A new regional cotton growth model based on reference crop evapotranspiration for predicting growth processes

Abstract

Meteorological conditions and irrigation amounts are key factors that affect crop growth processes. Typically, crop growth and development are modeled as a function of time or growing degree days (GDD). Although the most important component of GDD is temperature, it can vary significantly year to year while also gradually shifting due to climate changes. However, cotton is highly sensitive to various meteorological factors, and reference crop evapotranspiration (ET_O) integrates the primary meteorological factors responsible for global dryland extension and aridity changes. This paper constructs a cotton growth model using ET_O, which improves the accuracy of crop growth simulation. Two cotton growth models based on the logistic model established using GDD or ET_O as independent factors are evaluated in this paper. Additionally, this paper examines mathematical models that relate irrigation amount and irrigation water utilization efficiency (IWUE) to the maximum leaf area index (LAI_max) and cotton yield, revealing some key findings. First, the model using cumulative reference crop evapotranspiration (CET_O) as the independent variable is more accurate than the one using cumulative growing degree days. To better reflect the effects of meteorological conditions on cotton growth, this paper recommends using CET_O as the independent variable to establish cotton growth models. Secondly, the maximum cotton yield is 7171.7 kg/ha when LAI_max is 6.043 cm²/cm², the corresponding required irrigation amount is 518.793 mm, and IWUE is 21.153 kg/(ha·mm). Future studies should consider multiple associated meteorological factors and use ET_O crop growth models to simulate and predict crop growth and yield.

Introduction

With rapid population growth in recent decades, the demand for grain and cash crops has been constantly increasing. To improve crop yield and quality, it is necessary to accurately describe crop growth processes and better control the nutritional factors required for crop growth. A precise understanding of the complex interactions between crops and their surrounding environments is important to predict how environmental conditions affect crop growth processes. Towards this goal, crop growth models have been developed to simulate crop growth and development using complex mathematical functions and modelling techniques¹. These models can quantitatively and dynamically describe crop growth development and yield formation processes², which is useful for assessing the impact of drought on future crop yields. Additionally, crop growth models can provide detailed estimations of crop status, including phenological status, leaf area index (LAI), and yield of specific crop types³. Furthermore, these models can predict crop yields as a function of soil conditions, weather, and management practices⁴. Therefore, crop growth models have become important tools for quantitatively evaluating the relationships among soil, weather, and vegetation, and for facilitating the timely regulation of crop growth, which has attracted widespread attention.

Crop growth models have simple forms and are convenient to use⁴, and can be employed to simulate crop growth processes to guide agricultural production. Some models have been adapted to crop breeding to simulate the effects of changes in morphological and physiological characteristics of crops, which helps to identify optimal phenotypes in different environments⁵. In addition, some models have been used to simulate crop growth processes, including the Gompertz⁶, Richards⁷, and logistic^8,9 models. Pronk¹⁰ established a dynamic model of dry matter accumulation (D) and LAI based on the relationship between wheat and radiation-use efficiency, while Villegas¹¹ used the logistic equation to establish a model of D of durum wheat based on the crop growth characteristics in a Mediterranean climate. Wang¹² took the growing degree days (GDD) as the independent variable and simulated the dry matter accumulation process in grape fields using logistic, Richards, and Hoerl models. An evaluation of the model results showed that the simulated values using the logistic model were closest to the measured values, indicating that the logistic model was the most accurate.

Crop growth is influenced by many factors, including soil properties, meteorological conditions, nutritional status, and crop species¹³. While soil properties, nutritional status, and crop species can be easily adjusted, meteorological conditions cannot be controlled. They significantly impact agricultural production and show high spatial variability at regional and global scales. Numerous studies have explored the effects of climate variability on crop growth, development, and yield^14,15. Generally, growth days, such as days after sowing, are used as the independent variable when establishing crop growth models¹⁶. However, because of differences in meteorological conditions, crop seeding time varies among regions. Furthermore, crop growth is strongly influenced by temperature^17,18,19. Temperature plays a central role in the life cycles of crops, affecting growth, development, and yield²⁰. Nevertheless, temperature is a highly variable factor that exhibits clear diurnal cycles. To investigate how diurnal variation in temperature affects crop growth stages, the concept of GDD was proposed as an ecosystem indicator²¹. GDD not only measures accumulated heat but also characterizes the crop growth processes and development in a fixed planting area²². GDD can be used to analyze thermal conditions under certain meteorological conditions, which can determine the optimal sowing date, growth period, and corresponding physiological growth characteristics of crops²¹. Therefore, some scholars have used GDD instead of time to establish crop growth models^9,23,24. Logistic models based on GDD, rather than time, better describe the change processes in major crop growth indicators^{9,23,24,25,26}. However, GDD does not directly reflect the impacts of solar radiation, air humidity, and wind speed on crops, nor does it accurately indicate the effects of high temperatures on crop growth.

Reference crop evapotranspiration (ET_O), as the name suggests, is the evapotranspiration from a reference surface like well-watered grass. This value, when multiplied by the crop coefficient based on the crop and its growth stage, gives the value of crop evapotranspiration, which could be considered as the total water requirement, as such, is an important indicator of regional climate change and water cycle status^27,28. ET_O is calculated using the FAO-56 Penman–Monteith equation, which takes into account temperature, solar radiation, humidity, and wind speed^29,30. The ET_O has been shown to generally provide reasonable results under a variety of meteorology conditions. Compared to GDD, which primarily considers temperature, ET_O can more comprehensively characterize changes in meteorology factors. This means that changes in climate variables due to the effects of global warming will be reflected in the ET_O, so it is possible to comprehensively study how meteorology conditions affect crop growth processes at regional scales using ET_O.

Cotton (Gossypium hirsutum L.) production has far-reaching influences on global economics and social development. Cotton is an important oilseed crop³¹ and one of the most important raw materials for natural textile fibers in the world. Recently, numerous scholars have established cotton growth logistic models based on either time or GDD. For example, Wang²⁴ developed a cotton growth logistic model based on GDD, using the Xinjiang region as an example. Gao³² described the biomass accumulation process of main stem leaves, boll subtending leaves, capsule wall, and seed cotton weight using the logistic growth model, while Zhang¹⁶ used the logistic model to simulate the effects of planting densities and nitrogen rates on cotton biomass accumulation. Although many factors, such as the length of the growing season and soil moisture, affect cotton growth, cotton is especially sensitive to meteorological conditions²⁰. Cotton production has gradually shifted to arid and semi-arid regions of the world since the 1950s, which are suitable for growing cotton^33,34. Four main climate features are most important in cotton production: solar radiation, rainfall, humidity, and temperature³⁵. Global warming has accelerated cotton growth and cotton phenology, and the cotton growing period has shortened³⁶. Therefore, it is necessary to establish a new cotton growth model that comprehensively considers meteorological impacts on regional cotton growth processes. In this article, we develop a new cotton growth model based on ET_O at the regional scale and evaluate its accuracy using experimental data collected from published papers. This new method can provide insights for analyzing regional cotton growth and its influencing parameters.

Materials and methods

Data sources

The cotton growth index data used in this study were obtained from previously published studies on cotton growth in China. The portion of the data used for analyzing the characteristics of cotton growth index in Xinjiang came from Wang²⁴, and data from other regions are summarized in Table 1. The main cotton planting areas in China are concentrated in Central China, North China, and the Xinjiang region (Fig. 1). In Central China, the climate is characterized by cold winters and hot summers, with frequent drought and flooding. North China has a warm-temperate monsoon climate, with significant seasonal differences in temperature, precipitation, and evaporation. In Xinjiang, the climate is characterized by large temperature ranges throughout the year, low precipitation, and a dry continental climate. The sowing times for different varieties of cotton in different regions range from mid-April to mid-May, with harvest times from late September to late October.

Table 1 Sample sizes and data sources by region.

Figure 1

Map of the study area. The stars represent the main data sources. Note: ArcMap 10.5 was employed to generate the map of the study area.

Meteorological data were collected from the National Meteorological Information Center of the China Meteorological Data Service Center (http://data.cma.cn/) and the ERA5 hourly data on single levels from 1979 to present (https://doi.org/10.24381/cds.adbb2d47). The data were used to analyze the effects of meteorological conditions on regional cotton growth. Specifically, we collected data on temperature, solar radiation, humidity, wind speed, and precipitation. To collect cotton growth index data, we used the GetData Graph Digitizer to extract data directly from text or figures in published studies. More than three sets of data samples were selected from most regions, but a few areas had only 1–3 sets of data samples. Table 1 shows the sample sizes and data sources for cotton growth index data from in China, including North China and Central China region.

Data processing method and error analysis

All data were processed in Microsoft Office Excel (Microsoft Corporation) and MATLAB (MathWorks Inc., Natick, MA, USA) was used for model parameter calculation. ArcMap 10.5 was employed to generate the map of the study area. To address the potential errors or bad numbers that may have arisen during the data digitization process, we took the following methods to minimize their impact on our analysis: i) We carefully reviewed all digitized data points to check for any obvious outliers or inconsistencies, and corrected any that were identified. ii) We compared the digitized data with the original data source and conducted three repetitions of the same digitized data to ensure that there were no significant differences. Correlations were assessed using the R², and the accuracy was evaluated through RMSE and RE, as follows:

$$ R^{2} = \frac{{\left[ {\sum {\left( {x_{i} - \overline{x} } \right)\left( {y_{i} - \overline{y} } \right)} } \right]^{2} }}{{\sum {\left( {x_{i} - \overline{x} } \right)^{2} \sum {\left( {y_{i} - \overline{y} } \right)^{2} } } }} $$

(1)

$$ RMSE = \sqrt {\frac{{\sum {\left( {m_{vi} - c_{vi} } \right)^{2} } }}{n}} $$

(2)

$$ RE = \frac{{\sum {\left( {m_{vi} - c_{vi} } \right)^{2} } }}{{\sum {m_{vi}^{2} } }} $$

(3)

where, x_i is the independent variable; y_i is the dependent variable; $\overline{x}$ and $\overline{y}$ represent the average values of x_i and y_i, respectively; m_vi is the measured value; and c_vi is the calculated value.

Mathematical model development and construction

Cotton growth model based on cumulative growing degree days

Every crop has biologically defined upper and lower temperature limits beyond which its growth and development can be adversely affected. For cotton, the upper and lower temperature limits are 40 ℃ and 10 ℃, respectively, as demonstrated by numerous studies^24,45,46,47. Growing degree days (GDD) are a measure of heat accumulation used to estimate crop development and growth. In this study, we converted daily GDD data to cumulative GDD (CGDD) using the following equation:

$$ CGDD = \sum {(T_{avg} - T_{base} )} $$

(4)

where, CGDD is the cumulative growing degree days, ℃; T_avg is the mean daily temperature, ℃; and T_base is the minimum daily temperature required for crop activity, ℃. McMaster and Wilhelm proposed a method for calculating T_avg⁴⁸:

$$ T_{avg} = \left\{ \begin{gathered} \frac{{(T_{x} + T_{n} )}}{2} \, if \, T_{base} < T_{avg} < T_{upper} \hfill \\ T_{base} \quad \quad \, if \, T_{avg} \le T_{base} \hfill \\ T_{upper} \quad \quad \, if \, T_{avg} \ge T_{upper} \hfill \\ \end{gathered} \right. $$

(5)

where, T_upper is the maximum temperature at which crop activities continue, ℃; T_x is the highest daily temperature, ℃; and T_n is the lowest daily temperature, ℃.

A logistic model describing plant height (H), leaf area index (LAI), and dry matter accumulation (D) of a crop using CGDD as the independent variable was established. Equations to calculate the relative change of indicators of H, LAI, and D are used to minimize the differences in cotton growth indicators caused by irrigation methods or soil conditions in different regions. The relative cotton growth index and CGDD are obtained using these corrected indicators. The logistic models between the relative CGDD (R_CGDD) and relative H, relative LAI, and relative D, are expressed as:

$$ R_{H} = \frac{H}{{H_{\max } }} = \frac{1}{{1 + e^{{a_{1} + b_{1} R_{CGDD} }} }} $$

(6)

$$ R_{LAI} = \frac{LAI}{{LAI_{\max } }} = \frac{1}{{1 + e^{{a_{2} + b_{2} R_{CGDD} + c_{2} R_{CGDD}^{2} }} }} $$

(7)

$$ R_{D} = \frac{D}{{D_{\max } }} = \frac{1}{{1 + e^{{a_{3} + b_{3} R_{CGDD} }} }} $$

(8)

where, R_H is the relative plant height; H is the plant height, cm; H_max is the theoretical maximum H, cm; R_LAI is relative leaf area index; LAI is leaf area index, cm²/cm²; LAI_max is the theoretical maximum LAI, cm²/cm²; R_D is relative dry matter accumulation; D is dry matter accumulation, g/plant; D_max is the theoretical maximum D, g/plant; R_GDD is the relative CGDD; and a₁, a₂, a₃, b₁, b₂, b₃, and c₂ are model fitting parameters. The maximum measured value of each index in a field experiment may not actually reach the maximum value, so we increased the values using multiplication factors (for example, H_max was multiplied by an incremental factor of between 1.01 and 1.05) to reach the theoretical maximum value^8,24.

Cotton growth model based on cumulative reference crop evapotranspiration

The crop growth model based on GDD only accounts for the effect of temperature on crop growth, but cotton growth is influenced by other meteorological factors as well. Therefore, using GDD alone is insufficient to accurately model cotton growth. Instead, we used reference evapotranspiration (ET_O), which considers various meteorological conditions such as temperature, solar radiation, water vapor pressure, and wind speed. The Penman–Monteith method is a widely accepted method for calculating ET_O, as it incorporates both physiological and meteorological parameters. We used the FAO-56 Penman–Monteith method, which is a standard and reliable approach used worldwide. To account for cumulative evapotranspiration over time, we calculated cumulative reference crop evapotranspiration (CET_O) using daily ET_O data and a specific equation.

$$ CET_{O} = \sum {ET_{O} } = \sum {\frac{{0.408\Delta (R_{n} - G) + \gamma \left( {\frac{900}{{T + 273}}} \right)u_{2} (e_{s} - e_{a} )}}{{\Delta + \gamma (1 + 0.34u_{2} )}}} $$

(9)

where, R_n is the net radiation, MJ/m²/d; G is the soil heat flux, MJ/m²/d; $\gamma$ is the psychrometric constant, kPa/°C; Δ represents the slope of the saturation vapor pressure versus temperature curve, kPa/°C; e_s-e_a is the vapor pressure deficit, kPa; T is the average atmospheric temperature, °C; and u₂ is the wind speed at approximately 2 m above ground level, m/s.

Thus, the logistic model with CET_O as the independent variable is expressed as:

$$ R_{H} = \frac{H}{{H_{\max } }} = \frac{1}{{1 + e^{{d_{1} + e_{1} R_{{CET_{o} }} }} }} $$

(10)

$$ R_{LAI} = \frac{LAI}{{LAI_{\max } }} = \frac{1}{{1 + e^{{d_{2} + e_{2} R_{{CET_{o} }} + f_{2} R_{{CET_{o} }}^{2} }} }} $$

(11)

$$ R_{D} = \frac{D}{{D_{\max } }} = \frac{1}{{1 + e^{{d_{3} + e_{3} R_{{CET_{o} }} }} }} $$

(12)

where, R_CETo is relative CET_O; and d₁, d₂, d₃, e₁, e₂, e₃, and f₂ are model fitting parameters.

Model evaluation

Growth model of plant height

Plant height (H) is a useful indicator of crop growth and nutritional status⁴⁹, and it is closely related to CGDD and CET_O. Several studies have demonstrated that H can accurately estimate aboveground biomass and LAI of crops^50,51. We analyzed the relationship between H and CGDD and CET_O, using partial samples from different regions, as shown in Figs. 2 and 3. The growth curve of H for both indicators showed a similar "S" shape, indicating slow growth in the early stage, rapid growth in the middle, and slow growth in the later period. This pattern suggests that high temperatures and sufficient light promote rapid growth in the middle stage, whereas low temperatures and reduced sunshine in the later period lead to slower growth. In southern Xinjiang, H peaked at approximately 1800 ℃ CGDD and 600 mm CET_O, while in northern Xinjiang, H peaked at around 1600 ℃ CGDD and 700 mm CET_O. The difference in CGDD between southern and northern Xinjiang was mainly due to temperature, with low precipitation and high temperatures in the former and vice versa in the latter. In contrast, CET_O reflects various meteorological conditions, and thus the difference in CET_O between the two regions was smaller. This suggests that CET_O is a more comprehensive indicator for regulating cotton H, as it considers multiple meteorological factors.

Figure 2

Relationship between cotton plant height and cumulative growing degree days in different regions.

Figure 3

Relationship between cotton plant height and cumulative reference crop evapotranspiration in different regions.

The characteristics of H change were similar across all regions, indicating a consistent pattern of growth regulation. To normalize the H change characteristics, we established relative logistic models of cotton R_H with R_CGDD and R_CETo, using data from different regions. The R_H change curves, along with the fitting results, are presented in Fig. 4 (P < 0.05). These models provide a useful framework for analyzing the effect of different meteorological conditions on cotton growth, and can be used to identify the optimal conditions for maximizing R_H and other growth indicators. The R_H model were established as follows:

$$ R_{H} = \frac{H}{{H_{\max } }} = \frac{1}{{1 + e^{{2.82 - 5.424R_{CGDD} }} }} $$

(13)

$$ R_{H} = \frac{H}{{H_{\max } }} = \frac{1}{{1 + e^{{3.212 - 6.091R_{{CET_{o} }} }} }} $$

(14)

Figure 4

Relationships of relative plant height with relative cumulative growing degree days (A) and relative cumulative reference crop evapotranspiration (B).

Figure 4 shows the fitting results of R_H of cotton based on CGDD and CET_O in different regions, the R² ≥ 0.866, indicating that the model had a good fit and high degree of precision. Moreover, it can be seen that the fitting result where R_CETo was the independent variable is better than with R_CGDD, and the discrete level of R_CETo was smaller than that of R_CGDD. Those experimental data that were not included in the modeling were used to further validate the model (Fig. 5 (A and B); R² ≥ 0.891, RMSE ≤ 0.072, RE ≤ 0.970%). There was good agreement between the measured and calculated values of R_H. These results clearly showed that the model with R_CETo as the independent variable was more accurate than that with R_CGDD.

Figure 5

Model validation results of relative plant height and relative cumulative growing degree days (A), relative cumulative reference crop evapotranspiration (B).

Growth model characteristics of cotton leaf area index

As an important quantitative indicator of each growth stage of cotton, LAI plays a very important role in assessing growth status. The cotton LAI change processes with CGDD and CET_O are shown in Figs. 6 and 7, respectively. The LAI increased with CGDD and CET_O in the early cotton growth stage, but gradually decreased in the late growth period. Photothermal resources are highly important and cotton growth requires sufficient temperature and light. When the CGDD was about 1400 ℃, LAI reached its maximum value in Xinjiang. When the CGDD was about 1600 ℃, LAI reached its maximum value in Central China and North China. However, when CET_O was used to describe the LAI growth process, the CET_O values when the LAI reached its maximum in different regions varied greatly. This was because meteorological conditions varied greatly among regions, and while CGDD is only sensitive to temperature, CET_O integrates many meteorological factors, which creates a more realistic predictor of the growth process of LAI. This further showed that replacing CGDD with CET_O will improve the accuracy of a crop growth model.

Figure 6

Relationship between cotton leaf area index and cumulative growing degree days in different regions.

Figure 7

Relationship between cotton leaf area index and cumulative reference crop evapotranspiration in different regions.

It can be seen from Figs. 6 and 7 that there were differences in the LAI among different regions due to varying conditions, such as soil fertility, irrigation, and fertilization systems. Despite these differences, the change trends in LAI remained consistent throughout the growth period. The change characteristics of the LAI were analyzed and compared in different regions to determine the universal change characteristics. Thus, logistic models of cotton R_LAI related to R_GDD and R_CETo were established, as shown in Fig. 8. The curve's rate of change was relatively high in the early stages of cotton growth, indicating that appropriate meteorological conditions can significantly enhance leaf area growth during this period. The R_LAI was fit to the models as shown in Fig. 8 (P < 0.05). The R_LAI model was established as follows:

$$ R_{LAI} = \frac{LAI}{{LAI_{\max } }} = \frac{1}{{1 + e^{{4.553 - 16.03R_{CGDD} + 10.09R_{CGDD}^{2} }} }} $$

(15)

$$ R_{LAI} = \frac{LAI}{{LAI_{\max } }} = \frac{1}{{1 + e^{{5.024 - 15.66R_{{CET_{o} }} + 9.5R_{{CET_{o} }}^{2} }} }} $$

(16)

Figure 8

Relationships of relative leaf area index with relative cumulative growing degree days (A) and relative cumulative reference crop evapotranspiration (B).

The based on CGDD and CET_O model fitting and validation results are shown in Figs. 8 and 9, with R² ≥ 0.778, RMSE ≤ 0.133, and RE ≤ 3.627%. The fitting effect when using R_CETo as the independent variable was better than when using R_CGDD, and the discrete level of R_CETo was smaller than R_CGDD. If $\frac{{dR_{LAI} }}{{dR_{CGDD} }} = 0$, $\frac{{dR_{LAI} }}{{dR_{{CET_{O} }} }} = 0$, we know that the R_LAI maximum occurred when R_CGDD and R_CETo were 0.794 and 0.824, respectively. These results showed that R_CGDD and R_CETo were notably different when R_LAI peaked, further indicating that CET_O is a better indicator than CGDD.

Figure 9

Model validation results of relative leaf area index and relative cumulative growing degree days (A), relative cumulative reference crop evapotranspiration (B).

Growth model characteristics of cotton dry matter accumulation

Crop yield is based on dry matter production, which is determined by nutrient absorption⁵². Dry matter accumulation is an essential indicator of cotton growth and is crucial for achieving high-quality and high-yielding cotton crops. The trends in the increase and decrease of cotton dry matter were similar among different regions. The rate of change of dry matter accumulation peaked when CGDD was between 700 and 1000 °C (Fig. 10) and CET_O was between 200 and 500 mm (Fig. 11), during the budding stage, flowering stage, and boll development stage. During these stages, many leaves grew, and photosynthesis significantly increased due to sufficient temperature and light availability, leading to high cotton yields. Dry matter generally accumulated throughout the growing season, starting slowly during the seedling stage, gradually accelerating during the budding stage, peaking during the flowering to boll setting stages, and then stabilizing. However, due to the arid climate in Xinjiang, with high evaporation and little precipitation, dry matter accumulation peaked when CGDD was about 1900 °C and CET_O was about 750 mm. The peak CGDD for dry matter accumulation was lower in Xinjiang than in Central China and North China, while the peak CET_O was higher in Xinjiang than in Central China and North China.

Figure 10

Relationship between cotton dry matter accumulation and cumulative growing degree days in different regions.

Figure 11

Relationship between cotton dry matter accumulation and cumulative reference crop evapotranspiration in different regions.

The change characteristics of the cotton D in separate regions were analyzed and compared to reveal the universal change characteristics of D. Thus, the logistic models of the change processes of cotton R_D with R_CGDD and R_CETo were established, as shown in Fig. 12. The fitting result is shown in Fig. 12 (P < 0.05), with R_D calculated using the following formulas:

$$ R_{D} = \frac{D}{{D_{\max } }} = \frac{1}{{1 + e^{{2.877 - 5.764R_{CGDD} }} }} $$

(17)

$$ R_{D} = \frac{D}{{D_{\max } }} = \frac{1}{{1 + e^{{3.206 - 6.163R_{{CET_{o} }} }} }} $$

(18)

Figure 12

Relationships of relative dry matter accumulation with relative cumulative growing degree days (A) and relative cumulative reference crop evapotranspiration (B).

The based on CGDD and CET_O model fitting and validation results are shown in Figs. 12 and 13, with R² ≥ 0.674, RMSE ≤ 0.138, and RE ≤ 4.069%. Similar to H and LAI, the fitting effect when using R_CETo as the independent variable was better than that when using R_CGDD, and the discrete level of the measured values was smaller than that of R_CGDD. Therefore, by using R_CETo as the independent variable, a cotton growth index model with more accurate fitting results can be established. Such a model reflects the cotton growth process more precisely.

Figure 13

Model validation results of relative dry matter accumulation and relative cumulative growing degree days (A), relative cumulative reference crop evapotranspiration (B).

Growth model comparison

It can be seen from Fig. 14 that with the changes in R_CGDD and R_CETo, the rate of change in R_LAI was greater than in R_H and R_D in the early stage of cotton growth. Conversely, the rate of change in R_LAI was smaller than in R_H and R_D in the later stage of cotton growth. This proves that suitable meteorological conditions significantly impact the form of D in the later stages of cotton growth, during which the energy absorbed is mainly used for the growth of the cotton bolls. This growth pattern is similar to that of other crops, such as potatoes²³ and rice⁹. The rate of change in the CET_O-based model was greater than in the CGDD-based model in the middle and later stages of cotton growth. This could be attributed to the fact that cotton growth is sensitive to various meteorological conditions, such as solar radiation, temperature, humidity, etc., which are included in the CET_O calculation process. This further demonstrates that CET_O is preferable for describing the cotton growth process.

Figure 14

Cotton growth index logistic model and its first-order derivative change rate curve with relative cumulative growing degree days and relative cumulative reference crop evapotranspiration.

In addition, Eqs. (13)–(18) can be used to find the second derivative and obtain the change process of the first derivative function of the logistic model with R_CGDD and R_CETo. The figure shows that the change rate and inflection points based on the CGDD and CET_O logistic models were different. Therefore, although the CGDD calculation is simple and the CET_O calculation is complex, CET_O should be used when modeling to ensure that the model reflects the impacts of meteorological conditions on cotton growth, which CGDD is unable to do.

Cotton boll growth relies on the contributions from all parts of the plant, as reflected by the growth indexes of H, LAI, and D. To comprehensively examine the effects of meteorological conditions on cotton growth and compare with traditional methods, we analyzed the impacts of CGDD and CET_O on cotton H, LAI, and D. The results indicate that using CGDD alone to describe cotton growth yields a relatively simple relationship that does not fully reflect the impacts of meteorological conditions. Due to the spatial heterogeneity caused by meteorology conditions, cotton growth varies across regions, leading to spatial differences in yield. Therefore, it is crucial to consider meteorology conditions when describing the cotton growth process. Although differences in cotton H, LAI, and D, as well as irrigation water, directly affect yield, leaves are the main body for crop photosynthesis and transpiration. Therefore, further quantitative studies on the relationship between cotton LAI and yield, as well as irrigation water, are necessary.

Relationship of maximum leaf area index with cotton yield and irrigation amount

Cotton yield is determined by multiple factors, including LAI and irrigation water amount (W). Previous studies have reported that increasing leaf area contributes little towards yield, because dark respiration rate is enhanced correspondingly after leaf expansion, which was not conducive to biomass accumulation⁵³. A suitable LAI or an suitable increase in crop LAI is required to achieve higher yields. In particular, higher yields are related to the sustained photosynthetic activity of leaves. Therefore, it is important to establish the relationship of LAI_max with Y and W to understand cotton yields (Fig. 15). The LAI_max range was divided into seven categories (2–3, 3–4, 4–5, 5–6, 6–7, 7–8, and 8–9), and the W range was divided into six categories (400–450, 450–500, 500–550, 559–600, 600–650, and 650–700 mm). The average LAI_max and W were calculated for each category.

Figure 15

Variation trend of seed cotton yield with maximum leaf area index (A) and maximum leaf area index with irrigation amount (B).

The relationship of LAI_max with Y and W can be described by the quadratic polynomial functions:

$$ Y = - 36.9LAI_{\max }^{2} + 446LAI_{\max } + 5824 $$

(19)

$$ LAI_{\max } = - 1.319 \times 10^{ - 4} W^{2} + 0.1513W - 36.95 $$

(20)

The R² of the fitted curves of LAI_max with Y and W with LAI_max were determined to be 0.888 and 0.841, respectively. The first-order derivative of Eq. (19) was 0, while Y was maximized (7171.7 kg/ha) for LAI_max values of 6.043 cm²/cm². Similarly, the first-order derivative of Eq. (20) was 0 with a LAI_max of 4.938 cm²/cm², which corresponded to a W of 573.541 mm. The results suggested that if LAI_max values were close to 6.043 cm²/cm², the cotton yield would be higher.

Figure 16 presents the validation results from the comparison of measured and calculated values. The results of the Y and LAI_max validations revealed R² values of 0.791 and 0.704; RMSE values of 82.06 kg/ha and 0.404 cm²/cm²; and RE values of 0.014% and 0.431%, respectively.

Figure 16

Model validation results of maximum leaf area index and seed cotton yield (A), irrigation amount and maximum leaf area index (B).

Crop irrigation water utilization efficiency (IWUE) is a comprehensive physiological ecological index used to evaluate the degree of crop growth by examining the relationship between crop irrigation amount and yield. The IWUE is calculated as follows:

$$ IWUE = \frac{Y}{W} $$

(21)

We can then obtain Eq. (22) by combining Eqs. (19)–(21). Therefore, the IWUE can be calculated from W. Figure 17 presents the validation results of the measured and calculated values. The R², RMSE, and RE were determined to be 0.701, 0.972 kg/(ha·mm), and 0.299%, respectively.

$$ IWUE = \frac{{ - 6.4197 \times 10^{ - 7} W^{4} + 0.0015W^{3} - 1.2632W^{2} + 480.061W - 61035.362}}{W} $$

(22)

Figure 17

Validation results of the cotton irrigation water utilization efficiency mathematical model.

Discussion

This study demonstrates the important roles that CGDD and CET_O play in simulating crop growth, and highlights the need to investigate the relationship between them. Su²⁸ conducted an analysis of CGDD trends and the correlation between CGDD and CET_O in the Turpan area of China. Their findings indicate a strong correlation between R_CGDD and R_CETo, and suggest that a cubic polynomial function (Eq. (23), Fig. 18 (A)) can accurately describe the relationship between grape budding and maturity.

$$ R_{{CET_{O} }} = gR_{CGDD}^{3} + hR_{CGDD}^{2} + kR_{CGDD} + l $$

(23)

Figure 18

Relationships between relative cumulative growing degree days and relative cumulative reference crop evapotranspiration.

The logistic growth model can be expressed as follows:

$$ \left\{ \begin{gathered} R_{{y_{1} }} = \frac{{y_{1} }}{{y_{1_{\max } }}} = \frac{1}{{1 + e^{{d + e(gR_{CGDD}^{3} + hR_{CGDD}^{2} + kR_{CGDD} + l)}} }} \hfill \\ R_{{y_{2} }} = \frac{{y_{2} }}{{y_{2_{\max }} }} = \frac{1}{{1 + e^{{d + e(gR_{CGDD}^{3} + hR_{CGDD}^{2} + kR_{CGDD} + l) + f(gR_{CGDD}^{3} + hR_{CGDD}^{2} + kR_{CGDD} + l)^{2} }} }} \hfill \\ \end{gathered} \right. $$

(24)

where g, h, k, and l are model fitting parameters; y₁ is cotton H or D; and y₂ is cotton LAI.

However, the cubic polynomial function has many parameters and complex forms, making it difficult to calculate when performing the equation transformation. In this study, we found that the relationship between R_CGDD and R_CETo of different regions can be approximated using a linear function (Eq. (25), Fig. 18B). The linear function has fewer parameters, so it is simpler in form and easier to calculate. The linear relationship function between R_CGDD and R_CETo uses R_CGDD as the independent variable as follows:

$$ R_{{CET_{O} }} = mR_{CGDD} + n $$

(25)

where m and n are model fitting parameters.

Combining Eqs. (6) –(8), Eqs. (10) –(12), and Eq. (25), the relationship between the model parameters d₁, d₂, d₃, e₁, e₂, e₃, and f₂ and a₁, a₂, a₃, b₁, b₂, b₃, and c₂ can be expressed as:

$$ a_{1} + b_{1} R_{CGDD} = (d_{1} + e_{1} n) + e_{1} mR_{CGDD} $$

(26)

$$ d_{1} = a_{1} - \frac{{b_{1} n}}{m} $$

(27)

$$ e_{1} = \frac{{b_{1} }}{m} $$

(28)

$$ a_{2} + b_{2} R_{CGDD} + c_{2} R_{CGDD}^{2} = (d_{2} + e_{2} n + f_{2} n^{2} ) + (2f_{2} mn + e_{2} m)R_{CGDD} + f_{2} m^{2} R_{CGDD}^{2} $$

(29)

$$ d_{2} = a_{2} - \frac{{b_{2} mn - c_{2} n^{2} }}{{m^{2} }} $$

(30)

$$ e_{2} = \frac{{b_{2} m - 2c_{2} n}}{{m^{2} }} $$

(31)

$$ f_{2} = \frac{{c_{2} }}{{m^{2} }} $$

(32)

Likewise:

$$ d_{3} = a_{3} - \frac{{b_{3} n}}{m} $$

(33)

$$ e_{3} = \frac{{b_{3} }}{m} $$

(34)

We used the equations to simulate the relationship between R_CGDD and R_CETo in each region and presented the results in Fig. 19. The fitting effect had an R² ≥ 0.976, and we obtained the values of parameters m and n for each region. Table 2 shows the validation of parameters d₁, d₂, d₃, e₁, e₂, e₃, and f₂, which were calculated using Eqs. (26)–(34) to fit different regions. The validation of the model parameters H and D showed a good fit with RE < 0.10. However, the LAI model parameter validation showed unsatisfactory results. We used the modified logistic model to simulate the LAI change process, but this model was not well-suited for the task due to the quadratic polynomial exponential term exp, which increased the possibility of error during equation transformation. Nonetheless, it is possible to approximately calculate parameters d, e, and f from parameters a, b, and c for convenient calculations or situations where meteorological data are lacking to simulate the crop growth process.

Figure 19

Relationships between relative cumulative growing degree days and relative cumulative reference crop evapotranspiration in every region.

Table 2 Comparison of the fitting and calculation results of parameters d, e, and f.

Climate suitability is an important ecological characteristic for crops and forms the basis of the distribution of crop production⁵⁴. Cotton is sensitive to climate variability and is unable to resist damage caused by drought and cold⁵⁵. Numerous studies have reported quantitative predictions of the impact of future climate change on cotton production^{56,57,58,59,60}. These studies generally predict major changes in the trends of cotton production under future climate scenarios, and describe how cotton agricultural production can respond and adapt to climate change. However, discrepancies among research methods (such as crop models, climate models, and climate scenarios) create uncertainties regarding the extent and severity of the impacts that future climate change will have on cotton yields. Therefore, future research should focus on quantitatively exploring the relationship between climate change and cotton growth process.

Many mathematical models have been used to describe crop growth processes, such as the Logistic, Richards, Gompertz, Hoerl, and others. However, previous research has shown that the Logistic model has a better fitting effect when using time or CGDD as independent variables^11,12. While many studies have quantitatively analyzed the impact of CGDD on crop growth, including for winter wheat⁸, rice⁹, cotton²⁴, potato²³, maize⁶¹, and watermelon²⁶. It is important to note that these models have some limitations, as CGDD considers only a few meteorological factors, such as the highest daily temperature and the lowest daily temperature. However, meteorology factors such as temperature, wind, rainfall, relative humidity, and sunshine duration significantly affect the production of cotton flowers and bolls²⁰. For instance, temperature is the most critical meteorology factor that affects cotton yield, and heat stress can lead to reduced fruit retention, delayed crop maturity, and lower lint quality. Strong winds may also cause boll shedding, thereby reducing yield. Additionally, continuous rainfall during flowering and boll opening can impair pollination and potentially reduce fiber quality³¹. Moreover, climate change is a crucial issue worldwide, and its negative effects on cotton growth and development can result from an increased number and severity of days with very high temperatures during the cotton season. These events can lower cotton yields by decreasing daily photosynthesis and, at times, raising respiration at night^20,31,56,57. Further research is necessary to comprehensively consider these factors when simulating cotton growth. In this study, we present a new cotton growth model that try to overcomes some of these limitations and demonstrate its superiority over existing models through comparison and analysis. Specifically, we use the CET_O, which comprehensively considers meteorology conditions such as local solar radiation, humidity, temperature, and wind speed. Our results showed that the CET_O-based logistic model more accurately described the cotton growth process compared with the CGDD-based model. We deeply evaluated both CET_O and CGDD as effective predictors when simulating crop growth processes. In future studies, we plan to make further improvements to this method and expand our research to other crops. These research results and the constructed mathematical models can provide a scientific basis for improving agricultural production efficiency while also contributing to the improvement and development of crop growth models.

Conclusion

To provide a more accurate description of the effects of climate change on cotton growth, this paper proposes a novel logistic cotton growth model that uses CET_O instead of CGDD as the independent variable. Although crop growth and yields varied significantly across different regions and years, the overall trends in their developments throughout the year remained consistent. A comparison between models using CET_O and CGDD as independent variables tested the precision of the former in simulating crop growth. The results showed that CET_O can comprehensively reflect the impacts of meteorological factors on cotton growth and more accurately describe the cotton growth process. Additionally, this study established mathematical models relating cotton LAI_max to Y and W. When cotton Y reached its maximum value of 7171.7 kg/ha, the value of LAI_max was 6.043 cm²/cm². The required W was 518.793 mm, and the IWUE was 21.153 kg/(ha·mm). In the future, more attention should be paid to the impacts of meteorological conditions and irrigation on crop growth.