 Welcome back everyone. What I want to do next is talk about an alternative model to the natural growth We saw in the previous video here now as a model of population growth Natural growth is typically good only for a short amount of time. That is that equation P Equals P naught e to the kt. This works really only well when you don't have a really long time period So for short intervals, this is going to make a very good assumption So why does this why does this law of? Uninhibited growth not work in the grand scheme of things. Why does it become unrealistic over a long time period? Well, that's because it's based upon only one assumption the bigger we are the faster we grow There are additional factors that could affect the population's growth such as space restrictions Or maybe limited amounts of food Predators Immigration there. There's a lot of other things that could be affecting the population here And so well, let's take the idea of scarcity of resources. What if there is some type of scarcity? You can't necessarily have infinite Resources right we can't just open up our game and type in like Show me the money or glittering prizes or any other cheat code like that. What if our population? Has a certain carrying capacity. There is a maximum population For which that the ecosystem can support and beyond that it doesn't work So what if you have this maximum population in what would happen in such a situation like that? Well, what we can do is we can create a growth model that has this Maximum population built into it. Let us take the assumption Let's take the assumption that the more space or the so to speak The more elbow room one has The more elbow room the more elbow room the faster growth So what I mean by elbow room right here Well, imagine you are going to go watch a movie in a theater or you want to go out to dinner at a restaurant Well, when you go to these Attractions if you're the first one there, it's like hey, I can sit anywhere I want wonderful, but then when the next party comes they're gonna kind of sit but they're not gonna sit right next to you They're gonna go sit somewhere else and then the third party's gonna sit somewhere else and there and there and there and there And so as the theater or the restaurant starts filling up all the chairs start getting filled all the tables start getting filled People are gonna start to not want to go there anymore. It's like hey that restaurant looks kind of busy right now Maybe we should go to the movies or it's like hey this movie theater is kind of full Maybe let's just refund our tickets and let's go to a restaurant instead if there's no room if there's no room in the in the theater people aren't gonna They'll come into it and people's willingness to expand in the theater has a lot to do with how much space is there left So when there's a lot of elbow room and so by elbow room I mean take the maximum population subtract the current population This is what we mean by elbow room if there's a lot of elbow room We're gonna grow rapidly and then as things get closer and closer and closer to The maximum population the growth will tend to slow down down and down down So what we're saying here is what if the growth rate P prime is proportional to the elbow room? Things grow fast when there's a lot of elbow room and things will grow slowly when there's not a lot of elbow room We'll clear the denominator. We get the following equation right here P prime equals K times M minus P and this gives us the law of inhibited growth in this growth model We have built into the fact that the more we have the slower it grows There'll be a quick explosion of growth at the very beginning, but then it slows down over time All right, and so Just like with our law of uninhibited growth K is our growth factor when it's positive This is gonna be a growth model and when it's negative This would represent some type of decay Like so and so for the most part We're gonna make some assumptions that K is always gonna be a positive number and that P is gonna be less P can be less than M That is there is some room to grow because if we actually had P larger than M even with a positive growth model Things would decay because there's not enough. There's not enough space to support all these things So with this differential equation in hand, how does one solve this to give us? Well, give us a general a general form of us a general equation to work with here Now this right here is a separable differential equation We could try to separate the variables to help us out here So I would begin by times in both sides of the equation by dt and we're also gonna Divide both sides by M minus P so they cancel over here and then minus P over there So doing that we're gonna get dp over M minus P This is equal to K dt and now we want to integrate these things with respect to the variables on the two sides The right hand side is gonna be fairly straightforward. This is just gonna be t plus a constant Sorry kt times a constant on the on the left hand side in order to integrate with respect to P I'm gonna do a quick u substitution u equals M minus P Therefore du equals negative dp. So we need a double negative to correct that Then the left hand side would look like negative the integral of du over u This becomes the negative natural log of the absolute value of u Like so and then replacing u with its value We end up with negative the natural log of the absolute value of M minus P Now because we're assuming that P is always less than M. This actually does mean that M minus P is always gonna be a positive number. So it turns out we don't actually need the absolute values here A parenthesis is sufficient in this case So we have this negative natural log of M minus P equals kt plus C So let's times both sides by negative one. We get the natural log of M minus P This is equal to negative kt plus a constant now notice here We time C by a negative one as a digital add-in is cube. It just absorbs the negative sign makes no difference whatsoever Exponentiate both sides. We're gonna get M minus P is equal to E raised to the negative kt plus C and like we've seen before the plus C in The exponent can actually come out and as a C e to the negative x t. That's the magic of the gelatinous cube right there All right, and then let's see we're going to add P to both sides So the cancel at P. We're also going to subtract C e to negative kt on both sides And when we do so we will end up with the following equation. We have right here The population will equal the maximum population So P is the current population at a specific moment of time So the current population will be the maximum population minus some constant multiple of e to negative kt Where that k is the same? Growth constant we saw in the original differential equation. What can we say about this number C right here? Well, let's plug in the initial value P of zero equals P naught well if you do that you're gonna get P of zero This is equal to P naught like we said and then you're gonna get M minus C e to negative k times zero Well k times zero is going to be zero You get e to the zero and So you're gonna get M minus C. This is equal to P naught if we subtract C from both sides So I add C to both sides. We're gonna add C and we're gonna subtract P naught subtract P naught there We end up with This equation right here C equals M minus P naught So this right here gives us our equation for the law of of inhibited growth where C there Is measuring the difference between the maximum population and the initial population So this is what I like to think is the initial the initial elbow room Is what this number is computing? all right now this This formula some of you might have seen before This model has built into it the idea that As you get closer and closer and closer to this threshold value M The the the spread the growth will shrink over time, right? This is actually a really good model for example when one talks about Newton's law of cooling Newton's law of cooling In that context if you've seen this before you take M to be the The temperature of the environment you're taking so if you take like a really hot Cup of chocolate and you put in the refrigerator The the population could be its temperature which when you put it inside the refrigerator It can't get colder than the refrigerator it's in and you'll see that this This uh, this will cool start to its temperature will decay closer and closer and closer Towards the temperature of the refrigerator. It is really hot At first there'll be a rapid decay of of cooling But as it gets closer and closer and closer to the temperature of the environment it'll slow down dramatically Let's take a look at an example using actual populations here Uh, suppose a certain nature reserve can support no more than 4,000 mountain goats So this is going to represent a maximal population here So the reserve can only support 4,000 mountain goats Assume that the rate of growth is proportional to how close the population is to the maximum that here sounds like inhibited growth like Newton's law of cooling The growth is proportional to this maximum population or I should say it's proportional to the elbow room With a growth rate of 20% So what this is telling us is with respect to inhibited growth our k value is going to equal 20% Aka 0.2 Like so, so we want to write a differential equation for the rate of the growth of this population and solve it So in terms of the differential equation, we're going to get that p prime is equal to 0.2 times 4,000 minus p So remember here that this m minus p is the maximum population minus the current population And this was our growth factor of 0.2 So we get that pretty quickly By solving the differential equation We're going to get and we don't necessarily have to go through the solution again The solution will be identical to the separation of variables we did earlier We're going to get that p Is equal to the maximum population 4,000 minus c c was Recall the initial elbow room c equals the maximum minus the initial And so this is again equal to 4,000 minus I guess I never actually said what the current population is It says there are currently a thousand goats in the in the reserve So we take 4,000 take away a thousand. So that's going to be 3,000 this is the initial elbow room the goats start off with a lot of potential to grow You're going to have 4,000 minus 3,000 Times e to the negative 0.2 t And so this right here gives us A model for the growth of this goat population Assuming they grow by the assumptions of unedited growth So what will the target goat population be in five years? So if our t value if they're growing 20 per year The growth we just have to look at p of five p of five right here in which case you get 4,000 minus 3,000 Times e to the negative 0.2 times five That's all there is to that in which case 0.5 times sorry 0.2 times five is one So you're going to get 4,000 minus 3,000 Times e to negative one or one over e if you prefer And so we're going to have to estimate this use a calculator to help you out here No one has to be a hero and you're going to get an estimate of 2,896.4 Now, of course, you can't have 0.4 of a goat. So we'd have to round right? I mean you could round You could round say to just 2,896 that's sort of like the nearest goat there But one thing we should remember about about models is that these are only estimates the only predictions We don't expect these things to be perfect and therefore Um, actually it's quite it would be quite natural to be like just round this to 2,900 Now be aware if you're working on a homework question It might ask you to round to the nearest whole number follow the instructions there But in practice, we might not be like, oh, yeah, there's going to be 2,896 goats and after five years We'd be like, oh, yeah, there'll be about 2,900 goats. We as human beings. We like Rounded numbers so rounding to 2,900 is pretty nice We could be even round to 3,000 and we'd be pretty happy with such a thing