BIO 5910: Homework Assignment 4
Due on September 24, 2013
1. Suppose a snail starts crawling
across the sidewalk at 8:30 a.m. The velocity of the snail t
√
minutes after it starts is t centimeters per minute (it is speeding up to beat the heat).
a. Draw a graph of the speed of the snail as a function of time.
b. Write a differential equation for the position of the snail as a function of time (use initial
condition p(0) = 0).
c. What kind of differential equation is this?
d. Use integration to find the solution.
e. Draw a graph of the position of the snail as a function of time.
f. If the sidewalk is 0.5 meter wide, how long will it take this snail to cross? How fast is it
going when it reaches the other side?
g. Extra Credit: The snail has but 1 cm left to travel to reach the grassy refuge (it has
traveled 49 cm) when, out of nowhere, a cat lunges after the snail at a rate of 20 mph
(about 50000 cm/min) starting from 1 meter away! Will the snail escape into the grass
before the cat reaches it?
2. Suppose the population size of some species of organism follows the model
dN
3N 2
=
−N
dt
2 + N2
where N is measured in hundreds.
a. Find the equilibria.
b. Draw the phase-line diagram.
c. Which of the equilibria are stable, according to the stability theorem?
d. Interpret your diagram in biological terms. Why might this population behave as it does
at small values?