I have another question I would like to ask about, this is the question as follows

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The simplest useful model for a shery comes from the logistic model for population growth,

together with a harvest h which is proportional to the current population P, that is,

h = EP; (1)

where the constant E is called the eort. E measures the fraction of the population harvested,

so that 0 <= E <= 1. This gives the model

$\displaystyle \frac{dP}{dt}=kP(1-\frac{P}{a})-EP$

where P(t) is the number of these sh at time t years and k (the natural growth rate) and a (the

carrying capacity) are constants for a particular sh population. In what follows take k = 1 and

a = 4, for simplicity.

(a) Determine the equilibrium solution(s) for a given eort E.

(b) Plot the phase plane dP

dt versus P for general E, 0 E 1, and determine for which values

of P the population is increasing or decreasing with time.

(c) Make a qualitative sketch of P versus t for various initial population sizes P(0).

(d) Hence determine which equilibrium population size(s) are stable and which are unstable.

(e) The harvest h corresponding to the stable population size is called the sustainable yield

from the shery. Using equation (1), and your answer to part (d), nd the sustainable yield

as a function of the eort, plot it, and show it has a maximum at E =$\displaystyle \frac{1}{2}$; the harvest at

this eort is called the maximum sustainable yield (MSY). What is the MSY and the stable

population size at this eort?

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So I was able to do Question a to d, although I'm not certain if I did it correctly or not. How does one do question e?

for question a), i let $\displaystyle \frac{dP}{dt}=0$ because it is an equilibrium solution, thus $\displaystyle EP=P(1-\frac{P}{4})$ so $\displaystyle E=1-\frac{P^2}{4}$

for question b), I use the equation $\displaystyle \frac{dP}{dt}=P-\frac{P^2}{4}-EP$ where E =0 and 1 seperately

thus I have two inverse parabolic graph with one of them touch the origin and the other passed through P when P = 0 and 4, I then shade the area between these graphs

for C) I made a graph of P vs t, where when P(0) and P(4) is a straight line, because of the shape of the graph in question b), I assume that P(4) is stable while P(0) is unstable, this is also the answer for question 4

How would you do question e)?

Thank Junks