Chapter 3.1, Problem 35E

a.

To determine

To Show: The general solution to $P^{\prime }=rP-a{P}^2-h$ is $P\left(t\right)=\frac{1}{2a}\left[r+\sqrt{4ah-r^2}\times \tan \left(\frac{1}{2a}\left(C-at\right)\sqrt{4ah-r^2}\right)\right]$.

b.

To determine

The time at which the species become extinct if $r=0.03$, $a=0.0001$, $h=2.26$ and $C=-1501.85$.

c.

To determine

To graph: The function $P\left(t\right)$ if $r=0.03$, $a=0.0001$ and $P\left(0\right)=5.3$.

d.

To determine

The maximum allowable harvest rate to assure that the species survives.

e

To determine

The maximum allowable harvest rate that ensures survival of the species for arbitrary values of $a$ and $r$.