Nonlinear Modelling Estimates Potassium Sufficiency Ranges for Quinoa ( Chenopodium quinoa Willd.) Under Salinity Stress

Salinity is a major environmental constraint limiting crop productivity, particularly in arid and semi‐arid regions. Quinoa ( Chenopodium quinoa Willd.), a facultative halophyte, demonstrates remarkable tolerance to salinity; however, its potassium (K) nutrition dynamics under salt stress remain poorly characterized. This study investigated the role of K in alleviating salinity‐induced stress and estimated model‐derived K sufficiency ranges associated with quinoa's physiological and agronomic performance under contrasting salinity conditions. Field experiments were conducted over two consecutive years (2022–2023) using a split‐plot design with three salinity levels (10, 80, and 160 mM NaCl) imposed by irrigation treatments and seven K concentrations (5–11 mM). A suite of physiological (photosynthesis (Pn), transpiration (E), stomatal conductance (Gs), chlorophyll a ( Chl a), ) and chlorophyll b ( Chl b )), biochemical (proline, malondialdehyde (MDA)), ionic traits, and seed yield (SY) and relative seed yield (RSY) were measured. Linear, linear‐plateau, tent‐shaped, and negative exponential decay (NED) regression models were applied to the trait data to characterize response patterns and determine key K thresholds. Model robustness was assessed using leave‐one‐out cross‐validation (LOOCV), and the resulting validation statistics were compared with the calibration metrics. For the majority of traits, the LOOCV statistics supported the calibration results, confirming the robustness and predictive reliability of the selected models. Across the evaluated traits, the model‐derived responses generally supported an approximate K sufficiency range of ~8–10 mM under the experimental conditions. Physiologically, increasing K improved quinoa's photosynthetic performance, gas exchange, and chlorophyll content, with response patterns and K thresholds varying among traits and salinity levels. Under mild salinity, several physiological traits exhibited linear‐plateau responses with thresholds of approximately 8.4–9.2 mM K, whereas under moderate salinity, thresholds generally increased to approximately 9.4–10.7 mM K, and some traits, such as stomatal conductance, responded linearly across the tested K range. Increasing K also improved ionic homeostasis by increasing tissue K + and suppressing Na + accumulation. The decline in leaf and shoot Na + concentrations with increasing K supply was well described by negative exponential models, indicating that K supply was strongly associated with reduced tissue Na + accumulation and improved ionic balance. Both SY and RSY exhibited tent‐shaped responses under severe salinity: they peaked at an optimal K (~9–10 mM) and then declined at ~11 mM, indicating a supra‐optimal K threshold beyond sufficiency. The corresponding model‐derived RSY point estimates were 8.8, 9.8, and 9.2 mM K under 10, 80, and 160 mM NaCl, respectively; because confidence intervals were not available for these fitted optima, the values should be interpreted collectively as indicating an approximate 9–10 mM response range rather than as precisely distinct physiological thresholds. Most nonlinear models showed good predictive performance within the experimental dataset and captured the observed plateau and peak response patterns, thereby providing quantitative working estimates of K sufficiency and supra‐optimal response ranges under the salinity conditions evaluated. Moderate K enrichment within an approximate range of ~8–10 mM was associated with improved quinoa performance under the salinity conditions evaluated, whereas further K generally provided little additional benefit or was associated with declining responses in some traits. These model‐derived ranges represent experimentally derived working estimates that require independent validation across cultivars, environments, and management conditions before broader agronomic application.