The logistic model can also be changed by incorporating a constant continuous rate of decrease, such as due to hunting: dP/dt=kP(1-P/A)-H where H members of the population are removed per year (over the course of the year). Suppose a population of fish obeys the logistic differential equation with k = 0.08 and carrying capacity A = 1000. (a) Suppose c = 15 fish are caught by fishers per year. Draw a slope field for the resulting differential equation, and identify the equilibrium solutions (where y′ = 0). (b) Suppose instead c = 25 fish are removed each year. Draw the resulting slope field. What effect does this have on the fish population overall? (c) Find the maximum annual amount of hunting c > 0 that allows the population of fish to survive. Graph the slope field for this differential equation.
The logistic model can also be changed by incorporating a constant continuous rate of decrease, such as due to hunting:
dP/dt=kP(1-P/A)-H
where H members of the population are removed per year (over the course of the year). Suppose a population of fish obeys the logistic differential equation with k = 0.08 and carrying capacity A = 1000.
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(a) Suppose c = 15 fish are caught by fishers per year. Draw a slope field for the resulting differential equation,
and identify the equilibrium solutions (where y′ = 0).
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(b) Suppose instead c = 25 fish are removed each year. Draw the resulting slope field. What effect does this have
on the fish population overall?
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(c) Find the maximum annual amount of hunting c > 0 that allows the population of fish to survive. Graph the slope field for this differential equation.
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(d) If you worked for the forest service, make a recommendation for how much fishing to allow per year to ensure sustainable fishing.
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