Exponential and logistic growth in populations
- [Voiceover] Let's say that we're starting with a population of 1,000 rabbits, and we know that this population is growing at 10% per month. What I wanna do is explore how that population will grow, if it's growing 10% per month. So let's set up a little bit of a, let's 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 1,000 rabbits. Now let's think about what's gonna happen after one month. Well, our population's gonna grow by 10%, so we can take our population at the beginning of the month, and growing by 10%, that's the same thing as multiplying by one point one. You have your original population, and then you grow it by 10%, one plus 10% is one point one. So we can multiply it by one point one, and that math we can do in our head, it is 11 hundred, or 1,100, but let's just write this as 1,000 times one point, not one point five, times one point one. Now let's think about what happens as we go to month two. What's gonna have is gonna be the population that we started at the beginning of the month times one point one again, so it's gonna be the population at the beginning of the month, which was that, which we have right over there, but then we're gonna multiply by one point one again, or we can just say this is one point one squared. I think you see a pattern emerging. After another month, the population's gonna be 1,000 times one point one to the third power, we're just gonna multiply by one point one again. And so if you were to go n months into the future, well you can see what's going to be. It's gonna be 1,000 times, or being multiplied by one point one n times, or 1,000 times one point one to the nth power. And so we can set up an expression here, we can say look the population, let's say that the population is P. The population as a function of n, as a function of n, is gonna be equal to our initial population, our initial population, times one point one to the nth power. And you might say, "Okay this makes sense, "it doesn't look like we're getting crazy numbers", but just for kicks, let's just think about what's gonna happen in 10 years. So 10 years would be 120 months. So the population at the end of 120 months is gonna be 1,000 times one point one to the 120th power, and so let's, let me get a calculator out to do that. I can not calculate one point one to the 120th power in my head. One point one to the 120th power is equal to that, times our original population, so times 1,000, one two three, is going to be equal to roughly 93 million rabbits, let me write that down. So we started with 1,000, and we're gonna have approximately 93 million rabbits, 93 million, million rabbits. And so we grew by a factor of 93,000 over 10 years, so over another 10 years, we're gonna grow by 93,000 times this. And so you quickly realize 10% per month is quite fast, and this might seem extremely fast, but it's 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, let's say in months, and this axis you have your population, you have your population. This type of function, or this type of equation, let me see population I, population, this is an exponential function, and so your population as a function of time is gonna look like this, it's gonna have this kinda 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 they had enough space to do it. But if 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 it isn't the case that there are no predators, or competition for resources. And so there is actually a maximum carrying capacity for certain part of the environment for a certain type of species. And so what's more likely to happen, what we described right over here, is exponential growth, exponential growth, and why is it called exponential growth? Well you notice, we are growing by 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 can't just 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, is reasonable to model it with exponential growth, but as it get closer and closer to that carrying capacity, it is going to asymptote up towards it, so it's gonna 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 population, especially when they're thinking about the population once they approach the environment's carrying capacity. This is, this kinda s shaped curve, that is considered, that's called logistic growth, and there is a logistic function that describes this, but you don't have to know it in the scope of a kinda 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, or they start to get close to that ceiling, and overall the logistic, or logistic function, or logistic growth, is a better model for what is actually going to happen.
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