Per capita population growth and exponential growth | Ecology | AP Biology | Khan Academy

In a previous video, we started thinking about things like population growth rate and how it relates to the birth rate and the death rate within a population. We related that to some of the seemingly complex formulas that you might see on an AP Biology formula sheet. Now we're going to extend that conversation to discuss some of the other formulas you might see, but to realize that they really are just intuition using a little bit of fancy math notation.

So just as a little bit of review, we looked at an example where, in a population, the birth rate is 60 bunnies per year. We're talking about bunnies here; it's a population of bunnies. And the death rate is 15 bunnies per year. Well, what's the population growth rate? In a given year, you would expect 60 bunnies to be born, so that would add to the population, and you would expect 15 bunnies to die, so that would take away from the population, for a net increase of 45 bunnies per year.

To put that in the language of your AP Biology formula sheet, the notation they use for population growth rate uses a fancy notation. So, actually, let me just write it over here: they say ( n ) is equal to your population. ( n ) is equal to population, and then your population growth rate—they use calculus notation—so our change in population per change in time. This is really talking about something in calculus known as instantaneous change, but we don't have to get too bogged down with that just yet.

Your population growth rate, which you could use this notation for, is equal to your birth rate of 60 bunnies per year, and the notation they use for birth rate is just ( b ). They don't use the same rate notation for that; I probably would have, but that's fine. I'm just trying to make you familiar with what you might see, and then minus the death rate, minus ( d ). So this right over here is something that you would see on that formula sheet, but it makes fairly intuitive sense.

Now the next idea we're going to think about is something known as a per capita growth rate of population. Let me write it out in words first. So here we're going to think about a per capita growth rate or population growth rate per capita. Population growth rate.

Now, per capita means you could view it as, on average per individual, what is the average growth rate per individual? What is that going to be? Pause this video and try to think about it. Well, one way you could think about it is the total population growth rate divided by the population, divided by the number of people there are. So, it's going to be our population growth rate divided by our population.

Now, let's say that we have a population of 300 bunnies. Actually, let's make the math a little bit simpler. Let's say we have a population of 450 bunnies. So what is going to be our per capita population growth rate? Pause this video and try to figure that out.

Well, if we have a population of 450 bunnies, our population growth rate, per the number of people—we or number of bunnies, I should say—is going to be equal to our population growth rate of 45 bunnies per year. That's going to be for every 450 bunnies, which will get us to ( 45 \div 450 = 0.1 ), and then the units 'bunnies' cancel with 'bunnies,' so it's ( 0.1 ) per year.

Now, why is the per capita population growth rate interesting? Well, it tells us just how likely—most populations need at least a male and a female in order to reproduce—but there are some organisms that can just split and reproduce asexually. But it tells us, on average, per individual organism, how much they are going to grow per year, so it gives you a sense of that.

Now, connecting it to the notation that you might see on an AP Biology formula sheet, it would look like this: the per capita population growth rate is usually denoted by the lowercase letter ( r ), and then they would say that that is going to be equal to our population growth rate, which we've already seen that notation, the rate of change of our population with respect to time ( \frac{dn}{dt} ) divided by our population.

Now, we can algebraically manipulate this a little bit to get another expression. We could multiply both sides by our uppercase ( N ) times our population, and we're going to get ( \frac{dn}{dt} = N \times r ) or ( r \times N ). Let me rewrite it: we could rewrite this as ( \frac{dn}{dt} = ) our per capita population growth rate times our population.

Now, this once again makes sense if you say, "Okay, this is how many people—how many individuals I have." If, in a given year, they grow by this much on average, well, if you multiply the two, you'll know how much the whole population has grown.

So if we didn't know these numbers, and someone said, "Hey, well actually, we could think about this," let's think about now a population of a thousand bunnies.

So, if ( N ) was equal to 1000 and let's say they're the same type of bunnies that have the same probability of reproducing and the same likelihood, we know that ( r ) is equal to ( 0.1 ) per year for this population of bunnies. What is going to be our population growth rate? Pause this video and try to figure that out.

Well, in this situation, ( \frac{dn}{dt} ) is going to be our per capita population growth rate. So it's going to be ( 0.1 ) per year times our population times 1,000 bunnies. I'll keep my units here: bunnies.

So this is going to be equal to ( 1000 \times 0.1 = 100 ) bunnies per year. So hopefully you're getting an appreciation for why these types of formulas, which are fairly straightforward, are useful.

Now, this is also an interesting thing to look at because even though this is in fancy calculus notation, and they're saying that our rate of change of population is equal to ( r \times N ), this is actually a differential equation. If you were to think about what this population, the type of population this would describe, this would actually be a population that's just growing exponentially.

So this is often known as an exponential growth equation. Let me write that down: exponential growth. In other words, in your math classes, in your calculus classes or even in your pre-calculus classes, you will study exponential growth. In a biology class, you're really just thinking about how to manipulate this a little bit.

But just to give you a little sense of what's going on with exponential growth, if you have a population of bunnies with this type of exponential growth, what is happening here: this is time, and this is your population. You're going to have some starting population here, and it's just going to grow exponentially.

The higher the ( r ) is, the steeper this exponential growth curve is going to be. But this describes how populations can grow if they are not constrained by the environment in any way. They have just as much land, as much water, and as much food as they need. Eventually, the bunnies will fill the surface of the earth and the universe.

Now, obviously, we know that that is not a realistic situation. Any ecosystem has some natural carrying capacity. There's only so much food; there's only so much land. At some point, there's just going to be bunnies falling from trees, and it's going to be much easier for predators to get them and all these other things.

We will discuss that in the next video: how do we adapt the exponential growth equation right over here to factor in a little bit more of a real-world situation, where at some population, you're going to be getting up against the carrying capacity of the environment?