(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)?
a) To find the optimal time for the bee to leave one flower for the next in order to collect nectar at the maximum rate, we need to maximize the rate of nectar collection. The rate of nectar collection is given by the derivative of the nectar function F(t).
F'(t) = (d/dt)(t/(1+t))
= (1+t) - t/(1+t)^2
To find the optimal time, we need to find when F'(t) = 0. So we set (1+t) - t/(1+t)^2 = 0 and solve for t.
(1+t) - t/(1+t)^2 = 0
(1+t)(1+t)^2 - t = 0
(1+t)^3 - t(1+t) = 0
(1+t)^3 - t - t(1+t) = 0
(1+t)^3 - 2t - t = 0
(1+t)^3 - 3t = 0
Unfortunately, this equation does not have an easily solvable analytical solution. So we would need to use numerical methods or a computer/calculator to solve for t. The value of t that solves this equation will give us the optimal time for the bee to leave one flower for the next to maximize nectar collection.
b) Similarly, to find the optimal time for the bee to leave one flower for the next in order to collect pollen at the maximum rate, we need to maximize the rate of pollen collection. The rate of pollen collection is given by the derivative of the pollen function G(t).
G'(t) = (d/dt)(t/(2+t))
= (2+t) - t/(2+t)^2
Setting G'(t) = 0 and solving for t will give us the optimal time for the bee to leave one flower for the next to maximize pollen collection.
The two times for optimal nectar and pollen collection may be different because the rate functions F(t) and G(t) have different characteristics and dependencies on time.
c) If the bee values pollen twice as much as nectar, we need to find a function V(t) that gives the overall value of the resources collected by time t. Since the bee values pollen twice as much as nectar, we can assign a value of 1 to nectar and a value of 2 to pollen.
V(t) = 1 * F(t) + 2 * G(t)
By substituting the expressions for F(t) and G(t) into V(t), we can simplify and find the function V(t) that gives the value of resources collected by time t. Then we can find the optimal time for the bee to leave by setting the derivative of V(t) equal to 0 and solving for t.
d) If the bee first collects nectar for 1.0 second and then switches to pollen, we already know the time spent on nectar is fixed at 1.0 second. The remaining time available for pollen collection is t-1. Now we can maximize the rate of pollen collection by maximizing the derivative of the pollen function G(t-1). We can set G'(t-1) = 0 and solve for t-1 to find the optimal time for the bee to spend on pollen.
e) If the travel time between flowers increases to τ = 2.5 seconds, it will impact the answers in (b), (c), and (d). Since the travel time is longer, the overall time available for collecting resources will be reduced. This will affect the optimal time for the bee to leave for both nectar and pollen collection, as well as the optimal time for the bee to spend on pollen given a fixed time for nectar collection. The increased travel time will likely lead to different optimal solutions in these scenarios.