Hi Sibylle,
I can chime in a bit here. the half saturation constant can be used in two different ways - either to predict the abundance of bees visiting a crop (and then having a separate relationship between abundance and yield) or to use the yield equation directly with the pollinator visitation score itself.
In work I’ve collaborated on where bees have been sampled, we estimate the relationship bee between observed bee abundance and the pollination index. The half-saturation value indicates the pollinator index score that would lead to an estimate of half the maximum number of bees. In our past assessments, we’ve found that a value from 0.1 to 0.2 is a good estimate.
you could also look at the relationship between bee-contributed yield and the index directly. In this case, we’ve used the value of 0.1 again as an estimate.
with respect to Klein et al’s estimates: yes, this is separate. so if one were to connect pollinator-dependence, and the half saturation constant. A simpler way to see the function without subscripts used in the user guide is this:
Y = ( (1-v) + v * PA / (PA + hs) ) ; Y = relative yield; v = pollinator dependence (Klein); PA is pollinator abundance index; hs is the half-saturation constant. So for something like almonds with a 90% or more dependent on bees - an abundance index score of 0.1 would lead to an estimated relative yield of 0.55; while something more like coffee where only 20% of yield depends on external pollination, an abundance index of 0.1 would lead to yield of 0.90. the 2013 paper by ricketts and lonsdorf is perhaps a good example of this - Coffee in costa Rica. Hope this helps! - eric