# differential equation problem

• Nov 7th 2010, 08:11 AM
fionais
differential equation problem
Hi, sorry this is quite a long question, but struggling to think where to even start!
Thanks!

To help determine a possible strategy for future whaling the population is modelled
mathematically.
(a) In the absence of any shing occurring the population of humpback whales, Y (t),
can be approximated as obeying the logistic equation
dY
dt= rY (1 - Y/K) :
If Y (0) = K=3 and Y (t). Hence fi nd the time t =  at which the population has
doubled from its intial population.

(b) The model can be extended to take "harvesting" of the whales into account. A
simple model is to assume that the rate at which whales are caught, called the
yield, is proportional to the population of whales. Speci cally the yield is taken
to be EY (where E is a constant determined by the amount of resources devoted
to catching the whales) and the model is then
dY
dt= rY (1 - Y/k) - EY�- Y=K) 􀀀 E
(This is known as the Schaefer model.)

Show that if E < r there are two equilibrium points Y = 0 and Y = K(1 - E/r)
and that the first of these is unstable and the second stable.
From this solution find the yield (ie: EY ) that will occur after a long time (this
is called the sustainable yield).
Find the value of E which gives the maximum sustainable yield (and hence the
level of harvesting that will result in the maximum number of whales being caught
on a sustainable basis).
Comment on what might occur if we take E > r?
• Nov 7th 2010, 11:21 AM
mr fantastic
Quote:

Originally Posted by fionais
Hi, sorry this is quite a long question, but struggling to think where to even start!
Thanks!

To help determine a possible strategy for future whaling the population is modelled
mathematically.
(a) In the absence of any shing occurring the population of humpback whales, Y (t),
can be approximated as obeying the logistic equation
dY
dt= rY (1 - Y/K) :
If Y (0) = K=3 and Y (t). Hence fi nd the time t =  at which the population has
doubled from its intial population.

(b) The model can be extended to take "harvesting" of the whales into account. A
simple model is to assume that the rate at which whales are caught, called the
yield, is proportional to the population of whales. Speci cally the yield is taken
to be EY (where E is a constant determined by the amount of resources devoted
to catching the whales) and the model is then
dY
dt= rY (1 - Y/k) - EY�- Y=K) �� E
(This is known as the Schaefer model.)

Show that if E < r there are two equilibrium points Y = 0 and Y = K(1 - E/r)
and that the first of these is unstable and the second stable.
From this solution find the yield (ie: EY ) that will occur after a long time (this
is called the sustainable yield).
Find the value of E which gives the maximum sustainable yield (and hence the
level of harvesting that will result in the maximum number of whales being caught
on a sustainable basis).
Comment on what might occur if we take E > r?

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