Logistic growth versus exponential growth | Ecology | AP Biology | Khan Academy

Let's now think a little bit more about how we might model population growth. As we do so, we're going to become a little bit more familiar with the types of formulas that you might see in AP Biology formula sheet.

In a previous video, we introduced the idea of per capita growth rate of a population and we used the letter r for that. Let's say that the per capita growth rate for a population is 0.2. That means that on average, for every one individual in that population, a year later it would have grown by twenty percent, by two tenths. So for every one, you would now have one point two of that population a year later.

Now, as we mentioned, many populations that are the ones that reproduce sexually need at least two: a male and a female. But there are populations of certain things that can just reproduce on their own. They can just bud or they can divide if we're talking about especially unicellular organisms.

From this notion, we can get a related concept, which is our maximum per capita growth rate of a population. You could view this as your per capita growth rate if the population is not limited in any way; if there's ample resources: water, food, land, territory—whatever that population needs to grow. But that still is talking about per capita growth rate of population. We're just viewing this as the unfettered one. This is the maximum.

From that, we can set up an exponential growth equation. We've seen this in other videos where the rate of change of our population with respect to time n is our population. So dn/dt is our rate of change of population with respect to time or our population growth rate. Right over here, let me write this down: population growth rate.

If we're dealing with a population that in no way is being limited by its ecosystem—which, in reality, is not realistic—at some point you would be well, then the rate of growth of population is going to be your maximum per capita growth rate of population times your population itself.

We could see set up a little table here to see how these would relate to each other. So let me do that. Let’s think about what the rate of change of population will be—our population growth rate— for certain populations. Let's think about what it's going to be when our population is 100, when our population is 500, and when our population is 900.

Given these populations, what would be your population growth rate for each of them? Pause this video and try to answer that. Well, when our population is 100, our population growth rate is just going to be 0.2 times that. So let me write this down: this is just going to be dn/dt is just going to be 0.2, our maximum per capita growth rate of the population times our population times 100, which is equal to 20.

So we're going to grow per year by 20 when our population is a hundred. Now, what about when our population is 500? What is going to be our population growth rate? Pause the video again and try to answer that. Well, once again we just take our maximum per capita growth rate and multiply it times our population. So 0.2 times 500, our population growth rate is now 100.

If we're talking about bunnies, this would be—and if our time is in years—this would be 100 bunnies per year or 100 individuals per year. Let’s think about it: when our population is 900, what's our population growth rate then? Pause the video again. All right! Well, we're going to take 0.2 times 900, so it is going to be 180 individuals per year.

Now, as I just mentioned, this is talking about a somewhat unrealistic situation where a population can just grow and grow and grow and never be limited in any way. We know that land is limited, food is limited, water is limited, and so there's this notion of a natural carrying capacity of a given population in a given environment. To describe that, we’ll use the letter k.

Let’s say we for the organisms that we're studying here: let’s say they’re bunnies, and in their bunnies on a relatively small island, let's say that the natural carrying capacity for that island is one thousand—that the island really can't support more than one thousand bunnies.

So how would we change this exponential growth equation right over here to reflect that? What mathematicians and biologists have done is they’ve modified this. They multiply this times a factor to get us what's known as logistic growth.

This is exponential growth, and what we're going to now talk about is logistic growth. They start with the exponential growth, so my population growth rate you could view as your maximum per capita growth rate times your population. That’s exactly what we had right over here. But then they multiply that by a factor so that this thing slows down the closer and closer we get to the carrying capacity.

The factor that they add is your carrying capacity minus your population, over your carrying capacity. Now, let's see if this makes intuitive sense. So let’s set up another table here and I’ll do it with the same values. So let’s say we have n, our population. What’s going to be our population growth when our population's 100, when it's 500, and when it's 900?

I encourage you to pause this video and figure out what dn/dt is at these various times. Well, at 100, it's going to be—I’ll do this one, I’ll write it out: it's going to be 0.2 times 100 times our—the carrying capacity is 1000, so it's going to be 1000 minus 100, all of that over 1000.

So this is 900 over a thousand; this is going to be 0.9. And then 0.2 times 100 is 20. So 20 times 0.9, this is going to be equal to 18. So it's a little bit lower; it's being slowed down a little bit, but it's pretty close.

Now let’s see what happens when we get to n equals 500. Pause this video and figure out what dn/dt, our population growth rate would be at that time. So in this case, it's going to be 0.2 times 500 times 500, times this factor here, which is now going to be 1000 minus 500—that’s our population now—minus 500, all of that over 1000. Now what's this going to be?

This is 100, which we had there, but it’s going to be multiplied by 500 over 1000, which is 0.5. So we’re only going to grow half as fast as we were in this situation because once again we don't have an infinite amount of resources here. So this is going to be 100 times 0.5, which is equal to 50.

And then if you look at this scenario over here, when our population is 900, what is dn/dt? Pause the video again. Well, it is going to be 0.2 times 900, which is 180, times this factor, which is going to be 1000 minus 900, all of that over 1000.

So now this factor is going to be 100 over a thousand, which is 0.1. This part right over here is equal to 180 times 0.1, which is going to be equal to 18. So now our population growth has slowed down. Why is that happening?

Your population rate—the rate of growth—is growing and growing and growing because the more bunnies or whatever types of individuals you have, there's just more to reproduce and they're just going to keep growing exponentially. But here they're getting closer and closer to the carrying capacity of whatever environment they're at.

At 900, they're awfully close. So now you’re going to have some bunnies that are going hungry and maybe they're not in the mood to reproduce as much, or maybe they're getting killed, or they’re dying. This is very unpleasant thinking; they’re dying of starvation, or they're not able to get water—dehydration—who knows what might be happening.

We could also think about this visually. If we were to make a quick graph right over here, where if this is time and if this is population, our exponential growth right here would describe something that looks like this. So for exponential growth, our population will grow like this: the more our population is, the faster it grows. The more it is, the faster it grows.

It'll just keep going forever until it just—you know—there's no limit in theory, and obviously, we know that's not realistic.

Now, with logistic growth, I'll do this in red, and logistic growth in the beginning looks a lot like exponential growth. It's just a little bit slower. But then, as the population gets higher and higher, it gets a good bit slower and it's limited by the natural carrying capacity of the environment for that population.

So, k would be right over there; it would asymptote up to it but not quite approach it. And if you want to think at the limit, what would happen? Well, what happens at a population of a thousand in this circumstance? Well, then this factor right over here just becomes zero, so your population, at that point, just wouldn't grow anymore if you were to even get there.