frP=(frP,1−frP,3)∗[1−frPHU+exp(p1−p2∗frPHU)frPHU]+frP,3 5:2.3.19
where frP is the fraction of phosphorus in the plant biomass on a given day, frP,1 is the normal fraction of phosphorus in the plant biomass at emergence, frP,3 is the normal fraction of phosphorus in the plant biomass at maturity, frPHU is the fraction of potential heat units accumulated for the plant on a given day in the growing season, and p1 and p2 are shape coefficients.
The shape coefficients are calculated by solving equation 5:2.3.19 using two known points (frP,2,frPHU,50%) and (frP,3,frPHU,100%):
p1=1n[(1−frP,1−frP,3)(frP,2−frP,3))frPHU,50%−frPHU,50%]+p2∗frPHU,50% 5:2.3.20
p2=frPHU,100%−frPHU,50%(1n[(1−(frP,1−frP,3)(frP,2−frP,3))frPHU,50%−frPHU,50%]−1n[(1−(frP,1−frP,3)(frP,∼3−frP,3))frPHU,100%−frPHU,100%]) 5:2.3.21
where p1 is the first shape coefficient, p2 is the second shape coefficient, frP,1 is the normal fraction of phosphorus in the plant biomass at emergence, frP,2 is the normal fraction of phosphorus in the plant biomass at 50% maturity, frP,3 is the normal fraction of phosphorus in the plant biomass at maturity, frP,∼3 is the normal fraction of phosphorus in the plant biomass near maturity, frPHU,50% is the fraction of potential heat units accumulated for the plant at 50% maturity (frPHU,50%=0.5), and frPHU,100% is the fraction of potential heat units accumulated for the plant at maturity (frPHU,100%=1.0). The normal fraction of phosphorus in the plant biomass near maturity (frN,∼3) is used in equation 5:2.3.21 to ensure that the denominator term (1−(frP,1−frP,3)(frP,∼3−frP,3))does not equal 1. The model assumes (frP,∼3−frP,3)=0.00001
where bioP,opt is the optimal mass of phosphorus stored in plant material for the current growth stage (kg P/ha), frP is the optimal fraction of phophorus in the plant biomass for the current growth stage, and bio is the total plant biomass on a given day (kg ha−1).
Pup=1.5∗Min{bioP,opt−bioP4∗frP,3∗Δbio 5:2.3.23
where Pup is the potential phosphorus uptake (kg P/ha), bioP,opt is the optimal mass of phosphorus stored in plant material for the current growth stage (kg P/ha), and bioP is the actual mass of phosphorus stored in plant material (kg P/ha), frP,3 is the normal fraction of phosphorus in the plant biomass at maturity, and Δbio is the potential increase in total plant biomass on a given day (kg/ha). The difference between the phosphorus content of the plant biomass expected for the plant’s growth stage and the actual phosphorus content is multiplied by 1.5 to simulate luxury phosphorus uptake.
where Pup,z is the potential phosphorus uptake from the soil surface to depth z (kg P/ha), Pup is the potential phosphorus uptake (kg P/ha), βP is the phosphorus uptake distribution parameter,z is the depth from the soil surface (mm), and zroot is the depth of root development in the soil (mm). The potential phosphorus uptake for a soil layer is calculated by solving equation 5:2.3.24 for the depth at the upper and lower boundaries of the soil layer and taking the difference.
Pup,ly=Pup,zl−Pup,zu 5:2.3.25
where Pup,ly is the potential phosphorus uptake for layer ly (kg P/ha), Pup,zl is the potential phosphorus uptake from the soil surface to the lower boundary of the soil layer (kg P/ha), and Pup,zu is the potential phosphorus uptake from the soil surface to the upper boundary of the soil layer (kg P/ha).
Root density is greatest near the surface, and phosphorus uptake in the upper portion of the soil will be greater than in the lower portion. The depth distribution of phosphorus uptake is controlled by βp, the phosphorus uptake distribution parameter, a variable users are allowed to adjust. The illustration of nitrogen uptake as a function of depth for four different uptake distribution parameter values in Figure 5:2-4 is valid for phosphorus uptake as well.
Pactualup,ly=min⌊Pup,ly+Pdemand,Psolution,ly⌋ 5:2.3.26
where Pactualup,ly is the actual phosphorus uptake for layer ly (kg P/ha), Pup,ly is the potential phosphorus uptake for layer ly (kg P/ha), Pdemand is the phosphorus uptake demand not met by overlying soil layers (kg P/ha), and Psolution,ly is the phosphorus content of the soil solution in layer ly (kg P/ha).