MATH 106D - CALCULUS II
WINTER 2005
QUIZ 10
NAME:
Show ALL your work CAREFULLY.
Rabbits were brought to a remote deserted island some years ago. The
growth of the rabbit population is governed by the following logistic di๏ฌer-
ential equation.
dR
dt =0.1R(1000 โR)
Here, R=R(t) denotes the population of rabbits at time t.
(i) What will the rabbit population be in the long run? (What is the
equilibrium solution?)
We know that the logistic equation has a non-trivial (stable)
equilibrium when R๎=0and dR
dt =0. It follows that R= 1000 is the
stable solution. Thus, in the long run, R(t)โ1000.
Now, suppose bobcats were later brought to this island and a population
model for the rabbits and for the bobcats is given by the following system.
dR
dt =R๎2โRโB
3๎
dB
dt =B๎1โRโB
2๎
where Band Rdenote the population of the bobcats and of the rabbits (at
time t) respectively.
(ii) Give the equations of the nullclines.
The nullclines are precisely the lines in the BโRplane where
the slope fields are horizontal or vertical. By setting dR
dt =0,we
obtain R=0and B=3(2โR). By setting dB
dt =0,weobtainB=0
and B=2(1โR). Thus, the nullclines are given by R=0,B =
0,B =3(2โR)and B=2(1โR).
Date: April 8, 2005.
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