Introduction

Traditionally, plant quarantine programs have interpreted the possible impacts that the distribution of an exotic pest could have in a country, based on qualitative and intuitive decisions, making a simplification of the system. In this sense, the use of tools derived from remote sensing and Geographic Information Systems (SIG) can provide efficient elements for risk analysis on the potential of exotic and quarantine pests distribution to countries that assess the import of a given plant product (^{Dahlsten et al., 1989}; ^{LeVeen, 1989}). Thus, models of species distribution, understood as cartographic representations of the suitability of an environment to sustain the presence of a species (depending on environmental variables), can become a tool for decision making in potential scenarios invasion.

Several empirical development of insects, particularly phenological models degrees/day, models have been used to predict events development of populations of insects and other poikilothermic organisms (^{Higley et al., 1986}; ^{Milonas et al., 2001}). In this regard, studies on the pattern of insect f light is to establish systems alerted and represent a useful tool for conducting phytosanitary actions in a timely manner, facilitating prognosis and prevention (^{Savopoulou et al., 1996}). That is why studies are required to determine the spatial distribution of populations of thrips and thus to implement measures targeted control, this will contribute to reducing losses avocado production.

The avocado (Persea americana Mill.) Is the fourth largest in the world tropical fruit. World production of avocado is estimated at 4.2 million tons. Mexico is the largest producer of avocados in the world, with an average annual production of 1 467 837.35 t in 168 113.64 has obtained a yield of 10.18 t ha^-1 (^{SIAP, 2013}). The State of Mexico has 6 671.00 hectares planted (^{SIAP, 2013}). This presents fruit as any crop pests and diseases, for avocado one of the main damage is caused by insects and the largest group is the order belonging to Hemiptera (^{Moreno et al., 2010}). The order of the Thysanoptera, cause damage to prefer feeding on soft tissues such as leaf buds, flowers, young leaves and developing fruits (^{Castañeda and Johansen, 2011}).

Thrips injure avocado leaves and fruits as a result of their diet in the cell layers of the epidermis, creating pale areas or cafes, caused injuries can be entry points for pathogens.

In the fruits of cv. Hass, these insects come to cause scars or marks, reducing its market value. They can cause birth defects in fruit by causing alterations in the newly formed shell fruits, flowers inhibit fertilization and cause them to fall. The reported worldwide species associated with these damages are: Heliothrips haemorrohidalis, Selenothrips rubrocinctus, Scirtothrips persea, S. aceri, Frankliniella spp. and Liothrips perseae. F. occidentalis has presented maximum abundances in periods of vegetative growth, flowering and fruit set, in the period ranging from early January until the onset of rains; in conditions not appropriate these insects staying in weeds (^{De Villiers and Van den Berg, 1987}; ^{Fisher, 1989}; ^{Mc Murtry et al., 1991}; ^{Coria, 1993}; ^{Childers, 1997}; ^{Bender, 1998}).

Thrips control has been carried out with application of insecticides and weed removal (^{Coria, 1993}; ^{Mendez et al., 1999}). Knowledge of the distribution of thrips is indispensable for the development of integrated management programs, due to the importance of this pest, therefore, the objectives were to determine the spatial distribution of adult F. occidentalis in the cultivation of avocado in Coatepec Harinas and map the spatial distribution using spatial statistics, in particular geostatistics it provides a direct measure of spatial dependence, taking into account the two-dimensional nature of the distribution of organisms through its exact location space and display maps (^{Ramirez and Porcayo, 2010}).

Materials and methods

The municipality of Coatepec Harinas is located between parallels 18° 48' and 19° 05' north latitude, meridians 99° 43' and 99° 54' with an altitude of 1 600 and 3 900 meters; the study area consisted of three plots four hectares each. Using the method of quadrants (20 m x 20 m), took 40 of them at random, within each quadrant three trees were selected for sampling, thus having a total of 120 trees ten years old per plot. Biweekly counts of thrips were carried out by placing 12 waste cup trap to catch and count reference thrips adults taking the cardinal points of the tree. Each tree was georeferenced with a model view browser eTrex Vista HC x 2007. The vessels used were plastic yellow color No. 8, 10 cm high, 4.5 cm diameter and base diameter 7.5 cm top edge; this according to the tests made by (^{Gonzalez et al., 1998}). To place the glasses were asked a metal hook at the base, spreading over them shortening for thrips will stick around the outside of the glass, to make observations a magnifying glass 20 x was used, the sampling they started from July 3, 2014 to December 18, 2014.

Results and discussion

With the obtained data modeling populations of thrips and their representation it was made by population density maps. For semivariograms, most to a gaussian model was adjusted in the first plot, except for sampling one, five, seven, eight and twelve. (Table 1). For all samples, the nugget effect was zero, this value reflects the scale of sampling used was correct and that the sampling error was almost nil. The spatial dependence calculated for all semivariograms was high (Table 1, 2 and 3). The semivariograms were properly validated to be within acceptable ranges (Table 5).

Table 1 Parameters of theoretical models adjusted to the semivariogram of F. occidentalis, by sampling date in Coatepec Harinas, State of Mexico, summer-autumn 2014 (plot 1). 

Table 2 Parameters of theoretical models adjusted to the semivariogram of F. occidentalis, by sampling date in Coatepec Harinas, State of Mexico, summer-autumn 2014 (plot 2). 

Table 3 Parameters of theoretical models adjusted to the semivariogram of F. occidentalis, by sampling date in Coatepec Harinas, State of Mexico, summer-autumn 2014 (plot 3). 

In the phase of structural analysis of the data, the "variogram" function was chosen to characterize the spatial continuity. The variograms were built subsequently adjusted to theoretical models.

Regarding the plot two we can see that the models that were adjusted semivariograms are mostly spherical. The range for the sampling of the plot two ranges between 10.8 and 16.8 m, this data is indicative of the maximum distance that a relationship exists between the data. These data are reflected in the density maps (Figure 2), they can be seen that as many of these insects are in the months of July, August and September, which coincides with the f lowering "crazy" that It presented in these same months, reducing significantly from October to December due to the decrease in temperature (Table 4, Figure 1).

Figure 1 Thrips density maps in the cultivation of avocado (plot 1). 

Figure 2 Maps density of thrips in avocado cultivation (plot 2). 

Table 4 Values of statisticians semivariograms obtained cross-sampling validation of F. occidentalis, by quadrants in the municipality of Coatepec Harinas, State of Mexico (plot 1). 

MEE= media de los errores de estimación (MEE); ECM= error cuadrático medio (ECM); ECMA= error cuadrático medio adimensional (ECMA).

In the case of the plot three, the tilt range between 10.8 and 21.6, the semivariograms adjusted to spherical model and only two of them to the gaussian model (Table 3), all with high spatial dependence and nugget effect zero. Cross-validation for the three plots and the value of allowable values reflects nugget effect, which corroborates the efficiency of sampling scale and minimal sampling error (Tables 4, 5, 6).

Table 5 Values statisticians semivariograms obtained cross-sampling validation of F. occidentalis, by quadrants in the municipality of Coatepec Harinas, State of Mexico (plot 2). 

Table 6 Values statisticians semivariograms obtained cross-sampling validation of F. occidentalis, by quadrants in the municipality of Coatepec Harinas, State of Mexico (plot 3). 

The statistical parameters of the cross-validation (Tables 4, 5 and 6) allowed validation of experimental semivariograms adjusting to theoretical models, semivariograms of the plot 1 to gaussian model were adjusted almost mostly, but not for the first half of july, september, october and two fortnights the second half of december where adjusted to a spherical model. (Figures 1, 2, 3; Tables 1, 2 and 3) and the plot two was set mostly by the spherical model, the second half of august and the months of november and december a gaussian model model with a structure spatial aggregate (Tables 1, 2 and 3).

Figure 3 Map of thrips density avocado cultivation (plot 3). 

Plot for three most fit a spherical model only two gaussian for the second fortnight of october-december (Tables 1, 2 and 3). In the three plots of sampling aggregate spatial structures F. occidentalis in the study area as seen in Figures 1, 2, 3. This is also consistent with reports showing ^{Solares et al. (2011)} who observed a spatial distribution of thrips, type added at different sampling dates. Figure 1 shows the trend distribution of thrips in different sampling dates, for example, for the case of sampling one, two, three, six, ten are located on the right side of the plots to plot four infestation is given to the banks of the plot and in all other cases are located in the central part of the plot and left. The aggregation centers located within the plot two located on the right and central part of the plot in most sampling dates except for 1, 9, 10 and 12 which are located on the left side and sampling 8 which it is located across the board, mainly on the banks of the plot (Figure 2). On the plot three location of thrips on the left side of the plot most density maps (Figure 3) was evident, although in the case of sample 1 and 11 as many thrips was found in the right side of the map and sampling 7 and 8 were placed in the central part of the plot.

Based on the parameters of theoretical models adjusted to semivariograms a large majority of spherical models they are presented, indicating that within the plots there are areas with higher incidence of thrips, which follows in aggregation centers that are located in specific locations (Figures 1, 2, 3) this is largely due to agricultural management which gives the producer. For all models adjusted one nugget zero effect occurred, this value is indicative that the sampling scale used was correct and the sampling error was minimum which means that 100% of the variation of the distribution of F. occidentalis, is explained by the spatial structure established in semivariograms for each sample, this is consistent with those reported by ^{Rossi et al. (1992)}. As plateau values varied in the three study plots, plot one of 487.2 to 4.12, plot two 755.3 1.62 and plot three of 584.8 to 0.0018 (Table 1, 2 and 3) possibly response types aggregation in the three study plots.

In all models were validated successfully established aggregation maps (Figures 1, 2, 3) of pest incidence allowing viewing. The maps obtained from infested sampling quadrants showed that populations of thrips were located in centers of aggregation surface, indicating that the distribution of the pest, is located at specific points or foci of incidence distributed in the study area. This is consistent with those reported by (^{Jimenez et al., 2013}). For density maps, the red shows the geographical areas in which the largest number of recorded thrips was determined; unlike the white color which is an indication that there are no areas thrips. In plots 1, 2 and 3 (Figures 1, 2, 3) centers targeted aggregation observed.

As shown in Table 7, in the case of the plot one the largest area infested was 92% and 93% in the second half of august and early september; in the case of the plot two the maximum percentage of infestation was reached in the second half of august with 96% of the infested area. In the case of the plot 3, the most infested surface can be observed in the first half of September. While the lower surface with presence of thrips was given in the plot one sampling carried out on 18 december, as low temperatures naturally reduced thrips populations.

Table 7 Percentage of surface infested plot one, two and three. 

Conclusions

With the application of geostatistical techniques it was possible to determine the distribution of insect F. occidentalis in the cultivation of avocado, as these insects are centers scattered aggregation within crop orchards, a situation that was verified in the three study plots in the municipality of Coatepec Harinas, State of Mexico. It is noteworthy that the results show that there was a high spatial dependence between the data and the sampling error was minimal. The highest percentage of infestation was found in the plot two during the month of august with 96% infestation.