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