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?

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

(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 $\frac{dP}{dt} = \frac{1}{2}P(3-P)$.