189.
Draw the directional field of the threshold logistic
equation, assuming K = 10, r = 0.1, T = 2. When does
the population survive? When does it go extinct?
190. For the preceding problem, solve the logistic
threshold equation, assuming the initial condition
P(0) = P0.
191. Bengal tigers in a conservation park have a carrying
capacity of 100 and need a minimum of 10 to survive.
If they grow in population at a rate of 1% per year, with
an initial population of 15 tigers, solve for the number of
tigers present.
192. A forest containing ring-tailed lemurs in Madagascar
has the potential to support 5000 individuals, and the
lemur population grows at a rate of 5% per year. A
minimum of 500 individuals is needed for the lemurs to
survive. Given an initial population of 600 lemurs, solve
for the population of lemurs.
193. The population of mountain lions in Northern
Arizona has an estimated carrying capacity of 250 and
grows at a rate of 0.25% per year and there must be 25
for the population to survive. With an initial population of
30 mountain lions, how many years will it take to get
the mountain lions off the endangered species list (at least
100)?
The following questions consider the Gompertz equation,
a modification for logistic growth, which is often used for
modeling cancer growth, specifically the number of tumor
cells.
194. The Gompertz equation is given by
P(t)′ = αln⎛
⎝
K
P(t)
⎞P⎠
(t). Draw the directional fields for this
equation assuming all parameters are positive, and given
that K = 1.
195. Assume that for a population, K = 1000 and
α = 0.05. Draw the directional field associated with this
differential equation and draw a few solutions. What is the
behavior of the population?
196. Solve the Gompertz equation for generic α and K
and P(0) = P0.
197. [T] The Gompertz equation has been used to model
tumor growth in the human body. Starting from one tumor
cell on day 1 and assuming α = 0.1 and a carrying
capacity of 10 million cells, how long does it take to reach
“detection” stage at 5 million cells?
198. [T] It is estimated that the world human population
reached 3 billion people in 1959 and 6 billion in 1999.
Assuming a carrying capacity of 16 billion humans, write
and solve the differential equation for logistic growth, and
determine what year the population reached 7 billion.
199. [T] It is estimated that the world human population
reached 3 billion people in 1959 and 6 billion in 1999.
Assuming a carrying capacity of 16 billion humans, write
and solve the differential equation for Gompertz growth,
and determine what year the population reached 7 billion.
Was logistic growth or Gompertz growth more accurate,
considering world population reached 7 billion on October
31, 2011?
200. Show that the population grows fastest when it
reaches half the carrying capacity for the logistic equation
P′ = rP⎛
⎝1 - KP ⎞ ⎠.
201. When does population increase the fastest in the
threshold logistic equation P′(t) = rP⎛ ⎝1 - KP ⎞ ⎠⎛ ⎝1 - TP⎞ ⎠?
202. When does population increase the fastest for the
Gompertz equation P(t)′ = αln⎛
⎝
K
P(t)
⎞P⎠
(t)?
Below is a table of the populations of whooping cranes in
the wild from 1940 to 2000. The population rebounded
from near extinction after conservation efforts began. The
following problems consider applying population models
to fit the data. Assume a carrying capacity of 10,000
cranes. Fit the data assuming years since 1940 (so your
initial population at time 0 would be 22 cranes).
