Our starting point for the logistic equation is the simple model that I discussed in the last video, f of p equals rP. This has the rather unrealistic feature that populations grow without bound. ¶ The world would be completely taken over by rabbits in the previous example.¶ We know that's not the case, because we can move about the world without hitting rabbits, so we know in general that populations don't grow forever. So we'd like to modify this in some simple way, to account for the fact that there's going to be some limit to the number of rabbits, or whatever it is we're studying. We'll do that by adding a term to this equation. I've modified this equation by adding this term 1-P over A. The quantity A is known as an annihilation population; if the rabbits ever reach this population A, the next year there will be no rabbits. So this must be some level such that the rabbits will eat all their food, or fight with each other so badly that the next year there will be no rabbits. A is an annihilation population. You could also think of it as an "apocalypse" population. It's "we ever reach this population, the next year there will be no more rabbits." So let me show you that statement algebraiclally. If the population equals A, all the rabbits die. Let's see how that would work. The question is, "what's F of A?" Remember that this function tells you the population next year, if you know the population is P the current year. So if the population is A, f of A would give you the population next year. Lets plug in and see what happens. So I have r, I'm plugging in A for P A over A is 1, 1 minus 1 is zero, and rA times zero is zero. Indeed, if the population reaches A, there will be no rabbits next year. So A is the annihilation population, or the apocalypse population. There is another property of this equation that I want to mention, that I think make sense; and that is if the populaiton is small, it is very close to the original model we studied. Let me write that out. For small populations, that are very far away from the apocalypse or the annihilation, we should see the rapid growth that we saw earlier. It's only when P gets to be large that the rabbits start to run out of food, or are competing with each other for space or something, that the population growth might start to slow down. First I should say what I mean by small. By small I mean that P is much less than A... and if that's the case, P over A is approximately zero. So maybe A is ten thousand, ten thousand rabbits is the apocalypse number, and P might be ten or twenty. Ten or twenty over ten thousand is a very small number. It's close to zero. When that's the case, P over A is close to zero. One minus something close to zero is close to one, this will be just a little bit less than one. So this term in parentheses here is around one. This equation just becomes rP. For small populations, this logistic equation--this is a logistic equation the population will grow rapidly like the rabbit example from the previous video. But then, once the population gets large, this term starts to matter and the population growth will slow down and there's some absolute upper limit and annihilation or apocalypse number at which the population will completely crash. So these two properties, I hope, seem reasonable for a simple model whose only goal is to have some limit to the rabbit growth. Next, I'll plot this equation, I'll plot this function of P and we'll interpret it graphically. The logistic equation is in this form: f of P is rP times 1 minus P over A. And this equation, when iterated, would describe the growth of a population. The idea is down here, I'm going to plot this function. On the horizontal axis, I'll have Pn, the population this year. And on the vertical axis, this function tells me what the population will be next year. So this--next year's population--you could call it F of P, or Pn plus one. So let's make a quick sketch of this function, and it turns out that it looks like this: It's an upside-down parabola. Obviously, just a rough sketch, but let's see what it tells us. If the population is small, then we have population growth. These units are arbitrary... If I'm at one unit here, then the population the next year would be two. We're about doubling...if I'm at two, then the population the next year might be very roughly four. So here, for a small population, we have pretty rapid growth. Here, if we're at the annihilation or apocalypse population, we have a very large population this year, The purple curve (the function) goes through zero, so that means that there will be zero rabbits the next year, Hence, annihilation or apocalypse. And if we're close to this value, there will be close to zero the next time. If I'm here, I'm very close to the maximum value--the apocalypse value--then what's my population next year? I'd go up here; the height of the graph tells me it will be quite small. So, as we approach the apocalypse value, the population is going to get smaller. In any event, the logistic equation: Here it is, it looks like this; we'll see graphs of this again. It is a simple model designed to capture population growth where there is some limiting factor to the population. It can't grow forever, there is some maximum value that--if it ever reaches-- you suddenly lose all the population. I'll complete the derivation of the logistic equation by simplifying this equation a little bit, and putting it in a slightly different and more standard--and I think more general--form. Here is the logistic equation, rP times 1 minus P over A; A is the annihilation population, and r is a growth parameter, and this tells me the population next year, if I know the population this year. Let me write that in a slightly different way: Pn plus 1, the population next year, is r times Pn times 1 minus Pn over A. To simplify things a little bit, I'm going to divide both sides of the equation by A. I'm going to divide this by A, and I'm going to divide this by A. Mathematically, that's a legal move; I'm allowed to divide both sides of the equation by A, as long as A isn't zero, which it won't be. That will preserve the inequality. And then, notice I've got a P over A, a P over A, and a P over A and that will let me simplify things a little bit. So I'm going to define a new variable x, as follows. I'll define this new variable x as P divided by A. It's the population, but expressed as a fraction of the annihilation population. So, if x equals 0.5, that means we're halfway to the annihilation or apocalypse value; we're halfway to the maximum possible number. If x is point 8, then we're 80% of the way; if x is point one, we're just 10% of A. x, then, is a number that's always between zero and one. I can use this x to write this equation in a simpler form. P over A is x, P over A is x, P over A is x, So this tells me that next year's population is equal to rx times one minus x, where x is the population expressed as a fraction of this maximum possible value. Let me write this as a fuction, as well. The function we'll be working with that we'll be iterating a lot in the next couple of sub-units, is just this: F of x is Rx times one minus x. This is the logistic equation in the standard form that we'll work with. It's important enough that I'll put a red box around it. Just for completeness, r is a growth rate parameter, so r is something that will vary...will change, and we'll see how the behavior of this equation changes. Lastly, I can expand or multiply out the right-hand side of this, and I would get this: rx minus rx squared. So the logistic equation is just a second order polynomial, it's a parabola; it's a very simple function. You've studied parabolas in high school for sure, it's not an exotic or complicated function at all. In the next several subunits, we'll start iterating this function, and we'll see what the properties of the function are. I'll end this lecture with just a quick example, before you try one on your own. Let me do a quick example, just do review the idea of iterating the function. I'll iterate the logistic equation, and I'll let r equal 1.5. The function I'll be working with is f of x is 1.5x times one minus x. I need to choose a seed, so I'll see what happens if x equals zero point two. So the first iterate is obtained by applying the function to the seed, so that's 1.5 times 0.2 times one minus 0.2. So I'll do that on a calculator, let's see here... I get 0.24. OK, so then, the next iterate is F applied to 0.24, which is 1.5, times 0.24 times one minus 0.24. Let's do that on the calculator... I get 0.2736. We can keep going and going, and get the next iterate by applying the function again and again to the seed and then we can ask what its long term behavior is. We'll do that in the next sub-unit. Before you do so, I suggest you do the quiz that follows this lecture just to make sure you see how this goes.