Download Homework 7 Due Friday Apr 14

Homework 7 Due Friday Apr 14
1. Using the simple predator prey model (eqns 6.1 and 6.2) with r=0.2, =0.01, q=0.4,
and =0.02, draw the state space graph and zero growth isoclines for predators and
victims. For a predatory-victim system that starts with 10 of individuals each, show the
initial population trajectory.
victim isocline (dashed): predators = r/=20
pred. isoclind (dotted): victims = q/=20
Pedators
20
20
Victims
2. In the basic SIR model for infectious disease, the system of equations were:
dS/dt = -SI
dI/dt = SI – kI
dR/dt = kI
S, I, and R are the proportions of susceptible, infected and resistant (or recovered)
individuals,  is the disease transmission coefficient and k is the recovery rate of infected
individuals.
Use the excel spreadsheet that I demonstrated in class to model chicken pox in a
population of 1000 public school children. The infectious period for chicken pox is about
5 days, so k = 0.2. Studies have shown that  for chicken pox is approximately 1.3.
a. If you start the population with one infected child (I=0.001), how long is the
epidemic likely to last? What proportion of the children is predicted to become
infected during this outbreak?
The predicted number of infecteds is <1 individual (I<0.001) after 40 days (and
I<0.0001 after 53 d).
All 1000 children are predicted to get sick during the outbreak (R=0.9999 at end,
rounds to 1000 children).
b. There is now a vaccine for chicken pox. Its effect is to make some fraction of
the initial population resistant, prior to the outbreak of the disease. What
proportion of the population must be vaccinated in order to prevent the outbreak
of chicken pox?
Change the initial value of S and look at the effect on I and R. When the initial S
is <= 0.15 (85% vaccinated), then the proportion infected declines steadily and
never gets higher than the initial 1/1000 during the outbreak.
[You might have used a stricter criterion: that no additional children should be
predicted to get sick. That would require that R increase by only 1 individual
during the outbreak—the recovery of the initial sick child. In that case the
vaccination percentage must be 94%].
c. Now repeat that analysis assuming you have a less transmissible disease, say
SARS. There  is approximately 0.25. What fraction of an initially susceptible
population is likely to become infected during the course of an outbreak?
Look at the “recovered” column to find the fraction that had been sick at some
time during the outbreak: It stabilizes at 0.377.
What fraction of the population would have to be vaccinated in order to prevent
disease spread?
Once 20% of the population are vaccinated (initial S=0.80), the infection
percentage declines steadily and never increases above the initial 0.001.
(About 70% have to be vaccinated to prevent ANY of the other children from
getting sick. In that case R increases by only 1 individual, the initial sick child.)
3. MacArthur and Wilson’s theory of island biogeography predicts that large islands
should have more species than small islands. Using their immigration and extinction
graph show why that is true. Now modify that graph to show how it could be possible to
find cases where there are more species on
a small island than on a large one.
I or E rate
a. In their model, the extinction rate
(dashed) is lower in large islands than
small. For a constant immigration
rate(dotted) to the two islands, the large
island will have more species at
equilibrium than the small.
Species #
b. Immigration rate will be lower on
islands far from the mainland source of
species. It is possible for a large, far island
to have fewer species than a small, close,
island.
Species #
4. Subtidal marine communities can show regular patterns of succession just as plant
communities do. A simplified dataset showing successional transition probabilities in
marine invertebrates (Hill et al. 2004) has the following transition matrix. The species
studied were:
Hymedesmia sp. (a sponge) Hy
Crisia eburnea (a bryozoan) Cr
Myxilla fimbriata (a sponge) Myx
Mycale lingua (a sponge) Myc
Filograna implexa (a polychaete worm) Fil
Next
year
Hy
0.85
0.11
0.02
0.00
0.02
Hy
Cr
Mxy
Myc
Fil
This year
Cr
0.18
0.74
0.04
0.01
0.03
Mxy
0.06
0.07
0.84
0.00
0.03
Myc
0.02
0.06
0.01
0.90
0.01
If the community starts out with these initial abundances, what is the predicted
composition of the community after one year?
Species
Hy
Cr
Mxy
Myc
Fil
Initial
abundance
10
3
5
20
30
S(t+1) = A S(t)
E.g.:
Species
Predicted
abundance
next year
Hy
Cr
Mxy
Myc
Fil
14.24
13.27
5.92
18.33
16.24
Hy(t+1) = 10*0.85 + 3*0.18 + 5*0.06 + 20* 0.02 + 30*0.15 = 14.24.
Cr(t+1) = 10*0.11 + 3*0.74 + 5*0.07 + 20*0.06 + 30*0.28 = 13.27
Do the same for the other species.
Fil
0.15
0.28
0.04
0.01
0.52