North Carolina State University
Department of Mathematics
An Introduction to Applied Mathematics
MA 325
Assignment #1
DUE Date: April 3, 2017
1. A family of salmon ﬁsh living oﬀ the Alaskan Coast obeys the Malthusian law of population
growth
dp
(
t
)
/dt
= 0
.
003
p
(
t
), where
t
is measured in minutes. It is also assumed that at time
t
= 0 there are one million salmon.
(a) Using
SimBiology
, simulate the salmon population,
p
(
t
).
(b) Now, assuming that at time
t
= 0 a group of sharks establishes residence in these waters
and begins attacking the salmon. The rate at which salmon are killed by the sharks is
0
.
001
p
2
(
t
), where
p
(
t
) is the population of salmon at time
t
. Moreover, since an undesirable
element (the sharks) has moved into their neighborhood, 0
.
002 salmon per minute leave
the Alaska waters.
i. Modify the Malthusian law of population growth to take these two factors into ac
count. Using
SimBiology
to simulate the salmon population,
p
(
t
). What happens as
t
→ ∞
?
ii. Show that the above model is not realistic. Hint: Show, according to this model, the
salmon population decreases from one million to about one thousand very quickly (in
about one minute).
2. The population of New York City would satisfy the logistic law
dp
dt
=
1
25
p

1
25
×
10
6
p
2
,
where
t
is measured in years, if we neglected the high emigration and homicide rates. Assume
that population of New York City was 8
,
000
,
000 in 1970.
(a) Using
SimBiology
, ﬁnd the population,
p
(
t
). What happens as
t
→ ∞
?
(b) Modify the logistic equation to take into account the fact that 9
,
000 people per year
move from the city, and 1
,
000 people per year are murdered. Using
SimBiology
, ﬁnd the
population,
p
(
t
). What happens as
t
→ ∞
?
3. Consider a predatorprey model of foxes and rabbits. If we let
R
(
t
) and
F
(
t
) denote the number
of rabbits and foxes, respectively, then the LotkaVolterra model is:
dR
dt
=
a
*
R

b
*
R
*
F
dF
dt
=
c
*
R
*
F

d
*
F,
where
a
= 0
.
04,
b
= 0
.
0005,
c
= 0
.
00005, and
d
= 0
.
2.
(a) Using
SimBiology
with various initial conditions (including the cases where the number
of rabbits is larger than and less than foxes, initially), describe the dynamics of rabbits
and foxes from your simulations.
(b) Now, assuming that in the absence of foxes, rabbits grow according to a logistic law, then
we have the system
dR
dt
=
a
*
R

b
*
R
*
F

e
*
R
*
R
dF
dt
=
c
*
R
*
F

d
*
F
Assuming
e
= 0
.
002, describe the dynamics of rabbits and foxes (for various initial condi
tions) (Hint: logistic law). What values of parameter
e
resulting in periodic solutions?
In the report that you turn in, please include the followings:
•
diﬀerential equations that you exported from SimBiology for each problem.
•
the plots for each simulation
•
your writeups to the questions