Optimal control of organic matter applications

Abstract

Organic matter amendments appear to increase yield, but need to be sustained, as yield decreases when amendments cease. Here we mathematically devise optimal strategies for organic matter applications that take account of how quickly, in years of application, yields build up with amendments and how long these benefits persist. The empirical idea of a nutrient response curve is used and extended to include more than a single nutrient input as well as the effect of yield-enhancing factors such as organic matter that endure for more than one year. Nonlinear regression is used for the selection and the parameter identification for a reciprocal response curve working with a dataset from Rothamsted’s Woburn organic manuring long term experiment. Such a response curve is then treated analytically to develop economically optimum applications over a period of time. A simple static case is developed first and is shown to be equivalent to the well-known break-even ratio (BER) used in nitrogen fertiliser guidance by the Agricultural and Horticultural Development Board in the UK. The mathematical technique of optimal control is then employed to deduce dynamic strategies where the application of an amendment may change from year to year and for different time frames. Because this empirical modelling methodology can appear complex, we infer a rule-of-thumb for an equilibrium level of yield-enhancement rather like the equilibrium level of organic carbon that builds up over several years. This yield-enhancing power of organic matter is somewhat variable and probably does not persist in soil for as long as the organic matter from which it derives. It appears beneficial to apply amendments at a constant rate for much of the time-frame of interest but to begin with a large application to raise the fertility to the yield-enhancement equilibrium. After a transition year with reduced amendments, applications of organic matter are stopped for the final five years with the example amendment studied, farmyard manure. These conclusions depend on the persistence of the yield-enhancing power of organic matter in soil associated with the soil organic carbon kinetics.