The Marginal Value Theorem: Foraging for Food
or
Do Bees Know Calculus?
The Marginal Value Theorem can be applied to many different situations, including
economics. For this problem we encourage students to solve the given scenario involving
Foraging Bees to form a concrete mathematical understanding of the concept. They should then
find alternate real-world applications in business.
A simple way to describe the concept is to consider foraging animals. They have a basic
problem to solve, when their food supply is in patches (acorns under trees, flowers growing in
patches, worms in apples, etc). Food is easy to collect when animals first begin searching a new
area. As the animal continues feeding, food becomes more difficult to find and it has to search
longer and longer for each additional morsel. This suggests that the curve relating the time spent
foraging in the patch and the amount of food found (or energy gain) starts off with a steep slope
but then gradually levels off. Food is easy to find in another patch, but it takes time to find and
travel to the next patch, and the animal gains no food while it is traveling.
The choice about how long to stop for a feeding is one of the basic decisions for an
organism that is searching for resources among widely scattered patches. What is the optimal
“giving up time” (when an organism should leave a patch that it is exploiting). When should the
animal say enough is enough and move on to find the next patch? One crucial parameter that
governs this decision is the travel time between patches. Since an animal gains no energy while
traveling to a new feeding ground, it wants to delay leaving its present patch. But the longer the
animal stays in the patch, the harder it is to find food. Once it reaches a new patch, it will then
be easy to find food again. This is the classic max-min dynamic.
There are many other situations similar to foraging that apply:
1. Guppies are a small tropic fish found in freshwater streams in Trinidad. The males
court females and the female chooses whether or not to mate with male. The longer a male
courts a female, the greater his chance is of mating with her. The males are polygamous (they
may mate with many females) but they must spend time searching for each female. How do they
decide when to abandon the courtship and try another female?
2. Students preparing for the AP Calculus problem cram for the exam in the last couple
of days before the test but how should they utilize their time? If they decide to study for 24
consecutive hours, they will gain knowledge over the entire time but at a less efficient rate as
they become tired. Therefore at some point it may be advantageous to stop and nap. Clearly
there will be missing out on the opportunity to gain knowledge while they sleep but when they
wake they are refreshed and ready to begin studying at a higher rate. How many consecutive
hours should a student study before deciding to take a nap?
3. Traveling circuses are received with a lot of excitement (and sales!) when they arrive
to a new town but the interest level drops the longer they stay. The circus has a tough decision to
make. Should they continue to perform in a town with dropping sales or should they pack up
and travel to a new town where sales will again be initially high? Remember that they bring in
no income when they are packing up and traveling.
1
In this investigation, we will consider the bee foraging for pollen on flowers that are
dispersed across a yard. It gathers pollen rapidly when it first arrives at a new plant, but then
finds it more difficult to pick up additional pollen. At some point, the bee must decide when to
leave for “greener pastures”. Once you have solved the Bee Problem we encourage you to think
of a scenario in human economics that operates under the same principles and write a paragraph
describing how/why the marginal value theorem applies. Enjoy!
Bees Foraging for Food or Do Bees Know Calculus?
Foraging animals have a basic problem to solve, when their food supply is in patches
(acorns under trees, flowers growing in patches, worms in apples, etc). Food is easy to collect
when animals first begin searching a new area. As the animal continues feeding, food becomes
more difficult to find and it has to search longer and longer for each additional morsel. This
suggests that the curve relating the time spent foraging in the patch and the amount of food found
(or energy gain) starts off with a steep slope but then gradually levels off. Food is easy to find in
another patch, but it takes time to find and travel to the next patch, and the animal gains no food
while it is traveling.
The choice about how long to stop for a feeding is one of the basic decisions for an
organism that is searching for resources among widely scattered patches. What is the optimal
“giving up time” (when an organism should leave a patch that it is exploiting). When should the
animal say enough is enough and move on to find the next patch? One crucial parameter that
governs this decision is the travel time between patches. An animal gains no energy while
traveling.
Other situations are quite similar. Guppies are a small tropic fish found in freshwater
streams in Trinidad. The males court females and the female chooses whether or not to mate
with male. The longer a male courts a female, the greater his chance is of mating with her. The
males are polygamous (they may mate with many females) but they must spend time searching
for each female. How do they decide when to abandon the courtship and try another female?
In this activity, we will consider a bee
foraging for pollen on flowers that are
dispersed across a yard. It gathers pollen
rapidly when it first arrives at a new plant, but
then finds it more difficult to pick up
additional pollen. At some point, the bee must
decide when to leave for “greener pastures”.
Suppose the travel time between
flowers is 5 seconds and the graph of the
pollen collection function looks like the one
shown in Figure 1 at right, where time is
measured in seconds and the amount of pollen
is measured in micrograms.
Figure 1
Figure 2 below shows the results of two scenarios: one with a foraging time of 5 seconds
and one with a foraging time of 15 seconds where the travel time in each is 5 seconds.
2
Figure 2 illustrates the difference in total pollen
accumulation in a 40 second period if the bee
forages for 5 seconds before moving on and
foraging for 15 seconds before moving on.
Notice the total amount accumulated in a 40
second period is greater for 5 second foraging
(about 76  g ) than for 15 second foraging
(about 58  g ).
In this investigation, we want to determine the
length of foraging time that gives the largest total
pollen accumulation in a fixed period of time.
Figure 2: Comparing 5 and 15 second foraging times
The Effect of Travel Time and Rate of Collection
First, let’s check our intuition. Upon what does the optimal foraging time depend? For
each of the figures below, describe how the foraging time might depend on the time for travel
between patches, T, and the shape of the function describing the energy gains over time.
1)
For the two pollen curves shown below in which the travel times differ, would you expect
the bee to spend longer or shorter periods of time on a flower if the travel time is longer?
2)
For the two pollen curves shown below in which the travel times are the same, for which
Pollen curve would you expect the bee to spend the longer time foraging?
3
3)
As a first approximation, consider the pollen collection graph shown below. Use the
graph to compare foraging times of 2 seconds, 4 seconds, and 8 seconds. Note there is a fullpage copy of the graph provided at the end of the investigation handout.
4)
In comparing the foraging times in (3, which time did you find to be the best foraging
time, i.e. the time that produced the maximum accumulation of pollen? What aspect of the graph
indicates that the optimal time has been found? Based on this example, can you describe when it
appears to be to the bee’s advantage to leave a flower looking for easier gathering? After you
answer this question, talk to your teacher before you proceed to the next part of the investigation.
 0 if t  T

5)
Consider the collection function C  t   
.
A
t

T
if
t

T


What affect do the values of the constants A and T have on the location of the best foraging time?
Vary the values of A and then the values of T to explore this question.
6)
Use the results from (4 to find the optimal foraging location for each of the following
collection functions.
 0 if t  5
 0 if t  T


a) C  t   
b)
C
t



3
0.2 t 5
if t  5.


30  30e
 t  T if t  T .
Use Calculus to verify that you have found the optimal solution in each example. After you have
completed this portion of the investigation, ask your teacher for #7.
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Bees Foraging Part II
Ask for this only after you have completed #1 – 6.
7)
a) If we want to accumulate the maximum amount of pollen in a fixed period of time, we
can accomplish this by maximizing the rate at which we accumulate pollen, that is, maximize
C t 
R t  
. We must remember that our time t includes the traveling time T from patch to
t
patch. You may have used this method to find the optimal foraging times for the two collection
curves in 6 a) and b). If you did NOT use this method, is this method equivalent to your
method? Support your answer using algebraic methods.
b) The result you found in 4) is known as the Marginal Value Theorem and says that the optimal
foraging time is found when the instantaneous rate of accumulation is equal to the average rate of
accumulation. This point on the graph is sometimes called the “knee” of the graph. Compare
this theorem to the Mean Value Theorem. Will there always be a point at which the theorem is
satisfied?
c) Use the Marginal Value Theorem to find the optimal foraging time for each of the collection
function below. What does each of the parameters A, T, or k represent in the function? Which
parameter has the greatest effect on the foraging time? Support your answers using Calculus.
i)
0 if t  T


C t    k

 A t  T if t  T .
iii)
0 if t  T


C t   
 k  t T 
if t  T .

 A  Ae
0 if t  T


ii) C  t    A  t  T 
if t  T .

 t T  k
Note: You will need technology to find the optimal foraging time for this collection function in (iii)
As you answer the questions, notice that the parameters affect the graphs of these pollen
collection functions in specific ways. Go back and look at the graphs in (1 and (2. Was your
intuition correct in (1 and (2?
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