El propósito del presente estudio fue determinar el patrón de distribución espacial de los árboles presentes en la región de El Salto, Pueblo Nuevo, Durango. Para ello, se seleccionaron 12 comunidades representativas de las condiciones de los bosques de la región, donde se ubicaron, de manera aleatoria, 269 sitios circulares de 0.1 ha, distribuidos en un intervalo altitudinal de 1 500 a 3 000 m. El patrón de distribución de los árboles por nivel de altitud se determinó aplicando los índices de Cox y Morisita, mientras que el correspondiente para cada especie se probó con las pruebas de ajuste de la distribución de Poisson y binomial negativa. El grado de agregación se evaluó a través del parámetro de agregación K, proporción de la densidad media por taxón en relación a la media 1/K, el número promedio por individuo por especie, con respecto a los individuos presentes en la unidad de muestreo m*. Los índices de Cox y Morisita variaron de 12.05 a 17.39 y de 1.18 a 1.28, respectivamente, los cuales indican que en cada nivel de altitud los individuos se distribuyen de manera agregada. A su vez, el índice de Cox y el ajuste de la distribución binomial negativa (P>0.05) señalaron que los individuos por especie se distribuyen de manera agregada. Los valores del parámetro K, 1/K y m* variaron de 0.002 a 0.454, 2 a 500 y 9 a 69, respectivamente evidencian un alto grado de agregación entre individuos.

The purpose of this study was to determine the pattern of spatial distribution of trees present in the region of El Salto, Pueblo Nuevo, Durango. For this purpose, 12 communities that are representative of the conditions of the forests of the region were selected, where 269 round sites of 0.1 ha were randomly distributed at an altitude range of 1 500 to 3 000 m. The pattern of distribution of the trees by altitude level was determined by applying Cox’s and Morisita’s indices, and the pattern corresponding to each species was tested with the Poisson and negative binomial distribution adjustment tests. The level of aggregation was evaluated through the clustering parameter K, the ratio of the average density per taxon in relation to the mean 1/K, and the average number per individual per species in relation to the individuals present in the sampling unit m*. Cox’s and Morisita’s indices ranged from 12.05 to 17.39 and from 1.18 to 1.28, respectively, indicating that at each level of altitude individuals are distributed on an aggregate basis. At the same time, Cox’s index and the negative binomial distribution adjustment (P>0.05) pointed out that the individuals per species are distributed in clusters. The values of the parameter K, 1/K and m*, which ranged from 0.002 to 0.454, from 2 to 500 and from 9 to 69, respectively, exhibit a high degree of aggregation between individuals.

The organisms belonging to a population or community considered as a continuous inhabitable space can be distributed in a random, uniform or aggregated way. The random distribution implies that all the points in space have an equal possibility of being occupied and that the existence of an individual in a point does not affect the presence of another. In the uniform distribution, the individuals exhibit a negative interaction expressed by the competition for a resource, while the clustered distribution exhibits positive effects of attraction to form dense groups of individuals (Franco-López et al., 1989).

The study of the spatial dispersion makes it possible to identify mechanisms and factors that promote inter- and intraspecies coexistence and the plant diversity of ecosystems (Montañez et al., 2010). The analysis of the pattern of spatial distribution is an important action to determine the microenvironmental conditions required by each of the taxa that make up a community and thus, account for the establishment and renewal capacity, the skill development and growth, the probability of mortality and the competence of the species (Linares-Palomino, 2005).

In turn, the knowledge of the biological, ecological, biogeographic and anthropic factors that defines the distribution of species and their ecological affinities is important for the structuring of conservation plans (Maciel-Mata et al., 2015). The spatial distribution of the taxa within the landscape is due to a differential selection of the biotic and abiotic resources that favor their optimal development (Díaz et al., 2012).

The analysis of the distribution pattern of individual trees is supported by different indexes and probabilistic methods; the application of some of them have special requirements with respect to the distribution, the size or shape of the sampling units, the spatial location of individuals and the distance between them (Clark and Evans, 1954; Ripley, 1977; Aguirre et al., 2003). In particular, the analysis of Cox’s and Morisita’s indices, as well as the negative binomial and Poisson distribution adjustment tests make it possible to use non-contiguous samples and only require information about the abundance of individuals per sampling unit (Ledo et al., 2012; Ledo, 2013).

Regardless of the vast knowledge that exists about the identification, the volumetric assessment and the growth rates of the timber taxa in the region of El Salto, Durango, little has been documented about the pattern of spatial distribution, and the basic and relevant information that helps to explain the influence of the local environment in the presence of the species. As a contribution to this knowledge, the purpose of this study was to determine the spatial distribution pattern in terms of the degree of aggregation, uniformity or randomness of the tree species, through an altitude gradient.

The study was carried out in the forest region of El Salto, Durango, which is located in the massif of the Sierra Madre Oriental (Western Sierra Madre), to the southwest of the state (Figure 1). It covers approximately 507 127 ha; the altitudes above the sea level range from 1 400 to 3 000 m (INEGI, 2010).

12 communities that are representative of the conditions of the forests of the region were selected, where 269 round 0.1 ha sites were randomly placed in order to register the control variables of the site, as well as the location, presence and abundance of the species.

The distribution pattern of the species for each level of altitude was estimated using Cox’s and Morisita’s índices (Ip). The former (Cox, 1971), also known as Strand’s index (1953), is based on the estimation of the variance/mean ratio of the number of elements of the samples; while Morisita’s index (1959) is a measure of dispersion, independent of the size of the sampling unit or the density present in each sample (Krebs 1999; Badii et al., 2011). According to Ledo et al. (2012), both indexes are applicable when there is a lack of information regarding the location and the distance between the individuals and the necessary information comes from samples that are not necessarily contiguous -a situation which prevailed in this study.

Where:

= Mean

S 2 = Variance

n = Sample Size

x i = Number of individuals in quadrant i

If the result of Cox’s index is equal to 1, it is established that the distribution is random; if it is lower or higher, then it follows that individuals are evenly distributed. The decision rule of Morisita’s index is similar to that of Cox’s (Ledo et al., 2012), except that the reference value related to the random distribution is zero. Values below or above zero indicate the presence of a uniform or aggregated distribution, respectively.

The pattern of random distribution of individual trees by species within their area of occupation was tested using the Poisson distribution, and when this test was not significant, the negative binomial distribution was used to test whether or not they have a clustered distribution. The Poisson distribution function, P(x) is expressed as:

Where:

e =Base of the natural logarithm (2.7183)

λ = Mean

x ! = x factorial

The expected frequencies of Poisson, E(x) were estimated by multiplying the total number of individuals registered in the sampling (N) by the Poisson probability function P(x):

The implementation of the function of the probability of the negative binomial involved the calculation of a first approximation of a K value, which is considered to be a measure of aggregation (Badii et al., 2011):

Where:

= Average number of individuals

S 2 =Variance in the number of individuals

This first approximation of the K value is applicable when it is below or equal to 3 and the mean is less than or equal to 1. If the above condition is not met, then the value of K is reestimated using the following equation:

Where:

K = Level of influence to the aggregation

n = Total number of samples

n0= Number of samples with zero individuals

m = Average number of individuals

K is determined by the method of trial and error, starting with the previously estimated value of K.

After calculating the value of K, the expected frequencies for zero individuals, f0, was estimated; this estimation required knowing the probability of failure, (q), which depends on the probability of success, p.

Subsequently, the frequencies for Xi individuals, fe x , were estimated:

Where:

p = Probability of success or the occurrence of an event in a binomial distribution

q = Probability of failure or non-occurrence of the event in a negative binomial distribution

x = Class number or number of individuals or elements

In order to determine the goodness-of-fit of the Poisson or negative binomial distributions, the statistical test Chi-square test was used (X 2 ), at a significance level of 0.05:

Where:

Fx = Observed frequencies

Ex = Expected frequencies

If the value of the probability of X 2 is within the range established by the values in the table of X 2 (0.975 - 0.025), the distribution is random (Ludwig and Reynolds, 1998), but if the value of the result of X 2 is higher, the distribution will be clustered; and if it is lower, the distribution will be uniform.

The degree of clustering was estimated upon the basis of the proportion in which the mean abundance of individuals of each species exceeds the average of the total abundance

and the clustering index m* (Lloyd, 1967), which defines the average number per individual in relation to other individuals (m*).

Where:

m * = Clustering index

m = Average density

K = Parameter indicative of the clustering

Of the 12 Pinus species and 10 Quercus species recorded, P. strobiformis Engelm., P. cooperi C. E. Blanco, P. durangensis Martínez, P. lumholtzii Robins & Ferns, P. michoacana Martínez, P. teocote Schiede ex Schltdl., Q. crassifolia Humb & Bonpl., Q. rugosa Née, Q. sideroxyla Humb & Bonpl. and Q. candicans Née are located through the entire range of the studied altitude distribution (1 500 to 3 000 m). Juniperus deppeana Steud. is located at an altitude of 1 800 to 3 000 m; Alnus firmifolia Fern and Populus tremuloides Michx are located at an altitude of 2 400 to 2 700 masl. As for Populus tremuloides Martínez var. coahuilensis (I.M. Johnston) Martínez and Pseudotsuga menziesii (Mirb.) Franco, they are distributed between 2 400 to 2 700 and 2 700 to 3 000 masl, respectively.

Although Cox’s index (12.05<X<17.39) and Morisita’s index (1.18<IM<1.28) indicate that the species are clustered together, the degree of aggregation between levels was different. Table 1 shows that the level of aggregation was higher at altitudes of 1 800-2 100 (CI= 16.30 and MI=1.28) and 2 700-3 000 (CI=17.39 and MI=1.25).

The χ 2 test, applied to the Poisson distribution adjustment indicated that individuals of each species are distributed in a non-random way (P<0.05). On the other hand, the values of the clustering parameter K (0.002<K<0.454), the proportion of the average abundance of individuals in relation to the mean density 1/K (2<1/K<500), the mean clustering index m* (9<m*<62) and the variance/mean CI (4<CI<41) evidenced the presence of a high level of clustering between individuals of the same species (Table 2). The minimum and maximum value of K, and, therefore, the minimum and maximum proportion of average abundance in relation to the average density, corresponded to Q. sideroxyla Humb. and Q. urbanii Trel., respectively, while the minimum and maximum clustering index, expressing the average number per taxon in relationship to the other individuals of the same sampling unit (Lloyd, 1967), occurred in Q. durifolia Seemen and P. occarpa Schiede ex Schltdl., which had the minimum and maximum Cox’s index, respectively.

The presence of the clustered distribution was corroborated, in most species, with the goodness-of-fit test of the negative binomial distribution (P>0.05). The exception occurred in Pinus strobiformis Engelm. (P=0.003), Pinus cooperi C. E. Blanco (P=0.0003), Pinus douglasiana Martínez (P=0.002), Pinus durangensis Martínez (P=0.005), Pinus teocote Schiede ex Schltdl. (P=0.028) and Quercus durifoliai Seemen (p=0.002), for which the Poisson and negative binomial distribution tests did not exhibit a significant adjustment (P<0.05). In order to determine that the pattern of distribution of individuals corresponds to the added, using as a reference the estimators of K, 1/K and CI, whose values varied from 0.027 to 0.324, from 3 to 37 and 8 to 20, respectively.

According to Taylor (1961), the clustered distribution is the most common form in nature, especially in areas populated with trees in the initial stages of development, which, when their density decreases by effect of the mortality associated with the competition, tend to switch to a random distribution (Aldrich et al., 2003; Rozas and Camarero, 2005). In the present study, the clustered distribution pattern determined at each altitude level can be attributed to the fact that the analysis of the distribution pattern was determined on the full forest mass existing at each level, which -being mixed, non-coeval and heterogeneous- consists of a high density of trees of smaller dimensions, complemented with a smaller amount of trees that correspond to the intermediate and upper strata. The presence of trees that develop under the upper strata indicates that there is a positive relationship between individuals of the same or different species.

Montañez et al. (2010) hold that, although the clustered species require common environmental conditions for their establishment and development, the formation of this type of distribution is due to the interactions between individuals of the same and different species, as well as between the individuals and the environment. According to Badii et al. (2011) and Montañez et al. (2010), the physiological response of plants to the abiotic microenvironmental conditions such as temperature, relative humidity, direction and wind speed, physical and chemical properties of the soil, slope and exposure of the areas may account for the ability of different species to cluster together. In general, dispersal limitation (Hubbel, 2001) or habitat specialization (Tilman, 1982) are the mechanisms that determine the distribution pattern of species.

The results of this study, as well as those obtained by Rossi and Huguchi (1998) in a study of a tropical forest of the Amazon, prove that Cox’s and Morisita indices generate similar results, regardless of the estimation method, in the assessment of the distribution pattern of the species that make up the studied populations or communities. Malleux (1973), Linares-Palomino (1973), (2005), López (2008) and Zarco-Espinosa et al. (2010) documented the presence of species clusters in tropical forests; Ledo (2013), in tropical cloud forests; Aguirre et al. (2003), Domínguez-Calleros et al. (2014) and Montañez et al. (2010), in high mountain forests; Juárez-Sánchez et al. (2014) in temperate mixed forests, and Martínez-Antúnez (2014), in species of the temperate climate of the Western Sierra Madre across the state of Durango.

Cox’s and Morisita’s spatial distribution indexes indicate that the groups of tree species of the region of El Salto, Durango, tend to cluster, forming mixtures of species. Similarly, the clustering parameter K, the proportion in which the average density of each species exceeds the average density 1/K, the clustering index m* and Cox’s index, corroborated by the negative binomial distribution adjustment test, indicate that the individuals cluster by species.