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.
ADD ALT TEXT TO THIS IMAGE
(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.
ADD ALT TEXT TO THIS IMAGE
(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.
