# A New Pabs Model for Quantitatively Diagnosing Phosphorus Nutritional Status in Corn Plants
## 1 Introduction  
Phosphorus is essential for regulating plant physiological processes and enhancing tolerance to abiotic stresses, such as heat, salinity, drought, waterlogging, elevated CO\({}_{2}\), and heavy metal toxicity [1, 2]. Plants are capable of sensing changes in phosphorus availability and responding through specific signaling pathways, adjustments in root architecture, and modifications in stomatal morphology [3, 4]. A deeper understanding of how plants perceive and respond to phosphorus availability is vital for developing strategies to improve crop yield and stress resilience. Phosphorus also plays a critical role in key metabolic and physiological processes, such as energy metabolism, ATP, DNA synthesis, cell division, and phospholipid biosynthesis, primarily in the form of phosphate (Pi) or Pi esters [5]. A deficiency in Pi severely impacts root development, vegetative growth, fruit production,and quality, ultimately leading to a decrease in crop yields [6; 7; 8]. Therefore, precision in phosphorus fertilizer application and effective management strategies are paramount for optimizing production. Plant phosphorus concentration (P) is a widely recognized indicator of crop phosphorus nutritional status. To enhance the precision of phosphorus diagnostics, the critical phosphorus concentration (Pc) concept was introduced, representing the minimum phosphorus level required to achieve maximum aboveground biomass (Bio). Pc decreases as biomass increases, and their relationship is described by a negative power function, forming the critical P dilution curve [9; 10]. This curve allows for the determination of Pc at any given biomass. Researchers worldwide have developed critical phosphorus concentration dilution curve models for various crops in different regions, climate conditions, and environments. For example, critical phosphorus dilution models have been established for potato (_Solanum tuberosum_ L.) [9; 11], cotton (_Gossypium hirsutum_ L.) [12], and Timothy grass (_Phleum pratense_ L.) [13], with corresponding models as follows: Pc = 3.92 Bio\({}^{-0.50}\) and Pc = 5.23 Bio\({}^{-0.19}\), Pc = 7.84 Bio\({}^{-0.22}\) and Pc = 7.74 Bio\({}^{-0.20}\), and Pc = 5.23 Bio\({}^{-0.40}\) and Pc = 3.27 Bio\({}^{-0.20}\), respectively. The comparison of P to Pc, expressed as the phosphorus nutritional index (PNI), provides a robust assessment of phosphorus supply status. However, PNI calculations traditionally involve destructive sampling and chemical analysis, which are time-intensive, costly, and impractical for large-scale applications, akin to the nitrogen nutrition index (NNI) [14]. In nitrogen nutrition research, combining vegetation indices (VIs) with NNI has emerged as an efficient, non-destructive diagnostic approach. This method leverages remote sensing technologies to assess nitrogen content rapidly and accurately, facilitating effective management in large-scale agricultural systems, such as rice (_Oryza sativa_ L.), wheat (_Triticum aestivum_ L.), and maize (_Zea mays_ L.). For example, GreenSeeker NDVI and RVI explained 47% and 44% of the variability in winter wheat NNI across different sites and growth stages, respectively, demonstrating their effectiveness in nitrogen monitoring [15]. Additionally, using two vegetation indices--Green Re-normalized Difference Vegetation Index (GRDVI) and Modified Green Soil Adjusted Vegetation Index (MGSAVI)--was effective in estimating winter wheat NNI across various site years and growth stages, achieving an R\({}^{2}\) value between 0.77 and 0.78 [16]. For rice, the GreenSeeker system explained 25-34% of NNI variability at the stem elongation stage and 30-31% at the heading stage, indicating the relevance of growth-stage-specific nitrogen assessments [16].