0:00

Hello, welcome back to introductions to genetics and evolution.

Â In previous videos, we've been looking at concepts related to the amount of genetic

Â variation in populations relative to the amount of total phenotypic variation.

Â Now as I mentioned before, this genetic variation is that subset of variation,

Â which can be used by natural selection.

Â So let's talk about natural selection for a minute and

Â apply this to a couple of other areas.

Â And very soon, you'll see we'll be transitioning into population genetics.

Â 0:28

Well, natural selection is both noncontroversial and

Â as I mentioned before is a mathematical inevitability.

Â It's a mathematical inevitability if three simple conditions are met.

Â First, you have the phenotypic variation.

Â We talked about phenotypic variation just recently in the context of

Â quantitative genetics.

Â And that would be that Vp is not zero.

Â That basically, everybody doesn't look exactly the same in every way.

Â 0:52

Some of that phenotypic variation you see is actually inherited, or genetic.

Â In that regard, we're saying that the heritability is not zero.

Â So you see, I'm introducing the same terms we did before,

Â but now with some quantitative genetics behind it.

Â And last but not least, this variation that is inherited,

Â must affect either survival or reproduction.

Â 1:13

Let's walk through this very simple cartoon,

Â which illustrates this point nicely.

Â So here we have several peppers that vary in their mildness or hotness.

Â There are some that are mild, some that are hot.

Â So that there is this variation.

Â This variation is inherited.

Â The mild peppers give birth to other mild peppers.

Â The hot ones breed other hot ones.

Â 1:31

Here's an interesting point I haven't emphasized so far, but

Â this is relevant for today's lecture.

Â More individuals are born than will survive to reproduce.

Â Our capacity for population growth tends to be very high.

Â We'll come back to this in just a minute.

Â And related to this, some variants survive or reproduce at higher rates than others.

Â In this case, we can see that the milder peppers don't survive as well as the hot

Â peppers because the mild ones get eaten up by things like mice or

Â humans who likes the mild peppers.

Â The outcome of this is that the population changes over time.

Â That over time as this is iterated over and over again,

Â more of the surviving peppers will be hot, because you've eliminated them.

Â Not only the mild peppers themselves, but even that

Â 2:13

genetic contribution within the population that made the peppers hot.

Â Now if we go back in time, that original concept of natural selection was very

Â intrinsically tied to that of population growth.

Â And Darwin's ideas were very much influenced Malthus, who was an economist.

Â Malthus had pointed out that populations are actually kept

Â from growing by limited food and

Â resources, because again, our capacity to breed to breed is very, very great.

Â And if you provide more and more food to a particular species,

Â you'll tend to see that they will reproduce at a higher rate than is needed

Â just to recreate the number of individuals present.

Â That basically, the population will grow if it is able to grow, almost always.

Â Darwin pointed out that this limitation produces struggle, wherein some

Â subset that are better able to survive or reproduce will tend to spread.

Â But let's look at this concept of population growth and

Â why is it that natural populations grow so much?

Â Why do we see this potential for great growth?

Â Well the capacity for growth is huge in most species out there.

Â And imagine, to maintain a constant population assuming

Â the population was sexual, assuming you need a pair of organisms to breed,

Â each pair would only need to produce two surviving offspring.

Â Now, if conditions are favorable, how many offspring can most individuals produce?

Â We'll look at some plants, for example.

Â Let's see that you're looking at seeds or pollen.

Â Here's a picture of a tulip stamen and all the pollen on it.

Â Look at all those individual grains of pollen on there.

Â How many possible offspring could that tulip have?

Â Definitely a lot more than two.

Â Think about insect larvae, if you let an insect breed,

Â how many offspring can it have?

Â Think about things like cane toad eggs.

Â Cane toads as I mentioned a long time ago are an invasive species in Australia and

Â also in Hawaii.

Â And they can have hundreds, thousands of offspring just from one breeding pair.

Â And it's even true for

Â humans that if humans were allowed to in some way, we could potentially.

Â One couple could produce many, many, many kids easily 10, 20 something like that.

Â 4:23

Now we like to model the rate of increase and see how it's actually happening.

Â Well populations can be modeled

Â with what's referred to as a stable rate of increase.

Â This is assuming that a steady fraction of the population, or

Â the population increases by a steady proportion year after year.

Â To model this we need a couple of parameters.

Â We need a birth rate which can be modeled as for

Â example number of births per thousand per year.

Â We need a death rate, the number of deaths per thousand individuals per year.

Â And from this we can come up with what's referred to as the intrinsic rate

Â of increase of a population.

Â That is very simply the birthrate, so

Â the input minus the death rate which is basically the output from the population.

Â So in the United States as an example, the birth rate is about 14 per 1000,

Â the death rate is about 6 per 1000.

Â So that makes the intrinsic rate of increase about 0.008.

Â In this case, this is not considering immigration.

Â This is just looking naturally.

Â What this means is the population will grow naturally by 0.8% per year.

Â 5:22

Now if the birth rate is greater than the death rate, then the population grows.

Â Conversely, if the birth rate is lower than the death rate,

Â then the population will decline.

Â Now, so let's look at the effect on population size.

Â And we'll go through a little bit of math here just to work this out.

Â I hope you don't mind.

Â So, let's say that n is the population size and t is time in years.

Â Now we can identify a standard rate of population growth as dN over dt.

Â Or change in population size over change in time.

Â And that would be equal to that rate of increase times the population size.

Â 5:58

Now, if we wanted to get a algebraic solution, if we assumed that

Â this was a constant process, we can approximate this as N sub t,

Â so population size at time t is equal to N sub zero

Â which is that starting population size, times e.

Â E is the algebraic number you often see, I think it's approximately 2.71.

Â E to the power of rt, r being that intrinsic rate of increase and

Â t being time.

Â So time may be measured in years.

Â So, what is the population doubling time?

Â How long does it take for a population to double it's number of individuals, right?

Â Obviously, that's going to be related to R, it's going to be related to the rate of

Â increase that if R is very large the doubling time will be very short.

Â If R is very, very small then the doubling time would be very long,

Â but let's put some actual numbers on this just so you can see.

Â So this is the formula I showed you before.

Â N sub t = N sub 0, e to the rt.

Â And probably for this N sub t, is the population size at time t,

Â at the end of the period that you're studying.

Â What we want to solve for, is we want to solve for the doubling time.

Â The time it takes to go from a particular population size,

Â lets call it N sub 0 at the beginning, to double that size.

Â So, we're waiting for N sub 0 2N sub 0.

Â So, what we can do,

Â is we can basically solve this formula by putting 2N sub 0 here for N sub t.

Â Okay, and that what we're going to do is we're going to solve for t.

Â The idea here is to look at basically how many generations it takes to go from

Â 7:35

So we start off, as I said, with substituting to this formula for

Â N sub t we put 2N sub 0 and we have 2N sub 0 = N sub 0 = e to the rt.

Â Simple algebra, we just divide those side by n sub zero so

Â we have two = N to the rt.

Â Now how do we solve for t with this?

Â Because that's ultimately what we are going for.

Â What we have to do here is we have to use the natural log.

Â Remember, e is approximately equal to 2.71,

Â this is a factor that's used quite a bit.

Â You have a button on your calculator that probably says E to the X and

Â another that says ln, ln is the natural log.

Â So what we do is we take the natural log of both sides.

Â Take the natural log of both sides.

Â So we said natural log of 2, is equal to the natural log of e to the r t.

Â By the way, if you don't have a calculator that does this, you can just go to Google

Â and type in ln space 2, and it'll actually solve that for you.

Â So, we take the natural log of 2 and that calculates out to 0.693.

Â The natural log of e to any power is that power.

Â So in this case the natural log e to the rt is rt.

Â So we have now very simply 0.693 = rt.

Â Now again, we're solving for t, so what do we do?

Â We divide both sides by r and there we go.

Â The doubling time is now t which is 0.693 / r.

Â So, let's put some numbers into this.

Â Well, we have a US population at the time this was recorded of about 310 million.

Â I mentioned before that the population growth factor was about 0.008.

Â So all we have to do is put in this r into the formula right there.

Â So 0.693 divided by 0.008 and we have 86.6 years.

Â 9:17

Think about that, that is just a little bit longer than the average life time.

Â A lot of people live that long.

Â And that is how long it would take until the population of United States goes

Â 310 million to 620 million people.

Â That is a lot of people and that's fairly quick.

Â This is assuming this sort of growth as we drew it here, but it's not unrealistic.

Â Now let me get you to try one but I want you to notice ahead of time that you don't

Â actually need to know the population size to get the doubling time.

Â You basically don't need to know this n sub 0 factor.

Â All you need to know is r.

Â 9:54

Solomon Island, here are some numbers I got from Wikipedia.

Â The birth rate on Solomon Island is about 35/1000.

Â Death rate's about 5/1000.

Â Population size today is about 500,000, but you don't need to know that.

Â How long would it take to get to a million?

Â How many years would it take to get to a million?

Â 10:10

Well, I'll let you solve that problem here on the question online.

Â Well, I hope that one wasn't too challenging.

Â Very, very simple problem.

Â What we want to do is, is we want to calculate r.

Â R as I mentioned before for is equal to the birth rate minus the death rate.

Â So it would be 35/1000- 5/1000

Â which is in this case would come out to 0.03.

Â So that's your r.

Â So when we want to calculate the doubling time,

Â say doubling time in this case t would equal to r or

Â 0.693 divided by r, 0.03.

Â So in this case comes out to 23 years.

Â That's really dramatically fast isn't it?

Â That's changing the population on these island from 500,000 to a million.

Â So you're doubling the population size in 23 years.

Â 11:12

This first figure shows historic and projected future population growth.

Â So you notice our world population very slow, and

Â we started seeing here very very recently just in the last couple 100 years,

Â this dramatic increase in numbers of individuals.

Â So we hit 1 billion in the year 1800, we hit 3 billion in 1960,

Â 4 billion in 1974, 5 billion in 1987, 6 billion in 1999,

Â 7 billion in 2011, and it's projected we'll hit 8 billion by 2024.

Â This divides it up by some countries where you're looking at some of the developed

Â countries and less developed countries, Africa as a continent, China by itself and

Â India by itself.

Â So this is projections up to 2050, in this case, for world population growth.

Â Now, interestingly, our rate of population growth has actually gone down.

Â Now we are still increasing.

Â R is still positive.

Â The birth rate is still exceeding the death rate.

Â But the amount with which it's exceeding it has actually gone down

Â over the last couple of years.

Â But it's still positive and

Â that's why we're continuing to see this very rapid increase.

Â 12:26

this shows you the population growth in China over time.

Â And this shows you the age distribution as of 2009.

Â And you may be thinking like why is this still going right?

Â Because we have the One-Child policy instituted around 1979 1980, yet

Â we still see this increase Increase.

Â Interesting if you look at this,

Â the increase is actually closer to linear than it is to exponential.

Â Like it's not going up, it's not continuing to accelerate,

Â as you would expect.

Â So this probably has actually had some effect on slowing population growth,

Â but, again, the birth rate is continuing to exceed the death rate.

Â So we're still seeing an increase in the population size in China,

Â as in many other places in the world.

Â Well, why is that?

Â Well again, we have two causes of population growth.

Â There's high birth rate.

Â And there's low death rate.

Â So we can cut the birth rate, but we still see growth.

Â And what that means is the death rate is decreasing

Â faster than the birth rate is decreasing.

Â Now that's not intrinsically a bad thing.

Â Obviously, we like decreasing our death rate.

Â We don't want to have people just die willy nilly of course.

Â But this is a problem in terms of how many people are on the planet.

Â 13:33

Over the course of many countries and over the course of a lot of recent time,

Â we've seen this declining mortality or declining death rate.

Â And some of it is associated with things like introduction of the use of soap.

Â Improved sanitation, definitely things like antibiotics and

Â other modern medicine have really increase life expectancies.

Â I can show you how much too.

Â This is just showing the difference between 1950 and 2010.

Â Now look at this in Latin America and the Caribbean.

Â We have an increase in life expectancy from 51 years to 73 years.

Â In Africa from 38 to 55,

Â North America 68 to 78 and just over the course of the whole

Â world there we can now live expect to live 20 years longer than we could before.

Â And as a result of this we're seeing this just increased life expectancy at birth.

Â And this again, divides it up by several countries.

Â