The experiment was conducted during 2022–2023 in Yuwang Town (36°48′2″N, 106°21′53″E, altitude 1489 m), a typical area of the central arid zone in Ningxia, northwest China, as specifically shown in Figure 1a. The experimental area features a continental arid climate characterized by significant diurnal temperature variations, low precipitation, and intense evaporation. The multi-year average rainfall is approximately 270 mm, with annual evaporation reaching about 2,325 mm. The groundwater table lies below 10 m depth. The experimental soil is classified as sandy loam, with basic physical-chemical properties of the 0–40 cm tillage layer before sowing as follows: bulk density 1.41 g cm^-3, field water-holding capacity 21.8%, pH 8.57, total salt content 0.6 g kg^-1, organic matter content 6.65 g kg^-1, alkaline hydrolysis nitrogen content 38 mg kg^-1, available phosphorus content 3.94 mg kg^-1, available potassium content 130 mg kg^-1, total nitrogen content 0.27 g kg^-1, total phosphorus content 0.64 g kg^-1, and total potassium content 17.4 g kg^-1.

A two-year experiment was designed with two factors and three levels, involving 3 irrigation quotas and 3 nitrogen application rates, and a control group (CK) was set according to local management practices, which relied solely on rainfall without irrigation, with 15 t ha^-1 of cow dung organic fertilizer and 195 kg ha^-1 of nitrogen fertilizer applied. The maximum irrigation quota was implemented in accordance with the Technical Specification for Potato Drip Irrigation Cultivation in Ningxia issued by the Water Resources Department of Ningxia Hui Autonomous Region in 2017, while the maximum nitrogen application rate adopted the local conventional application amount. Irrigation treatments included three quotas: 2250 m³ ha^-1 (W3), 20% water-saving (1800 m³ ha^-1, W2), and 40% water-saving (1350 m³ ha^-1, W1). Nitrogen application treatments consisted of the recommended nitrogen rate (195 kg ha^-1, N3), 20% nitrogen reduction (156 kg ha^-1, N2), and 40% nitrogen reduction (117 kg ha^-1, N1), using urea (46% nitrogen content) as the nitrogen fertilizer. In total, there were 10 treatments with 3 replicates each, resulting in 30 plots, as detailed in Figure 1b. Irrigation and fertilization were conducted through integrated water and fertilizer management with multiple topdressings, which were applied 6 times during the seedling stage, tuber formation stage, tuber growth stage, and starch accumulation stage in a ratio of 1:2:2:1. Irrigation was carried out 10 times during the sprout growth stage, seedling stage, tuber formation stage, tuber growth stage, and starch accumulation stage in a ratio of 1:2:3:3:1, as shown in Table 1. The cumulative irrigation amounts and nitrogen application rates for potatoes are presented in Figure 2, and the rainfall and temperature during the potato growth stages are shown in Figure 3.

Potato planting dates were May 1, 2022, and April 30, 2023, with harvest dates on October 3, 2022, and October 2, 2023. The tested variety was “Qingshu 9”, which was bred by Qinghai Academy of Agricultural and Forestry Sciences. It is characterized by late-maturing and fresh-eating, luxuriant branches and leaves, concentrated tuber setting, oblong tubers, and high quality. The planting mode adopted was under-film drip irrigation, with “one film for two rows” planted at equal row spacing. The plastic film was 1.2 m wide, and the seeds were sown at a depth of 10 cm. The 1.2 m-wide plastic mulch covered seeds planted at 10 cm depth. Planting parameters included 50 cm plant spacing and 60 cm row spacing, with 10 plants per row and 40 plants per plot, resulting in a planting density of 33,345 plants ha^-1. Basal fertilizers included calcium superphosphate (12% P₂O₅ content, 82.5 kg ha^-1) and potassium sulfate (50% K₂O content, 150 kg ha^-1), applied as a single basal dose. Each experimental plot measured 17.6 m² (5.5 m length × 3.2 m width), surrounded by 1 m-wide buffer rows between plots and 2 m-wide peripheral buffer rows. The experimental design comprised 30 plots (3 replicates × 10 treatments). Each plot featured an independent irrigation control unit with a water meter, gate valve, and pressure gauge. The embedded drip tape (16 mm inner diameter, 0.15 mm wall thickness) operated at 0.1 MPa pressure, delivering 2 L h^-1 through emitters spaced 50 cm apart (one emitter per plant). Field management practices including weeding and pesticide application followed locally recommended agronomic protocols.

The technique for order preference by similarity to ideal solution (TOPSIS) was used to identify a solution from the feasible solution set by defining the positive ideal solution and the negative ideal solution for the decision problem so that it was closest to the positive ideal solution and farthest from the negative ideal solution (Yang et al., 2023).

1. Nine treatments were set as evaluation objects, with nine evaluation indicators including tuber yield, SPAD, Tr, Pn, Gs, Ci, PSN, PLN, PRN, PTN, PSP, PLP, PRP, PTP, PSC, PLC, PRC, PTC. The evaluation indicators were normalized to establish a normalized matrix Specific formulas see Equation 5:

where z_ij is the j index normalized value in i treatment; x_ij is the j index value in the i treatment. i = 1, 2,···, n; j = 1, 2,···, m;

2. The ideal solution (Z_ij⁺) and the negative solution (Z_ij⁻) were determined to form the ideal solution vector Z⁺ and the negative solution vector Z⁻, respectively Specific formulas see Equation 6, 7:

where Z_ij⁺ and Z_ij⁻ represent the maximum and minimum values of the evaluation object in the j-th index, respectively;

3. The Euclidean distances (D_i⁺ and D_i⁻) were determined Specific formulas see Equation 8, 9:

where w_j is the weight of indicator j;

4. The relative proximity coefficient R_i of each treatment was calculated; that is, the proximity between the evaluation object and the optimal scheme was calculated as follows Specific formulas see Equation 10:

Data collation was performed using Microsoft Excel 2019 (developed by Microsoft Corporation, USA); graphing was conducted with Origin 2021 (developed by OriginLab Corporation, USA); Duncan’s multiple comparison significance test (p< 0.05) was carried out via SPSS 26 (developed by International Business Machines Corporation, USA); and R (V.4.3.1, developed by Ross Ihaka and Robert Gentleman from the Department of Statistics, University of Auckland, New Zealand) was utilized for Pearson correlation analysis, plotting of fixed-model correlation analysis, as well as the construction of random forest and TOPSIS models.

The effects of water-nitrogen interaction on the relative chlorophyll content of potato leaves are shown in Figure 4. Consistent trends were observed across both experimental years. During the growth period, leaf SPAD values under water-nitrogen regulation exhibited an initial increase followed by a decline, reaching their maximum at the tuber formation stage. At constant nitrogen levels, SPAD values first increased and then decreased with rising irrigation quotas; similarly, under fixed irrigation, SPAD values initially rose before declining as nitrogen application rates increased. Among all treatments, the W2N2 regime resulted in the highest relative chlorophyll content, with an average increase of 2.87% compared to the CK treatment across all growth stages over both years. Statistical analysis confirmed significant interactive effects between water and nitrogen, with both factors significantly influencing leaf chlorophyll content (p < 0.05).

The interactive effects of water and nitrogen on total potato yield are shown in Figure 11. The variation trends of total potato yield under water-nitrogen regulation were generally consistent across the two experimental years. In 2022, under the same irrigation quota, total yield initially increased and then decreased with increasing nitrogen application rate. Under the same nitrogen application rate, total yield showed increasing trends with higher irrigation quotas for N1 and N3 treatments, while N2 treatment exhibited an initial increase followed by decrease. Compared with CK treatment, W1N1, W1N2, W1N3, W2N1, W2N2, W2N3, W3N1, W3N, and W3N3 increased yields by 9.03%, 36.67%, 19.39%, 40.43%, 64.71%, 48.53%, 49.33%, 62.45%, and 55.90% respectively. In 2023, under W1 treatment, total yield increased continuously with nitrogen application, while W2 and W3 treatments showed an initial increase followed by decrease. Under the same nitrogen application rate, all treatments exhibited an initial increase followed by decrease in total yield with increasing irrigation quota. Compared with CK treatment, W1N1, W1N2, W1N3, W2N1, W2N2, W2N3, W3N1, W3N, and W3N3 increased yields by 11.62%, 17.88%, 28.70%, 49.05%, 65.84%, 54.81%, 46.98%, 56.37%, and 52.62% respectively. Excessive irrigation and nitrogen application were detrimental to yield formation, while water and nitrogen demonstrated synergistic effects in enhancing potato productivity.

The physiological ecosystem of plant leaves exhibits significant spatial variability, and it is difficult for a single process of changes in potato physiological indicators and plant nutrients to reflect the scientific nature of potato growth. To further clarify the regulatory mechanism of water-nitrogen interaction on the photosynthetic stoichiometric balance of potatoes and the relationship between potato yield and various indicators, a Mantel test analysis was conducted on each potato indicator and potato yield (Y) (Figure 12). Since the potato tuber growth stage is the most critical period for yield formation, the correlation analysis was performed using the average values of the potato tuber growth stage. As shown in Figure 12a, the results of Mantel test analysis on soil indicators and potato yield in 2022 indicated that the overlap between SPAD index and potato yield was poor (Mantel r< 0.4), suggesting that the SPAD index had little impact on yield. Other indicators showed good overlap with potato yield (Mantel r ≥ 0.4) and had a strong correlation with potato yield (Mantel P ≤ 0.05), indicating that Tr, Pn, Gs, Ci, PSN, PLN, PRN, PTN, PSP, PLP, PRP, PTP, PSC, PLC, PRC, and PTC all had significant impacts on potato yield. As can be seen from Figure 12b, in 2023, the overlap between potato leaf SPAD, PRN, PLC indicators and potato yield was poor (Mantel r< 0.2), implying that SPAD, PRN, and PLC had little influence on potato yield. Except for soil Tr, PSN, and PTP, which had a weak correlation with potato yield (Mantel P< 0.05), other soil indicators showed a strong correlation with potato yield (Mantel P ≤ 0.05). There were correlations among various indicators of potato plants (except SPAD), indicating that there was a promoting effect between the physiological indicators of the plant. Therefore, Pearson correlation analysis clarified that there were strong correlations among various soil indicators under water-nitrogen interaction, and also illuminated the correlations among SPAD, leaf photosynthesis, and plant nutrients, providing a reference for clarifying the regulatory mechanism of water-nitrogen interaction on the photosynthetic stoichiometric balance of potatoes.

Random forest modeling predicted crop yield using SPAD, Tr, Pn, Gs, Ci, PSN, PLN, PRN, PTN, PSP, PLP, PRP, PTP, PSC, PLC, PRC, and PTC (Figure 13). The 2022 model (Figure 13a) identified PTP, PRN, PLC, PRP, and Tr as key yield determinants, while the 2023 model (Figure 13b) highlighted PRP, PLN, PSP, PSC, PTC, PTP, and PRC as major influencing factors.

Taking yield (Y) as the target dependent variable, with irrigation quota and nitrogen application rate as dependent variables, and irrigation water amount (W) and pure nitrogen application rate (N) as independent variables, regression analysis was performed on the data to establish a binary quadratic regression equation model (retaining three decimal places). In the 2022 experiment, the model formula for irrigation quota, nitrogen application rate, and yield was Y2022 = -128779.309 + 82.698W + 1174.116N - 0.019W² - 3.569N² - 0.017W·N, with R²=0.9788 and P = 0.0001. In the 2023 experiment, the model formula was Y2023 = -143778.716 + 155.395W + 595.990N - 0.037W² - 1.432N² - 0.059W·N, with R²=0.9650 and P = 0.0001. The R² values were all greater than 0.85, and predicted values showed significant correlation with measured values, indicating that the established models could effectively predict the interactive effects of water and nitrogen on yield. As shown in Figure 14, the irrigation amount and nitrogen application rate at maximum yield were 2107.84 m³ ha^-1 and 159.93 kg ha^-1 in 2022, and 1966.25 m³ ha^-1 and 167.86 kg ha^-1 in 2023. Comprehensive analysis recommended the W2 and W3 irrigation treatments and N2 nitrogen application treatment.

To further optimize the water-nitrogen regime, the TOPSIS model was employed using 18 indicators including yield, SPAD, Tr, Pn, Gs, Ci, PSN, PLN, PRN, PTN, PSP, PLP, PRP, PTP, PSC, PLC, PRC, and PTC to select the most efficient irrigation and nitrogen fertilization scheme for potatoes. Table 8 presents the score ranking table for each treatment. Analysis revealed that the W2N2 treatment achieved the highest total score, while the CK treatment had the lowest score. Under the experimental conditions, the W2N2 treatment is recommended as the optimal water-nitrogen coupling pattern for potato cultivation in the central arid zone of Ningxia.

A reasonable irrigation system helps maintain soil moisture balance (Xu et al., 2017). In this study, the W2N2 treatment showed the highest relative chlorophyll content, with an average increase of 2.87% compared to the CK treatment at all growth stages over the two-year experiment. This indicates that appropriate water and nitrogen management can increase the SPAD value (Ma et al., 2023) and maintain a relatively high chlorophyll content. When nitrogen supply is sufficient, the synthesis of chlorophyll in leaves increases, and the SPAD value rises accordingly. Transpiration in potato leaves is a process where water diffuses into the atmosphere in gaseous form. Through transpiration, plants lose over 97% of the water they absorb, highlighting the vital significance of water for plant life activities (Chaves et al., 2016; Griffani et al., 2024). The study found that under water-nitrogen interaction, the W2N2 treatment was significantly higher than other treatments, with an average increase of 228.49% compared to the CK treatment at all growth stages over the two years. This indicates that water-nitrogen interaction can enhance the absorption and transport of water by potato leaves and regulate the transpiration rate of potato leaves. Different water treatments affect the water status of potato plants, thereby influencing the absorption and transport of water by plants as well as the leaf transpiration rate (Zhang et al., 2018). Increased nitrogen application leads to increased biomass accumulation and excessive canopy leaf area, resulting in excessive increases in transpiration and water loss, and ultimately excessive consumption of soil moisture (Wang et al., 2020). The net photosynthetic rate of potato leaves plays a crucial role in plant organic matter accumulation, water regulation, and nutrient distribution, serving as a key physiological indicator of plants (Zahoor et al., 2017). Under water-nitrogen interaction, the W2N2 treatment showed a significantly higher net photosynthetic rate than other treatments, with an average increase of 150.09% compared to the CK treatment across all growth stages over the two years. This indicates that an appropriate supply of water and nitrogen can maintain the normal metabolic activities of leaf cells, ensuring the smooth progress of photosynthesis. Stomatal conductance of potato leaves refers to the degree of stomatal opening, which is a key factor affecting potato photosynthesis, respiration, and transpiration (Damour et al., 2010). Under water-nitrogen interaction, the leaf stomatal conductance of the W2N2 treatment was significantly higher than that of other treatments, with an average increase of 186.49% compared to the CK treatment across all growth stages over the two years. This indicates that when water is sufficient, the stomata of potato leaves usually remain relatively open, facilitating gas exchange. Nitrogen supply can promote the growth and development of potato leaves, increasing the number of stomata and their opening degree; however, either excessive or insufficient nitrogen may have adverse effects on stomatal conductance (Querejeta et al., 2022). The intercellular CO₂ concentration in potato leaves refers to the concentration of carbon dioxide within the leaf, which is essential for photosynthesis as potatoes absorb CO₂ to drive this process—directly influencing the rate and efficiency of photosynthesis, and thereby affecting potato growth and development (Shah et al., 2017). Under water-nitrogen interaction, the W2N2 treatment showed a significantly higher intercellular CO₂ concentration than other treatments, with an average increase of 63.09% compared to the CK treatment across all growth stages over the two years. When water and nitrogen supplies are sufficient, potato leaf stomata remain open, facilitating CO₂ entry into leaf cells, which in turn. The ratio of photosynthetic rate to transpiration rate provides insight into the water use efficiency (WUE) of potato leaves. This study revealed that WUE reflects the efficiency with which leaves utilize water during photosynthesis: higher WUE indicates stronger growth capacity of potatoes in arid regions. Thus, water-nitrogen interaction can regulate the WUE of potato leaves, thereby promoting potato growth (Wang et al., 2018).

The content, accumulation, and distribution of carbon (C), nitrogen (N), and phosphorus (P) in different organs are constrained by growth conditions and their own structural characteristics (Sanchez-Bragado et al., 2017). The concentrations and ratios of C, N, and P in plants are important indicators of ecological processes (Liu and Lu, 2024). Nitrogen (N) uptake is regulated by water availability. Water deficiency can limit crop nitrogen response by reducing nitrogen uptake and utilization. The complex and multi-faceted interactions between water supply and crop nitrogen response make it difficult to predict and quantify the impact of water deficiency on crop nitrogen status (Zhao et al., 2024a). This study found that water-nitrogen interaction significantly affects the uptake and distribution of C, N, and P nutrients in potato plants. The level of nitrogen supply directly affects carbon metabolism: appropriate nitrogen application (N2 treatment) enhances photosynthesis by promoting chlorophyll synthesis, thereby improving leaf carbon assimilation capacity. However, excessive nitrogen application (N3 treatment) stimulates excessive growth of aboveground parts, leading to over-distribution of photosynthates to stems and leaves and a reduction in carbon accumulation in tubers. Phosphorus uptake is greatly affected by water conditions. Increasing phosphorus application during the tuber expansion stage can improve nutrient use efficiency, and optimizing the nitrogen-phosphorus ratio can promote plant growth (Rosell et al., 2023). The interactive effect of nitrogen and water is obvious: under drought conditions, high nitrogen levels will exacerbate the imbalance of carbon and nitrogen metabolism, while appropriate irrigation (W2) can alleviate nitrogen stress and promote the transport of nitrogen and phosphorus to tubers. Water-nitrogen interaction significantly affects the accumulation of organic carbon in potato plants by altering the efficiency of photosynthetic carbon assimilation and distribution. Among all water-nitrogen interaction treatments, the W2N2 treatment showed the highest organic carbon accumulation. Reasonable water and nitrogen combination can optimize carbon metabolism pathways, increase the activity of starch synthesis-related enzymes, and improve the carbon conversion efficiency in tubers.

The synergistic effect of water and nitrogen affects the distribution of organic carbon by regulating plant metabolic processes. In 2022, the proportion of organic carbon in stems and roots was relatively high, which may be related to the enhanced supply of water and nitrogen promoting the transport of photosynthates to structural organs (stems and roots). In 2023, however, the increased proportion of organic carbon in leaves and roots reflects the improvement of photosynthetic carbon assimilation efficiency and the enhancement of root carbon storage capacity after water and nitrogen optimization. The characteristic that total nitrogen distribution is dominated by leaves is consistent with the function of leaves as the core organ of nitrogen metabolism. The interaction of water and nitrogen promotes the preferential enrichment of nitrogen in metabolically active leaves by affecting the pathways of nitrate absorption, reduction, and amino acid synthesis (Wang et al., 2022). The lowest proportion of nitrogen in stems is related to their lower nitrogen demand. The dynamic distribution of total phosphorus is closely related to the energy metabolism demand driven by water and nitrogen. The differences in the proportions of phosphorus in roots and stems among different treatments suggest that there is a threshold effect of water status on the radial transport of phosphorus.

The stoichiometry of carbon (C), nitrogen (N), and phosphorus (P) (C:N:P) in leaves, stems, and roots reflects the trade-offs between plant resource acquisition and their growth strategies (Yin et al., 2021). The water-nitrogen interaction drives nutrient allocation and metabolic coordination among organs by regulating the stoichiometric characteristics of C, N, and P (Zheng et al., 2023). The water-nitrogen ratio in the W2N2 treatment can maximize metabolic efficiency and yield formation, providing a theoretical basis for precise water and nitrogen management in potato production. The carbon-nitrogen ratio in stems decreased by 5.25% to 39.96%, indicating that the synergistic effect of water and nitrogen promoted nitrogen enrichment in stems, which may be related to the developmental needs of their vascular tissues and the improved efficiency of water transport. The relatively high carbon-nitrogen ratio in roots reflects the enhanced carbon deposition in roots under water regulation, while the carbon-nitrogen ratio in tubers ranged from 77.07% to 96.79%, suggesting that moderate water restriction combined with high nitrogen supply can optimize the reallocation of carbon to tubers. The increased fluctuation in the leaf carbon-nitrogen ratio indicates a dynamic balance between photosynthetic carbon assimilation and nitrogen metabolism. Among all treatments, the leaf carbon-nitrogen ratio in the W3N3 treatment increased significantly, which may be associated with the promotion of chlorophyll synthesis by high nitrogen levels. The nitrogen-phosphorus ratio in tubers was positively correlated with yield (with a 64.71% yield increase in the W2N2 treatment), further illustrating the contribution of efficient nitrogen and phosphorus utilization to yield formation. The water-nitrogen interaction had the most significant impact on potatoes during the tuber formation stage. This period is a critical turning point for nutrient allocation to tubers, and the synergy of water and nitrogen directly affects the number and initial size of tubers by regulating the transport of photosynthates, the enlargement of stolons, and nutrient accumulation. Adequate water can improve nitrogen use efficiency, and reasonable nitrogen supply can enhance water absorption; an imbalance between the two will hinder tuber development, exerting the most prominent restrictive effect on yield formation (Ma et al., 2023). The nitrogen-phosphorus stoichiometric ratio (N:P) is a key indicator for assessing plant nutrient limitations. For potato leaves, an N:P ratio<14 indicates phosphorus limitation, >16 indicates nitrogen limitation, and a ratio between 14 and 16 indicates nitrogen-phosphorus balance or co-limitation (Güsewell, 2004). Zebarth et al. (2004) reported that potatoes absorb nitrogen rapidly during the seedling stage, and the leaf N:P ratio is often higher than 16, suggesting a high risk of nitrogen limitation. Phosphorus limitation is rare at the seedling stage unless the soil phosphorus level is extremely low. Bélanger et al. (2002) found that an increase in the leaf N:P ratio during the maturity stage is a normal phenomenon, but a sustained ratio higher than 16 may indicate excessive nitrogen fertilization.

In potato planting systems, water and nitrogen supply are crucial factors controlling production levels, especially in arid and semi-arid regions (Badr et al., 2012). Excessive irrigation and nitrogen application can have negative impacts on the yield and quality of potato tubers; overly high levels of irrigation and nitrogen are not conducive to yield formation. The mutual promotion of water and nitrogen on potato yield is consistent with the results of this study (Shrestha et al., 2023). Yield showed a trend of first increasing and then decreasing with the water and nitrogen gradients, revealing that there exists an optimal threshold for water-nitrogen interaction. The W2N2 treatment achieved an average yield increase of 65.08% over the two years. Optimizing the water and nitrogen regime balances the relationship between photosynthesis-stoichiometry synergy and yield formation. The water-nitrogen interaction (W2N2) can simultaneously realize physiological regulation and yield maximization (Wang et al., 2024), providing both theoretical and practical support for precise water and nitrogen management of potatoes in arid regions.