Logistic growth versus exponential growth
- [Instructor] Let's now think a little bit more about how we might model population growth, and as we do so, we're gonna become a little bit more familiar with the types of formulas that you might see in an AP Biology formula sheet. So in a previous video, we introduced the idea of per capita growth rate of a population, and we used the letter r for that. And so 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 20%, by two-tenths. So for every one, you would now have 1.2 of that population a year later. Now, as we mentioned, many populations, or the ones that reproduced sexually, you'd need at least two, a male and a female, but there's 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. And from this notion, we can get a related notion, which is our maximum per capita growth rate of a population and let me just write that there. 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 as, this is the unfettered one, this is the maximum. And from that, we can set up an exponential growth equation, and 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. And we could see it set up a little table here to see how these would relate to each other, so let me do that, let me set up a table. And so let's think about what the rate of change of population will be, our population growth rate, for certain populations. So 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. So, 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 0.2, 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 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 100. 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 .2 times 500, our population growth rate is now 100. If we're talking about bunnies, and if our time is in years, this would be 100 bunnies per year, or 100 individuals per year. And let's think about it when our population is 900. What's our population growth rate then? Pause the video again. Alright, well we're just gonna 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. And to describe that, we'll use the letter K. And so let's say, for the organisms that we're studying here, let's say they're bunnies, and they're bunnies on a relatively small island, let's say that the natural carrying capacity for that island is 1000. The the island really can't support more than 1000 bunnies. So how would we change this exponential growth equation right over here, exponential, to reflect that? Well, what mathematicians and biologists have done is they've modified this, they multiplied this times the factor, to get us what's known as logistic growth. Logistic growth. So this is exponential growth, and what we're gonna now talk about is logistic growth. And what they do is they start with the exponential growth, so my population growth rate you could view as your maximum per capita growth rate, times your population, so that's exactly what we had right over here, but then they multiplied that by a factor so that this thing slows down the closer and closer we get to the carrying capacity. And 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, so our population, what's going to be our population growth when our population's 100, when it's 500, and when it's 900? So I encourage you, 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 the carrying capacity is 1000, so it's gonna be 1000 minus 100, all of that over 1000. So this is 900 over 1000, 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 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 .5. So we're only going to grow half as fast as we were in this situation. 'Cause once again, we don't have an infinite amount of resources here. So this is going to be 100 times .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's going to be, 100 over 1000, which is .1, 0.1. This part right over here is, this part right over here is 180. 180 times 1/10th is going to be equal to 18. So now our population growth has slowed down. Why is that happening? Here, 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 gonna keep growing exponentially. But here, they're getting closer and closer to the carrying capacity of whatever environment they're at. And so at 900, they're awfully close, so now you're gonna 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 of, 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? And 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, the more it is, the faster it grows, and it'll just keep going forever until it, just, there's no limit in theory. And obviously, we know that's not realistic. Now with logistic growth, I'll do this in red, in logistic growth, in the beginning it 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 wanted to get the limit, what would happen? Well, what happens at a population of 1000 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.
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