With these two assumptions about bee and plant populations, there is clearly a
tradeoff between bee yields and seed yields. Bee yields (purple line) reach a
peak when a relatively small bee population is introduced into the field. As
more bees are introduced, density dependent mortality causes the bee offspring
population to decline. The black line through the middle of the graph
indicates where the number of introduced bees equals the number of offspring.
Where the purple female offspring curve crosses the black line, the introduced
bee population results in a sustainable yield of offspring bees. In this
graph, the sustainable yield of bees is less than the maximum bee yield, and
seed yield (red line) is still increasing if more bees are introduced into the
field. Seed yields are maximized when a relatively large number of bees are
introduced into the field.
In summary, if there is density dependent bee mortality, then the optimum
introduced bee population for maximizing bee reproduction is lower than the
optimum bee population for maximizing seed yields.
Download a free
demo
copy of isee Player from isee Systems. You need
this software on your computer to run the program.

Right mouse click the hyperlink below and select "save target as..." :

With density dependent bee mortality

Without density dependent bee mortality

The figure above, and the figure on slide 4 were made by running the model multiple times, each time
increasing the number of introduced bees. Results at the end of the run
were copied from the table in the model to an excel file. Download the
excel file.