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lazystudent

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First post on these forums, thanks all for your help!

It is estimated that the growth rate of the fin whale population (per year) is rx(1 - x/K), where r = 0.08 is the intrinsic growth rate, K = 400,000 is the maximum sustainable population, and x is the current population, now around 70,000. It is further estimated that the number of whales harvested per year is about 0.00001 Ex, where E is the level of fishing effort in boat-days. Given a fixed level of effort, population will eventually stabalize at the level where growth rate equals harvest rate.

a. What level of effort will maximize the sustained harvest rate? Model as a one-variable optimization problem using the five step method.

b. Examine the sensitivity to the intrinsic growth rate. Consider both the optimum level of effort and the resulting population level.

c. Examine the sensitivity to the maximum sustainable population. Consider both the optimum level of effort and the resulting population level.

G(x) = rx(1 - x/K)

r = 0.08 is the intrinsic growth rate

K = 400,000 is the maximum sustainable population

x is the current population, now around 70,000

H(x) = 0.00001 Ex

F(x) = G(x) - H(x) <------ this F(x) isn't given explicitly in the original problem, but I think it's necessary to solve the problem.

Intuitively, I had thought you would differentiate F(x) with respect to E, and then solving for 0 (first derivative test to see when the slope = 0, therefore E would be maximized). However, that gives you dF/dE = .00001x which is basically useless in maxing H(x)

My next attempt will be to differentiate it with respect to x, and then perhaps run the first derivative test to see where population can be maxed? Because H(x) will be maxed when x is maxed (but E has to remain constant).

Whatever insight the reader of this post can provide to the writer will be greatly appreciated!

P.S. We are allowed to use Wolfram Mathematica to solve these problems.[/B][/B]

## Homework Statement

It is estimated that the growth rate of the fin whale population (per year) is rx(1 - x/K), where r = 0.08 is the intrinsic growth rate, K = 400,000 is the maximum sustainable population, and x is the current population, now around 70,000. It is further estimated that the number of whales harvested per year is about 0.00001 Ex, where E is the level of fishing effort in boat-days. Given a fixed level of effort, population will eventually stabalize at the level where growth rate equals harvest rate.

__EDIT- FORGOT A, B, AND C.__a. What level of effort will maximize the sustained harvest rate? Model as a one-variable optimization problem using the five step method.

b. Examine the sensitivity to the intrinsic growth rate. Consider both the optimum level of effort and the resulting population level.

c. Examine the sensitivity to the maximum sustainable population. Consider both the optimum level of effort and the resulting population level.

## Homework Equations

G(x) = rx(1 - x/K)

r = 0.08 is the intrinsic growth rate

K = 400,000 is the maximum sustainable population

x is the current population, now around 70,000

H(x) = 0.00001 Ex

F(x) = G(x) - H(x) <------ this F(x) isn't given explicitly in the original problem, but I think it's necessary to solve the problem.

## The Attempt at a Solution

Intuitively, I had thought you would differentiate F(x) with respect to E, and then solving for 0 (first derivative test to see when the slope = 0, therefore E would be maximized). However, that gives you dF/dE = .00001x which is basically useless in maxing H(x)

My next attempt will be to differentiate it with respect to x, and then perhaps run the first derivative test to see where population can be maxed? Because H(x) will be maxed when x is maxed (but E has to remain constant).

Whatever insight the reader of this post can provide to the writer will be greatly appreciated!

P.S. We are allowed to use Wolfram Mathematica to solve these problems.[/B][/B]

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