Simple nonlinear model for the relationship between maize yield and cumulative water amount

Journal of Integrative Agriculture 2017, 16(4): 858–866 Available online at www.sciencedirect.com

ScienceDirect

RESEARCH ARTICLE

Simple nonlinear model for the relationship between maize yield and cumulative water amount LIU Cheng1, 2, SUN Bao-cheng2, TANG Huai-jun2, WANG Tian-yu3, LI Yu3, ZHANG Deng-feng3, XIE Xiaoqing2, SHI Yun-su3, SONG Yan-chun3, YANG Xiao-hong1, LI Jian-sheng1 1

Maize Research Center, China Agricultural University, Beijing 100193, P.R.China

2

Institute of Grain Crops, Xinjiang Academy of Agricultural Sciences, Urumqi 830091, P.R.China Institute of Crop Sciences, Chinese Academy of Agricultural Sciences, Beijing 100081, P.R.China

3

Abstract Both the additive and multiplicative models of crop yield and water supply are polynomial equations, and the number of parameters increases linearly when the growing period is specified. However, interactions among multiple parameters occasionally lead to unreasonable estimations of certain parameters, which were water sensitivity coefficients but with negative value. Additionally, evapotranspiration must be measured as a model input. To facilitate the application of these models and overcome the aforementioned shortcomings, a simple model with only three parameters was derived in this paper based on certain general quantitative relations of crop yield (Y) and water supply (W). The new model, Y/Ym–Wk/(Wk+whk), fits an S or a saturated curve of crop yield with the cumulative amount of water. Three parameters are related to biological factors: the yield potential (Ym), the water requirement to achieve half of the yield potential (half-yield water requirement, wh), and the water sensitivity coefficient (k). The model was validated with data from 24 maize lines obtained in the present study and 17 maize hybrids published by other authors. The results showed that the model was well fit to the data, and the normal root of the mean square error (NRMSE) values were 2.8 to 17.8% (average 7.2%) for the 24 maize lines and 2.7 to 12.7% (average 7.4%) for the 17 maize varieties. According to the present model, the maize water-sensitive stages in descending order were pollen shedding and silking, tasselling, jointing, initial grain filling, germination, middle grain filling, late grain filling, and end of grain filling. This sequence was consistent with actual observations in the maize field. The present model may be easily used to analyse the water use efficiency and drought tolerance of maize at specific stages. Keywords: yield, water, model, maize, water sensitivity, drought tolerance

Received 3 June, 2016 Accepted 30 September, 2016 LIU Cheng, Tel: +86-991-4507086, E-mail: [email protected]; Correspondence LI Jian-sheng, Tel: +86-10-62732422, E-mail: [email protected]; WANG Tian-yu, Tel: +86-1062186632, E-mail: [email protected]; LI Yu, Tel: +86-1062186632, E-mail: [email protected] © 2017, CAAS. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/) doi: 10.1016/S2095-3119(16)61493-4

1. Introduction The relationship between crop yield and water quantity is the foundation for optimizing irrigation schemes and developing new methods for evaluating drought tolerance, and research in this field can be traced back 100 years (Briggs 1914). Because a primary model for the whole growing period was established (Wit 1958), many models have been developed and continuously improved, and they include simple and

LIU Cheng et al. Journal of Integrative Agriculture 2017, 16(4): 858–866

complex models as well as empirical and theoretical models (Stewart et al. 1973, 1976; Blank 1975; Hanks 1975; Barrett 1980; Morgan et al. 1980; Singh et al. 1987; Rao et al. 1988; Li et al. 1997; Mehdi et al. 2014; Paredes et al. 2014; Shabani et al. 2015). A simple model is more acceptable in application because of its convenience (Peng et al. 2000; Shen et al. 2001b). Additive and multiplicative models are relatively simple compared with complicated theoretical models (Xia et al. 2003). Typical additive models were developed by Blank (1975), Stewart et al. (1973, 1976) and Singh (1987), and multiplicative models were developed by Jensen (1973), Minhas et al. (1974), Hanks (1975), and Rao et al. (1988). Additive models often have been criticized because they do not consider the lag effect of drought stress in crop growth stages, which may cause the water sensitivity coefficient to be negative (Taylor et al. 1983; Shen et al. 2001a, b). Therefore, multiplicative models, such as the Jensen model, have been a focus of greater attention (Rajput and Singh 1986; Wang et al. 2001; Cong et al. 2002; Wei et al. 2002; Zhang 2009). However, increasing evidence has indicated that these multiplicative models can also result in negative water sensitivity coefficient values (Li 1999; Jiao et al. 2004; Wei 2004), which may explain why the multiplicative model has been improved by certain researchers (Guo 1994; Chen et al. 1998; Cong et al. 2002; Jiao et al. 2004, 2005; Wei 2004). In the mathematical structure of the models, both additive and multiplicative models include several parameters that are located at an equivalent position and have an equivalent meaning; thus, these parameters have the same statistical distribution (Wei 2004). A large change in one parameter may easily be obscured by slight variations of several other parameters if test errors are introduced (Jiao et al. 2004). When the growth stage is specified, the number of parameters in both models will increase, which also increases the risk of parameter offset. This offset may explain why a multi-parameter model can fit the experimental data well while containing unreasonable parameters. Compared with a linear model, a nonlinear model has few parameters that are not at equivalent positions, which allows this type of model to effectively avoid negative parameters. The objective of this paper is to establish a simple nonlinear model for the relationship between crop yield and water supply and validate the model performance using data from 24 maize lines obtained in this study and 17 maize hybrids published by others.

2. Materials and methods 2.1. Model derivation and description The derivation of the model was based on three basic

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assumptions. First, the crop yield (Y) does not increase indefinitely with water increases. With increases in accumulated irrigation (W), the relative increment of crop yield will decrease gradually. Therefore, the change rate of crop yield with the amount of water is proportional to 1/W. Second, the increasing rate of Y with the amount of water depends on the developmental status of the plant, and it is lower at the early growing stage but higher at the late growing stage because the plant is larger and needs much more water. Thus, the change rate of the crop yield with the amount of water is proportional to the yield itself or the growing quantity (biomass). Third, when the plant develops to maturity and the yield reaches the maximum (Ym), less water is required. Therefore, the change rate of the crop yield with the amount of water is proportional to (1–Y/Ym). According to the above three assumptions, we established the following differential eq. (1): Y 1 ∂Y λ= = k × ×Y ×(1− ) (1) ∂W W Ym After integral derivation, we obtained an exponentiation saturation model for crop yield and water accumulation: Y Wk = k (2) Ym W +Whk Where, Y is the yield (or dry matter) and W is the cumulative amount of water. k is a constant that is only related to the line/variety. For stage j, the cumulative amount of water is as follows: j

Wj =∑ωt t =1

Where, ωt is the actual individual amount of water at time t, which includes irrigation and rainfall. The present model (eq. (2)) only has three parameters (Ym, Wh, k), and they are located at different positions in the model structure and have different biological meanings. Ym is the maximum yield (an expression of productivity) if water is not a limiting factor. Wh is the cumulative amount of water required to reach half of the maximum yield (half-yield water requirement). Within the same maturity period, smaller parameters correspond to stronger drought tolerance. k is the water sensitivity coefficient. Lines/varieties with large k values have higher water use efficiency but weaker drought tolerance. Generally, a line/variety with higher drought tolerance has small Wh and k values. Therefore, the combination form Whk or logic form klnWh can be used for the evaluation of drought tolerance. The characteristics of the present model must be described because certain characteristics will be used in subsequent applications. 1) According to eq. (2), the model fits an S curve if k>1 and a saturated curve if k=1.

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2) Eq. (3) is equivalent to eq. (1) and can be used to calculate the water sensitivity coefficient λ, the instant water use efficiency (IWUE), or the yield change rate with the water supply: ∂Y kY (WW )k λ=IWUE= = m k h k 2 (3) ∂w W (W +Wh ) 3) The maximum sensitivity coefficient λmax is at the curve of the inflection point S (Ws, Ys) and can be calculated using eq. (4):

λmax=

∂Y ∂W

= max

k −1 Ws =Wh k +1

1 k

Ym 4kWh , Ys =Ym

( k+1) ( k −1) k +1

k −1 1/k

(4) k −1 2k

4) Eq. (2) can be converted to a linear form for convenience to determine the parameter values:

ln[Y / (Ym −Y )]= α + βlnW

k=β, Wh =exp (−α / β)

(5)

2.2. Experimental design The field experiments were conducted at the Anningqu station of the Xinjiang Academy of Agricultural Sciences (Urumqi, China), which is located at 87°36´E and 43°46´N at an elevation of 654 m. Throughout the entire growing period of maize (April to October) from 2012–2015, the daily average temperature was 17.3–29.1°C, the relative humidity was 22.8–42.6%, and the rainfall was 93.6–119.7 mm (936– 1 197 m3 ha–1). Maize was commonly irrigated 8 times with 75 mm water (750 m3 ha–1) per irrigation. Because of the dry climate in Xinjiang, a single rainfall of less than 10 mm is ineffective for maize growth. To obtain the data for model validation, 24 inbred maize lines were tested from 2012–2015 at the station. According to the assumptions of the present model, the water value should be a cumulative amount. Discontinuous irrigation resulting in drought stress would affect the normal cumulative process and cause the model parameter to shift. Therefore, the experiments were designed so that the irrigation times in the growing period were gradually reduced for different water treatments. The well-watered block, labelled as CK, was irrigated 8 times throughout the growing period, each time with 75 mm or 750 m3 ha–1 irrigation. Other treatments were labelled according to the times of irrigation as T1, T2, T3 and T4. T1 was irrigated once: at the germination stage; T2 was irrigated twice: at the germination and jointing stages; T3 was irrigated 3 times: at the germination, jointing and tasselling stages; T4 was irrigated 4 times: at the germination, jointing, tasselling, and pollen-shedding stages. The irrigation amount at each stage was 750 m3 ha–1.

Three-meter wide belts were arranged for water isolation among the treatments. In a treatment block, 150 maize plants were sown in an area of 15 m2, and there were 3 replicates. The soil moisture was measured every 2 days, with approximately 5 to 7 measurements in an irrigation cycle. The variation range of the soil moisture content at a depth of 40 cm was 18 to 23% (weight). From May to September, the average evaporation was 1 760–2 210 mm. Because of the dry climate of the station, variations in the soil water were not included in the model. Thus, the cumulative water amount (mm) was equal to the accumulated rainfall plus the accumulated irrigating water. Twenty-four maize lines were tested at different watering levels with 3 repetitions for 4 years, and each repetition included 60 plants. The yields for all of the treatments were measured after harvest to establish and validate the model. The 24 maize lines were B6, B8, Z58, CBL, PLD, Z31, b1, B2, C7-2, b7, b10, B12, 478, B20, b21, LY92, U9-2, U9-19, Q319, b26, b28, B31, B33, and 7922.

2.3. Statistical analysis method The model parameters were determined using the SOLVER tool of Microsoft Excel. The estimated parameters were Ym, Wh and k, and the objective for minimization was ∑(Y–Ya)2, where Ya is the actual yield and Y is the yield estimated by eq. (2). If high-precision results are not required, then the Excel Intercept and Slope functions can be used to calculate the parameters (see eq. (5)). The fitness of the model was validated by the determinant (r2) value or the normal root of the mean square error (NRMSE) value (Willmott et al. 1985; Mehdi et al. 2014): NRMSE (%)=100

n

2 ∑ (Οj –Εj ) /(nΟαve ) 2

0.5

j=1

Where, Oj and Ej are the observation and estimation values, respectively, n is the number of observed data, and Oave is the average of the observed data. The simulation was considered excellent and presented an NRMSE of less than 10%. NRMSE values greater than 10% and less than 20% are considered good, values greater than 20% and less than 30% are considered fair, and values greater than 30% are considered poor (Mehdi et al. 2014).

3. Results 3.1. Model fitting and validation The water supply information for the different treatments from 2012–2015 is listed in Table 1. The total amount of water for treatments T1, T2, T3, T4, and CK was 1 947, 2 697, 3 447, 4 197, and 7 197 m3 ha–1, respectively. These

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values were used to determine the parameters of the present model (eq. (2)), whereas the cumulative amount of water for the CK treatment, which was irrigated 8 times, was used to calculate the water use efficiency after the parameters of eq. (2) were determined. Our next goal was to establish a model according to the actual data obtained above. For example, in T1, T2, T3, T4, and CK, the total amount of water was W={1 947, 2 697, 3 447, 4 197, and 7 197, respectively}, the yield of the maize line T8920B6 was Y={1 095.7, 3 097.2, 4 132.6,

4 720.0, and 6 008, respectively}. The parameters used in model eq. (2) (determined by the Excel SOLVE tool) were Ym=6 008, Wh=2 823 and k=3.64. The modelled yield for maize line T8920B6 was Y–6 008W3.64/(W3.64+2 8233.64). All 24 maize lines and 17 maize varieties were treated in the same manner. Furthermore, the NRMSE values based on the 24 inbred lines and 17 hybrids were calculated and are listed in Tables 2 and 3, respectively. For the maize lines, the NRMSE ranged from 2.8 to 17.8% (lower than 20%), and

Table 1 Water supply amounts for the different treatments (means for 2012–2015) Days after sowing (d)

Growth stage

1 Germination 36 Jointing 43 Tasselling 55 Pollen shedding and silking 67 Initial grain filling 79 Middle grain filling 91 Late grain filling 103 End of grain filling Total amount of water 1)

T1 1 114 56 56 84 84 129 45 379 1 947

Amount of water for different treatments (irrigation and rainfall, m3 ha–1)1) T2 T3 T4 1 114 1 114 1 114 806 806 806 806 806 56 834 84 84 84 84 84 129 129 129 45 45 45 379 379 379 2 697 3 447 4 197

CK 1 114 806 806 834 834 879 795 1 129 7 197

Cumulative amount of water for CK (m3 ha–1) 1 114 1 920 2 726 3 560 4 394 5 273 6 068 7 197

T1 was irrigated once, at the germination stage; T2 was irrigated twice, at the germination and jointing stages; T3 was irrigated 3 times, at the germination, jointing and tasselling stages; T4 was irrigated 4 times, at the germination, jointing, tasselling, and pollen-shedding stages. Underlined numbers show the sum of irrigation and rainfall, whereas unlined numbers are only rainfall.

Table 2 Model parameters that were determined and validated using 24 maize lines1) Maize inbred line 01_T8920B6 02_T8920B8 03_Zheng58 04_CBL4-2009-2 05_PLD-2009-2 06_Zong31 07_b1 08_B2 09_Chang7-2 10_b7 11_b10 12_B12 13_478 14_B20 15_b21 16_Luyuan92 17_U9-2 18_U9-19 19_Qi319 20_b26 21_b28 22_B31 23_B33 24_7922 Minimum Maximum Mean 1)

Ym (kg ha–1) 6 008 6 016 6 021 7 081 4 351 4 574 5 460 3 078 4 774 6 214 4 972 4 147 6 086 5 494 4 279 5 829 6 047 4 520 5 777 6 095 8 041 8 122 6 784 8 179 3 078.4 8 179.4 5 747.9

Wh (m3 ha–1) 2 823 2 599 3 234 3 109 3 008 3 124 3 755 3 561 3 847 3 303 3 169 3 367 3 158 3 288 3 779 3 589 3 688 3 680 3 796 3 445 3 562 3 573 2 717 3 558 2 598.9 3 847.2 3 363.8

k 3.64 4.13 6.94 5.15 4.71 6.69 10.01 10.01 9.31 8.48 8.65 7.42 3.77 4.52 8.60 7.03 7.60 6.61 10.66 5.06 7.76 6.46 4.42 6.36 3.6 10.7 6.8

NRMSE (%) 5.3 2.8 6.8 7.6 17.8 15.3 4.0 5.5 7.7 3.1 4.3 7.6 6.2 6.9 4.4 5.8 7.6 8.3 10.6 7.9 7.6 9.7 4.6 4.3 2.8 17.8 7.2

r2 0.987 0.995 0.991 0.983 0.915 0.948 0.999 0.996 0.994 0.999 0.997 0.991 0.988 0.988 0.998 0.995 0.994 0.990 0.989 0.987 0.992 0.984 0.989 0.997 0.915 0.999 0.987

Ym, maximum yield of well watering; Wh, water requirement of half-yield; k, water sensitivity coefficient; NRMSE, normal root of the mean square error; r2, determination coefficient.

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the average was 7.2% (Table 2). For the maize hybrids, the NRMSE ranged from 2.7 to 12.7% (lower than 20%), and the average was 7.4% (Table 3). The results suggest that the present model is well fit to the data for the 24 maize lines and the 17 hybrids (Figs. 1 and 2). The relationship of maize yield with the amount

of water for all 24 maize lines could be depicted as an S curve (Fig. 3).

3.2. Model application The value of the change rate of crop yield with the amount

Table 3 Model parameters that were determined and validated using 17 maize hybrids (data were obtained by Fan (2011) Maize hybrid line Dongdan 60 Zhongyu 9 Tunyu 42 Danyu 86 Tunyu 38 Nongda 108 JIngyu 811 Shengyu 18 Zhengdan 958 Zhongdan 5458 Xianyu 335 Jingyu 62 Jingdan 55 Zhongdan 14 Qiangsheng 9 Ludan 6006 Dafeng 3 Minimum Maximum Mean

Yields (kg ha–1) of different treatments (water, m3 ha–1) No Low Mid High (4 884) (5 454) (6 324) (7 037) 6 221 6 924 8 700 4 866 6 249 6 602 8 126 4 343 6 145 6 970 7 472 4 188 6 222 7 276 8 272 4 314 6 095 7 047 8 545 4 171 6 024 7 357 8 732 4 442 6 519 7 149 8 924 3 959 6 483 7 980 9 163 4 238 7 284 8 994 10 007 3 697 6 284 7 696 8 576 3 138 7 102 8 517 10 501 4 087 6 160 7 630 9 315 3 629 6 669 8 055 10 088 3 862 5 446 5 787 7 544 2 586 5 977 7 034 8 514 2 705 6 149 7 003 9 421 3 405 5 325 6 988 8 683 2 003

Model parameters1) Ym Wh (kg ha–1) (m3 ha–1) 8 700 4 536 8 126 4 572 7 472 4 703 8 272 4 774 8 545 4 838 8 732 4 839 8 924 4 904 9 163 4 941 10 007 5 075 8 576 5 076 10 501 5 097 9 315 5 113 10 088 5 128 7 544 5 134 8  514 5 186 9 421 5 195 8 683 5 400 4 536 7 472 5 400 10 501 4 971 8 858

k 4.27 4.95 9.14 7.28 6.07 6.33 6.07 7.91 10.39 10.37 7.24 7.48 7.04 6.83 8.76 6.13 9.94 4.3 10.4 7.4

NRMSE (%)2) 9.0 8.6 2.7 4.2 6.9 5.7 9.2 4.2 4.5 4.9 7.7 6.9 8.2 12.7 9.0 11.9 9.0 2.7 12.7 7.4

r2 0.926 0.914 0.990 0.984 0.964 0.977 0.931 0.989 0.986 0.983 0.966 0.976 0.965 0.895 0.951 0.930 0.967 0.895 0.990 0.958

1)

Ym, maximum yield of well watering; Wh, water requirement of half-yield; k, water sensitivity coefficient. NRMSE, normal root of the mean square error. 3) 2 r , determination coefficient. 2)

T8920B6 T8920B8 Zheng 58 CBL4-2009-2 PLD-2009-2 Zong 31

b1 B2 Chang 7-2 b7 b10 B12

478 B20 b21 Luyuan 92 U9-2 U9-19

Dafeng 3 Danyu 86 Dongdan 60 Jindan 55 Jindan 62 Jinyu 811

Qi319 b26 b28 B31 B33

Estimated yiled (kg ha–1)

Estimated yield (kg ha–1)

NRMSE(%)=2.8–17.8 (r2=0.914–0.999)

6 000 4 000 2 000 0

0

2 000

4 000

6 000

8 000

Actual yield (kg ha–1)

Fig. 1 Model performance to fit 24 maize lines.

Xianyu 335 Zhengdan 958 Zhongdan 14 Zhongdan 5458 Zhongyu 9

12 000

10 000 8 000

Ludan 6006 Nongda 108 Qiangsheng 9 Shenyu 18 Tunyu 38 Tunyu 42

10 000

NRMSE(%)=2.7–12.7 (r2=0.895–0.990)

10 000 8 000 6 000 4 000 2 000 0

0

2 000

4 000 6 000 8 000 10 000 12 000 Actual yield (kg ha–1)

Fig. 2 Model performance to fit 17 maize hybrids.

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LIU Cheng et al. Journal of Integrative Agriculture 2017, 16(4): 858–866

Maize yield (×1 000 kg ha–1)

of water can be considered the water use efficiency (WUE) when the growing stage is sufficiently short. Moreover, the WUE is the IWUE at a given stage in the growing period. Concurrently, the change rate can also be considered the value of the water sensitivity. Therefore, eq. (1) or (3) in this model can be used to analyse either the IWUE or drought tolerance at a specific stage. As an example, the parameters of maize T8920B6 (as determined above) were Ym=6 008, Wh=2 823 and k=3.64. Using eq. (4) and the cumulative amount of water (W)

b28 8 6 4 2 0 U9-2 8 6 4 2 0 8 478 6 4 2 0 8 Chang7-2 6 4 2 0 PLD-2009-2 8 6 4 2 0 8 T8920B6 6 4 2 0 0 1 2 3 4

5

listed in Table 4, the maximum IMUE or the maximum water sensitivity coefficient (λmax) of this maize can be calculated as 2.09 kg m–3, where the cumulative amount of water reaches 2 418 m3 ha–1 (Ws) and the yield is equal to 2 719 kg ha–1 (Ys). According to Table 4, these values are observed at approximately June 20 at the beginning of the tasselling stage. The IWUE value of certain maize lines at different stages was depicted in Fig. 4, which indicates that the IWUE values of all 20 maize lines peaked at the tasselling stage, or

B31

B33

7922

U9-19

Qi319

b26

B20

b21

Luyuan 92

b7

b10

B12

Zong 31

b1

B2

T8920B8

Zheng 58

6 7 0 1 2

3 4

5

6 7 0 1 2

3 4

CBL4-2009-2

5

6 7 0 1 2

3 4

5

6 7

Cumulative water amount (irigation+rainfall, ×1 000 m ha ) 3

–1

Fig. 3 S curve tendency of maize yield and water supply fitted with the present model. Table 4 Actual amount of water used to calculate the value of instant water use efficiency (IWUE) (e.g., maize B8920B6) Date May 11 May 18 May 25 Jun. 5 Jun. 15 Jun. 22 Jul. 4 Jul. 16 Jul. 28 Aug. 5 Aug. 9 Aug. 21 Sep. 5 Sep. 15 Sep. 20

Days after Sowing (d) 1 8 15 26 36 43 55 67 79 87 91 103 118 128 133

–, not irrigated.

Growing stage Germination Germination Seedling Bell-mouthing Jointing Tasselling Pollen shedding and silking Initial filling Middle filling Middle filling Late filling End filling Milky maturity Full maturity Harvest

Water supply for CK (m3 ha–1) Rainfall Irrigation Rainfall+Irrigation 103 750 853 102 – 102 103 – 103 56 – 56 56 750 806 56 750 806 84 750 834 85 750 835 84 750 834 45 – 45 45 750 795 48 750 798 111 – 111 112 – 112 111 – 111

Cumulative amount of water (m3 ha–1) 853 955 1 058 1 114 1 920 2 726 3 560 4 394 5 228 5 273 6 068 6 863 6 974 7 086 7 197

IWUE (kg m–3) 0.32 0.43 0.55 0.62 1.80 2.00 1.29 0.69 0.36 0.35 0.20 0.12 0.11 0.10 0.09

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1

1

0

0

0

4 3

B20 b21 Luyuan 92

5 4 3

G

er m

in

at Jo ion in Po Ta tin lle s s g n e di lin sp g e In rsa it f l M illin id g La fillin te g En fillin d g fil lin g

2

1

er m

5

U9-2 U9-19 Qi319

3

1

1

0

0

0

er m

in

at

io

n

2

1

in at Jo ion in Po Ta tin lle ss g e n di ling sp e In rsa it l fil M lin id g La fillin te g f En illin d g fil lin g G er m in at i Jo on in Po Ta tin lle ss g e n di ling sp e In rsa l it f M illin id g La fillin te g f En illin d g fil lin g

2

2

er m

b26 b28 B31 B33 7922

4

G

G

IWUE (kg m–3)

2

2

5

G

3

T tin lle ass g e n di ling sp er In sa it l f M illin id g La fillin te g f En illin d g fil lin g

3

b7 b10 B12 478

4

Po

4

5

b1 B2 Chang 7-2

in

3

5

Jo

4

T8920B6 T8920B8 Zheng 58

in at i Jo on in t Po Ta in lle s s g e n di ling sp e In rsa it l f M illin id g La fillin te g f En illin d g fil lin g G er m in at i Jo on in Po Ta tin lle s s g e n di ling sp e In rsa l it f M illin id g La fillin te g f En illin d g fil lin g

IWUE (kg m–3)

5

Fig. 4 Change of the instant water use efficiencies (IWUEs, kg m–3) of the 20 maize lines based on the data obtained in 2012.

pollen-shedding, but were lower at other stages.

4. Discussion Generally, the water sensitivity of a crop is higher at middle stages and lower at earlier or later stages. The largest sensitivity usually appears during the reproductive stage (Wang 1988). A single peak curve of water sensitivity is more reasonable for maize grain production (Zhang 2009). For maize, the yield can be greatly reduced if drought occurs during the reproductive stages of tasselling, silking and pollination (Claassen and Shaw 1970). Robins and Domingo (1953) found that soil water depletion to the wilting point for 2 days during tasselling or pollination could lead to yield decreases of 22%. Drought stress at the tasselling and silking stages reduces the activity of maize pollen, delays silk emergence, and thus induces embryo abortion or a reduction of potential kernel size (Westgate 1994). Additive and multiplicative models are polynomial equations with many parameters. When a large shift of one parameter occurs in the model, the influence of these changes may be collectively obscured by many small changes of other parameters, which likely explains why these models can provide an excellent fit to data while including unreasonable parameters based on actual field practice. Compared with polynomial models, a nonlinear model has fewer parameters that do not occupy equivalent positions in the model structure, and they show less interaction.

Additionally, the number of parameters of a nonlinear model does not increase if the growing stages are refined. All of these ensure parameter stability. Therefore, an exponential saturated model was derived in the present paper based on certain general assumptions of the quantitative relationship between crop yield and water supply. The model only has three parameters and could fit an S curve or a saturated curve for the relationship between the yield and cumulative water amount. Moreover, evapotranspiration quantities are not required; thus, the model may be easily applied for the analysis of the water use efficiency and drought tolerance of maize. The present model was not set up based on complex biological theories. For simplicity and usability, the model was only constructed using basic assumptions of the quantitative relationship between yield and water supply, which is similar to the “black box” model of control theory. However, these three parameters still have certain biological meaning. Our results showed that the model was well fit to the data, and validation of the data from the 24 maize lines obtained in the study and the 17 maize varieties published by Fan (2011) showed that the parameters were stable. According to this study, the minimum half-yield water requirement (Wh) was 2 598.9 m3 ha–1 (or 260 mm). Therefore, to obtain reasonable parameters in the model simulation and fully understand the relationships between yield and water supply, experiments should be conducted in locations that experience less than 130 mm of rainfall throughout the

LIU Cheng et al. Journal of Integrative Agriculture 2017, 16(4): 858–866

maize growing period. Additionally, the cumulative amount of water in the model may be substituted by the cumulative soil moisture. Modelling the relationship between water supply and production has broad applications. Although computer simulations are developing steadily (Jamieson et al. 1991; Li 1997; Jiao et al. 2002; Shang et al. 2009; Liu and Xie 2010), a stable and simplified model is still required for researchers and crop producers. The results obtained in this study suggest that the nonlinear model may overcome the disadvantage of the parameters generated by the additive and multiplicative models. However, because of the complex nature and diversity of the maize germplasm, the present model must be validated and modified further. Although the coupling of fertilizer and water (amount and irrigating time) would affect maize yield, the model was still simplified by excluding these effects to keep the simplicity of model structure. But, studies on effects of water and fertilizer coupling to parameters are still expected.

5. Conclusion The present model was able to accurately describe the relationship between the yield and the cumulative water amount for 24 maize lines and 17 maize hybrids. Thus, this model has the potential for use in evaluations of maize drought tolerance after further validation.

Acknowledgements This work was supported by grants provided by the National Sci-Tech Key Program of Development of Transgenic Animals and Plants, Ministry of Science and Technology, China (2014ZX08003-004).

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