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PETER GARFIELD: This video is meant as a quick introduction
to phase plane analysis and the use of nullclines.
We're going to get a sense of what solutions to the system of differential
equations-- dx dt equals 0.2x minus 0.1 x y,
and dy dt equals minus 0.15y plus 0.05 xy--
look like.
This system is meant to be analogous to the systems of sardines and marlins
in the other videos.
But more mathematically familiar.
We'll be using x and y rather than s for sardines and m for marlins.
Let's start with what we can say about the solution curve
through a particular point.
As an example, let's pick the point x, y equals four, one.
What can we say about a solution curve passing through this point?
How is x changing?
The differential equation for x tells us that dx dt equals 0.2x minus 0.1 x y,
which is 0.2 times four minus 0.1 times four times one, which is 0.4.
It's not so critical at this point that we know the value of this derivative.
But it's nice to know the sign.
The fact that it is positive tells us that x is increasing with time.
Similarly, dy dt equals negative 0.15y plus 0.05 xy,
which is negative 0.15 one plus 0.05 times four times one.
This tells us y is increasing with time.
This means that any solution curve passing through the point four,
one has x increasing with time.
The curve is moving to the right.
It has y increasing with time.
The curve is moving up.
We represent this with an arrow on our graph.
Now we can continue in this way taking random points
to get a feel for our solution curves.
But we're going to try to be a little more thoughtful and systematic.
Where will our solution curves move to the right?
That is, where is dx dt positive?
And where will the curves move to the left?
Where is dx dt negative?
The easiest way to approach this is to first find where dx dt is zero.
Let's factor dx dt equals 0.2x minus 0.1 times x times y into 0.1 times x times
the quantity two minus y.
We see that dx dt is zero when x equals zero or when y equals two.
Let's draw these lines in.
These lines are where dx dt is zero.
So we've also drawn in vertical line segments
to indicate that the solution curves pass over these lines vertically.
Because x is not changing, there's no movement right or left.
These lines are called nullclines, where the null indicates
that the derivative is zero.
Now where are our solution curves moving to the right?
Where x is increasing or where dx dt is positive.
Similarly, our solution curves are moving to the left
where x is decreasing.
That is, where dx dt is negative.
These regions are actually determined by where dx dt is zero.
We can tell which region has positive derivative
and which has negative derivative by simply testing a sample point.
We've already seen that dx dt is positive at the point four, one.
And it is a simple computation to see that dx dt is negative
when x y equals one, three.
dx dt is 0.2 times x minus 0.1 times x times y,
which is 0.2 times one minus 0.1 times one times
three, which is 0.2 minus 0.3, which is negative 0.1.
We can do a very similar computation with dy dt equals negative 0.15y plus
0.05 x y, which equals 0.05 y times the quantity minus three plus x to get dy
dt zero on the lines y equals zero and x equals three.
As before, the nullclines represent where the derivative is zero.
In this case, the derivative is dy dt.
So where this is zero is where the solution curves are only
moving horizontally, not vertically.
Thus they have small horizontal line segments to represent this.
And again, we've tested points to determine which region the solution
curves go up--
that is, where dy dt is positive--
and in which region the solution curves go down--
that is, where dy dt is negative.
We finally put this all together to get a picture like this.
Now we're going to redraw this and simplify it in a few ways.
Every pair of arrows is going to be replaced by a diagonal arrow.
Also the short lines across the nullclines
are extended to be arrows to show which direction the solution curves travel.
Finally, points where both dx dt and dy dt
are zero, where nullclines from the two equations cross, are shown as dots.
These are equilibrium points.
This gives us an idea of what our solution curves might look like.
It looks like they travel counter-clockwise
around our equilibrium point at the point three, two.
They might be elliptical or circular.
Or perhaps they spiral outward or even spiral inward.
This phase plane analysis is an example of qualitative analysis.
While in general it's difficult to solve a system of equations like this,
the phase plane analysis allows us to understand
some of the general features of solution curves.
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