Analyzing soil soluble phosphorus transport with root-phosphorus-uptake applying an inverse method

Soil soluble phosphorus (P) transport with root-phosphorus-uptake (RPU) is a critical process for plant growth, cycling of P in soil-plant systems and environment protection. However, modeling soil soluble P transport is extremely challenging because it is difficult to measure the RPU distribution directly, especially in the field. In this study, an inverse method, which was utilized successfully to estimate the root-water-uptake (RWU) rate distribution by Zuo and Zhang (2002) and the source-sink term in the nitrate (NO⁻₃-N) transport equation by Shi et al. (2007), was applied to estimate the RPU rate distribution and analyze soil soluble P transport in the soil-plant systems. A soil column experiment (Exp. 1) and a field experiment (Exp. 2), respectively with winter wheat (Triticum aestivum L.) and summer maize (Zea mays L.) growth, were carried out to observe the dynamics of soil water and soluble P. Based on the experimental data in Exp. 1, the average RWU and RPU rate distributions during different irrigation periods were estimated using the inverse method. The relative errors of the total P extracted by wheat between the estimated and measured values during all periods were less than 10%. The estimated RPU rate distribution during the period of 10.5-15.5 days after planting (DAP) was used to optimize the dimensionless RPU factor δ to establish the RPU model (δ = 1.31), which helped to calculate the RPU rate distributions during other periods (from 16.5 to 57.5 DAP) in Exp. 1. The calculated RPU rate distributions were compared well with the estimated profiles by the inverse method, and the root mean squared error between them was less than 0.00005 mg cm⁻³ d⁻¹. Correspondingly, the calculated total P extracted by winter wheat was also comparable with the measured value, with the relative error less than 10%. Similarly, the procedures were employed for summer maize in Exp. 2. The estimated (using the inverse method) and calculated (through the RPU model with δ = 1.38) RPU rate distributions were in good agreement with the root mean squared error as less as 0.000031 mg cm⁻³ d⁻¹. According to the established RPU models (δ = 1.31 and 1.38 for Exps. 1 and 2, respectively), the distributions of soil water content and soluble P concentration were simulated, and compared well with the measured profiles, with the maximum root mean squared error of 0.0088 cm³ cm⁻³ and 0.0066 mg cm⁻³ in Exp. 1, and 0.023 cm³ cm⁻³ and 0.0015 mg cm⁻³ in Exp. 2, respectively. The inverse method should be effective and applicable for estimating the RPU rate distribution, establishing the RPU model and analyzing soil soluble P transport in soil-plant systems, either in laboratory or in the field.