Modeling Tomato Yield and Quality Responses to Water and Nitrogen Deficits with a Modified Crop Water Production Function

Abstract

Increasingly severe crises, such as climate change, water scarcity and environmental pollution, pose significant challenges to global food security and sustainable agricultural development. For efficient and sustainable tomato cultivation management under resource constraints, quantitatively describing the relationship between yield-quality harvest and water-nitrogen application is practically beneficial. Two successive greenhouse experiments with three irrigation levels (1/3 FI, 2/3 FI, and full irrigation (FI)) and four nitrogen fertilizer treatments (0 FN, 1/3 FN, 2/3 FN, and full nitrogen (FN)) were conducted on tomatoes during the whole phenological stage. The tomato evapotranspiration and nitrogen application amount, yield, comprehensive quality, solid–acid ratio, and lycopene content were measured. Based on crop water production functions, three equation forms of water-nitrogen production functions containing 20 models were established and evaluated to predict tomato harvest parameters. The results show that water increased tomato yield while decreasing fruit quality, and the effect of nitrogen was primarily contrary. Water most significantly impacted tomato formation, and the interaction of water and nitrogen changed among different harvest parameters. Tomato yield and quality formation was more sensitive to water and nitrogen at the flowering and fruit maturation stages. Model Singh-2 outweighed other models for yield estimates, with an R² of 0.71 and an RMSE of 0.11. Singh-Log, Singh-sigmoid and Rao-Root models were effective models for comprehensive quality, solid–acid ratio, and lycopene content prediction, with an R² of 0.41, 0.62, and 0.42, and an RMSE of 0.33, 0.50, and 0.16, respectively. Finally, models in the form of f(ET_i)·f(N) were ideal for tomato harvest prevision and are recommended for water and nitrogen management in tomato cultivation.

1. Introduction

Tomato (Solanum lycopersicum L.) is a popular vegetable worldwide due to its unique flavor, rich nutritional content, and health-promoting benefits [1,2]. As of 2022, the global cultivated area for tomatoes has reached 4.91 × 10⁶ ha, yielding a production of over 1.86 × 10⁸ kg [3]. Meanwhile, the quality of tomatoes has been increasingly valued by the market, as the growing yield has met consumers’ demands [4,5]. Moreover, the emerged water and soil resource scarcity [6], along with severe agricultural fertilizer waste and pollution [7], present significant challenges to the sustainable and efficient development of the tomato cultivation industry. Therefore, balancing tomato yield-quality harvests and irrigation-fertilizer regulation plays a pivotal role in modern sustainable agricultural research [8,9].

A moderate deficit could improve water productivity and fruit quality with an acceptable degree of yield decrease [10,11]. Lu et al. [10] reported that regulated deficit irrigation decreased tomato yield with a mean difference of 18.61 t·ha⁻¹ based on a meta-analysis. Zhang et al. [12] found that the irrigation treatment with 80% crop evapotranspiration (ET_c) obtained the highest tomato yield of 70–80 t·ha⁻¹ and led to an increase in fruit soluble solids and ascorbic acid contents. Nitrogen (N) is essential for the synthesis of proteins and chlorophyll, and it boosts plant maturation and photosynthesis capacity [13,14]. An optimal nitrogen fertilizing strategy can improve tomato yield and quality such as ascorbic acid content, sugar–acid ratio, soluble solids and soluble sugar content [15]. Hernández et al. [16] reported that lowering nitrogen supply from tomato flowering stage increased fruit lycopene and phytoene content without compromising yield, ascorbic acid and the main phenolic content. Li et al. [17] found that carbon-based urea application improved tomato yield, water-fertilizer productivity as well as fruit volume, soluble sugar and lycopene content. The coupling effect of water and nitrogen is considered to play a vital role in tomato photosynthesis capacity, yield accumulation, organics conversion, quality development and many other development processes [18,19]. Appropriate nitrogen application can decrease the biological stress caused by deficit irrigation and improve water use efficiency, while judicious irrigation can offset the negative effects of nitrogen stress on crop yield [20]. Different harvest parameters have been shown to exhibit varied reactivity and sensitivity to water and nitrogen application [21]. Li et al. [22] showed that water and nitrogen significantly affected tomato yield and quality including fruit soluble sugar, organic acid and ascorbic acid content, while the sugar–acid ratio remained unchanged. Additionally, Zhang et al. [23] reported a significant difference in ascorbic acid, total sugar and titratable acid content under different nitrogen stress treatments, while no significant difference was observed under different water stress treatments.

In terms of the complex effect of water and nitrogen on crop development and resource utilization [24], precise regulation in fertigation strategies is key in modern cultivation. Nevertheless, the fertigation strategies previously determined may not be appropriately applied due to varied plant species, soil texture, climatic condition, management regime and other factors [25,26], warranting the need for a quantitative simulation. The correlation between water consumption and yield production was initially described using regression functions of plant evapotranspiration (ET) or water deficit amount (1-ET) and crop yield [27,28], but the sensitivity variance of crop yield to water stress under different growth stages was not considered [29]. In the 1950s, separated crop water production functions (CWPFs) such as the Jensen [30], Minhas [31], Rao [32], Blank [33], Stewart [34] and Singh [35] models were proposed and extensively used in many species [36,37,38]. Mathematically, separated CWPFs can be divided into either cumulative or multiplicative functions. The former hypothesized that a plant’s sensitivity index in different growth stages does not have a mutual influence on each other, and thus the final effect on fruit harvest parameters is the sum of all growth stages. In contrast, the latter concerns the impact of each growth stage on subsequent growth, factoring in the comprehensive interaction among the growth stages [39]. Nowadays the developments of CWPFs have become relatively mature and are widely used in crop yield prediction, agriculture water allocation and water price policy formulation [40,41,42]. Furthermore, several previous studies attempted to modify CWPFs and expanded its application to fruit yield and quality simulations [39,43]. Saeidi et al. [44] used CWPFs to simulate maize yield under different levels of salinity and nitrogen stress, and reported a series of coefficients matched to four different conditions; Wu et al. [45] incorporated root soil water availability into CWPFs, and this improved model achieved a prediction R² of 0.84 on winter wheat (Triticum aestivum L.) and spring maize (Zea mays L.) yield in the North China Plain; Yuan et al. [46] built a crop-water-salt production function based on the Jensen model, producing a high-performing simulation with an RMSE lower than 800 kg·hm⁻¹ on maize yield.

Mainly previous studies about the utilization of CWPFs focus on crops, and more experimental data and model promotion for fruit and vegetation remain on the rise. For the utilization of CWPFs for tomato plants, Zhang et al. [47] changed evapotranspiration input in the Jensen model into the tomato soil water content and electrical conductivity; Chen et al. [39] modeled tomato yield and quality based on the Jensen, Stewart, and Minhas model. However, the prediction accuracy is unsatisfied, especially when referring to fertilizer input and quality simulation. Identifying more appropriate quality parameters and developing new formulations of CWPFs merit further investigations. Therefore, this study conducted two field experiments and aimed at (1) clarifying the effect of water and nitrogen deficit on tomato yield and fruit quality; (2) finding out the most important quality parameters to represent tomato fruit characteristics; (3) determining suitable CWPFs models to qualitatively describe the relationship between tomato water-nitrogen consumption, yield, and fruit quality. The findings can provide a simple and scientific method for fertigation regulation and resource allocation in tomato cultivation.

2. Materials and Methods

2.1. Experiment Site and Design

Two successive tomato field experiments were conducted in 2019 at an agricultural company (36°39′ N, 112°56′ E) located in Gaomi City, Shandong province of Northeastern China. Details of the location was described in a previous study [48]. The greenhouse experiment was conducted with a vegetable tomato variety Baoli 3 (Solanum lycopersicum L.) in two seasons from 13 April to 26 July (the first season, with daily average radiation of 9.93 MJ·d⁻¹, temperature of 24.61 °C and relative humidity of 71.67%) and from 13 August to 28 November (the second season, with daily average radiation of 5.43 MJ·d⁻¹, temperature of 21.62 °C and relative humidity of 76.80%) in 2019. Twelve treatments included three irrigation levels (1/3 FI, 2/3 FI, and full irrigation (FI)) and four nitrogen levels (0 FN, 1/3 FN, 2/3 FN, and full nitrogen fertilization(FN)) in both seasons (

Table 1. Irrigation and fertilizer amounts of different treatments.

		Irrigation (mm)					Nitrogen (g/m2)			Evapotranspiration (mm)
		Transplant	Stage I	Stage II	Stage III	Total	Stage I	Stage II	Total	Stage I	Stage II	Stage III	Total
First season	T1(CK)	20.0 (1)	10.0 (1)	180.0 (6)	50.0 (2)	260.0 (10)	2.7 (1)	12.0 (2)	14.7 (3)	55.61 ± 3.07 a	164.09 ± 1.38 a	79.71 ± 1.58 a	299.41 ± 0.11 a
	T2	20.0 (1)	10.0 (1)	180.0 (6)	50.0 (2)	260.0 (10)	1.8 (1)	8.0 (2)	9.8 (3)	55.46 ± 2.54 a	157.51 ± 1.84 b	88.02 ± 1.66 b	300.99 ± 2.72 a
	T3	20.0 (1)	10.0 (1)	180.0 (6)	50.0 (2)	260.0 (10)	0.9 (1)	4.0 (2)	4.9 (3)	53.73 ± 3.44 a	153.47 ± 2.11 bc	81.24 ± 2.35 b	288.44 ± 3.68 ab
	T4	20.0 (1)	10.0 (1)	180.0 (6)	50.0 (2)	260.0 (10)	-	-	0.0 (0)	56.13 ± 0.46 a	150.11 ± 2.84 c	76.08 ± 2.56 b	282.33 ± 4.95 b
	T5	20.0 (1)	6.7 (1)	120.0 (6)	33.3 (2)	180.0 (10)	2.7 (1)	12.0 (2)	14.7 (3)	54.68 ± 4.41 a	125.40 ± 1.88 d	65.72 ± 3.02 c	245.79 ± 9.31 c
	T6	20.0 (1)	6.7 (1)	120.0 (6)	33.3 (2)	180.0 (10)	1.8 (1)	8.0 (2)	9.8 (3)	52.50 ± 1.01 a	123.82 ± 3.40 d	66.93 ± 1.54 c	243.26 ± 3.94 c
	T7	20.0 (1)	6.7 (1)	120.0 (6)	33.3 (2)	180.0 (10)	0.9 (1)	4.0 (2)	4.9 (3)	58.03 ± 0.24 a	119.54 ± 2.33 d	62.33 ± 0.16 c	239.89 ± 2.74 c
	T8	20.0 (1)	6.7 (1)	120.0 (6)	33.3 (2)	180.0 (10)	-	-	0.0 (0)	54.11 ± 1.43 a	109.15 ± 0.99 e	61.78 ± 0.93 c	225.05 ± 1.37 d
	T9	20.0 (1)	3.5 (1)	63.0 (6)	16.7 (2)	103.2 (10)	2.7 (1)	12.0 (2)	14.7 (3)	57.05 ± 1.07 a	94.50 ± 1.69 f	53.05 ± 1.50 d	204.59 ± 2.11 e
	T10	20.0 (1)	3.5 (1)	63.0 (6)	16.7 (2)	103.2 (10)	1.8 (1)	8.0 (2)	9.8 (3)	52.81 ± 1.47 a	93.04 ± 0.18 f	44.98 ± 2.12 e	190.83 ± 3.41 f
	T11	20.0 (1)	3.5 (1)	63.0 (6)	16.7 (2)	103.2 (10)	0.9 (1)	4.0 (2)	4.9 (3)	55.16 ± 0.71 a	84.96 ± 0.95 g	41.15 ± 0.17 e	181.28 ± 1.49 fg
	T12	20.0 (1)	3.5 (1)	63.0 (6)	16.7 (2)	103.2 (10)	-	-	0.0 (0)	51.99 ± 2.22 a	82.36 ± 0.30 g	39.42 ± 0.80 e	173.77 ± 2.72 g
Second season	T1(CK)	20.0 (1)	20.0 (2)	100.0 (5)	40.0 (2)	180.0 (10)	9.3 (1)	11.4 (3)	20.7 (4)	66.01 ± 0.36 a	116.42 ± 0.07 a	57.37 ± 3.05 a	239.80 ± 3.33 a
	T2	20.0 (1)	20.0 (2)	100.0 (5)	40.0 (2)	180.0 (10)	9.3 (1)	7.6 (3)	16.9 (4)	64.50 ± 1.98 a	110.89 ± 0.55 a	54.20 ± 0.92 ab	229.59 ± 0.51 ab
	T3	20.0 (1)	20.0 (2)	100.0 (5)	40.0 (2)	180.0 (10)	9.3 (1)	3.8 (3)	13.1 (4)	62.27 ± 4.62 a	109.59 ± 3.11 a	53.59 ± 0.24 b	225.45 ± 7.49 b
	T4	20.0 (1)	20.0 (2)	100.0 (5)	40.0 (2)	180.0 (10)	9.3 (1)	-	9.3 (1)	65.14 ± 0.14 a	99.11 ± 2.09 b	47.61 ± 0.34 c	212.01 ± 2.43 c
	T5	20.0 (1)	14.0 (2)	70.0 (5)	32.0 (2)	136.0 (10)	8.2 (1)	11.4 (3)	19.6 (4)	66.80 ± 0.89 a	90.10 ± 2.06 bc	49.55 ± 0.12 c	206.46 ± 1.06 c
	T6	20.0 (1)	14.0 (2)	70.0 (5)	32.0 (2)	136.0 (10)	8.2 (1)	7.6 (3)	15.8 (4)	61.26 ± 3.82 a	95.69 ± 3.58 bc	49.74 ± 0.04 c	206.69 ± 0.20 c
	T7	20.0 (1)	14.0 (2)	70.0 (5)	32.0 (2)	136.0 (10)	8.2 (1)	3.8 (3)	12.0 (4)	69.44 ± 2.02 a	93.30 ± 1.26 c	42.77 ± 1.20 d	205.51 ± 0.44 c
	T8	20.0 (1)	14.0 (2)	70.0 (5)	32.0 (2)	136.0 (10)	8.2 (1)	-	8.2 (1)	60.63 ± 3.37 a	81.54 ± 4.96 d	41.39 ± 0.93 d	183.56 ± 7.40 d
	T9	20.0 (1)	8.0 (2)	40.0 (5)	16.0 (2)	84.0 (10)	8.2 (1)	11.4 (3)	19.6 (4)	66.84 ± 6.06 a	76.67 ± 3.31 d	39.65 ± 1.14 de	183.16 ± 8.23 d
	T10	20.0 (1)	8.0 (2)	40.0 (5)	16.0 (2)	84.0 (10)	8.2 (1)	7.6 (3)	15.8 (4)	65.08 ± 1.35 a	76.04 ± 2.10 d	36.63 ± 0.65 e	177.75 ± 2.80 d
	T11	20.0 (1)	8.0 (2)	40.0 (5)	16.0 (2)	84.0 (10)	8.2 (1)	3.8 (3)	12.0 (4)	64.96 ± 1.56 a	75.06 ± 1.23 d	36.75 ± 0.38 e	176.77 ± 2.41 d
	T12	20.0 (1)	8.0 (2)	40.0 (5)	16.0 (2)	84.0 (10)	8.2 (1)	-	8.2 (1)	67.39 ± 0.62 a	62.40 ± 0.21 e	31.71 ± 0.85 f	161.49 ± 0.44 e

Notes: The number in brackets represents the times of irrigation or fertilization, and the fertigation mount of each time was equal; lowercase letters following the data indicate significant differences by Duncan’s test at p ≤ 0.05 level.

). The irrigation source is groundwater, and the nitrogen source is urea (CO(NH₂)₂ with 46% N content). W1 treatments were irrigated to 90 ± 3% of θ_FC (field soil water capacity, measured as 0.33 cm³·cm⁻³) while soil moisture decreased down to 75 ± 3% of θ_FC. Additionally, the water deficit treatments were simultaneously irrigated by the rate of the full amount. The experimental substrate was local soil, a type of clay loam with dry bulk density valued 1.38 g·cm⁻³. Each treatment was replicated thrice, and all treatments were arranged in a randomized complete block design. Plants with 3–4 true leaves in similar heights were transplanted into the plots (9.6 m long and 6.0 m wide). Row spacing between plants was 0.4 m and planting spacing was 0.25 m. Each plot was separated by a 1 m depth acrylic flap to prevent the leakage of irrigation and fertilizer between the plots. The growth stage of tomatoes was divided into the seedling stage (stage I, from transplant to first fruit set), the flowering and fruit development stage (stage II, from first fruit set to first fruit maturity), and the fruit maturation and harvest stage (stage III, from first fruit maturity to uprooting plant after all fruits were harvested). All the other agronomic managements such as fertigation time, potassium (K₂SO₄ with 52% K content), phosphorus (Ca(H₂PO₄)₂ with 12% P content), insecticide application and pruning were consistent across all the treatments. Primarily for the control of whiteflies, 30% concentration of acetamiprid diluted 3000 times were applied every two weeks after transplanting. Prunings were conducted twice at flowering and fruit development stages, respectively, the secondary shoots were removed and only the main stem was left.

2.2. Measurements

2.2.1. Water and Nitrogen Consumption

The nitrogen application amount was defined as plant nitrogen consumption, and plant water consumption was defined as crop evapotranspiration (ET). ET was determined using the soil water balance method as follows:

$$\mathrm{E}\mathrm{T}=\mathrm{P}+\mathrm{I}+\mathrm{W}-\mathrm{R}-\mathrm{D}-∆\mathrm{W}$$

(1)

$$∆\mathrm{W}=1000\mathrm{H}\left({\mathsf{\theta }}_{\mathrm{i}}-{\mathsf{\theta }}_{\mathrm{i}-1}\right)$$

(2)

where P is precipitation (mm); I is irrigation amount (mm); W represents capillary rise supply water to root zones (mm); R is soil surface runoff (mm); D is drainage from root zones (mm). ΔW represents the change of soil water content in the root zone (mm), calculated by Equation (2); H is the depth of plant root zones (m), determined as 0.4 m at the seedling stage and 0.6 m in the other stages; θ_i and θ_i−1 are the mean water contents in root zones at time i and i − 1 (cm³·cm⁻³), respectively.

For greenhouse shelter, P was ignored. Similarly, R and D was also ignored due to the accurate irrigation amount controlled by drip irrigation, which did not generate runoff and drainage. W was negligible since the groundwater level was over 15 m in depth. Therefore, Equation (1) was simplified as follows:

$$\mathrm{E}\mathrm{T}=\mathrm{I}-1000\mathrm{H}\left({\mathsf{\theta }}_{\mathrm{i}}-{\mathsf{\theta }}_{\mathrm{i}-1}\right)$$

(3)

The soil water contents (θ) were tested by a portable water probe (Diviner 2000, Sentek Pty. Ltd., Adelaide, Australia) and PVC tubes set in plots every 5–7 days or before and after irrigation.

2.2.2. Fruit Yield and Quality Parameters

Tomato yield (Y) was calculated by the average fruit weight of 15 plants from each plot. The single fruit weight (SW) was determined by randomly selecting 15 tomato plants from each treatment. Tomato quality parameters, including the fruit’s total soluble solids (TSS), organic acid (OA), vitamin C (VC), lycopene (Lyc) and fruit firmness (Fn), were determined from 9 randomly selected tomatoes in each treatment. The comprehensive quality of tomato fruit (Q) was calculated by the AHP-TOPSIS method. Detailed measurement and calculation processes are available in pre-study [48].

2.3. Models

2.3.1. Water-Nitrogen Models for the Whole Growth Stage

Water-nitrogen models for the whole stage only consider the total water-nitrogen consumption, disregarding the variances among different growing stages. This results in more concise models with fewer input parameters needed. The simplest model for the whole growth stage is the binary regression function of water and nitrogen factors, proposed as follows:

Form 1, model 1: binary quadratic function (BQ)

$${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}=\mathrm{a}{\left({\displaystyle \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{m}}}}\right)}^2+\mathrm{b}{\left({\displaystyle \frac{{\mathrm{N}}_{\mathrm{a}}}{{\mathrm{N}}_{\mathrm{m}}}}\right)}^2+\mathrm{c}{\displaystyle \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{m}}}}·{\displaystyle \frac{{\mathrm{N}}_{\mathrm{a}}}{{\mathrm{N}}_{\mathrm{m}}}}+\mathrm{d}{\displaystyle \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{m}}}}+\mathrm{e}{\displaystyle \frac{{\mathrm{N}}_{\mathrm{a}}}{{\mathrm{N}}_{\mathrm{m}}}}+\mathrm{f}$$

(4)

where V_a denotes the Y or fruit quality parameters under different deficit treatments; V_m is the yield or fruit quality parameters under full irrigation and nitrogen treatment (CK); ET_a represents the ET under different deficit treatments and ET_m is the ET under CK; N_a is the total nitrogen application under different deficit treatments and N_m is the total nitrogen application under CK; a~f are coefficients that need to be calibrated.

Another model for the whole stage highlighted the importance of water consumption. Nitrogen factors were introduced as the coefficients of the water factor, resulting in a new compound function as follows:

Form 1, model 2: compound function (CF):

$$\left\{\begin{array}{c}{\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\mathrm{A}}_1{\left({\displaystyle \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{m}}}}\right)}^2+{\mathrm{A}}_2{\displaystyle \frac{{\mathrm{E}\mathrm{T}}_{\mathrm{a}}}{{\mathrm{E}\mathrm{T}}_{\mathrm{m}}}}+{\mathrm{A}}_3\\ {\mathrm{A}}_{\mathrm{i}}={\mathrm{a}}_{\mathrm{i}}{\left({\displaystyle \frac{{\mathrm{N}}_{\mathrm{a}}}{{\mathrm{N}}_{\mathrm{m}}}}\right)}^2+{\mathrm{b}}_{\mathrm{i}}{\displaystyle \frac{{\mathrm{N}}_{\mathrm{a}}}{{\mathrm{N}}_{\mathrm{m}}}}+{\mathrm{c}}_{\mathrm{i}}\end{array}\right.$$

(5)

where A_i, a_i, b_i, c_i are coefficients that need calibration.

2.3.2. Water-Nitrogen Models for the Separated Growth Stage

Comprehensively considering the characteristics and utilizations of each CWPF, Jensen, Minhas, and Rao models were chosen for multiplicative models, whereas Blank, Stewart, and Singh models were chosen for cumulative models. Based on these functions, two methods (Form 2 and Form 3) introducing a nitrogen assumption factor were used (

Table 2. The equations of form 2 and form 3 models.

Equation	S-CWPFs: f(ETi)	Form 2: f(ETi·Ni)	Equation	Form 3: f(ETi)·f(N)
Jensen	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{n}}}{\left({\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}\right)}^{{\mathsf{\lambda }}_{\mathrm{i}}}$	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{n}}}{\left({\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}·{\displaystyle \frac{{\mathrm{N}}_{\mathrm{ai}}}{{\mathrm{N}}_{\mathrm{mi}}}}\right)}^{{\mathsf{\lambda }}_{\mathrm{i}}}$	Sigmoid	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\mathrm{f}\left({\mathrm{E}\mathrm{T}}_{\mathrm{i}}\right)}_{\mathrm{\ast }}\times {\displaystyle \frac{\mathrm{a}}{1+{\mathrm{e}}^{\mathrm{b}-\mathrm{c}·\mathrm{N}}}}$
Minhas	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{n}}}{\left[1-{\left(1-{\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}\right)}^2\right]}^{{\mathsf{\lambda }}_{\mathrm{i}}}$	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{n}}}{\left[1-{\left(1-{\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}·{\displaystyle \frac{{\mathrm{N}}_{\mathrm{ai}}}{{\mathrm{N}}_{\mathrm{mi}}}}\right)}^2\right]}^{{\mathsf{\lambda }}_{\mathrm{i}}}$	Log	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\mathrm{f}\left({\mathrm{E}\mathrm{T}}_{\mathrm{i}}\right)}_{\mathrm{\ast }}\times \ln \left(\mathrm{a}\mathrm{N}+\mathrm{b}\right)+\mathrm{c}$
Rao	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{n}}}\left[1-{\mathrm{K}}_{\mathrm{i}}\left(1-{\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}\right)\right]$	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \prod _{\mathrm{i}=1}^{\mathrm{n}}}\left[1-{\mathrm{K}}_{\mathrm{i}}\left(1-{\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}·{\displaystyle \frac{{\mathrm{N}}_{\mathrm{ai}}}{{\mathrm{N}}_{\mathrm{mi}}}}\right)\right]$	Exp	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\mathrm{f}\left({\mathrm{E}\mathrm{T}}_{\mathrm{i}}\right)}_{\mathrm{\ast }}\times \exp \left(\mathrm{a}\mathrm{N}+\mathrm{b}\right)+\mathrm{c}$
Blank	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{K}}_{\mathrm{i}}\left({\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}\right)$	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{K}}_{\mathrm{i}}\left({\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}·{\displaystyle \frac{{\mathrm{N}}_{\mathrm{ai}}}{{\mathrm{N}}_{\mathrm{mi}}}}\right)$	Root	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\mathrm{f}\left({\mathrm{E}\mathrm{T}}_{\mathrm{i}}\right)}_{\mathrm{\ast }}\times \left(\mathrm{a}\sqrt{\mathrm{N}}+\mathrm{b}\right)$
Stewart	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}=1-{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{K}}_{\mathrm{i}}\left(1-{\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}\right)$	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}=1-{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{K}}_{\mathrm{i}}\left(1-{\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}·{\displaystyle \frac{{\mathrm{N}}_{\mathrm{ai}}}{{\mathrm{N}}_{\mathrm{mi}}}}\right)$	Line (1)	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\mathrm{f}\left({\mathrm{E}\mathrm{T}}_{\mathrm{i}}\right)}_{\mathrm{\ast }}\times \left(\mathrm{a}\mathrm{N}+\mathrm{b}\right)$
Singh	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{K}}_{\mathrm{i}}\left(1-{\left(1-{\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}\right)}^2\right)$	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{K}}_{\mathrm{i}}\left(1-{\left(1-{\displaystyle \frac{{\mathrm{ET}}_{\mathrm{ai}}}{{\mathrm{ET}}_{\mathrm{mi}}}}·{\displaystyle \frac{{\mathrm{N}}_{\mathrm{ai}}}{{\mathrm{N}}_{\mathrm{mi}}}}\right)}^2\right)$	Sqrt (2)	${\displaystyle \frac{{\mathrm{V}}_{\mathrm{a}}}{{\mathrm{V}}_{\mathrm{m}}}}={\mathrm{f}\left(\mathrm{E}\mathrm{T}\mathrm{i}\right)}_{\mathrm{\ast }}\times \left(\mathrm{a}{\mathrm{N}}^2+\mathrm{b}\mathrm{N}+\mathrm{c}\right)$

Notes: ET_ai represents plant evapotranspiration at i-th growth stage under deficit treatments, and ET_mi indicates the evapotranspiration under CK; n is the amount of growth stages, set as 3 for tomato plant; N_ai is nitrogen application amount at i-th growth stage under deficit treatments and N_mi denotes the nitrogen application under CK; λ_i and K_i are the sensitive index and coefficient at i-th growth stage. f(ET_i)_* is the best multiplicative and cumulative functions determined in Form 2, Rao and Singh; N is the nitrogen application amount at the whole growth stage; V_a presents the actual tomato harvest value (such as yield, SAR, lycopene content and so on) under deficit treatments and V_m indicates the value under CK.

). The former (Form 2) directly replaced the input of ET in the traditional CWPFs with the product of ET and N application. The latter (Form 3) considered the comparable slight effect of nitrogen, innovatively taking the N factor as an extra product term to estimate the effect of N application on fruit quality. Detailed equations were listed in Table 2.

2.4. Model Calibration and Evaluation

2.4.1. Model Calibration

Data in the second seasons were used to calibrate the models’ coefficients and indexes based on the least squares estimation method. Evapotranspiration and nitrogen application amount were set as inputs, and harvest parameters were set as the true value of outputs.

2.4.2. Model Evaluation

Data in the first season was used to verify the model performance as truth value. Statistic parameters including coefficient of determination (R²), mean absolute error (MAE), root mean square error (RMSE), relative error (RE), and their global performance indicator (GPI) were used to evaluate the performance of each model. In these 4 indicators, R² reflects model’s fit goodness, and the remaining three indicators reflect model’s simulated accuracy. Each side weighs half in GPI consideration. Detailed equations are described as follows:

$${\mathrm{R}}^2={\left[{\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left({\mathrm{O}}_{\mathrm{i}}-\overline{\mathrm{O}}\right)\left({\mathrm{P}}_{\mathrm{i}}-\overline{\mathrm{P}}\right)}{\sqrt{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{O}}_{\mathrm{i}}-\overline{\mathrm{O}}\right)}^2}\sqrt{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{P}}_{\mathrm{i}}-\overline{\mathrm{P}}\right)}^2}}}\right]}^2$$

(6)

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{P}}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\right|}{\mathrm{n}}}$$

(7)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{P}}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\right)}^2}$$

(8)

$$\mathrm{R}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{P}}_{\mathrm{i}}-{\mathrm{O}}_{\mathrm{i}}\right|}{{\mathrm{O}}_{\mathrm{i}}}}$$

(9)

$$\mathrm{G}\mathrm{P}\mathrm{I}={\sum }_{\mathrm{j}=1}^4{\mathsf{\alpha }}_{\mathrm{j}}\left({\mathrm{T}}_{\mathrm{j}}-{\mathrm{T}}_{\mathrm{j}\mathrm{n}}\right)$$

(10)

where P_i is the simulated value of i-th treatment; O_i is the observed value of i-th treatment; n is the treatment amount and in this study n = 12; $\overline{\mathrm{P}}$ and $\overline{\mathrm{O}}$ are the average value of P_i and O_i arrays, respectively; T_j is the orthogonalized and normalized value of evaluated indicators calculated above; T_jn is the median of each index; α_j is the weight of each index, set 0.5 for R² and 0.5/3 for MAE, RMSE, and RE equally.

2.5. Data Analysis

Calculated procedures including correlation cluster analysis, model calibration, model calculation and validation, and graphing were all performed by MATLAB 2022 (MathWorks Inc., Natick, MA, USA).

3. Results

3.1. Relationship between Water-Nitrogen Consumption, Yield, and Fruit Quality

Compared to tomato nitrogen (N) consumption (Figure 1), evapotranspiration (ET) at the whole growth stage had a more significant effect on fruit yield and quality parameters, especially on fruit yield (Y), single fruit weight (SW) and lycopene content (Lyc). The correlation indexes (R) of ET-Y, ET-Lyc and ET-SW reached 0.86, 0.55 and 0.59, respectively (|R| > 0.49, p ≤ 0.01). The absolute correlation index of ET on tomato fruit ranged from 0.29 to 0.86, while that of N ranged from 0.20 to 0.54. Phased ET and N consumption at the flowering and fruit development stage (stage II) and the fruit maturation and harvest stage (stage III) exerted a major influence on the formation of both tomato yield and quality. The R values of ET₂-Y, ET₂-organic acids content (OA), ET₂-Lyc, ET₂-solid–acid ratio (SAR), and ET₂-SW reached 0.85, 0.57, −0.59, 0.71, and 0.70 (p ≤ 0.01), while those of ET₃ were 0.82, 0.50, 0.58, −0.64, and 0.73 (p ≤ 0.01), respectively. As for N, only N₂-Lyc, N₃-TSS, OA, SAR, and vitamin C content (Vc) correlated relationships were significant, and the R value reached 0.57, 0.41, 0.64, −0.62 and 0.48 (|R| > 0.388, p ≤ 0.05), respectively.

The effects of ET and N on Y, Q, SAR, and Lyc are shown in Figure 2, and water consumption had a significantly positive effect on Y and Lyc, but had a negative effect on Q and SAR (p ≤ 0.01). Nitrogen application improved tomato Y and Lyc, while it decreased the SAR (p ≤ 0.05). The level of 1/3 full irrigation caused a 24.5% yield reduction under sufficient nitrogen (FN), and the reduction expanded to 37.6% under deficit nitrogen (0 FN). However, for Lyc, the decline caused by water deficit changed from 28.8% to 13.8% as nitrogen supply decreased. The contrary reactions indicated that the interaction of water and nitrogen could shift from synergistic to antagonism among different harvest parameters, and the complexity made it challenging to quantitatively simulate fruit parameters.

3.2. Coefficients Calibration of Water-Nitrogen, Yield, and Fruit Quality Models

Comprehensively considering the independence and importance of each harvest parameter, four harvest parameters were chosen as modeling targets, including Y as the only production parameter, fruit comprehensive quality (Q) representing total quality information, SAR as the fruit favor parameter, and Lyc representing the fruit nutrition parameter. The absolute correlation values of VC to Q and SW to Y were 0.72 and 0.67 (p ≤ 0.01), respectively, indicating that using Y and Q to represent VC and SW was appropriate. SAR reflected both TSS and OA characteristics; thus, the TSS and OA were disregarded. Lastly, fruit firmness (Fn) was eliminated from the simulated targets, as the fruit maturation and storage characteristics that Fn reflects are less considered by the market, and no significant correlation was found between Fn, ET and N.

Three forms of water-nitrogen-harvest parameter functions, including 20 model types, were used to estimate the yield and quality parameters. The coefficients of each water-nitrogen-yield and water-nitrogen-quality function were calculated using experiment data from the second season in 2019 (

Table 3. Tomato yield (Y), fruit comprehensive quality (Q), solid–acid ratio (SAR), and lycopene content (Lyc) amounts of different treatments.

	First Season				Second Season
Treatment	Yield (t·ha−1)	Q (a.u.)	SAR (ratio)	Lyc (mg·kg−1)	Yield (t·ha−1)	Q (a.u.)	SAR (ratio)	Lyc (mg·kg−1)
T1(CK)	66.09 a	0.344	10.54 d	26.40 a	59.92 a	0.404	11.76 d	46.08 a
T2	66.46 a	0.392	14.01 cd	24.00 a	57.55 ab	0.446	15.09 bc	44.01 ab
T3	65.33 a	0.403	16.53 bcd	22.03 a	54.32 ab	0.464	12.65 d	40.83 abc
T4	58.79 ab	0.182	12.93 cd	17.11 a	56.97 ab	0.353	13.76 cd	30.06 bc
T5	68.28 a	0.388	15.83 bcd	23.76 a	49.36 abc	0.470	11.46 d	35.10 abc
T6	63.84 a	0.421	15.34 bcd	20.20 a	56.86 ab	0.575	15.62 abc	37.12 abc
T7	48.23 bc	0.364	20.00 bc	18.46 a	51.77 abc	0.598	15.15 bc	35.67 abc
T8	47.98 bc	0.445	21.97 ab	20.70 a	42.14 bc	0.414	16.46 ab	31.39 abc
T9	40.23 c	0.599	18.90 bc	20.84 a	53.85 abc	0.641	17.83 a	29.25 bc
T10	38.48 c	0.770	22.59 ab	27.20 a	56.79 ab	0.563	16.94 ab	30.84 bc
T11	35.08 c	0.438	18.35 bc	17.01 a	41.46 bc	0.510	17.89 a	29.97 bc
T12	34.32 c	0.513	28.63 a	13.63 a	38.00 c	0.401	16.46 ab	27.86 c

Notes: T1–T4 were full irrigation (FN), T5–T8 were 2/3 FI, T9–T12 were 1/3 FI; from T1 to T4 (T5 to T8, T9 to T12), the nitrogen application levels were gradually reduced from full nitrogen (FN) to 2/3 FN, 1/3 FN and 0 FN. Detailed fertigation amounts can be found in Table 1. Lowercase letters following the data indicate significant differences by Duncan’s test at p ≤ 0.05 level.

). The calibration results were shown in

Table 4. The water and nitrogen sensitive coefficients of tomato yield, comprehensive quality, solid–acid ratio, and lycopene content.

Object	Model	Coefficient						R2	Sig.
		a/b/λ/K1	a/b/λ/K2	a/b/λ/K3	d/a	e/b	f/c
Y	F1: BQ	−8.31	0.46	2.44	14.02	−2.65	−4.87	0.44	0.02 *
	F1: CF	103.50	−140.07	46.33				0.89	0.00 **
		−172.39	228.68	−73.20
		70.92	−92.00	29.23
	F2: Jensen	0.12	0.07	0.22				0.41	0.06 ns
	F2: Minhas	1.52	0.16	0.58				0.41	0.06 ns
	F2: Rao	0.08	0.09	0.33				0.55	0.01 *
	F2: Blank	0.74	0.05	0.24				0.53	0.01 *
	F2: Stewart	0.10	0.08	0.31				0.57	0.00 **
	F2: Singh	0.20	0.15	0.63				0.57	0.00 **
	F3: Rao-Sig	−0.33	1.17	−1.34	3.05	1.17	0.48	0.75	0.00 **
	F3: Rao-Log	−0.31	1.16	−1.30	0.82	2.14	−0.07	0.75	0.00 **
	F3: Rao-Exp	−0.33	1.17	−1.35	0.33	−0.31	0.00	0.75	0.00 **
	F3: Rao-Rot	−0.29	1.14	−1.26	0.49	0.52		0.74	0.00 **
	F3: Rao-1	−0.32	1.16	−1.32	0.31	0.71		0.75	0.00 **
	F3: Rao-2	−0.34	1.17	−1.36	0.08	0.19	0.74	0.75	0.00 **
	F3: Sin-Sig	−0.04	1.15	−0.79	8.99	1.02	0.43	0.73	0.00 **
	F3: Sin-Log	−0.16	3.92	−2.65	0.58	2.00	−0.03	0.72	0.00 **
	F3: Sin-Exp	−0.03	0.85	−0.59	0.59	0.27	2.09	0.73	0.00 **
	F3: Sin-Rot	−0.04	0.87	−0.58	1.71	2.29		0.72	0.00 **
	F3: Sin-1	−0.48	−0.48	1.19	−0.07	4.27		0.43	0.02 *
	F3: Sin-2	−0.03	0.92	−0.64	0.64	0.24	3.40	0.73	0.00 **
Q	F1: BQ	20.44	−2.98	−6.14	−33.36	10.36	12.67	0.38	0.03 *
	F1: CF	338.19	−441.53	117.19				0.97	0.00 **
		−590.94	770.48	−206.38
		252.71	−328.09	89.37
	F2: Jensen	−1.17	−0.02	−0.46				0.52	0.03 *
	F2: Minhas	−8.45	−0.09	−1.37				0.38	0.07 ns
	F2: Rao	−1.11	−0.05	−0.68				0.10	0.32 ns
	F2: Blank	2.47	−0.10	−1.13				0.04	0.51 ns
	F2: Stewart	−1.25	−0.04	−0.76				0.12	0.28 ns
	F2: Singh	3.30	0.01	−2.09				0.01	0.80 ns
	F3: Rao-Sig	−0.69	1.00	−3.84	1.03	2.25	6.77	0.81	0.00 **
	F3: Rao-Log	−0.85	1.16	−4.81	1.84	1.47	−0.15	0.78	0.00 **
	F3: Rao-Exp	−0.86	1.19	−4.96	0.03	3.09	−21.66	0.77	0.00 **
	F3: Rao-Rot	−0.85	1.16	−4.81	1.09	−0.06		0.79	0.00 **
	F3: Rao-1	−0.86	1.19	−4.96	0.66	0.38		0.77	0.00 **
	F3: Rao-2	−0.64	1.03	−4.02	−1.23	2.36	−0.11	0.80	0.00 **
	F3: Sin-Sig	1.52	−0.31	−0.60	1.97	7.45	19.76	0.59	0.00 **
	F3: Sin-Log	0.07	−0.01	−0.03	21.17	−8.85	43.42	0.59	0.00 **
	F3: Sin-Exp	3.70	4.66	−6.85	0.02	3.02	−20.08	0.40	0.03 *
	F3: Sin-Rot	0.61	0.70	−1.07	4.56	0.75		0.43	0.02 *
	F3: Sin-1	−0.52	−0.52	1.29	−0.07	3.95		0.03	0.62 ns
	F3: Sin-2	3.34	−1.03	−0.95	−1.96	3.18	−0.36	0.57	0.00 **
SAR	F1: BQ	6.86	−1.46	1.62	−16.48	0.41	10.17	0.57	0.00 **
	F1: CF	−6.62	72.05	−45.86				0.90	0.00 **
		−15.98	−82.99	62.46
		16.39	20.94	−19.39
	F2: Jensen	−0.40	−0.03	−0.71				0.67	0.01 *
	F2: Minhas	−5.34	−0.08	−2.09				0.66	0.01 *
	F2: Rao	−0.14	−0.09	−1.03				0.59	0.00 **
	F2: Blank	2.76	−0.16	−1.43				0.15	0.22 ns
	F2: Stewart	−0.22	−0.08	−1.13				0.61	0.00 **
	F2: Singh	4.25	−0.03	−3.07				0.38	0.03 *
	F3: Rao-Sig	−1.16	0.53	−2.16	0.99	−109.29	−64.83	0.71	0.00 **
	F3: Rao-Log	−1.44	0.64	−2.73	0.56	3.10	−0.31	0.77	0.00 **
	F3: Rao-Exp	−1.44	0.64	−2.73	0.02	2.23	−8.44	0.77	0.00 **
	F3: Rao-Rot	−1.44	0.63	−2.71	0.27	0.73		0.77	0.00 **
	F3: Rao-1	−1.44	0.64	−2.73	0.16	0.84		0.77	0.00 **
	F3: Rao-2	−1.29	0.50	−2.32	−0.60	0.99	0.59	0.78	0.00 **
	F3: Sin-Sig	1.26	−0.10	−0.68	2.48	−87.07	−84.48	0.54	0.01 *
	F3: Sin-Log	0.05	0.02	−0.05	3.8 × 106	−1.5 × 106	62.27	0.54	0.01 *
	F3: Sin-Exp	5.80	2.55	−6.37	0.00	2.57	−12.56	0.52	0.01 *
	F3: Sin-Rot	1.10	0.48	−1.21	0.34	2.80		0.52	0.01 *
	F3: Sin-1	−0.39	−0.39	0.96	−0.09	5.30		0.53	0.01 *
	F3: Sin-2	1.32	−0.63	−0.24	−3.52	4.97	0.94	0.60	0.00 **
Lyc	F1: BQ	1.69	−0.40	−0.49	−2.09	1.39	0.87	0.50	0.01 *
	F1: CF	−17.44	34.09	−14.15				0.99	0.00 **
		21.32	−43.59	19.37
		−6.51	13.93	−6.02
	F2: Jensen	0.77	−0.03	0.83				0.94	0.00 **
	F2: Minhas	6.09	0.04	2.46				0.84	0.00 **
	F2: Rao	0.79	−0.04	0.84				0.73	0.00 **
	F2: Blank	0.14	−0.04	0.85				0.76	0.00 **
	F2: Stewart	0.74	−0.03	0.79				0.73	0.00 **
	F2: Singh	−1.18	−0.03	2.09				0.54	0.01 *
	F3: Rao-Sig	−0.12	0.52	0.33	0.93	3.96	15.09	0.88	0.00 **
	F3: Rao-Log	−0.16	0.64	0.20	0.31	2.12	0.06	0.86	0.00 **
	F3: Rao-Exp	−0.15	0.64	0.21	0.05	0.96	−1.80	0.86	0.00 **
	F3: Rao-Rot	−0.16	0.64	0.21	0.22	0.73		0.86	0.00 **
	F3: Rao-1	−0.15	0.64	0.21	0.13	0.82		0.86	0.00 **
	F3: Rao-2	0.07	0.45	0.42	−0.74	1.15	0.51	0.88	0.00 **
	F3: Sin-Sig	−0.23	0.79	−0.22	2.68	0.08	4.22	0.73	0.00 **
	F3: Sin-Log	−0.04	0.16	−0.05	342.40	−109.84	7.99	0.73	0.00 **
	F3: Sin-Exp	−1.70	7.70	−3.36	0.01	1.98	−6.95	0.72	0.00 **
	F3: Sin-Rot	−0.20	0.88	−0.36	1.22	1.70		0.72	0.00 **
	F3: Sin-1	−0.59	−0.59	1.47	−0.06	3.46		0.55	0.01 *
	F3: Sin-2	−0.37	1.27	−0.34	−0.72	1.41	0.92	0.73	0.00 **

Notes: The coefficient means a–f for model BQ (in Equation (4)), a₁–a₃/b₁–b₃/c₁–c₃ for model CF (in Equation (5)), λ₁–λ₃ for F2 Jensen/Minhas/Rao models (in Table 2), K₁-K₃ for F2 Blank/Stewart/Singh models (in Table 2), λ₁–λ₃/a-c for F3 Rao models (in Table 2), and K₁–K₃/a–c for F3 Sin models (in Table 2). R² and sig are the determination coefficient and significant value for model fitting, ** means highly significant, * means significant, and ns means not significant.

. Regression coefficients of a-f in the BQ model represented the sensitivities of each tomato parameter to water and nitrogen levels. For all the chosen characteristics, the absolute values of a (1.69–20.44) and d (2.09–33.36) were higher than b (0.40–2.98) and e (0.41–10.36). The significant effect of water stress on tomato yield and quality was consistent with the analysis in 3.1. Coefficients of λ_i and K_i in forms 2 and 3 were defined as the sensitivity indexes since they reflected the responses of yield and quality to water and nitrogen stress at different growth stages. In form 2 functions, relatively high absolute values of λ_i/K_i were observed at stages I and III, suggesting these two periods to be the most sensitive periods. In contrast, form 3 functions indicated stages II and III as the sensitive periods instead. Taking the above findings together, stage III was determined to be the most sensitive period, where water and nitrogen stress had more important effects on tomato yield and quality.

Overall, the estimating performances of the models varied for different harvest parameters, but models in the same form usually expressed similarity, except models in form 1. The CF model of form 1 obtained the highest R², ranging from 0.89 to 0.99 (p ≤ 0.01), while the BQ model of form 1 produced a suboptimal result, with R² ranging from 0.38 to 0.57 (p ≤ 0.05). The estimating performances of form 2 functions were relatively poor, especially for Q (p > 0.05), and the best result obtained was for Lyc using the Jensen model, with an R² of 0.94 (p ≤ 0.01). Except the Sin-line (1) model for Q, all other functions of form 3 had satisfactory performances, with R² values achieving statistical significance for the estimation of harvest parameters (p ≤ 0.05). The multiplicative models generally showed more favorable performances over accumulative models.

3.3. Prediction Performance of Water-Nitrogen Yield and Fruit Quality Function

The simulation results for models are shown in Figure 3 and Figure 4. Model BQ of form 1 functions produced a better interpretation for tomato yield, SAR, and Lyc, compared with form 2 functions. Among the multiplicative models of form 2, model Rao excelled in the prediction of Y, SAR, and Lyc parameters. For the accumulated models of form 2, model Stewart performed well for Y and SAR, whereas model Singh had stronger performances for Q and Lyc, with a slightly decreased precision for Y. Finally, multiplication model Rao and accumulation model Singh were chosen as f(ET_i)_* for structuring form 3 functions.

Considering the correlation coefficient and RMSE of models, functions of Form 3 had better performance for the four harvest parameters (Figure 3). For tomato yield prediction, the distance between model Sin-Sig, Log, Exp, Sqrt (2) and observation point was the shortest, indicating that the RMSE prediction results ranging in 0.11–0.12 were the closest to the true values, and the variance among those models was minimal. The same equation form of nitrogen factor also obtained decent precision when added into model Rao, with the RMSE ranging from 0.15 to 0.16.

For tomato Q and SAR, the results of all the water-nitrogen yield and quality functions were subpar. In terms of tomato fruit comprehensive quality, the similarity among models in the same equation forms noticeably decreased compared to that of the other three parameters. Only model Sin-Log of form 3 resulted in a comparably low RMSE value of 0.33. For tomato SAR, accumulation models Sin-Sig, Log, Exp, and Rot of form 3 were the best simulation models, and for Lyc, multiplicative models Rao-Exp, Rot, and (1) of form 3 were the most promising.

Synthetically comparing the model evaluation indicators, Form 3 functions were the best-fit models for tomato harvest parameter simulation. The Sin-2, Exp, and Sig models were the best-suited for Y according to the GPI rank, with an R² of 0.71, 0.69 and 0.68, respectively (Table 4). The best models ranked by GPI for tomato Q, SAR, and Lyc were model Sin-Log with a R² of 0.41, model Sin-Sig with a R² of 0.62, and model Rao-Rot with an R² of 0.42 (

Table 5. The simulated precision of models for tomato yield, comprehensive quality, solid–acid ratio, and lycopene content.

Object	Index	BQ	CF	Jensen	Minhas	Rao	Blank	Stewart	Singh	Rao-Sig	Rao-Log	Rao-Exp	Rao-Rot	Rao-1	Rao-2	Sin-Sig	Sin-Log	Sin-Exp	Sin-Rot	Sin-1	Sin-2
Y	R2	0.65	0.02	0.30	0.19	0.29	0.13	0.27	0.33	0.52	0.50	0.52	0.37	0.51	0.53	0.68	0.67	0.69	0.52	0.69	0.71
	MAE	0.18	0.35	0.15	0.28	0.15	0.26	0.16	0.18	0.13	0.13	0.13	0.17	0.13	0.12	0.10	0.10	0.10	0.13	0.14	0.10
	RMSE	0.21	0.59	0.18	0.36	0.18	0.33	0.19	0.22	0.15	0.16	0.15	0.20	0.15	0.15	0.12	0.12	0.12	0.16	0.17	0.11
	RE	0.26	0.56	0.48	0.61	0.20	0.29	0.20	0.21	0.15	0.16	0.15	0.21	0.16	0.15	0.13	0.13	0.13	0.17	0.21	0.13
	GPI	12	20	17	18	15	19	16	14	7	9	6	13	8	5	3	4	2	10	11	1
Q	R2	0.45	0.09	0.03	0.05	0.04	0.14	0.03	0.48	0.27	0.25	0.26	0.19	0.26	0.21	0.21	0.41	0.27	0.17	0.33	0.14
	MAE	0.66	1.20	2.28	178.64	1.20	0.49	0.96	0.26	0.55	0.52	0.47	0.61	0.47	0.62	0.59	0.23	0.34	0.45	0.57	0.65
	RMSE	0.89	2.32	3.47	330.68	1.56	0.58	1.21	0.33	0.67	0.62	0.56	0.76	0.56	0.78	0.74	0.33	0.42	0.59	0.79	0.93
	RE	0.60	0.97	1.94	112.76	1.23	0.51	0.98	0.32	0.44	0.41	0.35	0.50	0.35	0.51	0.49	0.23	0.25	0.35	0.42	0.57
	GPI	5	7	14	6	18	20	19	2	11	12	8	17	9	16	15	1	3	10	4	13
SAR	R2	0.57	0.23	0.23	0.07	0.33	0.00	0.33	0.26	0.61	0.41	0.42	0.26	0.42	0.10	0.62	0.52	0.43	0.35	0.49	0.01
	MAE	0.38	0.68	1.09	55.28	0.34	0.68	0.33	0.45	0.48	0.55	0.54	0.58	0.54	0.63	0.40	0.42	0.45	0.47	0.92	0.64
	RMSE	0.49	0.96	1.67	107.96	0.47	0.84	0.44	0.53	0.57	0.64	0.64	0.70	0.64	0.77	0.50	0.51	0.55	0.57	1.15	0.87
	RE	0.22	0.42	0.57	23.97	0.22	0.35	0.21	0.25	0.26	0.30	0.30	0.32	0.30	0.34	0.21	0.22	0.24	0.25	0.48	0.34
	GPI	2	17	15	14	7	20	6	12	3	10	8	16	9	19	1	4	5	11	13	18
Lyc	R2	0.57	0.29	0.32	0.23	0.46	0.36	0.51	0.26	0.51	0.34	0.33	0.42	0.33	0.50	0.54	0.26	0.43	0.51	0.13	0.53
	MAE	0.15	0.23	0.36	0.46	0.41	0.31	0.61	0.30	0.27	0.10	0.10	0.10	0.10	0.16	0.18	0.15	0.14	0.16	0.21	0.17
	RMSE	0.21	0.35	0.44	0.57	0.48	0.36	0.75	0.39	0.36	0.15	0.15	0.16	0.15	0.20	0.23	0.19	0.18	0.20	0.27	0.21
	RE	0.21	0.31	0.40	0.50	0.55	0.48	0.86	0.61	0.37	0.12	0.12	0.13	0.12	0.20	0.24	0.19	0.17	0.21	0.28	0.21
	GPI	6	13	17	20	15	14	18	16	12	8	10	1	9	4	7	11	5	3	19	2

).

4. Discussion

4.1. The Reaction of Tomato to Water-Nitrogen Deficit

Tomato is a characteristically water-demanding vegetation with an average fruit water content of over 95% [49] and a water productivity of about 22–25 kg·m⁻³ [48]. Evapotranspiration was the most dominant factor with direct influences on tomato fruit yield and qualities [50]. In this study, the evapotranspiration showed positive effects on tomato Y, OA, Lyc, and SW, but it showed negative effects on Q, TSS, SAR, VC and Fn (Figure 1). Similar results were also reported in northwestern China [51], the Ahvaz region of Iran [52], and southern Italy [53]. Deficit irrigation can restrain the growth of tomato height, leaf area and branches as well as fruit yield [6]. Nevertheless, a certain degree of water deficit can promote fruit quality, as decreased water makes solute accumulation and concentration rise in fruit [54]. In addition, water deficit can affect tomato gene expression and galactose metabolism, as well as starch and sucrose metabolism, promoting small molecule content such as reducing sugar, ascorbic acid, and lycopene in fruit [55].

Water deficit under maximum nitrogen levels resulted in a more prominent effect on fruit yield and some quality features. This might be due to the fact that water stress could restrict the plants’ health and reaction sensitivity to other environmental factors, which was commonly observed in multi-factors field experiments [56]. The highest absolute values of sensitivity coefficients in form 3 models were all found in the flowering and maturation period, demonstrating that the phenological stages II and III were the key period for tomato fruit, in agreement with previous reports [39].

As an essential constituent of amino acids, proteins, nucleic acids, and chlorophyll, nitrogen significantly determines plant photosynthesis process, growth development and resistance formation [57]. N application showed a positive effect on tomato yield and quality parameters, except SAR, in this study, consistent with previous experimental results in Nepal [58] and Italy [59].

4.2. Evaluation of the Three Forms of Tomato Production Models

As an empirical model, the water crop production functions used in this study hardly reflect the mechanisms behind plant growth and fruit formation. However, the data input demands, such as ET and nitrogen application amount, were general and cheap to observe, as well as more acceptable to farms with limited investments. In the three forms of functions, the ad and disadvantages were discussed as follows:

Form 1 models with the lowest input demand took the water and nitrogen consumption at whole stages as inputs without considering the sensitivity variance among different stages and performed comparably poor. Model CF in form 1 produced the best-fit model but its predictive performance was suboptimal, due to the overfitting caused by having too many coefficients [60]. For the BQ model with fewer coefficients, the fitting accuracy declined and prediction precision slightly improved. The coefficients set balance between fitting and prediction was still a problem in model establishments. The same issues of overfitting and coefficient balance were also observed in other form models, compromising impacted models’ simulation outcomes.

Form 2 models with the highest input demand took the product of water and nitrogen applications for every growing stage to represent the stress in each growing stage. Although the model theoretically considered stage variances, simply taking the product of water and nitrogen amount as the input was reckless and may wrongly expand the degree of stress. For example, the multiplicative models in form 2 such as model Jensen would return an obviously false result of 0 for 0 level nitrogen treatments. Even though we had excluded those impracticable points in item 3.3, the prediction precision of form 2 models still performed not well in this study. Possibly, using the plant’s actual nitrogen uptake could enhance the efficacy of form 2 models. Nevertheless, the additional processes and expenditures entailed in ascertaining actual nitrogen uptake must be balanced under specific conditions.

Form 3 models took separable evapotranspiration and whole-period nitrogen application as input and performed best in tomato harvest parameter simulations. Form 3 models were established based on form 2. Since the combination of water and nitrogen factors as an independent variable in form 2 f(ET·N) exhibited low precision, we divided the two factors apart and established a composite function as form 3 f(ET)·f(N). In form 3, the growth period variances were only considered on water, and the basic function forms of Rao and Singh were chosen. For the nitrogen application, to avoid those 0 points, mathematic unary functions f(N) that set the whole nitrogen application amount as the only input was built. Finally, form 3 models were established by individually multiplying f(ET) and f(N). Form 3 models showed the best prediction results in all of the simulated parameters with averagely 5–6 fitted coefficients, indicating the ascendency of composing separated water and full-term nitrogen. Similar results were also reported in the modified maize water-nitrogen production function [61] and tomato crop-water-salt production function [47].

Compared with yield and lycopene content, simulation results for comprehensive quality and solid–acid content ratio were less favorable. Both Q and SAR were composite parameters containing multiple information, hence the fitness capabilities of only one set of coefficients were insufficient, resulting in poor performance of the models. Since the interactions between water and nitrogen applications on fruit yield and qualities are very intricate [62], the semiempirical models are better suited for simulating individual parameters [39,63].

5. Conclusions

Water and nitrogen significantly affected tomato yield and quality formation. Compared to nitrogen application, water assumption, especially during the flowering and fruit maturation periods, was of the utmost importance in determining harvest parameters. Among the tomato harvest parameters, four indicators, including Y, Q, SAR, and Lyc, exhibited a more significant relation with water-nitrogen input and representation for fruit characteristics.

To quantitatively measure the relationship between water-nitrogen input and tomato yield and quality output, 3 equation forms including 20 models were built based on crop water production functions. Form 1 had the least input demand, form 2 f(ET_i·N_i) had the least calibration coefficients, and form 3 f(ET_i)·f(N) function showed the best simulation performance. Models of Singh-2, Singh-Log, Singh-sigmoid and Rao-Root, with R² ranging from 0.41 to 0.71 and RMSE ranging from 0.11 to 0.50, were recommended for the simulations of tomato yield, comprehensive quality, solid–acid ratio, and lycopene content, respectively. Our study provides a relatively precise and low-input method to determine irrigation and N fertilization strategy while also considering the balance between yield and fruit quality.