Modeling Seed Longevity and Percentile Prediction: A Sigmoidal Function Approach in Soybean, Maize, and Tomato

Abstract

This study aims to evaluate the behavior of seed longevity in soybean, maize, and tomato stored under controlled conditions using Logistic and Boltzmann sigmoidal models. Additionally, it seeks to determine the performance of these models in predicting $P_{50}$, $P_{85}$, and $P_{25}$. The models were fitted to the experimental longevity data, and their performance in predicting the percentiles was evaluated. The Logistic model showed better performance in predicting $P_{50}$ (time for viability to drop to 50%), $P_{85}$ (time for viability to drop to 85%), and $P_{25}$ (time for viability to drop to 25%), estimating the parameters more frequently within the experimental range (obtained from the initial viability data). The results of this study suggest that some cultivars exhibited different patterns in deterioration rates, with some showing abrupt declines in viability, highlighting differences in the speed and nature of seed deterioration. The Logistic model proved to be superior, with an accuracy of 83% in estimating the $P_{85}$ and $P_{25}$ percentiles, while the Boltzmann model achieved an accuracy of 54%. The tomato cultivar Gaucho showed the greatest loss in germination, reaching $P_{25}$ quickly, while the soybean cultivar M 7119 IPRO and maize cultivar MAM06 maintained high germination for a longer period. These findings emphasize the importance of using viability percentiles to optimize storage practices, minimize economic losses, and prevent genetic erosion in conservation programs. Modeling seed longevity using sigmoidal models can significantly contribute to determining various viability percentiles, supporting storage practices and providing valuable insights for strategic decision-making in seed management, proving useful in both commercial and species conservation contexts.

1. Introduction

Soybean (Glycine max), maize (Zea mays), and tomato (Solanum lycopersicum) are crops of great economic importance in Brazil, with a significant impact on both domestic and international markets. Soybean is the world’s leading oilseed and Brazil’s most valuable agricultural product, accounting for approximately 40% of global production [1,2]. Maize plays a fundamental role in food production, animal nutrition, and the biofuel industry, being essential for food security and various industrial applications. Tomato, widely consumed in Brazil and worldwide, is a rich source of vitamins and antioxidants, contributing to a healthy diet [3]. Regardless of the crop, the physiological quality of seeds is crucial for the successful establishment of seedlings and optimal field performance [4]. Germination is one of the main indicators of seed quality, as it directly influences plant establishment, uniformity, and field distribution [5]. Brazilian legislation, through IN 42/2019 and IN 45/2013, sets minimum germination standards of 80% for soybean and tomato and 85% for maize to ensure the quality of seeds available on the market [6,7].

The production of high-quality seeds, as well as maintaining their quality during storage, is essential for achieving high agricultural productivity [4,8]. These aspects are intrinsically linked to seed longevity, which plays a fundamental role in preserving quality over time and ensuring the success of both commercial and conservation initiatives.

Orthodox seeds have the ability to remain viable for extended periods due to their high tolerance to desiccation. In this dry state, metabolic processes are significantly reduced without losing germination capacity for a considerable time. The desiccation tolerance mechanism is a multifactorial property closely related to seed longevity [9,10]. Longevity is a physiological quality parameter defined by the time required for viability to be reduced to sigma during storage [11]. In other words, longevity represents the period during which the seed remains viable under dry storage conditions [5,12,13].

As storage progresses, seeds gradually undergo a deterioration process and loss of vigor, eventually leading to germination failure [14]. The dynamics of germination capacity loss during storage typically follow a sigmoidal pattern. In some populations, this process starts with little to no germination loss, followed by a phase where mortality sharply increases, causing a significant decline in the curve [15,16].

One of the widely used models to describe seed mortality during storage is the linear model proposed by Ellis and Roberts (1980) [15], which predicts the influence of environmental variables on seed longevity. However, this model has limitations as it assumes a normal distribution of mortality over time and a common deterioration rate for seed lots stored under identical conditions [17]. Additionally, for seed lots that initially experience limited deterioration, maintaining high germination percentages for an extended period, the model may produce inaccurate predictions at the extremes of the curve due to the difficulty of linear functions to accurately capture the natural behavior of the data in these ranges [16].

According to Yin (2003) [18], the sigmoidal pattern can be represented in a segmented manner using exponential, linear, and convex models. However, a more effective approach to describing the sigmoidal pattern is by using curvilinear models, such as sigmoidal ones, which allow for a smooth transition between different phases.

Sigmoidal models have been widely used to describe growth curves, such as plant height, weight, leaf area index, fertilizer and herbicide application rates, and even seed germination over time [19]. These models are notable for their ability to describe a wide range of mean functions, capturing complex and dynamic behaviors in the data, such as exponential growth and decay, in a simplified manner, with fewer parameters, making interpretation easier. Exploring this aspect, Cantão et al. (2023) [20] developed POMONA, a software that uses sigmoidal models for predicting and analyzing seed longevity. The software allows data input, automatically calculates parameters such as $P_{50}$, and graphically demonstrates the modeling, providing a practical and efficient tool for researchers and professionals.

$P_{50}$, one of the parameters calculated by the software, represents the storage or aging period in which germination drops to 50% during storage. This parameter is traditionally used to determine seed longevity. Thus, $P_{50}$ prediction models are important tools for providing insights into germplasm bank management and seed physiology [20,21,22,23]. Although $P_{50}$ is the most commonly used, predicting other percentiles ($P_x$) can also be a valuable tool for monitoring the loss of germination capacity during storage.

In the standards for germplasm banks for plant genetic resources for food and agriculture, the Food and Agriculture Organization (FAO) (2014) [24] sets guidelines to avoid genetic erosion and excessive viability loss, which could compromise accession renewal and efficient labor management, as well as prevent the excessive consumption of biological material since the analyses are destructive. The use of viability prediction models is recommended to determine the time required for sample viability to drop to 85% ($P_{85}$). Based on this, it is established that new monitoring of the accession should be conducted at one-third of the predicted time, considering a reasonable safety margin to reduce the chances of loss.

Furthermore, the use of other percentiles can be valuable in a commercial context, as they can also optimize seed use in quality control analyses, improve resource management, and support Logistical decision-making. Seed lots with initial stability that take longer to reach $P_{85}$ can be allocated for later shipment or directed to meet strategic demands. On the other hand, seed lots that lose their initial viability more quickly may be prioritized for shipment, as they are more likely to lose vigor over time.

Thus, the objective of this study is to evaluate the applicability and suitability of the Logistic and Boltzmann sigmoidal models in describing seed longevity for agriculturally important species such as soybean, maize, and tomato, which exhibit distinct physiological characteristics. This study aims to compare the performance of these models in predicting key percentiles of viability loss ($P_{50}$, $P_{85}$, and $P_{25}$) and to determine whether they can capture common patterns of seed deterioration despite the differences in storage behavior among the analyzed species. By focusing on widely cultivated crops, this research seeks to provide a broader understanding of the model’s robustness and generalizability.

2. Materials and Methods

2.1. Biological Material

This study utilized seeds from three species of significant agronomic importance: soybean (Glycine max L.), tomato (Solanum lycopersicum L.), and maize (Zea mays L.). For soybean, five commercial cultivars were selected: BRS 6979 IPRO, 7677 RSF IPRO, M7119 IPRO, 8579 RSF IPRO, and 8473 RSF RR. Two tomato cultivars—LA1509 and LA1511—were provided by the Tomato Genetics Resource Center (TGRC, https://tgrc.ucdavis.edu/), while the commercial cultivar Gaucho was sourced from the market. The LA cultivars were cultivated following the methodology described by Batista et al. [25]. For maize, four cultivars were selected: MAM03, MAM06, MAM07, and MAM08. The physiological quality of these seeds was characterized based on their water content, germination, and longevity.

2.2. Water Content

The water content was determined using two replicates of 10 seeds each, which were placed in an oven at 105 ± 3 °C for 24 h.

2.3. Germination

For germination, the roll towel method was used, with the paper towels moistened with distilled water in an amount equivalent to 2.5 times the weight of the dry paper. Four replicates of 50 seeds each were used, and the rolls were kept at a constant temperature of 25 °C in the dark. The percentage of normal seedlings (with complete, proportional, and healthy essential structures) was determined as described by ISTA (2020) [26].

2.4. Longevity

Longevity was assessed by exposing the seeds to a controlled environment with a constant temperature of 35 °C and 75% relative humidity [12] using hermetically sealed boxes containing a supersaturated NaCl solution. The seeds were evaluated at time intervals according to the germination methodology previously described. The experimental $P_{50}$ was calculated to determine the time at which 50% of the seeds lost viability during storage.

2.5. Statistical Methodology

Figure 1 illustrates the main procedures for obtaining and analyzing the data. The longevity data obtained were first subjected to the Kolmogorov–Smirnov normality test. Subsequently, non-linear regression models were fitted to evaluate the performance and parameters of the models in predicting $P_{50}$. The Boltzmann and Logistic models were analyzed over the time interval x = 0 to x = m, where x represents time and m varies depending on the cultivar and species analyzed. The Boltzmann model can be defined by the equation:

$$y=A_2+{\displaystyle \frac{(A_1-A_2)}{1+e^{\frac{(x-x_0)}{dx}}}}$$

(1)

where

• $A_1$ = initial frequency;
• $A_2=$ final frequency;
• $x=$ time at moment $i$;
• $x_0=$ time at which the frequency is 50%;
• $dx=$ time constant;
• $e=$ base of the natural logarithm.

The Logistic model is defined by:

$$y=A_2+{\displaystyle \frac{(A_1-A_2)}{1+\left({\frac{x}{x_0}}^p\right)}}$$

(2)

where

• $p=$ is a parameter controlling the slope of the curve.

2.6. Fitting Longevity Predictors

Model fit was determined using the standard deviation of residuals (SDR) as a parameter:

$$SRD=\sqrt{{\displaystyle \frac{MSR}{(n-p)}}}$$

(3)

where

• $MSR$ = the mean square of residuals;
• $n$ = number of observations;
• $p=$ number of model parameters.

Additionally, the adjusted coefficient of determination was used, defined as:

$$R_{adj}^2=1-\left[{\displaystyle \frac{(1-R^2)(n-1)}{(n-p)}}\right]$$

(4)

where

$$R^2=1-{\displaystyle \frac{SSR}{ SST}}$$

(5)

• $SSR=$ is the sum of squares of the regression;
• $SST=$ is the total sum of squares.

Furthermore, specific parameters for each cultivar, such as initial viability, total longevity, and the interval in days containing the experimental $P_{50}$ (interval of interest), were analyzed.

2.7. Percentile Estimation

To estimate the $P_{85}$ and $P_{25}$ percentiles from the data distribution, algebraic operations were performed using the fundamental equations, allowing these percentiles to be calculated from the equation of the statistical model fitted to the longevity data. For this, the variable x was isolated in the Boltzmann and Logistic models. For the Boltzmann model, the equation was defined as:

$$x=x_0+d_{x}ln{\displaystyle \frac{2 {(A}_1+A_2-y)}{y}}$$

(6)

For the Logistic model, the equation was defined as:

$$x=x_0{\left|{\displaystyle \frac{{(A}_1-A_2)}{(y-A_{2)}}}\right|}^{(\frac{1}{p})}$$

(7)

where x is the time at which the frequency equals the $P_{85}$ and $P_{25}$ percentiles of the distribution. For each percentile studied, an interval of interest (II) was defined based on the initial viability data to assess whether the estimated $P_x$ values fell within the interval of interest. Moreover, for each model, the number of times the percentile estimates fell within or outside the interval of interest was computed and subsequently subjected to the chi-square test with p < 0.05. All analyses were performed using OriginPro software, version 2024b (OriginLab Corporation, Northampton, MA, USA).

3. Results

The decline in viability over time during storage at a constant temperature of 35 °C and 75% relative humidity is presented in Figure 2, Figure 3 and Figure 4 for soybean, tomato, and maize seeds, respectively.

The longevity of freshly harvested seeds generally exhibits high viability rates. According to Brazilian legislation, the minimum required value must be equal to or greater than 80% for the species under study (

Table 1. Parameters for evaluating viability over time (days).

Species	Cultivar	Initial Viability (%)	Total Longevity (Days)
Soybean	BRS 6979 IPRO	98	83
	7677 RSF IPRO	98	83
	M 7119 IPRO	99	70
	8579 RSF IPRO	98	70
	8473 RSF RR	100	70
Tomato	Gaucho	66	55
	LA1509	74	55
	LA1511	97	97
Maize	MAM03	88	91
	MAM06	90	154
	MAM07	92	161
	MAM08	95	84

). However, an exception was observed for samples from two tomato cultivars, with 66% and 74% viability. The total longevity ranged from 55 to 161 days (Table 1), and it is noteworthy that tomato cultivars Gaucho and LA1509, which had lower initial viability, were the least long-lived, with a total longevity of only 55 days for both.

The $P_{50}$ values obtained from the fitting of the Logistic and Boltzmann models, along with their confidence intervals and ranges of interest, as well as the adjusted determination coefficients, are presented in

Table 2. Seed longevity of soybean, tomato, and maize over time (days) estimated by sigmoid models.

Species	Cultivar	Model	Range of Interest	UL ≤ P50 ≥ LL	P50	R² Aj.
Soybean	BRS 6979 IPRO	Logistic	34–49	[37.33; 37.28]	37.31	0.99
		Boltzmann		[37.81; 36.56]	37.19	0.98
	7677 RSF IPRO	Logistic	49–58	[61.63; 55.14]	58.38	0.96
		Boltzmann		[56.43; 53.76]	55.09	0.97
	M 7119 IPRO	Logistic	58–70	[65.09; 63.89]	64.49	0.99
		Boltzmann		[65.28; 64.12]	64.70	0.99
	8579 RSF IPRO	Logistic	49–58	[54.24; 53.59]	53.92	0.99
		Boltzmann		[54.41; 53.77]	54.09	0.99
	8473RSF RR	Logistic	49–58	[57.41; 56.74]	57.07	0.98
		Boltzmann		[57.51; 56.82]	57.16	0.98
Tomato	Gaucho	Logistic	15–20	[24.39; 21.67]	23.03 *	0.91
		Boltzmann		[24.04; 21.38]	22.71 *	0.91
	LA1509	Logistic	10–20	[55.69; 19.29]	37.49 *	0.87
		Boltzmann		[27.84; 18.34]	23.09 *	0.87
	LA1511	Logistic	27–35	[33.33; 31.61]	32.47	0.94
		Boltzmann		[33.47; 31.81]	32.64	0.94
Maize	MAM03	Logistic	35–42	[38.52; 35.14]	36.83	0.93
		Boltzmann		[37.43; 33.68]	35.56	0.93
	MAM06	Logistic	63–77	[88.23; 81.72]	84.97 *	0.93
		Boltzmann		[88.41; 81.85]	85.13 *	0.93
	MAM07	Logistic	63–77	[75.30; 70.65]	72.97	0.93
		Boltzmann		[74.72; 70.21]	72.46	0.93
	MAM08	Logistic	56–71	[60.65; 56.46]	58.55	0.96
		Boltzmann		[60.07; 56.24]	58.15	0.96

* Indicates parameter estimates outside the range of experimental interest (II).

.

The maize cultivars MAM06 and MAM07 exhibited the highest $P_{50}$ values for both the Logistic and Boltzmann models and also had the longest total longevity among the crops studied (Table 1 and Table 2). On the other hand, the lowest $P_{50}$ for the models was observed for tomato cultivar Gaucho, which was also the shortest-lived and had the lowest initial viability among the cultivars analyzed.

Overall, the models estimated $P_{50}$ within the experimental confidence interval. However, it is important to note that for tomato cultivars Gaucho and LA1509, as well as maize cultivar MAM06, both the Logistic and Boltzmann models estimated $P_{50}$ values outside the range of experimental interest, though within the confidence limit.

For most cultivars, both the Logistic and Boltzmann models showed similarly adjusted determination coefficients and provided adequate fits with adjusted R² values greater than 0.7, demonstrating the robustness of both models for estimating seed survival. However, tomato cultivar LA1509 had the lowest fit, with an adjusted R² of 0.87 (Table 2), which was also reflected in the largest discrepancy between the $P_{50}$ values estimated by the two models, with a difference of more than 14 days between the estimates (Table 2).

Figure 5, Figure 6 and Figure 7 show the graphical fitting of the Logistic and Boltzmann models to seed longevity during storage at a constant temperature and relative humidity of 35 °C and 75%. It can be observed that there are no great distinctions between the curves of the models for species longevity, which supports the satisfactory values of the adjusted determination coefficients and the similar $P_{50}$ estimates between the models, indicating the adequate fit of the sigmoid functions under study to the experimental data.

Percentile Estimation

The performance of the models in predicting seed longevity percentiles of $P_{85}$ (15% decrease in viability during storage) and $P_{25}$ (75% decrease in viability during storage), using the algebraic method and their respective intervals of interest for each cultivar, are presented in

Table 3. Model performance in predicting the percentiles of seed longevity curves stored at 35 °C and 75% relative humidity, concerning the range of interest.

Model	Error (%)	Precision (%)
Logistic	17	83
Boltzmann	46	54

and

Table 4. P₂₅ and P₈₅ estimates of seed longevity of soybean, tomato, and maize over time (days), calculated using the algebraic method.

Species	Cultivar	Model	P85	P25
		Range of Interest (II)	21–34	34–49
Soybean	BRS 6979 IPRO	Logistic	27.85	44.06
		Boltzmann	39.07 *	48.86
		II	21–34	58–70
	7677 RSF IPRO	Logistic	31.69	81.49 *
		Boltzmann	57.43 *	74.34 *
		II	58–70	58–70
	M 7119 IPRO	Logistic	60.00	66.94
		Boltzmann	65.27	69.17
		II	49–58	49–58
	8579 RSF IPRO	Logistic	49.22	56.62
		Boltzmann	54.68	58.70 *
		II	49–58	58–70
	8473RSF RR	Logistic	53.00	60.04
		Boltzmann	57.81	61.25
		II	-	20–27
Tomato	Gaucho	Logistic	17.04	26.44
		Boltzmann	20.38	29.55 *
		II	-	35–55
	LA1509	Logistic	9.33	34.64 *
		Boltzmann	35.81	39.75
		II	27–35	35–69
	LA1511	Logistic	27.00	39.16
		Boltzmann	33.47	38.86
		II	7–21	42–49
Maize	MAM03	Logistic	14.80	44.54
		Boltzmann	36.49 *	43.55
		II	63–77	112–154
	MAM06	Logistic	50.43 *	104.78 *
		Boltzmann	87.41 *	96.60 *
		II	42–49	77–91
	MAM07	Logistic	43.97	87.02
		Boltzmann	73.09 *	82.82
		II	28–42	70–84
	MAM08	Logistic	29.51	72.63
		Boltzmann	59.68 *	66.81 *

* Indicates parameter estimate outside the experimental range of interest (II).

. The direct calculation performed by the Logistic model achieved an accuracy of 83%, estimating the percentiles within the experimental range of interest more frequently compared to the Boltzmann model. On the other hand, the Boltzmann model had an accuracy of 54% in estimating the percentiles within the experimental range of interest (Table 3).

The maize cultivar MAM03 stood out for showing the fastest reduction in germination by 15% during storage under experimental conditions (the shortest experimental range of interest), as evidenced in the $P_{85}$ estimate using the direct calculation method for the Logistic model. It is noteworthy that this model was the only one to estimate the parameter within the experimental range of interest for this cultivar (Table 4).

Conversely, the cultivars that maintained germination above 85% for the longest time were the soybean cultivar M 7119 IPRO (Figure 4 and Table 4) and the maize cultivar MAM06 (Figure 7 and Table 4). However, for the MAM06 estimates, neither of the studied models’ estimated $P_{85}$ within the experimental range of interest.

It is also important to note that $P_{85}$ was not experimentally reached for the tomato cultivars Gaucho and LA1509. The crop that most rapidly lost 75% germination ($P_{25}$) was the tomato cultivar Gaucho, both within the experimental range of interest and for the $P_{25}$ estimate using the Logistic model (Table 4). The Boltzmann model did not estimate $P_{25}$ within the experimental range of interest for this cultivar.

An analysis of the model performance by percentile is detailed in

Table 5. Model performance by percentile in predicting the longevity of seeds stored at 35 °C and 75% relative humidity.

Model	Parameter	Error (%)	Precision (%)
Logistic	P85	8	92
Boltzmann		50	50
Logistic	P25	25	75
Boltzmann		42	58

. Notably, the Logistic model demonstrated superior accuracy in estimating most percentiles within the experimental range of interest, with an error of only 8% for $P_{85}$ and 25% for $P_{25}$. In contrast, the Boltzmann sigmoid model exhibited an approximately 50% error in estimates for both percentiles under study.

These results clearly indicate the poor performance of the Boltzmann model in estimating both $P_{85}$ and $P_{25}$ using the algebraic method (Table 5) and through the chi-square test (

Table 6. Chi-square test results (p-value) for the frequency of P₂₅ and P₈₅ estimated by the Logistic and Boltzmann models within the range of interest (II).

	Boltzmann
Logistic	0.1258

), which showed that the chosen model leads to differences in percentile estimation, with the Logistic function being the best predictor of these parameters (p-value < 0.05) compared to the Boltzmann model.

4. Discussion

Seed longevity, the time during which seeds remain viable and vigorous after reaching physiological maturity, is a crucial physiological attribute essential for both species conservation and commercial factors. Over time, the loss of vigor and viability can lead to significant economic losses by reducing the period during which seed lots remain marketable, as evidenced in Figure 2, Figure 3 and Figure 4, which illustrate the decline in seed longevity during storage time. In addition to time, seed longevity can be influenced by several factors, including the seed’s chemical composition, maturation stage, initial viability, genetic traits inherent to the genetic material, and, most notably, environmental conditions such as temperature and moisture content, which play a pivotal role in determining seed longevity [12,27]. Even under similar storage conditions and initial viability, longevity can vary, as shown in Table 1, reflecting different rates of deterioration over time (Figure 2, Figure 3 and Figure 4). Similar results have also been observed in soybean [23,28], maize [17], tomato [29], rice [30], and other 18 agricultural species [31]. Therefore, understanding the dynamics of this attribute is crucial for maximizing efficiency in production, commercialization, and species conservation.

The estimation of seed longevity using mathematical models is valuable for seed management during storage. If accurate, the determination of this index can serve as a basis for commercial Logistics, conservation strategies, and cost reduction [11].

The $P_{50}$ is the primary parameter observed in seed longevity studies, representing the time point at which 50% of the initial viability is lost. However, the analysis of other percentiles, such as $P_{85}$ and $P_{25}$, can also be useful tools, as evaluated in this research. These additional percentiles allow for a more detailed assessment of seed deterioration over time under storage conditions, thereby optimizing seed use in analyses, improving quality assessments and resource management, and better identifying longevity patterns that are crucial for conservation and storage Logistics [30,32].

As observed in Table 2, the experimental intervals of interest for the cultivars 7677 RSF IPRO, 8579 RSF IPRO, and 8473 RSF RR are similar, as are their initial viability and total longevity (Table 1). The $P_{50}$ values for these cultivars are also close (Table 2). However, as shown in Figure 4 and the $P_{85}$ results (Table 4), it is evident that although similar, cultivars 8579 RSF IPRO and 8473 RSF RR took longer to reach $P_{85}$ compared to cultivar 7677 RSF IPRO, indicating that the rate of deterioration differs between cultivars, information that is not traditionally measured.

Additionally, analyzing other percentiles alongside the graphs showing the fitted curves from the sigmoidal longevity models reveals how rapid the decline in longevity can be in certain accessions, such as the cultivars M 7119 IPRO, 8579 RSF IPRO, MAM06, and MAM07. These cultivars display a similar behavior, where viability remains high for a longer period ($P_{85}$ ranging from 43 to 60 days). However, the decline is abrupt, especially when considering the evolution of the experimental interval of interest and the $P_{50}$ and $P_{25}$ values that overlap. This can lead to economic losses and, in species conservation, significant genetic erosion due to the excessive and sudden loss of viability, compromising the renewal of accessions if not properly managed.

It is also worth mentioning that using other percentiles can be relevant in a commercial context, as it may optimize seed use in analyses, therefore reducing the number of tests needed in internal quality control and consuming fewer samples, improving human resource management, and supporting Logistical decision-making. Seed lots with initial stability that take longer to reach $P_{85}$ can be reserved for later dispatch or allocated to meet specific strategic demands. In contrast, lots with a faster decline in initial viability may be prioritized for delivery, as they tend to lose vigor more quickly over time.

A study conducted by Guzzon et al. [33] on maize seed longevity over 60 years highlights the critical impact of such viability losses. The study found that nearly half of the 1000 accessions analyzed required regeneration, as their viability fell below the $P_{85}$ threshold in active chambers (−3 °C; intended for seed distribution), while 14% of the same accessions stored in base chambers (−15 °C; for long-term conservation) also required regeneration.

Under constant storage conditions, the seed viability curve demonstrates a sigmoidal shape [34]. However, Roberts (1972) [35] suggests that the asymmetry caused by the linear increase in the logarithm of the relative death chance over time does not reflect the conditions reported in many experiments, where viability curves are symmetrical, conforming to a cumulative normal distribution. That is, when seeds are stored under constant conditions, most individuals have a viability period randomly distributed around the mean, resulting in a symmetrical pattern, justifying the use of the Probit function and other linear functions. Nevertheless, nonlinear models, such as sigmoidal ones, can be employed due to their ability to represent the biological behavior of longevity loss over time, highlighting patterns such as the initial decline in viability, which is gradual at first, followed by a phase of rapid decay, and eventually stabilizing as most seeds become non-viable, accurately reflecting the biological reality of the process.

Nonlinear sigmoidal models, such as the Boltzmann and Logistic models, are traditionally used to describe biological data. This popularity is partly due to their high plasticity in fitting observed data and the availability of accessible software that enables modeling these data without requiring an in-depth statistical understanding of the involved parameters capturing complex and dynamic behaviors in the data, such as exponential growth and decay, in a simplified manner, with fewer parameters, making interpretation easier [36,37,38].

5. Conclusions

The analysis of viability percentiles ($P_{85}$, $P_{50}$, and $P_{25}$) provides valuable insights for strategic decision-making in seed management, proving useful in both commercial and species conservation contexts. The results of this study suggest that sigmoidal models, particularly the Logistic model, are highly effective in capturing the natural patterns of seed deterioration and estimating longevity percentiles for agriculturally important species such as soybean, maize, and tomato. Despite their distinct physiological and storage characteristics, the models proved robust and adaptable, supporting their generalizability across diverse crops.

However, some limitations must be acknowledged. The study utilized a limited number of tomato seed lots, and their suboptimal germination quality may have influenced the accuracy of the models for this species. Addressing these limitations could form the basis for future research, such as expanding the analysis to include a greater number of tomato seed lots or exploring additional species and varying qualities to further validate and refine the models’ applicability.