Exponential and logistic growth in populations | High school biology | Khan Academy

Let's say that we were starting with a population of 1,000 rabbits, and we know that this population is growing at 10% per month. What I want to do is explore how that population will grow if it's growing at 10% per month. So, let's set up a little table here, a little table.

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 going to happen after 1 month. Well, our population is going to grow by 10%. So, we could take our population at the beginning of the month and grow it 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 + 10% is 1.1.

So, we can multiply it by 1.1, and that math we can do in our head. It is 1,100. But let’s just write this as 1,000 times 1.1. Now let's think about what happens as we go to month two. Well, it's going to be the population that we started at the beginning of the month times 1.1 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 1.1 squared. 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.

So, if you were to go n months into the future, well, you can see what's going to be; it's going to be 1,000 multiplied by 1.1 n times, or 1,000 times 1.1 to the nth power. We can set up an expression here. We could 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 going to be equal to our initial population, our initial population times 1.1 to the nth power.

You might say, okay, well, 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. The population at the end of 120 months is going to be 1,000 times 1.1 to the 120th power.

So, let me get a calculator out to do that. I cannot calculate 1.1 to the 120th power in my head. 1.1 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 rabbits. And so we grew by a factor of 93,000 over 10 years. So, over another 10 years, we'd grow by 93,000 times this, and so you quickly realize 10% per month is quite fast, and this amount seems extremely fast.

But this actually is 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 is time, say in months. In this axis, you have your population. This type of function, or this type of equation—let's see—population. I. Population. This is an exponential function. 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 it isn’t the case that there are no predators or competition for resources.

So, there is actually a maximum carrying capacity for a certain part 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—why is it called exponential growth? Well, you notice we are growing by our—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 is reasonable to model it with exponential growth.

But as it gets closer and closer to that carrying capacity, it is going to asymptote 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 populations, once they approach the environment's 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 this. But you don't have to know it in the scope of kind of an introductory biology.

There's 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.

Overall, the logistic function or logistic growth is a better model for what is actually going to happen.