The growth of populations is held in check by several factors. These can include predators, food and other resources, and density. Population density affects growth rate by determining how likely is it that an organism will interact with a member of its own species compared to an organism of a different species. Population growth studies rely on the mathematics of logs and exponents.
Cotgreave, Peter and Irwin Forseth. Introductory Ecology, chapter 6
March 30, 2009
Professor Stephen Stearns: Population growth is one of those things in ecology that is good to get kind of an intuitive feeling about. And the main thing that you need to grasp about population growth is that it is multiplicative. So that if you have a certain number of things, and they are making more things, then the amount you have next time is proportional to the number you have this time, and it multiplies by itself. It turns out that this kind of process takes a little bit of playing around with to internalize.
What we'll do today is start by talking about density independent growth, and that is basically compound interest. So compound interest yields exponential growth, and if it only went on forever, we would all be very wealthy.
When I was a professor in Switzerland, at one point I suggested to the Swiss that if they had simply invested 100 Swiss francs, when the university was founded in 1460, that they would have an endowment greater than the gross domestic product of the planet, at this point; at only a nice conservative Swiss 4% interest. And they would, but they looked at me knowingly and they said, "Ah, you've forgotten Napoleon" who wiped them out. [Laughs] Okay?
So the basic point about--let's see, have I gotten through that? Yes--the basic point that I want you to get about compound interest is that it will very rapidly produce a whole lot of stuff, and that the rate of multiplication is quite a bit faster than the arithmetic difference in the interest rate. The idea of this whole chart is to show you that a small difference in interest rate can make a huge difference in the outcome.
So I've used basically a bank account to attract your interest, with dollars, but this will work with organisms and populations just as well. The time to double for money invested at 1% is 70 years; 5% is 14 years; and 10% is 7 years. And the yield after 70 years, on $100.00, looks like this. You get $200.00 at 1% interest, and you get $100,000.00 at 10% interest. Okay? So a ten times increase in the interest rate leads to a 512 times increase in the outcome.
Small differences in interest rates make big differences in outcomes. And as we'll see in a sec, the interest rate on a population is the difference between the birth and the death rate. So you can jiggle the birthrate a little bit, you can jiggle the death rate a little bit, you'll get a fairly substantial difference in the growth rate, the interest rate on the population, and something like this can happen.
So I want to take you just through the math of doubling time, so you understand that previous chart. And you also need to start getting familiar with this kind of notation, because in population dynamics and ecology it is used pervasively. So B is often used for the birthrate, D for the death rate, and the per capita growth rate is the birthrate minus the death rate. Okay? That's birthrate per capita, death rate per capita, and this is growth rate per capita.
So if you've got N organisms, then you have the simplest differential equation you can write down practically, which is that the rate of change of the population is equal to the growth rate, times the number of organisms that are present. Okay?
That means that the slope of population growth is RN. So if we have N on the y-axis, and time on the x-axis, it looks something like this. And I put this in to indicate that for a given change in time unit, the amount of growth you get in the population just keeps on going up and up and up. Okay? It keeps jumping. So that the rate of increase in an exponential process is proportional to the growth rate times the amount of stuff that's there; and that's the slope of the relationship.
So if you write this down and rearrange it a little bit, then you can integrate both sides, and you get that the natural log of the number of organisms in the population is equal to the growth rate times the amount of time that's elapsed since they started growing, plus some constant. You can exponentiate both sides. So if you raise E to the log, you get N. Okay? And rearrange to get that.
So the number of organisms you have T-time units later, is equal to the number that you started with, times E, the base of the natural logarithms, raised to R times T. And, in fact, that's the basic compound interest formula as well. So you can use that for bank accounts, just as well as you can use it for organisms.
So you ask yourself, "Well how long does it take for that population to double?" Where did I come up with that chart that I showed you on the second slide? Well the doubling time is the time that it takes for whatever that number is, divided by what you started with, to equal 2. And that will happen when 2 = eRT. So the log of 2 is equal to R, times the doubling time; and the doubling time is about 0.69 divided by R, because that's the log of 2.
So you can multiply that by 100, if you want to use percent, and you get the simple rule of thumb that you can divide 69 by the interest rate to get the doubling time. Or, as a rule of thumb, if you're comfortable with approximations, it's a little easier to divide 72 by the interest rate to get the doubling time, because you can divide 72 by 3 and by 4 and by 8, and things like that. Okay?
Now, I'd like you to get a little bit familiar with this sort of thing. And here is a voluntary homework assignment. Some of you will enjoy this, some of you will not. This can easily be done on a spreadsheet. You will have to look up a few numbers on the Web; they are readily available.
I'd like you to consider an asexual rabbit population consisting only of females, and each individual reproduces at an age of just three months; gives birth to two offspring and dies. I did that, I set it up that way so that you would have a doubling every three months. All of the offspring survive to reproduce; that preserves the doubling. You can play around with these assumptions. Adult rabbits weigh one kilo. There's no density dependence. I didn't give you this, but let's say they cover 100 square centimeters, something like that; so about 1/10th of a meter.
How long will it take to cover Science Hill two meters deep in rabbits? I think you'll be surprised, it doesn't take very long. We can just start with one, and you're going to have Science Hill six feet deep in rabbits pretty quick. But you can go farther than that. Okay? How long will it take for the leading edge of the expanding ball of fur to exceed the speed of light, and what will be the diameter of the ball at that point, measured in earth orbits? I think the answer is roughly 750 years, and you're somewhere about the diameter of Jupiter. Okay? About; if I recall correctly. I did this once.
And then you can ask how long will it take to reach the point of gravitational collapse into a black hole? Well actually I think it will happen before the ball of fur hits the speed of light. Okay? Because you're well out beyond the diameter of the sun. The point of this ridiculous example is exponential growth rapidly makes a heck of a lot of stuff. Okay? I have left unexplained where the rabbits got their food. [Laughs] Okay? So a small change in growth rate will make a big difference in the outcome.
Remember how to integrate this differential equation; it's pretty straightforward. Doubling times are a very convenient way to conceptualize growth rates. So you can divide 69 by the growth rate to get the doubling time; divide it by the growth rate in percent to get the doubling time, for any process.
Now in summarizing this, there was no age structure. All of those numbers that we were counting up represented the population as though every single organism in it had the same probability of reproducing or dying; and that's not true.
One of the most obvious things about populations is that organisms of different ages reproduce at different rates and have different risks of dying. So it would be nice to be able to do population dynamics, with a bit more realism, by sticking in all the difference that age structure makes; in other words, doing demography.
Pre-adult organisms don't give birth. Old organisms have high probabilities of dying. Sometimes newborn have very high probabilities of dying. And if we look around the world, we can see that there are roughly speaking three different kinds of survivorship curves. We have age on the x-axis here, in an arithmetic scale, and we have survivorship on the y-axis, in a logarithmic scale.
So when you see something like a type-3 survival curve, that is a huge drop in the number that are present say, that are all born here, at age 0. And on this scale there's only one of them left alive after a short period of time.
Here you have a straight line, and if you have a semi-log graph, and you have a straight line, what the straight line means is that a constant proportion are dying at a given time interval. So if you are looking say at this point, on the type-2 survival curve, you have 1/10th of what you had at the start, and if you look at this point on the type-2 survival curve, you have 1/10th of what you had there. So the straight line is a constant proportion dying in each time interval.
And up here, life looks pretty good, until you get to be a somewhat aged adult, and then you have rapid aging. So the kinds of things that have type-1 survival curves are humans, elephants and albatrosses. Small birds have type-2 survival curves, as do hydra. And oysters and trees have type-3 survival curves; and orchids. They make hundreds of millions, or billions of seeds; most of the seeds die; and once you have made it to adulthood, then your prospects are pretty good. This curve is pretty flat here. It's almost like this curve is up here.
So when we look around the world, we see that different kinds of organisms have a--well they actually have quite a bit of continuous variation in demography, but if we want to stick them into categories, we can see some illustrative extreme cases, that help us to understand the diversity of population dynamics that organisms encounter, as well as the diversity of selection pressures that they encounter.
Now in order to analyze that, we need a bit of demographic notation. And I'll step through this, just pointing out some of the key things where people can stumble when they first encounter this notation, and points that you want to make sure you keep in your memory as important distinctions.
The first one is the distinction between age and time, and that is the distinction between being 62-years-old and 21-years-old in 2009, and what's happened over the last 20 years. Okay? So you can have people of different ages at the same time, and you can have people of the same age at different times. So X keeps track of age and T keeps track of time.
Then there are two different kinds of ways of thinking about survival. One is the probability of surviving from now to next year, or now to the next time unit, however you choose to scale your time unit.
And the other is the probability of surviving from birth until now, which would be the probability here, LX, probability of surviving from birth to beginning of age class X, whatever X is. And that is just the product of all of the PXs up to that point. So it's the product of surviving from birth to age 1; age 1 to age 2; age 2 to age 3; etcetera, up to now.
Then once you get to be X years old, MX is the symbol that keeps track of how many babies a female would have, that survived to that age. Alpha is age at maturity; Omega is age at last reproduction. And age at maturity doesn't mean, in demography, age at acquisition of secondary sexual characteristics or acquisition of the ability to reproduce, it means the actual age at which a baby arrives; okay, when offspring are born.
Little-r is the population growth rate. And here we have a collection of three different growth rates, and they mean somewhat different things. And if you're thinking about zero population growth, the value of these things is going to be different for zero population growth.
r, as I mentioned before, is B minus D, birthrate minus death rate, and it's an instantaneous per capita population growth rate. Okay? So it is comparable say to the interest rate that your bank tells you that you're getting on your savings, or that you're being charged on your credit card.
Big Ro is the lifetime expectation of female offspring. In demography we tend to keep track only of females, because they are the rate-limiting sex. They are normally, as you remember from sexual selection lectures, the limiting resource, and it is their reproductive rate that actually determines the rate at which the population will grow. So this is the rate of growth per generation, and it's Ro; and Lambda, which is er is the multiplicative rate of growth per time unit. And you'll see how these numbers compare as we go through the next slides.
These things are calculated on the basis of life tables. So a life table is an accounting tool, and frankly life tables are a bit like natural selection in the following way; they actually do imply a kind of natural selection, but that's not the point I want to emphasize. They are deceptively simple.
Life tables are basically a rather boring kind of accounting of births and deaths, but they have very deep implications. Just as natural selection is based on four very simple conditions that you can write down, but nevertheless creates all of the complexity of brains and livers and everything else, life tables have that kind of deceptive simplicity.
So LX--we're going to make a simple one--LX is the probability of surviving from birth to age X. BX or MX is the number of female offspring, born to females of age X. And LXMX is the probability both of surviving to X and of having MX offspring.
So the sum, LXMX, over all ages X, is the expected number of female offspring per female per lifetime; that's R0. So you can just put an equals sign in here, that's R0. Okay? So that's a measure of population growth, an important one.
And if we take a simple example where we have this survival rate from one year to the next--and I've set this equal to 0, so that nobody makes it to age 3--starting at birth everybody--we're keeping track of things that get born--so everybody has a probability of 1 of being born--because those are the only ones we're counting--and 50% of making it to age 1, and 25% of making it to age 2. Just P0 times P1 gives us .25 here.
This is the birthrate. Okay? So these organisms mature at age 1 and they get better at reproducing; perhaps they continue to grow and they have 2.4 offspring per female at age 2. And these are their contributions to fitness: .5 here; so half of them make it to age 1 and have 1 offspring, and a quarter of them make it to age 2 and have 2.4 offspring. We multiply those numbers together, sum them up, we get R0 = 1.10. The important thing about R0 is that it's greater than 1. This population is growing. Okay?
Now if we want to look ahead and ask what's going to be the age distribution in future years? If you have a life table for humans--you might be interested in knowing will there be enough people around, who are young, to pay for your Social Security when you're old?--you can use this procedure to do so, and that's what demographers do with it. Okay?
So the number of newborn is the sum across age classes; so the number of females alive that year, times their expected fecundity. That's pretty straightforward. Just take the number of females, take the average number of babies they'll have, and that's the number of newborn.
The number alive in any older age class is the number that were alive in the younger age class in the previous year, times the probability they survived. That's pretty straightforward. This is the deceptive simplicity of life tables; just straightforward accounting.
So here we have again a …. this is the same life table we had before. Now I've put in 10 organisms in each of the three age classes. Okay, we have 10 newborn, 10 1-year-olds, and 10 2-year-olds. And this is how the population will develop. Half of these survive, half of these survive. Half of these survive, half of these survive. I've rounded these off to get whole organisms. So that's the survival part of it.
How about the births? Well these 10 here are going to be giving birth to 5 offspring, and these 10 here are going to be giving birth--excuse me, these 10 here are going to be giving, each of these are going to be giving birth to 1 offspring. So that's 10. And these 10 here give birth to 2.4. That adds up to 34. Okay? This times this, plus this times this, is that. Five offspring come from this group and 12 offspring come from this group, and we get 17. Okay?
And that just keeps going. So in each generation you can carry that out as a matrix transformation. It's called the Leslie matrix. You can use all of the properties of linear algebra to project things into the future. You can use mathematics packages, like MatLab or Mathematica, and even Excel spreadsheets, to carry this process out.
Now if you study that process, it turns out these are its critical take-home messages. Any such process, where you have a constant rate of birth and death, will produce a population that attains a stable age distribution in which the proportion of individuals in each class remains constant. So that means like the ratio of five-year-olds to ten-year-olds remains constant. And it will get there fairly quickly; not immediately but fairly quickly.
When it gets into stable age distribution, both the entire population and each age class in it are growing exponentially at the constant rate, r, and that r has exactly the same meaning as it did when we were back dealing with a simple population in just that simple differential equation. Okay, so that's this state. The growth rate of the age-structured population is the same as the growth rate of a density independent population.
So, let's do an example. Here's a life table that actually is roughly that of a small bird. So the guys that fly into my bird feeder in Hamden, the house sparrows, the chickadees and so forth, they could have a life table that looks something like this. They don't live very long.
So here are the probabilities of surviving to age 1, 2 and 3. Here are the birthrates. I've set them constant for the three age classes. Here's LXMX. This is our population growth rate. It's 1.2 per generation.
So just based on this, we can make several interesting statements. Their population is growing. Okay? And that's because 1.2 is greater than 1.0. And it is multiplying actually 1.2 times per generation; not per year, but per generation.
We can calculate the generation time with this formula. So we just divide sum of LXMX times X, by R0. That turns out to be 1.67 years. And the meaning in words of generation time, in demography--it's a technical meaning--is the average age of the mother of a newborn. So that's the average age of the mother of a newborn.
The little-r, can be estimated--and this is an estimate, it's not precise--by taking the log of big-R and dividing it by the generation time; and that's about 0.11. So this population is growing at a compound interest rate of about 11% per year, and we can use that to go back and use our doubling time calculations to figure out, oh, it's doubling about every 6.3 years. If I'm putting the bird seed out there in Hamden, I had better be ready for large expenditures.
And I'd like you to note that R0 is calculated on a different time period, generation time, than little-r. So when big-R0 is equal to 1, little-r is equal to 0, and the population is then--the word is used, stationary; it's just replacing itself. A stationary population is replacing itself, but a population with age structure, that is growing, can be in stable age distribution. Stable age distribution refers to the ratios between the age classes; stationary refers to whether or not it's just replacing itself, or whether it's growing or declining. Okay?
So that's a rough sketch of simple demography, and an introduction to the different ways that ecologists and demographers conceptualize growth rates. Now I want to criticize--not really criticize, but comment on one of the basic assumptions of this.
The first way that you can complicate simple dynamics is by putting in age structure, but the second way you can complicate it is by putting in density. Here is growth of a bird that was introduced to Great Britain, the turtledove, and it took off from a very small population, and it grew exponentially. Remember, if you have a straight line on a semi-log plot--so y-axis here; numbers is on a log scale; x-axis is on an arithmetic scale. So this is a straight line, and it's going like gangbusters, it's growing exponentially.
And about every, let's see, one, two, three--it's increasing in size about ten times every three years. So these doves are really pumping out the babies and they're surviving pretty well. But something happens up here. This is a real point here; this is a real point here, and it starts to level off.
So the question is, what does increasing population density do to the demography, the birth probabilities, and the death probabilities of individuals? Can we understand that in terms of the kinds of concepts we're already covered this morning?
Well you've seen this plot before. So as density--we can think of this being a high density population and this being a low density population, with rapid growth, rapid individual growth--now not population growth, but rapid individual growth, increase of size with age at low density, and slow individual growth at high density.
And I think by now you're familiar with the idea that there is a reaction norm for age and size at maturity. And if you move from low density to high density, basically what happens, in many cases, is you get organisms that mature later, at a smaller size. So one of the very basic characteristics that determines population growth is age at maturity.
That's basically the interval over which the compound interest is being calculated, and that's responding plastically to density. So as the density of the population goes up, the organisms delay maturity, they mature later--here, rather than here--and they mature smaller--here, rather than here. So they are less capable of making babies, and they're doing so later in life and at a slower rate.
The other dramatic thing that happens with density is that mortality increases. Here is an experiment that was done with trout. This is a log scale here and a log scale here, and the flat line here basically indicates that growth has stopped, or that density has stopped increasing.
And if you just take an experimental stream bed, and you seed it with baby trout, and you do that a number of times, at different initial densities per square meter--okay? So this would be 10 raised to the 2.5 per square meter. So this is about 300 trout per square meter, and this is 10 trout per square meter over here. So in arithmetic terms, we're going from 10 per square meter, up to about 300, and what we get out, at the end of the experiment, is roughly 10 per square meter.
So there's an enormous amount of mortality which is going on, over on this end. And you can think of that as sort of runt of the litter kinds of things, baby pigs competing in a litter, or you can think of it simply as interspecific competition. These trout are competing for food, and the ones that are best at finding it are going to be the ones that survive.
There've been lots of experiments done with plants. Here again we have a log scale on this axis and a log scale on this axis, and this is the density of surviving plants here. Okay? So this is the number per square meter.
On this axis we have the number that were planted, and that's going to range from--down here we're probably at about 12 or 15, up to about 1000 per square meter, were planted. And these different lines here are showing you what the density of plants is 22 days after planting, 39 days after planting, 61 days after planting, and 93 days after planting.
And this is the demographic process that basically leads to constant final biomass. So if you go on, all the way through to harvest, you're going to end up with about 100 plants per square meter, and the--by the way, the way that plants regulate their individual growth, you essentially get the same amount of leaf area or root area covering a square centimeter of ground.
So if you plant a lot, you're going to end up with fewer leaves and less roots, and you're going to get smaller plants, but the total amount of plant material sitting on that square meter will be roughly constant, irrespective of initial planting density, if you just let this process go on long enough.
The other thing that happens is that as density goes up, fecundity goes down, and that's because as density goes up, competition for food increases, individuals are not getting fed as much, and they are therefore less able to make babies.
In this panel over here, we see basically natural variation in song sparrows on Mandarte Island, which is a small island off the coast of Vancouver Island; it's not too far from Victoria. If you take the ferry from Vancouver, over to Victoria, you go within about a kilometer of Mandarte Island. Okay?
And this is the number of young that females were able to fledge. These are the years. So that's 1980, 1981, 1982. This is the number of breeding females on the island. And what you see is that as the number of breeding females increases, the number they are able to fledge starts to decrease.
And in the year 1985 they did an experiment where they added food--so they wanted to see whether this natural variation could be manipulated, just by putting food out on the island--and those who were fed were able to rear nearly four offspring; and at high density, up here, in 1985, those who were not fed were only able to rear one.
So there was a difference per female of nearly three offspring, and that was what the population density was doing to them, and the manipulation experiment showed that it was the density and that food was the mechanism.
If you look at a dune plant in the Netherlands--this is Vulpia, which is a grass, and this is growing near Leiden, on the dunes on the North Sea in the Netherlands. The number of seeds per plant, number of flowing plants per quarter square meter; the number goes down. So the more crowded they get, the fewer seeds they can make.
So this is variation that's really going on in Nature, and it shows you that the effects of density are quite real, they're pretty dramatic. These effects can be combined. So if we want to combine number of births and numbers dying, this would be density, over here, going from 0 up to a number K.
The numbers dying increase as we go across this, and the numbers being born decrease, and at some intermediate point you have the greatest difference between the births and the deaths, and that difference between births and deaths is the recruitment. Okay? So the recruitment inferred from this graph looks like this, and at 0--you get zero recruitment when you hit this number K; and that number K is called carrying capacity. So the population, under density-dependent regulation, will increase until it hits carrying capacity, and then it will stop growing.
Now as density is taking its effect, the size distribution of individuals in the population often shifts. So what I'm basically doing here is I'm showing you all the things that density does to the individual lives of the organisms that are experiencing this density increase. And this is a very interesting one, because basically what's going on here--and this is measured in flax.
If you start at low, medium or high density, two weeks after emergence the distribution of body sizes in the population is pretty similar; it's a nice bell-shaped curve. This is now six weeks from emergence. So a bunch of other replicates have now been gathered.
And you can look at plant weight here, you can see that they're bigger, the scales are changing. But you'll notice that at the medium and at the high-density treatments you're starting to get quite a bit of skew. You're starting to get a few big ones and lots of little ones; that's what this kind of histogram means here, lots of little ones and a few big ones.
And at the final harvest, which is probably about 12 weeks after planting, you're starting to get a little bit of skew at the low density treatment, but at the medium and high density treatments, you've got a lot of skew. Okay?
Now this is something that was going on, by the way, in those trout experiments; you were getting a few big ones and a lot of little ones. In the plants living in the Dutch dunes, you were getting a few big ones and a lot of little ones. It's a pretty widespread pattern that happens as density increases.
There's going to be intra-specific competition. Some individuals are going to be better at it than others. Okay? And it's going to result in these kinds of size and growth distributions. So the more competition there is, the greater the skew will be between lots of little ones and a few big ones.
So that means that competition is producing an asymmetry, and this asymmetry, where you have a really skewed size distribution, just basically means that a few individuals are going to be doing most of the surviving and reproducing, and a lot of individuals are going to be doing most of the dying, and not reproducing.
In many cases the number of large individuals is relatively constant, while the number of small individuals varies much more widely. That's the law of constant final yield. Okay? Or the flat line that we saw with the trout that were planted at- were seeded into the streams, at many different densities, but you always ended up with roughly the same number. That was probably the number of the large surviving competitive individuals, and the small individuals are being squeezed out.
That has an evolutionary consequence. This is a direct tie now between population dynamics and natural selection. Not all those who are born will survive to reproduce; some are starving or are being crowded out. And not all who survive to reproduce are large and in good shape. Many of them, in fact, are in poor condition and have few offspring.
So intra-specific competition produces great variation in capacity to reproduce and in reproductive success, and that means that wherever you have reproductive and mortality skew, that's being produced by intra-specific competition, you are generating the conditions for natural selection. It's as though there's an internal process in ecology that just hands you the conditions for natural selection on a dinner plate.
Now I previously mentioned the case of the collapse of the sardine fishery in California. And the sardine fishery, which collapsed in California between about 1948 and 1955--and still hasn't come back by the way--it collapsed because of conditions that were affecting juvenile sardines that were out in the plankton in the Pacific. It did not collapse because of a fishery that was operating on the adult sardines; and this is pretty well established now.
When fewer and fewer juveniles were being recruited into the adult population, the individual adults were essentially experiencing a release from intra-specific competition. So their response was to grow faster; they could eat more, they could grow faster and they could make more babies. And just as the last boats in that fleet were pulling in and deciding to stop fishing, they were catching a very few, very big sardines; the sardines were about a meter long.
They had gone from that big to that big, and that was what the release from intra-specific competition had done to them. And if they were able to re-establish the fishery and have lots of babies and have the babies recruit into the population properly, and grow up, the size would shrink right back down. That would be the impact of density dependent population growth on the California sardine population.
Okay, populations are held in check by lots of things. They're not just held in check by food. They are held in check by breeding sites, by space, by lots of limiting resources. As populations increase in density, the individuals shift along reaction norms. They reduce growth, they are smaller; when they become adults they experience more variation in adult size; they have lower fecundity and more variation in fecundity; and they have higher mortality and more variation in mortality, as they encounter density dependence.
Now if you have a successful species--okay? So it's a dominant species in its local habitat, in its ecosystem, then intra-specific competition, where these kinds of things are going on, is often a more important brake on its growth than inter-specific competition. And that's because, because it's dominant, it's numerically abundant, and the average interaction that an individual has is with another individual of the same species, rather than with some individual from another species.
So that would be a circumstance under which we could expect that intra-specific competition, generating all of these effects, is likely to be important. If the species are rare, or if they're often at a low population density, then the opposite might be the case; then the interactions with other species may be generating these kinds of effects.
But whether the effects are generated by interactions with other individuals of your own species, or individuals of another species, the impact of increased overall density is likely to be qualitatively similar. It's going to produce reduced growth, smaller adult size, lower fecundity, higher mortality, and more variation in all of these parameters.
Okay, next time we're going to do competition between species. But we have a few minutes this morning, because I managed to get through this one a little more quickly than usual, and so I would be happy to take questions on anything, if you wanted to ask; and we have about five minutes. [Pause]
You know, a lack of questions means that everything is just so stunningly clear that everybody could explain it perfectly well themselves. Fantastic. I once had the experience, when I was in my first year of graduate school in British Columbia, of being asked to be a TA. I had arrived in January, and I was asked to be a TA in Spring semester, and I was a TA in a statistics course.
And I stepped into this statistics course and discovered that even though the Fall semester had been taught by one of the greatest teachers at the University of British Columbia, a man who was revered for the clarity of his lectures and for the engagement of his speaking style, that the students, in fact, knew very little statistics.
And the problem was that they had thought, because he was so clear, that they understood everything. That was not the case. They couldn't understand it until they explained it themselves, and they discovered they couldn't explain it themselves. Any questions? Have a good day.
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