyi 1974). Other methods are based on variable radii, often weighted by tree dimensions of the subject tree and its competitors, for example dbh or height (Daniels 1976; Ford and Diggle 1981), and others combine two or more different criteria (Bigging and Dobbertin 1992). In this work, we tested 11 selection competitor methods named with the codes M1 to M11. Some of them were obtained from other studies (e.g. Pukkala and Kolström 1987; Holmes and Reed 1991; Biging and Dobbertin 1992; Mäkinen 1997; Alvarez et al. 2003) and some are proposed by the authors. The M1 identifies as competitors all those trees that are considered in a variable plot radius sampling, the socalled ‘‘Bitterlich method’’ whose centre is the subject tree with a basal area factor (BAF) equals to 4 m2 ha
1. Mack and Harper (1977) suggested that the competitive influence of neighbors is greater if they are evenly distributed around a target plant, rather than aggregated on one side. In order to take this effect into account, M2 considers as competitors all the trees selected with the M1 method and the nearest tree that is placed inside each one of the four quadrants defined by the cardinal points, even though they are not within the variable plot radius sampling. M3 selects as competitors of the subject tree its four nearest neighboring trees. 137 Fig. 3 a For M5, competitors are chosen with a height angle of 60 from the subject tree’s crown base. b A hypothetical example illustrating the procedure to select active competitors within the subject tree’s influence zone, considering an angle gauge of 90. Among the 14 potential competitors, trees numbers 1–4 are active ones Considering that a tree’s capacity to intercept available light has been shown to be important in growth dynamics (Doyle 1983 in Biging and Dobbertin 1992), the M4, first proposed by Biging and Dobbertin (1992), is based on considering active competitors those trees whose total height is greater than an imaginary inclined line traced from the base of the subject tree with a 60 angle from the horizontal (Fig. 2b). The M5 is similar to M4 but in this case the imaginary inclined line begins at the subject tree’s live crown base. The competitors are those trees whose height exceeds the projected angle (Fig. 3a). The M6 is based on the influence-zone concept initially proposed by Staebler (1951). The potential influence-zone of a tree is usually defined as a circle with a radius whose value depends on the size of the subject tree. Competition is assumed to exist when the zones of two trees overlap. We defined as active competitors those trees that fulfil the following condition: distij  rZi þ rZj the first competition elimination sector and facing the reference tree is the active competitor number 2. Trees behind competitor 2 and within the second competition elimination sector are passive competitors. The procedure concludes when all the active competitors have been identified. The elimination angle gauges used in this work were 90, 75, 60, 45 and 30 for M7 to M11, respectively. Not all the distance-dependent competition indices are compatible with the eleven competitor selection methods. CI7, CI8 and CI9 indices can be only computed when combined with M6, because they are based on the influence zone concept. The other indices can be calculated with all competitor selection methods tested in this paper. When taking into account the different ways of defining the competition measures, the competition situation of the trees was evaluated by 108 different variables (nine single competition measures (i.e. CI1–CI6, CI7, CI8 and CI9) plus 99 combinations of the other competition indices (i.e. CI10–CI18) with a competitor selection method). Influence of sample plots number on competition To evaluate whether competition is effected by the sample plot, the non-linear extra sum of squares method (Bates and Watts 1988) was used. This method requires the fitting of a reduced and a full model. The reduced model corresponds to a linear model with the individual tree basal area increment (Dgi) as the dependent variable and the diameter at breast height (di) as independent variable and the same set of parameters for all the sample plots (Dgi = b0 + b1Ædi). In full model the parameter set is different sets for each sample plot and it is obtained by expanding each parameter (b0 and b1), including an associated parameter and a dummy variable to differentiate the sample plot: ð2Þ where rZi and rZj are the influence zone radii of the subject tree i and the competitor j respectively and distij is the distance between the subject tree i and competitor j. In this work we have considered as influence zone radii the maximum value that could be reached by the tree’s crown width if it was open-grown (Eq. 1). The M7 to M11 take into account the competition elimination angle concept (Lee and Gadow 1997). Each neighbor of a given subject tree may be an active or a passive competitor, based on a competition elimination sector defined by a specific elimination angle. An example illustrating the iterative procedure is presented in Fig. 3b, using a 90 competition elimination angle. The competition zone around the subject tree contains a defined number of potential competitors. The nearest neighbor of the reference tree is the active competitor number 1. The trees located behind competitor 1 and within the first competition elimination sector defined by a 90 competition elimination angle centred in competitor number 1 are passive competitors. The next neighbor located outside bi ¼ 8 X bij  Ij i ¼ 1; 2 ð3Þ j¼1 where bij are coefficients to be estimated and Ij is a dummy variable which value is equal to 1 for sample plot j and 0 for the remaining sample plots. The appropriate test statistics is given by: F ¼ (SSEðRÞ
SSEðF ÞÞ=ðdfR
dfF Þ SSEðF Þ=dfF ð4Þ where SSE(R) is the error sum of squares of the reduced model; SSE(F) is the error sum of squares of the full model; dfR and dfF are the error degrees of freedom of the reduced and full model, respectively; and F* follows an F-distribution. Individual tree basal area growth models Individual tree basal area growth depends on a number of factors such as normal section at the beginning of the 138 analyzed growth period, age, crown spread diameter, micro-environment (i.e. site index), genetic characteristics, and the competition among trees for light, water, nutrients, and physical growing space. Hence, an individual tree basal area growth model should consider all these factors. However, many of the details of how these processes are regulated are still unknown. It is difficult, if not impossible, to generate a growth model for individual trees which includes the cited factors that affect the tree growth. Generally researchers have accepted that the development of individual tree growth models, which include competition indices, is a more comprehensive attempt to estimate potential growth (Pukkala and Kolström 1987; Biging and Dobbertin 1995; Tomé and Burkhart 1989; Schröder and Gadow 1999; Pretzsch 2001; Nagel et al. 2002; Alvarez et al. 2003). In the present paper, two individual tree basal area growth models were fitted: a reduced and a complete model. The reduced model (Eq. 5) was formulated as a control to study the implications of modelling individual tree basal area growth without considering competition. This model is based on the hypothesis that individual tree basal area growth (Dgi) is a function of the diameter at breast height at the beginning of the analyzed growth period (di), the site quality defined by the site index (S), the initial stand density defined by the number of stems per hectare (N), the initial stand basal area (G) and the historical tree vigour (cri). If age is not available or intentionally not used, the ratio between crown width and tree height (crown ratio) may also be a useful predictor for describing the vigour of trees of similar stem size (Monserud and Sterba 1996; Hökkä and Groot 1999; Schröder et al. 2002). The complete model (Eq. 6) is similar to the reduced model, but considers the contribution of a competition index included as a new predictor variable: Dgi ¼ a1 þ a2  di þ a3  S þ a4  N þ a5  G þ a6  cri R2adj ¼ 1
ðn
1Þ   n X n X ðyi
^y i Þ2 =ðn
pÞ i¼1 2 ðyi
y i Þ ð8Þ i¼1 where yi, ^y i and y i are the measured, predicted and average values of the dependent variable, respectively; n is the total number of observations used to fit the model and p is the number of model parameters. The variance inflation factors (VIF) of all the independent variables in Eqs. 5 and 6 were calculated for detecting multicollinearity; values up to 10 were accepted (Draper and Smith 1998; Belsey 1991; Soares and Tomé 2001): VIFðiÞ ¼ 1 1
R2 ðiÞ ð9Þ where R2(i) is the multiple correlation coefficient obtained when the ith independent variable Xi is regressed against all the remaining independent variables in the individual tree growth model. Finally, a partial F-test was carried out to test the significance of including each competition index comparing the reduced model with the complete one. Moreover, to evaluate how well a competition index performs, the mean square error reduction (MSER) relative to no-competition was calculated using the following expression:   MSE6 MSER ¼ 1
100 ð10Þ MSE5 where MSE6 is the mean square error of the complete model (Eq. 6) and MSE5 is the mean square error of the reduced model (Eq. 5). ð5Þ Results and discussion Dgi ¼ a1 þ a2  di þ a3  S þ a4  N þ a5  G þ a6  cri þ a7  CIi ð6Þ where Dgi is the basal area increment of tree i (cm2), during the analyzed period of time (5 years in this case); di is the initial dbh of tree i (cm); S is the site index (m); N is the initial density (stems ha
1); G is the initial stand basal area (m2 ha
1); cri is the initial crown ratio of tree i; CI is a competition index, and ai are coefficients to be estimated. Equations 5 and 6 were fitted by ordinary least squares using the REG procedure of the SAS/STAT system (SAS Institute 1999). To evaluate the performance of the fits the root mean square error (RMSE) and the adjusted coefficient of determination (R2adj) were estimated: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X RMSE ¼ ðyi
^y i Þ2 =ðn
pÞ ð7Þ i¼1 The results of the non-linear extra sum of squares method (Bates and Watts 1988) revealed that the sample plot has a significant effect on the competition (F=6.18; P>F=0.0001). Therefore, the inclusion in Eqs. 5 and 6 of independent variables representing the initial stand status (site index, number of stems per hectare or stand basal area) should be considered. The values of the VIF were always lower than 10, except for the stand basal area (G). Therefore, this variable was finally excluded of Eq. 5 to avoid multicollinearity and to ensure that the model solution was global rather than local. From a total of 108 competition index varieties, only 31 increased the value of the adjusted coefficient of determination (see Tables 3, 4). The best combinations for each competition index are shown in Table 4. The three first stand level density measures (CI1–CI3) did not show a significant reduction of MSE when compared with Eq. 5. These results come across with 139 Table 3 Mean square error reduction of the combinations of distance-dependent competition indices and competitor selection methods that showed an improvement of the adjusted coefficient of correlation comparing to using no competition index (Eq. 5) The line indicates no improvement Table 4 Contribution of competition indices to individual tree basal area growth models (multiple linear regressions) From the 108 fitted models, only the best variation of each competition index is shown here a Significant at 0.05 level; b Significant at 0.01 level; R2E5 and R2E6 are the adjusted coefficient of determination of the reduced (Eq. 5) and the complete model (Eq. 6), respectively Distance Competitor selection method dependent M2 M3 M4 M5 M1 M6 M7 M8 M9 M10 M11 CI10 CI11 CI12 CI13 CI14 CI15 CI16 CI17 CI18 3.1357 0.1764 0.0786 – – – – – – – – – – – – 0.5613 – – – – – – – – 1.7280 – – – 0.0322 – – – – 1.9680 – – – – – – – – 1.9735 0.4698 0.0369 – – – – 0.0315 0.0421 2.1844 0.5512 – 1.3674 – – – – – – – – 1.1662 – 0.4025 – – – 0.7074 – – – – – – – – 1.7285 – – – – – – – – – – – 1.1723 – 0.4132 – – – 0.7624 – Competition index n R2E5 R2E6 Partial-F (index term) RMSE MSER (%) CI1 CI2 CI3 CI4 CI5 CI6 CI7M6 CI8M6 CI9M6 CI10M11 CI11M6 CI12M6 CI13M5 CI15M11 CI16M11 CI17M11 CI18M11 310 310 310 310 310 310 289 289 289 310 289 289 310 310 310 310 310 0.6792 0.6792 0.6792 0.6792 0.6792 0.6792 0.6823 0.6823 0.6823 0.6792 0.6823 0.6823 0.6792 0.6792 0.6792 0.6792 0.6792 0.6795 0.6804 0.6805 0.6843 0.6868 0.6816 0.6842 0.6861 0.6838 0.6801 0.6934 0.6840 0.6816 0.6802 0.6803 0.6871 0.6821 0.26 1.12 1.25 4.94a 7.37b 2.32 1.73 3.43 1.37 1.01 10.21b 1.52 2.29 0.92 1.03 7.65b 2.73 8.08 8.06 8.05 8.01 7.98 8.04 7.92 7.90 7.93 8.06 7.81 7.93 8.05 8.06 8.06 7.98 8.04 0.0204 0.0786 0.0927 1.2734 2.0573 0.4221 0.2444 0.8317 0.1162 0.0369 3.1357 0.1764 0.4132 0.0315 0.0421 2.1844 0.5512 those found by Biging and Dobbertin (1995) who in general, did not find any improvement for the diameter squared and height growth models, they tested, when traditional stand density measures were used as competition indices. Thus, the results found in this study make evident the main problem of the use of the stand variables as competition indices, i.e. all counted trees contribute equally to the competition estimate, despite size or proximity (Opie 1968; Moore et al. 1973; Alemdag 1978; Mäkinen 1997). On the other hand, the distance-independent indices that consider tree basal area or crown parameters of selected competitors around a subject tree (CI4–CI6) showed a higher reduction of the MSEs comparing to using no competition index, specially those indices based on the sum of the basal areas of the trees larger than the subject tree (BAL and BALMOD). The contributions of the distance-independent indices: BAL (CI4) and BALMOD (CI5) showed highly significant values in the partial F-test (Table 4). These results are consistent with those obtained in similar studies with other conifer species (e.g. Biging and Dobbertin 1985; Álvarez et al. 2003). As can been seen in Table 4, none of the selected influence-zone overlap competition indices (CI7, CI8 and CI9) showed a significant contribution (Partial-F test) to the growth model compared to using no competition index. The results of the size-ratio indices that use the dbh as indicator of size (CI10, CI11, CI12 and CI13) depend on the use of the competitor selection method. The best results were obtained for the index proposed by Alemdag (CI11) combined with the M6 that is based on the influence-zone concept defined as a circle with a radius equal to the crown width of an open grown tree. This competitor selection method showed also the best results for indices CI12 and CI13, although, in these cases the results of partial-F test were not significant. The size-ratio index based on the total height (CI14) did not show any contribution to the individual tree basal area growth model, no matter which competitor selection method was used. These results differ with those obtained with size-ratio indices based on dbh dimensions when the partial contribution of competition index depends on the competitor selection method. In order to find an explanation for this effect, the spatial quantities diameter and height differentiation were calculated, using the variable size differentiation proposed by Gadow and Füldner (1995) 2 based on the three nearest neighbors of a given reference tree. Average 2 The differentiation of a size variable (Ti) for a given tree i and its n nearest P neighbors j (j=1, ..., n) is defined as follows: Ti = 1
(1/ n) nj=1 (min (Xi, Xj))/(max (Xi, Xj)) with X = a size variable and 0 £ Ti £ 1 140 values of size differentiation of 0.2413 and 0.1342 for dbh and total height were observed respectively, indicating a greater homogeneity in total height when compared with dbh. Hence, the standard deviation of the size-ratio indices based on total height for any competitor selection method is smaller than those obtained for size-ratio indices based on dbh. Biging and Dobbertin’s size-ratio indices that use the crown dimensions as indicator of size (CI15 and CI16) did not present an improvement of the adjusted coefficient of determination of Eq. 5, except when they are combined with the M11 competitor selection method which corresponds to the competition elimination angle concept, using an angle gauge of 30. However, the results of the partial F-test indicate that these competition indices varieties do not contribute significantly to the individual tree basal area growth when they are included as competition indices. The distance-dependent competition index CI17 proposed by Valles et al. (1998) for P. cooperi showed, in general, good results among the competition indices analyzed. The inclusion of this competition index in Eq. 5 results in reduction of the MSE for 8 of the 11 competitor selection methods tested in this study. The best combination for this index in terms of Partial-F test (7.65; significant at a 99% level) was obtained using competition elimination angle of 30 (M11). On the other hand, the best combination for competition index CI18 was found to be CI18M11. This competition index was proposed also by Valles et al. (1998) and is currently used in the forest growth simulator SICREMAS developed for the forest region of San Dimas, Durango. This variety corresponds as well to the M11 competitor selection method which derived from the competition elimination angle concept, using an angle gauge of 30. However, CI18M11 variety performed slightly worse than the best distance-independent indices (CI4 and CI5) or some of the size-ratio indices based on dbh (CI11) or based on dbh and total height (CI17). In relation to the competitor selection methods, in general, the best results were obtained with those based on the influence-zone concept defined as a circle with a radius equal to the crown width of an open grown tree (M6) or based on the competition elimination angle concept, using an angle gauge of 30 (Table 3). Therefore, the use of small angle gauges for mature stands of P. cooperii with low density is more suitable to select an adequate number of competitor trees since more competitor trees are chosen than using large angle gauges. Alvarez et al. (2003) found also good results for the competition elimination angle concept but using angle gauges of 45 or 60 in their competition study with young Pinus radiata stands. In general, the selected distance-dependent competition indices performed best, in terms of adjusted R2, RMSE, partial F-tests and the mean square reduction (MSER) relative to no competition when they are combined with the influence-zone concept (M6) or competition elimination angle concept (M11; angle of 30) competitor selection methods. Seven of the selected distance-dependent indices (CI10, CI12, CI13, CI14, CI15, CI16 and CI18) did not show a significant contribution to the individual tree basal area growth model (see Table 4). However, taking into account that the sample plot data are derived from mature and thinned stands (with less than 238 stems ha
1) where the competition effect is not great, further studies in younger stands are needed in order to evaluate their suitable contribution to individual basal area growth models for P. cooperi. As mentioned above, the distance-dependent index proposed by Alemdag (1978) combined with a competitor selection method based on the influence-zone concept (CI11M6) and the distance-dependent index proposed by Valles et al. (1998) combined with a competition elimination angle of 30 (CI17M11) performed slightly better than BAL (CI4) and BALMOD (CI5) indices. However, the BAL and BALMOD indices can be derived using variables that are readily available (number of trees per hectare, basal area and dominant height) for each sub-compartment in the study area. Since one of the objectives of this work was to find a competition index which had a significant contribution to the individual tree basal area growth model and does not require too many field measurements, the BALMOD is recommended to be incorporated into an individual tree basal area growth model for P. cooperi in the forest region of El Salto, Durango Mexico. Figure 4 shows a scatterplot illustrating the relationship between individual tree basal area growth and BALMOD-index. The BALMOD index showed important reductions in the MSEs for growth models in previous competition studies of Pinus pinaster (5.13%; Schröder and Gadow 1999) and P. radiata (24.3%; Alvarez et al. 2003) in Spain. In these studies, the evaluated stands were younger and with a higher number of stems per hectare than the ones used in our analysis. Hence, a considerable improvement of the individual tree basal area growth model for P. cooperi could be attained using the Fig. 4 The relationship between individual tree basal area growth and BALMOD-index 141 BALMOD index with a more complete database including more combinations of different ages, site qualities and stand densities. Conclusions The distance-independent competition indices BAL (CI4) and BALMOD (CI5) performed as well as the best distance-dependent competition indices (CI11M6, and CI17M11) based on the mean square error reduction they induced when added to an individual basal area growth model without competition effect. When these four competition indices were included in the model, the increase in the adjusted multiple coefficient of determination was not very high in magnitude. However, the variables BALMOD (CI5), CI11M6, and CI17M11 were highly significant while BAL (CI4) was significant. The results indicate that, for P. cooperi growing in even-aged stands, the BALMOD index is the most suitable one to be incorporated into an individual tree basal area growth model for the forest region of El Salto, Durango, Mexico. This competition index performs as well as the other indices analyzed, including index (CI18) proposed by Valles et al. (1998), for this pine species in the forest region of San Dimas, Durango. The BALMOD index is much easier to calculate using data that are usually available. When comparing the different competitor-selection methods, we conclude that the influence-zone concept (M6) defined by a circle with a radius equal to the crown width of an open grown tree and the competition elimination angle concept using an angle gauge of 30 are the most reasonable criteria for defining neighboring trees as active competitors for this species, based on the data used. However, the competition elimination angle concept has the advantage that is very easy to use in the field and for other stand conditions, different angles could be tested to take into account the stand density and thus to control the number of competitors. In the pine stands evaluated in this study, about 33% of the individual tree basal area growth variation could not be explained. Part of this variation may be caused by genetic differences between trees (Perry 1985) or by the spatial variation in the site attributes (Pukkala and Kolström 1987) that are difficult to model. An unexplained part may be due to competition effects that are not expressed by the chosen indices. Therefore, as has been mentioned, additional research on tree competition effects is required for a better understanding of the competition dynamics and to search for a more physiological competition measure. 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