Equations for describing sigmoid yield responses and their application to some phosphate responses by lupins and by subterranean clover

Responses to nutrients are sometimes sigmoid. A series of equations is proposed to describe such curves and to test whether the sigmoid component is significant. These equations are then applied to responses to freshly applied, and to incubated, phosphate by three species of lupin and by subterranean clover. The responses by Lupinus angustifolius, and especially by subterranean clover, were sigmoid on a log scale; the response by L. luteus was not significantly sigmoid and the sigmoid component of the response by L. cosentinii was small and only barely significant. The sigmoid response by subterranean clover occured even though it was strongly infected with vesicular arbuscular mycorrhiza. Including the sigmoid component enabled a closer description of the response and therefore a more precise estimation of the effectiveness of the incubated phosphate relative to that of fresh phosphate. This produced increased confidence in the conclusion that there was no statistically significant differences (P < 0.01) in the relative effectiveness of incubated and fresh phosphate amongst the four species of legume.