Study area

The study was conducted in the Forest Management Unit 0808 in Guadalupe y Calvo, Chihuahua (Figure 1), whose area covered by species of the Pinus and Quercus genera is 642 551 ha (^{Chávez et al., 2009}), located north of the Western Sierra Madre and the south of the state of Chihuahua, within the R10, the Sinaloa Watershed. The predominant climate according to Köppen’s classification modified by ^{García (2004)} is temperate semi-cold and sub-humid, with a rainy, cool summer C(e)(w₂)(x’). The soils exhibit an albic B subsurface horizon with a medium to thick texture and moderate organic matter content; these soils are classified as Pheozem, Cambisol, Regosol, Litosol, Planosol and Acrisol. The mosaic of temperate plant communities is formed by pure coniferous forests and mixed coniferous and broadleaf forests (^{Chávez et al., 2009}).

Figure 1 Geographical location of the study area. 

Sampling

A sample of 26 P. lumholtzii and P. strobiformis trees, and 30 P. leiophylla trees was collected through targeted sampling in order to include three levels of seasonal quality (low, medium and high); the selected individuals were dominant and co-dominant and exhibited no physical damage. The minimum age for P. lumholtzii was 42 years; while those for P. strobiformis and P. leiophylla were 52 and 75 years, respectively. The selected trees allowed the reconstruction of 219 profiles of P. lumholtzii, 249 of P. strobiformis, and 385 of P. leiophylla, with the stem analysis technique (^{Klepac, 1983}) for estimating the growth and annual increment in diameter, height, basimetric area and volume. Periods of ten years were considered for this purpose.

The reconstruction of the profiles was carried out by measuring and analyzing different diameters and recording the respective ages of the slices, which were obtained from each tree at a height of 0.30 m and 1.30 m from the ground, and then at every 2.60 m, up to the tip of the tree, in order to commercially exploit each of the logs that were extracted.

Variables

The variables of interest were species; normal diameter with bark; diameter without bark of each slice, measured with a Lufkin^® rule, starting at the stump; the height of each section, and the number of rings of each slice. The annual growth rings were grouped into 10-year age classes in order to facilitate the estimation of the growth of the normal diameter and the basimetric area. The prediction of the true height per section of the slices was estimated using the formula by ^{Carmean (1972)} subsequently modified by ^{Newberry (1991)}, as expressed in Equation 1.

Where:

h_ij = Estimated slice height of the i ^th slice, corresponding to growth ring j

h _i = Accumulated height at the top cut of the i ^th slice

h_i+1 = Accumulated height up to the upper cut of the previous i ^th slice

r _i = Number of growth rings of each slice

j = Growth rings from the pith in each slice

The equations for cylinder (2), Smalian (3) and cone (4), respectively, were used to estimate the volume of each section per tree (i.e. of the stump, logs and tip of each tree).

Where:

V_i = Log volume

D_t = Upper diameter of the stump

D_M = Largest diameter of each log

D_m = Smallest diameter of each log

D_b = Diameter of the base of the tip (m)

l = Length of the log (m)

Statistical analysis

The quality of the statistical regression adjustment of four growth models was evaluated in order to estimate growth in diameter, basimetric area, height and bark-free volume (Table 1). The respective expressions of these models for the aim of estimating the current annual increment (CAI) and the mean annual increment (MAI) of individual trees (^{Kiviste et al., 2001}) was also assessed, which are obtained by applying the first derivative to the growth function, and dividing the growth function by the age, respectively.

The adjustment was made by applying the PROC MODEL procedure of the SAS statistical software (^{SAS, 2002}). The goodness of fit of each model was determined by considering the adjusted coefficient of determination ( _Adj R ² ), the root mean square error (RMSE), and the significance of the parameters for selecting the most efficient model. Also, it was considered the graphic analysis of the growth trend curves, CAI and MAI.

The observed data of the growth of the analyzed variables kept a serial relationship through time, so that the observations are correlated. This causes the errors of the adjusted regression functions to be correlated as well. Therefore, the significance of this dependence was evaluated for each adjusted model by means of the Durbin-Watson autocorrelation test (d) (^{Sharma et al., 2011}; ^{Quiñonez et al., 2018}), which is expressed as follows (5):

Where:

et = Residual t of each observation

et-1 = Residual of each observation before the residual et

The autocorrelation between the errors of the best models was corrected by applying the auto-regression model CAR(X) developed by ^{Zimmerman et al. (2001)}. This model allows to adjust the parameters of each of the growth models with those of the autoregressive model together. The structure of the autoregressive model is:

Where:

eij = The j ^th ordinary residual in observation i

di = 1 for j >1

di = 0 for j = 1

hij-hij-1 and hij-hij-2 = These are the separation distances between the j ^th and the j-1 ^th observations and between the j ^th and the j-2 ^th observations

ϵij= Independent error that follows a normal distribution with zero mean and constant variance

After selecting the best model for each analyzed species and variable, the maximum annual increase (CAI _max ), the maximum annual increase (MAI _max ) and the point of intersection CAI-MAI (node) were estimated.

Growth models

Table 2 shows the statistics for the goodness-of-fit of the growth functions evaluated by species. Determination coefficients above 0.95 and standard errors below 2.4 cm for P. leiophylla and P. lumholtzii indicated a good fit for the four models tested for growth in diameter. For P. strobiformis, these same indicators suggested that only Chapman-Richards, Hossfeld I and Schumacher models had a good fit.

In a similar way, the fit of all the models for growth in the basimetric area was good. In general, the R ² varied between 0.9614 and 0.9854; while the RSME ranged between 0.00407 and 0.00651 m²; the best fits were for P. leiophylla, followed by P. strobiformis and P. lumholtzii.

The fit of the models for estimating the growth in height was also acceptable for all species (0.82223<R²<0.9433 and 2.9375 m<RMSE<1.7992 m). The observed difference of the R ² and RMSE statistics indicated that no model was superior for P. leiophylla and P. lumholtzii, while the best models for P. strobiformis were those of Chapman-Richards, Schumacher and Weibull. The goodness of fit of the models indicated that they all exhibit the best fit in P. lumholtzii, followed by P. leiophylla and P. strobiformis.

Regarding the fit of the models for estimating growth in volume, the statistics R ² and RMSE indicated that all the models exhibited a good fit in each of the species (0.9676< R ² <0.9821 and 0.0475< RMSE<0.0690), which suggests that any of them can be applied.

Table 3 shows the values of the estimators for each of the parameters of the models evaluated and selected for growth in diameter, basimetric area, total height and volume, as well as the standard error (SE) associated with each of them.

The parameter estimators of the growth models in diameter, basimetric area, height and volume of Chapman-Richards, Hossfeld I and Schumacher were significantly different from zero (Pr<0.0001) in all three species studied. In the case of the Weibull height growth model, the parameters were significantly different only for P. leiophylla and P. strobiformis; in P. lumholtzii they were significant only in the asymptote parameters a and c. Based on the values of R ² and RMSE for selecting the best model of growth in normal diameter, basimetric area, total height and volume, all models showed good fits in each of the assessed species; however, when considering the significance of the parameters, the Weibull model was discarded.

As a complement to the analysis of the results, the trends in the growth curves of the three previously selected models allowed us to determine that those of Schumacher and Chapman-Richards best represented biological growth in normal diameter, basimetric area and volume of P. leiophylla; while those of Hossfeld I and Chapman-Richards corresponded to the best trends in height growth. Schumacher’s model best characterized the growth in normal diameter, basimetric area, height and volume of both P. lumholtzii and P. strobiformis. In the latter species, the Chapman-Richards model also exhibited a good trend in height growth (Figures 2, 3 and 4).

Figure 2 Trends in the average growth of normal diameter, basimetric area, height and volume of the Chapman-Richards, Hossfeld I and Schumacher models of Pinus leiophylla Schiede ex Schltdl. & Cham. 

Figure 3 Trends in the average growth of normal diameter, basimetric area, height and volume of the Chapman-Richards, Hossfeld I and Schumacher models of Pinus lumholtzii B.L.Rob. & Fernand. 

Figure 4 Trends in the average growth of normal diameter, basimetric area, height and volume of the Chapman-Richards, Hossfeld I and Schumacher models of Pinus strobiformis Engelm. 

Current and average annual increment

Pinus lumholtzii was the species that exhibited the highest CAI _max in each one of the studied variables, followed by P. strobiformis and P. leiophylla. In P. lumholtzii, the CAI _max in diameter was presented at the age (t) of 30 years, much earlier than in P. leiophylla (t=55) and P. strobiformis (t=35). The age at which the CAI _max was recorded in the basimetric area of P. leiophylla was higher (t=85 years) than that calculated for P. lumholtzii (E=25 years) and P. strobiformis (t=25 years) (Table 4).

The age at which the CAI _max in height was estimated with the Hossfeld, Schumacher and Chapman-Richards models in the three species was 25 years, while the estimated age for the CAI _max in volume was highest in P. leiophylla, followed by P. lumholtzii and P. strobiformis. Considering that the optimal age for defining the physical rotation of forest species happens when the CAI and the MAI in volume are equal (^{Mendoza, 1983}), the rotation age for P. leiophylla estimated with Schumacher’s equation was 170 years, in P. lumholtzii 145 years and in P. strobiformis 115 years.

Growth models

Similar studies have documented that the Chapman-Richards and Schumacher models are well suited for estimating the growth of individual trees. For example, ^{Arteaga (2000)} evaluated several height growth models for estimating the site index for P. radiata D. Don, P. oaxacana Mirov and P. pseudostrobus Lindl., and ^{Monárrez and Ramírez (2003)} for P. durangensis Martínez reported that Chapman-Richards’ model had the best fit.

Likewise, ^{Corral and Návar (2005)} indicated that this model is the best for estimating growth and increase in diameter, basimetric area, height and volume in five pine species in the state of Durango; while ^{Quiñonez et al. (2015)} record that this same model, in the form of algebraic differential equations, is the best for estimating growth in normal diameter of P. lumholtzii. ^{Hernández et al. (2014)}, when adjusting various regression models to estimate growth in height and determine the site index for P. gregii Engelm. ex Parl., point to Schumacher’s model as the one that presented the best fit; this model also exhibited a good fit for estimating the growth in normal diameter in P. durangensis (^{Monárrez and Ramírez, 2003}).

Current and mean annual increase

Based on the estimations of the CAI _max , MAI _max , the age at which the maximum annual current increase and the physical shift of each variable are attained, we infer that the species analyzed herein achieve lower CAI _max or require more time than other species of the same genus to reach the maximum increases and rotation age. As evidence of the above, we cite the CAI _max for normal diameter (0.75 cm), basimetric area (0.009 m²), height (0.74 m) and volume (0.048 m³) in P. herrerae Martínez (^{Calvillo et al., 2005}); and for height in P. cooperi C.E.Blanco (0.39 m), P. durangensis (0.35 m), P. engelmanii Carr. (0.40 m, P. herrerae (0.43 m) (^{Corral and Návar, 2005}) were superior to P. leiophylla, P. lumholtzii and P. strobiformis, species evaluated in this study.

When comparing the CAI _max registered by ^{Corral and Návar (2005)} for P. leiophylla with those observed in the present research, we conclude that the CAI _max in height (0.32 m vs 0.197 m) and volume were higher (0.008 m³ vs 0.007 m³), while that of the diameter (0.32 m) was lower (0.32 cm vs 0.35 cm).

When considering the age at which CAI _max occurs in order to evaluate the growth of P. cooperi, P. durangensis, P. engelmanii, P. herrerae and P. leiophylla in the state of Durango, ^{Corral and Návar (2005)} determined that CAI _max in diameter occurred between one and 16 years, and in height, between 19 and 24 years ―ages lower than those estimated for those variables in the three species evaluated in this study. However, the age at which the CAI _max occurred in volume (70 to 105 years) was similar to that estimated by ^{Corral and Návar (2005)} for P. leiophylla, P. lumholtzii and P. strobiformis (100, 75 and 60 years, respectively).

The CAI-MAI intersection of the volume estimated by ^{Corral and Návar (2005)} for P. leiophylla is lower than that documented herein for the same species (155 years vs 160-170 years). ^{Monárrez and Ramírez (2003)} indicate that the ages at which CAI _max and the physical rotation age in the diameter (t=17 years and t=35 years) and height (t=15 years and t=28 years) of P. durangensis occur were also lower than those determined for the species assessed in this study. ^{Quiñonez et al. (2015)} report that among six Pinus species, P. lumholtzii and P. leiophylla show the slowest growth in normal diameter and basimetric area; the authors argue that the slow growth that characterizes these two taxa explains the lack of interest in exploiting them economically.