# Section 7 - The Dynamics of the System

## What path does the system take to get to this equilibrium? What does it
mean?

The a/y phase portrait can be used to explain the dynamics of this system. Set a on the
horizontal axis and y on the vertical axis, and use {b,g,d,n,u} = {2,2,1,1,1}.
Begin looking at the graph at a point where y' = 0 -- i.e. the curve's tangent line has a
slope of 0. Here, there is a peak number of predators and n/u adult prey. At this point in
time, the predator population begins to decrease, because the prey population is no longer
sufficient to support that peak number. [This happens in the "second quadrant" of the
system, if you place the origin of these imaginary axes at the equilibrium.]

When the system reaches the point on the left side where a' = 0 -- i.e. the curve's tangent
line is vertical -- the predator population has decreased so much that the effect of
predation on the adult prey population is minimal. During the time when the population of
the predators is so small, the children who had been maturing while the adults were being
eaten begin to "grow up" and become officially adults. [This happens in the "third
quadrant"]

At the second point where y' = 0 -- the lowest point of this cycle -- the children have
sufficiently repopulated the prey population that the y population can begin to increase
again (y' goes from being negative to being positive). [This happens in the "fourth
quadrant"]

At the point on the right side where a' = 0, the predator population is again large enough
to be a drain on the prey population (a' goes from positive to negative) and the prey
population begins to decrease. [This is the "first quadrant"]

When the system reaches a horizontal tangent line again, the predators have eaten so many
of the prey that there is no longer enough food to support them all, and they begin to die
off. This begins the "second quadrant" phase again, though on an orbit closer to the
equilibrium than the last cycle was.

One can give a similar explanation for each of the three 2-D graphs.

Note that when a' = 0, y is = to the y-coordinate of the
equilibrium, and when y' = 0, a' is
= the a-coordinate of the equilibrium.

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