Optimizing the establishment of bean and maize varieties in tropical environments

The successful establishment of any crop is the initial indication of its productivity. Optimizing the establishment of a crop implies ensuring generalized, fast and concentrated emergence. This work studies optimal temperature ranges, under non-limiting water conditions, for both germination and emergence of two bean (Phaseolus vulgaris L.) varieties (catarina and ervilha) and two maize (Zea mays L.) varieties (matuba and sam3). Experiments used a thermogradient plate. Petri dishes were used for germination experiments. Emergence experiments were performed in aluminium containers filled with packed portions of a sandy loam clay textured soil. Size, speed and spread of both germination and emergence were measured at different temperatures by Cu-CuNi thermocouples. Thermal ranges with optimal counts of both germination and emergence [To1 sz, To2 sz] were identified using a flattened bell curve function. Speed was maximized for either germination or emergence over thermal ranges [To1 sp, To2 sp] defined using the plateau model to relate either germination or emergence rates with temperature. Ranges along which the spread of both germination and emergence are nearly minimized [To1 sd, To2 sd] were identified with the aid of even-degree polynomials. The intersection of all three thermal ranges gave rise to optimal temperature ranges [To1, To2] for germination (OTRG) of the four varieties in study and for emergence (OTRE) of three of them. In general, the lower thermal limit of OTRg was determined by speed (To1 = To1 sp) and the upper thermal limit by size (To2 = To2 sz). OTRe begins at To1 sp for ervilha and sam3 and at To1 sd for catarina and ends at To2 sz for catarina and at To2 sd for the others. The endpoints and length of both the OTRG and OTRE were also found to be crop-dependent. Thus, farmers can choose between crops and optimize their establishment. The identification of these parameters may also be useful in assessing weather forecasts and for warning systems and agro-climatic zoning. The influence of the substrate used in each experiment was also discussed.

populations germinate or emerge (large size), as quickly as possible (maximum speed) and in as short a time span as possible (minimal spread). To achieve this goal, the size, speed and spread of both germination and emergence, as a function of temperature, were previously modelled. The research also sought to improve knowledge of the four crops thermal requirements.

Seeds
Seeds were used for germination and emergence experiments of two bean (Phaseolus vulgaris L.) varieties (catarina and ervilha) and two maize (Zea mays L.) varieties (matuba and sam3). These varieties are very common in sub-Saharan Africa and are important nutritionally, as well as from an economic and social point of view. They are the basic "cash crops" for many rural communities. The four selected varieties have high potential yields, despite having different characteristics in terms of rusticity and resistance to pests and disease.
One hundred seeds of catarina and ervilha, weighed 36.6±0.8g and 34.2±0.9g, respectively; 100 seeds of matuba and sam3 weighed 29.8±1.0g and 37.0±1.1g, respectively. Each of these means ± standard deviations were calculated from ten 100-seed samples. The seeds used were provided and certified by the Estação Experimental Agrícola of Chianga, Huambo, an appropriate authority in Angola. The seeds were selected by visual inspection after being immersed for 2 min in a sodium hypochlorite solution (1%) to minimize the risk of bacterial and fungal infection. They were then washed in distilled water.

Thermogradient plate
The germination and emergence experiments were carried out on a temperature-controlled thermogradient plate built in the Agrometeorology Laboratory of Instituto Superior de Agronomia, at the University of Lisbon, Portugal [35]. Uniform thermal gradients along an aluminum alloy plate were achieved by heating one end with electrical resistances connected in series and by cooling the other end with a coolant (ethylene glycol) pumped from a refrigeration unit. Changing the energy inputs and outputs on the plate endpoints provided different thermal ranges. Target temperature ranges were chosen based on the known thermal tolerance range of each crop in the substrate that was used. Type-T thermocouples in direct contact with the substrate for each experiment were used to measure temperatures in twelve transversal bands. Two additional ones measured the temperature at both ends. The plate had a stable behaviour once thermal equilibrium was achieved (Table 1). Details regarding materials, design, modus operandi, behavior, accuracy and reproducibility of the thermogradient plate have been described in [32]. Table 1. Examples of temperature means (Tmean, in ºC) and standard errors (SE, in C) on the thermogradient plate used in the germination (7ºC-40ºC) and emergence (9ºC-42ºC) experiments performed with Zea mays var. matuba. Letters A-L denote the transversal bands of the thermogradient plate; (A) and (L) are the cold and hot ends, respectively.

Germination and emergence experiments
In germination experiments, seeds were placed on filter paper soaked in water, inside sixty (60) glass Petri dishes distributed across the plate as shown in Figure 1a (12 transverse bands × 5 dishes per band). Twenty seeds per dish were placed, totalling 100 seeds for each temperature/band. First, a target basic range of temperatures (7ºC-40ºC) was imposed for all the varieties under study. In a second step, another target range (25ºC-42ºC) was imposed twice to obtain additional information on germination at other temperatures, namely close to those where germination rates were found to be high and/or in the thermal range over which the final percentage of germinated seeds is tending to decrease.
Details on the characteristics of the filter paper and the plates used, on the procedures that kept the seeds permanently moistened (filter paper method), as well as on the criteria for germination and counting frequency were described in [36]. Total germination at each temperature was taken as percentage of the seeds sown and considered the corresponding final germination (Gf). Germination experiments will sometimes be denoted as GFP mainly when the importance of the substrate used must be highlighted. used in the emergence experiments (two per band); Letters A-L stand for the transversal bands of the thermogradient plate (adapted from [36]).
Twenty-four (24) parallelepiped-shaped aluminium containers (257 mm long, 90 mm wide and 65 mm deep), filled with packed portions of soil (fine earth fraction), were distributed along the aluminium plate (12 transversal bands × 2 containers per band) (Figure 1b). Ten seeds per container were placed at 2 cm depth about 2.5 cm apart, totalling 20 seeds for each temperature/band. Two target basic ranges of temperature (9º-40ºC for bean varieties and 9-42ºC for maize varieties) were used considering the results obtained in the germination experiments. Here too, another thermal range (25ºC-42ºC) was imposed to obtain additional information about the emergence at other temperatures. Thermocouples were placed at sowing depth. Details regarding the type of soil used and corresponding characteristics (texture, bulk density, wilting point and field capacity), procedures that maintained optimal water conditions throughout the experiment and avoided damping off of seedlings (soil sterilization) as well as emergence and counting frequency criteria were described in [36].
Total emergence at each temperature was taken as percentage of the seeds sown and considered the corresponding final emergence (Ef). Emergence will sometimes be denoted as Esoil, especially when the importance of the substrate used must be highlighted.

Analytical procedures
The size, speed and dispersion of both germination and emergence were modelled separately, applying the models that were discussed in greater detail and successfully fitted to Mediterranean crops in [26]. The results will be discussed jointly, so as to highlight a successful establishment in terms of both germination or emergence.

Size (Sz)
Germination size (SzG) and emergence size (SzE) were measured at each temperature and denoted by Gf and Ef, respectively. High germination and/or high emergence are considered to occur whenever agronomically acceptable minimum (a.a.m.) values for Gf or Ef are achieved. This is a concept that varies depending on the crop [37]. As FAO Quality Declared Seed (QDS) [38] states that the percentage of seeds that germinate and develop should be at least 80% for maize and 70% for beans, these were the values of a.a.m. considered in this work.
High Gf and Ef values depend essentially on temperature, if water is not a limiting factor [16]. Values of Gf tend to be high and more or less constant along a fairly broad thermal range, but are significantly lower at both lower and higher extreme temperatures [17,26,39]. The variation of Ef with temperature follows a similar pattern to that of Gf [5,40]. A kernel function of the type f k (x)= e − x 2 k was proposed in [26] to model Gf as a function of temperature and was used here to model both SzG and SzE. The positive integer k is a shape parameter. When k = 1, a Gaussian (bell-shaped) curve is obtained. For k > 1, the curve becomes flatter at the top. Three additional parameters make this function more flexible: Szmax (representing either Gmax -maximum values of Gf, or Emax -maximum values of Ef), Tmax (the midpoint of the plateau) and cs (controlling the plateau width). The model equation to simulate either germination or emergence sizes (Sz), as a function of temperature (T), was therefore: (%) (1) Model (1) was fitted using a standard non-linear regression approach. Since k must be a positive integer (to ensure an appropriately shaped curve), different (small) fixed values of k were considered and the parameters were estimated by the least-squares approach. Goodness-of-fit was measured by the Residual Sum of Squares (RSS) and also Akaike's Information Criterion (AIC). By setting Sz(T) = a.a.m. for each variety in Eq. (1), two temperatures are defined as the lower limit (To1 sz ) and the upper limit (To2 sz ) of a thermal range, across which the Gf or Ef are considered sufficiently high. For each variety, this thermal range [To1 sz , To2 sz ] optimizes the respective germination and emergence sizes.

Speed (Sp)
Germination and emergence speeds are often expressed by the reciprocal of the chronological time (tG and tE) necessary to achieve Gf or Ef (or some given fraction of them) and are denoted RG and RE.
The daily and annual thermal fluctuations make it more practical to identify a thermal range that maximizes either germination or emergence, rather than to calculate a single optimum temperature [32]. Furthermore, a model that gives more importance to the rate than to the optimum temperature and directly optimizes both germination and emergence (i.e., without the need to subsequently resort to the prior imposition of a minimum rate) is consistent with farmers practical needs [26]. In this context, both germination and emergence speed were evaluated by using the Piper et al. model ([41]), successfully applied in the relationship between RG [18,32] or RE [23,42] (in day −1 ) and temperature (T), which is a three-segment continuous piecewise-linear function. Specifically, this plateau-shaped broken-stick function is defined by equation (2) on the interval [Tb, Tc] with an upward-sloping line segment that unites the points (Tb, 0) and (To1, Rmax); a horizontal line segment at height Rmax, between the two temperatures To1 and To2; and finally a third, downward-sloping line segment uniting points (To2, Rmax) and (Tc, 0): The parameters Tb (base temperature), Tc (ceiling temperature) and To1 and To2 (optimal temperatures) are the cardinal temperatures, and parameter Rmax is the maximum rate. The interval [To1, To2] provides an optimal range along which both germination and emergence rates are nearly maximum. The model generalizes the classic triangular model as parameterized by [43], which is a special case when To1 = To2. In this work (optimizing germination or emergence speed), To1 and To2 will be denoted as To1 sp and To2 sp .
Standard linear regression techniques were used to fit both the ascending and descending segments (associated with the sub-optimal and supra-optimal ranges, respectively) and the horizontal segment (optimal range), by previously specifying a partition of the data points into groups corresponding to each interval. This rudimentary method always provides solutions, but it does not allow statistical inference for the cardinal temperatures and the maximum rate for either germination or emergence [32]. In order to overcome this drawback, the model (Eq. (2)) was also fitted with standard non-linear regression methods [44] using the full data set. However, the 'broken-stick' nature of the model, with points at which the function is not differentiable, means that sometimes optimal solutions cannot found. The fitting algorithms only converge when the fitted values of To1 and To2 partition the same three groups of points used to fit each line segment. In such instances, standard nonlinear regression theory provides approximate confidence intervals for all model parameters. The initial values fed to the numerical algorithms used were the best estimates of the plateau-shaped (PS) model parameters obtained by the linear regression techniques proposed above for both germination and emergence. When convergence was not possible, no classical statistical inference is available. Whenever available, solutions that resulted from the nonlinear regression fitting algorithms were used. In all cases, the solutions chosen had the smallest value of RSS and the smallest difference between observed and estimated Rmax.
Goodness-of-fit was assessed by the global R 2 glb=1-(RSSglb/TSS), where RSSglb is the total sum of squared residuals (determined, for each point, in relation to the corresponding fitted line segment, as discussed in [32]) and TSS is the overall total sum of squares. In order to avoid the usual difficulties in fitting the initial part of the cumulative germination curve [45], the five parameters in Eq. (2) (To1, To2, Tb, Tc, and Rmax) were estimated for the 0.2 fraction of Gf. The parameters were also estimated for the fraction 0.8 Gf, which is an agronomically satisfactory final proportion for most crops, above which the number of germinated seeds per unit of time decreases considerably [7]. Frequent errors of observation associated with the small counts of initial and final germinations can also be avoided when considering these two percentiles of Gf [46]. Whenever no statistically significant differences are found between the values for both percentiles, the value of each cardinal temperature for either germination or emergence was taken to be their mean.
The thermal time for a given vegetative process is also a useful indicator to measure speed [47]. For both germination and emergence in the suboptimal ranges, thermal time (θ) -the accumulated temperature (in °Cd) above the respective base temperatures (Tb) required by a given fraction of either Gf (to germinate) or Ef (to emerge) [24] -is assumed to be constant between Tb and To1 and was estimated as the reciprocal of the slope in the ascending portion of model (Eq. (2)), that is, θ1 = (To1−Tb)/Rmax. (ºCd) (3) Equation (3) was also used to calculate the thermal time as a function of both tG and tE for temperatures T between To1 and To2 (the optimal range), where θ is no longer constant (it increases with temperature). This means that further increases in temperature in the optimal range made little difference for both germination and emergence processes [26]. Thermal times corresponding to 0.8Gf and 0.8Ef were, in this work, considered indicative of optimal germination and emergence, respectively. Since both Gf and Ef are nearly constant over a relatively wide thermal range (see previous subsection) and the time course of cumulative percentage for either germination [39] or emergence [48] processes approximately follow a sigmoidal (S-shaped) curve regardless of temperature, both times tG and tE (in days) required for the germination or the emergence of a fraction of Gf or Ef, respectively were obtained by linear interpolations between observed germination or emergence percentiles [32].

Spread (Sd)
The three phases of S-shaped curves for both cumulative germination and emergence curves (lag phase, near-linear growth phase and the final asymptotic plateau phase) vary over the thermal tolerance range of any crop [16,22,49].
It is thus useful to know for each crop a temperature range that ensures minimum dispersion values or acceptable values from the farmer's point of view, rather than finding only a single temperature that minimizes dispersion. Also, in the case of dispersion, the relevance of this issue increases with both daily and annual variations in soil temperature [49].
Both germination and emergence dispersions (SdG and SdE, respectively) were assessed by the differences between the respective germination or emergence times necessary for 0.2Gf or 0.2Ef (t20) and 0.8Gf or 0.8Ef (t80) at each temperature tested [26,49]. Reasons for using chronological time instead of thermal time can be found in [16] or [49]. The option for these two percentiles was largely justified in the previous subsection (speed). The difficulties arising from the use of TSG (Time Spread of Germination) should also be overcome by using them [50]. Thus, the dispersion analysis focuses on the fastest phase of each process. This means that, for each temperature, it will depend only on the slope of the line segments connecting the points (t20, 0.2Gf) and (t80, 0.8Gf), for the cumulative germination curves and (t20, 0.2Ef) and (t80, 0.8Ef) for the cumulative emergence curves. Steeper slopes are associated with less dispersion along time of either germination or emergence. Plots of germination or emergence times versus temperature [51,52] suggested that, for all crops, both times were minimal along a fairly broad thermal range and increased toward the more extreme temperatures considered, regardless of the germination or emergence fraction. A polynomial function of even degree (2k, k ϵ ℕ) was proposed in [49] to model the relationship between Sd (either SdG or SdE, both expressed by t80−t20) and temperature (T): Here, the positive integer k controls the width of the interval where the function values are close to the minimum, Sdmin is the minimum value of t80−t20 (minimum dispersion for either germination or emergence, in hours), Tmin (in °C) is the central (midpoint) value of temperatures in the range corresponding to the basin around Sdmin and cd (in °C) is a parameter associated with the basin width.
For given values of k, this model was fitted using standard non-linear regression procedures, and estimates of Sdmin, Tmin and cd, were obtained. The solutions selected were those which, among the values of k considered, minimized RSS and AIC, which were again the measures of goodness of fit used. Unlike for size (both Gf and Ef), recommended admissible maximum values for the spread of both germination and emergence were not found in the literature. The value of m.a.s. =1.05Sdmin was considered as the maximum acceptable spread [49]. Setting, in Eq. (4), Sd(T) = m.a.s. for each variety, estimates for the lower and the upper endpoints of a thermal range [To1 sd , To2 sd ] across which the spread of both germination and emergence are minimal were obtained.

Optimal thermal range for both germination and emergence
The optimization of germination, based on the thermal conditions to which the seeds are exposed requires the identification of a thermal range that maximizes both size and speed, with minimal spread. For an optimal emergence, these same requirements must also be present in a thermal band that defines the conditions to which the seedlings are exposed while rising towards the soil surface.
Therefore, optimal thermal ranges for either germination (OTRG) or emergence (OTRE), ([To1, To2]G and [To1,To2]E, respectively) can be obtained by intersecting the three ranges defined for speed, size and spread of each process, i.e.: [ No optimal thermal range will be defined whenever the above intersection is an empty set. When [To1 sp , To2 sp ] varies with the percentile considered, two OTRG and OTRE were defined for each crop.
The R software [53] was used to fit the above models for size, speed and spread of germination. The statistical significance of differences between the estimates of the various parameters was assessed by possible overlaps of their 95% confidence intervals.

Results
GFP below 10ºC and above 35ºC was not relevant for any of the studied varieties (Table 2). Only the germination of sam3 was still noticeable at temperatures close to 40ºC (Gf = 46% at 39ºC). Thermal ranges along which Esoil occurred were narrower than those found for GFP. Only matuba emerged at about 12ºC (11.7ºC) and residually (Ef =5%), whereas at about 37.5ºC only maize varieties emerged (both with low Ef). Table 2. Final germinations (Gf) and emergences (Ef) and corresponding standard errors (mean±SE, in%) at the mean temperatures (T) used to model their size, speed and spread for two bean varieties (catarina and ervilha) and two maize varieties (matuba and sam3).
Variations in the values of Sz (SzG or SzE), R (RG or RE) and Sd (SdG or SdE) as a function of temperature (T) for both germination and emergence corroborate the assumptions that justified the use of the three proposed models: (a) both Gf and Ef were high (generally above the a.a.m) over a relatively wide thermal range, significantly decreasing for the highest and the lowest temperatures ( Figure 2); (b) both the germination and the emergence speeds increased to a maximum value (Rmax), remaining at this level over a more or less long interval, and then decreased to zero ( Figure 3); (c) dispersions (DG and DE) were small over a broad thermal range and increased visibly toward the most extreme temperatures that were studied (Figure 4).

Optimizing size
The relationships between either Gf or Ef and temperature, for the four varieties (Figure 2), were generally well described by Eq. (1). Thus, for all varieties a thermal plateau along which either Gf or Ef are high was identified. Goodness-of-fit measures (RSS and AIC) are generally good for all varieties, regardless of the stage of development considered (germination or emergence). The lowest values of RSS were obtained when the germination size of both ervilha and matuba varieties and the emergence size of bean varieties were modelled. AIC values were generally lower for emergence than for germination (the exception was matuba).
Best fits for germination size were obtained with different exponents than those found for emergence size ( Table 3). The powers corresponding to the best fits depended on the variety used. In GFP experiments they ranged from k=1 for catarina to k=3 for matuba and sam3; for Esoil, they ranged from k=3 for catarina and ervilha to k= 9 for matuba. Estimated values of Gmax ranged from 86.5% to 93.5% (corresponding to a Tmax = 22.6ºC for ervilha and a Tmax= 24.4ºC for sam3, respectively) whereas Emax ranged from 86.1% to 100% (corresponding to a Tmax=26.3ºC for sam3 and a Tmax=24.1ºC for ervilha, respectively). The 95% confidence intervals for both Gmax and Emax values in the different varieties overlapped (that is, the values are therefore not significantly different). However, the values in the confidence intervals for Tmax were always larger for Esoil than for GFP. Furthermore, maize varieties had higher germination Tmax than bean varieties, whereas no clear trend existed for emergence (Tmax values remained significantly lower for ervilha only). Tmax for GFP ranged from 21.8ºC (catarina) to 24.4ºC (sam3) whereas Tmax for Esoil ranged from 24.1ºC (ervilha) to 26.3ºC (sam3).  1)), for two bean varieties (catarina and ervilha) and two maize varieties (matuba and sam3) and corresponding measures of "goodness-of-fit" (RSS and AIC). The values of Gmax, Tmax and cs (model parameters) are shown in Table 3.
The width of the thermal plateau that ensures near-maximum levels of Gf or Ef (expressed by cs) depends not only on the type of crop but also on the stage of development considered. Three of the varieties studied showed values of cs for germination significantly greater than for emergence (only ervilha was an exception). Both for germination and for emergence, the cs values were higher for maize varieties than for bean varieties.
Both the values of To1 sz and To2 sz and those of the length of the thermal plateau that guarantees either high Gf or Ef (To2 sz -To1 sz ) differ with the stage of development considered. Furthermore, the values for each stage depend on the varieties studied. To1 sz for germination were about 14-15ºC in all cases whereas for emergence they ranged from 13ºC (ervilha) to 17.6ºC (catarina). Values of To2 sz for germination ranged from 28.7ºC (catarina) to 35ºC (sam3) whereas those for emergence ranged from 33.8ºC (matuba) to 35.8ºC (sam3). Maize varieties had longer interval lengths (To2 sz -To1 sz ) for germination than for emergence, while the reverse was found for bean varieties. Differences between interval lengths (To2 sz -To1 sz ) obtained for germination and emergence were greater for ervilha (about 7.1ºC) than for the others (about 2-3ºC).   Table 3. Parameters estimated by fitting Eq. (1) to final germination and final emergence for two bean varieties (catarina and ervilha) and two maize varieties (matuba and sam3) as a function of temperature: k is a (fixed) shape parameter; Gmax and Emax are the maximum germination and emergence, respectively; Tmax is the plateau midpoint; cs controls the plateau width; To1 sz and To2 sz are the lower and upper thermal limits for both high germination and emergence. Point estimates are given by est. and, when possible, the corresponding 95% confidence intervals are denoted by conf. int.

Optimizing speed
Relationships between the rates of both germination and emergence and temperature for each of the four varieties in study were well-described by Eq. (2) irrespective of the percentile (20 th and 80 th ) or the phase considered, germination or emergence (Figure 3). When using this plateau-shaped broken-stick, R 2 glb values exceeded 0.90 in all cases, and even 0.98 in most cases. Maximum rates (Rmax) and cardinal temperatures (Tb, To1 sp , To2 sp and Tc) varied with both the crop and the stage considered. Variations in cardinal temperatures with the fraction considered (20% or 80%) in each stage were generally small and without any defined trend model. Only in the case of To1 for germination of catarina and To2 for germination of sam3 were there larger differences, of around 7-8ºC, which could be considered statistically relevant because their confidence intervals did not overlap (Table 4). Thus, cardinal temperatures were taken as constant (a single temperature was considered for each variety/stage), except in those two cases where the values obtained for each of the two percentiles were considered for To1 and To2 ( Table 5).
The size of the thermal tolerance intervals (Tc-Tb) for germination were greater than for emergence (Tb for germination were always smaller than Tb for the emergence whereas Tc showed an inverse trend for three out of four varieties). Maize varieties tolerate wider thermal ranges than beans (the ranges for germination and emergence of maize varieties were greater than 40ºC and 30ºC, respectively, and smaller than those values for bean varieties). Variation in ranges throughout the population were only relevant (greater than 2-2.5ºC) for the germination of the maize varieties. The values of Tb were larger for emergence than for germination whereas Tc values were larger for germination than for emergence (with statistic relevance in both cases). For germination, sam3 had the largest Tb (9.0ºC) and matuba the largest Tc (53.1ºC) whereas the latter had the lowest Tb (3.8ºC) and ervilha the lowest Tc (38.1ºC). For emergence, differences between varieties were much smaller: Tb varied between about 10ºC (matuba) and 12ºC (catarina), whereas Tc were around 37.5ºC for the bean varieties and 43-45ºC for maize varieties.
The minimum time required (i.e. the reciprocal of the maximum rates estimated for the optimal thermal range) to upshoot 80% of Gf ranged from 30h (ervilha) to 46.2h (sam3). On the other hand, catarina and ervilha take longer to reach 0.8Ef (82.8h and 58.2h, respectively) than matuba (55.8h) and sam 3 (48h). Differences either between 0.8Gf and 0.8Ef or 0.2Gf and 0.2Ef were more relevant for bean varieties than for maize varieties: the former ranged from 1.8h (sam3) to 42 Figure 3. Germination (RG) and emergence (RG) rates as a function of temperature (T) for 0.2Gf or 0.2Ef (    ) and 0.8Gf or 0.8Ef (---o---) and the corresponding measures of goodness-of-fit (R 2 glb and RSSglb) for two bean varieties (catarina and ervilha) and two maize varieties (matuba and sam3), using the plateau-shaped model of Piper et al. [41] (Eq. (2)). G -germination; Eemergence.   Table 4. Coefficients estimated by applying Eq. (2) to the relationship between rate and temperature, cardinal temperatures (Tb-base temperature, T01 and To2optimal temperatures and Tcceiling temperature) and maximum rates (Rmax) for two percentiles (20 th and 80 th ) of both final germination and emergence of two bean varieties (catarina and ervilha) and two maize varieties (matuba and sam3), with the corresponding 95% confidence intervals (conf. int.) Exp.-Experiment; R -Regression techniques used (1-nonlinear; 0linear) Table 5. Cardinal temperatures (mean values) for both germination and emergence of two bean varieties (catarina and ervilha) and two maize varieties (matuba and sam3), and thermal times at To1 (To1) and To2 (To2) for 0.8Gf and 0.8Ef.
The differences between the values of To1 sp or To2 sp for germination and for emergence were not relevant in four cases (differences did not exceed 1ºC for To1 sp in the case of maize varieties and for To2 sp in the cases of matuba and catarina varieties) ( Table 5). On the contrary, the To1 sp of the bean varieties were noticeably larger for germination than for the emergence (the differences reached almost 15ºC when the value of the 20 th percentile for the germination of the catarina was compared with the respective value obtained for the emergence) whereas the To2 sp for ervilha and sam3 showed a different trend (in any case, the differences were around 3-4ºC).
Hence, the length (To2 sp -To1 sp ) of the optimal ranges [To1 sp , To2 sp ] varied with both the variety used and its stage (germination or emergence) and, in two cases (catarina and sam3 germinations), with Gf fractions (Table 5). Catarina and sam3 (the varieties with the heaviest seeds) presented the most extensive optimal ranges for germination (10.8ºC and 10.6ºC for the 80 th percentile, respectively, and both about 2.5ºC for 20 th percentile) whereas ervilha and matuba had much narrower optimal ranges (about 1-1.5ºC in extension). As a rule, the range lengths found for emergence were greater than for germination (sam3 is the exception when the optimal range for 0.8Gf was considered). These differences seem to be clear for the bean varieties (higher by at least 5ºC) but irrelevant for the matuba variety. This trend was also evident for the 20 th percentile of sam3, but not for the 80 th percentile.
Germination speed was optimized at lower temperatures for catarina (24.8ºC, when 0.2Gf is considered) than for sam3 ( (32.1ºC for matuba and 28.7ºC in the case of sam3).
The accumulated temperature above Tb(θ1) was constant up to To1. For GFP, catarina, ervilha and sam3 needed at least 33.2ºCd, 31.5ºCd and 35.8ºCd to complete 0.8Gf, respectively, whereas matuba needed much more (51.9ºCd). To complete 0.8Ef in soil matuba also needed to accumulate more temperature (catarina, ervilha and sam3 required 23.4ºCd, 35.4ºCd and 34.4ºCd, respectively) ( Table  5). Above the optimum range [To1 sp , To2 sp ] defined for germination, the accumulated temperature increased by 18ºCd for catarina and 20ºCd for sam3, but only about 1-3ºCd for ervilha and matuba. On the other hand, the increase over the interval [To1 sp , To2 sp ] defined for emergence was greater for the bean varieties (54.9ºCd and 21.4ºCd for catarina and ervilha, respectively) than for maize (3.1ºCd and 11ºCd for matuba and sam3, respectively).

Optimizing spread
Given the observed U-shaped pattern in the relation between dispersion of either GFP or Esoil and temperature, the success of the application of the model expressed in Eq.4 was not surprising. This shape means that the dispersion (t80-t20) is minimal along a fairly wide thermal range and increases noticeably toward the most extreme temperatures (Figure 4). The increases observed in the thermal extremes for the three varieties (ervilha, matuba and sam3) seem to be more visible in emergence than in germination, and for germination, more evident in the coldest than in the warmest thermal range.  Table 5.
Measures of goodness-of-fit (RSS and AIC) are generally good for all crops. They are better for emergence than for germination in the case of bean varieties and worse in the case of maize. For germination they were obtained with polynomials of degree 2k=6 for bean varieties and 2k=4 for maize varieties, whereas for emergence they were obtained with lower k values for three varieties (2k=2) and with a greater value (2k=8) for ervilha. Values of Dmin, and Tmin, and c were both crop and stagedependent (Table 6) (10.5 and 15.4 hours, respectively) whereas the maize varieties had the lowest values for emergence (about 7 hours). In both stages (germination and emergence), differences for the other varieties were relevant. The estimated Tmin value for the germination of catarina was significantly lower (24.7ºC) than those found for other varieties, which ranged from 38.2ºC to 45.8ºC. Also, the Tmin value for the emergence of ervilha (24.7ºC) was significantly lower than those of other varieties, whose values ranged from 26.3ºC to 28.1ºC. The 95% confidence intervals of Tmin for the emergence of catarina and sam3 also did not overlap.
The intersection of each even-degree polynomial function (Eq. (4)) with horizontal lines representing 1.05Dmin delimited a thermal band along which the dispersion is nearly minimal [To1 sd , To2 sd ]. Both the interval sizes (To2 sd -To1 sd ) and their thermal endpoints were crop-dependent. For any variety, the range was always larger for germination than for emergence (Table 6). Differences in ranges for both stages were less relevant for catarina (12.5ºC) than for other varieties (always greater than 20ºC). Ervilha had the widest interval for both germination and emergence. Maize varieties had very similar ranges, both for germination (about 26-27ºC) and for emergence (about 4ºC). Table 6. Parameters estimated by applying Eq. (4) to simulate dispersion of both germination and emergence for two bean varieties (catarina and ervilha) and two maize varieties (matuba and sam3) as a function of temperature (k-shape parameter; Dmin -minimal dispersion; Tmin -the plateau midpoint; c-extent of the plateau), with the corresponding 95% confidence intervals (conf. int.), and the lower (To1 sd ) and upper (To2 sd ), thermal limits of optimal thermal ranges that ensure minimal dispersions [To1 sd , To2 sd ] when a maximum dispersion of 1.05 Dmin was accepted.

Optimizing establishment
To1 sz values were always lower than either To1 sp or To1 sd , irrespective of the stage. To2 sp values were greater than both To2 sd and To2 sz for Esoil of the four varieties and in two out of the six cases for GFP (in the other cases, To2 sd was lower than To2 sp ). The range for low spread [To1 sd , To2 sd ] was the largest for the germination of the four varieties((To2 sd -To1 sd )/(Tc-Tb)≥0.44), whereas the maximum speed interval [To1 sp , To2 sp ] was the narrowest ((To2 sp -To1 sp )/(Tc-Tb)≤0.28). Thermal ranges that maximize the final emergence counts [To1 sz , To2 sz ] were larger (0.47≤(To2 sz -To1 sz )/(Tc-Tb)≤0.83) than those that ensure the fastest emergence (0.04≤(To2 sp -To1 sp )/(Tc-Tb)≤0.62) or minimize dispersion (0.12≤(To2 sd -To1 sd )/(Tc-Tb)≤0.50). The near-minimum dispersion range for emergence was generally larger than the maximum speed range [To1 sp , To2 sp ] (catarina was the exception). Thermal ranges that maximize either germination or emergence rates are often the narrowest in most cases (only intervals for the speed of emergence in catarina and sam3 [To1 sd , To2 sd ] are narrower). Table 7 contains optimal thermal ranges ([To1, To2], thermal bands that combine high, fast and sparsely dispersed germinations or emergences) for both GFP (OTRG) and Esoil (OTRE), estimated for the four varieties and for a level of m.a.s. =1.05Dmin. In most cases, it was possible to calculate them. Only OTRE of matuba and OTRG for the 20 th percentile of catarina were empty sets. In the former case, the intersection would only be possible if the required Gf falls to an unacceptable level (a.a.m., below 50%). In the second case, the maximum acceptable value for the dispersion would have to double (m.a.s. = 2Dmin). In cases where it was possible to compare OTRG with OTRE, they overlapped to a greater or lesser extent. In general, the lower thermal limit of OTRG was determined by speed (To1 = To1 sp ) and the upper limit by size (To2 =To2 sz ). The OTRG for sam3 was the exception in both cases (To1= To1 sd and To2=To2 sp ), but only for the 20 th percentile. The lower limits of OTRG (To1) were around 28.5°Cd for sam3 (both percentiles), 33°Cd for matuba, 30°Cd for ervilha and about 25°Cd for catarina (80 th percentile). Sam3 and catarina had the largest OTRG (from 2.4ºC for the 20 th percentile of sam3 to 6.4ºC for its 80 th ) whereas matuba presented the narrowest (1ºCd). OTRG for ervilha was reduced to a single temperature (=30.3°C, corresponding to both To1 sp and To2 sz ). The lower and the upper thermal limits of OTRE for both ervilha and sam3 were determined by speed (To1 = To1 sp ) and by dispersion (To2= To2 sd ), respectively whereas those for catarina were determined by dispersion (To1= To1 sd ) and by size (To2= To2 sz ), respectively. The lower thermal limits of OTRE were about 24-25ºC for bean varieties and about 29ºC for sam3. Bean varieties had the largest OTRE (about 5-6ºC) whereas sam3 had the narrowest (1.5ºC). In these intervals, optimal chronological durations are therefore expected for 0.8Gf or 0.8Ef corresponding to the estimated Rmax for the respective percentile. Table 8 contains thermal times for the 80 th percentile at the lower (1 at To1) and upper (1 at To2) thermal limits of both OTRG and OTRE for the four varieties. To optimize germination, bean varieties needed to accumulate less temperature (above the base temperature) than maize varieties. Catarina needed to accumulate 33.2°Cd at least (at To1) for 0.8Gf to be reached, whereas ervilha needed 31.6ºCd; maize varieties always need more than 40ºCd (at least 41.1ºCd for sam3 and 51.9ºCd for matuba). The range of thermal times that optimize germination is narrower for matuba (1.8ºCd) than for catarina (6.6ºCd) and sam3 (12.5ºCd).
Bean varieties needed to accumulate more temperature above Tb to reach 0.8Ef in soil than to guarantee the same percentage of seeds germinated on filter paper, regardless of the thermal limit considered (To1 or To2). This was not, however, the case with sam3 (no results were obtained for matuba), which makes the substrate used an issue to be discussed in future work. Catarina and ervilha needed, at least (To1), around 8ºCd and 3.7ºCd more to emerge in optimal conditions, whereas sam3 needed 6.6ºCd less for this. These trends were accentuated over their optimal ranges (more than 18ºCd for bean varieties and less than 16 Table 8. Thermal times corresponding to the lower (To1) and the upper (To2) thermal limits for the Optimal Thermal Ranges (OTRG) of four varieties (catarina, ervilha, matuba and sam3) obtained, verified at both 0.8Gf and 0.8Ef, for a maximum acceptable spread (m.a.s.) =1.05Dmin.

Discussion
The size, speed and spread of both germination and emergence in two varieties of beans (catarina and ervilha) and two varieties of maize (matuba and sam3) were plotted against temperature of the substrate used (filter paper for germination and soil for emergence). The successful use of a plateau-shaped piecewise-linear function to simulate speed (expressed as rates) as a function of temperature was consistent with the results obtained by [21,32] and [42]. Both a flat Gaussian-type function to simulate size and an even degree polynomial function to simulate dispersion (expressed as t80-t20) were effective, thus validating results in [26] and [49], respectively. These results suggest that these models may be promising also for other crops, under similar conditions. The good fits identify three thermal intervals along which size, speed and spread were optimized: [To1 sz , To2 sz ], [To1 sp , To2 sp ] and [To1 sd , To2 sd ], respectively. Nevertheless, the definition of each interval is not automatic. Only Eq. (2) defines, for a given fraction of Gf or Ef, a thermal range associated with the maximum speed of each of the stages. With both the flat Gaussian model (Eq. (1)) and the even-degree polynomial model (Eq. (4)) it is necessary to specify minimum requirements for successful percentages of final germination and emergence (a.a.m.) and maximum acceptable spreads (m.a.s.), respectively. These requirements depend on the crop or variety used, on local climate and local agronomic criteria [37,54].
The ranges that minimized the dispersion [To1 sd , To2 sd ] for germination were larger than those that maximize size and speed ([To1 sz , To2 sz ] and [To1 sp , To2 sp ]). These results differed not only from those obtained for Esoil but also from those found in [26] (GFP of Mediterranean crops). In all cases (GFP, Esoil and [26]) the thermal intervals that maximized size, [To1 sz , To2 sz ], were the largest. The tropical varieties used in this study thus seem to guarantee minimum SdG at temperatures close to the ceiling temperatures (i.e., maximizing the germination growth rate is still possible at the greatest tolerated temperatures) which may represent a relevant adaptive value. This issue deserves further investigation, not least because the results regarding the emergence of the same varieties in clayey soil did not reflect the same trend.
As found in [26] for the germination of seven Mediterranean crops, it was in most cases possible (emergence of matuba was the exception) to define thermal ranges that optimized both germination and emergence, i.e., along which Gf or Ef were high, RG or RE maximum and SdG or SdE minimum. However, the upper and lower limits of either OTRG or OTRE were not often determined by the same parameters that defined the OTRG of the crops studied in [26]. The best-defined difference concerns the parameter that determines the upper limit of OTRG. In fact, [To1 sd , To2 sd ] found for tropical varieties was large enough that the upper limit of OTRG is dependent on size (To2 = To2 sz ) and not on dispersion (To2 = To2 sd ), as in the case of Mediterranean crops.
The results obtained in the germination and emergence experiments for each variety are not concordant in most cases, either in the corresponding OTR thermal limits (and the limits for the thermal time associated to each of them) and respective ranges or in the parameters (size, speed and spread) that guide such ranges. In fact, results regarding size (k, Gmax, Tmin, c), speed (cardinal temperatures, to2 sp -to1 sp ) and dispersion (k, Dmin, Tmin, c) parameters for the germination of each variety were not the same as for the subsequent emergence. In many cases the differences were even relevant (e.g., only for catarina, the OTRG and OTRE are they similar).
There are factors that affect one process but not the other. For example, hypocotyl/coleoptile elongation (cell multiplication that depends essentially on temperature) and cell growth (which depends mainly on water) as the seedling moves towards the soil surface can influence the different results obtained, with possible repercussions in the greater or lesser extent of the thermal range that optimizes speed [To1 sp , To2 sp ], especially in bean varieties. On the other hand, several factors affect germination and emergence to a different degree, namely oscillations in both soil temperature and soil moisture [6,10,13] and the residence time of seeds or seedlings in the soil [4,55], and can to a large extent explain those differences. The increasing thermal amplitudes to which the seedling is exposed as it moves towards the surface [56] and the influence that temperature has on the circulation of water in the soil, thus influencing emergence more than the germination, are an example of such factors. The excessive residence time of seeds and seedlings in the soil, usually due to more extreme temperatures, exposes them to soil pathogens, slowing or even preventing not only germination but also the upward trajectory towards the soil surface of the seedlings constituted in the meantime [4,55]. The larger this residence time, the greater the chances that soil pathogens will limit the survival of seeds and seedlings. The combined action of these two factors clearly increases the likelihood that not only will Ef be less than Gf but also that the difference between Ef and Gf will grow with lower temperatures. It may even imply that the base temperature for emergence is greater than Tb for germination, as [57] and [55] have already suggested, although without data to back up the claim.
As the substrate used in the germination and emergence experiments was different, its nature can become an important factor to consider when discussing the observed differences. The results obtained seem to corroborate the importance of such influence (e.g., final germination with filter paper was sometimes lower than final emergence in the soil, thermal times for emergence were sometimes lower than those for germination), but debating the influence of the substrate used is not an easy task because the methodology used did not include the study of germination in the soil. Even so, the results obtained in [58], where seed losses of the varieties studied here were evaluated as a function of temperature and the substrate used, may help to discuss Ef-Gf values. Those authors showed that, for lower temperatures, germination in soil is more limited than on water-soaked filter paper (GFP), and suggested that this may be due to the different levels of contact between water and seeds and/or to the interaction between temperature, moisture and permeability from soil to air, that is, the conditions that affect the rate of oxygen diffusion in the soil [58]). Discussion on crop-dependent differences between germination and emergence regarding speed (e.g., RG-RE) and dispersion (SdG vs. SdE) are limited due to the different types of substrate used.
Considering the base temperature in both processes (always greater in Esoil than in GFP) the thermal times tended to be close for any of the varieties. There were cases in which the thermal times were even greater for Gsoil than for Esoil (catarina and sam3 for temperatures that optimize speed and sam3 for temperatures that optimize the respective processes), which reinforces the potential influence of the type of substrate.
Not surprisingly, the results showed relevant differences between the behavior of the crops studied (and even between varieties of the same species), allowing for clearer choices depending on the agroclimatic conditions present (options between crops to be installed, sowing time or even sowing depth). The consistency of the options should be based mainly on the differences between the parameters of OTRG and/or OTRE (limits and lengths) found for each crop or variety. Thermal ranges that optimize either germination or emergence allow agroclimatic zoning in tropical areas depending on the altitude. On the other hand, their corresponding lengths can highlight the role of proximity to the sea as an influential climatic factor in that zoning. Differences between OTRG and OTRE (more visible for ervilha and matuba than for catarina and sam3) may also indicate different levels of adaptability of each variety to the substrate used, thus suggesting future work on germination and emergence in soils with different textures. Results for sam3 only converge when OTRG for 0.8Gf is considered, for ervilha they converge in To2 but have completely different extensions, whereas for catarina T01 and T02 they are almost coincident. Catarina (and ervilha if only the results obtained for the emergency are considered) seem to adapt better to tropical climates of medium-altitude than maize varieties. Although the lengths of both ORTE and OTG are relatively small (never exceeding 6ºC), the establishment of catarina still seems to be more favoured in regions further from the sea than the other varieties (the greater the thermal interval length (To2 -To1) of a crop, the greater its adaptability to regions further away from the coast). When the results for the germination of ervilha can be compared with those for emergence, we find apparently contradictory results with regard to the adaptability of this variety, both for more inland and highland zones (those for emergence suggest greater adaptability to areas further away from the coast and/or greater altitudes, whereas those of germination suggests the opposite), which again indicates the influence of the substrate used. Sam 3 will be favoured in warmer areas but is more sensitive to large thermal variations, that is, more inland regions.
Although no OTRE has been defined for matuba (no comparison with OTRG is possible), it is possible to predict sub-optimal thermal conditions for its establishment between 28.8ºC (To2 sd ) and 32.1ºC (To1 sp ). With an assured high percentage of seedlings emerged (Ef> a.a.m.), the closer the temperature is to 28.8ºC, the lower the dispersion over time, the closer to 32.1ºC (temperature very close to To1 for germination) the faster the emergence will be. Considering this sub-optimized range and the estimated OTRG, matuba may be more suitable for warmer areas and with less thermal amplitudes. These results to some extent counter suggested guidelines from studies based only on the size parameter for the same varieties [27].

Conclusions
Thermal ranges for two varieties of beans (catarina and ervilha) and two varieties of maize (matuba and sam3) were defined by fitting functions relating germination or emergence counts with temperature (a flatter version of a Gaussian curve), germination or emergence rates with temperature (plateau-shaped piecewise-linear function) and germination or emergence dispersions with temperature (even degree polynomial). Across these thermal ranges, both germination and emergence are high ([To1 sz , To2 sz ]), their speeds are maximal ([To1 sp , To2 sp ]) and their dispersions are minimal ([To1 sz , To2 sz ]), respectively.
The intersection of these three temperature intervals usually resulted in an interval of optimal temperature ranges [To1, To2] for germination (OTRG) and emergence (OTRE). For most OTRG, speed determined the lower thermal limit (To1=To1 sp ) and size the upper limit (To2=To2 sz ). For OTRE, this trend was not so clear. The endpoints (To1 and To2) and lengths of both OTRG and OTRE are crop-dependent. For the varieties that were considered, OTRG and OTRE do not always coincide. Differences found between values of the germination and emergence parameters suggested not only that different factors act differently in both processes, but also that the type of substrate used in the experiments may need to be taken into account for its interpretation.
This three-interval approach can provide farmers with an important tool to increase the chances of establishment success under favourable soil water conditions. This approach allows decisions regarding either the crops or varieties to be installed, or the sowing times of a particular crop or variety, to be made with a view to optimizing their establishment. Farm weather forecasts, warning systems of various kinds and agro-climatic zoning can also benefit from the knowledge of the parameters obtained.