Introduction

Mexican peach cultivars cover an area of 43942ha with a production rate of 4ton·ha-1. The yield per ha varies ac­cording to region, variety, ag­ronomic handling techniques, climate, insects and diseases. Within Mexico, the states of Zacatecas, Michoacán, México, Puebla, Chihuahua, Morelos and Durango are the primary peach producers. Mexico’s per capita peach intake is 1.5kg, it is the twelfth largest peach producer in the world, after Chile, Argentina and Brazil in Latin America, and production has increased in recent years (SAGARPA, 2013). However, problems with handling, nutri­tion, phytosanitation and com-mercialization have hindered increases in the quantity and quality of peaches. Knowledge of peach phenology and agro­climatic variables will confer the potential for local adapta­tion of peach crops.

Zucconi (1986) established the existence of three stages of peach fruit growth and de­velopment. The first stage encompasses full flower de­velopment through endocarp hardening, in which mitosis takes place during the first three weeks and then rapidly declines. The second stage is characterized by slow growth of the mesocarp, cessation of general elongation and lignifi­cation of the endocarp. During the third stage the fruit grows rapidly; cellular elongation continues and the intercellular spaces almost disappear (Marini and Reighard, 2008). Peach fruit phenology, its mathematical growth models and agrocli­matic variables are important tools because they describe the rate of growth during sprouting, f lowering and fruiting. This information is vital for handling peach or­chards and for the incorpora­tion of new varieties. Several studies have focused on peach fruit phenology. Interestingly, Medina-Torres (2000) found a mean of chill units and a thermal time accumulation (TTA) of 226.8 and 1207 of the peach variety CP-9216 from January 20th to May 20th, with a period of 120 days. Lott (1942) character­ized the phenology of the ‘Hale Haven’ peach in three stages in addition to describ­ing the effect of applying ni­trogen soda to peach trees. Similarly, a full description of this peach variety from germ sprout through flowering and fruit development was provid­ed by Donoso et al. (2008) for certain regions of Chile; however, they did not develop mathematical models. Casierra- Posada et al. (2004) studied peach phenology, measured wet and dry weights and cal­culated PD/ED ratios over time using third degree polynomials and found correla­tions >0.97. Gutiérrez-Acosta et al. (2008) carried out a characterization of the ‘Ana’ peach cultivar quantifying fruit number, PD, ED and fruit stone and pulp thickness from 2000 to 2004 using descriptive statistics, correlation coeffi­cients and a simple linear re­gression among variables. However, although Gutiérrez- Acosta et al. (2008) and Casierra-Posada et al. (2004) used mathematical models, they did not recognize the three stages described by Zucconi (1986).

The morphological character­ization and simulation of dif­ferent peach fruits has also been reported by Álvarez and Boche (1999), who measured the perpendicular equatorial diameter of late-season nec-tarines (c.v. Sun Grand) and modeled the diameters with monomolecular, logistic and Gompertz models (Paine et al., 2012). They found that the lo­gistic model had the lowest mean square error (MSE) with a correlation coefficient be­tween 0.993 and 0.997. In con­trast, in Mexico State, Rojas- Lara et al. (2008) applied four non-linear regressions includ­ing the double sigmoid logistic, exponential, logistic, Michaelis- Menten and monomolecular models to estimate ‘manzano’ hot pepper (Capsicum pubes­cens R & P) growth under greenhouse conditions during two sampling periods in 2004 and 2005, using fresh fruit weight as the dependent vari­able and fruit growth time as the independent variable. There were significant differences between the two periods, and the monomolecular model pro­vided the best estimate of fresh fruit weight for both periods. Pear fruit growth was evaluat­ed by Arenas Bautista et al. (2012) under two drip irriga­tion systems; there were no significant differences between treatments and the logistic re­gression provided the best fit for fruit growth. Pear (Pyrus communis L) growth and phys­ical and physiological charac­terization without mathematical modeling was conducted by Parra-Coronado et al. (1998). In tomatoes, a growth analysis of three hybrid fruits (Solanum L and S. copersicum L.) under greenhouse conditions was conducted by Ardila et al. (2011) using logistic models; the independent variable was the TTA with a base tempera-ture of 10ºC. A description of the full growth of sweet or­ange fruit (Citrus sinensis, Valencia variety) was reported by Avanza et al. (2004) using logistic, Gompertz and mono­molecular models. The authors concluded that the monomolec­ular model was the most suitable one.

Non-linear models have been used to describe the phenology of nectarines, hot peppers, to­matoes, pears and oranges. Thus, double sigmoid models such as the logistic, monomo­lecular and Gompertz equa­tions are useful and adequate tools for simulating peach fruit phenology. The objective of this study was to compare these three different double sigmoid models: logistic, monomolecular and Gompertz, to describe the phenology of peach fruit growth as a func­tion of growth rate and growth diameters in three peach tree handling systems.

Materials and Methods

The experimental data used for model simulations was ob­tained from three mixed Creole peach orchards with three dif­ferent handling systems in the community of San Nicolás de Arriba, Santiago Papasquiaro, Durango, México. This commu­nity is located at 25º02’ 38’’N and 105º25’09’’W, at an eleva­tion of 1713masl (Figure 1). It has a dry temperate sub-humid seasonal climate with an aver­age precipitation of 450mm dis­tributed primarily during the summer, with only 5-10% of precipitation occurring during the winter and mean tempera­ture of 17.7ºC (García, 1990).

The harvest year was 2013. In orchard T1, the agronomic handling techniques included a winter pruning system with the removal of dead, diseased or broken branches in February, followed by an application of Bordeaux mixture at a concen­tration of 250g of copper (II) sulfate (CuSO4) and 2kg of slaked lime (Ca(OH)2) diluted in 5l of water; the mixture was used as a fungicide and applied to trees at a height of 1m above ground. A bowl for auxiliary irrigation was constructed in March. Fertilizer was applied in April. Composted manure of medium composition from the same region was applied to the trees at a rate of 7kg·m2 in ad­dition to a chemical fertilizer with a 25-25-25 NPK ratio di­vided into three bimonthly doses. The first dose was in May; beginning in June it was combined with a foliar fertilizer at a dilution of 250ml of fertil­izer in 15l water. The summer pruning was conducted in June to eliminate vigorous growth that causes shading.

Peach cultivars were irrigated to avoid a moisture deficit from March until the rainy season in July at a rate of 100l water ev­ery 10 or 12 days per tree. Alternative methods were used to control insects. Delta traps with species-specific lures (male pheromones) were set out in June to prevent reproduction and fertilization of female in­sects; these traps had a longev­ity of 35 days. In addition, in­sect control methods included biodegradable or environmental­ly friendly products such as extracts of vegetable oils (e.g., neem oil) and pesticides of bo­tanical origin (garlic and orega­no); these products were applied every 15 days from the begin­ning of June to the end of September. For aerial control of disease, 1kg of Captan® per ha was applied every 15 days from June to September; this prevent­ed disease during the rainy sea­son. Orchard T2 received the same general treatment as or­chard T1, but there was no pro­tection against insects and no aerial disease control. Orchard T3 was the control. Although crop management, pruning, chemical fertilization, and in­sect and disease control were not applied in T3, the planned irrigation was regularly per­formed. Five trees were selected from each treatment and from each tree, five fruit were ran­domly chosen.

Measurements began 21 days after full bloom (DAFB), which occurred on April 6th 2013, and continued until fruit were ready for harvesting. Fruit growth, measured as po­lar and perpendicular equatori­al diameters, was recorded. The measurement period lasted for 18 weeks. The final mea­surement occurred on August 24th 2013, when the fruit reached the required ripening point for harvesting. TTA with a base temperature of 10ºC from full bloom until fruit were ready to harvesting, pre­cipitation from January to August and chill units from January to February were ob­tained from the climatic station Las Margaritas, Santiago Papasquiaro of INIFAP.

A repeated measures analysis of variance (ANOVA) with a 3×5 factorial design was con­ducted. The first factor was the treatment (handling systems); the second factor was the trees. The Gauss-Markov postulates for this test were met. The least significant difference (LSD) was used for pairwise compari­sons. The growth kinetics of fruit diameters were obtained for the logistic (1), monomolec­ular (2), and Gompertz (3) mod­els. In each case, f(t) represents the fruit diameter measurement in mm and t the time in DAFB. The constant a is related to the final diameter and the constants b, c, d and e are associated with growth rates. For this analysis, the Gauss-Newton al­gorithm with the Marquardt correction was used. The itera­tive computing method de­mands the introduction of initial values for the coefficients; thus, it was necessary to provide ap­proximate estimates of the coef­ficients. To measure the good­ness of fit and compare the models, it was necessary to calculate the SSE, the Akaike criterion and the coefficient of determination (R2). Mean model coefficients were compared us­ing the Student’s t test for each coefficient of each model. The first derivative of each model was calculated to determine the growth rate and the inflec­tion point (IP); the diameter growth rates of PD/ED were calculated for each individual model. The software used was STATISTICA 7 (StatSoft, Inc. 2004, Tulsa, OK, USA) and MATLAB 7.0 (The MathWorks, Inc. Natick, MA, USA).

Results and Discussion

Table I, shows the polar and equatorial diameters of the peach fruit for the three treat­ments. Measurements were taken every two weeks for 140 days beginning on April 27th 2013, 21 days after full blos­som, and ending on August 24th when the peach fruit achieved their maximum size and were ready to be harvest­ed. The proposed model for the experimental design met the Gauss-Markov assumptions of normality, homogeneity of vari­ance and independence. The ANOVA showed significant differences among treatments and a significant interaction between treatment and time for both diameters. Significant dif­ferences were observed among the different time periods and treatments, with the exception of the equatorial diameter of T1, which corresponded to the second growth stage and was characterized by slow growth. Significant differences were exhibited among the three treatments and the two diame­ters at day 140. This result in­dicated that both the agronom­ical and pest management treatments were effective.

Table II shows the model coefficients for the polar and equatorial diameters of peach fruit after flowering for T1, T2 and T3. Based on the SSE, the Akaike criterion and the R2 values, the best model for the polar and equatorial diameters of T1, T2 and T3 was the mono-molecular model. Therefore, the chosen model to calculate the growth rate and the equato­rial and polar diameter ratio was the monomolecular model (Figure 2a).

Table III compares the coef­ficients of the growth models (t=2.11, df=16 and p≤0.05). Values of t >2.11 show signifi­cant differences between coeffi­cients. Values of t in a coeffi­cient between T1 and T2 as well as between T1 and T3 showed differences in all han­dling systems in the three mod­els. However, there were no significant differences between T2 and T3. No significant dif­ferences for coefficients b, c, d or e were found. Based on the model comparisons, T1 was significantly different from both T2 and T3. Polar and equatorial diameters were larger in the T1 treatment due to the agronomi­cal practices and protection from pests (Table II).

Figure 2b shows the IP in the minimum and maximum of first derivative from the mono­molecular model obtained in MATLAB 7.0. These points show the change in the peach fruit growth rate for the PD (continuous lines) and ED (dot­ted lines). The PD values re­vealed sustained growth until day 63 for T1, T2 and T3. Subsequently, there was a peri­od of lower growth on average, from day 63 until day 107, 104 and 110 for T1, T2 and T3, respectively, with a mean of 107. Later, a rapid growth phase continued until day 140 when T1 reached its maximum diameter followed by T2 and T3. Equatorial diameter main­tained sustained growth until day 66, 60 and 64 for T1, T2 and T3, respectively, with a mean of 63. Subsequently, a period of slow growth contin­ued until day 109, 107 and 105 for T1, T2 and T3, respective­ly. Rapid growth then contin­ued until day 140 when T1 reached its maximum diameter followed by T2 and T3. It is notable that in both diameter types (PD and ED), T1 exhibit­ed the largest diameter followed by T2 and T3. TTA from 6th April to 24thAugust were 1474, chill units 233 and precipitation 250mm (Figure 2a).

The double sigmoid models and agroclimatic variables are adequate to simulate peach fruit phenology as measured by growth rate and growth diame­ter. Agroclimatic variables TTA 1474 and chill units 233 coincide with the study of Medina-Torres (2000) with 1207 and 226.8.

With respect to the f inal polar and equatorial diame­ters, Gutiérrez-Acosta et al. (2008) reported a PD of 48.9 to 65.1mm; the average of the actual data in this study was 54.92 and 51.6mm for T1 and T2, respectively. These values are within the range reported by Gutiér rez-Acosta et al. (2008); however, T3 had an average PD of 47.8mm, which is below this range. In the case of ED, only the average value of ED for T1 falls with­in the range of 52.23 to 69.1mm reported by Gutiérrez- Acosta et al. (2008). However, Lott (1942) reported higher PD and ED values of 60.5 and 62.8mm, respectively, compared with those found in this study.

The similarity between the results of this study and the results of Gutiérrez-Acosta et al.(2008) is likely because the peach is of the same vari­ety; the difference compared with Lott (1942) is likely be­cause the peach is a different variety, and nitrate of soda was applied to the trees. The differences among treatments in this study are related to pruning (T3), pruning, irriga­tion, fer tilization (T2) and pruning, irrigation, fertiliza­tion and application of pest treatments (T1) (Table I).

Zucconi (1986) settled the presence of three growth stag­es in the peach fruit. The first is distinguished by an incre­ment in fruit mitosis. Then, a slower growth than the first, and the third stage is charac­terized by an accelerated growth of the mesocarp (Marini and Reighard, 2008). A third degree polynomial sim­ulation with an R2>0.98 was reported by Casierra-Posada et al. (2004), which yielded a weight of 0% at day 60 of DAFB (Figure 2 in Casierra- Posada et al. (2004)) and sus­tained growth until day 112. Subsequently, growth slowed until day 146. Between days 146 and 194, rapid growth oc­curred. As a consequence of the third degree polynomial model, growth continued; the first stage was 112-60=52 days, the second, 146-112=34 days, and the third of 194-146=48 days, for a total period of 134 days. The f lowering period was reported by Gutiérrez- Acosta et al. (2008) as ranging from January 28th (Julian day 28) to February 8th (39), and the harvest season was report­ed as ranging from August 9th (221) to August 16th (220). However, Lott (1942) deter­mined an average growth peri­od for the ‘Hale Haven’ peach of 56 days with the first stage beginning on April 14th, a sec­ond stage of 28 days and a third stage of 39 days with full maturation on July 7th. Moreo-ver, Donoso et al. (2008) de­scribed the equality of the first and third stages. In this study, the first, second, third and total growth periods were 63, 44, 33 and 140 days, respectively (Figure 2b). The results report­ed by Casierra-Posada et al. (2004) yielded values of 52, 34, 48 and 134, and the results re­ported by Lott (1942) yielded 56, 28, 39 and 123 days. Goodness of fit from a c2 test was calculated using the data from the present study as ob­served values and data from Lott (1942) and Casierra-Posada et al. (2004) as expected values. The analysis showed that c2= 10.94, df=3, p=0.0042 and c2= 9.95, df=2, p=0.006, respec­tively, indicating that there are significant differences between the data in this study and the data reported by Casierra- Posada et al. (2004) and Lott (1942). However, between these two authors there are no signif­icant differences in growth pe­riods. Despite this, the full blossoming dates from this study differed from Lott (1942) due to the differences in lati­tude, and also differed from Gutiérrez-Acosta et al. (2008) because of the difference in peach variety. The results of Gutiérrez-Acosta et al. (2008) are very similar to those of Donoso et al. (2008) as a con­sequence of the similarities be­tween the first and the third stages.

Figure 2c shows a progres­sive decrease in the ratio of PD/ ED, implying an initial oblong shape in the first and second stage that becomes spherical in the third stage, and in coinci­dence with the data reported by Lott (1942) and Casierra-Posada et al. (2004). Even following the application of agronomical and pest control measures in T1 and agronomical measures in T2, there was no influence on time and growth rate but there was an influence on the fruit diameter.

The best fit among the dou­ble sigmoidal models in this study was the monomolecular model, according to largest values of R2 and lowest SSE and Akaike coefficients. The present study, therefore, is con­sistent with reports of the sweet orange, C. sinensis (Avanza et al., 2004), in which the MSE was used for model comparison and the Student’s t test was used to test for differ­ences between the coefficients. In addition, Rojas-Lara et al. (2008) found that the best fit for the ‘Manzano’ hot pepper (C. pubescens) data were the monomolecular model; for model comparisons, R2, c2 and MSE parameters were evaluat­ed. However, Álvarez and Boche (1999) concluded that the best model for the simula­tion of late nectarine growth was the logistic model, and for the model evaluation they used R2 and MSE. For the modeling of three tomato hybrids, Ardila et al. (2011) used the logistic model and only used MSE for comparisons. It is worth noting that Casierra-Posada et al. (2004) performed peach model­ing, but they only employed third degree polynomials, which did not permit the com­parison of final fruit diameter with the modeling estimates. Physical and physiological characteristics of the pear vari­ety ‘Triumph of Vienna’ were measured by Parra-Coronado et al. (1998), but they did not model the data under the same conditions as in this study. Moreover, Donoso et al. (2008) conducted a peach fruit growth analysis without modeling. Even when fruit differs, the double sigmoid model is ade­quate to simulate fruit phenol­ogy, particularly the monomo­lecular model, and the statistics R2, c2, t and MSE are useful for comparison.

Conclusion

The double sigmoidal models describe peach fruit growth with sufficient precision, and they are useful because they predict the appropriate timing for agronomical labor such as pruning, thinning, fertilization and insecticide application. The use of models based on thermal time accumulation is recom­mended, as is the measurement of additional characteristics such as the weight and the length of branches, as well as other climatic variables such as precipitation, humidity and solar radiation. These models could be successfully applied to the growth of other fruit from warm temperate climates.

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