Task
An important example of a model often used in biology or ecology to model
population growth is called the logistic growth model. The general
form of the logistic equation is
$$
P(t) = \frac{KP_0e^{rt}}{K+P_0(e^{rt}1)}.
$$
In this equation $t$ represents time, with $t = 0$ corresponding to when the population in question is first measured; $K,P_0$ and $r$ are all real numbers
with $K$ being called the ''carrying capacity'' while $r$ is a growth rate and
is normally a positive number.

Explain why the value $P_0$ represents
the population when it is first measured.

Explain why, as time elapses, the population stabilizes, approaching the value $K$.

Explain how the behavior of $P$ changes if the growth rate $r$ is increased or decreased.

Below is the graph of a particular logistic function $P$, showing the growth of a
bacteria population. Using the graph, identify $P_0$ and $K$.

Using the values of $P_0$ and $K$ from the previous part, sketch the graph of the logistic function $Q$ given by
$$
Q(t) = \frac{KP_0e^{2rt}}{K+P_0(e^{2rt}1)}.
$$
Note that $Q$ is the same as $P$ except that the growth rate $r$ has been doubled.
IM Commentary
This task is for instructional purposes only and students should already
be familiar with some specific examples of logistic growth functions
such as that given in ''Logistic growth model, concrete case.'' This
is an important example of a function with many constants: $P_0$ the
initial population, $K$ the carrying capacity, and $r$ the growth rate. Each
of these has a specific meaning which determines the shape of the graph
and, in case of $P_0$ and $K$, can be readily estimated using the graph.
The goal of this task is to have students appreciate how the different
constants ($P_0$, $K$, and $r$) influence the shape of the graph. Only
$r$ has been changed here, in part (e), because it is the most abstract
of these numbers. If the instructor wishes to change the other numbers,
the function used to generate this particular graph is
$$
P(t) = \frac{5}{1 + 10e^{t}}.
$$
Note that this is not given in the form of the logistic equation given above
with $K,P_0,$ and $r$. It corresponds, after algebraic manipulation, to the
case where $r = 1$, $K = 5$, $P_0 = \frac{5}{11}$. Showing this identity
is a worthwhile algebraic exercise which requires careful manipulation of
fractions and exponential functions.