standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: 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  $K$.
[IMG]
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.


aligns: no