Hello, welcome back to introductions to genetics and evolution. In previous videos, we've been looking at concepts related to the amount of genetic variation in populations relative to the amount of total phenotypic variation. Now as I mentioned before, this genetic variation is that subset of variation, which can be used by natural selection. So let's talk about natural selection for a minute and apply this to a couple of other areas. And very soon, you'll see we'll be transitioning into population genetics. Well, natural selection is both noncontroversial and as I mentioned before is a mathematical inevitability. It's a mathematical inevitability if three simple conditions are met. First, you have the phenotypic variation. We talked about phenotypic variation just recently in the context of quantitative genetics. And that would be that Vp is not zero. That basically, everybody doesn't look exactly the same in every way. Some of that phenotypic variation you see is actually inherited, or genetic. In that regard, we're saying that the heritability is not zero. So you see, I'm introducing the same terms we did before, but now with some quantitative genetics behind it. And last but not least, this variation that is inherited, must affect either survival or reproduction. Let's walk through this very simple cartoon, which illustrates this point nicely. So here we have several peppers that vary in their mildness or hotness. There are some that are mild, some that are hot. So that there is this variation. This variation is inherited. The mild peppers give birth to other mild peppers. The hot ones breed other hot ones. Here's an interesting point I haven't emphasized so far, but this is relevant for today's lecture. More individuals are born than will survive to reproduce. Our capacity for population growth tends to be very high. We'll come back to this in just a minute. And related to this, some variants survive or reproduce at higher rates than others. In this case, we can see that the milder peppers don't survive as well as the hot peppers because the mild ones get eaten up by things like mice or humans who likes the mild peppers. The outcome of this is that the population changes over time. That over time as this is iterated over and over again, more of the surviving peppers will be hot, because you've eliminated them. Not only the mild peppers themselves, but even that genetic contribution within the population that made the peppers hot. Now if we go back in time, that original concept of natural selection was very intrinsically tied to that of population growth. And Darwin's ideas were very much influenced Malthus, who was an economist. Malthus had pointed out that populations are actually kept from growing by limited food and resources, because again, our capacity to breed to breed is very, very great. And if you provide more and more food to a particular species, you'll tend to see that they will reproduce at a higher rate than is needed just to recreate the number of individuals present. That basically, the population will grow if it is able to grow, almost always. Darwin pointed out that this limitation produces struggle, wherein some subset that are better able to survive or reproduce will tend to spread. But let's look at this concept of population growth and why is it that natural populations grow so much? Why do we see this potential for great growth? Well the capacity for growth is huge in most species out there. And imagine, to maintain a constant population assuming the population was sexual, assuming you need a pair of organisms to breed, each pair would only need to produce two surviving offspring. Now, if conditions are favorable, how many offspring can most individuals produce? We'll look at some plants, for example. Let's see that you're looking at seeds or pollen. Here's a picture of a tulip stamen and all the pollen on it. Look at all those individual grains of pollen on there. How many possible offspring could that tulip have? Definitely a lot more than two. Think about insect larvae, if you let an insect breed, how many offspring can it have? Think about things like cane toad eggs. Cane toads as I mentioned a long time ago are an invasive species in Australia and also in Hawaii. And they can have hundreds, thousands of offspring just from one breeding pair. And it's even true for humans that if humans were allowed to in some way, we could potentially. One couple could produce many, many, many kids easily 10, 20 something like that. Now we like to model the rate of increase and see how it's actually happening. Well populations can be modeled with what's referred to as a stable rate of increase. This is assuming that a steady fraction of the population, or the population increases by a steady proportion year after year. To model this we need a couple of parameters. We need a birth rate which can be modeled as for example number of births per thousand per year. We need a death rate, the number of deaths per thousand individuals per year. And from this we can come up with what's referred to as the intrinsic rate of increase of a population. That is very simply the birthrate, so the input minus the death rate which is basically the output from the population. So in the United States as an example, the birth rate is about 14 per 1000, the death rate is about 6 per 1000. So that makes the intrinsic rate of increase about 0.008. In this case, this is not considering immigration. This is just looking naturally. What this means is the population will grow naturally by 0.8% per year. Now if the birth rate is greater than the death rate, then the population grows. Conversely, if the birth rate is lower than the death rate, then the population will decline. Now, so let's look at the effect on population size. And we'll go through a little bit of math here just to work this out. I hope you don't mind. So, let's say that n is the population size and t is time in years. Now we can identify a standard rate of population growth as dN over dt. Or change in population size over change in time. And that would be equal to that rate of increase times the population size. Now, if we wanted to get a algebraic solution, if we assumed that this was a constant process, we can approximate this as N sub t, so population size at time t is equal to N sub zero which is that starting population size, times e. E is the algebraic number you often see, I think it's approximately 2.71. E to the power of rt, r being that intrinsic rate of increase and t being time. So time may be measured in years. So, what is the population doubling time? How long does it take for a population to double it's number of individuals, right? Obviously, that's going to be related to R, it's going to be related to the rate of increase that if R is very large the doubling time will be very short. If R is very, very small then the doubling time would be very long, but let's put some actual numbers on this just so you can see. So this is the formula I showed you before. N sub t = N sub 0, e to the rt. And probably for this N sub t, is the population size at time t, at the end of the period that you're studying. What we want to solve for, is we want to solve for the doubling time. The time it takes to go from a particular population size, lets call it N sub 0 at the beginning, to double that size. So, we're waiting for N sub 0 2N sub 0. So, what we can do, is we can basically solve this formula by putting 2N sub 0 here for N sub t. Okay, and that what we're going to do is we're going to solve for t. The idea here is to look at basically how many generations it takes to go from N number of individuals to 2N number of individuals. So let's walk through. So we start off, as I said, with substituting to this formula for N sub t we put 2N sub 0 and we have 2N sub 0 = N sub 0 = e to the rt. Simple algebra, we just divide those side by n sub zero so we have two = N to the rt. Now how do we solve for t with this? Because that's ultimately what we are going for. What we have to do here is we have to use the natural log. Remember, e is approximately equal to 2.71, this is a factor that's used quite a bit. You have a button on your calculator that probably says E to the X and another that says ln, ln is the natural log. So what we do is we take the natural log of both sides. Take the natural log of both sides. So we said natural log of 2, is equal to the natural log of e to the r t. By the way, if you don't have a calculator that does this, you can just go to Google and type in ln space 2, and it'll actually solve that for you. So, we take the natural log of 2 and that calculates out to 0.693. The natural log of e to any power is that power. So in this case the natural log e to the rt is rt. So we have now very simply 0.693 = rt. Now again, we're solving for t, so what do we do? We divide both sides by r and there we go. The doubling time is now t which is 0.693 / r. So, let's put some numbers into this. Well, we have a US population at the time this was recorded of about 310 million. I mentioned before that the population growth factor was about 0.008. So all we have to do is put in this r into the formula right there. So 0.693 divided by 0.008 and we have 86.6 years. Think about that, that is just a little bit longer than the average life time. A lot of people live that long. And that is how long it would take until the population of United States goes 310 million to 620 million people. That is a lot of people and that's fairly quick. This is assuming this sort of growth as we drew it here, but it's not unrealistic. Now let me get you to try one but I want you to notice ahead of time that you don't actually need to know the population size to get the doubling time. You basically don't need to know this n sub 0 factor. All you need to know is r. Solomon Island, here are some numbers I got from Wikipedia. The birth rate on Solomon Island is about 35/1000. Death rate's about 5/1000. Population size today is about 500,000, but you don't need to know that. How long would it take to get to a million? How many years would it take to get to a million? Well, I'll let you solve that problem here on the question online. Well, I hope that one wasn't too challenging. Very, very simple problem. What we want to do is, is we want to calculate r. R as I mentioned before for is equal to the birth rate minus the death rate. So it would be 35/1000- 5/1000 which is in this case would come out to 0.03. So that's your r. So when we want to calculate the doubling time, say doubling time in this case t would equal to r or 0.693 divided by r, 0.03. So in this case comes out to 23 years. That's really dramatically fast isn't it? That's changing the population on these island from 500,000 to a million. So you're doubling the population size in 23 years. Well we've seen a lot of population growth throughout the world and I'll show some pictures from a recent article in science. This first figure shows historic and projected future population growth. So you notice our world population very slow, and we started seeing here very very recently just in the last couple 100 years, this dramatic increase in numbers of individuals. So we hit 1 billion in the year 1800, we hit 3 billion in 1960, 4 billion in 1974, 5 billion in 1987, 6 billion in 1999, 7 billion in 2011, and it's projected we'll hit 8 billion by 2024. This divides it up by some countries where you're looking at some of the developed countries and less developed countries, Africa as a continent, China by itself and India by itself. So this is projections up to 2050, in this case, for world population growth. Now, interestingly, our rate of population growth has actually gone down. Now we are still increasing. R is still positive. The birth rate is still exceeding the death rate. But the amount with which it's exceeding it has actually gone down over the last couple of years. But it's still positive and that's why we're continuing to see this very rapid increase. Now this is something that several different countries have attempted to address, one example would be China. China as many of you know back in 1979 instituted the One-Child policy, this shows you the population growth in China over time. And this shows you the age distribution as of 2009. And you may be thinking like why is this still going right? Because we have the One-Child policy instituted around 1979 1980, yet we still see this increase Increase. Interesting if you look at this, the increase is actually closer to linear than it is to exponential. Like it's not going up, it's not continuing to accelerate, as you would expect. So this probably has actually had some effect on slowing population growth, but, again, the birth rate is continuing to exceed the death rate. So we're still seeing an increase in the population size in China, as in many other places in the world. Well, why is that? Well again, we have two causes of population growth. There's high birth rate. And there's low death rate. So we can cut the birth rate, but we still see growth. And what that means is the death rate is decreasing faster than the birth rate is decreasing. Now that's not intrinsically a bad thing. Obviously, we like decreasing our death rate. We don't want to have people just die willy nilly of course. But this is a problem in terms of how many people are on the planet. Over the course of many countries and over the course of a lot of recent time, we've seen this declining mortality or declining death rate. And some of it is associated with things like introduction of the use of soap. Improved sanitation, definitely things like antibiotics and other modern medicine have really increase life expectancies. I can show you how much too. This is just showing the difference between 1950 and 2010. Now look at this in Latin America and the Caribbean. We have an increase in life expectancy from 51 years to 73 years. In Africa from 38 to 55, North America 68 to 78 and just over the course of the whole world there we can now live expect to live 20 years longer than we could before. And as a result of this we're seeing this just increased life expectancy at birth. And this again, divides it up by several countries. Now Again, intrinsically it's not a bad thing that populations grow. However, there is some maximum number of individuals that can be sustained on the planet. We'll come back to this question in the next video. Thank you.