A Steady-State Model of Nutrient Uptake Accounting for Newly Grown Roots

A model of solute uptake that accepts root growth, water uptake, and soil solution concentration as time-varying input is required to interactively link plant and soil processes. The advantage of the steadystate approach to solute uptake over more exact numerical solutions lies in the independence of the mathematical solution to previous conditions. Uptake thus calculated can accommodate unpredictable changes in root growth and mortality, root density, water uptake rates, and sources and sinks of nutrients such as decomposition and leaching, as required in simulating plant growth for multiple seasons in a dynamic soil environment. Previous steady-state models were improved by including nonlinear uptake kinetics and the contribution of new root growth to uptake. The correction for new root growth is most important for relatively fast-growing plants and immobile nutrients. The importance of each model parameter, as indicated by sensitivity analysis, depends on the values of other parameters. For example, root surface area and uptake kinetics are important when solution concentrations at the root surface are high, while root length, water uptake rate, and diffusion become important when delivery of solute to the root surface is limiting. Because the limiting factors can vary with environmental and plant conditions, it is important to represent these aspects of nutrient uptake in modeling plant-soil interactions. A consistent derivation of the improvements and the original model is appended. M PLANT UPTAKE of dissolved soil constituents is essential to predicting plant growth under nutrient limitation. Solute uptake by plants can also be important in explaining changes in the chemistry of soil and drainage waters. Many existing models of forest growth, such as FORTNITE (Aber et al., 1978, 1982), FORCYTE (Kimmins and Scoullar, 1984), and FOREST-BGC (Running and Coughlan, 1988), and of soil chemistry at the ecosystem scale, such as the ILWAS model (Gherini et al., 1985; Davis et al., 1987), the "magic" model (Cosby et al., 1986), and STEADYQL (Furrer et al., 1989, 1990), do not include mechanistic representations of nutrient uptake. This omission occurs partly because solute uptake by plants is not entirely within the domain of either soil chemistry or plant growth models, but also because a suitable model has not been available. To be useful in long-term simulations of vegetated ecosystems, a nutrient uptake model should be capable of beginning the simulation with a fully established root system and it should allow root growth and mortality. To serve as a link between soil and plant simulators, it should accept time-varying inputs of soil solution chemistry, transpiration rates, and root dynamics during the simulation. None of the existing nutrient uptake models has all of these properties. Bouldin (1961) formulated the mathematics of diffusion of solutes through an infinite and stationary soil solution to a cylindrical sink, assuming that the rate of Boyce Thompson Inst. for Plant Research, Cornell Univ., Ithaca, NY 14853. Received 1 Sept. 1993. *Corresponding author (ruth.yanai® cornell.edu). Published in Soil Sci. Soc. Am. J. 58:1562-1571 (1994). uptake is proportional to the solution concentration at the root surface. Olsen et al. (1962) devised a similar model, exploring different assumptions about the boundary condition at the root surface, namely (i) constant rate of uptake and (ii) constant concentration at the root surface. These diffusion-only models were applied to explain phosphate movement and uptake. For more mobile ions, mass flow of the solution can also be an important mechanism of solute movement. Nye and Spiers (1964) presented the equations for simultaneous mass flow and diffusion, and solved them for the steady-state condition. To describe the solute uptake and concentration profile of the root with time required solving these same equations in the non-steady-state condition; Nye and Marriot (1969) and Claassen and Barber (1976) solved them numerically, allowing Michaelis-Menton uptake kinetics. Claassen and Barber (1976) further allowed for a distribution of root ages, assuming an exponentially growing root system. None of these models included interroot competition. Cushman (1979) and Barber and Cushman (1981) included interroot competition as an outer boundary condition of no nutrient movement, although the distance to this boundary was not affected by changing root density. The Barber-Cushman model has been reprogrammed for microcomputers and is easy to use (Gates and Barber, 1987). It is not, however, suitable for linking to a plant or soil simulator because, like its predecessor numerical models, it cannot accept time-varying input. The pattern of root growth (linear or exponential) must be specified in advance, and the rate of water influx and the average distance between root axes are parameters that cannot vary during the course of a simulation. Further, there can be no other sources or sinks for solute besides plant uptake, an untenable situation for long time scales. The steady-state approach originated by Nye and Spiers (1964) offers a solution to the problem of time-vary ing input. This approach assumes that the concentration profile around the root can be considered to be in a steady state; change with time is accommodated by recalculating the solution at each iteration of the model. The advantage of the steady-state approach lies in the independence of the mathematical solution to previous conditions. This makes it ideal for linking plant and soil simulation models, where feedback between plant and soil makes it impossible to specify changes in root growth and soil status in advance of running the models. By calculating solute uptake at each time step, changes in root density can dictate a changing radius for the zone of influence of a root, water uptake rates can be varied with time, and the amount of solute in the system can be changed at each iteration. The concentration profile around the root develops stepwise in a manner similar to that predicted by the more exact models (Baldwin et al., 1973). One weakness of the steady-state models to date has been the omission of the nutritional benefit incurred by new roots entering unexploited soil. Nye et al. (1975)