Assignment 5

AMATH 383 Winter 2014
Homework 5
Due: Thursday, February 13, 2014
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1. Harvesting problem
Obtain, by hand, the explicit solution to the harvesting problem
N
dN
= rN 1 −
− H(N ), H = qEN
dt
K
subject to the initial condition N (0) = N0 . Here N = N (t). The parameters r, K, q, and
E are positive constants. Deduce the two limiting behaviors as t → ∞.
2. Fish population
For a fish population modeled by a depensation growth model, we have, when there is harvesting,
dN
= F (N ) − H(N )
dt
where
N
N
F (N ) = rN
−1
1−
NC
K
and H(N ) = qEN .
(a) Find and plot, freehand, the sustained (nontrivial stable) yield H(N3∗ ), the unsustainable (unstable) yield H(N2∗ ), and the trivial stable yield H(N1∗ ) as a function of effort
E on the same figure.
(b) Find the E(= Emax ), where the two curves in (a) merge (this happens when N2∗ = N3∗ ).
What happens to the fishery and the fish population when E = Emax ?
(c) What happens if harvesting is done at an effort level E > Emax ? What will happen if,
after the effort level was raised above Emax , it is then lowered below Emax again?
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3. Mouse population cycles
It has been observed that plagues appear in mice populations whenever the population becomes too large. Further, a local increase of density attracts predators in large numbers.
These two factors will succeed in destroying 97-98% of a population of small rodents in two
or three weeks, and the density then falls to a level at which the disease cannot spread. The
population, reduced to 2% of its maximum, finds its refuges from the predators sufficient,
and its food abundant. The population therefore begins to grow again until it reaches a level
favorable to another wave of disease and predation. Assume the following:
During “Phase 1,” the growth phase, the population grows from the minimum value q
until it reaches the maximum value Q according to
N 0 = aN.
During “Phase 2,” the epidemic phase, and the population shrinks from the maximum
value Q until it reaches the minimum value q according to
N 0 = −BN 2 .
After the population shrinks to q, it enters Phase 1 and the cycle begins again.
(a) Show that time spent in Phase 1, T1 , and time spent in Phase 2, T2 , are given by
Q
Q−q
1
,
T2 =
.
T1 = ln
a
q
qQB
(b) Given T1 = 4 years, T2 = 2 weeks, Q = 100, and q = 2, determine values for a and B.
(c) Sketch or plot the population over one period of the cycle: T1 + T2 , using the values
from (b).
(d) Describe your results from parts (b) and (c) in terms of the system being modeled.
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4. Smallpox
In 1760 Daniel Bernoulli developed a model for the spread of smallpox, which at the time was
a major threat to public health. His goal was to determine whether or not a controversial
inoculation program would be effective. His model applies equally well to any disease where
survival after contraction of the disease results in lifelong immunity.
Consider the cohort of individuals born in a given year (ie: at time t = 0). Let n(t) be the
total number of these individuals who are still alive t years later. Let x(t) be the number
of members of the cohort who have not contracted smallpox by year t, and are therefore
still susceptible. Let β be the rate at which susceptibles contract smallpox, and let ν the
chance that people who contract smallpox die from the disease. Finally, let µ(t) be the death
rate from all causes other than smallpox. Then the population is governed by the following
differential equations:
dx
= − (β + µ(t)) x
dt
dn
= −νβx − µ(t)n
dt
(a) Explain why these equations govern the population described above.
(b) Let z =
x
n
and show that z satisfies the initial value problem
dz
= −βz (1 − νz) ,
dt
z(0) = 1.
(c) Describe what z represents in terms of the population being modeled.
(d) Solve this new equation for z(t). (Hint: The equation now resembles the differential
equation for the logisitic model. You may cite the formula for the solution to the logistic
model from the lecture notes.)
(e) Bernoulli estimated that ν ≈ 81 yr−1 and β ≈ 18 . Using these values, sketch or plot
z(t), describe your plot, and determine the proportion of 20-year-olds who have not
had smallpox.
Note: Based on the model described above and using the best mortality data available
at the time, Bernoulli calculated that if deaths due to smallpox could be eliminated (if you
could somehow make ν = 0), then approximately 3 years could be added to the average life
expectancy in 1760: 26 years and 7 months. Therefore Bernoulli supported the proposed
inoculation program.
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