Lab 11 – Marginal Value Theorem and Maximizing Nectar Intake
Date: November 8, 2011
Assignment Due Date: November 15, 2011
Goal: Today we will use the Marginal Value Theorem to consider maximum nectar intake of bees in
different circumstances. Using R, we will explore changes in parameter values to optimize strategies for
bees.
Maximizing Nectar Intake
Suppose a bee is out collecting nectar from flowers. Once the bee lands on a flower the amount of
nectar it is able to consume is a function of the amount of time it spends on that flower. However, as
the bee remains on that flower the amount of available nectar decreases which decreases the rate at
which the bee can collect nectar. Therefore, it may not be the most efficient to consume all of the
nectar on a single flower before moving on to a new flower. Flying takes time and energy though. We
will consider the following model for nectar intake and discuss how the bee can best maximize nectar
intake.
Rate at which nectar is collected:
Food per visit:
Cost of flying:
Use R to plot F(t). Why does the shape of F(t) make sense?
In order to maximize the rate of nectar intake, the bee needs to know how long to spend on each
flower. We now know that we can find where minimums and maximums occur by using the first
derivative of R(t). As we have discussed in lab and the lecture, we need to find the critical points,
specifically where the first derivative is zero. By differentiating the function R(t) and considering where
R’(t) is zero, we are able to derive the Marginal Value Theorem. This is shown in detail in the book on
page 272 for the case where there is no cost associated with flying.
Differentiate R(t) and set it equal to zero in order to obtain the following result of the Marginal Value
Theorem for this model in which we consider the cost of flying.
This theorem is useful as it allows us to differentiate F(t) instead of R(t) which is a simpler function.
Using the F(t) and k(τ) functions specified above solve by hand for the value of t which will maximize
R(t).
You should obtain the following result for the value of t which maximizes R(t).
Now write code in R to do the following:
 Define the parameters tau, w, and b. (Begin with tau = 1, w = 0, and b = .5 which makes this the
model used in the book without cost of flying.)
 Define the functions F(t) and R(t).
 Plot the functions F(t) and R(t) side by side on the same figure.
 Define tmax as shown above and print this value to the console.
 Put a point on your R(t) plot at (tmax,R(tmax))
Comparing Species of Bees
Now that we have our code working we will compare two species of bees: Bombus terrestris (BIG bees)
and Bombus pascuorum (little bees). Suppose these two species cohabitate the same meadow full of
flowers. The BIG bees are able to quickly move from flower to flower but at a high cost. The little bees
are slower in moving between flowers but are able to fly with low cost.
What does this mean about the values of tau and w for each species? Which species has a larger value
of tau? Which species has a larger value of w?
In addition, the BIG bees are able to collect nectar more quickly from the flowers while the little bees
collect nectar more slowly. This means the function F(t) is different for the two species. The value of b
controls this. Larger values of b correspond with slower consumption of nectar.
For the BIG bees, set w = .3, tau = .5, and b = .5. For the little bees, set w = .05, tau = 2, and b = 2.
To maximize nectar intake, how long should the BIG bees stay on each flower? How long should the
little bees stay on each flower?
If both species are using their optimal strategy for collecting nectar, which species has a higher rate of
collecting nectar?