(Bees collecting pollen and nectar - Optimization)
Many bees collect both pollen and nectar. Pollen is used for protein, and nectar is
used for energy. Suppose the amount of nectar harvested during t seconds on a
flower is: F(t)=t/(1+t)
and that the amount of pollen harvested during t seconds on a flower is:
G(t)=t/(2+t)
.
The bee collects pollen and nectar simultaneously. Travel time between flowers is
τ = 1.0 seconds.
a) What is the optimal time for the bee to leave one flower for the next in order to
collect nectar at the maximum rate?
b) What is the optimal time for the bee to leave one flower for the next in order to
collect pollen at the maximum rate? Why are the two times different?
c) Suppose that the bee values pollen twice as much as nectar. Find a single
function V(t) that give the value of resources collected by time t. What is the
optimal time for the bee to leave? [You may need to use a computer or
calculator to solve this equation.]
d) Suppose that the bee first collects nectar and then switches to pollen. Assume it
spends 1.0 second collecting nectar. How long should it spend on pollen?
e) Suppose that the flower patch is not as populated and it takes τ = 2.5 seconds to
travel from one flower to the next. How does this impact your answers in (b),
(c), and (d)?