The population P(t) of cod in a region of the North Sea (at time t) satisfies
$\frac{dP}{dt}$ = $kP(1-\frac{P}{M})-aP$
where k>0, M>0 and a>0 are constants, with k representing the breeding rate, M the maximum sustainable population size in the absence of fishing and a the rate of fishing.
A government minister asks your advice; fishermen are lobbying her to increase the rate of fishing (i.e. to increase a) but environmentalists are concerned about what may happen to cod stocks if there is overfishing. By solving the differential equation, can you provide appropriate advice? In particular, you may wish to consider questions such as:
- What is the largest possible value of a for which the fish population does not die out as t -> infinity?
- What is the best value of a to choose in order to maximise aP, the total size of the catch, as t -> infinity?
These questions seem to be answerable without any attempt to solve the equation.
Consider the phase plane of the differential equation...
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