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On-line version ISSN 2521-9766Print version ISSN 1405-3195

Agrociencia vol.52 n.7 México Oct./Nov. 2018


Natural Renewable Resources

Density funcions: an application for delimiting optimal intervals of climate and physiography for forest species

Pablo Antúnez1 

Carmen Z. Quiñones-Pérez2 

Wenceslao Santiago-García1 

Mario E. Suárez-Mota1  * 

1 División de Estudios de Postgrado-Instituto de Estudios Ambientales, Universidad de la Sierra Juárez, Avenida Universidad S/N, Ixtlán de Juárez, 68725 Oaxaca, México.

2 Instituto Tecnológico del Valle del Guadiana. Carretera Durango-México Km 22.5 Villa Montemorelos, C.P. 34371, Durango, México.


The space a species occupies in a natural system can be delimited by the physical-geographic medium or by the environmental conditions that define it. The objective of this study was to delimit climate intervals in which the maximum presence rate occurs of three tree species native to the Sierra Norte of Oaxaca, Mexico (Pinus pseudostrobus Lindl (var. Apulcensis), Pinus patula Schl. et Cham, and Quercus macdougallii Martínez), in function of nine environmental variables using the Weibull density function and the finite Gaussian mixture model. To this end, we used data from 634 plots measuring 1,000 m2, which were established systematically in the study area. The results showed that high dispersion of the two pines species is related to mean precipitation from April to September. In contrast, the scarce presence of Quercus magdougalli, an endemic species, seems to be related to the reduced intervals of winter precipitation and to altitude. The two density functions tested allowed definition of optimal environmental intervals for each species. The finite mixture model was more flexible than the Weibull function when identifying bimodal distributions, particularly for the two pines species, whose observed dispersion pattern was more heterogeneous than that of Quercus. The results obtained will serve to prioritize areas for purposes of conservation or commercialization.

Key words: Gaussian mixture model; Weibull function; Quercus magdougalli; Sierra Norte of Oaxaca; temperate forest


El espacio que ocupa una especie en el sistema natural puede delimitarse por el medio físico-geográfico o por las condiciones ambientales que lo definen. El objetivo de este estudio fue delimitar intervalos climáticos en los que ocurre la tasa de presencia máxima de tres especies arbóreas (Pinus pseudostrobus Lindl (var. Apulcensis), Pinus patula Schl. et Cham y Quercus macdougallii Martínez) nativas de la Sierra Norte de Oaxaca, México, en función de nueve variables ambientales usando la función de densidad de Weibull y el modelo de Gauss de mezclas finitas. Para lo anterior, se usaron datos de 634 parcelas de 1,000 m2 las cuales se establecieron sistemáticamente en el área de estudio. Los resultados mostraron que la alta dispersión de dos de las especies estudiadas (ambas de pino) está relacionada con la precipitación media de abril a septiembre; en contraste, la escasa presencia de Quercus magdougalli (especie endémica) parece estar relacionada con los intervalos reducidos de la precipitación en el invierno y la altitud. Las dos funciones de densidad probadas permitieron definir los intervalos ambientales óptimos para cada especie. El modelo de mezclas finitas fue más flexible que la función de Weibull al identificar distribuciones bimodales, en particular para las dos especies de pino cuyo patrón de dispersión observado fue más heterogéneo que el de Quercus. Los resultados obtenidos podrían servir para priorizar áreas con fines de conservación y comercialización.

Palabras clave: Modelo de Gauss de mezclas finitas; función de Weibull; Quercus magdougalli; Sierra Norte de Oaxaca; bosque templado


Several analytical tools were explored for studying distribution and abundance of live organisms, mainly statistical models or these in combination with geographic information systems, to characterize species’ habitats (Austin, 1987; Segurado and Araujo, 2004; Elith et al., 2006) or to evaluate the response of a specific species in function of change in environmental variables that define the bioclimate niche (Antúnez et al., 2017a). Models based on maximum entropy algorithms are also used to predict the potential distribution of organisms (Brotons et al., 2004; Phillips et al., 2009; Franklin, 2010).

The space a species occupies can be delimited by the physical-geographic medium or by environmental conditions (Pearman et al., 2008; Elith y Leathwick, 2009). Maps are a valuable tool for illustrating the physical-geographic space where a species can find ideal conditions. But delimiting and representing only the environmental space is not easy because each variable of the natural system is a variable in magnitude and intensity (Martínez-Antúnez 2013; Antúnez et al., 2017a). This task could be facilitated if we knew the optimal intervals of those variables whose effect can limit or potentiate the abundance of a species in a location. That is, it is easier to delimit the optimal space of a species in function of the most relevant variables that delimit the space defined by all the variables (multi-dimensional space) (Hutchinson, 1957; Austin and Smith, 1990).

Statistical likelihood functions are used to describe the relationship between living organisms and the environment parting from observed patterns, to explain the relationship between the species and their area of greatest abundance, or to determine the spatial pattern and identify optimal climate values (Borda-de-Água et al., 2002; Magurran, 2004; Gowda, 2011; Verberk, 2012; Martínez-Antúnez, 2015).

Given that it is possible to model the largest concentration of data in a probabilistic space, it is also possible to use the same principle to define an interval of any environmental variable based on maximum likelihood of a density function (Antúnez et al., 2017a). In this sense, a probability density function can be a useful tool in defining climate values in which the maximum probability that an abundance of a species would occur.

The objective of this study was to determine the environmental intervals where the maximum abundance of individuals of three forest species native to the Sierra Norte of Oaxaca, Mexico, would occur using the Weibull density function and the finite Gaussian mixture model. The hypothesis was that these functions permit defining the width of the partial niche with each of the climate variables whose effects are significant to the distribution and abundance of forest species.

Materials and Methods

Study area

Santiago Comaltepec is located in the Sierra Norte region of Oaxaca (17°33’35” N and -99°26’32” W) southeast of Mexico City and has an area of approximately 26.5 km2 (Figure 1). Altitude varies between 1700 and 3000 m. Mean annual high temperature is 13.4 ºC, the mean annual low temperature is 4.7 ºC and summer rainfall is 600 to 1200 mm (CNA, 2017; INEGI, 2015).

Figure 1 Map of the study area. 

Sampling and studied variables

In the study of live organisms, the most conventional abundance indicators are dominance, frequency and density (Schweik, 2017). In our study, relative density of each plot was used as the indicator of abundance, which is defined as the relationship between the number of individuals of each species registered in each plot and the total of individuals of the same species in all the plots. Systematic sampling was used to establish the sampling plots, and each sampling unit had an area of 1000 m2, where individuals with a diameter at breast height larger than or equal to 7.5 cm were counted. In the study area, 634 plots were established.

The tree species studied were Pinus pseudostrobus Lindl (var. Apulcensis), Pinus patula Schl. et Cham and Quercus macdougallii Martínez. The first species is often used (in the study region) to reforest areas with degraded soils or sites without vegetation because it is a fast-growing species. The second is of high demand in sawmills, furniture factories, and cellulose and paper industries (Muñoz et al., 2011). Quercus macdougallii is a species endemic to the Sierra Juárez and was registered in 33 of the 634 plots; it has no commercial use and is in the red list of endangered species in the category of “vulnerable” of the International Union for Conservation of Nature (IUCN, 2017).

The variables selected for the study were altitude above sea level of each site (ALT, m), dominant slope of each plot (PEN, %), geographic exposure (EXP: zenithal (1), north (2), northeast (3), east (4), southeast (5), south (6), southwest (7), west (8), and northwest (9), mean winter precipitation (Nov+Dec+Jan+Feb) (WINP, mm), Julian date of the last freezing date of spring (SDAY, day), balance of precipitation summer/spring (Jul+Aug)/Apr/May) (SMRSPRPB), precipitation from April to September (GSP, mm), annual aridity index (BHH) whose value was estimated using the square root of the sum of degree-days above 5 ºC divided by mean annual precipitation (Rehfeldt et al., 2006; Sáenz-Romero et al., 2012), and mean summer precipitation (Jul+Aug) (SMRP, mm). These variables were selected by a multivariate correlation analysis using the bootstrapping method (Yoder et al., 2004), selecting the variables that had the highest coefficients (< 0.8 with at least one species) of a total of 22 available variables that include temperature measurements (high, low, average), precipitations in specific periods and frosts (Rehfeldt et al., 2006). Physiographic variables were recorded in the field with a GPS receiver (global positioning system) for altitude and a Suunto® clinometer for exposure and slope. The other variables were obtained with the ANUSPLIN® modeler of the Forest Service of the US Department of Agriculture (Rehfeldt et al., 2006; Crookston et al., 2008; Sáenz-Romero et al., 2010), whose algorithms are based on historical climate information from more than 4,000 weather stations in Mexico, southern US, Guatemala, Belize and Cuba, from 1961 to 1990. These variables were used in similar studies because of their importance for forest species (Tchebakova et al., 2005; Martínez-Antúnez et al., 2015; Rehfeldt et al., 2015).

Data analysis

To estimate the value of an environmental variable in which the maximum abundance rate of a species occurs, two probability density functions were tested: 1) the two-parameter Weibull function (W2p), and 2) the finite Gaussian mixture model, using the density of each species expressed in relative terms as the variable of interest. The Weibull function and the finite Gaussian mixture model generate robust, flexible models. The Weibull function allows analytical expression of the value of the integral using the functions of accumulated distribution (Torres, 2005). The Gaussian model offers satisfactory results because of the contributions of each Gaussian mixture in terms of likelihood (e.g. Bilmes, 1998; Yang and Ahuja, 1998; Paalanen et al., 2006).

The two-parameter Weibull likelihood density function is expressed as follows:

fxc,b=cbcxc-1e-(x/b)c (1)

And its accumulated function is:

fxc,b=cbcxc-1e-(x/b)c (2)

where c >0 is the form parameter and b >0 is the scale parameter. Goodness of fit of the Weibull model was verified with the Kolmogorov-Smirnov (K-S) test at a 0.2 significance level. This technique is based on the absolute maximum difference between the accumulated distributions of the observed values and of the expected (theoretical) values (Marsaglia et al., 2003). Moreover, to obtain consistent, asymptotically efficient estimators, the final estimation of the Weibull parameters was done with the maximum likelihood method (MLE) (Zarnoch and Dell, 1985; Borders et al., 1987; Seguro and Lambert, 2000).

The finite Gaussian mixture model is expressed as follows:

p(x|Ci)=k=1MWik×p[xt|μik,ik] (3)

where Wik are the contributions of each Gaussian mixture in terms of probability from the kth mixture to M total Gaussian distributions, whose sum Σk=1M is equal to 1 (Biolmes, 1998), and the probability density function p[xt|μik,ik] corresponds to the non-singular multivariate normal distribution of a random D-dimensional variable (Paalanen et al., 2006). The initial expressions of the non-singular multivariate normal distribution of a random variable with D-dimensions and its expression to describe the probability density function of a random vector, as well as their derivations of the original expression using the normal distribution can be consulted in Bilmes (1998), Xuan et al. (2001) and Paalanen et al. (2006).

Probability densities of the finite mixture model p(x|Ci) were estimated with the maximum likelihood method, with the training algorithm of maximum expectancy given it is highly sensitive (Dempster et al., 1977), according to the methodology of Fraley et al. (2012) with the mclust package in R (R Core Team, 2017).

The optimal abundance interval for each species was delimited using a probabilistic cluster defined by the density of the finite mixture model whose space can be classified into tauth probabilities (Chen et al., 2006; Fraley et al., 2012; Fraley et al., 2017). In our study, tau is a standardized measure of probability and takes any possible value (elementary successions) of the probabilistic space (between 1 and 100), the zone near the centroid of the cluster being that of greatest probability. A tau of 0.35 is used because 98 % of the maximum probabilities defined by both models were distributed between the limits of this probabilistic region (from the center outward). For the two pines species, the Gaussian models with two mixed components were used (Chen et al., 2006) in order to identify distributions with multi-modal tendencies, and for Q. macdougallii a Gaussian model with a single component was adjusted because a smaller number of individuals was recorded in the study area.

Results and Discussion

The density curves projected by the two functions used revealed the environmental values in which the maximum likelihood of abundance of each species occurred. For example, the optimal abundance rate of P. pseudostrobus occurred when the balance of summer/spring precipitation (SMRSPRPB) has a value near 6.6 (Figure 2A), and the optimal abundance rate of Q. macdougallii occurs near 2,775 m altitude (Figure 2B).

Figure 2 Density curves of (A) Pinus pseudostrobus as a function of the summer/spring precipitation balance; (B) density curves of Quercus macdougallii as a function of altitude. 

The distance between the upper and lower limits, referred to in our study as the optimal abundance interval (IOA), varied for each species, although they grow in the same eco-graphic region (Table 1). For example, P. patula had an IOA at sites whose slopes fluctuated between 8 and 80%, with a broader interval in relation to slope, followed by Q. macdougallii (50) and P. pseudostrobus (34). Regarding altitude, Q. macdougallii had a narrower IOA than the other two species with a width of only 550 m. In contrast, P. patula had a broader IOA with limits at 2200 to 2900 m (a width of 700 m) (Table 1).

Table 1 Values of climate and physiographic variables in which the maximum probability occurs and limits of the optimal interval for each species studied. 

Especies WINP (mm) PEND (%) SDAY (días)
Pinus patula 150 185.4 447 297 18 50.6 70 52 8 12.9 80 72
Pinus pseudostrobus 180 252.1 447 267 28 52.5 62 34 18 56 68 50
Quercus macdougallii 165 336.4 425 260 10 15.5 60 50 10 55.6 79 69
Pinus patula 0.02 0.032 0.046 0.026 5.3 5.4 5.8 0.5 2200 2263 2900 700
Pinus pseudostrobus 0.017 0.026 0.034 0.017 5.4 5.5 5.8 0.4 2300 2613 2890 590
Quercus macdougallii 0.01 0.019 0.035 0.025 5.3 5.6 5.8 0.5 2350 2775 2900 550
Pinus patula 1100 1118.7 2200 1100 450 489.6 1000 550 oeste y noroeste
Pinus pseudostrobus 1150 1385.3 2100 950 500 615.1 980 480 noreste, noroeste
Quercus macdougallii 1150 1782.2 2100 950 500 807.5 950 450 suroeste, noroeste, noreste (predomina

WINP: Winter precipitation; PEN: dominant slope; SDAY: Julian date of the last freezing date of spring; BHH: annual aridity index; SMRSPRPB: summer/spring precipitation balance (Jul+Aug)/(Abr+May); ALT: altitude over sea level; GSP: precipitation from April to September; SMRP: summer precipitation; EXP: geographic exposure; LI: lower limit of the optimal abundance interval; LS: upper limit of the optimal abundance interval; MAX: value of the respective variable where the maximum abundance rate occurs; and IOA: optimal abundance interval.

Similar observations can be made regarding other variables such as precipitation recorded in specific periods, annual aridity index and day of the last frost in spring. For example, the optimal amount of precipitation for P. patula from April to September (GSP) was 1,100 to 2,200 mm, the optimal in summer was 450 to 1,000 mm, and in winter 150 to 447 mm, but Q. macdougallii had narrower IOA in the same periods: 1,150 to 2,100 mm from April to September (a width of 950 mm), 500 to 950 mm in summer and 165 to 425 in winter. The narrowest IOA of the aridity index was observed in P. pseudostrobus, 0.017 to 0.034, followed by optimal intervals of Q. macdougallii and P. patula whose limits were 0.01 to 0.035 and 0.02 to 0.046, respectively (Table 1).

During data collection we observed variations in density of individuals that depended on the predominating physiographic variables, particularly the exposures of each unit of sampling. Thus, the maximum abundance rate of P. patula was observed in sites with west and northwest exposures, for P. pseudostrobus in northeast and northwest exposures, and Q. macdougalli with greater presence in southwest, northwest and northeast exposures (Table 1).

The two probability density functions used robustly modeled maximum abundance of the three species studied, with greater sensitivity in the projections generated with the finite Gaussian mixture models. This model detected bimodal trends of several species in the face of variation in an environmental variable, such as the case of P. pseudostrobus in function of summer precipitation whose values of maximum probability were observed when precipitation was 615.1 mm (Table 1) and 872.8 mm (Figure 3A and 3B). Likewise, Q. macdougalli behaved in a similar way with altitude above sea level (Figure 2B) when a second smaller scale vertex of the density curve was observed around 2200 masl, as a response to the concentration of sample data between 2000 and 2500 m altitude (Figure 2B). The greatest plasticity of the finite Gaussian mixture model could correspond to the larger number of parameters in its structure and, above all, to the individual contribution of each Gaussian mixture (Bilmes, 1998; Xuan et al., 2001; Paalanen et al., 2006). However, despite the reduced number of parameters of the Weibull function (Equations 1 and 2), this function also projected a maximum abundance probability similar to the mixed model (Figures 2A and 2B).

Figure 3 Bidimensional (A) representation and in perspective of the optimal abundance Interval (B) for Pinus pseudostrobus as a function of summer rainfall and land slope at a tau of 0.35. The two triangles in the bidimensional figure represent the points at which the maximum probabilities of a two-component finite Gaussian mixture model occur. 

The value of a variable at which the probability of maximum abundance of a species occurs does not always remain in the center of the distribution, given that in most cases they do not follow normal distributions (Figures 2B and 3A). Moreover, abundance does not follow a unique pattern of distribution because of changing environmental variables and, when more environmental variables are added, the resulting space will not have a geometric form, nor could it be modeled with the Gaussian standard normal function (Antúnez et al., 2017b).

The results of our study suggest that the scarce distribution of Q. macdougallii in the study area could be related to narrow optimal intervals (IOAs) of summer and winter precipitation and altitude, whose intervals were small compared with those of Pinus patula and P. pseudostrobus (Table 1). The broad distribution of the latter seems to correspond to broad IOA of the number of frost and rainfall events from April to September, variable whose effect is significant on several conifers and latifoliate species in northwestern Mexico such as Abies durangensis, Pinus maximinoi, Quercus resinosa, Q. acutifolia and Q. urbanii (Martínez-Antúnez et al., 2013).

In our study, optimal intervals of the species were not identified in function of the annual aridity index, due to the small values of this variable. However, like precipitation from April to September and degree-days above 5 ºC, this index has a significant effect on forest species diversity (Silva-Flores et al., 2014) and on their distribution and abundance, according to Sáenz-Romero et al. (2010) and Sáenz-Romero et al. (2012).

When field data were being collected, evidence of forest fire was observed on adult tree trunks and, particularly, in areas where more Q. macdougallii were present. The fire could have altered the density of this species, and abundance of plants is affected by other factors not considered in our study, such as edaphological characteristics or human activity (Clark et al., 1998; Rajakaruna, 2004). It should also be taken into account that the absence of a species in a given location is not necessarily due to scarcity of resources or absence of optimal environmental conditions, but that the species has not explored that location (Soberón and Peterson, 2005; Soberón and Miller, 2009).

Because the optimal intervals of abundance delimited with density functions are not similar to any geometric figure (Figure 3A), particularly when two or more mixed components are included, our study could be complemented with other analytical tools that would allow study of the undefined shapes that species IOA take on, for example, using tools from differential geometry.


The density functions tested in our study allowed definition of the optimal interval of a relevant environmental variable for a species. In this interval, the highest probability of abundance of the entire spectrum of values of any variable occurs, for example, to establish plantations of these or other species of ecological interest in the face of a climatic contingency or one caused by different types of biological factors or agents.

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Received: July 2017; Accepted: January 2018

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