It is estimated that the growth rate of the fin whale population $($per year$)$

is $rx(1-\frac{x}{K}),$ where $r=0\mathrm{.}08$ 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 $0\mathrm{.}00001Ex,$ where $E$ is the level of fishing effort in boat$-$

days. Given a fixed level of effort, population will eventually stabilize 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. Con$-$

sider both the optimum level of effort and the resulting population level.