An intermediate-resolution analytical model of nitrogen, phosphorus, and potassium utilization in the plant-soil system was developed and tested. Starting from specified natural or artificial sources in the soil, element transport to root absorption surfaces was modeled in terms of diffusion, mass flow, and soil buffering mechanisms. Element uptake was described by carrier theory formalism, and assimilation was based on four premises about the roles of N, P, K, and photosynthate in cell chemistry. There were three main objectives of the model. The first was to predict the first-order interactive growth response of particular plant species to any combination of these macronutrients supplied in the soil medium. Species parameters required by the model include root absorption rate and certain cell chemistry reaction rates. The second objective was to make the model sufficiently general to describe a broad range of species. It was built upon common denominator principles of physiology condensed from available experimental data on corn (Zea mays L.), bean (Phaseolus vulgaris), pine (Pinus elliottii Var. elliottii), etc. In this generic sense it is a measure of what plants have in common. The third objective was to use the model to test several well-known theories of plant growth.
The model was validated against reported experiments on ryegrass (Lolium perenne L.), oat (Avena sativa), a legume (Stylosanthes humilis), and rutabaga (Brassica napobrassica, Mill.), in which dry matter yield was measured as a function of factorial application of N, P, and K to the soil. The model shows that much of the deficient, optimal, toxic, and interactive response of plants to N, P, and K can be explained in terms of strong linear response of cell chemistry to low nutrient concentrations and inhibition by N, P, and K at high nutrient concentrations. Applying the model in a test of plant response to suboptimal nutrient concentrations, the model strongly confirms the Liebig Law of the Minimum and refutes the opposing multi-limiting-element Baule Product Law. The model also shows that the Liebig theory of linear growth response to nutrient concentration and the opposing, nonlinear Mitscherlich Law of Diminishing Return are not necessarily in disagreement, but rather may apply to different parts of the nutrient concentration range. And the model confirms that the effects on growth of nitrogen, phosphorus, and potassium levels are more often additive on the reciprocal scale than on either the logarithmic or untransformed scales.