In this optional video, I'll give a
derivation, or motivation,

of the Lotka-Volterra equations.

So the Lotka-Volterra Equations describe
two populations and I'll discuss them in

terms of rabbits and foxes, but the key
thing is that the rabbits are the prey,

things that are eaten, and the foxes
are the predators.

These are often called 
predator-prey dynamics.

This is a particular example of a 
predator-prey dynamic and it

applies to lots of things beyond
rabbits and foxes.

In fact, I don't know if it even
applies to rabbits and foxes,

who knows -- people apply these
models to all sorts of things.

So let me give a motivation for the model,
and then say a little bit more about it.

So let's start by thinking about the
fate of the rabbits.

So our starting point, think about the
growth rate of the rabbits.

So we could start with a population model
like this: this says that the dR/dt

(that's the growth rate of the rabbits) 
is a function of the number of rabbits.

It's "a," some constant, times
the number of rabbits.

So for "a" positive, the growth rate is
always positive, the more rabbits there

are the faster the growth rate is, 
the faster the growth rate is,

the more rabbits there are, and so on.

And this leads to exponential growth.

The solution to this differential
equation is an exponential function.

However, that's not the end of the story, 
there are also the presence of foxes

and that could enter in like this.

So "a" is a model parameter, it's the
rabbit growth rate if there are no foxes.

And "b" is another model parameter,
we'll take "b" to be positive and so this

term is negative, and "b" is some measure
of how deadly the foxes are to rabbits.

Let's look at this term by term.

The minus sign-that means that this
term tends to decrease the growth rate.

The presence of foxes makes the growth
rate smaller, perhaps even negative

if this term is bigger than that.

This term on the right, R times F is the
simplest sort of interaction term that has

the following properties:

If we're to double the number of foxes,
looking at this term here,

if we double the number of foxes,
this term gets twice as large.

So that makes sense--twice as many foxes
should have twice the impact on the

growth rate of the rabbits.

Additionally, the number of rabbits also

has an impact on the growth
rate of the rabbits.

If there are a lot of rabbits, it's easy
for the foxes to find them and so there

will be more rabbits eaten, leading
to a greater impact here.

So, separately, this term is linear in 
both R and F. If you double the number

of rabbits, you double the number
of rabbits that get eaten.

That's what this says - if you double the
number of foxes,

you double the number
of rabbits that get eaten.

And then "b" is some parameter that can be
some measure of the deadliness of foxes.

OK, so that's this term for the rabbits...
let's look at the foxes.

So dF/dt, that's the fox growth rate, 
and I'll start with this term here.

This says that the growth rate of 
foxes depends on cRF.

"c" again is model parameter, 
some constant that we could change.

In this case it represents the nutritional
value of rabbits (from the point

of view of a fox) or how easy
it is for foxes to catch rabbits.

Some sort of interaction strength term.

Let's look at this as similar properties
to this term, this says if I double the

number of rabbits, I double the growth
rate of the foxes, that makes sense.

The more rabbits there are,
the more the foxes can eat.

We double the number of rabbits,
we double the fox nutrition.

That's going to double this growth rate.

Also the more foxes there are, the more
foxes there are to catch rabbits.

So that will also increase
this growth rate.

So RF is an interaction term. It's large
when both R and F are large.

So if this was the end of this equation,
the foxes would take over the universe.

They would keep growing and
growing and growing.

We know that foxes don't take over the
universe, so we add one more term, dF.

"d" here is a parameter (this isn't a
differential d) another constant that

represents the death rate of the foxes.

So if there were no rabbits,
if we got rid of the cRF term,

then we get rid of the interaction term
between foxes and rabbits.

This says that foxes die at rate "d." 
The fox population sadly would

exponentially decay down to zero
because they have nothing to eat.

So we have these two terms, rabbit growth
moderated by being eaten by foxes and fox

growth because they're eating the rabbits
moderated by some natural death rate.

So these two terms taken together have
the form of a Lotka-Volterra model,

put forth by Lotka and Volterra 
about a century ago.

Again "a", "b", "c", and "d" are 
parameters that can be adjusted.

I'm not sure anybody takes this model
tremendously seriously as a model of

rabbit and fox dynamics, but like the
Logistic Equation, either the Logistic

map or the differential equation,
this is more of a cartoon-style or

sketch designed to give a general feel 
for what sort of dynamics might happen

if we have two interacting 
populations this way.

In the field of mathematical ecology this
is one of the very basic models and one

uses this as a building block for more
complex and more realistic models.

So again, these are Lotka-Volterra
equations, and they're an example of

predator-prey dynamics. And lastly, I
should mention these can also be used

to model certain situations in economics,
where you have two industries or sets

of firms. One that's going to tend to grow
but maybe there is something else that

is hostile to it, and then this would be 
some firm or industry that depends

on 'rabbits' and if there aren't any 
rabbits, whatever that firm would be,

they'd disappear, they would die.

So one can see the use in economics also
to model all sorts of cyclic behavior.

I know less about economics than I do
about biology, so I won't even try to

make up a story for that but the bottom
line is: these are very simple equations

that model some sort of
predator-prey interaction.