We're looking at differential equations of this form. So we have two variables, I'll call them x and y now
instead of r and f for rabbits and foxes, any old variables, and this is a dynamical system
these could be different functions, and note that x depends on y in general, and y depends on x
And then one gets two solution curves. So let me show an example of that:
lets's think what the phase plane might look like for this.
And I just want to get a general picture of the shape of this. So I start: Initially X is -7 and Y is -3
So X is -7, Y is -3, that's going to put me somewhere over here..
because X is going DOWN here, the value of X goes from -7 to -8, so X goes from -7 to -8, but Y is increasing,
pulled in and we have this sort of spiral thing...we can't really spiral in quite the same way
So we have something spiraling in to the origin
So here's another example - suppose we have these two solution curves: X is a function of t and Y is a function of t.
So X is -7, Y is -3. I am going to start here...
As I start here X is going UP, that means I expect this blue thing to move to the right
Then, Y starts increasing, that means I'm going to be going UP in this direction, X is still increasing
So X is going from roughly -10 to 10 (maybe that's 9.5)
these are the min and max values for X. Y is going roughly between 4.5 and -4.5.
...sinusoidally on a phase plane, will be some sort of an ellipse or oval.
There it is. I'll put arrows on.