Recall that for one-dimensional
differential equations,
we formed a phase line, and
this lets us summarize the
behavior of all solutions to the equation.
We see anything between 0 and a
100 goes up to a 100
and anything bigger than a 100
goes down, and so on.
So what I'd like to do next is generalize
this idea of a phase line to 2 dimensions.
So as before, here are the solutions to
the Lotka-Volterra Equation.
There are two solutions because we're
keeping track of two different things,
the rabbits and the foxes.
So if it was a one-dimensional situation
and we just had rabbits growing,
logistically perhaps, we would just have
a one-dimensional phase line because
there's one thing to keep
track of, rabbits.
Now there are two, so we'll plot
this on a phase plane.
So what I'm going to do is take the
rabbit population and plot them
against the fox population.
So I'm going to remove time,
just like we did when we made
the one-dimensional phase lines.
So let me show you what this would
look like for this solution.
So if I plot the red curve against the
blue curve, I end up with this cycle.
And it's easiest to make sense of this by
by talking through this cycle using
rabbit and fox stories as I did here.
So let's start here in the lower left.
This is the rabbit axis-this is how many
rabbits there are in tons of rabbits.
This is how may tons of foxes there are
-- something like that.
We start down here; rabbits are small,
foxes are small, the populations small.
So that's good news for the rabbits.
The rabbits like it when there
aren't too many foxes.
So the rabbit population increases,
moving to the right on this diagram
means that the rabbit
populations are increasing.
As the rabbit populations increase,
the fox population starts to
increase as well.
Moving up on this diagram,
up on this phase plane,
means the fox population
is getting larger.
So the fox population is getting
larger and larger and larger,
and here the rabbit population
starts decreasing.
The rabbit population is decreasing
because we are moving to the left.
I should probably put some arrows here to
make it clearer which way the motion.
So we move this way and then the rabbit
population starts to decrease but the fox
population is still increasing. Why?
Because we are going up- this curve is
moving up in the "fox" direction,
the y direction. So the rabbits are
decreasing but the foxes are still
increasing until we get to here.
And here the fox population starts to
crash because there aren't enough rabbits
and we are going to have
hungry foxes again.
So in this segment of the graph, the fox
population is decreasing, the rabbit
population decreases a little more, then
we have this sharp, sharp drop in the fox
population, rabbits are more or less
constant and then we begin again.
So we could put arrows on this just like
we put arrows on the phase line and
we know that the population will
cycle around like this. It doesn't tell us
the time information, we don't know how
long it will take to cycle around,
it could take 1 time unit, 10, or 100.
We lose that information when we go to a
phase plane just like we lose time
information when we go to a phase
line in one dimension.
I hope the story that I told about the
cycle here is familiar-
- in fact it's the same story I told here
when we were tracking these two together.
Let me mention one more
way to look at this:
if we think just in terms of the foxes,
if we ignore the rabbits,
imagining we're compressing
this in or something,
the foxes go down to about 1 and
their maximum is about 11.
And that's what we see here - the foxes
go down to about 1 and up to about 11.
It won't look quite the same because the
scales are different, but if I wanted to
look at the rabbits, the rabbits go down
to a little bit less than 1 and their
maximum value is a
little bit less than 11.
So rabbits are always between these two
values. And that's what we see here -
the rabbits go down to less than 1,
almost a 1/2, and their maximum
is a little bit less than 11.
Again, to summarize, we took two
solution curves: "F" as function of time
and "R" for rabbits as a function of time
and we plotted them against each other.
In so doing we lose the time information
but are able to see much more clearly how
the rabbit and fox populations cycle
together and how those cycles are related.
In the next unit I'll do a few more
examples of curves in phase space and
I'll talk a little bit more generally
about the sorts of solutions we can and
can't see in two dimensions.
That will be in the next sub-unit.