Preview Activity 7.6.1.
Recall that one model for population growth states that a population grows at a rate proportional to its size. In symbols: \(\frac{dP}{dt} = kP\text{.}\)
(a)
We begin with the differential equation
\begin{equation*}
\frac{dP}{dt} = \frac12 P\text{.}
\end{equation*}
Sketch a slope field as well as a few typical solutions on the axes provided.
(b)
Find all equilibrium solutions of the equation \(\frac{dP}{dt} = \frac12 P\) and classify each as stable or unstable.
(c)
If \(P(0)\) is positive, describe the long-term behavior of the solution to \(\frac{dP}{dt} = \frac12 P\text{.}\)
(d)
Let’s now consider a modified differential equation given by
\begin{equation*}
\frac{dP}{dt} = \frac 12 P(3-P)\text{.}
\end{equation*}
As before, sketch a slope field as well as a few typical solutions on the following axes provided.
(e)
Find any equilibrium solutions and classify them as stable or unstable.
(f)
If \(P(0)\) is positive, describe the long-term behavior of the solution.
