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

Abstract

A generalized cumulative uptake formula of nutrient uptake by roots following our previous formula (Reginato-Tarzia, Comm. Soil Sci. and Plant., 33 (2002), 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) 961–964) and Cushman (Cushman, Soil Sci. Soc., 43 (1979) 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) 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.