Recall that one model for population growth states that a population grows at a rate proportional to its size.

(a) We begin with the differential equation $\displaystyle{\frac{dP}{dt} = \frac{1}{2} P }$. Find an equilibrium solution:
$P =$

Is this equilibrium solution stable or unstable?

A. stable
B. unstable

Describe the long-term behavior of the solution to $\frac{dP}{dt} = \frac{1}{2}P$.

A. The value of $P$ increases without bound.
B. The value of $P$ oscillates and does not approach a limit.
C. The value of $P$ approaches zero.
D. The value of $P$ approaches a nonzero constant.
E. None of the above

(b) Let's now consider a modified differential equation given by $\displaystyle{\frac{dP}{dt} = \frac{1}{2} P(3-P) }$.

Find a stable equilibrium solution:
$P =$

Find an unstable equilibrium solution:
$P =$

If $P(0)$ is positive, describe the long-term behavior of the solution to $\frac{dP}{dt} = \frac{1}{2}P(3-P)$.

A. The value of $P$ oscillates and does not approach a limit.
B. The value of $P$ increases without bound.
C. The value of $P$ approaches a nonzero constant.
D. The value of $P$ approaches zero.
E. None of the above