The crop water production function,namely the model of crop response to water,can predict the influence of water deficit of different degrees in different stages on the yield in the crop growth process. This function analyzes the relationship between water and yield on assumption that other agricultural technical and agricultural factors are consistent,reflecting the response of crop yield to water change^{[1, 2]}. It is the most basic function to regulate deficit irrigation,and also an important basis for reasonable allocation of the water resources,to achieve the maximum crop yield by optimizing the irrigation system^{[1, 2]}. Water sensitive index (λ or K) of crop water function is the key for guidance and implement of water-saving irrigation and water management,making different crops adapt to water deficit in different periods.

Up to now,crop water production function has been intensively studied in the world. SINGH et al.^[3],BLANK^[4] and STEWART et al. ^[5] established the additive model to describe the relationship between water and yield. JENSEN^[6] put forward the multiplicative model in 1986. In China,KANG et al.^[7]carried out the relevant researches on distinguishing methods for crop water deficit status and irrigation index. TANG^[8] discussed the water requirement law and characteristic of rice cultivated in aerobic soil with different water deficit levels in different growth periods,analyzed the change law of different water deficit periods and water deficit degrees with the yield,and calculated the water sensitive coefficient and water production function of the rice. The research of SUN et al.^[9] showed that Jensen model could better reflect the relationship between water and yield of winter wheat. SUN^[10] analyzed the influence of drought on the yield in different periods,and calculated the water sensitive coefficients of wheat,corn and cotton in different growth periods by regression analysis method. ZHANG^[11] analyzed the water sensitive coefficients of corn cultivated under different soil conditions in different growth periods,fit the growth curve through distribution fitting method,and analyzed the accumulation function of sensitive indexes in Jensen model. WANG^[12] analyzed and obtained the water requirement law of super rice through the insufficient irrigation test data,and made calculations with several crop water production functions,thinking that Jensen model was the best crop water production function for Yongji,Jilin Province. However,CHENG et al.^[13]thought that Jensen model could better reflect the relationship between rice water and yield in East Jilin Province,and the relationship model between number of days after rice transplanting and sensitive index was established through the insufficient irrigation test. CHEN^[14] analyzed the influence of different water stresses on the final yield and the applicability of frequently-used water production functions in central Liaoning,and the crop water sensitive index in each period was calculated. SONG^[15] solved the water production function of irrigation quota and evapotranspiration with crop yield in Shihezi,and preliminarily obtained the suitable drip irrigation system for silage corn and oil sunflower.

The crop water production functions have been intensively studied all over the world,but not in Northern Xinjiang. At present,crop water production function was unavailable for this region,especially the water production function for alfalfa and Sudan grass. In this paper,the relationships were investigated among forage crop yield treated with different irrigations,the total evapotranspiration in the whole growth period and the water consumption in each stage,to develop the sensitive index of artificial grassland in different growth periods,to calculate the crop-water model of artificial grassland in arid desert area of Northern Xinjiang,aimed at providing technical support and theoretical basis for efficient and reasonable allocation and utilization of the water resources,in the process of developing,managing and utilizing irrigated forage grass fields in the area.

The experimental area located inside the Altay Experiment Station,Institute of Water Resources for Pastoral Area of Ministry of Water Resources in Fuhai County,Altay Prefecture,Xinjiang (87°40′22″ E,46°10′45″ N). The field surface elevation is 558 m above sea level. In this area,the terrain is flat with barren soil. The soil layer is 40 cm thick,and is mainly composed of sandy-loam soil. The annual average sunshine duration is 2 881 hours,and the total annual average solar radiation amount is 546.7 kJ/cm². The annual average temperature is 3.4 ℃,and the annual accumulated average temperature above 10 ℃ is 2 904.9 ℃. The annual average frost-free period is about 150 days. The experimental area is a Gobi desert area,with arid climate insufficient in rainfall. The annual average evaporation is 1 830 mm,and the annual average precipitation is 112.7 mm.

The low pressure pipe irrigation is adopted in this study. Seventeen irrigation treatments were set up as shown in Table 1. Every treatment is repeated three times,and the plot layout is conducted according to irrigation experiment standard^[16]. The area of the experimental plot is 10 m×6 m=60 m²; between two experimental plots,a 1 m-wide protection zone was set up to avoid interaction. In the middle of the field,there is an underground water observation well. The experimental grasses are alfalfa (Algonquin),Sudan grass (Qitai Sudan grass) and silage corn (Xinyu 10). The silage corn should be reaped and stored in good time during the milk stage,and the alfalfa should be reaped and stored in good time at the end of the flowering period; after harvest of the alfalfa,water and fertilizer management should be strengthened to resume the growth as soon as possible.

Table 1

Table 1 Design of experimental treatments of alfalfa, Sudan grass and silage corn

Table 1 Design of experimental treatments of alfalfa, Sudan grass and silage corn Treatments No. Field moisture capacity in different periods/% Returning green stage Branching stage Squaring-flowering stage Alfalfa     Medium drought in returning green stage MX1 50 70 70     Medium drought in branching-bud stage MX2 70 50 70     Medium drought in bud-flowering stage MX3 70 70 50     Medium drought in the whole growth period MX4 50 50 50     Sufficient irrigation (CK) MX5 70 70 75 Treatments No. Field moisture capacity in different periods/% Seedling stage Tillering-jointing stage Booting-flowering stage Filling-first mature stage Sudan grass     Heavy drought SD1 45 45 45 45     Medium drought SD2 50 50 50 50     Light drought SD3 60 60 60 60     Sufficient irrigation (CK) SD4 70 75 75 75 Silage corn     Light drought in seedling stage YM1 50-60 70 75 75     Light drought in jointing stage YM2 70 55 75 70     Light drought in booting-flowering stage YM3 70 70 55-65 70     Heavy drought in booting-flowering stage YM4 70 70 No irrigation 70     Light drought in filling-milk stage YM5 70 70 75 60     Light drought in seedling and jointing stages YM6 50-60 55 75 75      Light drought in jointing and booting-flowering stages YM7 70 55 55 75     Sufficient irrigation (CK) YM8 70 75 75 75 The water contents listed in the table are all the lower limit values and the upper limit ones of the field moisture capacity.

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The observation contents of the experiment include meteorological data,increment of underground water,soil physical properties,soil water content and crop yield.

The meteorological data are from the field meteorological station inside the HOBO U30 Station,including daily maximum,minimum and average temperatures,evaporation,sunshine duration,wind speed at 2 m height,maximum,minimum and average relative humidities,precipitation,radiation and so on. The increment of underground water was measured through the observation well,once every three days. The soil water diffusivity (D) was measured through taking back undisturbed soil from the project area to make horizontal soil column; the soil water characteristic curve was obtained by adopting 1600-type 5 Bar pressure film instrument and then the soil hydraulic conductivity (K) was deduced. A positioning flux method was adopted to calculate the increment of underground water to the planned moisture layer. Soil physical properties included the water content,field moisture capacity,soil bulk density,and porosity before the experiment start. Soil water contents at the depths of 0,10,30 and 50 cm were measured in every five days,using TRIME-PICO TDR portable soil water measurement instrument. Crop yield was achieved by measuring the yield at the final stage of every growth period using quadrat method.

The water balance equation was adopted to calculate actual evapotranspiration (ET_a) of the artificial grassland (the computing process was not presented). According to the experimental data,we could obtain maximum evapotranspiration (ET_m),the actual yield (Y_a) and the maximum yield (Y_m) of different forage grasses in the whole growth period,as shown in Table 2.

Table 2

Table 2 ETa, ETm, Ya and Ym of alfalfa, Sudan grass and silage corn under different treatment conditions

Table 2 ETa, ETm, Ya and Ym of alfalfa, Sudan grass and silage corn under different treatment conditions Artificial grasslands Treatment No. ETa/mm ETm/mm Ya/(kg/hm2) Ym/(kg/hm2) Alfalfa MX1 422 690 13 770.0 14 235.0 MX2 422 690 11 895.0 14 235.0 MX3 440 690 12 705.0 14 235.0 MX4 405 690 10 245.0 14 235.0 MX5 460 690 14 235.0 14 235.0 Sudan grass SD1 138 302 19 176.3 24 330.3 SD2 184 302 20 449.6 24 330.3 SD3 245 302 21 116.6 24 330.3 SD4 277 302 24 330.3 24 330.3 Silage corn YM1 302 379 34 484.0 54 527.2 YM2 267 379 43 688.6 54 527.2 YM3 281 379 52 292.8 54 527.2 YM4 194 379 24 012.0 54 527.2 YM5 298 379 45 822.9 54 527.2 YM6 238 379 22 878.2 54 527.2 YM7 262 379 19 109.6 54 527.2 YM8 379 379 54 527.2 54 527.2 ETa is the actual evapotranspiration under different water deficit treatments; ETm is the maximum evapotranspiration when the irrigation is sufficient; Ya stands for the actual yield of grass under different water deficit treatments; Ym stands for the maximum yield when the irrigation is sufficient.

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In the growth and developmental process of alfalfa,water stress could lead to a series of bad consequences. The most obvious manifestation was that the plant became lower and that the crop yield decreased. The results of different treatments showed that the crop yield decreased obviously with the decrease of soil water content. According to the Table 2,under the sufficient irrigation condition,the yield of alfalfa did not decrease. When the soil water content was controlled at 50% of the field moisture capacity,the reduction rate of yield reached 29.77%.

The yields of Sudan grass varied with different water content conditions (Table 2). The highest yield was observed under the sufficient irrigation condition,and the lowest one was under the heavy drought,with the yield reduction rate of 21.18%. Moreover,the yield reduction and its rate varied along with different drought degrees,which were linearly dependent on the soil water content.

According to the Table 2,water stress could lead to a series of bad consequences for silage corn at different stages of the growth and developmental process,and the most obvious influence was reduction of the crop yield. In case of drought in a single stage,the heavy drought in the booting stage was the most serious (the yield reduced by 55.96%). That is because this stage is the reproductive growth stage of the corn; the heavy drought in this stage will seriously affect the yield at later stage. Among the seedling stage,jointing stage and filling stage,the seedling stage was the most sensitive one (the yield reduced by 36.76%),because silage corn in the project area was sowed in early summer,and dry,hot and rainless days with high-temperature came after sowing. Therefore,in the seedling stage,irrigation should be conducted to supplement sufficient water. Besides,the water physiological activity of silage corn was different from other crops,the key cultivation management of which was to promote tillering and jointing rather than “hardening of seedling” for other crops. If suffering from drought in this stage,the yield may reduce by 19.88%. The project area was mainly composed of Gobi desert and soil with fast evaporation and poor water retention capacity,so the water on the surface would infiltrate or evaporate quickly after irrigation. Therefore,“more times with less amount” irrigation system should be adopted to avoid the state of being dry surface and wet inner layer so as to promote tillering.

At present,the crop water production function models mainly reflected the whole growth period and various growth stages. Based on the results achieved by previous research,combining the objective conditions of the experimental station,Jensen model,Minhas model,Blank model,Singh model and Stewart model were selected to analyze and study the response of crop yield of artificial grassland to water deficit levels.

Jensen model: Water deficit in a certain growth stage does not only affect this growth stage,but also will exert a hysteresis effect on the next stage; it is a multiplicative model established with a staged relative evapotranspiration as an independent variable.

where Y_a stands for the actual yield of grass under different water deficit treatments,kg/hm² ; Y_m stands for the maximum yield when the irrigation is sufficient,kg/hm²; ET_a is the actual evapotranspiration under different water deficit treatments; ET_m is the maximum evapotranspiration when the irrigation is sufficient; i is the number of the stages; n is the total number of stages of the model established; λ is the water sensitive index,reflecting the sensitivity of grass to the influence of water deficit in different stages on the crop yield. ET_m/ET_a≥1.0,λ_i≥0. The larger λ_i is,the smaller Y_a/Y_m is after the multiple multiplication,showing that the influence of stage i on the yield is larger. On the contrary,the smaller λ_i is,the smaller the influence on the yield is. λ_i is the key parameter of the multiplicative model. Jensen model was originally derived from the yield of maize seeds.

Minhas model: It is a crop water production model with the relative water deficit in different growth stages as an independent variable and is also a multiplicative model.

where the meanings of Y_a,Y_m,ET_a,ET_m and λ_i are the same with the equation 1; α₀ stands for correction factor of other factors rather than water deficit to Y_a/Y_m; in the single-factor water production function,α₀=1^[17]; b₀ stands for power exponent of the independent variable; usually,b₀=2.0.

Blank model: It is an additive model with a staged relative evapotranspiration as an independent variable,and it is represented by its product and sensitive coefficient of corresponding stage.

where the meanings of Y_a,Y_m,ET_a and ET_m are the same with the equation 1; K_i stands for sensitive coefficient of water deficit of artificial grass in different growth stages to the grass yield. Because ET_a/ET_m≤1.0,K_i>0,the larger K_i is,the larger Y_a/Y_m is after multiple addition of various stages,showing that the influence on the yield is smaller.

Singh model: It is a crop water production model with relevant water deficit of various growth stages as a variable,and it is represented by its product and sensitive coefficient K_i of corresponding stage. It is an additive model.

where the meanings of Y_a,Y_m,ET_a and ET_m are the same with the equation 1; b₀ stands for empirical coefficient (b₀=2).

Stewart model: It is an additive model with relevant water deficit of various growth stages as a variable,and it is represented by its product and sensitive coefficient K_i of corresponding stage.

where the meanings of Y_a,Y_m,ET_a and ET_m are the same with the equation 1.

The crop sensitive indexes represented the influence of water deficit on the crop yield in different stages,and they changed along with the environment. Besides,the sensitivity to water deficit changed for different crops,as well as different growth stages. Larger sensitivity in a stage would bring higher yield reduction by unit water deficit. The crop sensitive indexes of alfalfa,Sudan grass and silage corn were deduced according to the selected five models. The models were all transformed to be unified linear equations by formal transformation,taking logarithm for both sides of the equations; the least square method was adopted,to obtain a normal equation system about water deficit sensitive indexes; the matrix method was used to solve the above normal equation system,and then get the crop stage sensitive indexes or coefficients. The calculation results were shown in Tables 3-5.

Table 3

Table 3 Crop water deficit sensitive indexes or coefficients of alfalfa in different water production function models

Table 3 Crop water deficit sensitive indexes or coefficients of alfalfa in different water production function models Models 1st harvest 2nd harvest Returning green-branching stage Branching-bud stage Bud-flowering stage R2 Returning green-branching stage Branching-bud stage Bud-flowering stage R2 Jensen 0.037 1 0.563 0 0.404 4 1.000 0 0.160 5 0.512 7 0.363 9 1.000 0 Minhas -0.212 3 1.474 3 5.808 5 0.993 4 55.970 8 12.997 6 -52.524 9 0.992 8 Blank 0.060 3 0.546 7 0.393 0 1.000 0 0.003 9 0.482 1 0.528 4 1.000 0 Singh -0.211 2 1.095 5 5.699 8 0.990 8 51.633 0 12.700 0 -49.809 2 0.990 8 Stewart 0.060 3 0.546 7 0.393 0 1.000 0 0.003 9 0.482 1 0.528 4 1.000 0

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Table 4

Table 4 Crop water deficit sensitive indexes or coefficients of Sudan grass in different water production function models

Table 4 Crop water deficit sensitive indexes or coefficients of Sudan grass in different water production function models Models Seedling stage Tillering-jointing stage Booting-flowering stage Filling-milk stage R2 F Jensen 1.556 2 2.642 8 1.625 7 2.521 4 0.838 2 < F0.05 Minhas 0.539 1 0.955 8 0.882 1 1.462 6 0.646 0 < F0.05 Blank 0.852 7 0.767 2 0.746 7 0.736 5 0.872 4 >F0.05 Singh 1.133 0 1.034 5 1.045 4 0.921 8 0.669 6 < F0.05 Stewart 1.252 7 1.510 2 1.746 7 1.828 2 0.882 9 >F0.05

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Table 5

Table 5 Crop water deficit sensitive indexes or coefficients of silage corn in different water production function models

Table 5 Crop water deficit sensitive indexes or coefficients of silage corn in different water production function models Models Sowing-seedling stage Seedling stage Tillering-jointing stage Booting-flowering stage Filling-milk stage R2 F Jensen 0.035 2 0.633 2 0.081 2 0.904 4 0.512 7 0.893 3 >F0.05 Minhas 0.512 1 1.589 6 0.551 7 2.235 8 1.448 2 0.830 3 >F0.05 Blank 0.001 2 0.025 5 0.381 0 0.332 5 0.311 5 0.823 5 >F0.05 Singh 0.042 4 0.680 8 0.344 9 0.005 6 -0.412 2 0.495 8 < F0.05 Stewart 0.252 4 0.715 7 0.367 9 0.382 2 0.414 7 0.904 5 >F0.05

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Next,the crop sensitive indexes according to the calculation results in the Tables 3-5 were analyzed as follows:

For alfalfa,in Singh model a negative value appeared in the returning green stage. According to its physical significance,the water deficit in this growth stage posed an certain promoting rather than inhibiting effect on the yield,which was contradictory to the actual situation. However,this may have something to do with the crop physiology,or the constraint that should be kept between positive and negative terms,when searching optimization in the least square method was adopted; in such a case,the existence of negative value was reasonable. However,the correlation coefficient of this model R²=0.495 8,which was relatively low; and F < F_0.05,showing that the regression effect was not obvious. Therefore,this model should not be adopted.

Generally,the appearance of negative values in the model can be explained from the following two aspects. First,in the early growth period,negative values may appear easily. Studies showed that the water deficit in early stage could make the crops adapt to water deficit and gain higher yield. Therefore,it is reasonable for the water sensitivity index to be negative in the early growth stage. The super compensation effect,that the yield of alfalfa can increase by water deficit in early growth stage,needs further study in the future. Second,CHEN et al.^[17] pointed out that multiple regression analysis was a statistical method which could be adopt to solve the water sensitive index. In the model deduction process,according to the requirement of processing data by least square method,when residual sum of squares is minimum because of positive and negative coordination,negative values may appear.

For Sudan grass,in Jensen model,Minhas model and Singh model,F < F_0.05,showing not obvious regression effect; in Blank model and Stewart model,F>F_0.05,showing obvious regression effect. The sensitive index in the seedling stage was the largest in Blank model. According to its definition,the sensitivity is the smallest for the stage where the sensitive coefficient K_i is the largest. The model reflected that the influence of water deficit on the yield was the smallest in the seedling stage,which was inconsistent with the experience in harvesting of Sudan grass. In Stewart model,the order of sensitive coefficients was filling-milk stage > booting-flowering stage > tillering-jointing stage > seedling stage. Therefore,the Sudan grass in filling-milk stage was most sensitive to water.

For silage corn,in Jensen model,the water sensitive index λ in booting-flowering stage was the highest; in the growth period,the order for λ was booting-flowering stage > seedling stage > filling-milk stage > tillering-jointing stage > sowing-seedling stage. According to the definition of Jensen model,higher yield reduction rate (lower Y_a/Y_m) came along with larger λ after water deficit. Therefore,the stage order about sensitive degree of water deficit to the yield,which was reflected in Jensen model in the Table 5,should be consistent with the water physiological theory and local actual irrigation experience for silage corn. The correlation coefficient of this model R²=0.893 3,and F>F_0.05,so the regression effect was obvious. Hence,Jensen model was more suitable for predicting the yield of silage corn in arid desert area in Northern Xinjiang. In Stewart model and Blank model,F>F_0.05,so the regression effect was obvious. Meanwhile,the sensitive coefficients in Stewart model and Blank model were largest in the seedling stage and tillering-jointing stage,respectively. According to their definitions,the sensitivity is the smallest for the stage where the sensitive coefficient K_i is the largest. Namely,the two models reflected that the influence of water deficit on the yield was the smallest in the seedling stage and tillering-jointing stage,which was inconsistent with the experience in harvest of silage corn. The sensitive indexes in three stages of Minhas model were larger than 1,which was unreasonable.

According to the above discussion and analysis,the water production function model built for alfalfa and silage corn in arid desert area in Northern Xinjiang is Jensen model,and for Sudan grass is Stewart model.

Water production function of alfalfa was expressed by

Water production function of Sudan grass was expressed by

Water production function of silage corn was expressed by

In order to test the reliability of the models,we substituted the group data of evapotranspiration and yield of alfalfa,Sudan grass and silage corn in Xinjiang experimental area under different treatments into the above models,and then the predicted yield and relative errors were calculated and presented in Table 6. The test result showed that the maximum relative errors of the yield for alfalfa,Sudan grass and silage corn simulated by the models were 11.71%,15.47% and 22.27%,respectively; and the average relative errors were 6.51%,9.24% and 9.25%,respectively. Therefore,the models have a relatively high accuracy.

Table 6

Table 6 Results of crop water production model simulating the yield of artificial grassland

Table 6 Results of crop water production model simulating the yield of artificial grassland Artificial grasslands Treatment No. Actual yield/(kg/hm2) Predicted yield/(kg/hm2) Relative error/% Models Alfalfa MX1 45.53 50.87 11.71 Jensen MX2 63.27 58.13 -8.11 MX3 61.20 55.40 -9.48 MX4 52.87 52.47 -0.76 MX5 56.47 55.07 -2.48 Sudan grass SD1 85.23 93.17 9.32 Stewart SD2 90.89 81.04 -10.83 SD3 93.85 108.37 15.47 SD4 108.13 106.68 -1.34 Silage corn YM1 153.26 161.45 5.34 Jensen YM2 194.17 181.07 -6.74 YM3 232.41 214.19 -7.84 YM4 106.72 130.49 22.27 YM5 203.66 215.75 5.94 YM6 101.68 116.75 14.82 YM7 84.93 92.26 8.63 YM8 242.34 248.28 2.45

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For alfalfa,the drought in the returning green-branching stage led to the minimum yield reduction rate and the minimum water production efficiency; the drought in the whole growth period led to the minimum yield reduction rate and the maximum water production efficiency. The rule of Sudan grass was similar with alfalfa; a linear relation was found between the crop yield reduction rate and drought situations. For silage corn,the yield under the light drought in the booting-flowering stage was lower than sufficient irrigation; its water production efficiency was maximum among all the insufficient irrigation treatments,and the water consumption was only 76% of the sufficient irrigation. And the drought in jointing-heading stage led to the greatest influence on the yield and the maximum yield reduction rate. Whereas the drought in the seedling-jointing stage led to the minimum water production efficiency.

Jensen model,Stewart model and Jensen model were respectively adopted for alfalfa,Sudan grass and silage corn in arid desert area of Northern Xinjiang,and the relationship between the water consumption and the yield could be reflected relatively correctly; the average relative errors of the models for alfalfa,Sudan grass and silage corn were 6.51%,9.24% and 9.25%,respectively. The sensitive indexes of alfalfa in different growth stages under Jensen model were 0.037 1,0.563 0,0.404 4,0.160 5,0.512 7 and 0.363 9,respectively; the sensitive coefficients of Sudan grass in different growth stages under Stewart model were 1.252 7,1.510 2,1.746 7 and 1.828 2,respectively; the sensitive indexes of silage corn in different growth stages under Jensen model were 0.035 2,0.633 2,0.081 2,0.904 4 and 0.512 7,respectively. The most sensitive growth stages of the three crops were branching-bud stage (1st harvest),filling-milk stage and booting-flowering stage,respectively.

The crop sensitive indexes change not only along with crop types,growth period,agriculture and its technical measures,but also related to climate zones and different hydrological years. The values of λ and K in this study were based on the experimental results of Altay Prefecture in Xinjiang from 2012 to 2014,showing the general law of water deficit in different growth stages on the yield. For application in the production as an annual average value,further studies should be conducted to find out the change law of λ (or K) along with hydrological years. Besides,λ (or K) is also related to soil water potential,potential evapotranspiration,rainfall and temperature. Therefore,we can further analyze the interrelation of λ or K with other factors,discuss the internal mechanism of λ or K change and then better apply it in the actual production.