This predator-prey model was orginally proposed and studied by Volterra in 1926 in an effort to explain the oscillatory levels of certain fish harvests in the Adriatic Sea.
Here's the idea:
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). This gives you two simultaneous differential equations:
        ,
        .

Simultaneous differential equations are just the ticket when you want to see how one process interacts with another.

B.2.a)

Examine what happens for the sample choice of proportionality constants
         , , , and
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.

Answer:

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

Hey! A cyclic pattern.
And the predator population as a function of time :

Another cyclic pattern. Here they are together:

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

Look at those populations reacting with each other.

B.2.b)

Look at:

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

Discuss the relationships between the curves.

Answer:

Cyclic oscillations of both populations are evident as all get out.
Feast your eyes on the vivid relationship between the crests and dips of the predator population and those of the prey population! The cycles are slightly out of sync. When you think about it, they should be slightly out of sync. Look again:

When there are enough prey to sustain strong predator growth, the predators begin to grow so much that they outstrip their food supply. This puts both populations into a decline until the point at which the predators are not a powerful menace; then the prey become numerous enough to support strong predator growth. The cycles go on and on.

B.2.c)

Estimate the length of one cycle in time units.
Plot one cycle, two cycles, three cycles, and four cycles.

Answer:

Put in horizontal lines shooting out from the vertical axis reflecting the information that
          and .

A closer look:

One cycle is about by eyeball estimate.
Use Mathematica to improve on this:

Check:

The eyeball was not so bad.
A plot of one cycle:

Two cycles:

Three cycles:

And four cycles:

To get the full effect, grab all four plots of the cycles,
animate and run forward slowly.

Nice excursion into mathematical ecology.
As William E. Boyce and Richard DiPrima point out in their book, Elementary Differential Equations (Wiley, 1977), "Cyclic variations of predator and prey as [predicted above] are often observed in nature."
In fact E.P. Odum [Fundamentals of Ecology, Saunders, 1953] was able to use the fur catch records of the Hudson Bay Company from 1850 to 1930 to estimate the Canadian lynx and snowshoe hare populations during these years. Odum's curves are not too different in character from the curves you saw above.
All mathematical ecologists agree that the predator-prey model is not an end in itself but is important as a tool in the quest for asking the right questions.

B.2.d)

Why all this fooling around with fake plots?
Wouldn't actual formulas for and be more convenient?

Answer:

Maybe you've seen this question before.

It all depends on what you want to do.  
If what you want is a plot, then the fake should be as good as the actual formula.  
If you want to do theoretical analysis of the true solution, then the actual formula is what you want. The only trouble is that the actual formula is available only in special situations; the fake is always available.
In the case of the predator-prey model, the formulas so treasured in old-fashioned math classes are not available.
Try it:

Mathematica failed to come up with formulas for and . You can't chalk this up as a fault of Mathematica's, because no scientist has ever succeeded in finding formulas for and . For predator-prey, the fake is the only game in town.

B.2.e)

Where do you go for further reading on predator-prey models?

Answer:

Predator-prey models have been (and still are) under heavy study by biologists and mathematicians. For further reading, here are some good places to begin:

1. The next lesson on parametric plotting.
2. E. Batschelet,Introduction to Mathematics for Life Scientists, Springer-Verlag, 1979. (introductory and clear)
3. Peter Lax,Samuel Burstein and Anneli Lax, Calculus with Applications and Computing,Springer-Verlag,1976. (serious mathematics)
4. J. D. Murray, Mathematical Biology,Springer-Verlag,1990 (serious mathematics and serious biology).
5. William E. Boyce and Richard DiPrima, Elementary Differential Equations, Wiley, 1977 (serious math although somewhat dated in presentation)

Few traditional calculus or differential equations courses attempt to touch this topic, and you know why.

B.2.f)

What version of Euler's method underlies Mathematica's fake plots above?

Answer:

None of the instructions below can be activated.

To fake a plot of the solution of , Euler's method uses
        
        
       

To fake a plot of the solution of the simultaneous differential equations
          
          ,
Euler's method uses:
                              
.
       
The fake points for as a function of are fished out from the first two slots of , and the fake points for as a function of are fished out from the first and the third slots of .
You could program it yourself, but why bother?