Introduction

The use of trilinear corn hybrid (Zea mays) in Mexico, including Valles Altos (2 200-2 600 masl), is widespread nowadays; its beginning dates back to 1986, when identifying advantages to this type of hybrids: facilitate seed production; fewer parents are required to reach the production of certified seed; control of genetic quality is accessible as it is easy to demix the male parent line, thus ^{Rowe and Andrew (1964)} overall mention that homogeneous maize populations (inbred lines and single crosses) show lower stability than heterogeneous populations (double and trilinear crosses).

In the period 1943-1985, double cross hybrids from low inbreeding parents, was the formation of hybrid most commonly used in Mexico, based on an objective related to the intent that this had greater adaptability; however, overall showed low heterosis, difficulty for seed production, more effort for its production in crop cycles, and higher production costs to obtain final hybrid (^{Espinosa et al., 1986}). The single cross hybrids that have been generated for the production area Valles Altos have not had the expected production potential, but neither has highly yielding lines homozygous females; these hybrids have high genotype_*environment interaction (GE). These simple crosses are the key to obtaining trilinear hybrid, because the single cross that participates as female has high productivity (^{Espinosa et al., 1998}; ^{Sierra et al., 2006}; ^{Márquez, 2009}).

In breeding programs, it is important to define which genotypes are outstanding in yield, adaptation and stability, assessing these properties in multi-environments to recommend its use commercially (^{Crossa et al., 2006}). Adaptation broadly refers to the outperformance of a genotype in most test environments, while is specific aspects, the sample genotype with outperformance in a given test environment (^{Cooper et al., 1999}; ^{Fuentes et al., 2005}). In this regard, ^{Hanson (1970)} defines stable genotype as that having the lowest possible variability, when grown under different environments.

The study of genotype_*environment (GE) interaction in plant breeding is very important because this is the response of each genotype against environmental variations (^{Crossa et al., 2006}). To make further progress in genetic breeding of a species, it is necessary to properly establish the methodologies to be used for the evaluation of the genotype_*environment (GE) interaction, to accurately estimate the differential response of genotypes across test environments.

Among the various methodologies for the study of crop stability, highlights the ^{Eberhart and Russell (1966)} model, modified from ^{Finlay and Wilkinson (1963)} method, using the arithmetic mean of actual data and noting that the regression coefficient from environmental effects on phenotypic effects can be used as an estimator to measure the response of each cultivar to environmental indices, and production stability is possible to measure the magnitude of deviation from the linear regression; that is, by the mean square deviation of regression. In this model a variety with high average, regression coefficient ßi = 1 and deviation not significantly different from zero (δdi2=0) is considered the ideal stable variety.

Moreover, in multivariate methods, the additive main effect and multiplicative interaction model (AMMI), consists essentially of combining the techniques of analysis of variance and principal component analysis (PCA) in a single model, here the analysis of variance allows to study the main effects of genotypes and environments, while the genotype x environment interaction (IGA) is treated in a multivariate way by the ACP, where a reconfiguration of the regression model is performed to improve the interpretation of the interaction (^{Zobel , 1988}).

To interpret the results obtained from AMMI analysis, according to ^{Yan et al. (2000)}, it is to use the combined effects of genotypes (G) and the interaction genotype x environment (IGE) in yield evaluation, obtaining the graphics called GGE biplot (genotype G + genotype_*environment interaction GE), which facilitate visual identification of genotypes and environment evaluation. Generally, GGE biplot graphics are made using the first two principal components (CP1 and CP2). Thus, the genotype that is at the vertex is the one that responds better in environments of evaluation (^{Yan et al., 2001}).

Based on the above, the objective of this study was to identify trilinear corn hybrids that have better grain yield stability, using two models to assess the stability parameters stability from Eberhart and Russell and AMMI, thus define the model that best describe the genotype_*environment interaction. Based on the results, it is intended to recommend the subsequent commercial release in the case of the best trilinear hybrids identified in this work.

Materials and methods

This work was carried out in the spring-summer 2012, in two locations; the first was FESC-UNAM (Cuautitlán-UNAM), municipality of Cuautitlan Izcalli, State of Mexico, at an altitude of 2 274 m; the second location was on the field Santa Lucia de Prías from the Campo Experimental Valle de Mexico (CEVAMEX), from the Instituto Nacional de Investigaciones Forestales, Agrícolas y Pecuarias (INIFAP), municipality of Texcoco, State of Mexico, at an altitude of 2 240 masl. In both locations, two experiments were planted in two sowing dates. In Cuautitlán-UNAM the first planting date was May 21, 2012 and second on June 1, 2012, while in CEVAMEX-INIFAP, the first planting date was May 18, 2012 and the second May 29, 2012.

Trilinear hybrid Puma 1183 AEC1 and Puma 1183 AEC2 from UNAM, H-53 (^{Espinosa et al., 2012a}) and H-57 (^{Espinosa et al., 2012b}) from INIFAP were used, all in their male-sterile version (AE) and fertile (F) in two population densities, 55 000 and 70 000 plants per hectare; densities that are recommended by the phenotypic characteristics of plants genotypes evaluated by competition. Trilinear hybrids the two planting dates and the two population densities in the two locations were individualized to form environments, thus creating eight environments (two locations x two sowing dates x two population densities).

In all environments, at different planting date, irrigation was applied after planting. For weed control Gesaprim (atrazine) was used at a dose of 2 kg ha^-1 and Hierbamina (24D amine) at a dose of 2 L ha^-1. The harvest was performed on December 10 and 17, 2012 in Cuautitlan Izcalli and Santa Lucia respectively. In both localities a complete randomized block design with three replications was used, where the experimental plot was a groove of 5 m long and 0.80 m between rows, the useful plot had an area of 4 m². A factorial design for statistical analysis of yield, where variation factors for trilinear hybrids were environments, genotypes and interactions between variation factors was used; comparison of means was performed with Tukey at 0.05; to assess genotype_*environment interactions, the variable yield from trilinear hybrids was used as variable.

To evaluate the response of environments, genotypes and IGA the additive main effects and multiplicative interaction (AMMI) was used, an analysis of variance was performed to determine the effects of genotypes, environments, and their interactions. The first two principal components were considered to explain genotype_*environment interaction. Graph representations were obtained by a Biplot, which showed the pattern of genotypes_*environments interaction.

For the analysis AMMI programming routines described by ^{Vargas and Crossa (2000)} were employed, using the following mathematical model:

Where: Y_ij= yield of the ith genotype in the jth environment; μ= general mean; g_i= effect of the ith genotype; e_j= effect of the j-th environment; λ_k= square root of the characteristic vector from the kth axis of ACP; α_ik= ACP scoring for the kth axis of the ith genotype; γ_jk= ACP scoring for the kth axis from the jth environment; E_ij= value of the error.

^{Eberhart and Russell (1966)} stability parameter were obtained to measure the genotype x environment interaction, and the classification of varieties adaptability proposed by ^{Carballo and Márquez (1970)} was used, where b_i is the regression coefficient and δdi2 is the deviation of regression, which is based on the stability definition proposed by ^{Eberhart and Russell (1966)}, which says that an ideal stable variety is one that shows a regression coefficient equal to unit (b_i = 1) and deviations of regression equal to zero (δdi2=0), with a high average yield. The statistical model is as follows:

Where: Y_ij= average from genotype i in the j environment; μ_i= average from genotype i in all environments; β_i= regression coefficient that measures the response from genotype i in varying environments; I_j = environmental index of the jth environment, which is calculated as the deviation from genotype average in a given environment from the overall average; δ_ij= deviation of regression.

The data used in this methodology were analyzed by applying a computational algorithm to obtain stability indicators from ^{Mastache and Martínez (1996)}.

All statistical analyzes were performed using SAS version 9.0 (^{SAS Institute, 2002}).

Results and discussion

The combined analysis of variance from AMMI (Table 1) detected highly significant differences between environments and between genotypes for yield, while for genotype_*environment interaction was not significant. The coefficient of variation was 12.7%, considered an acceptable value as to the condition of the experiment, for the values obtained by r-square and application of the experimental design. The difference between environments for yield indicates that environmental conditions and their effects on genotypes were different in all test environments. The difference in yield between genotypes was the result of genetic divergence among hybrids evaluated, i.e., the yield from genotypes was not the same because these had a different genetic origin, because the progenitors were derived from different maize populations. The absence of significance of genotype x environment interactions (IGE) for yield, which is a highly quantitative trait and therefore greatly influenced by environmental conditions, indicates that statistically there was not a differential response from genotypes yield through the different test environments.

Regarding the amount of the total sum of squares, the effect of environments contributed 14.2%, the genotypes x environments interaction in 11.2%, and 42.4% of the total corresponded to genotypes effect. This indicates that the genotypes, as a whole, contributed in higher proportion to yield variation compared to environmental factors and genotype_*environment interaction, even when there were highly significant differences between environments. This result contrasts with that obtained in other researches, in which the environmental factor and genotype_*environment interaction were higher than the genotypes effect (^{Alejos et al., 2006 Palemón et al., 2012}).

The genotypes that had the highest yield were H-53 and H-57 in all test environments. Environments with higher yields were Cuautitlán in the second planting date (June 1, 2012), with a population density of 50 000 plants per hectare and Cuautitlán in the second planting date (June 1, 2012), with a population density 70 000 plants per hectare; as result of beneficial environmental conditions in the locality of Cuautitlán, of rainfall and soil properties to retain available moisture for the plant. Genotypes that scored lower absolute values from CP1, i.e., that interacted less with the environment, were, H-57, with 14.72, and Puma 1183 AEC2 with -17.93, when presenting the values of CP1 closer to zero, were considered the most stable across environments (^{Medina et al., 2002}; ^{Alejos et al., 2006}; ^{Palemón et al., 2012}).

Environments with better yield (we cannot say that an environment is stable, it is wrong therefore it is added to have good behavior) with respect to absolute values of CP1 were: Cuautitlán in the second planting date, with population density of 70 000 plants per hectare (A5), with -0.28; Santa Lucia in the first planting date, with a population density of 50 000 plants per hectare (A3), with 5.12; Santa Lucia in the second planting date, with population density of plants per hectare (A7), with 9.15, and Cuautitlán in the first planting date, with population density of plants per hectare (A2), with -9.3 (Table 2).

Hybrid H-53 showed a higher average yield than average,while the other three hybrids had lower yields than average (Figure 1). Puma 1183 AEC1 showed the lowest average yield. Hybrids with low values i n CP1 and that interacted less with the environment were, H-57 and Puma 1183 AEC2, while genotypes with greater interaction with the environment for its CP1 high values were, H-53 and Puma AEC1. More precisely, it was possible to identify environments A5, A3, A7 and A2 as those with the best behavior, for its yield and CP1 values, for being closer to zero, i.e., those that showed the least variation between them, and considering that each environment influenced population density, locality and planting date on environments A5, A3, A7 and A2 having the best conditions for the best yields.

The circles represent the genotypes and diamonds environments.

Figure 1 CP1 biplot based on the average yield of four trilinear maize hybrids evaluated in eight environments in the FESC-UNAM and Santa Lucía INIFAP, spring - summer 2012.  

In the AMMI analysis, the first principal component (PC1) explained 60.5% sum of squares, and the second principal component (PC2) observed 26.8% of the sum of squares, so between the two main components described 87.3% the effect of genotype x environment interaction. Furthermore, with respect to environments, ^{Yan et al., (2000)} point out that the environments whose angles are less than 90 ° will classify the genotypes in the same manner. In this case, in particular two environments groups with similar behaviors were formed; in the first group environments, A1, A2, A5 and A6 were found; in the second A3, A4, A7 and A8, being environments A1, A4 and A6, those that best discriminated genotypes.

Meanwhile, ^{Eeuwijk (2006)} indicates that a smaller angle of 90° or greater than 270° between crop vector and site vector indicate that the cultivar has a positive response to the test site. A negative response of the cultivar is indicated with angles greater than 90° and below 270°. This coincides with the results from biplot (Figure 2), where H-53 responded better in the A7, A8, A3 and A4 environments, while H-57 showed a better response in the A4, A8 and A6 environments. Meanwhile, Puma 1183 AEC2 improved in the A5, A6 and A2 environments, while Puma 1183 AEC1 showed a better response in A5, A2 and A1. Thus, the genotypes closest to the origin in the biplot were the least interacting with the environment and therefore the most stable; on the contrary, the farthest showed greater variation in their behavior (^{Yan et al., 2000}). These results show that the most stable genotypes were Puma AEC2 1183 and H-57, while genotypes that most interacted with the environment were H-53 and Puma 1183 AEC1.

Figure 2 AMMI biplot for four trilinear maize hybrids; the points represent the trilinear hybrids and vectors represent environments. 

The analysis of variance of ^{Eberhart and Russell (1966)} stability parameters, for yield, showed no significance between environments, genotypes, and genotype_*environment interaction. This indicates that between environments, environmental conditions had similar behavior, which caused no significant effect on yield, whereas for genotypes, even though came from different genetic origin, their average behavior in all environments was very similar. The genotype_*environment interaction, not being significant indicates that there was a differential response of genotypes yield across different test environments.

According to estimated stability parameters, the regression coefficient (b_i), deviation of regression (δdi2) (Table 3), and using the adaptability classification from ^{Carballo and Márquez (1970)}, it was determined that the four genotypes can be considered as stable, since the regression coefficients were not significantly different from 1, which coincides with the results of ^{Mejía and Molina (2003)}. On the other hand, deviations of regression showed no significant differences and was considered that their values were zero, coinciding with the results obtained by ^{Rodríguez et al. (2002)}, who interpreted that the linear model is not appropriate to describe the response of genotypes based on the environmental effect because, as it is known, the effect of genotype_*environment interaction is a multiplicative effect, which is always is present, even when the interaction turns to be non-significant.

To define the best methodology to characterize the most stable genotypes and environments where genotypes have better yield response, for this, the values of the first principal component (PC1) corresponding to AMMI methodology and the environmental indices values for the methodology of stability parameters were compared. Table 4 shows that AMMI model identified as environments with better response from genotypes for yield to A5, A3, A7 and A2, for having CP1 values closer to zero, while the environments with poor response of genotypes in yield were, A1, A4, A6 and A8, for having CP1 values farthest from zero. The stability parameters model identified as the best environments for yield to A6, A5, A8 and A2, by the values of its positive environmental indices; i.e., higher than the overall average, and environments with bad behavior were, A1, A3, A7 and A4, for having negative environmental indices; i.e., lower than overall average. The correspondence between the two models was 50%, since both models coincide in determining two stable and two unstable environments, from the eight test environments.

Comparing AMMI with Eberhart and Russell model to identify the most stable genotypes (Table 5), with the AMMI model the most stable genotypes were H-57 and Puma 1183 AEC2, for their CP1 values closest to zero; with the stability parameters model, all genotypes were stable. The correspondence between both models was 50%, since only two of the four genotypes evaluated match as stable with both models, similar results to those from ^{Cordova (1991)}, who obtained 40% match when using both methodologies.

Conclusions

Trilinear hybrids H-57 and Puma 1183 AEC2 had the best stability and higher yield. Environments with best genotypes behavior in yield of trilinear hybrids were, Cuautitlán in the second planting date, planting density of 50 000 plants per hectare (A5); Santa Lucía in the first planting date, planting density of 50 000 plants per hectare (A3); Santa Lucía on the second date of planting with planting density of 50 000 plants per hectare (A7) and Cuautitlán in the first planting date, with planting density of 70 000 plants per hectare (A2).

The best model to analyze the genotype_*environment interaction was AMMI, compared with the stability parameters of Eberhart and Russell. This is because the interaction genotype_*environment should be analyzed as a multiplicative effect and not just as a linear effect, in addition to the use of Biplot, is easily explained genotype_*environment interaction, using the values of the principal components and genotypes yield.