This has been asked a couple of time on here but it either wasn't answered or the answer doesn't give any steps for how it was solved which is less than helpful.
Suppose a fish population experiences logistic growth with growth rate r per year and environmental carrying capacity C. The population is harvested at a rate of h per year (h might be a constant or it might be variable depending on the population). This situation is described by the differential equations
dpdt=rp(1−p/C)−h
Where p is the population and t is time (in years). It is desired to harvest the fish to reach the maximum sustainable yield. That is, if you harvest too few fish, then you aren't getting all the fish you could. But if you harvest too many fish, you will drive the population too low (perhaps to extinction) and so your harvest won't be sustainable. The next 4 problems take you through the calculations of the maximum sustainable yield for two simple harvesting models.
1. Suppose the growth rate is r=0.2 (20%), the carrying capacity is C=25,000, and the harvest is a constant H fish per year.
What are the equilibrium for this model? Are they stable or unstable?
What is the maximum value of H so that the model has an equilibrium with p>0?
Hint: as always when determining equilibrium, drawing a graph of dp/dt vs. p may be helpful.
2. Suppose the harvest is a constant H fish per year. What is the maximum value of H so that the model has an equilibrium with p>0? Your answer this time will have H as a function of r and c, in contrast to the previous problem where you were given specific values for r and C.
3. Suppose the harvest is not a constant but is a fixed proportion of the total population, so h=Ep where E is the proportion of the population that is harvested.
What are the equilibrium for the model now?
Which equilibrium is stable and which is unstable?
Note that the equilibrium populations will depend on the value of E, so your answers will be functions E as well as r and C
4. Assuming you end up at the stable equilibrium in problem 19, what proportion E maximizes the harvest (recall h=Ep where p is now a function of E, r, and C)? What is the maximum sustainable yield h for this approach?