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# Exponential growth

## Video transcript

hi it's mr. Andersen and in this video I'm going to talk about exponential growth which is how populations can explode most students understand exponential growth but the math sometimes it gets a little tricky and so I'm going to step you through that in a couple of ways and so let's start with this rabbit right here let's say it's part of a population we don't only have one rabbit but we have a number of rabbits in our population we refer to that in all of the equations as n n is going to be the population size now that's going to change as we go through time but this is going to be our original population which is going to be n let's stack those rabbits up so we can count them so our end to start is going to be 10 so we have 10 rabbits at time 0 now that population is here they're going to increase it's going to decrease or it's going to stay the same and what things are determining that it's going to be our growth rate and so this is the second letter you should remember and that's our R is going to refer to how much it's changing over time and there's really only two things that are going to change that population we're gonna have new rabbits that's going to be births and then we're gonna have dead rabbits and that's going to be deaths and so those two things are going to contribute to the change in the population but it's a per capita in other words we have to divide by the N which is going to be the original population size and so let me make some baby rabbits so if I click here we've got five baby rabbits and so our births would be five and then let's say I want to kill a couple of rabbits let's kill that guy don't worry they're okay they're just virtual rabbits and so I killed that one as well and so we've got births of five we've now got deaths of two and what was our end to begin with it was ten and so if we figure out our R value that's going to be five minus two divided by 10 which is three over 10 which is going to be 0.3 and so our R value is 0.3 or our growth rate is 0.3 what does that really mean it's the factor at which our population is increasing and so if I take 10 times 0.3 I'm going to get 3 and that's how much our population increased and one thing you should know about that growth rate is that if the ecosystem is stable the growth rate is essentially going to stay the same it's not going to change over time and so you might think well if the growth rate stays the same isn't the population just going to increase along a consistent amount not really and so let's watch what happens now we're going to take point three growth rate for the next generation and we're going to instead of multiplying at times ten we now have to multiply it times thirteen and if we do that and we'll use this equation right here this is the change in n or the change in T we're gonna take our growth rate which is 0.3 and now multiply it times 13 well we don't get three anymore we get three point nine and so I'm going to round that sounds a lot like for rabbits so I'm going to add four rabbits and now our population is up to 17 so even though our stayed the same since we multiplied two times a larger value we're going to get more rabbits the next generation so let's do a generation three we're now taking point three times 17 and I get 5.1 which is a lot like five rabbits and I'm going to add those two rabbits or if we now multiply that growth rate times 22 I get six point six which is pretty close to seven rabbits so we're gonna add those seven rabbits and so we now have got a population of 29 and so you can see that the population is increasing but if I were to ask you a question I could ask you some hard questions the first ones not so hard what's the population going to be in year five well to do that you take 29 times 0.3 and then we add that to 29 but what if I asked you 10 or even 30 well this problem gets pretty hard and so you're quickly gonna want a little bit of help and for me when I want help the first place I go to is to a spreadsheet and let's go to the spreadsheet so we're gonna go to excel kind of remember those numbers there and so let's kind of rebuild that chart so on the left side we're gonna have 0 as our first time and 1 as our second time if you didn't know how to do this in Excel I can select both of those grab this little corner here and I can increase and it'll do the counting for me and so let's go up to population time 30 okay so I've filled that in now what's our original population that's gonna be 10 so I'm going to just put that in to start with now what's the next population so I got to put a formula in this box and to do that I'm just going to put an equal sign so I'm going to put an equal sign you can see the formula right here and so what did we do remember we're gonna take the original population so let me click on that so that's going to be 10 and then we're going to add the to that we're gonna add our growth rate which was 0.3 times and then we're gonna click on that again and so let's see what we get so we get 13 and so what we did is again we took what was in this cell and we added it to what was in this cell times 0.3 and so again through the magic of a spreadsheet I can simply grab this and it's just gonna use that same thing over and over it's gonna iterate on that and so what we're gonna get is it's gonna do all the math for us and so what did we have in the first one 13 17 22 this sounds familiar now it's obviously not rounded off and so this isn't the correct number of rabbits but we see now this exponential curve here or this J shaped curve or sometimes we call it like a hockey stick curve because it's quickly turning up like that and so now we could quickly answer those questions at time 5 we should have around 37 rabbits what was the next one I think 10 we should around 138 rabbits and if we go all the way down here to time 30 we're gonna have two hundred and twenty six thousand one hundred and 99 rabbits so that's a lot of rabbits really really quickly and so you can see how exponential growth takes off but what's fun about a spreadsheet is I could play around with it a little bit let's say instead of 0.3 if I change my growth rate to 0.1 so if I do that what are we gonna get well if I move this all the way down again we're gonna get another j-shaped curve now it's not going to be as steep as that one was and we didn't have as many rabbits at the end but we're still gonna be exponential growth we refer to go edit that variable again let's make it zero so let's say we take it times zero what would we get then well it's ten in other words if I go all the way down here what are we gonna get for a value well we're not increasing at all and so it's gonna be ten it's gonna stay ten the whole time a really hard question that I asked the students is this let's say we get a negative growth rate so let's make it negative 0.3 well what are we gonna get it for a value there well we get seven here but if I go all the way down what do we get well that's weird we're gonna approach kind of a limit we're gonna approach zero and that's because we're gonna take off larger amounts to begin with and then we're going to take off the less amounts as as the population gets smaller and smaller and smaller so again that's spreadsheets but you don't have these have a spreadsheet with you sometimes you just need a calculator and a little bit of algebra so let's go to the algebra this is going to be the algebraic solution to this and so we've got an equation for exponential growth and so changing n over T is going to be equal to n which is going to be our population size times 1 plus R where R is going to be the growth rate and then we're going to raise that to T where T is going to be equal to time and so let's make sure that this works so sorry right here is zero and so let's plug in our numbers so we're gonna put 10 in here for n that's that original population 1 and then R is going to go right here it's going to be 0.3 and then we raise it to the 0 power because our time is going to be 0 well if we simplify that a little bit anything raised to the 0 power is always 1 and so that's gonna be 10 and so that works out so far but don't trust me let's keep going let's go to the next one let's go to 1 so if we now put in 1 for time it's gonna be n is still 10 this is 1 plus R again there R is not changing but we're raising it to the 1 power anything raised to the first power is going to be itself so that's going to be 1.3 and we get 13 let's try that again with 2 so if we go to the second power again we plug in 2 here it's the only thing that we're changing so we get 1 point 3 to the second power so we're gonna have to square one point 3 which is 1 point 6 9 and we get 16 point 9 which is a lot like 17 or I could just throw out another time so let's say we go time 30 so then we're gonna raise it to the 30th power and so I get 26,000 199 which is gonna match up exactly with our spreadsheet we're gonna have a lot of rabbits really really quickly and so that's kind of an algebraic solution um it's a quick way if you'd be given a time and then figure out how much it's going to grow and so um a good question I could ask you is this let's look at bacteria rather than rabbits okay and so let's say we have one bacteria e.coli can reproduce in about 20 minutes in other words one can make an exact copy of itself in about 20 minutes and so let's say that none of them died let's say we get rid of the death rate so we've got our births over N and so how many births would we have if we're just dividing in half we're going to have 1 nikohl i what was our original population it was going to be one as well and so now we're going to give an R of one and so instead of increasing by 0.3 we're now increasing by 1 which is really increasing by a hundred percent so we had one bacteria and now we have a hundred percent as many bacteria twice as many or in 100% in addition to that original and so now we have two and so what are we going to have on the next round we're gonna have 4 and 8 and 16 and then 32 and you can see how exponential growth gives us a huge amount of bacteria really really quickly and so the question I might ask is is the sky the limit and so if we're looking at your exponential growth you know after 20 rounds like this we're gonna be way up in here I can't even read this 5 million something like that and so does it just keep going and going and going no because what I told you is kind of a lie R is not gonna stay the same forever as it starts to grow they're gonna run out of food resources space and so our R is going to start to change and so then we're gonna start to move into what's called logistic growth and that's going to be a totally set a different set of equations and I'm going to include that in another video and so that's exponential growth and I hope that was helpful