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MA333: Homework #6 Due Friday, February 25 Read Chapter 2 sections 5 and 6!! Carefully!
MA333: Homework #6
Due Friday, February 25
Read Chapter 2 sections 5 and 6!! Carefully!
1. High levels of cholesterol in the blood are known to be a risk factor for heart
disease. Cholesterol is manufactured by the body for use in the construction
of cell walls and is absorbed from foods containing cholesterol. The following
is a very simple model of blood cholesterol levels. Let C(t) be the amount of
cholesterol in the blood (in milligrams per deciliter) of a particular person at
time t days. Then
dC
dt = k
1
(C
0
C) + k
2
E
where C
0
is the persons natural cholesterol level, k
1
is a production param-
eter (set by experiment and meant to quantify how fast the body is producing
cholesterol), E is the amount of cholesterol eaten per day, and k
2
is an absorp-
tion parameter (set by experiment and meant to quantify how fast the body
is absorbing cholesterol from the food).
a. Suppose C
0
= 200, k
1
= 0.1, k
2
= 0.1, E = 400, and C(0) = 150. What will
the persons cholesterol be after two days on this diet?
b. With the initial conditions from part a, what will the persons cholesterol be
after ve days on this diet?
c. With the initial conditions from part a, what will the persons cholesterol be
after a very long time on this diet?
d. Suppose that, after a very long time on the high cholesterol diet described
above, the person goes on a very low cholesterol diet, so E changes to E = 100.
(Take your answer to part c above as your initial cholesterol level for this second
diet.) What will the persons cholesterol be after one day on this second diet?
After ve days on this second diet? After a very long time on this second diet?
e. Suppose the person stays on the high cholesterol diet but takes drugs that
block some of the uptake of cholesterol from food, so k
2
changes to k
2
= 0.075.
With the cholesterol level from part c as your initial cholesterol level, what will
the persons cholesterol level be after one day taking the drug? After ve days
on the drug? After a very long time on the drug?
2. For the dierential equations below nd all their equilibria and determine
their stability (stable vs unstable). Explain your reasoning. You may use a
direction eld plot to check yourself but give reasons that do not cite the plot.
a.
dp
dt = p
2
2p 8
1 b.
dp
dt = p
2
16
p
2
3. The Gompertz model: In 1825 Gompertz proposed the following popula-
tion model:
dp
dt = rp ln(
K
p )
with p(0) = p
0
where p(t) is the population size at time t, r > 0 is the intrinsic
growth rate, K > 0 is the carrying capacity for the population, and p
0
> 0 is
the initial population. Note: does this model have the three properties that our
book says we should look for in a population model (top of page 76): Is r ln(
K
p
)
close to r for small p? Does r ln(
K
p
) decrease as p get larger? Is r ln(
K
p
) less
than 0 for p suciently large?
a. Show that p = K is an asymptotically stable equilibrium.
b. Solve the initial value problem above. Hint: use the substitution u = ln(
K
p
)
and derive a dierential equation for u(t). If you do this right you will get a very
familiar linear ODE. Solve this ODE for u and from that obtain your solution
p(t).
c. Using your solution from part b, show that as t , p(t) K, as long as
p
0
> 0.
d. In class (and in the book) we solved and analyzed solutions of the logistic
population growth model:
dp
dt = rp(1 p/K)
where p(0) = p
0
. Now we can compare the Gompertz model with the logistic
model. Take K = 1000 and r = 1 in both models. Choose a couple of values
for p
0
< 1000 and a couple of values for p
0
> 1000 and plot the solutions of the
Gompertz and the logistic DE with these values. Comment on the dierences
and the similarities in the behavior of the solutions. What biological insight do
you gain from the comparisons?
4. Logistic model with constant yield harvesting: Consider a population
that grows logistically, but is subjected to harvesting (or removal) in which H
individuals are harvested per unit time, H > 0 (H is a constant). The Initial
value problem (IVP) describing this situation is
dp
dt = rp(1 p/K) H
where p(0) = p
0
. Here r is the intrinsic growth rate and K is the carrying
capacity.
2 a. Find the two equilibria for this model?
b. Actually, the two equilibria that you found above only exist (ie are real
numbers) under certain conditions on the model parameters H, K, and r. What
are these conditions? Can you make sense of the conditions from a biological
point of view?
c. Show that when no equilibria exist,
dp
dt
< 0 for all t.
What happens in this case is that the population hits 0 (ie becomes extinct)
in nite time. Thus, constant yield harvesting can have catastrophic conse-
quences on the population if the conditions found in part b for the existence of
positive equilibria are violated. One example to which this might apply is to
the over-shing of some species of sh. The point is one has to be careful not
to catch the sh at too great a rate.
d. Now examine the case in which positive equilibria do exist. Examine the
stability of each of the equilibria.
e. Use the phaseportrait command from Maple to plot some solutions for this
model when K = 1000, r = 1, p
0
= 100 and several values for H that satisfy the
conditions you found in part b. Also plot solutions for the logistic equation for
these parameter values (but with H = 0 so there is no harvesting). Compare an
comment on the results. Do your plots of the solutions agree with your results
from part d about stability?
Extra credit f. Note that you have done this whole problem without solving
the harvesting ODE. You can actually solve it analytically with partial fractions.
For extra credit solve this IVP in the case when the conditions of part b are
met.
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