Integrating the production functions of Liebig, Michaelis-Menten, Mitscherlich and Liebscher into one system dynamics model

Any agricultural production process is characterized by input—output relations. In this paper we show that the production functions of Liebig, Mitscherlich and Liebscher for the relation between nutrient supply and crop production can be regarded as special variants of one ‘integrated model’. The model is elaborated for two nutrients, nitrogen and phosphorus, and is based on the Michaelis-Menten hyperbolic equation. This basic equation has two main terms and one multiplicative interaction term. The parameter values determine which one of the aforementioned functions is generated. ‘Greenwood's variant of the Michaelis-Menten function’ is approached if the main terms dominate. ‘De Wit's variant of the Mitscherlich function’ is approached if the multiplicative term dominates. Liebig's function emerges from any of these variants if nutrient supply is constrained to such an extent that nutrient uptake continually exhausts the nutrient stock. The ‘Liebscher variant’ — considered the most appropriate for most empirical situations — is intermediate between those of Liebig, Michaelis-Menten and ‘De Wit's Mitscherlich’, and can be obtained by parameter calibration. Generally, these functions result in ‘decreasing marginal returns’ with increasing nutrient supply. However, if interacting nutrients are supplied in precisely the required proportion, the variant with a multiplicative term does show ‘increasing marginal returns’, but only in conditions of low nutrient supply rates, low nutrient affinities and / or high nutrient losses.