Introduction

The intensive use of agave ash (Agave durangensis Gentry and A. salmiana) to obtain ethyl alcohol in the Durango region dates back more than 50 years. Previously, the use of this plant was done domestically, since there were no vinases that used considerable quantities for commercial exploitation. It is at the end of the seventies when some mezcal-producing plants were established in the region of the Mountain range of Registrillo, which consumed and hoarded the agave ash, for the production of beverages such as mezcal (^{Díaz, 2002}).

The Official Mexican Standard 007, version 1997 of the Ministry of Environment and Natural Resources (^{NOM-007-SEMARNAT-1997}), states that this species must be exploited under techniques that ensure its development and sustainability. However, this is rare, since management programs lack technical information to quantitatively and qualitatively measure its potential (^{Díaz, 2002}). On the other hand, ^{Velasco and Marquez (2003)} point out that with the increase in demand for mezcalero agave to satisfy the regional, national and international market, the resource has been under pressure.

Currently, it has become a rule to implement methods based on the productive potential that classify forest areas that make estimates for the future growth of the resource to be harvested. ‘The site index is the unit of measurement with which the productivity of the forest areas is determined, this is defined as the maximum height reached by the dominant or codominant trees at a base age under the current conditions presented by the site’ (^{Pritchett, 1991}). This parameter of productivity can be applied and estimated in other non-timber plants, such as agaves, since it is also possible to classify these plants by their dominance, proposed by ^{Kraft (1884)}.

The knowledge of this factor contributes in the decision making for the elaboration of an adequate management program, when determining a cutting intensity that does not put at risk the sustained yield and at the same time, avoid the deterioration of a forest area underutilization (^{Corral- Rivas et al., 2014}). The objective of the present work was to evaluate the feasibility of mathematical models to determine the site quality for A. durangensis in the Mountain range of Registrillo region, municipality of Durango, Mexico, by using the dominant-age height relationship.

Materials and methods

Location of the study area

This investigation was carried out in the Mountain range of Registrillo in the municipality of Durango, Mexico, located within the geographic coordinates 23º 55’ and 23º 41’ north latitude and 104º 25’ and 104º 33’ west longitude with respect to the Greenwich meridian; specifically, in lands of the commons San Francisco of Manzanal, Tomas Urbina, Jose María Pino Suarez, Nicolás Romero, Valle Florido, Antonio Gaxiola, Primero of Mayo and Colonia Minerva. The access to the land is through the Durango-Mezquital road or the Durango-Mexico highway (Figure 1) at a distance of 25 km from the city of Durango, Durango.

Figure 1 Location of the study area. 

Sampling methods

A directed sampling with circular sites of 1 000 m² was used, with the purpose of covering all ranges of station quality, age, altitude, exposure and slope, considering also the biometric characteristics of A. durangensis and its spatial distribution within of the study area for the elaboration of the site index curves. This type of sampling is common in the development of works related to the estimation of quality curves (^{Corral-Rivas et al., 2014}). The agaves used to carry out this work corresponded to those whose phenotypic characteristics (height and vigor) have the appearance of being dominant within the different sampling sites and growing free of suppression (Figure 2).

Figure 2 Field-directed sampling method. 

Results and discussion

The estimation of the parameters, in all cases, was significant or different from zero. The results indicate that all models predict similarly the dominant height of A. durangensis. However, models 1 and 2 (Weibull and Chapman-Richards) presented better adjustments, since the values of the Akaike information criterion were the lowest (Table 3).

Also, these two models had slightly higher coefficients of determination and their errors were low. The bias values (0.099424 and 0.188395 cm for model 1 and 2, respectively) indicate only a minor underestimation. Based on the principle of selecting the simplest model, Model 2 or Chapman-Richards is recommended for determining the site index of the species A. durangensis in the studied area. The expression to be used is determined by the following equation:

Where: H= dominant height (in cm); t= age (in years).

The graphic analysis of the residuals for this model showed an adequate distribution, by adequately correcting the heterocedasticity problem of variance present in the dominant-age height relationship of the data analyzed (Figures 3 and 4). The Figure 3 is the result of the residuals obtained from the adjustment of the model without weighted regression, where a problem of inequality of variance is clearly shown. The correction of this problem is observed in Figure 4, which was derived from the studentized residuals obtained from the weighted adjustment.

Figure 3 Residues against the predicted dominant height without correcting the variance heterocedasticity for A. durangensis. 

Figure 4 Studied residuals against the predicted dominant height with corrected heteroskedasticity for A. durangensis. 

The value of k obtained from the residuals generated by this equation was 1.17. This value was then included within the weighting factor wi=1dstik to obtain the homocedastic residuals of Figure 4.

Finally, the qualities of the site index were determined in relation to a base age of 10 years, leaving 45 cm, 70 cm, 95 cm, 120 cm and 145 cm for the site qualities (CS) 1, 2, 3, 4 and 5 respectively (Figure 5). The CS 3 corresponds to the guide curve superimposed on the dominant-age height data pairs. It was decided to use a base age of 10 years, because it is estimated that this value corresponds to the physiological turn of the species. In the graphic representation of the anamorphic curves, it can be observed how they fit very well to the data cloud of dominant height and age for A. durangensis (Figure 6).

Figure 5 Site index curves obtained after solving the model by the parameter β ₀ . 

Figure 6 Anamorphic curves of site index for A. durangensis obtained with the Chapman-Richards model at a base age of 10 years. 

Also, the Chapman-Richards model showed good flexibility for the estimation of the site index of A. durangensis explaining more than 62% of variance observed in the dominant height. This result indicates that the Chapman-Richards model also shows good flexibility for modeling the site index of non-timber species such as maguey, as well as timber species (^{Corral-Rivas et al., 2004}). Until the writing date of this work, the mathematical tools developed for the management of maguey in the study region were very few. The work of ^{Díaz et al. (2007)}; ^{Ortiz (2007)}, who propose some mathematical models to elaborate production tables of A. durangensis; however, they have the disadvantage of not considering site quality as an input variable and therefore assume the same productivity levels for all stands.

^{Loera et al. (2012)} conducted a study on the ecological factors of an agave community in the same region, and found a significant relationship between age and height of the agaves, a result that is consistent with the present in the use of the variable age as a variable of predictive importance in the modeling of the growth of this species; on the other hand, ^{Aguirre et al. (2001)} report relationships between mezcalero agave dimensions in San Luis Potosí, Mexico and site conditions. The use of season quality curves developed in this work will allow forest managers to make technical decisions that do not compromise A. durangensis persistence, so that in the quality of site 5 (poor) the possibility of harvesting should be lower than in those areas that enjoy a quality of site 1 (good).

Conclusions

The dominant height-age relationship of the species A. durangensis was found to be a valid parameter to determine the quality of the site in the study area. Of the models analyzed, the Champman-Richards model was the most adequate to construct the system of anamorphic curves for A. durangensis in the studied area, providing adequate estimates of the site index and the height growth dominant for said species.

The results of this work represent a very useful tool for forest managers in the field of non-timber forest resources, since there is almost no research in this regard. By now having a measure of the quality of the site, the dasocratic division of the productive potential of each of the stands and subroads can be made; through, its incorporation into the management programs of the area.