Study area and data

The study area comprised 446 permanent forest research sites, located in the mountain system known as the Western Sierra Madre, in the state of Durango, between 22°54' and 26°35' N, and 104°40' and 107°06' W, with altitude intervals ranging between 1 200 and 2 8 00 m (Figure 1).

Figure 1 Study area with the distribution of the permanent forest research sites. 

The sites were located through a systematic sampling in a mesh of points with a variable separation of 3 to 5 km. They measured 50 × 50 m, and for each tree with a normal diameter (at a height of 1.3 m above soil level) of 7.5 cm or more, the following variables were considered: species, dominance, normal diameter (cm), total height (m), commercial height (m), cover (m), sanity, physical damage, azimuth (°), distance from the trees to the center of the site (m), and spatial location through coordinates X and Y (Table 1). Normal diameter was calculated from the average of two crosswise measurements taken with a 650 mm Haglöf Mantax forest caliper; height was measured with a Haglöf Vertex IV hypsometer, and the azimuth, with a Tandem Suunto 360PC/360R clinometer.

Characterization of the spatial structure

The spatial distribution of trees, the interaction between groups of species, and the size of the trees were determined according to the methodology described below.

Spatial distribution of the trees. Ripley’s K(r) function was applied; it is probably one of the most widely used tools for characterizing point patterns (Equation 1) (^{De la Cruz, 2013}), since it allows knowing in a graphic manner, the type of distribution of the trees within the sites at different distances or scales. The K(r) function represents the expected number of trees separated by a distance equal to or shorter than r, and the number of trees that exist in a circle with a variable radius (r) around a particular tree. The empirical function estimated based on the real data was compared to the one generated in simulated sites through a random process; the typologies of the structures present at all the sites at different scales were determined in this manner.

Where:

A = Study area

n = Number of points in the study area

lij = Distance between the i ^th and j ^th trees

Wij = Correction factor of the border effect for points i-j

^{Diggle (1983)}, in his discussion of ^{Ripley’s article (1977)}, proposes using the square root of the K(r) function because it stabilizes the variances and facilitates interpretation (Equation 2). The following transformation has a mean (μ) equal to zero and an approximately constant variance (σ) when the distribution of the trees is random:

At a site with a random distribution, L(r) is equal to zero at any distance. If L(r) has a negative value (i.e. if it has a lower value than expected for a random distribution), then the trees are somewhat distant from one another, which suggests age evenness or inhibition. Conversely, positive L(r) values (i.e. above the value expected for a random distribution) indicate that the trees are closely grouped or that there is a certain attraction between them. Therefore, the L(r) function was utilized in the present study to characterize the type of point pattern of the permanent sites as a function of the scale.

Interaction between tree marks

Interaction between groups of species. The spatial relationship or interaction between trees of the Pinus and Quercus genera was examined with the K ₁₂ (r) function (Equation 3), which determines the number of points with characteristic 2 and distance (r) between the points with characteristic 1. It is defined as follows (^{Moeur, 1993}):

Where:

K ₁₂ (r) = Ripley’s bivariate function

A = Surface area of the sampling site

n ₁ and n₂ = Number of trees with characteristics 1 and 2, respectively, of all the pairs of trees separated by a distance l _ij ≤ r

w _ij (r) = Border correction factor equal to the inverse of the proportion of the circumference of a circle centered in tree i and passing through tree j, located within the site

(w_ij(r) = 1 for circles that are completely within the sampling site w _ij (r) > for those cases that require correction for border effect

Its normalized function indicates the intensity, type, and distribution interval exhibited by those points with a particular characteristic (^{Soto et al., 2010}). The value expected for function L ₁₂ (r) is null when the distribution is independent; positive, when the points tend toward attraction, and negative, when the patterns tend toward repulsion. Thus, it is possible to define whether two marks (Pinus and Quercus, in this case) attract or repel one another, or are independent from one another:

Where:

L ₁₂ (r) = Normalized Ripley’s bivariate function

K ₁₂ (r) = Ripley’s bivariate function

ϖ = Pi value

r = Distance

Interaction between the tree diameters. The mark correlation function Kmm(r) defined by Equation 5 was utilized (^{Stoyan et al., 1987}; ^{Ávila-Flores et al, 2014}). In this study, x is the location of the tree, and m(x), the normal diameter used as a characteristic of that tree (^{Stoyan et al., 1987}). Therefore, Kmm(r) is defined as:

Where:

E = Expectations

m ₁ and m ₂ = Marks associated to two points of the process by a distance r

m ₁ , m' = Independent realizations of a marginal distribution of marks

In the present study, the function Kmm(r) relates the diameter of a pair of trees located at a given distance (^{Penttinen et al., 1992}). Values under 1 suggest a negative correlation or mutual repulsion; values above 1 indicate a positive correlation or mutual attraction; and those that remain at the mean indicate independence between these marks (^{Martin, 1994}).

The spatial analyses described above were performed for all 446 permanent sites available for the state of Durango. The analysis of the information was performed with the statistical software R Studio^™ (^{Rstudio Team, 2015}), through the ggplot2 and spatstat libraries (^{Baddeley and Turner, 2005}).

Spatial distribution of the trees

Figure 2 shows the current spatial distribution of the trees in two of the analyzed permanent sites. The observed distribution is presented on the left side (images a and b), while the estimation of the L(r) values in relation to the expected L(r) value for a fully random distribution, based on the lower and upper limit estimated with a 95 % confidence interval in 199 random simulations, can be seen on the right side (charts c and d). It shows how the empirical distribution of the L(r) values can be used to examine the repulsion of a random distribution in terms of the distance.

Estimation of L(r) (continuous line) at scales of 0 to 12 m. The gray shaded area represents the lower and upper limit, both of which regenerated with a 95 % confidence interval based on 199 random simulations. Image a) corresponds to a clustered distribution (Site ME_10_00069), and image b) to an even-aged, random and clustered mixture (Site ME_10_00169).

Figure 2 Current spatial distribution of trees in two characteristic sites of the study area (images a and b). 

Although no details are expressed for each of the 446 sites, the results of the two examples are characteristic of the mixed, uneven-aged forests in the state of Durango.

Table 2 summarizes the information of the spatial distribution of all the sites analyzed in terms of the scale. A random points pattern was observed to dominate at distances of less than 2 m, as the estimated L(r) curve remains within the confidence interval determined for a random points pattern, while a clustering tendency prevails in most of the studied sites at distances of over 4 m: the estimated L(r) values are positive and are situated above the upper confidence limit defined for a fully random points pattern.

The even distribution of the trees was practically not registered within the study area because these forests regenerate naturally, not through the use of forest plantations (^{Castellanos-Bolaños et al., 2010}).

Regardless of the method used to characterize the spatial distribution of the permanent forest research sites, the results are similar to those cited by ^{Hernández et al. (2018)}, ^{Solís-Moreno et al. (2006)} and ^{Aguirre-Calderón et al. (2003)}, who describe distribution patterns with a degree of clustering in mixed, uneven-aged forests in the state of Durango. According to ^{Vallejo (2009)}, the clustered distribution in forests is related to a good quality of the habitat available for the fauna, regeneration in forest clearings, a limited dispersion of the seeds, and the topographic conditions of the land.

On the other hand, studies have been made with Ripley’s K function that describes random distribution patterns in mixed forests of Mexico. This is the case of ^{Ávila-Flores et al. (2014)}, ^{Ruiz-Aquino et al. (2015)}, and ^{Rubio-Camacho et al. (2017)}, in whose papers these authors make reference only to a limited number of experimental sites (2 to 9), and, therefore, their results cannot be generalized.

Interaction between pine and oak trees

The results of Ripley’s bivariate L ₁₂ (r) function are shown in Table 3 and indicate that, in most sites (an average of 82 %), no significant attractions or interactions exist between Pinus and Quercus trees at all the considered distances; the estimated L ₁₂ (r) curve was kept within the confidence interval determined for a pattern that was independent from the marks of the points. However, L ₁₂ (r) was observed to be positive in more than 11 % of the sites at distances of over 6 m; this indicates a significant positive attraction between pine and oak individuals.

Sites with repulsion at all the assessed distances were recorded, for example, site ME_10_00018, where L ₁₂ (r) was below the lower limit of the confidence interval determined by a mark-independent points pattern at all the assessed distances.

The results evidenced the dominance of a pattern of independence between the positions of the Pinus and Quercus individuals, as these had a primarily random distribution. According to ^{Chen and Bradshaw (1999)}, this condition is characteristic of mixed forests in maturity or second-growth phases, as in most of the studied sites. Cases of attraction between the Pinus and Quercus trees increase in relation to the distance and can be attributed to the existence of low-efficiency seed dispersion processes (^{Stoyan and Penttinen, 2000}).

^{Rozas and Camarero (2005)} point out that the L ₁₂ (r) function is a very useful tool in forest ecology, as it allows the study of the role of coexistence among the species.

Figure 3 shows the result of Ripley’s bivariate L ₁₂ (r) function for two of the sites analyzed for the purpose of assessing spatial interactions between pine and oak individuals.

Estimation of L ₁₂ (r) (continuous line) at scales of 0 to 12 m. The gray shaded area represents the lower and upper limits of the confidence interval generated with a 95 % confidence level based on 199 random simulations. Image a) reveals independence between the individuals of the Pinus (+) and Quercus (x) genera (ME_10_00099), and b) shows repulsion (ME_10_00018).

Figure 3 Results of the bivariate L ₁₂ (r) function, showing the interaction between pine and oak individuals in two characteristic sites of the study area (images a and b). 

Size differentiation

The prevalence of an independent pattern was made evident in the diameter sizes of the trees (in 90 % of the sites in average), at all the studied distances. The estimated curve remained within the confidence interval determined for a mark-independent pattern for each site, Kmm(r) = 1 (Table 4). According to this result, it is common to observe trees of all diameter categories (>7.5 cm DBH) distributed within the research sites displaying hardly any significant attractions or inhibitions at the assessed scales.

A negative correlation (Kmm(r) < 1) at scales of 0 to 4 m was obtained at certain sites; this indicates a significant repulsion between the individuals, caused by the existence of competition for growth space between them. On the other hand, only in a few isolated cases were positive attractions observed between the diameters of the trees (Kmm(r) > 1) at distances greater than 4 m.

In regard to the values obtained with the mark correlation function Kmm(r) for two sites, trees of different sizes were observed to mix independently within the sampling site (ME_10_00352), while at another site the repulsion was significant (ME_10_00477) (Figure 4).

The continuous line represents the estimated Kmm(r). The dotted line shows the theoretical value of the random distribution, and the gray shaded area represents the interval of confidence for a mark-independent pattern. Image a) depicts an example of independence between the diameter sizes of the individuals, and image b) shows a good example of a significant repulsion.

Figure 4 Correlation function for two sites, based on the diameter of the trees as a characteristic mark. 

According to ^{Soto et al. (2010)}, a random correlation between the diameter sizes is characteristic of uneven-aged forests, as trees with various sizes tend to be distributed in this manner. On the other hand, certain authors have described negative correlations at small scales in temperate mixed forests as a result of the competition process imposed by a minimum distance between the individuals (^{Szwagrzyk and Czerwczak, 1993}; ^{Réjou et al., 2011}).