BIO 5910: Homework Assignment 1
Due on September 3
1. Let’s work out a specific case of the power-velocity problem. Suppose that power as a
function of velocity is P (v) = c + (v − 5)2 where c is a constant that depends on the
size of the bird.
a. What velocity minimizes power use?
b. What velocity minimizes energy used to fly a distance d if c = 1 for a small bird?
c. What velocity minimizes energy used to fly a distance d if c = 25 for a large bird?
d. Can you make sense of the difference between the answers for small and large
birds?
2. We can include energy costs of travel and harvesting into the framework for the
marginal value theorem. That is, an animal has gained F (t) units of energy after
harvesting in a patch for time t, but it uses energy at rate c1 while collecting food and
at rate c2 while traveling. If you get stuck trying to do everything in general, feel free
to make up a convenient increasing function for F (t) and to pick simple values for c1
and c2 .
a. Find the net energy collected (subtracting the energy used) after being in a patch
for time t and traveling for time T .
b. Find the net rate at which the animal gains energy.
c. Show that the optimal time to remain in the patch if c1 = c2 is the same as in
the model from class where we ignored energy costs. Why is this?
d. Do you think the optimal time to remain in the patch would be longer time or
shorter if c1 > c2 ?
e. Extra credit: Prove it.
f. The efficiency is the energy taken in divided by the energy used. Find an expression for the time to remain in the patch that maximizes efficiency.
g. More extra credit: Does this give the same answer as maximizing the rate of
energy intake?