Hello, and welcome to Unit Seven.

So far in this course, we've looked almost exclusively at one-dimensional dynamical systems.

We've looked at how a single number - often population or maybe temperature - changes over time.

In this unit, we'll look at two- and three-dimensional dynamical systems, and I'll introduce these through several examples.

The Lotka-Volterra system, the Lorenz systems, those are both differential equations, and then also the Henon map, which is a two-dimensional iterated function.

One of the main concepts or ideas that will emerge from this is a notion of phase space, a generalization of the idea of a phase line,

and we'll see that this is a really useful technique for analyzing these higher dimensional systems.

There won't be much chaos in this unit.

We'll bump up against it a few times, but the higher dimensional chaos, two- and three-dimensional chaos,

we'll save for the next unit, Unit Eight, where we'll encounter strange attractors.

The first example I'll do in this unit, is the Lotka-Volterra system, and I'll start by quickly reminding us about one-dimensional differential equations.

So, let's get started.

In previous units, we've looked at one-dimensional differential equations.

Let me remind you about that quickly here.

So we've looked at a equations of this form. dP/dT = f(p). So this says that the growth rate of P is some function of P.

Thinking in terms of a population, if I know the population P, this tells me how fast the population is changing.

An example that we looked at in some detail is the logistic differential equation, and here is one version of that [equation].

So this might say that the derivative of P, the growth rate of P, is 2 times P times 1 minus P over 100.

And so we've interpreted 2 as some type of a growth rate and 100 as the carrying capacity.

So remember we're viewing this as a dynamical system; a dynamical system is a rule that tells us how some quantity changes in time.

Here the quantity we're interested in is P. We want to know how P changes in time, and a differential equation is a rule that specifies a P of t,

it just does it a little bit indirectly rather than by directly specifying the values of the P, it tells us how P changes.

We've seen how to analyze this. I can make a sketch of the right-hand side. Might look like this, so this is dP/dT.

This is P. So this tells us how the growth rate depends on population. This would be 100.

And let's see, solutions to this [function] look something like this [graph].

I'll draw a dotted line and say this is P, this is t, dotted line here at 100, solutions - I'll do those in a different colour.

If I start in here, I grow - the population increases until it reaches 100. That might look something like this.

If I start above, the population decreases, until it hits 100.

So we have an unstable fixed point at 0, and a stable fixed point at 100.

And we can summarize that behaviour in a phase line [draws]. Two fixed points, or two equilibria, at 0 and 100...

and it would look something like that [graph].

So the fixed point at 0, the equilibrium at 0, that's unstable. The equilibrium at 100, that's stable.

If we're in between 0 and 100, we move up. If we're above 100, we move down approaching 100.

And again, just to underscore this, this is a dynamical system. There's a rule that specifies how P changes,

and we can imagine this line, these P values, if we know the P value, we can use the equation - follow the rule - that tells us how to change,

and that tells us which direction to go, and how big this area - what direction to move and how fast we should be moving.

So these are one-dimensional differential equations, and we've analyzed them quite a bit.

What we'll do next is look at some two-dimensional differential equations, and we'll see how they're different.

For two-dimensional differential equations, we're interested in how two quantities change, not just one.

And to be concrete, I'll introduce this in the context of population models, much like we did for logistic growth.

So now we'll have two populations. Let's say rabbits and foxes, and the key thing here is that foxes eat rabbits.

So R is going to be number of rabbits, and F will be the number of foxes.

And we won't worry about units, we'll end up with funny fractional units.

So I'm not going to really worry about calibrating those, so as usual, don't take these too seriously as a model of rabbits and foxes.

Okay, so the idea for this type of differential equation is as as follows: we might want to know, "How does the rabbit population change?"

This is the growth rate of the rabbits, and the idea is that this depends now on the number of rabbits, and the number of foxes.

So in order to figure out how fast the rabbit population is growing or shrinking, I need to know not just the number of rabbits, but also the number of foxes.

And similarly, the growth rate of the foxes is a function now also of rabbits and foxes.

These are different functions; the fox growth rate will be larger the more rabbits there are, because there's more for the foxes to eat,

and the opposite story will be true from the rabbits' point of view. The growth rate will be smaller and perhaps negative if there are a lot of foxes.

There are lots of possible examples for this, and I'll do the simplest and one of the most famous, which is called the Lotka-Volterra equation,

introduced I think in the 1920s, independently by Lotka and Voletrra, and here's one example of this.

...and the fox growth rate is going to be 0.2 Rf minus 0.6F (foxes).

So in an optional video in this subunit, I'll give a rough derivation of where this comes from, a loose argument for why we might expect to have terms like this.

For now, let's just accept this as a model and we'll try to understand it and consider it as a mathematical object.

Before we analyze solutions to this, let me again highlight that this is a dynamical system.

Now we're trying to keep track of two things: the population of rabbits and the population of foxes, and this is a rule that tells us how that works -

how the rabbits and foxes change over time.

This says that to know how the rabbits change, i need to know the rabbits and foxes, and then to know how the foxes change, I need to know the number of rabbits and the number of foxes.

So it's very similar to the one-dimensional logistic equation or any of the one-dimensional equations we've done before.

We're specifying the rate of change of something as a function of that thing.

The only difference now is that we have two things we're keeping track of instead of one.

So the solution methods we'd learned for the one-dimensional differential equations carry over, more or less, to two dimensions.

In particular, most of what I'll do in this unit will be showing you numerical solutions to these equations.

That means I'll be doing Euler's method or something like it.

So remember Euler's method that says that we have some initial value, an initial number of rabbits, and an initial number of foxes,

and we want to know what's the number of rabbits and foxes a little later on, so we plug in to the right-hand side and we get the rate of change

of these rabbits and foxes.

And that rate of change is changing all the time, but we pretend that that rate of change is constant for a time interval,

and then update the number of rabbits and foxes, and then re-evaluate the derivative and so on.

So that's Euler's method. We talked about it a bunch in Unit Two.

I don't want to go through it again here; I'll just present you the results of Euler's method.

The main thing is [that] Euler's method works because this is an unambiguous way of describing how a population changes of time.

If I tell you the initial number of rabbits, the initial number of foxes and say, "Here's the rule",

there's one and only one solution to this differential equation and Euler's method, or things like it, will get closer and closer to that solution.

So let me show you some solutions to the Lotka-Volterra equations.

Again, here are the differential equations. [There are] two different equations, one each for R and F. and we'll say that these are coupled, by the way,

because R depends on F and F depends on R. The fates of the rabbits and the foxes are intertwined.

They're coupled - linked - together.

And then in order to solve a differential equation, I need to chose a starting value.

I'll imagine that my initial value for R is 10 and F is 6. So that means I have 10 units of rabbits and 6 units of foxes.

Who knows, I wouldn't think of these as 10 actual rabbits and 6 foxes. Maybe this is measuring total biomass, so this is 10 tons of rabbits and

there are 6 tons of foxes.

Again, this is not meant to be at all biologically correct.

But the main this is we've got the rule, we've got the starting point that uniquely determines the solutions R(t) and F(t):

the rabbits as a function of time and the foxes as a function of time.

So I can have a computer do that using Euler's method, and the result would look something like this.

So here is a solution for the rabbits. So this is time (t), measured in months or years, who knows.

And here is the rabbit population measured in tons of rabbits or something.

So we start with 10, that's what we see there, and then very quickly the rabbit population crashes. It looks like times are bad for the rabbits.

But then they recover! And spike up to a little bit more than 10, almost 11, tons of rabbits.

And then the population crashes again, spikes up, crashes again, and spikes up and so on.

So we see a cycle. There is some cyclic behaviour to the rabbit population.

Okay, so those are the rabbits. What about the foxes? Let's look at them.

So here's the fox solution. This again, is time (t), and this is the fox population, again measured in tons of foxes or something like that.

Starts at 6, grows very quickly to right around 11, then the fox population crashes - bad time for the foxes - but then good times again.

Many, many foxes. And it cycles.

So we also see cyclic behaviour for the foxes.

So we have rabbits - the population oscillates, and the foxes - the population oscillates.

You might wonder, "How are these oscillations related?", and one way to see that is to plot these two on the same graph.

I can almost do that here, you can kind of see it, but it's easier to it this way.

So we'll spend a little while analyzing this. So here we have rabbits and foxes plotted on the same graph.

So we have time (t) and we have population here, measured in tons of animals, for foxes or rabbits.

I decided to colour the rabbits blue and foxes red. Foxes are often a kind of reddish colour and I don't know, there aren't blue rabbits.

It would be cool if there were, but anyway, so red foxes and blue rabbits!

Let's start analyzing this situation, figuring out what's going on with this initial decline here.

So there's a decline in the rabbit population to a low here, and that's not so good for the foxes.

What are the foxes going to eat? They're not going to eat much, so you have a lot of hungry foxes, and it's sad, the fox population crashes down to here.

But now, there aren't so many foxes, so things are looking up for the rabbits.

There aren't all those foxes eating them all the time, so the rabbit population grows very rapidly because there are no foxes around to control the rabbits.

Then we have lots and lots of rabbits and the fox population starts to grow as well.

it's good to be a fox if you have a lot of rabbits to eat.

So then the fox population grows and grows, and then up here (or a little before here), we have lots and lots of foxes,

and that starts to be bad news for the rabbits again, so the rabbit population crashes - down it goes - and that's bad news

for the foxes, because they have nothing to eat. The foxes are hungry; that population crashes, and the cycle goes again.

So we can see that the cycles have the same period, which sort of makes sense, and that the rabbit population spikes up and that's followed by

a spike in the fox population. Again, rabbits are blue, foxes are red.

So the blue rabbits spike up because there aren't so many foxes, but then the foxes come back - they're doing really well because they're eating all the rabbits.

But then the rabbits don't do so well because the foxes are eating them and so on.

So we see these cycles.

Okay, in the next video I'll show you another way to view this that is in some ways even more (perhaps) helpful,

and is really useful in dynamical systems, but first there'll be a quick quiz that you can try if you want to experiment a little bit

and think about how you might do Euler's method for these equations.