 Welcome back everyone. In this video, we're going to take a look at a third growth model, commonly referred to as logistic growth. That's going to be the best of both worlds between the inhibited growth and the uninhibited growth. So we like about the inhibited growth model as it does have this idea of a maximum population. Your population can't just grow, grow, grow, grow, grow without bounds, right? It has a carrying capacity built into it. But the inhibited growth still is insufficient to grow, to model every growth model. While the growth shrinks, when you get close to the maximum population, for inhibited growth, you have rapid, rapid growth when you're far away from the, when you're far away from the maximum population. That works really well when you're like cooling temperatures and such, but for a population of organisms, that's not exactly the best model. If your population is so small, it actually might be difficult for organisms to find each other, reproduce, mate, spread, all of that. In which case, the more, the more the more population is, the more you can grow, but also you can't grow too big. So what we want to do is look for a growth model that's a compromise, and that's this logistic growth model. So consider the following differential equation. We'll give you some explanation of why this was chosen in just a second. So take the equation dp over dt equals kp times 1 minus p over m. So what is this thing doing right here? Well, the first thing to note is, as p goes to 0, right, if your population is really, really small, and that's how I want you to interpret this, if you have small population, what that means is, this expression 1 minus p over m, well, this would converge, well, this will, this will converge to 1 minus 0 over m. Well, that's gonna be 1 minus 0. That is 1. And that situation, we then get that our derivative p prime, it will be approximately kp times 1, the 1 minus p over m becomes 1, and thus you get kp. And so what we see right here is that when our population is small, this right here, kp, this is gonna look like uninhibited growth. It looks like there's no end to how big this thing can grow. All right, so when it's small, it'll look like natural exponential growth. On the other hand, as the population gets closer to the maximum population, we see that we get 1 minus p over m. This thing will converge towards 1 minus m over m, which is 1 minus 1, which is now 0. And so in this situation, we see that p prime will be approximately kp times 0, that is 0. And so what's happening here is that when p gets closer to m, this looks like there's no growth whatsoever. And that's because this is now inhibited growth. And so this model, this logistic model here, then in fact, finds this hybrid between inhibited growth and uninhibited growth. And how did we come up with this thing? Well, this model is based upon the following assumptions. All models here are based upon assumptions. And this model, p prime, we're assuming is jointly, it's jointly proportional to two things. We have the size, the current size, which is p, and we have the so-called relative, relative elbow room, which is this quantity 1 minus m over p. Let's give some explanation. Oh, I wrote that upside down. 1 minus p over m. To give some explanation of what that thing means here, if we take the elbow room we talked about before, m minus p, that's how much capacity is there to grow towards the maximum. Divide that by the maximum. And this is going to give us a percentage. This will tell us what percentage we have growth left. And so this is this relative growth, a relative elbow room we were talking about. And this would be, could be written as 1 minus p over m. Like so. So we see that this differential equation has built into it. These assumptions about when it's small, it'll grow slowly. When it's big, it'll grow slowly. But when you're in the middle, it'll grow rapidly. This is the Goldilocks of growth models. Not too hot, not too cold. That's when it'll grow rapidly. So great. We came up with a differential equation. That's sort of like the first part of this. But now how do we solve this differential equation? Well, just like inhibited growth, this differential equation, p prime, we'll write dp over dt. This is a separable differential equation. kp times 1 minus m over p. We can separate the variables, in which case we're going to get dp over p times 1 minus m over p. Like so. And this will equal k dt. Integrating both sides, once you've separated the variables, the right-hand side is pretty much a cinch. It's going to look like kt plus a constant. The left-hand side is going to take a lot more effort. So to begin with, we actually have this rational expression for which to integrate it, we need to have to do a partial fraction decomposition. So we get 1 over p times 1 minus m over p. I don't really like fractions inside of fractions. So to begin with, oh, did I write, I wrote the fraction upside down again. I'm sorry. I keep on doing that. Let me fix this real quick. So this should be p over m. And I'll also kill off the other erroneous instances of this. So we have p over m. And then we have here p over m. Great. So to clear the baby fractions inside of the mom fraction there, we're going to times the top and bottom by m. And so to distribute the m onto this piece right here, this would give us m over p times m minus p. And so this thing we want to decompose into partial fractions, we're going to get a over p plus b over m minus p. If we cleared the denominators, this will give us m is equal to a times m minus p plus bp. And so let's try some cool annihilating values. If we take p to equal zero, that'll annihilate the b. And we're left with m is equal to a m. That is to say a equals 1. And then likewise, if we take p to equal m, that'll annihilate a. And you're going to get that m equals bm. Divide both sides by m. You get b equals 1, like so. And so our partial fraction decomposition would then become 1 over p plus 1 over m minus p, like so. And so returning to the integral, we now have to integrate 1 over p dp plus the integral of 1 over m minus p dp. This is equal to kt plus a constant. The first one, the first integral, 1 over p, that's going to give us the natural log of p. I'm ignoring the absolute value here because the population does have to be positive here. For the second one, you're going to get minus the natural log of m minus p. Again, we can't be bigger than the maximum, so m minus p is positive. This equals kt plus c. Combining the logarithms, we're going to get the natural log of p over m minus p, kt plus c. And so for matters of convenience, we're actually going to flip this thing upside down. Actually, I guess another way of saying it is if we actually times both sides of the equation by negative 1 over here, this gives us the natural log of m minus p over p, and this is equal to negative kt plus c. The gelatinous cube absorbs the negative sign, and we have the following equation here. So then to get rid of the natural log, we'll take the natural exponential of both sides. We get m minus p over p, and this is equal to e to negative kt plus a constant. And like we've seen before, a gelatinous cube in the exponent actually becomes a coefficient out front. Cross-multiply, that is times both sides by p. So the p's cancel over here. We're going to get m minus p is equal to c, e to negative kt times p. Add p to both sides. We now are going to have m equals p plus p, c, e to the negative kt, like so. Act around the p on the right. So you get one plus e, sorry, c, e to negative kt. This is equal to m. And so divide both sides by the coefficient that we see right here. And now we get the logistic model we were looking for, p equals big fraction m on top, one plus c, e to the negative kt on the bottom. All right, it takes a little bit of effort, but let's go back up to what we had already written on the screen, and here we go. p equals m over one plus c, e to the negative t. As we've seen in other examples, we do have to investigate what is this constant c right here. If you plug in p or t equals zero, you're going to get p zero equals the maximum population on top, one plus c, e to the zero, that is m over one plus c. Cross-multiply, you're going to get p times one plus c is equal to m. In that case, I guess we could, well, we want to solve for c, so let's just divide by m. We're going to get one plus c equals m over p, that is c equals m over p minus one. And I guess I should be writing p here as p naught. This is the initial population. To find a common denominator, the one becomes a p naught over p naught. And so this thing would then become what we see over here. c equals m minus p naught over p naught. And so this would be the initial relative elbow room. That's this coefficient. So these coefficients c always have to do with some type of initial value. Let's look at a quick example of how one could use this to try to make predictions about a population. So let's write the solution of the initial value problem given by the following differential equation. So we have p prime equals 0.08 p times one minus p over 1,000. And we have an initial value of 100 organisms in this population. So we should hopefully recognize that this right here is the logistic differential equation. Our k value is 0.08. We see it right here. And our population, maximum population is 1,000, which we see right here. So we don't have to go through driving the derivative, or the general solution all over again, because we just saw that. We use the logistic model the way we saw just a moment ago. The population is going to equal the maximum population over one plus c e to the negative kt. So the maximum population was 1,000. So 1,000 over one plus. We got to calculate the c value. c was the maximum minus p naught over p naught. This initial relative elbow room, we get 1,000 minus the initial population of 100 divided by 100. So you're going to get 900 over 100. We end up with a coefficient of 9. We're going to go in right here. So we get 9 times e to the negative k, negative 0.08t. And so this right here gives us the logistic equation that we want to model this data with. And for many science classes or college algebra classes, this is where things start. They give you the natural exponential growth model. They give you Newton's law of cooling. They give you the logistic model. And they don't give you the darndest clue on where it came from. And that's because there's some impressive differential equations that show us where it comes from. But once we know it, we can just use it by plugging in the initial conditions here. And so now let's make some calculations about p 40 and p 80. So if we were to do that p 40, this is going to look like 1,000 on top, 1 plus 9 e to the negative 0.08 times 40. And this is some number we're going to shove into a calculator. I'm not going to go through all the details of this, but this would be approximately 731.6. And so again, rounding here, we probably would say something like, oh, this would be 731 or 730, something like that. If we do the 80 value, p of 80, same basic idea of 1,000 over 1 plus 9 e to the negative 0.08 times 80. Again, this is a number we're going to throw into a calculator here. And the calculator will estimate this thing to be approximately 985.3. And so again, we probably would round this to like 985, 980, 990, something like that because we can't have a percentage. Another question to ask here is when will it reach 900? So when will the population equal 900? Well, that one's a little bit more challenging to do so, but we can do it. So we're asking ourselves, when is the population equal to 900? So equal 1,000 over 1 plus 9 e to the negative 0.08. So we can use a calculator to help us out here, but algebraically, if you go through this, you're going to cross-multiply these things together. So you end up with 900 times 1 plus 9 e to the negative 0.08 t. This is equal to 1,000. Like so, I'm just going to divide both sides by 900 at this moment, in which case we then get 1 plus 9 e to the negative 0.08 t. This is equal to 10 ninths or 1 and a ninth. In which case then if we subtract 1 from both sides, we end up seeing that we get 9 e to the negative 0.08 t equals just 1 ninth. Divide both sides by 9. Do that over here. You're going to get 1 over 81. And then we have to take the natural log of both sides. You're going to get the natural log, sorry, you're going to get negative 0.08 t is equal to the natural log of 1 over 81, which is the same thing as negative natural log of 81, like so. And then if you divide both sides by negative 0.08, we get t equals, it's a double negative. So you're going to get 1 over 0.08 times the natural log of 81. Now 81 of course is the same thing as 3 to the fourth. We can use that. I also don't really like dividing by decimals. It just kind of feels a little uncocher. So you can move that over. You can move that over. So moving the decimal places and bringing this four out in front. This fraction here would simplify to be 50. There's a little bit of details going on there. I'm skipping, but really we're not doing any calculus at this moment. This is just some algebra that's now devolved and just just arithmetic. This would be approximately 54.9. So around 55 time units is when this population will reach 900. And so once we have this algebraic equation in hand, we can use it to answer questions about what's the population at this time, when will the population reach this, and other types of analyses that we do in an algebra class. And that's why these functions like these models like logistic growth and natural exponentials are given to an algebra student is that with the function in hand, they can analyze it using their algebraic tools. But they're nowhere prepared to understand why one uses that function. And now something always baffled me when I was an algebra student. Where did the world that this logistic model come from? But when you see the solution as a differential equation, it actually is quite natural. When you look at the original logistic model, it's like, oh, we want to be jointly proportional to the current size in the current elbow room. Great, that makes sense a lot. And so those assumptions are very natural. And that leads to what seems like an unnatural equation, but it's actually quite natural. This is why the number E is such a natural number, so to speak, is that although it's an irrational number, so from like an arithmetic point of view, it seems completely unnatural. But from a differential equation point of view, the assumption was very natural, and it leads to these exponential like expressions. And so as we finish off this lecture, lecture 27, I just want to mention that these three growth models we've seen, uninhibited growth, inhibited growth, and logistic growth, these do not account for every growth model, not every decay model that one could have. Other factors that we haven't considered could be like seasonal growth or cyclical growth. Maybe there's some type of cycle or pattern to when things grow. You could have things like migration or an exodus, things are leaving. Maybe there's a minimum population that if our population gets too small, then you can't grow anymore. You actually decay, right? Think of like the white rhino or something. There's sort of a limit that if it gets too small, the thing will just die off. There's issues like predator and prey. That's actually kind of an interesting problem, right? If you have a large population of prey, it becomes easier for the predators to eat them. But then when they start eating them too much, the predator population will get bigger because there's a lot of opportunity for food. But then the prey population gets small and then it's too hard for the predators to find enough food for everyone. So some of the predators fall down, but then the prey goes up. And there's a lot of reasons, a lot of context, a lot of assumptions that go into finding a good growth model. We've only seen some of the very basic examples. I don't want you to think this is all of it. But hopefully this gets you excited. So in the news, when you read about the spread of an infectious disease, are you an infectious disease called Facebook? Or when you look at the other type of statistical analysis about things growing and decaying, you can have some appreciation of where those things came from. It came from solving differential equations much like the logistic model. Now, we should also mention that all the examples we saw in this lecture do, they were solved as several different, separable differential equations. Of course, not every differential equation is separable. There are many times where it will be here, you know, inseparable. And so we're going to learn some more techniques of solving differential equations in our next lecture. See you then.