Water–potassium interactions determine potato (Solanum tuberosum L.) yield and water productivity: Evidence from response surface modeling and PLS-PM structural equation analysis

In the arid sandy region of northern Shaanxi, China, precise water–potassium (K) coupling is crucial for improving potato (Solanum tuberosum L.) productivity and resource-use efficiency, but the optimal management thresholds and underlying physiological mechanisms under different hydrological years remain unclear. In this study, a two-year field experiment was conducted during 2022–2023 in the wind-blown sandy grassland area of northern Shaanxi, China. Three irrigation levels (W1: 60% ETc, W2: 80% ETc, W3: 100% ETc) and seven K application rates (K0–K6: 0–420 kg K ha−1) were imposed, and key indicators including canopy structure (leaf area index and Chlorophyll content), photosynthetic gas-exchange parameters (Pn, Tr), plant K nutritional status, dry matter accumulation, and yield components were systematically measured. Bivariate quadratic regression models and a Partial Least Squares Path Model (PLS-PM) were then used to analyze the quantitative thresholds and driving pathways of water–K coupling. The results showed that water–K interactions significantly regulated canopy structure, photosynthetic traits, and yield formation. In the wet year (2022), moderate irrigation combined with a high K rate (W2K5) achieved the highest tuber yield (53.37 t ha−1) and water productivity (WPc, 12.43 kg ha−1 mm−1), whereas in the dry year (2023), full irrigation combined with a moderate K rate (W3K4) performed best, with a yield of 45.88 t ha−1 and a WPc of 9.44 kg ha−1 mm−1. The bivariate quadratic regression models quantified the threshold-shift phenomenon of optimal management: to simultaneously maximize yield and WPc (>95% of their theoretical potential), the recommended irrigation range increased from 132.5 to 183.0 mm in the wet year to 259.9–282.3 mm in the dry year, while the optimal K application range narrowed from 222.9 to 420.0 kg ha−1 to 240.0–342.9 kg ha−1. The PLS-PM revealed that water and K inputs independently drove evapotranspiration (ET, β = 0.91) and plant K concentration (PKC, β = 0.95), respectively. These factors jointly enhanced canopy function (Leaf parameters), which in turn drove dry matter accumulation (β = 1.00) and yield formation (β = 0.97). Crucially, the positive effect of yield increase on WPc (β = 1.19) was stronger than the negative effect associated with increased ET (β = −0.85), confirming that a high-yield strategy can simultaneously achieve high efficiency. These findings provide a theoretical basis and quantitative operational window for adaptive water–K management in potato cultivation on sandy soils under climate variability.