Remember the predator-prey model from the previous lesson.  In case you've forgotten about it, here's the scoop again:

This mathematical model was orginally studied by Lotka and Volterra.
For a critical analysis of it, get hold of J. D. Murray, Mathematical Biology, Springer-Verlag, New York,1990 and read.

Two species coexist in a closed environment. One species, the predator, feeds on the other, the prey. There is always plenty of food for the prey, but the predators eat nothing but the prey.
Put
            .
Put  
            .

It's reasonable to assume that there are positive constants and such that:
           
because the abundance of food for the prey allows the birth rate of the prey to be proportional to their current number, and the death rate of prey is proportional to both the current number of prey and the current number of predators.

It also makes some sense to assume that there are positive constants and such that:
             
because it's reasonable to assume that the death rate of the predators is likely to be proportional to the current population of predators and that the birth rate of the predators is proportional to both the current number of the predators and the size of the food supply (the prey).

Here is what happens for a sample choice of with the prey starting off with a population of and the predators starting out with a population of .

Here the units could be thousands or millions so that a fractional population can make sense.

Here is Mathematica's fake plot of the prey population as a function of for the first time units:

And the predator population as a function of time :

Here they are together:

The plot of the prey population is thicker than the plot of the predator population.

Note the relationship between the predator population's crests and dips and prey population's crests and dips.  Just after the predators start growing, the prey is just about eaten up.

T.3.a)

Use parametric plotting to display the predator population as a function of the prey population and describe what you see.

Answer:

Here it is:

A distorted ellipse that some folks call a closed curve.
This is a little hard to interpret.
To get a grasp on it, see what happens as goes from to :

Now see what happens as goes from to :

Now see what happens as goes from to :

Now see what happens as goes from to :

To get the full effect, grab all three plots, animate, and run forward.

Holy smoke!
This is the same as:

The above plot shows what happens as goes from to .
Review the plots:

As time advances, the parametric points advance around the closed curve in a counterclockwise way.  As time advances on and on, the parametric points cycle around this curve over and over again.
Take another look but this time setting the axes' origin to :

Again the closed curve tells you that predator and prey populations are periodic; they go through repeated cycles. And the time cycle for each is the same. The four sectors above defined by the axes indicate four phases depicting trends that reverse themselves as the curve crosses the axes at and .

T.3.b.i)

Wait a minute!
Take another look at the last plot.

Running this plot successfully requires that the cells in part a) are active.

Those axes pierce the curve at the maximum and minimum values of prey and predator populations.
Is this just an accident?

Answer:

Stupid question.

In mathematics, there are no accidents.

T.3.b.ii)

Explain why those axes pierce the plot at the maximum and minimum values of prey and predator populations.

Answer:

Look at the original differential equations
        ,
        .
Take the first:
        .
At the times at which the prey population is at its maximum or at its minimum, you gotta have
        .
Consequently, at the times at which the prey population is at its maximum or at its minimum,
          ,
            .
This tells you that at the times at which the prey population is at its maximum or at its minimum,
             .

Similarly, use
           
to see that for a maximum or minimum predator population, you have
           
             ,
so when the predator population is largest or smallest,
           .

This explains why when you use ; the  axes pierce the plot at the maximum and minimum values of prey and predator populations.  

Math happens again.