Study area

The study was carried out in the Pueblo Nuevo ejido located southwest of the municipality of the same name in the state of Durango, Mexico (Figure 1).

Figure 1 Study area: Polygon of the Pueblo Nuevo ejido. 

The natural forest that belongs to the ejido is made up of mixed pine-oak forest stands; however, for the present study only the data corresponding to the Pinus species present in the inventory areas were included.

The species of interest are Pinus cooperi C. E. White, P. durangensis Ehren, P. leiophylla Schltdl. & Cham., P. ayacahuite Ehren, P. teocote Schiede ex Schltdl., Quercus sideroxyla Humb. & Bonpl. and Arbutus spp. (Table 1).

Origen of the data

The data was provided by the Unidad de Manejo Forestal (UMAFOR) 1008 El Salto, Durango (Forest Management Unit (Umafor) 1008 El Salto, Durango). They refer to 196 plots of the forest inventory located in the Pueblo Nuevo ejido, Durango. According to ^{Castedo-Dorado et al. (2007)}, the idea is to cover the maximum density existing in pure and regular pine stands.

From 1 000 m² plots, the information regarding height (m) and normal diameter (cm) at 1.30 m high of each individual was taken, with a range of 7.5 to 45 cm and a minimum density of 100 trees ha^-1 per plot, since the proportion of pines covers 90 % of the area of the sampling sites.

Mensuration estimations

For each selected plot, the following variables were estimated: number of trees per hectare (N), average diameter (DN), average height (H), mean square diameter (dg), dominant height (H0), basimetric area per hectare (G), total volume (V) and index of Hart-Becking (IH) whose mathematical expressions are shown in Table 2.

The descriptive statistics of the sampling plots is summarized in Table 3.

Methodology

The methodology proposed by ^{Barrio and Álvarez (2005)} and ^{Pérez et al. (2009)}; for the construction of a density management diagram (DMD) the following steps are followed:

The Hart-Becking index is defined as the ratio between the average spacing of the trees and their dominant height (H0), expressed as a percentage; the average distance between trees depends on the theoretical distribution of these in the terrain that is assumed (^{Diéguez et al., 2009}) according to the following formula:

Where:

IH = Hart-Becking Index expressed as a percentage (%)

K = Constant (4/3)^0.25, under the assumption that it is a natural or three-tree distribution of the individuals in the forest, or if the distribution of the trees in the land assumes a square mesh (forest plantations), then the constant = 1.

N = Number of trees per hectarea

H0 = Dominant height per plot

3. A system of two non-linear equations.

The first proposed equation allows to predict the mean square diameter (dg), from the number of trees per hectare and the dominant height of the stand. This equation is based on the relationship between the average size of the trees, the density and an indicator of productivity.

The second equation estimates the productivity of the stand expressed as the relationship between the volume of mass and the volume of a representative tree (given by the product of the mean square diameter and the dominant height) and the number of feet per hectare.

Where:

dg = Mean square diameter (cm^-2)

V = Mass volume (m³ha^-1)

H0 = Dominant height (m)

N = Number of trees per hectarea

β0…β6 = Estimated parameters

Adjustment of equations

The DMD was built from the Hart-Becking Index, and two base functions, non-linear models that through an iterative process allow to estimate the parameters to make the representation of the graph.

The adjustment of the described models was done in the statistical software R Project 3.4.0. Rstudio 1.0.143 (^{R Core Team, 2017}), with the nls function, which allows the iterative process to which initial starting values are added to estimate the values of β0... β6, based on a positive convergence of the process.

When performing the simultaneous adjustment of the base equations (2) and (3) as a first step, N and H0 function as exogenous variables (their values are determined completely independently of the system); V is an endogenous function (it depends on other variables, that is, the model tries to explain it) and dg is an endogenous instrumental variable (it appears on both sides of equality), due to the correlation between the components of the system (^{Pérez et al., 2011}).

In order to evaluate the reliability of the model at a level of 95 %, the coefficient of determination of R² adjusted and the square root of the mean square error RMSE (for its acronym in English) was estimated.

Where:

Y _i = Observed value of the dependent variable

ŷ _i = Predicted value of the dependent variable

y = Average value of the dependent variable

p = Number of parameters to be estimated in the adjustment

N = Number of data used in the adjustment

Due to the correlation of the components of the error existing in the variables that make up the system of the two base equations, a simultaneous adjustment of the nonlinear equations is necessary, which was achieved by means of the maximum likelihood estimation method of the FIML procedure (Full Information Maximum Likelihood). The system of equations was adjusted simultaneously using the MODEL procedure of the SAS / ETS^® statistical system (^{SAS, 2004}).

Graphic representation of DMD

Once the parameters of equations (2) and (3) have been estimated, it is possible to obtain the expressions that allow the values for the Hart-Becking Index IH, the quadratic mean diameter dg, and the volume of the V stand to be plotted.

1. The level of thickness of the mass is expressed by the Hart-Becking Index. The representation of the isolines results from clearing N from equation (1).

2. The representation of the isolines of the mean square diameter is made by means of equation (2) and is set to dg as a constant, N is cleared and values are assigned to the dominant height H0 for each value of the mean square diameter.

3. The isolines for the volume in the graph are derived from clearing the number of trees per hectare N of equation (3) and dominant height values are given for each value of volume V.

Stand data

The distribution of the data that represent a normal distribution for the adjustment of the density models is shown in Figure 2, with a mean square diameter that ranges from 11.5-45.5 cm and a dominant height that ranges from 7.3 to the 25.2 m

Diámetro medio cuadrático = Mean square diameter; Altura dominante = Dominant height; Gráfico de dispersión = Dispersion graphic.

Figure 2 Dispersion graphic of the stand variables. 

Adjustment parameters of the models

The adjustment parameters of the models used in the development of the DMD chart are shown below, in Table 4. The final formulas adjusted with the parameters are:

When inspecting it is posible to state that their fit was highly significant.

Density management diagram for the genus Pinus in the Pueblo Nuevo ejido, Durango

The graphic representation of the Density Management Diagram is shown in Figure 3, in which the superposition of the key elements for the forest management of a stand through isolines allows to interpret the behavior of a forest mass, and facilitates the taking of decisions in silvicultural development. It takes the Hart-Becking Spacing Index as the first option and serves as the basis for the cut approach, since this index is independent of station quality.

Densidad =Density; Diagrama de manejo de densidad = Density management diagram; Diámetro medio cuadrático = Mean square diameter.

Figure 1 Density management diagram for Pueblo Nuevo ejido, Durango. 

The parameters obtained through the simultaneous adjustment of the equations were highly significant, which makes effective the decision making in the area of management when using DMD as a basis in the cutting scheme.

Practical application

The use of DMD allows the planning of different cutting schemes that best suit the development of the stand; in this case, it is to obtain timber with the right dimensions for sawing with a mean square diameter of 30 cm. Figure 4 illustrates the cutting scheme based on the Hart-Becking index in the form of an ascending staircase. Table 5 shows the production table derived from the diagram whose final objective is short to obtain 250 trees per hectare, making short every 3 or 4 meters without exceeding a thickness level of 30 %.

Densidad = Density; Diagrama de manejo de densidad = Density management diagram; Diámetro medio cuadrático = Mean square diameter.

Figure 4 Cutting sheme for the production of sawmill timber. 

The development of the Diagrams of Management of Density based on the Index of Hart-Becking, allows to carry out the silvicultural planning in the determination and intensity of the cuts according to the level of desired spacing that avoid the high existence of the natural mortality and that allows an adequate occupation of the land in common agreement with what was concluded by ^{Diéguez et al. (2009)}. This allows the end user to plan the cutting scheme based on two factors that are of great importance: i) the objective of cutting the stand at the end of the turn; and ii) the level of thickness desired for forest management.

According to ^{Corral-Rivas et al. (2015)} and ^{Escobedo (2014)}, who developed a DMD for natural forest stands and proposed a cutting scheme based on the Reineke density index (IDRR) or maximum density, conclude that the Thickness Index ranges from 25 to 30 %, which conffirms the reliability of using only the Hart-Becking index as a basis for the production scheme.