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Hello, and welcome to Unit Seven.
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So far in this course, we've looked almost exclusively at one-dimensional dynamical systems.
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We've looked at how a single number - often population or maybe temperature - changes over time.
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In this unit, we'll look at two- and three-dimensional dynamical systems, and I'll introduce these through several examples.
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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.
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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,
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and we'll see that this is a really useful technique for analyzing these higher dimensional systems.
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There won't be much chaos in this unit.
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We'll bump up against it a few times, but the higher dimensional chaos, two- and three-dimensional chaos,
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we'll save for the next unit, Unit Eight, where we'll encounter strange attractors.
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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.
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So, let's get started.
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In previous units, we've looked at one-dimensional differential equations.
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Let me remind you about that quickly here.
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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.
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Thinking in terms of a population, if I know the population P, this tells me how fast the population is changing.
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An example that we looked at in some detail is the logistic differential equation, and here is one version of that [equation].
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So this might say that the derivative of P, the growth rate of P, is 2 times P times 1 minus P over 100.
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And so we've interpreted 2 as some type of a growth rate and 100 as the carrying capacity.
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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.
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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,
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it just does it a little bit indirectly rather than by directly specifying the values of the P, it tells us how P changes.
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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.
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This is P. So this tells us how the growth rate depends on population. This would be 100.
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And let's see, solutions to this [function] look something like this [graph].
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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.
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If I start in here, I grow - the population increases until it reaches 100. That might look something like this.
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If I start above, the population decreases, until it hits 100.
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So we have an unstable fixed point at 0, and a stable fixed point at 100.
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And we can summarize that behaviour in a phase line [draws]. Two fixed points, or two equilibria, at 0 and 100...
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and it would look something like that [graph].
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So the fixed point at 0, the equilibrium at 0, that's unstable. The equilibrium at 100, that's stable.
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If we're in between 0 and 100, we move up. If we're above 100, we move down approaching 100.
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And again, just to underscore this, this is a dynamical system. There's a rule that specifies how P changes,
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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,
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and that tells us which direction to go, and how big this area - what direction to move and how fast we should be moving.
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So these are one-dimensional differential equations, and we've analyzed them quite a bit.
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What we'll do next is look at some two-dimensional differential equations, and we'll see how they're different.
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For two-dimensional differential equations, we're interested in how two quantities change, not just one.
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And to be concrete, I'll introduce this in the context of population models, much like we did for logistic growth.
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So now we'll have two populations. Let's say rabbits and foxes, and the key thing here is that foxes eat rabbits.
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So R is going to be number of rabbits, and F will be the number of foxes.
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And we won't worry about units, we'll end up with funny fractional units.
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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.
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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?"
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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.
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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.
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And similarly, the growth rate of the foxes is a function now also of rabbits and foxes.
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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,
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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.
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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,
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introduced I think in the 1920s, independently by Lotka and Voletrra, and here's one example of this.
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...and the fox growth rate is going to be 0.2 Rf minus 0.6F (foxes).
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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.
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For now, let's just accept this as a model and we'll try to understand it and consider it as a mathematical object.
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Before we analyze solutions to this, let me again highlight that this is a dynamical system.
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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 -
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how the rabbits and foxes change over time.
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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.
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So it's very similar to the one-dimensional logistic equation or any of the one-dimensional equations we've done before.
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We're specifying the rate of change of something as a function of that thing.
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The only difference now is that we have two things we're keeping track of instead of one.
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So the solution methods we'd learned for the one-dimensional differential equations carry over, more or less, to two dimensions.
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In particular, most of what I'll do in this unit will be showing you numerical solutions to these equations.
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That means I'll be doing Euler's method or something like it.
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So remember Euler's method that says that we have some initial value, an initial number of rabbits, and an initial number of foxes,
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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
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of these rabbits and foxes.
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And that rate of change is changing all the time, but we pretend that that rate of change is constant for a time interval,
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and then update the number of rabbits and foxes, and then re-evaluate the derivative and so on.
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So that's Euler's method. We talked about it a bunch in Unit Two.
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I don't want to go through it again here; I'll just present you the results of Euler's method.
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The main thing is [that] Euler's method works because this is an unambiguous way of describing how a population changes of time.
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If I tell you the initial number of rabbits, the initial number of foxes and say, "Here's the rule",
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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.
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So let me show you some solutions to the Lotka-Volterra equations.
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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,
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because R depends on F and F depends on R. The fates of the rabbits and the foxes are intertwined.
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They're coupled - linked - together.
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And then in order to solve a differential equation, I need to chose a starting value.
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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.
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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
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there are 6 tons of foxes.
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Again, this is not meant to be at all biologically correct.
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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):
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the rabbits as a function of time and the foxes as a function of time.
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So I can have a computer do that using Euler's method, and the result would look something like this.
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So here is a solution for the rabbits. So this is time (t), measured in months or years, who knows.
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And here is the rabbit population measured in tons of rabbits or something.
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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.
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But then they recover! And spike up to a little bit more than 10, almost 11, tons of rabbits.
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And then the population crashes again, spikes up, crashes again, and spikes up and so on.
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So we see a cycle. There is some cyclic behaviour to the rabbit population.
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Okay, so those are the rabbits. What about the foxes? Let's look at them.
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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.
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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.
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Many, many foxes. And it cycles.
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So we also see cyclic behaviour for the foxes.
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So we have rabbits - the population oscillates, and the foxes - the population oscillates.
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You might wonder, "How are these oscillations related?", and one way to see that is to plot these two on the same graph.
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I can almost do that here, you can kind of see it, but it's easier to it this way.
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So we'll spend a little while analyzing this. So here we have rabbits and foxes plotted on the same graph.
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So we have time (t) and we have population here, measured in tons of animals, for foxes or rabbits.
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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.
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It would be cool if there were, but anyway, so red foxes and blue rabbits!
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Let's start analyzing this situation, figuring out what's going on with this initial decline here.
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So there's a decline in the rabbit population to a low here, and that's not so good for the foxes.
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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.
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But now, there aren't so many foxes, so things are looking up for the rabbits.
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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.
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Then we have lots and lots of rabbits and the fox population starts to grow as well.
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it's good to be a fox if you have a lot of rabbits to eat.
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So then the fox population grows and grows, and then up here (or a little before here), we have lots and lots of foxes,
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and that starts to be bad news for the rabbits again, so the rabbit population crashes - down it goes - and that's bad news
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for the foxes, because they have nothing to eat. The foxes are hungry; that population crashes, and the cycle goes again.
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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
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a spike in the fox population. Again, rabbits are blue, foxes are red.
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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.
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But then the rabbits don't do so well because the foxes are eating them and so on.
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So we see these cycles.
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Okay, in the next video I'll show you another way to view this that is in some ways even more (perhaps) helpful,
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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
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and think about how you might do Euler's method for these equations.