WEBVTT 00:00:00.000 --> 00:00:06.000 Today, once again, a day of solving no 00:00:03.000 --> 00:00:09.000 differential equations whatsoever. 00:00:05.000 --> 00:00:11.000 The topic is a special kind of differential equation, 00:00:10.000 --> 00:00:16.000 which occurs a lot. It's one in which the 00:00:13.000 --> 00:00:19.000 right-hand side doesn't have any independent variable in it. 00:00:18.000 --> 00:00:24.000 Now, since I'm going to use as the independent variable, 00:00:23.000 --> 00:00:29.000 t for time, maybe it would be better to write the left-hand 00:00:28.000 --> 00:00:34.000 side to let you know, since you won't be able to 00:00:32.000 --> 00:00:38.000 figure out any other way what it is, dy dt. 00:00:37.000 --> 00:00:43.000 We will write it this time. dy dt is equal to, 00:00:42.000 --> 00:00:48.000 and the point is that there is no t on the right hand side. 00:00:50.000 --> 00:00:56.000 So, there's no t. There's a name for such an 00:00:56.000 --> 00:01:02.000 equation. Now, some people call it time 00:01:01.000 --> 00:01:07.000 independent. The only problem with that is 00:01:07.000 --> 00:01:13.000 that sometimes the independent variable is a time. 00:01:12.000 --> 00:01:18.000 It's something else. We need a generic word for 00:01:17.000 --> 00:01:23.000 there being no independent variable on the right-hand side. 00:01:24.000 --> 00:01:30.000 So, the word that's used for that is autonomous. 00:01:29.000 --> 00:01:35.000 So, that means no independent variable on the right-hand side. 00:01:37.000 --> 00:01:43.000 It's a function of y alone, the dependent variable. 00:01:41.000 --> 00:01:47.000 Now, your first reaction should be, oh, well, 00:01:45.000 --> 00:01:51.000 big deal. Big deal. 00:01:47.000 --> 00:01:53.000 If there's no t on the right hand side, then we can solve 00:01:53.000 --> 00:01:59.000 this by separating variables. So, why has he been talking 00:01:58.000 --> 00:02:04.000 about it in the first place? So, I admit that. 00:02:03.000 --> 00:02:09.000 We can separate variables, and what I'm going to talk 00:02:07.000 --> 00:02:13.000 about today is how to get useful information out of the equation 00:02:12.000 --> 00:02:18.000 about how its solutions look without solving the equation. 00:02:17.000 --> 00:02:23.000 The reason for wanting to do that is, A, it's fast. 00:02:21.000 --> 00:02:27.000 It gives you a lot of insight, and the actual solution, 00:02:25.000 --> 00:02:31.000 I'll illustrate one, in the first place, 00:02:29.000 --> 00:02:35.000 take you quite a while. You may not be able to actually 00:02:35.000 --> 00:02:41.000 do the integrations, the required and separation of 00:02:40.000 --> 00:02:46.000 variables to get an explicit solution, or it might simply not 00:02:47.000 --> 00:02:53.000 be worth the effort of doing if you only want certain kinds of 00:02:53.000 --> 00:02:59.000 separations about the solution. So, the thing is, 00:02:58.000 --> 00:03:04.000 the problem is, therefore, to get qualitative 00:03:03.000 --> 00:03:09.000 information about the solutions without actually solving -- 00:03:11.000 --> 00:03:17.000 Without actually having to solve the equation. 00:03:14.000 --> 00:03:20.000 Now, to do that, let's take a quick look at how 00:03:18.000 --> 00:03:24.000 the direction fields of such an equation, after all, 00:03:22.000 --> 00:03:28.000 it's the direction field is our principal tool for getting 00:03:27.000 --> 00:03:33.000 qualitative information about solutions without actually 00:03:32.000 --> 00:03:38.000 solving. So, how does the direction 00:03:35.000 --> 00:03:41.000 field look? Well, think about it for just a 00:03:39.000 --> 00:03:45.000 second, and you will see that every horizontal line is an 00:03:44.000 --> 00:03:50.000 isocline. So, the horizontal lines, 00:03:47.000 --> 00:03:53.000 what are their equations? This is the t axis. 00:03:51.000 --> 00:03:57.000 And, here's the y axis. The horizontal lines have the 00:03:56.000 --> 00:04:02.000 formula y equals a constant. Let's make it y equals a y zero 00:04:01.000 --> 00:04:07.000 for different values of the constant y zero. 00:04:06.000 --> 00:04:12.000 Those are the horizontal lines. And, the point is they are 00:04:11.000 --> 00:04:17.000 isoclines. Why? 00:04:12.000 --> 00:04:18.000 Well, because along any one of these horizontal lines, 00:04:17.000 --> 00:04:23.000 I'll draw one in, what are the slopes of the line 00:04:22.000 --> 00:04:28.000 elements? The slopes are dy / dt is equal 00:04:25.000 --> 00:04:31.000 to f of y zero, but that's a constant because 00:04:30.000 --> 00:04:36.000 there's no t to change as you move in the horizontal 00:04:35.000 --> 00:04:41.000 direction. The slope is a constant. 00:04:39.000 --> 00:04:45.000 So, if I draw in that isocline, I guess I've forgotten, 00:04:44.000 --> 00:04:50.000 as our convention, isoclines are in dashed lines, 00:04:47.000 --> 00:04:53.000 but if you have color, you are allowed to put them in 00:04:51.000 --> 00:04:57.000 living yellow. Well, I guess I could make them 00:04:55.000 --> 00:05:01.000 solid, in that case. I don't have to make a dash. 00:05:00.000 --> 00:05:06.000 Then, all the line elements, you put them in at will because 00:05:04.000 --> 00:05:10.000 they will all have, they are all the same, 00:05:08.000 --> 00:05:14.000 and they have slope, that, f of y0. 00:05:11.000 --> 00:05:17.000 And, similarly down here, they'll have some other slope. 00:05:16.000 --> 00:05:22.000 This one will have some other slope. 00:05:18.000 --> 00:05:24.000 Whatever, this is the y zero, the value of it, 00:05:23.000 --> 00:05:29.000 and whatever that happens to be. 00:05:25.000 --> 00:05:31.000 I'll put it one more. That's the x-axis. 00:05:30.000 --> 00:05:36.000 I can use the x-axis. That's an isocline, 00:05:33.000 --> 00:05:39.000 too. Now, what do you deduce about 00:05:36.000 --> 00:05:42.000 how the solutions must look? Well, let's draw one solution. 00:05:41.000 --> 00:05:47.000 Suppose one solution looks like this. 00:05:44.000 --> 00:05:50.000 Well, that's an integral curve, in other words. 00:05:48.000 --> 00:05:54.000 Its graph is a solution. Now, as I slide along, 00:05:52.000 --> 00:05:58.000 these slope elements stay exactly the same, 00:05:56.000 --> 00:06:02.000 I can slide this curb along horizontally, 00:06:00.000 --> 00:06:06.000 and it will still be an integral curve everywhere. 00:06:06.000 --> 00:06:12.000 So, in other words, they integral curves are 00:06:09.000 --> 00:06:15.000 invariant under translation for an equation of this type. 00:06:14.000 --> 00:06:20.000 They all look exactly the same, and you get them all by taking 00:06:20.000 --> 00:06:26.000 one, and just pushing it along. Well, that's so simple it's 00:06:25.000 --> 00:06:31.000 almost uninteresting, except in that these equations 00:06:30.000 --> 00:06:36.000 occur a lot in practice. They are often hard to 00:06:34.000 --> 00:06:40.000 integrate directly. And, therefore, 00:06:38.000 --> 00:06:44.000 it's important to be able to get information about them. 00:06:42.000 --> 00:06:48.000 Now, how does one do that? There's one critical idea, 00:06:45.000 --> 00:06:51.000 and that is the notion of a critical point. 00:06:48.000 --> 00:06:54.000 These equations have what are called critical points. 00:06:52.000 --> 00:06:58.000 And, what it is is very simple. There are three ways of looking 00:06:57.000 --> 00:07:03.000 at it: critical point, y zero; 00:06:59.000 --> 00:07:05.000 what does it mean for y0 to be a critical point? 00:07:04.000 --> 00:07:10.000 It means, another way of saying it is that it should be a zero 00:07:08.000 --> 00:07:14.000 of the right-hand side. So, if I ask you to find the 00:07:11.000 --> 00:07:17.000 critical points for the equation, what you will do is 00:07:15.000 --> 00:07:21.000 solve the equation f of y equals zero. 00:07:18.000 --> 00:07:24.000 Now, what's interesting about them? 00:07:21.000 --> 00:07:27.000 Well, for a critical point, what would be the slope of the 00:07:25.000 --> 00:07:31.000 line element along, if this is at a critical level, 00:07:28.000 --> 00:07:34.000 if that's a critical point? Look at that isocline. 00:07:33.000 --> 00:07:39.000 What's the slope of the line elements along it? 00:07:37.000 --> 00:07:43.000 It is zero. And therefore, 00:07:39.000 --> 00:07:45.000 for these guys, these are, in other words, 00:07:43.000 --> 00:07:49.000 our solution curves. But let's prove it formally. 00:07:48.000 --> 00:07:54.000 So, there are three ways of saying it. 00:07:52.000 --> 00:07:58.000 y zero is a critical point. 00:07:55.000 --> 00:08:01.000 It's a zero of the right-hand side, or, y equals y0 is a 00:08:00.000 --> 00:08:06.000 solution to the equation. Now, that's perfectly easy to 00:08:05.000 --> 00:08:11.000 verify. If y zero makes this 00:08:10.000 --> 00:08:16.000 right-hand side zero, it's certainly also y equals y0 00:08:14.000 --> 00:08:20.000 makes the left-hand side zero because you're 00:08:18.000 --> 00:08:24.000 differentiating a constant. So, the reasoning, 00:08:21.000 --> 00:08:27.000 if you want reasoning, is proof. 00:08:24.000 --> 00:08:30.000 Maybe we can make one line out of a proof. 00:08:28.000 --> 00:08:34.000 To say that it's a solution, what does it mean to say that 00:08:33.000 --> 00:08:39.000 it's a solution? It means to say that when you 00:08:38.000 --> 00:08:44.000 plug it in, plug in this constant function, 00:08:42.000 --> 00:08:48.000 y0, the dy0 dt is equal to f of y0. 00:08:47.000 --> 00:08:53.000 Is that true? Yeah. 00:08:49.000 --> 00:08:55.000 Both sides are zero. It's true. 00:08:52.000 --> 00:08:58.000 Now, y0 is not a number. Well, it is. 00:08:56.000 --> 00:09:02.000 It's a number on this side, but on this side, 00:09:00.000 --> 00:09:06.000 what I mean is a constant function whose constant value is 00:09:06.000 --> 00:09:12.000 y zero, this function, 00:09:09.000 --> 00:09:15.000 and its derivatives are zero because it has slope zero 00:09:14.000 --> 00:09:20.000 everywhere. So, this guy is a constant 00:09:20.000 --> 00:09:26.000 function, has slope zero. This is a number which makes 00:09:25.000 --> 00:09:31.000 the right-hand side zero. Well, that's nice. 00:09:30.000 --> 00:09:36.000 So, in other words, what we found are, 00:09:33.000 --> 00:09:39.000 by finding these critical points, solving that equation, 00:09:38.000 --> 00:09:44.000 we found all the horizontal solutions. 00:09:41.000 --> 00:09:47.000 But, what's so good about those? 00:09:44.000 --> 00:09:50.000 Surely, they must be the most interesting solutions there are. 00:09:50.000 --> 00:09:56.000 Well, think of how the picture goes. 00:09:53.000 --> 00:09:59.000 Let's draw in one of those horizontal solutions. 00:09:57.000 --> 00:10:03.000 So, here's a horizontal solution. 00:10:02.000 --> 00:10:08.000 That's a solution. So, this is my y0. 00:10:05.000 --> 00:10:11.000 That's the height at which it is. 00:10:08.000 --> 00:10:14.000 And, I'm assuming that f of y0 equals zero. 00:10:13.000 --> 00:10:19.000 So, that's a solution. Now, the significance of that 00:10:18.000 --> 00:10:24.000 is, because it's a solution, in other words, 00:10:22.000 --> 00:10:28.000 it's an integral curve, remember what's true about 00:10:27.000 --> 00:10:33.000 integral curves. Other curves are not allowed to 00:10:31.000 --> 00:10:37.000 cross them. And therefore, 00:10:35.000 --> 00:10:41.000 these things are the absolute barriers. 00:10:38.000 --> 00:10:44.000 So, for example, suppose I have two of them is 00:10:42.000 --> 00:10:48.000 y0, and let's say here's another one, another constant solution. 00:10:47.000 --> 00:10:53.000 I want to know what the curves in between those can do. 00:10:52.000 --> 00:10:58.000 Well, I do know that whatever those red curves do, 00:10:56.000 --> 00:11:02.000 the other integral curves, they cannot cross this, 00:11:00.000 --> 00:11:06.000 and they cannot cross that. And, you must be able to 00:11:06.000 --> 00:11:12.000 translate them along each other without ever having two of them 00:11:12.000 --> 00:11:18.000 intersect. Now, that really limits their 00:11:16.000 --> 00:11:22.000 behavior, but I'm going to nail it down even more. 00:11:21.000 --> 00:11:27.000 So, other curves can't cross these. 00:11:24.000 --> 00:11:30.000 Other integral curves can't cross these yellow curves, 00:11:29.000 --> 00:11:35.000 these yellow lines, these horizontal lines. 00:11:35.000 --> 00:11:41.000 But, I'm going to show you more, and namely, 00:11:39.000 --> 00:11:45.000 so what I'm after is deciding, without solving the equation, 00:11:45.000 --> 00:11:51.000 what those curves must look like in between. 00:11:49.000 --> 00:11:55.000 Now, the way to do that is you draw, so if we want to make 00:11:55.000 --> 00:12:01.000 steps, everybody likes steps, okay, so step one is going to 00:12:00.000 --> 00:12:06.000 be, find these. Find the critical points. 00:12:06.000 --> 00:12:12.000 And, you're going to do that by solving this equation, 00:12:11.000 --> 00:12:17.000 finding out where it's zero. Once you have done that, 00:12:16.000 --> 00:12:22.000 you are going to draw the graph of f of y. 00:12:21.000 --> 00:12:27.000 And, the interest is going to be, where is it positive? 00:12:27.000 --> 00:12:33.000 Where is it negative? You've already found where it's 00:12:32.000 --> 00:12:38.000 zero. Everywhere else, 00:12:35.000 --> 00:12:41.000 therefore, it must be either positive or negative. 00:12:39.000 --> 00:12:45.000 Now, once you have found that out, why am I interested in 00:12:43.000 --> 00:12:49.000 that? Well, because dy / dt is equal 00:12:46.000 --> 00:12:52.000 to f of y, right? 00:12:49.000 --> 00:12:55.000 That's what the differential equation says. 00:12:52.000 --> 00:12:58.000 Therefore, if this, for example, 00:12:55.000 --> 00:13:01.000 is positive, that means this must be 00:12:57.000 --> 00:13:03.000 positive. It means that y must be 00:13:00.000 --> 00:13:06.000 increasing. It means the solution must be 00:13:03.000 --> 00:13:09.000 increasing. Where it's negative, 00:13:07.000 --> 00:13:13.000 the solution will be decreasing. 00:13:09.000 --> 00:13:15.000 And, that tells me how it's behaving in between these yellow 00:13:13.000 --> 00:13:19.000 lines, or on top of them, or on the bottom. 00:13:15.000 --> 00:13:21.000 Now, at this point, I'm going to stop, 00:13:18.000 --> 00:13:24.000 or not stop, I mean, I'm going to stop 00:13:20.000 --> 00:13:26.000 talking generally. And everything in the rest of 00:13:23.000 --> 00:13:29.000 the period will be done by examples which will get 00:13:26.000 --> 00:13:32.000 increasingly complicated, not terribly complicated by the 00:13:30.000 --> 00:13:36.000 end. But, let's do one that's super 00:13:34.000 --> 00:13:40.000 simple to begin with. Sorry, I shouldn't say that 00:13:38.000 --> 00:13:44.000 because some of you may be baffled even by here because 00:13:42.000 --> 00:13:48.000 after all I'm going to be doing the analysis not in the usual 00:13:47.000 --> 00:13:53.000 way, but by using new ideas. That's the way you make 00:13:52.000 --> 00:13:58.000 progress. All right, so, 00:13:53.000 --> 00:13:59.000 let's do our bank account. So, y is money in the bank 00:13:58.000 --> 00:14:04.000 account. r is the interest rate. 00:14:02.000 --> 00:14:08.000 Let's assume it's a continuous interest rate. 00:14:06.000 --> 00:14:12.000 All banks nowadays pay interest continuously, 00:14:11.000 --> 00:14:17.000 the continuous interest rate. So, if that's all there is, 00:14:16.000 --> 00:14:22.000 and money is growing, you know the differential 00:14:21.000 --> 00:14:27.000 equation says that the rate at which it grows is equal to r, 00:14:26.000 --> 00:14:32.000 the interest rate times a principle, the amount that's in 00:14:32.000 --> 00:14:38.000 the bank at that time. So, that's the differential 00:14:38.000 --> 00:14:44.000 equation that governs that. Now, that's, 00:14:41.000 --> 00:14:47.000 of course, the solution is simply an exponential curve. 00:14:46.000 --> 00:14:52.000 There's nothing more to say about it. 00:14:49.000 --> 00:14:55.000 Now, let's make it more interesting. 00:14:52.000 --> 00:14:58.000 Let's suppose there is a shifty teller at the bank, 00:14:56.000 --> 00:15:02.000 and your money is being embezzled from your account at a 00:15:01.000 --> 00:15:07.000 constant rate. So, let's let w equal, 00:15:06.000 --> 00:15:12.000 or maybe e, but e has so many other uses in mathematics, 00:15:11.000 --> 00:15:17.000 w is relatively unused, w is the rate of embezzlement, 00:15:16.000 --> 00:15:22.000 thought of as continuous. So, every day a little bit of 00:15:21.000 --> 00:15:27.000 money is sneaked out of your account because you are not 00:15:26.000 --> 00:15:32.000 paying any attention to it. You're off skiing somewhere, 00:15:31.000 --> 00:15:37.000 and not noticing what's happening to your bank account. 00:15:38.000 --> 00:15:44.000 So, since it's the rate, the time rate of embezzlement, 00:15:42.000 --> 00:15:48.000 I simply subtract it from this. It's not w times y because the 00:15:47.000 --> 00:15:53.000 embezzler isn't stealing a certain fraction of your 00:15:51.000 --> 00:15:57.000 account. It's simply stealing a certain 00:15:54.000 --> 00:16:00.000 number of dollars every day, the same number of dollars 00:15:59.000 --> 00:16:05.000 being withdrawn for the count. Okay, now, of course, 00:16:03.000 --> 00:16:09.000 you could solve this. This separates variables 00:16:07.000 --> 00:16:13.000 immediately. You get the answer, 00:16:10.000 --> 00:16:16.000 and there's no problem with that. 00:16:12.000 --> 00:16:18.000 Let's analyze the behavior of the solutions without solving 00:16:17.000 --> 00:16:23.000 the equation by using these two points. 00:16:20.000 --> 00:16:26.000 So, I want to analyze this equation using the method of 00:16:24.000 --> 00:16:30.000 critical points. So, the first thing I should do 00:16:28.000 --> 00:16:34.000 is, so, here's our equation, is find the critical points. 00:16:34.000 --> 00:16:40.000 Notice it's an autonomous equation all right, 00:16:38.000 --> 00:16:44.000 because there's no t on the right-hand side. 00:16:42.000 --> 00:16:48.000 Okay, so, the critical points, well, that's where ry minus w 00:16:48.000 --> 00:16:54.000 equals zero. In other words, 00:16:51.000 --> 00:16:57.000 there's only one critical point, and that occurs when y is 00:16:56.000 --> 00:17:02.000 equal to w over r. So, that's the only critical 00:17:01.000 --> 00:17:07.000 point. Now, I want to know what's 00:17:05.000 --> 00:17:11.000 happening to the solution. So, in other words, 00:17:09.000 --> 00:17:15.000 if I plot, I can write away, of course, negative values 00:17:14.000 --> 00:17:20.000 aren't of particularly interesting here, 00:17:17.000 --> 00:17:23.000 there is definitely a horizontal solution, 00:17:21.000 --> 00:17:27.000 and it has the value, it's at the height, 00:17:25.000 --> 00:17:31.000 w over r. That's a solution. 00:17:30.000 --> 00:17:36.000 The question is, what do the other solutions 00:17:33.000 --> 00:17:39.000 look like? Now, watch how I make the 00:17:35.000 --> 00:17:41.000 analysis because I'm going to use two now. 00:17:39.000 --> 00:17:45.000 So, this is step one, then step two. 00:17:41.000 --> 00:17:47.000 What do I do? Well, I'm going to graph f of y. 00:17:45.000 --> 00:17:51.000 Well, f of y is ry minus w. 00:17:47.000 --> 00:17:53.000 What does that look like? 00:17:50.000 --> 00:17:56.000 Okay, so, here is the y-axis. Notice the y-axis is going 00:17:55.000 --> 00:18:01.000 horizontally because what I'm interested in is the graph of 00:17:59.000 --> 00:18:05.000 this function. What do I call the other axis? 00:18:04.000 --> 00:18:10.000 I'm going to use the same terminology that is used on the 00:18:08.000 --> 00:18:14.000 little visual that describes this. 00:18:11.000 --> 00:18:17.000 And, that's dy. You could call this other axis 00:18:14.000 --> 00:18:20.000 the f of y axis. That's not a good name for it. 00:18:18.000 --> 00:18:24.000 You could call it the dy / dt axis because it's, 00:18:22.000 --> 00:18:28.000 so to speak, the other variable. 00:18:24.000 --> 00:18:30.000 That's not great either. But, worst of all would be 00:18:28.000 --> 00:18:34.000 introducing yet another letter for which we would have no use 00:18:32.000 --> 00:18:38.000 whatsoever. So, let's think of it. 00:18:37.000 --> 00:18:43.000 We are plotting, now, the graph of f of y. 00:18:40.000 --> 00:18:46.000 f of y is this function, 00:18:43.000 --> 00:18:49.000 ry minus w. Well, that's a line. 00:18:48.000 --> 00:18:54.000 Its intercept is down here at w, and so the graph looks 00:18:53.000 --> 00:18:59.000 something like this. It's a line. 00:18:55.000 --> 00:19:01.000 This is the line, ry minus w. 00:18:59.000 --> 00:19:05.000 It has slope r. Well, what am I going to get 00:19:05.000 --> 00:19:11.000 out of that line? Just exactly this. 00:19:08.000 --> 00:19:14.000 What am I interested in about that line? 00:19:13.000 --> 00:19:19.000 Nothing other than where is it above the axis, 00:19:18.000 --> 00:19:24.000 and where is it below? This function is positive over 00:19:23.000 --> 00:19:29.000 here, and therefore, I'm going to indicate that 00:19:28.000 --> 00:19:34.000 symbolically, this, by putting an arrow here. 00:19:35.000 --> 00:19:41.000 The meeting of this arrow is that y of t is 00:19:39.000 --> 00:19:45.000 increasing. See where it's the right-hand 00:19:43.000 --> 00:19:49.000 side of that last board? y of t is increasing when f of 00:19:48.000 --> 00:19:54.000 y is positive. f of y is positive here, 00:19:52.000 --> 00:19:58.000 and therefore, to the right of this point, 00:19:55.000 --> 00:20:01.000 it's increasing. Here to the left of it, 00:19:59.000 --> 00:20:05.000 f of y is negative, and therefore over here it's 00:20:03.000 --> 00:20:09.000 going to be decreasing. What point is this, 00:20:08.000 --> 00:20:14.000 in fact? Well, that's where it crosses 00:20:11.000 --> 00:20:17.000 the axis. That's exactly the critical 00:20:15.000 --> 00:20:21.000 point, w over r. Therefore, what this says is 00:20:19.000 --> 00:20:25.000 that a solution, once it's bigger than y over r, 00:20:24.000 --> 00:20:30.000 it increases, and it increases faster and 00:20:28.000 --> 00:20:34.000 faster because this function is higher and higher. 00:20:34.000 --> 00:20:40.000 And, that represents the rate of change. 00:20:37.000 --> 00:20:43.000 So, in other words, once the solution, 00:20:40.000 --> 00:20:46.000 let's say a solution starts over here at time zero. 00:20:44.000 --> 00:20:50.000 So, this is the t axis. And, here is the y axis. 00:20:48.000 --> 00:20:54.000 So, now, I'm plotting solutions. 00:20:51.000 --> 00:20:57.000 If it starts at t equals zero, above this line, 00:20:55.000 --> 00:21:01.000 that is, starts with the value w over r, 00:20:59.000 --> 00:21:05.000 which is bigger than zero, a value bigger than w over r, 00:21:04.000 --> 00:21:10.000 then it increases, and increases faster and 00:21:07.000 --> 00:21:13.000 faster. If it starts below that, 00:21:12.000 --> 00:21:18.000 it decreases and decreases faster and faster. 00:21:16.000 --> 00:21:22.000 Now, in fact, I only have to draw two of 00:21:20.000 --> 00:21:26.000 those because what do all the others look like? 00:21:24.000 --> 00:21:30.000 They are translations. All the other curves look 00:21:29.000 --> 00:21:35.000 exactly like those. They are just translations of 00:21:35.000 --> 00:21:41.000 them. This guy, if I start closer, 00:21:39.000 --> 00:21:45.000 it's still going to decrease. Well, that's supposed to be a 00:21:47.000 --> 00:21:53.000 translation. Maybe it is. 00:21:50.000 --> 00:21:56.000 So, these guys look like that. Let's do just a tiny bit more 00:21:57.000 --> 00:22:03.000 interpretation of that. Well, I think I better leave it 00:22:06.000 --> 00:22:12.000 there because we've got harder things to do, 00:22:13.000 --> 00:22:19.000 and I want to make sure we've got time for it. 00:22:19.000 --> 00:22:25.000 Sorry. Okay, next example, 00:22:23.000 --> 00:22:29.000 a logistic equation. Some of you have already solved 00:22:31.000 --> 00:22:37.000 this in recitation, and some of you haven't. 00:22:39.000 --> 00:22:45.000 This is a population equation. This is the one that section 00:22:44.000 --> 00:22:50.000 7.1 and section 1.7 is most heavily concerned with, 00:22:48.000 --> 00:22:54.000 this particular equation. The derivation of it is a 00:22:52.000 --> 00:22:58.000 little vague. It's an equation which 00:22:55.000 --> 00:23:01.000 describes how population increases. 00:22:58.000 --> 00:23:04.000 And one minute, the population behavior of some 00:23:02.000 --> 00:23:08.000 population, -- -- let's call it, 00:23:06.000 --> 00:23:12.000 y is the only thing I know to call anything today, 00:23:10.000 --> 00:23:16.000 but of course your book uses capital P for population, 00:23:15.000 --> 00:23:21.000 to get you used to different variables. 00:23:18.000 --> 00:23:24.000 Now, the basic population equation runs dy / dt. 00:23:22.000 --> 00:23:28.000 There's a certain growth rate. Let's call it k y. 00:23:26.000 --> 00:23:32.000 So, k is what's called the growth rate. 00:23:29.000 --> 00:23:35.000 It's actually, sometimes it's talked about in 00:23:33.000 --> 00:23:39.000 terms of birthrate. But, it's the net birth rate. 00:23:38.000 --> 00:23:44.000 It's the rate at which people, or bacteria, 00:23:42.000 --> 00:23:48.000 or whatever are being born minus the rate at which they are 00:23:47.000 --> 00:23:53.000 dying. So, it's a net birthrate. 00:23:49.000 --> 00:23:55.000 But, let's just call it the growth rate. 00:23:53.000 --> 00:23:59.000 Now, if this is the equation, we can think of this, 00:23:57.000 --> 00:24:03.000 if k is constant, that's what's called simple 00:24:01.000 --> 00:24:07.000 population growth. And you are all familiar with 00:24:05.000 --> 00:24:11.000 that. Logistical growth allows for 00:24:08.000 --> 00:24:14.000 slightly more complex situations. 00:24:13.000 --> 00:24:19.000 Logistic growth says that calling k a constant is 00:24:18.000 --> 00:24:24.000 unrealistic because the Earth is not filled entirely with people. 00:24:26.000 --> 00:24:32.000 What stops it from having unlimited growth? 00:24:32.000 --> 00:24:38.000 Well, the fact that the resources, the food, 00:24:36.000 --> 00:24:42.000 the organism has to live on gets depleted. 00:24:41.000 --> 00:24:47.000 And, in other words, the growth rate declines as y 00:24:46.000 --> 00:24:52.000 increases. As the population increases, 00:24:50.000 --> 00:24:56.000 one expects the growth rate to decline because resources are 00:24:56.000 --> 00:25:02.000 being used up, and they are not indefinitely 00:25:01.000 --> 00:25:07.000 available. Well, in other words, 00:25:05.000 --> 00:25:11.000 we should replace k by a function with this behavior. 00:25:09.000 --> 00:25:15.000 What's the simplest function that declines as y increases? 00:25:14.000 --> 00:25:20.000 The simplest choice, and if you are ignorant about 00:25:17.000 --> 00:25:23.000 what else to do, stick with the simplest, 00:25:20.000 --> 00:25:26.000 at least you won't work any harder than you have to, 00:25:24.000 --> 00:25:30.000 would be to take k equal to the simplest declining function of y 00:25:29.000 --> 00:25:35.000 there is, which is simply a linear function, 00:25:32.000 --> 00:25:38.000 A minus BY. So, if I use that as the choice 00:25:37.000 --> 00:25:43.000 of the declining growth rate, the new equation is dy / dt 00:25:41.000 --> 00:25:47.000 equals, here's my new k. The y stays the same, 00:25:45.000 --> 00:25:51.000 so the equation becomes a minus by, the quantity times y, 00:25:50.000 --> 00:25:56.000 or in other words, 00:25:53.000 --> 00:25:59.000 ay minus b y squared. 00:25:55.000 --> 00:26:01.000 This equation is what's called the logistic equation. 00:26:01.000 --> 00:26:07.000 It has many applications, not just to population growth. 00:26:05.000 --> 00:26:11.000 It's applied to the spread of disease, the spread of a rumor, 00:26:11.000 --> 00:26:17.000 the spread of many things. Yeah, a couple pieces of chalk 00:26:16.000 --> 00:26:22.000 here. 00:26:28.000 --> 00:26:34.000 Okay, now, those of you who have solved it know that the 00:26:34.000 --> 00:26:40.000 explicit solution involves, well, you separate variables, 00:26:40.000 --> 00:26:46.000 but you will have to use partial fractions, 00:26:44.000 --> 00:26:50.000 ugh, I hope you love partial fractions. 00:26:48.000 --> 00:26:54.000 You're going to need them later in the term. 00:26:53.000 --> 00:26:59.000 But, I could avoid them now by not solving the equation 00:26:59.000 --> 00:27:05.000 explicitly. But anyway, you get a solution, 00:27:03.000 --> 00:27:09.000 which I was going to write on the board for you, 00:27:06.000 --> 00:27:12.000 but you could look it up in your book. 00:27:09.000 --> 00:27:15.000 It's unpleasant enough looking to make you feel that there must 00:27:13.000 --> 00:27:19.000 be an easier way at least to get the basic information out. 00:27:16.000 --> 00:27:22.000 Okay, let's see if we can get the basic information out. 00:27:20.000 --> 00:27:26.000 What are the critical points? Well, this is pretty easy. 00:27:23.000 --> 00:27:29.000 A, I want to set the right-hand side equal to zero. 00:27:26.000 --> 00:27:32.000 So, I'm going to solve the equation. 00:27:30.000 --> 00:27:36.000 I can factor out a y. It's going to be y times a 00:27:35.000 --> 00:27:41.000 minus by equals zero. 00:27:39.000 --> 00:27:45.000 And therefore, the critical points are where y 00:27:43.000 --> 00:27:49.000 equals zero. That's one. 00:27:47.000 --> 00:27:53.000 And, the other factor is when this factor is zero, 00:27:52.000 --> 00:27:58.000 and that happens when y is equal to a over b. 00:27:59.000 --> 00:28:05.000 So, there are my two critical points. 00:28:03.000 --> 00:28:09.000 Okay, what does, let's start drawing pictures of 00:28:08.000 --> 00:28:14.000 solutions. Let's put it in those right 00:28:12.000 --> 00:28:18.000 away. Okay, the critical point, 00:28:15.000 --> 00:28:21.000 zero, gives me a solution that looks like this. 00:28:18.000 --> 00:28:24.000 And, the critical point, a over b, 00:28:21.000 --> 00:28:27.000 those are positive numbers. So, that's somewhere up here. 00:28:25.000 --> 00:28:31.000 So, those are two solutions, constant solutions. 00:28:29.000 --> 00:28:35.000 In other words, if the population by dumb luck 00:28:32.000 --> 00:28:38.000 started at zero, it would stay at zero for all 00:28:35.000 --> 00:28:41.000 time. That's not terribly surprising. 00:28:39.000 --> 00:28:45.000 But, it's a little less obvious that if it starts at that magic 00:28:43.000 --> 00:28:49.000 number, a over b, it will also stay at that magic 00:28:47.000 --> 00:28:53.000 number for all time without moving up or down or away from 00:28:50.000 --> 00:28:56.000 it. Now, the question is, 00:28:52.000 --> 00:28:58.000 therefore, what happens in between? 00:28:54.000 --> 00:29:00.000 So, for the in between, I'm going to make that same 00:28:58.000 --> 00:29:04.000 analysis that I made before. And, it's really not very hard. 00:29:03.000 --> 00:29:09.000 Look, so here's my dy/dt-axis. I'll call that y prime, 00:29:09.000 --> 00:29:15.000 okay? And, here's the y-axis. 00:29:12.000 --> 00:29:18.000 So, I'm now doing step two. This was step one. 00:29:16.000 --> 00:29:22.000 Okay, the function that I want to graph is this one, 00:29:21.000 --> 00:29:27.000 ay minus b y squared, or in factor form, 00:29:26.000 --> 00:29:32.000 y times a minus by. 00:29:29.000 --> 00:29:35.000 Now, this function, we know, has a zero. 00:29:32.000 --> 00:29:38.000 It has a zero here, and it has a zero at the point 00:29:37.000 --> 00:29:43.000 a over b. At these two critical points, 00:29:43.000 --> 00:29:49.000 it has a zero. What is it doing in between? 00:29:46.000 --> 00:29:52.000 Well, in between, it's a parabola. 00:29:49.000 --> 00:29:55.000 It's a quadratic function. It's a parabola. 00:29:52.000 --> 00:29:58.000 Does it go up or does it go down? 00:29:55.000 --> 00:30:01.000 Well, when y is very large, it's very negative. 00:29:59.000 --> 00:30:05.000 That means it must be a downward-opening parabola. 00:30:04.000 --> 00:30:10.000 And therefore, this curve looks like this. 00:30:07.000 --> 00:30:13.000 So, I'm interested in knowing, where is it positive, 00:30:12.000 --> 00:30:18.000 and where is it negative? Well, it's positive, 00:30:16.000 --> 00:30:22.000 here, for this range of values of y. 00:30:19.000 --> 00:30:25.000 Since it's positive there, it will be increasing there. 00:30:24.000 --> 00:30:30.000 Here, it's negative, and therefore it will be 00:30:28.000 --> 00:30:34.000 decreasing. Here, it's negative, 00:30:32.000 --> 00:30:38.000 and therefore, dy / dt will be negative also, 00:30:36.000 --> 00:30:42.000 and therefore the function, y, will be decreasing here. 00:30:41.000 --> 00:30:47.000 So, how do these other solutions look? 00:30:44.000 --> 00:30:50.000 Well, we can put them in. I'll put them in in white, 00:30:49.000 --> 00:30:55.000 okay, because this has got to last until the end of the term. 00:30:54.000 --> 00:31:00.000 So, how are they doing? They are increasing between the 00:30:59.000 --> 00:31:05.000 two curves. They are not allowed to cross 00:31:04.000 --> 00:31:10.000 either of these yellow curves. But, they are always 00:31:08.000 --> 00:31:14.000 increasing. Well, if they're always 00:31:11.000 --> 00:31:17.000 increasing, they must start here and increase, 00:31:15.000 --> 00:31:21.000 and not allowed to cross. It must do something like that. 00:31:19.000 --> 00:31:25.000 This must be a translation of it. 00:31:22.000 --> 00:31:28.000 In other words, the curves must look like that. 00:31:26.000 --> 00:31:32.000 Those are supposed to be translations of each other. 00:31:32.000 --> 00:31:38.000 I know they aren't, but use your imaginations. 00:31:35.000 --> 00:31:41.000 But what's happening above? So in other words, 00:31:38.000 --> 00:31:44.000 if I start with a population anywhere bigger than zero but 00:31:42.000 --> 00:31:48.000 less than a over b, it increases asymptotically to 00:31:46.000 --> 00:31:52.000 the level a over b. What happens if I start above 00:31:50.000 --> 00:31:56.000 that? Well, then it decreases to it 00:31:52.000 --> 00:31:58.000 because, this way, for the values of y bigger than 00:31:56.000 --> 00:32:02.000 a over b, it decreases as time increases. 00:32:00.000 --> 00:32:06.000 So, these guys up here are doing this. 00:32:04.000 --> 00:32:10.000 And, how about the ones below the axis? 00:32:06.000 --> 00:32:12.000 Well, they have no physical significance. 00:32:09.000 --> 00:32:15.000 But let's put them in anyway. Whether they doing? 00:32:12.000 --> 00:32:18.000 They are decreasing away from zero. 00:32:15.000 --> 00:32:21.000 So, these guys don't mean anything physically, 00:32:18.000 --> 00:32:24.000 but mathematically they exist. Their solutions, 00:32:21.000 --> 00:32:27.000 they're going down like that. Now, you notice from this 00:32:25.000 --> 00:32:31.000 picture that there are, even though both of these are 00:32:29.000 --> 00:32:35.000 constant solutions, they have dramatically 00:32:32.000 --> 00:32:38.000 different behavior. This one, this solution, 00:32:37.000 --> 00:32:43.000 is the one that all other solutions try to approach as 00:32:41.000 --> 00:32:47.000 time goes to infinity. This one, the solution zero, 00:32:45.000 --> 00:32:51.000 is repulsive, as it were. 00:32:48.000 --> 00:32:54.000 Any solution that starts near zero, if it starts at zero, 00:32:52.000 --> 00:32:58.000 of course, it stays there for all time, but if it starts just 00:32:58.000 --> 00:33:04.000 a little bit above zero, it increases to a over b. 00:33:02.000 --> 00:33:08.000 This is called a stable 00:33:05.000 --> 00:33:11.000 solution because everybody tries to get closer and closer to it. 00:33:11.000 --> 00:33:17.000 This is called, zero is also a constant 00:33:14.000 --> 00:33:20.000 solution, but this is an unstable solution. 00:33:17.000 --> 00:33:23.000 And now, usually, solution is too general a word. 00:33:21.000 --> 00:33:27.000 I think it's better to call it a stable critical point, 00:33:26.000 --> 00:33:32.000 and an unstable critical point. But, of course, 00:33:30.000 --> 00:33:36.000 it also corresponds to a solution. 00:33:34.000 --> 00:33:40.000 So, critical points are not all the same. 00:33:37.000 --> 00:33:43.000 Some are stable, and some are unstable. 00:33:40.000 --> 00:33:46.000 And, you can see which is which just by looking at this picture. 00:33:46.000 --> 00:33:52.000 If the arrows point towards them, you've got a stable 00:33:51.000 --> 00:33:57.000 critical point. If it arrows point away from 00:33:55.000 --> 00:34:01.000 them, you've got an unstable critical point. 00:33:58.000 --> 00:34:04.000 Now, there is a third possibility. 00:34:03.000 --> 00:34:09.000 Okay, I think we'd better address it because otherwise 00:34:09.000 --> 00:34:15.000 you're going to sit there wondering, hey, 00:34:13.000 --> 00:34:19.000 what did he do? Suppose it looks like this. 00:34:18.000 --> 00:34:24.000 Suppose it were just tangent. Well, this is the picture of 00:34:24.000 --> 00:34:30.000 that curve, the pink curve. What would the arrows look like 00:34:31.000 --> 00:34:37.000 then? What would the arrows look like 00:34:35.000 --> 00:34:41.000 then? Well, since they are positive, 00:34:38.000 --> 00:34:44.000 it's always positive, the arrow goes like this. 00:34:41.000 --> 00:34:47.000 And then on the side, it also goes in the same 00:34:45.000 --> 00:34:51.000 direction. So, is this critical point 00:34:47.000 --> 00:34:53.000 stable or unstable? It's stable if you approach it 00:34:51.000 --> 00:34:57.000 from the left. So, how, in fact, 00:34:53.000 --> 00:34:59.000 do the curves, how would the corresponding 00:34:57.000 --> 00:35:03.000 curves look? Well, there's our long-term 00:35:00.000 --> 00:35:06.000 solution. This corresponds to that point. 00:35:05.000 --> 00:35:11.000 Let's call this a, and then this will be the 00:35:10.000 --> 00:35:16.000 value, a. If I start below it, 00:35:14.000 --> 00:35:20.000 I rise to it. If I start above it, 00:35:18.000 --> 00:35:24.000 I increase. So, if I start above it, 00:35:22.000 --> 00:35:28.000 I do this. Well, now, that's stable on one 00:35:27.000 --> 00:35:33.000 side, and unstable on the other. And, that's indicated by saying 00:35:35.000 --> 00:35:41.000 it's semi-stable. That's a brilliant word. 00:35:40.000 --> 00:35:46.000 I wonder how long it to do think that one up, 00:35:43.000 --> 00:35:49.000 semi-stable critical point: stable on one side, 00:35:46.000 --> 00:35:52.000 unstable on the other depending on whether you start below it. 00:35:50.000 --> 00:35:56.000 And, of course, it could be reversed if I had 00:35:53.000 --> 00:35:59.000 drawn the picture the other way. I could have approached it from 00:35:57.000 --> 00:36:03.000 the top, and left it from below. You get the idea of the 00:36:03.000 --> 00:36:09.000 behavior. Okay, let's now take, 00:36:08.000 --> 00:36:14.000 I'm going to soup up this logistic equation just a little 00:36:16.000 --> 00:36:22.000 bit more. So, let's talk about the 00:36:21.000 --> 00:36:27.000 logistic equation. But, I'm going to add to it 00:36:28.000 --> 00:36:34.000 harvesting, with harvesting. So, this is a very late 20th 00:36:36.000 --> 00:36:42.000 century concept. So, we imagine, 00:36:39.000 --> 00:36:45.000 for example, a bunch of formerly free range 00:36:43.000 --> 00:36:49.000 Atlantic salmon penned in one of these huge factory farms off the 00:36:49.000 --> 00:36:55.000 coast of Maine or someplace. They've made salmon much 00:36:54.000 --> 00:37:00.000 cheaper than it used to be, but at a certain cost to the 00:37:00.000 --> 00:37:06.000 salmon, and possibly to our environment. 00:37:05.000 --> 00:37:11.000 So, what happens? Well, the salmon grow, 00:37:08.000 --> 00:37:14.000 and grow, and do what salmon do. 00:37:11.000 --> 00:37:17.000 And, they are harvested. That's a word somewhere in the 00:37:15.000 --> 00:37:21.000 category of ethnic cleansing in my opinion. 00:37:19.000 --> 00:37:25.000 But, it's, again, a very 20th-century word. 00:37:23.000 --> 00:37:29.000 I think it was Hitler who discovered that, 00:37:27.000 --> 00:37:33.000 that all you had to do was call something by a sanitary name, 00:37:32.000 --> 00:37:38.000 and no matter how horrible it was, good bourgeois people would 00:37:37.000 --> 00:37:43.000 accept it. So, the harvesting, 00:37:42.000 --> 00:37:48.000 which means, of course, picking them up and 00:37:47.000 --> 00:37:53.000 killing them, and putting them in cans and 00:37:52.000 --> 00:37:58.000 stuff like that, okay, so what's the equation? 00:37:58.000 --> 00:38:04.000 I'm going to assume that the harvest is at a constant time 00:38:05.000 --> 00:38:11.000 rate. In other words, 00:38:08.000 --> 00:38:14.000 it's not a certain fraction of all the salmon that are being 00:38:13.000 --> 00:38:19.000 caught each day and canned. The factory has a certain 00:38:17.000 --> 00:38:23.000 capacity, so, 400 pounds of salmon each day 00:38:21.000 --> 00:38:27.000 are pulled out and canned. So, it's a constant time rate. 00:38:25.000 --> 00:38:31.000 That means that the equation is now going to be dy/dt is equal 00:38:31.000 --> 00:38:37.000 to, well, salmon grow logistically. 00:38:35.000 --> 00:38:41.000 ay minus b y squared, so, that part of the 00:38:39.000 --> 00:38:45.000 equation is the same. But, I need a term to take care 00:38:44.000 --> 00:38:50.000 of this constant harvesting rate, and that will be h. 00:38:48.000 --> 00:38:54.000 Let's call it h, not h times y. 00:38:50.000 --> 00:38:56.000 Then, I would be harvesting a certain fraction of all the 00:38:55.000 --> 00:39:01.000 salmon there, which is not what I'm doing. 00:39:00.000 --> 00:39:06.000 Okay: our equation. Now, I want to analyze what the 00:39:03.000 --> 00:39:09.000 critical points of this look like. 00:39:05.000 --> 00:39:11.000 Now, this is a little more subtle because there's now a new 00:39:09.000 --> 00:39:15.000 parameter, there. And, what I want to see is how 00:39:12.000 --> 00:39:18.000 that varies with the new parameter. 00:39:15.000 --> 00:39:21.000 The best thing to do is, I mean, the thing not to do is 00:39:19.000 --> 00:39:25.000 make this equal to zero, fiddle around with the 00:39:22.000 --> 00:39:28.000 quadratic formula, get some massive expression, 00:39:25.000 --> 00:39:31.000 and then spend the next half hour scratching your head trying 00:39:29.000 --> 00:39:35.000 to figure out what it means, and what information you are 00:39:33.000 --> 00:39:39.000 supposed to be getting out of it. 00:39:37.000 --> 00:39:43.000 Draw pictures instead. Draw pictures. 00:39:40.000 --> 00:39:46.000 If h is zero, that's the smallest harvesting 00:39:45.000 --> 00:39:51.000 rate I could have. The picture looks like our old 00:39:50.000 --> 00:39:56.000 one. So, if h is zero, 00:39:52.000 --> 00:39:58.000 the picture looks like, what color did I, 00:39:56.000 --> 00:40:02.000 okay, pink. Yellow. 00:40:00.000 --> 00:40:06.000 Yellow is the cheapest, but I can't find it. 00:40:03.000 --> 00:40:09.000 Okay, yellow is commercially available. 00:40:06.000 --> 00:40:12.000 These are precious. All right, purple if it's okay, 00:40:11.000 --> 00:40:17.000 purple. So, this is the one, 00:40:13.000 --> 00:40:19.000 our original one corresponding to h equals zero. 00:40:17.000 --> 00:40:23.000 Or, in other words, it's the equation ay minus b y 00:40:21.000 --> 00:40:27.000 squared. h is zero. 00:40:24.000 --> 00:40:30.000 Now, if I want to find, I now want to increase the 00:40:28.000 --> 00:40:34.000 value of h, well, if I increase the value of h, 00:40:32.000 --> 00:40:38.000 in other words, harvest more and more, 00:40:35.000 --> 00:40:41.000 what's happening? Well, I simply lower this 00:40:41.000 --> 00:40:47.000 function by h. So, if I lower h somewhat, 00:40:45.000 --> 00:40:51.000 it will come to here. So, this is some value, 00:40:49.000 --> 00:40:55.000 ay minus b y squared minus h1, 00:40:54.000 --> 00:41:00.000 let's say. That's this curve. 00:40:57.000 --> 00:41:03.000 If I lower it a lot, it will look like this. 00:41:03.000 --> 00:41:09.000 So, ay minus b y squared minus h a lot, h twenty. 00:41:08.000 --> 00:41:14.000 This doesn't mean anything. 00:41:11.000 --> 00:41:17.000 Two. Obviously, there's one 00:41:14.000 --> 00:41:20.000 interesting value to lower it by. 00:41:17.000 --> 00:41:23.000 It's a value which would lower it exactly by this amount. 00:41:22.000 --> 00:41:28.000 Let me put that in special. If I lower it by just that 00:41:27.000 --> 00:41:33.000 amount, the curve always looks the same. 00:41:32.000 --> 00:41:38.000 It's just been lowered. I'm going to say this one is, 00:41:36.000 --> 00:41:42.000 so this one is the same thing, except that I've subtracted h 00:41:42.000 --> 00:41:48.000 sub m. Where is h sub m on 00:41:46.000 --> 00:41:52.000 the picture? Well, I lowered it by this 00:41:49.000 --> 00:41:55.000 amount. So, this height is h sub m. 00:41:53.000 --> 00:41:59.000 In other words, if I find the maximum height 00:41:57.000 --> 00:42:03.000 here, which is easy to do because it's a parabola, 00:42:01.000 --> 00:42:07.000 and lower it by exactly that amount, I will have lowered it 00:42:07.000 --> 00:42:13.000 to this point. This will be a critical point. 00:42:12.000 --> 00:42:18.000 Now, the question is, what's happened to the critical 00:42:16.000 --> 00:42:22.000 point as I did this? I started with the critical 00:42:20.000 --> 00:42:26.000 points here and here. As I lower h, 00:42:22.000 --> 00:42:28.000 the critical point changed to this and that. 00:42:26.000 --> 00:42:32.000 And now, it changed to this one when I got to the purple line. 00:42:32.000 --> 00:42:38.000 And, as I went still further down, there were no critical 00:42:36.000 --> 00:42:42.000 points. So, this curve has no critical 00:42:38.000 --> 00:42:44.000 points attached to it. What are the corresponding 00:42:42.000 --> 00:42:48.000 pictures? Well, the corresponding 00:42:44.000 --> 00:42:50.000 pictures, well, we've already drawn, 00:42:47.000 --> 00:42:53.000 the picture for h equals zero is drawn already. 00:42:51.000 --> 00:42:57.000 The pictures that I'm talking about are how the solutions 00:42:55.000 --> 00:43:01.000 look. How would the solution look 00:42:57.000 --> 00:43:03.000 like for this one for h one? 00:43:00.000 --> 00:43:06.000 For h1, the solutions look like, here is a over b. 00:43:06.000 --> 00:43:12.000 Here is a over b, but the critical points aren't 00:43:10.000 --> 00:43:16.000 at zero and a over b anymore. They've moved in a little bit. 00:43:15.000 --> 00:43:21.000 So, they are here and here. And, otherwise, 00:43:18.000 --> 00:43:24.000 the solutions look just like they did before, 00:43:22.000 --> 00:43:28.000 and the analysis is the same. And, similarly, 00:43:25.000 --> 00:43:31.000 if h two goes very far, if h2 is very large, 00:43:29.000 --> 00:43:35.000 there are no critical points. h, too large, 00:43:33.000 --> 00:43:39.000 no critical points. Are the solutions decreasing 00:43:38.000 --> 00:43:44.000 all the time or increasing? Well, they are always 00:43:42.000 --> 00:43:48.000 decreasing because the function is always negative. 00:43:46.000 --> 00:43:52.000 Solutions always go down, always. 00:43:49.000 --> 00:43:55.000 The interesting one is this last one, where I decreased it 00:43:54.000 --> 00:44:00.000 just to (h)m. And, what happens there is 00:43:57.000 --> 00:44:03.000 there is this certain, magic critical point whose 00:44:01.000 --> 00:44:07.000 value we could calculate. There's one constant solution. 00:44:07.000 --> 00:44:13.000 So, this is one that has the value. 00:44:10.000 --> 00:44:16.000 Sorry, I'm calculating the solutions out. 00:44:13.000 --> 00:44:19.000 So, y here and t here, so here it is value, 00:44:17.000 --> 00:44:23.000 (h)m is the value by which it has been lowered. 00:44:20.000 --> 00:44:26.000 So, this is the picture for (h)m. 00:44:23.000 --> 00:44:29.000 And, how do the solutions look? Well, to the right of that, 00:44:29.000 --> 00:44:35.000 they are decreasing. And, to the left they are also 00:44:33.000 --> 00:44:39.000 decreasing because this function is always negative. 00:44:37.000 --> 00:44:43.000 So, the solutions look like this, if you start above, 00:44:42.000 --> 00:44:48.000 and if you start below, they decrease. 00:44:45.000 --> 00:44:51.000 And, of course, they can't get lower than zero 00:44:49.000 --> 00:44:55.000 because these are salmon. What is the significance of 00:44:53.000 --> 00:44:59.000 (h)m? (h)m is the maximum rate of 00:44:56.000 --> 00:45:02.000 harvesting. It's an extremely important 00:44:59.000 --> 00:45:05.000 number for this industry. If the maximum time rate at 00:45:05.000 --> 00:45:11.000 which you can pull the salmon daily out of the water, 00:45:09.000 --> 00:45:15.000 and can them without what happening? 00:45:12.000 --> 00:45:18.000 Without the salmon going to zero. 00:45:15.000 --> 00:45:21.000 As long as you start above, and don't harvest it more than 00:45:20.000 --> 00:45:26.000 this rate, it will be following these curves. 00:45:23.000 --> 00:45:29.000 You will be following these curves, and you will still have 00:45:28.000 --> 00:45:34.000 salmon. If you harvest just a little 00:45:31.000 --> 00:45:37.000 bit more, you will be on this curve that has no critical 00:45:36.000 --> 00:45:42.000 points, and the salmon in the tank will decrease to zero.