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.