Analysis and use of cumulative nutrient uptake formulas in plant nutrition and the temporal-weight-averaged influx

A generalized cumulative uptake formula of nutrient uptake by roots following our previous formula (Reginato-Tarzia, Comm. Soil Sci. and Plant., 33 (2002 Reginato, J. C., and D. A. Tarzia. 2002. An alternative formula to compute the nutrient uptake for roots. Communications in Soil Science and Plant Analysis 33 (5&;6):821–30.Taylor &; Francis Online], Web of Science ®] , Google Scholar]), 821-830) is developed. Cumulative nutrient uptake obtained by this formula is compared with the simulated results obtained by the Claassen and Barber (Claassen and Barber, Agronomy J., 68 (1976 Claassen, N., and S. A. Barber. 1976. Simulation model for nutrient uptake from soil by a growing plant root system. Agronomy Journal 68:961–64.Crossref], Web of Science ®] , Google Scholar]) 961–964) and Cushman (Cushman, Soil Sci. Soc., 43 (1979 Cushman, J. H. 1979. An analytical solution to solute transport near root surfaces for low initial concentrations: I. Equation development. Soil Science Society of America Journal 43:1087–90.Crossref], Web of Science ®] , Google Scholar]) 1087–1090) formulas. A mass balance is analyzed for the three formulas of cumulative nutrient uptake in order to decide which of them is correct. Moreover, the mass balance is also verified through a computational algorithm using data obtained from literature, and we compute the potassium (K) uptake for maize for low and high soil concentrations using the three mentioned formulas. The theoretical analysis shows that Claassen and Barber, and Cushman formulas do not verify, in general, the mass balance condition. The Claassen and Barber formula only verifies this condition when the influx is constant and root grows linearly. The Cushman formula verifies the mass balance when the influx is constant regardless of the law of root growth. Reginato and Tarzia formula always verifies the mass balance whatever be the representative functions for the influx and the law of root growth. Moreover, we propose a redefinition of the averaged influx from which the Williams formula (Williams, J. Scientific Res., 1 (1948 Williams, R. F. 1948. The effect of phosphorus supply on the rates of intake of phosphorus and nitrogen upon certain aspects of phosphorus metabolism in gramineous plants. Australian Journal of Scientific Research 1:333–61. Google Scholar]) 333–361) can be deduced. We remark that Williams formula is a consequence of our definition of temporal-weight-averaged influx for all root growth law expressions. Also, we present a comparison of influx and cumulative uptake of cadmium (Cd) with data extracted from literature. Cumulative uptake is obtained through the Barber–Cushman model and our moving boundary model by using the redefinition of averaged influx on root surface and the correct cumulative uptake formula presented in this paper.