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 find 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. Specically 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?