So this is the solution to question 4 from the advanced homework for unit 7.
A little bit of analytics to get a qualitative picture of what's going on
with the Lotka-Volterra equations.
So, here are the equations.
They're familiar by now, I hope, and I've written them in this factored form
that's a little bit more useful for us.
And in a previous problem we found a fixed point for this.
We find a fixed point by looking for R and F values that make
both derivatives equal to zero.
When both derivatives are at zero then we have a fixed point.
So one fixed point is if R and F are both zero.
That's not very interesting biologically or mathematically .
It just says that if there are no rabbits and foxes there will remain
no rabbits and foxes in this model.
The other solution, the other equilibrium is a little more interesting.
That would be obtained if we set each of these terms in parentheses equal to zero.
So we did that, we can see that F = 4 solves this and R = 3 solves that.
So let me just write that we have a fixed point and that's at R = 3, F = 4.
R = 3, F = 4.
And let's see. Let's put that on the phase plane.
Here are the rabbits, here we have foxes. 1,2,3. 1,2,3,4.
So I will draw that with a red dot.
So now I've set the stage to think about what are called nullclines.
So, let's imagine that we only have this condition met.
In other words that F = 4 but R is not necessarily 3.
If F = 4, then dR/dt = 0. And this is called a nullcline.
If we're at a nullcline. This nullcline is dR/dt=0 and that's at F=4.
So F=4, that's a line here. I'll draw it like this. So this is dR/dt=0.
Here, solution curves must be going straight up or straight down.
Because they can't go side to side, the rabbit population is constant.
So the solutions have to go either up or down.
Let's look at the other nullcline. That would happen if we made this term in
parentheses zero, equal to zero and we didn't worry about this.
So that would be at R = 3. So I'll write that here. If R = 3 then dF/dt = 0.
So this is another nullcline. And R=3 on this plot is a vertical line like this.
So let's look at what's going on in each quadrant.
So first, let's look at this term again.
If F > 4 then the term in parentheses is negative which means dR/dt is negative.
Let me write that down.
If F > 4, dR/dt < 0. So F > 4 above this line, and it's less than 4 below.
So in this quadrant and this quadrant dR/dt < 0.
dR/dt < 0. dR/dt < 0.
Down here dR/dt > 0.
A positive dR/dt means the rabbits are increasing.
A negative dR/dt means the rabbits are decreasing.
This says that if there are more than 4 units of foxes, 4 tons or whatever we
decide we're measuring foxes in, then the rabbit population will decrease.
If there are less than 4 tons of foxes in this little universe then the
rabbit population will increase.
Increasing rabbits is motion in this direction.
Decreasing rabbits is motion in that direction.
So now let's look at what this tells us.
So this tells me that if R < 3 then dF/dt < 0. Let me write that down.
If R < 3, dF/dt < 0.
If R < 3, that means I'm to the left of this nullcline, then dF/dt = 0.
Ok, let me write that. dF/dt, sorry, is is less than 0 and dF/dt > 0, oops.
Hang on. Oh dear. I knew I was going to make a mess of this.
Alright. dF/dt > 0. When I'm to the left of this line dF/dt < 0.
If R < 3, that means I'm over here, this is less than 0.
dF/dt < 0 here, dF/dt > 0.
Let me summarize this and then we'll start drawing some arrows.
This analysis says that as far as the foxes are concerned, if I'm to the left
of this, so I'm somewhere over here, the fox population will decrease.
If I'm over here, the fox population will increase.
Then looking at things from the point of view of the rabbits, this says that if I'm
above this line, there's more than 4 tons or 4 units of foxes, the rabbit population
decreases. If I'm below this line the rabbit population increases.
If I'm over here, R is increasing, F is increasing, that means motion is going
to be in this direction. In the direction of increasing R and increasing F.
If I'm over here, the rabbits are decreasing but the foxes are still
increasing. So a fox increase, that's up, rabbit decrease is to the left.
So it'll move like this.
Here both rabbits and foxes are decreasing and here foxes are decreasing, rabbits
increasing. Rabbits increasing means I'm moving to the right. Foxes decreasing
means I'm moving down.
This gives us a picture, a sense that we're going to expect some sort of cyclic
behavior. These blue lines are called nullclines.
They're a line, they don't always have to be vertical.
They could even be curves for more complicated
differential equations where one of the derivatives is zero.
So along this nullcline I've got motion in this direction.
Because along this nullcline the fox increase is zero.
Along this nullcline the rabbit increase is zero. That means I'm either going
straight up or straight down.
One way to analyze differential equations, or a couple differential equations like
this in two dimensional space, of course you could do Newton's method to plot
solutions but one can also solve for these nullclines and then draw the arrows along
the nullclines. Then in different regions you can figure out which direction
the general flow is. You can also solve for fixed points. That gives you a general
picture of how the equation behaves.
This sort of analysis is beyond the scope of this course. Obviously it makes use of
a bit more algebra and calculus than we normally use but it's a very standard
topic. You can find a chapter on this in most modern or dynamical systems-y books
on differential equations.
That's a little bit of analytical analysis for the Lotka-Volterra equations.
One can do more and analyze the nature of this fixed point.
That requires a little bit of linear algebra so it's more advanced still.
The main point of all this is hopefully not to get lost in too many details,
but just to point out that there are some analytics one can do to get a general
picture of solutions in a phase plane even if you can't solve in closed form for what
those curves might look like.