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Exponential and logistic growth in populations

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Video transcript

let's say that we we're starting with a population of 1,000 rabbits and we know that this population is growing at ten percent per month what I want to do is explore how that population will grow if it's growing ten percent per month so let's set up a little bit of set up a little table here a little table and on this left column let's just say this is the number of months that have gone by and on the right column let's say this is the population so we know from the information given to us that at zero months we're starting off with one thousand rabbits now let's think about what's going to happen after one month well our population is going to grow by 10% so we could take our population at the beginning of the month and growing by 10% that's the same thing as multiplying by 1.1 you have your original population and then you grow it by 10% 1 plus 10% is 1.1 so we can multiply it by 1 point 1 and that math we can do in our head it is 1,100 or 1100 but let it let's just write this as 1,000 times 1 point we're not 1.5 times 1.1 now let's think about what happens as we go to month 2 well it's going to have it's going to be the population that we started at the beginning of the month times 1 point 1 again so it's going to be the population at the beginning of the month which was that which we have right over there but then we're going to multiply by 1.1 again or we can just say that this is 1 point 1 squared and I think you see a pattern emerging after another month the population is going to be 1,000 times 1.1 to the 3rd power we're just going to multiply by 1.1 again and so if you were to go n months into the future well you could see what's going to be it's going to be 1,000 times or being multiplied by 1.1 n times or 1,000 times 1.1 to the nth power and so we can set up an expression here we could say look the pop let's say that the population is P the population as a function of n as a function of n is going to be equal to our initial population our initial population times 1 point 1 to the nth power and you might say okay well this this makes sense it doesn't look like we're getting crazy numbers but just for kicks let's just think about what's going to happen in 10 years so 10 years would be 120 months so the population at the end of 120 months is going to be 1,000 times 1.1 to the 120th power and so let's let me get a calculator out to do that I cannot calculate 1 point 1 to 120th power in my head 1 point 1 to the 120th power is equal to that times our initial population so times 1,000 1 2 3 is going to be equal to roughly 93 million rabbits let me write that down so we started with 1,000 and we're going to have approximately 93 million rabbits 93 million million rabbits and so we grew by a factor of 93 thousand over 10 years so over another 10 years would grow by 93 thousand times this and so you quickly realize 10 percent per month is quite fast and this might seem extremely fast but this is actually not outlandish for a population of rabbits that are not limited by space or predators or food and if you were to plot something like this out if you were to plot the rabbit population with respect to time you would see a graph that looks let me draw it so this axis it is time say in months and this axis you have your population you have your population this type of function or this type of equation see population I population this is an exponential function and so your population as a function of time is going to look like this it's going to have this kind of hockey stick J shape right over here and if you let these rabbits reproduce long enough they would frankly take over the planet if they had enough food and if they had enough space to do it but as you notice I keep saying if they have enough food and if they have enough space the reality in the world is that there is not infinite food and infinite space and there are nope and it isn't the case that there are no predators or competition for resources and so there is actually a maximum carrying capacity for a certain part of of the environment for a certain type of species and so what's more likely to happen what we just described right over here is exponential growth exponential growth and why is it called exponential growth well you notice we are growing by our the the the input which is time is being thrown into our exponent and so that is exponential growth but obviously you can't have an infinite number of rabbits or you just can't grow forever there is going to be some natural maximum carrying capacity that the environment can actually sustain and so the actual growth that you would see when the population is well below that carrying capacity it's reasonable to model it with exponential growth but as it gets closer and closer to that carrying capacity it is going to ask them to up towards it so it's going to get up towards it but not cross it and that's just a model there are other situations where maybe it goes up to it and it crosses it and then it cycles around it so these are all different ways of thinking about it but the general idea is you wouldn't expect something to just grow unfettered forever now this blue curve which people often use to model populations especially when they're thinking about things at populations once they approach the the environments carrying capacity this is this kind of s-shaped curve that is considered that's called logistic growth and there is a logistic function that describes but you don't have to know it in the in the in the in the scope of a kind of an introductory biology there's a logistic logistic growth and it's described by the logistic function if you're curious about it we do have videos on Khan Academy about logistic growth and also about exponential growth and we go into a lot more detail on that but the general idea here is when populations are not limited by their environment by food by resources by space they tend to grow exponentially but then once they get close that exponential growth no longer models it well once they start to really saturate their environment and they start to get close to that ceiling and overall the logistic logistic function or logistic growth is is a better model for what is actually going to happen
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