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Video transcript

let's think a little bit about modeling population and what I have pictures here are some of the the most Nolan wraps actually this gentleman right over here might be the most known person when it's when people think about population and the limits to growth of population this is Thomas Malthus he was a a British cleric and and and and writer and scholar at the end of the 1700s at the end of the 18th century early 19th century and he really challenged the notion that you know population could grow indefinitely and that we would always through technology be able to feed ourselves and that really that the environment would eventually put some caps on how much or where the population could grow to and PF Verhulst and I'm sure I'm mispronouncing his name here he is a he was a a Belgian mathematician who read Malthus his work and tried to model the behavior that Malthus was talking about that okay when there aren't environmental constraints maybe population does grow somewhat exponentially but then as it approaches kind of the limits set by the environment it's going to essentially asymptote towards some type of some type of population Malthus in particular he actually doesn't think it's just going to be a nice clean asymptote he actually thinks that the population would kind of go above the limit and you would have these catastrophes and then they would go crashing below the limit and you kind of oscillate right around the the limit through these these these catastrophes as you could tell Malthus was a fairly optimally optimistic guy but let's go through a little bit of the math and-and-and a little bit of the differential equations also it's not too these aren't overly hairy differential equations to think about population so the first way to think about population and I'll express it as a differential equation and so let's just actually let me just set some variables here let's say that n is our population so that's our population and we are going to assume that n is a function of T so n as a function of T is what we're going to be thinking about in this and frankly the next series of videos so one way to think about how to model this it's just so how does what is the rate of change of population with respect to time how does that relate to things and so we could say okay well what is the rate of change of population with respect to time so D at capital n DT well one way to think about it is it's going to be proportional to the population so you could say well maybe this is going to be some proportionality constant times the population times the population itself and this makes sense if the population is smaller then you're not going to have as much as much change per unit time as if the population is larger the larger the population the more the more it's going to grow in a particular unit of time and actually this is actually a very fairly simple to solve differential equation you might have done it before I encourage you to pause this video if you feel inspired to do so but I'll solve it right here and you'll see that we get an exponential we get an exponential function here for n so let's let's do that so let's solve this and we'll essentially separate the variables separate the end from the t's although we don't we only see a DT here but I'll do that in a second so if I divide both sides by n I get 1 over N and if I multiply both sides by DT if you kind of think about the DT is something that you can multiply so I'm going to multiply both side armer divide both sides by and multiply both sides by T DT so I'm going to get 1 over N DN on the left hand side and on the right hand side I'm going to get R times DT R times R times DT notice I got the DT on to the right hand side by multiplying both sides of that and then I divided both sides by and I got the 1 over n right over here now what we can do is we can take we can take the antiderivative of both sides we can take the antiderivative of both sides and what do we get on the left hand side well this is just going to be the natural log this is just going to be the natural log of the absolute value of our population and actually if we assume that the population is always going to be zero then we can actually take this these absolute value off but I'll do that in a second and that's going to be equal to R R times T R times T and then we could have added a constant here but I'm just going to do it on one side so it's R times T plus R times T plus C and now if we actually want to solve a for n we could take we could take if this is equal to this then e to this power should be the same as e to this power or another way of thinking about it e the natural log of the absolute value of n is equal to this is another way of saying that e to this is going to be equal to that lecture let me just do it this way let me just sake let me just take E to both the to both make to this power and to that part that's equal to that then e to that power should be the same as e to that power and so we're going to be left with e the e to the natural log e to the natural log of the absolute value of n that's going to give you the absolute value of n let's just assume let's just assume and positive assume let's just assume population is greater than zero and so then we could this left-hand side this left-hand side right over here will just simplify to N and then our right-hand side it's going to be well it could be e to the RT plus C or this is the same thing this is this thing right over here is the same thing as e to the RT e to the R times T times actually let me do that in that same color e to the RT times e to the C times e to the C right I'm just taking e to the some of these two exponents that's going to be e to the RT times e to the C and if we want to we could just say hey you know what this is still just going to be some arbitrary constant here so actually let's just call this let's just call this C so it's e to the RT times C or we could say C times e to the RT C times E e to the r e to the R T and notice so we have solved the differential equation we haven't gotten to this kind of less than optimistic reality of mouths is where we're limiting it this is just if we just assume population is going to grow the rate of change of population with respect to time is going to be proportional to population when we solve that differential equation we get that population as a function of time actually let me make it explicit that this is a function of time so let me let me just move the N over a little bit so let me write it this way and of n of T is going to be equal to this this was our solution to our differential to this differential equation and once again this is going to just going to grow forever and if we know the initial conditions and so let's say that we knew let's say that we knew that n of 0 when time is equal to 0 let's just say that's n sub-nought so what would C be well n of 0 and of 0 is going to be equal to C C times e to the 0 power e to the 0 power is just 1 so it's just going to be equal to C so C is equal to n sub naught is equal to n sub-nought so now we can even write it that the solution to this thing right over here is n as a function of T and as a function of T is going to be equal to C times or be careful and not our initial population times e to the e to the R T now once again this is an exponential essentially our population is going to look like this if I were to graph it it's going to look it's going to look if that's my time axis that's my n axis right over here if I see oh I could say it's y equals an axis what however I want however I want to however I want to denote it so that would be a nought and just going to grow exponentially from there and there you know the rate of this exponential function is going to be dictated by this by this constant right over there it's going to look something like this and it's just going to grow faster and faster and faster for ever and ever and ever and ever now as I mentioned at the beginning of the video Malthus Malthus does not does not believe that this is going to be true he thinks that we're going to hit some natural limits that are going to start to constrain the population and so in his mind a more natural or more realistic function that to model population would look something would look something like this or even potentially something that you know you keep growing crashing around that around that kind of limit and what we'll see in the next video is that PF Verhulst actually came up with a very good one differential equation and solution to the differential equation that does model a lot better to describe that better describes the reality that Malthus believed that we were in