WEBVTT 00:00:00.000 --> 00:00:04.000 And, we're going to make a major shift. You're going to feel like 00:00:04.000 --> 00:00:09.000 this is a whole different class compared to what we were talking 00:00:09.000 --> 00:00:14.000 about last time, because were jumping from the 00:00:14.000 --> 00:00:19.000 biogeochemical cycles, or looking at the biosphere as 00:00:19.000 --> 00:00:23.000 essentially a large biochemical machine, to studying individual 00:00:23.000 --> 00:00:28.000 populations of organisms, and the communities that they make 00:00:28.000 --> 00:00:33.000 up when they come together. So, before we were really talking 00:00:33.000 --> 00:00:38.000 about organisms as they function in the biosphere. 00:00:38.000 --> 00:00:44.000 Mentally, we're grinding them all up and thinking of them as a 00:00:44.000 --> 00:00:49.000 collective biochemistry basically. And now we are going to stop 00:00:49.000 --> 00:00:54.000 grinding them up, mentally, and think of them as 00:00:54.000 --> 00:00:59.000 individual organisms. So, the next series of lectures, 00:00:59.000 --> 00:01:05.000 we're going to talk about population ecology. 00:01:05.000 --> 00:01:12.000 If you remember the first lecture I gave we talked about the hierarchy 00:01:12.000 --> 00:01:19.000 of organization within ecological systems, and then we are going to 00:01:19.000 --> 00:01:26.000 talk about competition between organisms with a population, 00:01:26.000 --> 00:01:33.000 and between organisms of different species, and were going to talk 00:01:33.000 --> 00:01:40.000 about predation, and mutualism. These are all interactions between 00:01:40.000 --> 00:01:47.000 organisms that affect the fitness of organisms. And then we'll, 00:01:47.000 --> 00:01:54.000 at the end, talk about community structure. So this is sort of the 00:01:54.000 --> 00:02:01.000 outline for the rest of my lectures, not for this lecture. 00:02:01.000 --> 00:02:11.000 So, today we are going to talk about properties of populations. 00:02:11.000 --> 00:02:22.000 We're going to analyze how we measure growth rate, 00:02:22.000 --> 00:02:32.000 growth and death in populations, and this will include populations 00:02:32.000 --> 00:02:43.000 that have an age structure, and populations that don't. 00:02:43.000 --> 00:02:47.000 And this is all in preparation for the next lecture where we will talk 00:02:47.000 --> 00:02:52.000 about human population growth. So, in this field of population 00:02:52.000 --> 00:02:56.000 ecology, which is as I told you in the first lecture, 00:02:56.000 --> 00:03:01.000 and some universities you could take three courses in population ecology, 00:03:01.000 --> 00:03:06.000 and you could get a Ph.D. in population ecology. 00:03:06.000 --> 00:03:11.000 I mean, this is a whole field that we're going to cover in two lectures. 00:03:11.000 --> 00:03:16.000 But what population ecologists worry about fundamentally, 00:03:16.000 --> 00:03:21.000 well, they don't worry about it. This is what they study, is what 00:03:21.000 --> 00:03:27.000 regulates the density of populations? Obviously, it's a function of how 00:03:27.000 --> 00:03:32.000 fast they're growing, the birth rate, and how fast they're 00:03:32.000 --> 00:03:37.000 dying, the death rate. But what are the factors that 00:03:37.000 --> 00:03:41.000 actually influence those rates? Is it competition with other 00:03:41.000 --> 00:03:45.000 organisms? Is it the entire structure of the community? 00:03:45.000 --> 00:03:49.000 Is it the availability of food? Is it the various abiotic 00:03:49.000 --> 00:03:53.000 properties of the environment: temperature, etc. 00:03:53.000 --> 00:03:57.000 So, they analyze these and basically try to model the 00:03:57.000 --> 00:04:02.000 population growth as a function of these various parameters. 00:04:02.000 --> 00:04:06.000 The other questions they ask, is how are populations distributed 00:04:06.000 --> 00:04:10.000 in the environment? Are they clustered? 00:04:10.000 --> 00:04:14.000 Are they evenly distributed? This has specific meanings about 00:04:14.000 --> 00:04:19.000 their ecology. And, the other thing that people 00:04:19.000 --> 00:04:23.000 are really fascinated by, which is a really tough question, 00:04:23.000 --> 00:04:27.000 is why are some species' populations extremely abundant, while 00:04:27.000 --> 00:04:32.000 others are rare? And one of the discussions we always 00:04:32.000 --> 00:04:36.000 have in my lab, we work on an organism that's 00:04:36.000 --> 00:04:40.000 extremely abundant, this prochlorococcus, 00:04:40.000 --> 00:04:44.000 which I told you briefly about, is the most abundant photosynthetic 00:04:44.000 --> 00:04:49.000 cell on the planet. So, my students tend to keep saying 00:04:49.000 --> 00:04:53.000 why is it so successful? And I keep saying, it's successful 00:04:53.000 --> 00:04:57.000 but there are thousands of other species who are also successful. 00:04:57.000 --> 00:05:02.000 Abundance does not equal success. Endurance equals success. 00:05:02.000 --> 00:05:07.000 If you're here in the next generation, you're successful. 00:05:07.000 --> 00:05:13.000 If you're not, if your species is disappearing, then you're not 00:05:13.000 --> 00:05:19.000 successful. So, speaking of abundance, 00:05:19.000 --> 00:05:25.000 let's talk about how we measure abundance, population ecologists. 00:05:25.000 --> 00:05:30.000 And this is just one example. Obviously, for microorganisms, 00:05:30.000 --> 00:05:35.000 or some microorganisms it's really easy because they're tiny relative 00:05:35.000 --> 00:05:40.000 to their habitats. So for the prochlorococcus that we 00:05:40.000 --> 00:05:44.000 work on, there are 10^5 cells per milliliter. So, 00:05:44.000 --> 00:05:49.000 we can go take a milliliter of water and measure how many cells there. 00:05:49.000 --> 00:05:54.000 But for some organisms, larger ones, that are widely distributed, 00:05:54.000 --> 00:05:59.000 it's not that easy. So, one method is mark and recapture. 00:05:59.000 --> 00:06:04.000 That's used a lot for things like birds and butterflies. 00:06:04.000 --> 00:06:09.000 For a bird, the mark would be putting a band on the bird. 00:06:09.000 --> 00:06:14.000 For a butterfly, they often take a magic marker and put a mark on the 00:06:14.000 --> 00:06:19.000 wing. Well, that's largely what they do. You try to mark 00:06:19.000 --> 00:06:24.000 individuals in some way that would not influence their survivorship 00:06:24.000 --> 00:06:29.000 rate. So, if N equals the population size, 00:06:29.000 --> 00:06:34.000 that is, that's our unknown, what we're going to do is capture, 00:06:34.000 --> 00:06:39.000 say, for butterflies or moths, you use a butterfly net, or moths 00:06:39.000 --> 00:06:44.000 you can use a light to track them; for birds, you put up these big mist 00:06:44.000 --> 00:06:49.000 nets. They fly into them; they get tangled up a little bit but they 00:06:49.000 --> 00:06:54.000 don't get hurt. Then you band them, 00:06:54.000 --> 00:06:59.000 and that we let them go. That's the way you mark them. 00:06:59.000 --> 00:07:07.000 So, we're going to say n1 equals the total number of marked individuals 00:07:07.000 --> 00:07:15.000 released. So you capture them, you mark them, you release them. 00:07:15.000 --> 00:07:24.000 n2 is equal to, and then you go out sometime later and you recapture as 00:07:24.000 --> 00:07:32.000 many individuals as you can find, and this would be the total number 00:07:32.000 --> 00:07:41.000 [SIREN] that doesn't sound like a fire drill, does it? 00:07:41.000 --> 00:07:50.000 I assume we're good to go here. So, n2 is the total number of 00:07:50.000 --> 00:08:00.000 recaptured. And we're going to say m2 is equal to the numbers 00:08:00.000 --> 00:08:09.000 recaptured that are marked. OK, and then we assume that the 00:08:09.000 --> 00:08:18.000 fraction of the recaptured that are marked represent the fraction in the 00:08:18.000 --> 00:08:27.000 total population that was marked. So, we say m2 over n2 is equal to 00:08:27.000 --> 00:08:34.000 n1 over N. And the number that we're looking 00:08:34.000 --> 00:08:40.000 for, population size, is equal to n1, n2 divided by m2. 00:08:40.000 --> 00:08:45.000 So, of course, this assumes that there's no effect of the marking of 00:08:45.000 --> 00:08:51.000 the individuals. It assumes that there's no bias in 00:08:51.000 --> 00:08:56.000 the trapping for the marked or not marked individuals. 00:08:56.000 --> 00:09:02.000 There's all kinds of assumptions that underlie this. 00:09:02.000 --> 00:09:07.000 It's a start for assessing the population size. 00:09:07.000 --> 00:09:13.000 OK, so how do we measure population growth? We're going to first start 00:09:13.000 --> 00:09:19.000 with looking at populations that have age structure. 00:09:19.000 --> 00:09:25.000 Now, I hope you printed out the slides that were on the Web, 00:09:25.000 --> 00:09:31.000 because I'm depending on these overheads a lot for this lecture 00:09:31.000 --> 00:09:37.000 because we wouldn't get through any of it if I wrote all this 00:09:37.000 --> 00:09:44.000 stuff on the board. So, we're going to talk about 00:09:44.000 --> 00:09:53.000 populations that have an age structure. And the data I'm going 00:09:53.000 --> 00:10:03.000 to show you here is for human populations. 00:10:03.000 --> 00:10:12.000 But this applies to any population that has differential birth and 00:10:12.000 --> 00:10:21.000 death rates as a function of the age of the organism, 00:10:21.000 --> 00:10:30.000 OK? So, in these populations if birth rate and death rate are high, 00:10:30.000 --> 00:10:40.000 the population is dominated by young people. 00:10:40.000 --> 00:11:00.000 And, we'll look at this in a minute. And, if B and D are low, 00:11:00.000 --> 00:11:06.000 dominated by old people, or older I should say, since I now 00:11:06.000 --> 00:11:12.000 fit into the old category. OK, so here's a typical population 00:11:12.000 --> 00:11:19.000 age distribution for developed countries, where each slice here, 00:11:19.000 --> 00:11:25.000 these are females on the right, males on the left, 00:11:25.000 --> 00:11:32.000 and each slice is an age category: zero to 10 years, 10 to 20. 00:11:32.000 --> 00:11:36.000 And you can see that in these kinds of populations, 00:11:36.000 --> 00:11:41.000 you have a fairly even age distribution. Long periods of no 00:11:41.000 --> 00:11:45.000 net growth in a population lead to this. In these developed countries, 00:11:45.000 --> 00:11:50.000 and we're going to examine why this is, there's basically an even 00:11:50.000 --> 00:11:55.000 replacement rate of children for adults. And one of the things we 00:11:55.000 --> 00:11:59.000 worry about when you see this kind of age distribution, 00:11:59.000 --> 00:12:04.000 although it's good in terms of population growth, 00:12:04.000 --> 00:12:09.000 is when you have few young people and a lot of older people, 00:12:09.000 --> 00:12:13.000 who's going to take care of them, which is what's behind the Social 00:12:13.000 --> 00:12:18.000 Security crisis. But we won't get into that. 00:12:18.000 --> 00:12:23.000 Since you're the young people and I'm the old people, 00:12:23.000 --> 00:12:28.000 I don't want to dwell on that. OK, so what demographers do for 00:12:28.000 --> 00:12:33.000 human populations is project what the population will look like in the 00:12:33.000 --> 00:12:38.000 future based on the reproductive rates of the present. 00:12:38.000 --> 00:12:42.000 And you can see for the US here, it's reasonably stable if you look 00:12:42.000 --> 00:12:47.000 at these three snapshots. We're going to go backwards 00:12:47.000 --> 00:12:52.000 starting with 1950, and show you what the population has 00:12:52.000 --> 00:12:57.000 been doing since 1950. And I'm just going to walk through 00:12:57.000 --> 00:13:02.000 this. You only have one in your handouts, but I'll show you how it's 00:13:02.000 --> 00:13:07.000 moving along. Moving along, you can think of this 00:13:07.000 --> 00:13:13.000 as generations moving through the population. And this is the date up 00:13:13.000 --> 00:13:19.000 here. So, this is 1950, 1955, you can see this red cohort. 00:13:19.000 --> 00:13:25.000 A cohort is a group of individuals that were born at roughly the same 00:13:25.000 --> 00:13:31.000 time. So, you can see that red cohort there. And we are going 00:13:31.000 --> 00:13:37.000 along, 1965. This lip here, that we can now see, is the postwar 00:13:37.000 --> 00:13:42.000 baby boom. That's what I'm a member of. 00:13:42.000 --> 00:13:47.000 If you can see it in this bulge in this population. 00:13:47.000 --> 00:13:53.000 And now were marching along. Here's my cohort, and I just put 00:13:53.000 --> 00:13:58.000 these lines on to keep you oriented. And here comes you guys. I think 00:13:58.000 --> 00:14:03.000 those are you guys, 1985. That's roughly right, 00:14:03.000 --> 00:14:09.000 because I never know when I've last updated these slides. 00:14:09.000 --> 00:14:14.000 So, and here you go. See, here's the big bulge of all of 00:14:14.000 --> 00:14:19.000 these baby boomers that you guys are going to have to take care of. 00:14:19.000 --> 00:14:24.000 And now, we can actually see an echo. This is what's called the 00:14:24.000 --> 00:14:29.000 baby boom echo. These are the kids of the baby 00:14:29.000 --> 00:14:34.000 boomers, which is you guys. But you can only see that as we 00:14:34.000 --> 00:14:40.000 march through it. So, here we are at 2020. 00:14:40.000 --> 00:14:46.000 But you get the impression that it's a fairly stable, 00:14:46.000 --> 00:14:52.000 now, even age distribution in the US and these developed countries. 00:14:52.000 --> 00:14:58.000 Oops, here we go a little but more. Sorry. 2035, 2045, OK. 00:14:58.000 --> 00:15:03.000 Now, in less developed countries, the birth rate's high and the death 00:15:03.000 --> 00:15:09.000 rate's low. We see a much different age distribution. 00:15:09.000 --> 00:15:14.000 And here's Uganda, with a very high reproductive rate 00:15:14.000 --> 00:15:20.000 showing the projections to 2050. And here, we can march through from 00:15:20.000 --> 00:15:25.000 1970. You can see that this huge expansion, do you know what that 00:15:25.000 --> 00:15:31.000 noise is? OK. Does anybody have a hypothesis for 00:15:31.000 --> 00:15:37.000 what that noise is that we could test? 00:15:37.000 --> 00:15:40.000 Oh, OK, I guess we can't do anything about that. OK, 00:15:40.000 --> 00:15:44.000 so here's Uganda. And you can see the dramatic 00:15:44.000 --> 00:15:48.000 difference in a population where there is large birthrates, 00:15:48.000 --> 00:15:52.000 and reducing death rates. And we're going to get into analyzing that in 00:15:52.000 --> 00:15:56.000 the next lecture. I just want to show you this here 00:15:56.000 --> 00:16:00.000 so you have a feeling for what we are talking about in age 00:16:00.000 --> 00:16:04.000 structured populations. So, let's now look at how are going 00:16:04.000 --> 00:16:09.000 to analyze these populations to try to quantify growth rates or 00:16:09.000 --> 00:16:14.000 replacement rates. And to do this, we set up life 00:16:14.000 --> 00:16:19.000 tables. And this is basically what insurance agencies do for human 00:16:19.000 --> 00:16:24.000 populations. But we do the same thing for populations of ecological 00:16:24.000 --> 00:16:30.000 interests. We use the same techniques. 00:16:30.000 --> 00:16:34.000 In this lecture, going to use a unicorn is my example, 00:16:34.000 --> 00:16:39.000 because I can make up the numbers because they don't exist. 00:16:39.000 --> 00:16:44.000 But in a textbook there are examples for real organisms like 00:16:44.000 --> 00:16:49.000 lizards and things like that. OK, so we need to define an age 00:16:49.000 --> 00:16:54.000 interval, X, and then this is the number of intervals in the original 00:16:54.000 --> 00:16:59.000 cohort. Again, a cohort is a group of individuals 00:16:59.000 --> 00:17:04.000 that are born within a defined age interval. 00:17:04.000 --> 00:17:08.000 I mean, I think of you guys as a cohort. DX is the number dying 00:17:08.000 --> 00:17:13.000 during that interval. All of this is on the Web. 00:17:13.000 --> 00:17:18.000 These slides are on the Web. So, you don't need to write it down, 00:17:18.000 --> 00:17:23.000 but you can. And, NX is that number of individuals 00:17:23.000 --> 00:17:28.000 surviving to age X. LX is the portion of individuals 00:17:28.000 --> 00:17:33.000 surviving to age X. So, that's just equal to NX divided 00:17:33.000 --> 00:17:39.000 by N0. And, we're going to look at a table that shows this in a minute. 00:17:39.000 --> 00:17:45.000 And MX is something that's measured. It's the per capita 00:17:45.000 --> 00:17:51.000 births during age interval X to X plus one. And this is also called 00:17:51.000 --> 00:17:57.000 age-specific fecundity. And you can think of it as the 00:17:57.000 --> 00:18:03.000 number of female offspring produced per female in a particular 00:18:03.000 --> 00:18:11.000 age category. OK, is everybody comfortable with 00:18:11.000 --> 00:18:21.000 that? So, with these definitions, we're going to build a life table 00:18:21.000 --> 00:18:31.000 that will allow us to actually calculate some things of interest. 00:18:31.000 --> 00:18:37.000 And, what do we want to calculate? We want to calculate the 00:18:37.000 --> 00:18:50.000 survivorship probability, 00:18:50.000 --> 00:19:02.000 LX. We want to calculate the net replacement rate. 00:19:02.000 --> 00:19:10.000 No it's not really a rate, net replacement of population per 00:19:10.000 --> 00:19:19.000 generation, which we are calling R0. It's basically the number of 00:19:19.000 --> 00:19:27.000 children people have to replace who's there per generation. 00:19:27.000 --> 00:19:36.000 And then, for now, this is what we are going to look at. 00:19:36.000 --> 00:19:48.000 And to do that, we are going to generate what's 00:19:48.000 --> 00:20:00.000 called a cohort life table. And to do this, we follow a cohort 00:20:00.000 --> 00:20:11.000 of individuals throughout lifetime. Or, we can also generate a static 00:20:11.000 --> 00:20:19.000 life table because it's not that easy sometimes to have a group of 00:20:19.000 --> 00:20:27.000 organisms that are born at the same time to follow them throughout their 00:20:27.000 --> 00:20:34.000 entire lifetime. So there is a static life table of 00:20:34.000 --> 00:20:40.000 taking a snapshot at one time of the population, and calculating the age 00:20:40.000 --> 00:20:47.000 structure. So, you take a snapshot, 00:20:47.000 --> 00:20:53.000 and we look at the age structure. And, we are going to do this in a 00:20:53.000 --> 00:21:00.000 second so it will make more sense. OK, so we've defined our terms. 00:21:00.000 --> 00:21:06.000 And now, we are going to start by calculating LX. 00:21:06.000 --> 00:21:13.000 So, this is a cohort life table for unicorns. We're going to start out 00:21:13.000 --> 00:21:19.000 with a hundred baby unicorns that we have in our imaginary unicorn pen. 00:21:19.000 --> 00:21:26.000 So, this is a cohort size of 100. And, we find that after a year there 00:21:26.000 --> 00:21:34.000 are 50 of them left. 50 of them die in the first year. 00:21:34.000 --> 00:21:42.000 So, the probability here, the proportion surviving is 0. 00:21:42.000 --> 00:21:50.000 , NX over N0, and then a year later, .4, .3, and then by four years older, 00:21:50.000 --> 00:21:58.000 no unicorns left. They don't live very long. 00:21:58.000 --> 00:22:06.000 All right, so this is what's called the survivorship probability, 00:22:06.000 --> 00:22:12.000 and what we can do is look at there. Different types of organisms have 00:22:12.000 --> 00:22:16.000 different, what we call, survivorship curves. And this is 00:22:16.000 --> 00:22:20.000 discussed in your textbook. We'll just describe the extremes. 00:22:20.000 --> 00:22:25.000 These are just theoretical survivorship curves. 00:22:25.000 --> 00:22:29.000 But some organisms have a very high probability of survival as a 00:22:29.000 --> 00:22:34.000 function of age until they reach an old age. 00:22:34.000 --> 00:22:40.000 And then, they have a very low probability of survival. 00:22:40.000 --> 00:22:46.000 There are other organisms whose survivorship probability drops very 00:22:46.000 --> 00:22:53.000 fast, right after they're born. But if they make it through that 00:22:53.000 --> 00:22:59.000 interval, they're pretty good to go. And then there are some that have a 00:22:59.000 --> 00:23:08.000 steady probability of dying. So, where are humans, 00:23:08.000 --> 00:23:20.000 do you think, on this? Two? No, but that's OK. 00:23:20.000 --> 00:23:32.000 Let me ask you the other way; where our frogs, do you think? 00:23:32.000 --> 00:23:37.000 Yeah, OK, so you got that image. Tons of frogs' eggs: everybody eats 00:23:37.000 --> 00:23:42.000 them. Or for that matter, the video I showed towards the end 00:23:42.000 --> 00:23:47.000 of the last class where there were all those eggs of, 00:23:47.000 --> 00:23:53.000 what was that? Remember all those eggs that everybody was eating? 00:23:53.000 --> 00:23:58.000 Herring, thank you. So, any organism that puts out just tons of 00:23:58.000 --> 00:24:03.000 fertilized eggs, and knowing that most of them will 00:24:03.000 --> 00:24:09.000 be eaten, but some of them will survive, falls here. 00:24:09.000 --> 00:24:14.000 And, humans actually fall here. Any organism that has a high 00:24:14.000 --> 00:24:19.000 investment in the care of offspring, they have few offspring but they 00:24:19.000 --> 00:24:24.000 invest a lot into the care of those offspring, would fall here. 00:24:24.000 --> 00:24:30.000 And then this, actually birds and things fall here. 00:24:30.000 --> 00:24:34.000 So, here's some real but idealized survivorship curves. 00:24:34.000 --> 00:24:38.000 These are humans. And males and females are different. 00:24:38.000 --> 00:24:42.000 I'm not sure whether we understand that completely yet. 00:24:42.000 --> 00:24:46.000 Does anybody know whether that's socially constructed? 00:24:46.000 --> 00:24:51.000 Now that there's more women experiencing equal stress in the 00:24:51.000 --> 00:24:55.000 workplace as there are men that will probably even out. 00:24:55.000 --> 00:24:59.000 But, I think there are more women born, or girl babies. 00:24:59.000 --> 00:25:03.000 Anyway, there's some interesting biology behind this, 00:25:03.000 --> 00:25:07.000 but I don't know. I don't remember. 00:25:07.000 --> 00:25:11.000 And, here's grass, of course grass spew out all these 00:25:11.000 --> 00:25:14.000 seeds everywhere, and very few of them survive, 00:25:14.000 --> 00:25:18.000 also these frogs, etc. and birds are commonly like this, 00:25:18.000 --> 00:25:22.000 where they're somewhere in between. Why do we care so much about 00:25:22.000 --> 00:25:25.000 survivorship curves? Who cares? Well, I mean they're 00:25:25.000 --> 00:25:29.000 inherently interesting to population ecologists, but there are 00:25:29.000 --> 00:25:33.000 also uses for them. For example, if you want to conserve 00:25:33.000 --> 00:25:38.000 a species, if you're worried about a species going extinct, 00:25:38.000 --> 00:25:44.000 you want to figure out whether it's better to conserve the young ones or 00:25:44.000 --> 00:25:49.000 the old ones. For example, turtle species, you would pick a 00:25:49.000 --> 00:25:54.000 certain age group where the probability of survival is high, 00:25:54.000 --> 00:26:00.000 and decide to target the conservation of that age group. 00:26:00.000 --> 00:26:05.000 So, let's continue with, we are building our life table here. 00:26:05.000 --> 00:26:11.000 So, we have the survivorship probability, but what we really want 00:26:11.000 --> 00:26:17.000 to get at is understanding whether or not the population that we are 00:26:17.000 --> 00:26:23.000 describing is replacing itself with each generation. 00:26:23.000 --> 00:26:29.000 So, maybe we should define, when R0 is equal to one, that meets 00:26:29.000 --> 00:26:35.000 the population is exactly replacing itself. 00:26:35.000 --> 00:26:45.000 So, this is replacing, so the actual growth rate of the 00:26:45.000 --> 00:26:55.000 population would be steady. If R0 is less than one, the number 00:26:55.000 --> 00:27:02.000 of individuals is declining. And R0 of greater than one, 00:27:02.000 --> 00:27:08.000 it's increasing. So, we want to know for our unicorns what that is. 00:27:08.000 --> 00:27:14.000 And to get to that, we have to know something about the birth rates. 00:27:14.000 --> 00:27:20.000 So, MX is the average offspring per female of age X. 00:27:20.000 --> 00:27:26.000 So, this is called the age-specific fecundity. And that's something 00:27:26.000 --> 00:27:32.000 that's a known property of the population. 00:27:32.000 --> 00:27:41.000 Whoops, oh, my, my, my, my, I'm missing a slide. 00:27:41.000 --> 00:27:50.000 Oh, there we go. They're out of order. OK, so we have MX. 00:27:50.000 --> 00:28:00.000 So, how do we calculate R0? Well, R0 is the sum of LX MX. 00:28:00.000 --> 00:28:17.000 With the sum of the survivorship 00:28:17.000 --> 00:28:25.000 times the age-specific fecundity, and in this case, it sums up to 00:28:25.000 --> 00:28:33.000 three. So, what's happening to our unicorn population? It's growing. 00:28:33.000 --> 00:28:39.000 Yeah, we are getting three unicorns in each generation for every one 00:28:39.000 --> 00:28:46.000 that existed before. So, in our imaginary unit of our 00:28:46.000 --> 00:28:53.000 population, we're going to be knee deep in unicorns pretty fast. 00:28:53.000 --> 00:29:00.000 OK, so I forgot my watch, so I have to look at my computer. 00:29:00.000 --> 00:29:06.000 What if we can't follow cohort? Oh, thank you. 00:29:06.000 --> 00:29:11.000 How do we create the same kind of analysis for a population that we 00:29:11.000 --> 00:29:16.000 can't follow through time, but can only look at as a snapshot? 00:29:16.000 --> 00:29:22.000 OK, this is where we go to the slide. If you don't have it in your 00:29:22.000 --> 00:29:27.000 handout, it doesn't matter. I just got off the web this morning. 00:29:27.000 --> 00:29:32.000 I couldn't find a skeleton of the 00:29:32.000 --> 00:29:37.000 unicorn because, of course, that's totally imaginary, 00:29:37.000 --> 00:29:42.000 but I found a mastodon. So, just imagine that this is a unicorn, 00:29:42.000 --> 00:29:47.000 and I couldn't find a unicorn horn, so this is a sheep's. But, all 00:29:47.000 --> 00:29:52.000 these principles apply. I just discovered Images in Google, 00:29:52.000 --> 00:29:57.000 which is really exciting. So, you're going to get subjected 00:29:57.000 --> 00:30:02.000 to this for awhile. So, OK, so what you can do, 00:30:02.000 --> 00:30:06.000 and this has actually been done with mountain sheep, 00:30:06.000 --> 00:30:11.000 is you go out you find dead sheep, you find skeletons of sheep that 00:30:11.000 --> 00:30:15.000 have died for whatever causes. And you go out, and you sample 00:30:15.000 --> 00:30:19.000 until you have, say, 100 skeletons. 00:30:19.000 --> 00:30:24.000 And that's your cohort that you're looking at, at one point in time. 00:30:24.000 --> 00:30:28.000 And from their horn, you can actually tell how old they 00:30:28.000 --> 00:30:33.000 were when they died. You can count the number of rings, 00:30:33.000 --> 00:30:38.000 so that's what's here, annual horn rings. This is for a dall mountain 00:30:38.000 --> 00:30:43.000 sheep. So, you can say well now it died when it was two. 00:30:43.000 --> 00:30:47.000 That one died when it was 10. That one died when it was whatever 00:30:47.000 --> 00:30:52.000 age. And then you can create the same kind of life table, 00:30:52.000 --> 00:30:57.000 a static life table, where you have a hundred skeletons. 00:30:57.000 --> 00:31:02.000 That is your cohort. You look at the number dying of age 00:31:02.000 --> 00:31:06.000 zero to one, the number of one year olds, the number that died when they 00:31:06.000 --> 00:31:11.000 were one year old, the number that died when they were 00:31:11.000 --> 00:31:16.000 a two-year-old etc. And so, from these data, 00:31:16.000 --> 00:31:20.000 these are the data that you collected, you can calculate this 00:31:20.000 --> 00:31:25.000 column, NX, so NX is DX, or NX minus DX equals NX plus one. 00:31:25.000 --> 00:31:30.000 Does that make sense? I can never tell whether. 00:31:30.000 --> 00:31:35.000 I know if I write this on the board it might be easier, 00:31:35.000 --> 00:31:40.000 but it's so obvious isn't it? We are just saying that this is the 00:31:40.000 --> 00:31:45.000 number that died at the age. This is the number you started with, 00:31:45.000 --> 00:31:50.000 so that's how many are going to have that age, that age, 00:31:50.000 --> 00:31:55.000 and that age. And then, once you have this column, 00:31:55.000 --> 00:32:00.000 your proportion surviving LX, you can calculate LX. LX equals NX 00:32:00.000 --> 00:32:05.000 divided by N0, OK? So, we are doing exactly the same 00:32:05.000 --> 00:32:10.000 thing as we did before. It's just that we're getting the NX 00:32:10.000 --> 00:32:15.000 column instead of getting it by following the cohort. 00:32:15.000 --> 00:32:20.000 We're getting it by calculating it based on how old dead organisms were 00:32:20.000 --> 00:32:25.000 when they died. And in my ecology class that I 00:32:25.000 --> 00:32:30.000 teach, some years we actually go out to the Mount Auburn Cemetery. 00:32:30.000 --> 00:32:37.000 And you can do this from human gravestones. You can go to the 00:32:37.000 --> 00:32:44.000 cemetery, and pick out a number of gravestones, and see the age at 00:32:44.000 --> 00:32:51.000 which humans died. You create yourself a cohort, 00:32:51.000 --> 00:32:58.000 and you can create a life table. And you can do that for different 00:32:58.000 --> 00:33:06.000 eras, and see how replacements have changed. 00:33:06.000 --> 00:33:13.000 OK, now so that's the analysis for populations that have an age 00:33:13.000 --> 00:33:21.000 structure. Now we are going to go more into simpler type of population, 00:33:21.000 --> 00:33:29.000 and that is a population with a stable age distribution. 00:33:29.000 --> 00:33:46.000 And to do this, 00:33:46.000 --> 00:33:51.000 you're going to help me, and we're going to use your calculus 00:33:51.000 --> 00:33:55.000 that you've all been studying. So, instead of the unicorn now, 00:33:55.000 --> 00:34:00.000 have your imaginary population be a population of microbes 00:34:00.000 --> 00:34:06.000 that divide in half. They multiplied by dividing in half. 00:34:06.000 --> 00:34:12.000 So, each one of these is a microbe that's dividing in half. 00:34:12.000 --> 00:34:18.000 This is your mental image. This is what's called exponential 00:34:18.000 --> 00:34:25.000 growth. It's obvious how that happens. And we're going to model 00:34:25.000 --> 00:34:31.000 this population, we're going to first assume 00:34:31.000 --> 00:34:42.000 unlimited resources. OK, so we're going to say that the 00:34:42.000 --> 00:34:56.000 rate of population increase is equal to the average birth rate minus the 00:34:56.000 --> 00:35:11.000 average death rate times the number of cells. 00:35:11.000 --> 00:35:19.000 OK, so we are going to now turn this into math, and that is to say the 00:35:19.000 --> 00:35:28.000 dN/dt, the increase in population where N is the population number is 00:35:28.000 --> 00:35:36.000 equal to the birth rate minus the death rate times N which is the 00:35:36.000 --> 00:35:46.000 number of cells, OK? And then, we're going to let B minus 00:35:46.000 --> 00:35:58.000 D, the birth rate minus the death rate, be what we call r. 00:35:58.000 --> 00:36:08.000 And, this is what's called the intrinsic rate of increase of a 00:36:08.000 --> 00:36:19.000 population. OK, what are the units of r? 00:36:19.000 --> 00:36:30.000 One over time, exactly, time to the minus one. 00:36:30.000 --> 00:36:36.000 So, let's look at that more carefully. And also, 00:36:36.000 --> 00:36:43.000 it's a little misleading to say it's the rate of increase because r can 00:36:43.000 --> 00:36:49.000 be positive or negative, however it turns out. It can be 00:36:49.000 --> 00:36:56.000 positive or negative, but that's what it's called. 00:36:56.000 --> 00:37:02.000 So, we have the dN/dt equals rN. We're substituting r in this 00:37:02.000 --> 00:37:09.000 equation for one over N times dN/dt equals r. 00:37:09.000 --> 00:37:15.000 OK, so ours has the unit time to the minus one. And so, 00:37:15.000 --> 00:37:21.000 let's ask a question. Given N0 I give you the population 00:37:21.000 --> 00:37:27.000 density at some time which we're going to call T equals zero. 00:37:27.000 --> 00:37:33.000 Given a population growing according to this, 00:37:33.000 --> 00:37:39.000 which is exponential growth, what if we want to know the 00:37:39.000 --> 00:37:45.000 population, what N is at any time T? 00:37:45.000 --> 00:37:51.000 We want an equation that will give us, given N0 what would the 00:37:51.000 --> 00:37:57.000 population density be at some time, T? What do you have to do to this 00:37:57.000 --> 00:38:03.000 to get that? Yeah, so who wants to do that for me? 00:38:03.000 --> 00:38:10.000 Come on. You guys did this freshman year. It's the easiest thing there 00:38:10.000 --> 00:38:17.000 is, right? Every class I've had has had somebody who was willing to come 00:38:17.000 --> 00:38:24.000 up and do this. OK, so we'll just add a T there. 00:38:24.000 --> 00:38:31.000 So, N at sometime T is equally to N0 e to the rT. 00:38:31.000 --> 00:38:38.000 And so, We could say, then, r equals natural log of NT 00:38:38.000 --> 00:38:46.000 minus natural log of N0 divided by T. And I like to write it that way 00:38:46.000 --> 00:38:53.000 because then, we know what this looks like, right? 00:38:53.000 --> 00:39:01.000 Let's plot that. This is N and this is T. What does 00:39:01.000 --> 00:39:08.000 that look like? I know this is really rudimentary 00:39:08.000 --> 00:39:14.000 but remember we're modeling population growth. 00:39:14.000 --> 00:39:20.000 So here, if we plot the log of N, and this is what we do with cultures 00:39:20.000 --> 00:39:26.000 of microorganisms. That's a flask. Those are a lot of 00:39:26.000 --> 00:39:33.000 microbes in there. And what we do is we sample it at 00:39:33.000 --> 00:39:39.000 various points in time, and if you take the log we get a 00:39:39.000 --> 00:39:46.000 nice straight line that we can draw a regression through. 00:39:46.000 --> 00:39:53.000 And what's the slope of that line equal to? r. Exactly. 00:39:53.000 --> 00:40:00.000 The growth rate in the units: N to the minus one. 00:40:00.000 --> 00:40:12.000 OK, what's the Y intercept? N0. OK, now suppose we want to 00:40:12.000 --> 00:40:25.000 calculate the doubling time of the population, the time it 00:40:25.000 --> 00:40:37.000 takes to double. How would we do that? 00:40:37.000 --> 00:40:48.000 Let's first define it. It's the time, T, that it takes for 00:40:48.000 --> 00:40:59.000 NT to equal to N0, right? If we start with N0 the 00:40:59.000 --> 00:41:10.000 population doubles. Then, that's the time at NT. 00:41:10.000 --> 00:41:22.000 So, we want to solve for that T for the time it takes for the 00:41:22.000 --> 00:41:33.000 population to double. Since natural log of NT over N0 00:41:33.000 --> 00:41:45.000 equals rT, then the natural log of, sorry, 2N0 over N0 equals rT, and T 00:41:45.000 --> 00:41:57.000 equals the natural log of two divided by r equals our 00:41:57.000 --> 00:42:05.000 doubling time. Does that make sense? 00:42:05.000 --> 00:42:09.000 I'll put this out there so you can see it better. 00:42:09.000 --> 00:42:14.000 What's the natural log of two? 0.69, thank you, always a handy 00:42:14.000 --> 00:42:18.000 thing to have in our repertoire. So, that's just the way, it's 00:42:18.000 --> 00:42:23.000 easier to think about the time it takes for a population to double 00:42:23.000 --> 00:42:26.000 often, then the instantaneous growth rate.