 In this video we're going to talk about probably the most useful of all of the exponential growth models we're going to talk about in this lecture series. From an arithmetic point of view it is of course the most challenging, it's the most complicated, but at the same time it also is the most useful of all of them. So we should mention that the growth models that we've talked about previously, so the idea previously we had y equals you know some constant e, c times e to like kt, like so this this exponential model we had before sometimes we use different letters of course for these for these for the parameters k and c right here but there just there's parameters right. This is what's often referred to as uninhibited growth that when you look at the model in terms of growth the growth model will just get bigger bigger bigger bigger bigger bigger bigger bigger bigger right and that might seem a little bit ridiculous right that just continues to grow without bound right now if you're talking about like radio active of k then sure it's going to decay towards zero zero makes sense of some limiting value or with like Newton's law of cooling we had before it cools down towards the surrounding environments uh temperature which also makes sense so in that regard you're bounded by this asymptote but with growth models you're not really you're you're kind of going away from the asymptote and so what's the limit over here the idea is it's going to be infinity right just continues to grow grow grow grow grow grow and while this makes a very simple formula to use in comparison to the logistic growth we're going to see in a little bit it does seem a little unrealistic that we can just continue to go go grow grow that there eventually might be some type of scarcity of resources of some kind that inhibits the ability for the population to continue to grow without bounds so for example what if our model has built into it some lower bound like you can't have a population less than zero but would have had some upper bound some like it had two different you know horizontal asymptotes whatever for example our ecosystem has some type of carrying capacity when i was a kid i always thought it was that this was called a carrying capacity right carrying capacity you know you can only care for this many bunnies right that that's not what it was it's carrying capacity so what if there's like a limit there so that you know there's some lower bound that our model cannot decay below that but then there's some upper bound also and so perhaps at the beginning of growth it looks like exponential growth right that when you start getting away from this lower bound there's like this explosion of growth but then as you get close to the maximum population it kind of tapers off so it kind of switches from like exponential growth to exponential decay sort of not not really decay because it's still increasing but you're kind of like exponentially growing away from one asymptote but then you start to slow down as you get closer to the other asymptote can we get a function that looks something like this and this is an example of what we call the logistic growth model for which a function formula would look something like this right here the amount after a specific time so a of t it'll look like c over one plus a e to the negative k t so this is it's like combining a rational function with a exponential function that is we compose rational functions and exponentials together we can we can make this logistic function kind of like breeding a zerg with a protoss or something like that we have this hybrid creature that has properties from both exponential and rational functions all right and so of course the the ratio comes into play mostly to force this second this second asymptote to be inside the graph so let's figure out what the in behavior of this logistic function looks like to kind of motivate why we're using it in the first place so notice as t approaches negative infinity what happens to this function let's first look at this exponential for a moment you have a negative k so this is actually a decay model and so as you look at this graph you see something like the following right so as t goes towards negative infinity you're going this way on the graph this thing is pointing up this thing as as t goes to negative infinity we see that e to negative k t is going to approach infinity so our quantity a here is going to look like you have this constant c over one plus infinity that is c over infinity which when you have a fraction the bigger the denominator gets the smaller the ratio gets so if you take sort of like the biggest possibility ever infinity this thing is going to become a zero all right and so that's suggestive of this asymptote right here that as t goes to negative infinity the left hand side is going to go to zero it has this horizontal asymptote there all right so let me just kind of summarize that right here that as t approaches negative infinity we see that a is going to approach zero matching up with what we expect but on the other hand if we send t equals if we tend send t off towards infinity here in this situation we're now asking well what happens as we go to the far right of this graph in this case a is going to go towards zero and we see that on the graph you're going to have that sorry not a here the y-coordinate of the exponential function is going to go towards zero but the amount with this logistic model here this thing is going to look like c over one plus zero which then becomes a c which then suggests this in behavior right here this logistic model will range from will range from zero to c so in fact the domain the domain of this logistic function will be c excuse me zero towards c it'll range from zero towards the carrying capacity it'll be an increasing function and you do see this explosion of exponential growth near the middle of the graph and so that's why this model why this model actually turns out to be the right thing to this function family is the right one to get this type of growth here now what about this coefficient a in order to get it to behave the way we want it to this a right here is actually going to be c minus a not over a not or a not we're going to call the initial value like the initial amount inside of the population and so i want you to think of this right here as our as some type of like relative relative elbow room what i mean by that is like if you were to go to like a movie theater right when if you're the first one there the whole theater is open you can go sit anywhere want no big deal but when you get if you're the second person there he's like well i could sit anywhere but i don't want to sit right next to that person that's kind of awkward i want as much space as i can so you're going to go to a different spot and as more and more people come into the system the ability to kind of sit down is hindered and so your ability to grow will slow down the more and more things that come into it and so if you take c minus the initial population divided by the initial population this kind of tells you percentage wise like how much opportunity is there to grow inside of this ecosystem so let's do it let's do a specific example here suppose a bacteria culture with an initial population of a 100,000 bacteria grows logistically at a rate of 8 per hour so let's see some things we let's let's identify what we know already the initial population a not is going to be 100,000 i'm just going to say 100k there and we know that it grows at eight percent per hour so our growth rate k is equal to 0.08 per hour we're going to be measuring growth with hours here if the carrying capacity of the petri dish is one million right so there's only so much that this petri dish can grow can hold before there's not enough room for any more bacteria to grow there right so that's going to be a million is our carrying capacity to make things a little bit easier i'm going to say it's a thousand thousand right so thousand k find the population sizes after 40 hours and 50 hours when will the population reach 0.9 million and so let's look at this using our logistic model so what we need to do is we need to fill in the numbers right here we have the carrying capacity of a thousand and now we have to compute this a value so a by the form of above take our carrying capacity which will be a thousand minus our current which is a hundred and again these are all multiplied by a thousand divided by a hundred right here so we're going to get 900 divided by 100 so we end up with the coefficient of nine and we should think of this as nine thousand but we'll just multiply whatever finance or we get by a thousand in the end so our formula is going to look like the amount after a given time will look like 1000 divided by one plus nine times e to negative 0.08 t where t is being measured in hours so if we want to figure out what is the population after 40 hours that simply is just asking you know if we're doing 40 hours right here that's asking what is the amount with the population after 40 hours so it's just a plug and chug right here so we have 1000 over one plus nine times e to the negative 0.08 times 40 at some point we're going to have to take a power of e in which case then a calculator will be very useful here this is just something you throw in a calculator you take negative 0.8 times by 40 you get that you raise e to that power times it by nine plus one and then divide all that by 1000 so without going into details of how you actually plug something like this in your calculator I'll assume my viewers can use their scientific calculators this would give us approximately 731.6 thousand bacteria if we want to know how much bacteria there would be after 80 hours well then we take a of 80 so again we're just going to plug in 80 into our formula right here we end up with 1000 divided by one plus nine e to the negative 0.08 times 80 this time and so it'll look a little bit different but like I said this is something we're going to throw into our calculator when you put in the calculator and approximate this thing you end up with 985.3000 bacteria so it's after 80 hours we've almost reached our carrying capacity now notice in the first 40 hours our population went from 100,000 to 730,000 approximately there was a huge explosion of growth right we had we had about 600,000 bacteria that were grown in the first 40 hours but in the second 40 hours we didn't have another 600 bacteria that grew we only had about 200 about 250,000 bacteria that grew in that second 40 hours why such a decline and that's because of the growth model we see right here right that at the beginning like you know the first half there was like this large explosion but in the second half although it continued to grow we didn't have the same type of growth right because we didn't have as much elbow room as we did the first 40 hours then the last question is when will it reach when will it reach 900,000 or 0.9 million right so that's asking us in this situation we have to then solve we have to solve the equation 900,000 equals 1000 over 1 plus 9 e to the negative 0.08 times t we have to solve for t in this situation now this one takes a little bit more detail so although we will throw our answer in the character at the end let's go through the steps of this what's going on notice where our variable is our variable is in the exponent of a denominator so we have to kind of free our variable from its prison the first thing to do is to clear the denominators and as we just have 900 equal fraction you can think of as 900 over 1 and so you just cross multiply to get this thing forward so we're going to end up with 900 times 1 plus 9 e to the negative 0.08 t that's equal to 1000 and although it is tempting to just distribute the 900 through you could do that if you want to we could also just divide both sides by 900 right um after all because we have to solve for t right and that's and that's the that's the direction I'm going to go the 900s cancel right there and so we end up with 1 plus 9 e to the negative 0 that excuse me negative 0.08 t that equals then well 10 9ths like so we're going to subtract 1 from both sides subtract 1 and that's going to give us 9 e to the point our negative 0.08 t that equals now 1 9th uh that is 10 9ths take away 9 9ths is 1 9th uh and so now we're going to divide both sides by 9 notice we're dividing both sides by 9 again we're not timing both sides by 9 so you might be tempted to think the right hand side becomes 1 it actually becomes 181st just right now the left hand side now 1 over 81 is where we're at right here then to get rid of the base e we need to take the natural log of both sides take the natural log of both sides and we end up with our rate negative 0.08 t is equal to well the natural log of 1 over 81 there's a couple things you could do here you could write this as the natural negative the natural log of 81 because a reciprocal is a negative 1 power and a net power is inside the logarithm to come out as coefficients and also since since 81 is 3 to the 4th you could also write this as negative 4 times the natural log of 3 if you want again this is all just could be in our calculator no and it's one more second anyways so to finish off we need to divide by the negative 8 percent we had before like so and so in the end we end up with t would equal negative 4 times the natural log of 3 over a negative 0.08 and this is actually the reason why there's a negative sign in front of the k inside of the logistic formula going back up the screen right here why was there a negative sign right here in the first place and that's because we anticipate that's kind of actually correcting to the fact that there's going to be a negative later on like we have right here so you actually get a double negative and this then gives us so these are these right here will actually cancel out great and so we end up with our then our final result now if you did want to write this as a common fraction you can move the decimal place over you know two places here and simplify if you wanted to you'll end up with 50 times the natural log of 3 as the exact answer again as we're just going to put this in a calculator let's let's approximate that you get 54.9 this would be an hours right do notice that about 55 hours that so it should happen before it should happen before 80 hours but after 40 hours as we saw right here where it should hit 900,000 and so again that happens around 55 hours after the initial growth of the bacteria