WEBVTT 00:00:00.080 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high quality, educational resources for free. 00:00:10.730 --> 00:00:13.340 To make a donation or view additional materials 00:00:13.340 --> 00:00:17.236 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.236 --> 00:00:17.861 at ocw.mit.edu. 00:00:20.900 --> 00:00:22.400 PROFESSOR: Today, what we want to do 00:00:22.400 --> 00:00:25.720 is talk about something at a much higher scale than what 00:00:25.720 --> 00:00:28.100 we've thought about through most of this semester. 00:00:28.100 --> 00:00:30.642 And that's probably by design. 00:00:30.642 --> 00:00:32.100 Over the course of the semester, we 00:00:32.100 --> 00:00:35.300 started with kind of enzyme kinetics 00:00:35.300 --> 00:00:37.220 or molecular binding kind of events, 00:00:37.220 --> 00:00:40.680 and we slowly built our way up the larger and larger scales. 00:00:40.680 --> 00:00:43.110 Now there's always this question about whether we're 00:00:43.110 --> 00:00:46.820 claiming that we really understand how the higher 00:00:46.820 --> 00:00:49.974 levels of organization result from the lower level 00:00:49.974 --> 00:00:50.515 interactions. 00:00:50.515 --> 00:00:54.070 And I'd say, we definitely don't understand all of it. 00:00:54.070 --> 00:00:56.990 So you shouldn't come away with that as the notion. 00:00:56.990 --> 00:00:59.100 But at least one thing that I think 00:00:59.100 --> 00:01:01.690 is fascinating about this area of systems biology 00:01:01.690 --> 00:01:05.129 is that much of the framework that we use to understand, 00:01:05.129 --> 00:01:08.220 let's say, molecular scale interactions or stochastic gene 00:01:08.220 --> 00:01:11.420 expression, so these dynamics at the smaller scale, 00:01:11.420 --> 00:01:14.304 much of those ideas and such certainly transport up 00:01:14.304 --> 00:01:16.470 to these higher scales or translate up to the higher 00:01:16.470 --> 00:01:17.990 scales, where, in this case, we're 00:01:17.990 --> 00:01:19.900 using kind of master equation type formulas 00:01:19.900 --> 00:01:22.314 to try to understand relative species abundance. 00:01:22.314 --> 00:01:23.730 And so I think part of what I like 00:01:23.730 --> 00:01:28.420 about this topic of neutral theory versus niche theory 00:01:28.420 --> 00:01:30.780 and so forth in ecology is that you can just 00:01:30.780 --> 00:01:34.530 see how very, very similar ideas, that we applied 00:01:34.530 --> 00:01:36.400 for studying stochastic gene expression, 00:01:36.400 --> 00:01:38.358 can also be used to try to understand why it is 00:01:38.358 --> 00:01:41.800 that some species are more common than others when you go 00:01:41.800 --> 00:01:46.610 and you count them, in this case, on an island in Panama. 00:01:46.610 --> 00:01:50.230 Now, the subject is, by its nature, 00:01:50.230 --> 00:01:52.840 less experimentally focused than much 00:01:52.840 --> 00:01:54.840 of what we've done over the course the semester. 00:01:54.840 --> 00:01:56.680 And this is really a topic the tends 00:01:56.680 --> 00:02:00.200 to be a combination of mathematical theory 00:02:00.200 --> 00:02:04.680 with kind of careful counting of species in some different areas 00:02:04.680 --> 00:02:07.199 and trying to understand what that means. 00:02:07.199 --> 00:02:08.740 But it's an area that there have been 00:02:08.740 --> 00:02:11.880 a number of physicists involved in over the last 10 years. 00:02:11.880 --> 00:02:13.690 And I think that it's fascinating, 00:02:13.690 --> 00:02:17.010 because it does get to the heart of what we are looking 00:02:17.010 --> 00:02:19.530 for from a theory, what kind of evidence 00:02:19.530 --> 00:02:23.040 do we use to support a theory or to refute it. 00:02:23.040 --> 00:02:27.272 So I think there are a lot of very basic issues about science 00:02:27.272 --> 00:02:28.730 that come up when we start thinking 00:02:28.730 --> 00:02:31.630 about this question of neutral theory in ecology. 00:02:31.630 --> 00:02:34.832 And since it's, for many of us, a totally new area 00:02:34.832 --> 00:02:36.290 that we don't know very much about, 00:02:36.290 --> 00:02:40.570 you can come to it with maybe fresh eyes. 00:02:40.570 --> 00:02:42.320 And you don't have the same preconceptions 00:02:42.320 --> 00:02:44.981 that you would have for many other models that you might 00:02:44.981 --> 00:02:47.230 be more familiar with in the context of molecular cell 00:02:47.230 --> 00:02:47.730 biology. 00:02:50.410 --> 00:02:53.640 So the basic question that we're going 00:02:53.640 --> 00:02:56.610 to try to talk about today is just 00:02:56.610 --> 00:02:59.950 the question of why is it that, when you look out at the world, 00:02:59.950 --> 00:03:02.510 you see that there are some species that 00:03:02.510 --> 00:03:06.850 seem to be abundant and some that seem to be rare? 00:03:06.850 --> 00:03:10.350 Are there other patterns that are somehow universal? 00:03:10.350 --> 00:03:14.010 And what kind of sort of lower scale processes 00:03:14.010 --> 00:03:17.390 might lead to the patterns that we observe? 00:03:17.390 --> 00:03:22.220 And I think that this paper that we read is-- I mean, 00:03:22.220 --> 00:03:24.510 it's not that it's. 00:03:24.510 --> 00:03:30.930 Well, can somebody say what the actual scientific contribution 00:03:30.930 --> 00:03:33.801 of this paper was? 00:03:33.801 --> 00:03:34.300 Yes? 00:03:34.300 --> 00:03:35.330 AUDIENCE: They did a calculation. 00:03:35.330 --> 00:03:36.940 PROFESSOR: They did a calculation. 00:03:36.940 --> 00:03:39.350 But it's a little bit more specific than that. 00:03:39.350 --> 00:03:39.850 What is it? 00:03:39.850 --> 00:03:43.160 AUDIENCE: They came up with the closed form equation? 00:03:43.160 --> 00:03:44.160 PROFESSOR: That's right. 00:03:44.160 --> 00:03:47.230 Basically, there was a model of this neutral theory in ecology 00:03:47.230 --> 00:03:49.920 that we're going to explain or try to understand. 00:03:49.920 --> 00:03:51.680 You can simulate the model, but then there 00:03:51.680 --> 00:03:54.700 are possible issues associated with convergence or something 00:03:54.700 --> 00:03:55.410 of those. 00:03:55.410 --> 00:03:56.826 Although it's hard to believe that 00:03:56.826 --> 00:03:58.200 that's really such a concern. 00:03:58.200 --> 00:03:59.970 But you can simulate that model. 00:03:59.970 --> 00:04:01.680 What they did is they just showed 00:04:01.680 --> 00:04:05.919 that you could get an analytic-y kind of expression for it. 00:04:05.919 --> 00:04:07.460 It's not a super analytic expression, 00:04:07.460 --> 00:04:09.920 but, at least, it's not a straight up simulation. 00:04:09.920 --> 00:04:11.892 You kind of numerically do something, 00:04:11.892 --> 00:04:13.600 integrate something, as compared to doing 00:04:13.600 --> 00:04:14.683 the stochastic simulation. 00:04:17.844 --> 00:04:19.510 So it's not that that, in and of itself, 00:04:19.510 --> 00:04:23.469 is what you feel like-- it's not what we necessarily 00:04:23.469 --> 00:04:24.260 care so much about. 00:04:24.260 --> 00:04:28.880 But I think that it's still just a nice, short description 00:04:28.880 --> 00:04:31.681 of the model and the assumptions that go into it. 00:04:31.681 --> 00:04:33.180 And you get a little bit of a window 00:04:33.180 --> 00:04:34.990 into the debate that's going on between these two 00:04:34.990 --> 00:04:36.910 communities of kind of the neutral theory 00:04:36.910 --> 00:04:39.780 guys and the niche theory community. 00:04:45.650 --> 00:04:48.080 So there's only one figure in this paper. 00:04:48.080 --> 00:04:50.880 And it's an example of the kind of data 00:04:50.880 --> 00:04:53.110 that we want to try to understand. 00:04:53.110 --> 00:04:55.330 So there's a particular pattern in terms 00:04:55.330 --> 00:04:57.400 of the relative species abundance. 00:04:57.400 --> 00:05:00.150 And we want to understand what kind of models 00:05:00.150 --> 00:05:03.280 might lead to that observed pattern. 00:05:03.280 --> 00:05:05.940 But given that there's just one figure in the paper, 00:05:05.940 --> 00:05:07.640 we have to make sure that we understand 00:05:07.640 --> 00:05:09.560 exactly what is being plotted. 00:05:09.560 --> 00:05:12.210 And what I've found from experience-- 00:05:12.210 --> 00:05:17.000 and, actually, even the answer to the email question 00:05:17.000 --> 00:05:19.514 that was sent out, I think, was incorrect on one 00:05:19.514 --> 00:05:20.180 of these things. 00:05:20.180 --> 00:05:21.880 So we'll talk about that some more. 00:05:21.880 --> 00:05:24.587 So beware. 00:05:24.587 --> 00:05:25.420 We'll figure it out. 00:05:25.420 --> 00:05:28.040 But I think it's actually surprisingly tricky 00:05:28.040 --> 00:05:32.560 to understand what this figure is saying. 00:05:32.560 --> 00:05:36.020 But first of all, can somebody describe not 00:05:36.020 --> 00:05:38.620 what the figure is saying but just what 00:05:38.620 --> 00:05:40.160 the data is supposed to be? 00:05:56.550 --> 00:05:59.640 Where do they get the data? 00:05:59.640 --> 00:06:02.896 Anything that's useful? 00:06:02.896 --> 00:06:05.692 AUDIENCE: They were on an island ecosystem. 00:06:05.692 --> 00:06:06.900 PROFESSOR: There's an island. 00:06:06.900 --> 00:06:12.752 It's called BCI, Barro Colorado Island. 00:06:12.752 --> 00:06:13.668 AUDIENCE: [INAUDIBLE]. 00:06:25.080 --> 00:06:27.015 PROFESSOR: So it's a 50 hectare plot. 00:06:31.380 --> 00:06:33.504 Does anybody know what a hectare is? 00:06:33.504 --> 00:06:35.420 AUDIENCE: It's a lot more than a square meter. 00:06:35.420 --> 00:06:39.550 PROFESSOR: It's a lot more than a square meter, yes, indeed. 00:06:39.550 --> 00:06:40.200 Yeah. 00:06:40.200 --> 00:06:44.921 Is this an English unit of measure? 00:06:44.921 --> 00:06:46.920 This is the kind of thing that I have to Google. 00:06:46.920 --> 00:06:50.970 But it's one hectare is equal to 10 to the 4 meters squared. 00:06:54.032 --> 00:06:55.365 That's a good thing to memorize. 00:06:59.041 --> 00:06:59.540 I 00:06:59.540 --> 00:07:02.180 AUDIENCE: Exactly or approximate? 00:07:02.180 --> 00:07:03.430 PROFESSOR: I think it's exact. 00:07:03.430 --> 00:07:04.680 I think I think it's an exact. 00:07:04.680 --> 00:07:06.805 AUDIENCE: Then it's a metric unit. 00:07:06.805 --> 00:07:08.930 PROFESSOR: Yeah, so apparently it is a metric unit. 00:07:08.930 --> 00:07:12.150 So the idea is that if you take a 100 meters by 100 meters, 00:07:12.150 --> 00:07:13.070 this is a hectare. 00:07:13.070 --> 00:07:15.080 And there's 50 of them. 00:07:15.080 --> 00:07:17.435 It's about like a half a square kilometer 00:07:17.435 --> 00:07:21.110 to give you a sense of what we're talking about. 00:07:21.110 --> 00:07:23.331 And what do they do on this plot? 00:07:23.331 --> 00:07:28.930 AUDIENCE: They count a certain number as canopy trees. 00:07:28.930 --> 00:07:32.209 So the trees that are, like, really big. 00:07:32.209 --> 00:07:34.500 PROFESSOR: And how do they decide which trees to count? 00:07:34.500 --> 00:07:36.190 Did they count every tree? 00:07:36.190 --> 00:07:40.890 AUDIENCE: No, just the ones that like formed the top layer. 00:07:40.890 --> 00:07:44.987 PROFESSOR: I think that the way that they decide-- OK. 00:07:44.987 --> 00:07:47.070 Does anybody remember how many trees were counted? 00:07:50.283 --> 00:07:51.660 AUDIENCE: [INAUDIBLE]. 00:07:51.660 --> 00:08:06.390 PROFESSOR: So there are 21,457 trees in this 50 hectare plot. 00:08:06.390 --> 00:08:10.770 They identify the species for each one of these 21,000 trees. 00:08:10.770 --> 00:08:11.670 And they assign them. 00:08:11.670 --> 00:08:14.610 And they found that there were 225 distinct species. 00:08:19.590 --> 00:08:23.552 So this is really quite an amazing data set. 00:08:23.552 --> 00:08:27.960 Because I can tell you that I would not be able to do this. 00:08:31.080 --> 00:08:35.559 This was highly skilled biologists 00:08:35.559 --> 00:08:38.880 that can distinguish 225. 00:08:38.880 --> 00:08:40.834 If they can identify these 225, that 00:08:40.834 --> 00:08:43.250 means they have to be able to identify other ones as well. 00:08:43.250 --> 00:08:45.340 And they did it for 20,000 trees. 00:08:45.340 --> 00:08:47.620 And indeed, Barro Colorado Island 00:08:47.620 --> 00:08:54.560 is one of the major Smithsonian research institutes, 00:08:54.560 --> 00:08:56.032 where they've been tracking. 00:08:56.032 --> 00:08:57.740 They do this like every five years or so, 00:08:57.740 --> 00:09:00.630 where they do a census, where they count all of the trees. 00:09:00.630 --> 00:09:05.929 And they're also tracking many other-- it's not just trees. 00:09:05.929 --> 00:09:07.220 They're doing everything there. 00:09:07.220 --> 00:09:08.512 AUDIENCE: Is there only plants? 00:09:08.512 --> 00:09:09.470 PROFESSOR: What's that? 00:09:09.470 --> 00:09:10.890 AUDIENCE: Is it only plants? 00:09:10.890 --> 00:09:11.473 PROFESSOR: No. 00:09:16.090 --> 00:09:18.285 So actually, I visited BCI, and it 00:09:18.285 --> 00:09:20.410 seemed like they were studying all sorts of things. 00:09:20.410 --> 00:09:22.760 And there were nice looking birds there. 00:09:22.760 --> 00:09:24.589 AUDIENCE: No, I mean in this census. 00:09:24.589 --> 00:09:26.380 PROFESSOR: In this census, it's only trees. 00:09:26.380 --> 00:09:32.150 And the way that they decide which of the trees to do, 00:09:32.150 --> 00:09:36.880 it's the ones that are more than 10 centimeters DBH. 00:09:36.880 --> 00:09:38.940 Anybody can guess what DBH might mean? 00:09:50.050 --> 00:09:52.240 It's actually diameter at breast height. 00:10:00.908 --> 00:10:05.925 So what they do is they walk up to the tree with a ruler, 00:10:05.925 --> 00:10:07.800 and then, if it's larger than 10 centimeters, 00:10:07.800 --> 00:10:11.020 then they count it. 00:10:11.020 --> 00:10:13.200 You need to have some threshold at the lower end, 00:10:13.200 --> 00:10:17.140 otherwise you're in trouble, right? 00:10:17.140 --> 00:10:20.131 And there were plenty of trees that satisfied this requirement 00:10:20.131 --> 00:10:20.630 here. 00:10:27.380 --> 00:10:30.420 Then what they do, for all of these trees, 00:10:30.420 --> 00:10:34.295 it's assigned to some species. 00:10:38.020 --> 00:10:42.350 The basic goal of this branch of biology or ecology 00:10:42.350 --> 00:10:46.817 is to try to understand the pattern, 00:10:46.817 --> 00:10:48.650 from this sort of data, where it comes from. 00:10:48.650 --> 00:10:51.170 Or first describe it, and then once you 00:10:51.170 --> 00:10:52.950 have a description of it, then you 00:10:52.950 --> 00:10:56.377 can try to understand what microscale processes might 00:10:56.377 --> 00:10:57.210 lead to the pattern. 00:10:57.210 --> 00:11:00.030 And the pattern is what's plotted in figure 1. 00:11:00.030 --> 00:11:01.610 It's the only figure in the paper. 00:11:01.610 --> 00:11:05.660 I have reconstructed a rough version of it, 00:11:05.660 --> 00:11:07.120 here, for you on the board. 00:11:07.120 --> 00:11:08.960 But if you want a more accurate version, 00:11:08.960 --> 00:11:12.600 you can look at your paper. 00:11:12.600 --> 00:11:15.120 Now, we want to make sure that we understand 00:11:15.120 --> 00:11:16.430 what the figure is saying. 00:11:16.430 --> 00:11:21.880 So we will ask the following question. 00:11:21.880 --> 00:11:23.860 What is the most common number of individuals 00:11:23.860 --> 00:11:26.066 for a species in this data set? 00:11:30.040 --> 00:11:39.660 The most common/frequent number of individuals for a species 00:11:39.660 --> 00:11:49.340 to have in this data set. 00:11:49.340 --> 00:11:51.594 Now, it's maybe worth just saying 00:11:51.594 --> 00:11:52.760 something a little bit more. 00:11:52.760 --> 00:11:54.710 So you notice that they were not trying 00:11:54.710 --> 00:11:58.780 to count the total number of species, altogether. 00:11:58.780 --> 00:12:01.426 And in general, all of this field of relative species 00:12:01.426 --> 00:12:03.467 abundance, to try to understand them, what you do 00:12:03.467 --> 00:12:05.792 is typically take one trophic level. 00:12:05.792 --> 00:12:07.250 So some of the classic studies were 00:12:07.250 --> 00:12:11.000 of beetles in the Thames River. 00:12:11.000 --> 00:12:13.050 The idea is that it's some set of species 00:12:13.050 --> 00:12:14.850 that you think are going to be interacting, 00:12:14.850 --> 00:12:16.350 maybe competing, with each other, 00:12:16.350 --> 00:12:17.390 in some way, in the sense that they're 00:12:17.390 --> 00:12:20.180 maybe eating related things and being eaten by related things. 00:12:20.180 --> 00:12:24.070 And so in this case, these are the trees 00:12:24.070 --> 00:12:25.230 in Barro Colorado Island. 00:12:25.230 --> 00:12:30.537 And you can imagine that this is useful. 00:12:30.537 --> 00:12:32.620 The fact that it's trees instead of something else 00:12:32.620 --> 00:12:35.251 means that you can actually track the individuals 00:12:35.251 --> 00:12:35.750 over time. 00:12:35.750 --> 00:12:37.166 And when you go to the island what 00:12:37.166 --> 00:12:40.929 you see is that all the trees, they're wrapped by some tag. 00:12:40.929 --> 00:12:42.470 And presumably, they have some system 00:12:42.470 --> 00:12:45.590 to tell you which species that is so that they 00:12:45.590 --> 00:12:47.880 keep records of everything. 00:12:47.880 --> 00:12:53.870 But the question is, what's the most common number 00:12:53.870 --> 00:12:55.790 of individuals for species in the data set? 00:12:55.790 --> 00:12:57.506 Do you understand what I'm trying to ask? 00:13:03.960 --> 00:13:07.120 And we're going do approximate, so we'll say. 00:13:20.230 --> 00:13:21.640 Or this, can't determine. 00:13:28.530 --> 00:13:32.600 We want to know, what is the mode of this distribution 00:13:32.600 --> 00:13:36.360 of the number of individuals for each of these species? 00:13:39.140 --> 00:13:41.540 Do you understand the question? 00:13:41.540 --> 00:13:45.220 I'm going to give you 20 seconds to look at this. 00:13:55.659 --> 00:13:58.110 AUDIENCE: Should we just hold a blank piece of paper? 00:13:58.110 --> 00:14:00.516 PROFESSOR: Oh, we don't have our-- ah. 00:14:00.516 --> 00:14:02.310 AUDIENCE: [INAUDIBLE]? 00:14:02.310 --> 00:14:05.530 PROFESSOR: You know, the TA always lets me down. 00:14:05.530 --> 00:14:08.310 All right, yeah. 00:14:08.310 --> 00:14:13.634 So you can do A, B, C, D, E. Are we ready? 00:14:13.634 --> 00:14:14.550 AUDIENCE: [INAUDIBLE]? 00:14:21.570 --> 00:14:23.680 PROFESSOR: You can just do this if you're not. 00:14:23.680 --> 00:14:26.100 But given this was the only figure in the paper, 00:14:26.100 --> 00:14:28.470 and that this is a basic property of the distribution, 00:14:28.470 --> 00:14:31.407 I'm sure that you figured that out last night, anyways, right? 00:14:31.407 --> 00:14:33.240 Especially since it was one of the questions 00:14:33.240 --> 00:14:34.031 in the [INAUDIBLE]. 00:14:34.031 --> 00:14:36.410 So you presumably already thought about this question, 00:14:36.410 --> 00:14:36.940 right? 00:14:36.940 --> 00:14:38.430 OK. 00:14:38.430 --> 00:14:39.253 Yes? 00:14:39.253 --> 00:14:40.800 AUDIENCE: Yes. 00:14:40.800 --> 00:14:43.725 PROFESSOR: Ready, three, two, one. 00:14:47.590 --> 00:14:50.060 I'd say we got a lot of B's. 00:14:50.060 --> 00:14:51.935 So it seems like B is the most. 00:14:55.284 --> 00:14:56.950 So this, we'll put a question mark here. 00:15:00.860 --> 00:15:04.440 Can somebody verbally say why their neighbor 00:15:04.440 --> 00:15:08.740 said that the mode of the distribution is around 30? 00:15:11.810 --> 00:15:12.427 Yeah? 00:15:12.427 --> 00:15:13.510 AUDIENCE: The tallest bar. 00:15:13.510 --> 00:15:16.620 PROFESSOR: The tallest bar there is around 30. 00:15:16.620 --> 00:15:18.270 That's a very practical definition. 00:15:18.270 --> 00:15:21.240 So that's normally what we mean by the mode. 00:15:21.240 --> 00:15:23.550 There is a slight problem in all of this, 00:15:23.550 --> 00:15:25.520 which is that this thing is plotted 00:15:25.520 --> 00:15:28.415 in a very kind of funny way. 00:15:28.415 --> 00:15:30.290 So if you look at the figure, what you'll see 00:15:30.290 --> 00:15:31.748 is that it's number of individuals. 00:15:31.748 --> 00:15:33.375 And down here, it says, log2 scale. 00:15:40.610 --> 00:15:44.530 Now, when we say the mode, what we're wondering about 00:15:44.530 --> 00:15:50.080 is that, if you just take the most typical kind of species 00:15:50.080 --> 00:15:52.080 of tree that's there, how many individuals do 00:15:52.080 --> 00:15:53.371 we think there should be there? 00:15:55.570 --> 00:15:57.560 Of course, typical is hard to define. 00:15:57.560 --> 00:16:00.650 We can talk about mode, median, mean, et cetera. 00:16:00.650 --> 00:16:03.210 But the most common number of individuals 00:16:03.210 --> 00:16:06.890 for a species of the data set ends up not being 30. 00:16:06.890 --> 00:16:08.190 It ends up being 1. 00:16:12.220 --> 00:16:16.430 And we will try to reconstruct this right now. 00:16:16.430 --> 00:16:18.870 Because you have to do a little bit of digging 00:16:18.870 --> 00:16:21.330 to figure out what is being plotted here. 00:16:21.330 --> 00:16:23.340 But it's not the raw data. 00:16:23.340 --> 00:16:26.360 The problem here is that this is on this log scale, where 00:16:26.360 --> 00:16:29.720 the bins here are growing kind of geometrically 00:16:29.720 --> 00:16:33.310 or exponentially, whatever, as you move to the right. 00:16:33.310 --> 00:16:39.975 So over here, this thing only contains one real bin. 00:16:39.975 --> 00:16:41.350 And actually, we're about to find 00:16:41.350 --> 00:16:44.040 it's half a bin, which is even weirder. 00:16:44.040 --> 00:16:46.330 Whereas out here, this is maybe 30 bins. 00:16:46.330 --> 00:16:50.550 So the number of species that we're going to put in this bin 00:16:50.550 --> 00:16:56.990 is everything between around 20 something up to 50 or so. 00:16:56.990 --> 00:16:59.701 The number of kind of true bins that 00:16:59.701 --> 00:17:01.200 end up in each of these plotted bins 00:17:01.200 --> 00:17:05.480 is going to grow geometrically as we move to the right. 00:17:05.480 --> 00:17:09.730 So this is a very funny transform of the data. 00:17:09.730 --> 00:17:14.690 And indeed, I think it's always nice to just, in life, 00:17:14.690 --> 00:17:17.890 you always plot the raw data first. 00:17:17.890 --> 00:17:20.720 And then what you can do is then you can do funny. 00:17:20.720 --> 00:17:23.510 There's a reason to plot it this way. 00:17:23.510 --> 00:17:27.770 Because this is where they get this idea that this might 00:17:27.770 --> 00:17:30.530 described as described as a log normal. 00:17:30.530 --> 00:17:32.550 The idea is, if you take a log of the data, 00:17:32.550 --> 00:17:34.550 then you get something that looks like a normal. 00:17:34.550 --> 00:17:38.105 But you always plot the raw data first. 00:17:38.105 --> 00:17:40.480 So let's try to figure out what the raw data looked like. 00:17:45.260 --> 00:17:46.680 And now what we're going to do is 00:17:46.680 --> 00:17:49.141 we're going to have real scalings, honest to goodness 00:17:49.141 --> 00:17:49.640 numbers. 00:17:53.130 --> 00:17:55.170 Now the number of species you get still. 00:17:57.800 --> 00:18:00.580 So this is asking, how many different species 00:18:00.580 --> 00:18:04.100 do we see with one member or with two members 00:18:04.100 --> 00:18:05.980 or with three, four, et cetera? 00:18:11.715 --> 00:18:13.340 And I don't know how far we're actually 00:18:13.340 --> 00:18:14.860 going to be able to get. 00:18:14.860 --> 00:18:21.440 But in this one figure, in our paper, 00:18:21.440 --> 00:18:23.640 they tell us what the histogram means. 00:18:23.640 --> 00:18:26.600 So the first histogram bar represents what 00:18:26.600 --> 00:18:29.680 they call phi 1 divided by 2. 00:18:29.680 --> 00:18:36.920 Phi 1 was the number of species observed with one member, which 00:18:36.920 --> 00:18:42.980 means that even this first plot bar is not 00:18:42.980 --> 00:18:45.940 the number of species observed with a single individual. 00:18:45.940 --> 00:18:48.810 It's half of that. 00:18:48.810 --> 00:18:50.350 You can argue about the consistency 00:18:50.350 --> 00:18:52.230 of how these things should be, but that's 00:18:52.230 --> 00:18:54.000 what this thing's plotted. 00:18:54.000 --> 00:18:58.000 And it looks like it was nine, here, so this should be 18. 00:18:58.000 --> 00:19:02.500 So I'm going to put up here, here's a 20 and here's a 10. 00:19:05.950 --> 00:19:07.140 Right, so here is an 18. 00:19:14.510 --> 00:19:17.670 Now, what do they say? 00:19:17.670 --> 00:19:23.270 This bin represents phi 1 divided 00:19:23.270 --> 00:19:26.894 by 2 plus phi 2 divided by 2. 00:19:26.894 --> 00:19:28.310 So they took the number of species 00:19:28.310 --> 00:19:30.940 where they saw just a single individual plus the number 00:19:30.940 --> 00:19:32.690 of species where they saw two individuals, 00:19:32.690 --> 00:19:35.829 and they added those and they divided by 2. 00:19:35.829 --> 00:19:36.620 That's this number. 00:19:38.799 --> 00:19:40.840 We're not going to go through this whole process, 00:19:40.840 --> 00:19:42.350 because it's a little bit tiresome. 00:19:42.350 --> 00:19:46.410 But I've already done it for you. 00:19:46.410 --> 00:19:50.670 So I'm going to plot a few of things to get you there. 00:19:58.240 --> 00:20:02.490 And so I calculated it was 19, 13, 9, 6. 00:20:02.490 --> 00:20:05.696 It becomes ill-determined once you get out here, 00:20:05.696 --> 00:20:07.320 in the sense that we don't have enough. 00:20:07.320 --> 00:20:10.240 It's not uniquely specified going from that to that 00:20:10.240 --> 00:20:12.980 as it has to be. 00:20:12.980 --> 00:20:13.970 But I calculated it. 00:20:13.970 --> 00:20:17.565 It's around 5, in here, for a few. 00:20:17.565 --> 00:20:21.120 And somewhere in here, it's going to go into 4. 00:20:21.120 --> 00:20:27.095 And then this might go down to 3, and then deh, deh, deh. 00:20:32.550 --> 00:20:42.950 Now, if you look at this and the rapid rapid fall-off, 00:20:42.950 --> 00:20:46.665 do you think that you're going to find any species that 00:20:46.665 --> 00:20:47.915 have more than 20 individuals? 00:20:51.440 --> 00:20:52.340 We're going to vote. 00:20:52.340 --> 00:20:53.857 So you see this falling-off? 00:20:53.857 --> 00:20:55.690 So let's say that I've just showed you this, 00:20:55.690 --> 00:20:58.930 and I haven't yet calculated the rest, 00:20:58.930 --> 00:21:01.540 do we think that there's going to be any species with more 00:21:01.540 --> 00:21:04.230 than 20 individuals? 00:21:04.230 --> 00:21:08.430 Greater than 20 individuals, question mark? 00:21:08.430 --> 00:21:10.760 1 is yes. 00:21:10.760 --> 00:21:12.245 2 is no. 00:21:14.945 --> 00:21:16.230 It's going to be yes, no. 00:21:16.230 --> 00:21:18.835 Ready, three, two, one. 00:21:21.560 --> 00:21:25.190 So we got some 2s. 00:21:25.190 --> 00:21:27.300 So I'd say that most people are saying, no. 00:21:27.300 --> 00:21:28.532 Look at this fall-off. 00:21:28.532 --> 00:21:29.990 They're not going to be any species 00:21:29.990 --> 00:21:33.480 with more than 20 individuals. 00:21:33.480 --> 00:21:35.630 Although we already know that there 00:21:35.630 --> 00:21:38.480 are many species with more than 20 individuals. 00:21:38.480 --> 00:21:41.930 So this plot is useful for something. 00:21:41.930 --> 00:21:43.560 You can see that there are. 00:21:43.560 --> 00:21:45.450 And we know exactly the number of species 00:21:45.450 --> 00:21:47.930 that have more than 20 individuals, roughly. 00:21:47.930 --> 00:21:50.760 So those ones are all in these. 00:21:50.760 --> 00:21:52.985 So you can see that there are hundreds of species 00:21:52.985 --> 00:21:54.235 with more than 20 individuals. 00:21:57.630 --> 00:22:00.830 And indeed, it looks like there were two or three species that 00:22:00.830 --> 00:22:03.186 had more than 1,000 individuals or 1,500 00:22:03.186 --> 00:22:04.560 or whatever the cutoff there was. 00:22:07.920 --> 00:22:14.050 So this distribution starts out rather high 00:22:14.050 --> 00:22:15.590 but then falls quickly. 00:22:15.590 --> 00:22:19.019 And out here, it's going to be very, very sparse. 00:22:19.019 --> 00:22:21.060 So there's going to be a bunch of numbers in here 00:22:21.060 --> 00:22:24.160 where there's not any species in the histogram. 00:22:24.160 --> 00:22:26.340 And then out there, there's going to be one, right? 00:22:26.340 --> 00:22:28.680 And indeed, you have to go really far out. 00:22:28.680 --> 00:22:30.490 Because there's one species out there 00:22:30.490 --> 00:22:33.870 that has a couple thousand. 00:22:33.870 --> 00:22:40.660 And indeed, the mean number of individuals per species 00:22:40.660 --> 00:22:44.880 has to be around 100. 00:22:44.880 --> 00:22:47.160 We know how to calculate a mean. 00:22:47.160 --> 00:22:50.810 This divided by this is just short of 100. 00:22:50.810 --> 00:22:54.150 So the mean number of individuals in a species 00:22:54.150 --> 00:22:56.280 is around 100. 00:22:56.280 --> 00:22:59.090 The mode is one. 00:23:01.446 --> 00:23:02.070 And the median? 00:23:08.630 --> 00:23:11.200 Well, ready? 00:23:11.200 --> 00:23:14.387 We decided this was the mode. 00:23:14.387 --> 00:23:15.720 Where is the median going to be? 00:23:15.720 --> 00:23:18.260 Is it going to be A, B, C, D? 00:23:18.260 --> 00:23:20.935 Ready, three, two, one. 00:23:23.760 --> 00:23:27.060 Indeed, this tells you pretty clear where the median is. 00:23:27.060 --> 00:23:28.854 This thing is indeed around the median. 00:23:28.854 --> 00:23:31.020 Because you can say, oh, it's about the same numbers 00:23:31.020 --> 00:23:32.240 to either side. 00:23:32.240 --> 00:23:33.740 So the median is around here. 00:23:36.310 --> 00:23:39.370 And I told you where the mean was, again. 00:23:39.370 --> 00:23:40.330 You guys remember? 00:23:40.330 --> 00:23:42.710 Ready, three, two, one. 00:23:42.710 --> 00:23:43.960 Mean, uno. 00:23:43.960 --> 00:23:44.460 Mean. 00:23:50.220 --> 00:23:53.070 So this is a very, very funny distribution. 00:23:53.070 --> 00:23:54.730 I guess I want to highlight that. 00:23:54.730 --> 00:23:57.530 And I think it's not at all what you 00:23:57.530 --> 00:24:00.070 would have expected somehow. 00:24:00.070 --> 00:24:04.799 At least, if you had described this measurement process to me, 00:24:04.799 --> 00:24:06.590 if you told me that you went to this island 00:24:06.590 --> 00:24:10.545 and you counted 20,000 trees, I don't know how many species 00:24:10.545 --> 00:24:11.420 I would have guessed. 00:24:11.420 --> 00:24:16.255 But OK, 220, it's reasonable. 00:24:16.255 --> 00:24:18.630 Well, I would have guessed it would have looked something 00:24:18.630 --> 00:24:22.380 like this on a linear scale, maybe, right? 00:24:22.380 --> 00:24:25.430 You know, that there would be a bunch of them around 50 to 100 00:24:25.430 --> 00:24:28.340 and some would go couple hundred, some of them. 00:24:28.340 --> 00:24:32.480 So I guess I would have thought that the mean, mode, 00:24:32.480 --> 00:24:35.070 median would all be kind of a more similar thing. 00:24:35.070 --> 00:24:38.130 But this is just not the way the world is. 00:24:38.130 --> 00:24:40.230 It's not just on BCI. 00:24:40.230 --> 00:24:44.150 People, for hundreds of years, have been 00:24:44.150 --> 00:24:45.520 studying these distributions. 00:24:45.520 --> 00:24:51.100 And things that look like this, with extremely long tails, 00:24:51.100 --> 00:24:54.810 this is what people see. 00:24:54.810 --> 00:24:57.720 Now you can argue about exactly how fast it falls off 00:24:57.720 --> 00:25:00.040 and whether it's different on a mainland or an island. 00:25:00.040 --> 00:25:05.210 But this basic feature, that rare species are common, 00:25:05.210 --> 00:25:09.140 this seems to be just that's what you always see. 00:25:09.140 --> 00:25:11.220 This is the thing that you have to remember, 00:25:11.220 --> 00:25:12.425 rare species are common. 00:25:24.520 --> 00:25:28.680 And I think that this is the basic, surprising thing 00:25:28.680 --> 00:25:30.530 in this whole field. 00:25:30.530 --> 00:25:32.480 And the ironic thing is that even 00:25:32.480 --> 00:25:35.580 after spending all this time reading about theories 00:25:35.580 --> 00:25:39.140 to describe these distributions, it's still very possible-- 00:25:39.140 --> 00:25:41.780 and I would say, based on the statistics, this year 00:25:41.780 --> 00:25:44.380 and past years, it's not just possible, 00:25:44.380 --> 00:25:46.060 but it is the standard outcome-- is 00:25:46.060 --> 00:25:47.476 that after reading this paper, you 00:25:47.476 --> 00:25:51.230 do not realize that the distribution looks like this. 00:25:51.230 --> 00:25:54.000 You somehow still think that it looks-- you kind of still 00:25:54.000 --> 00:25:59.060 think it's like a linear scale, where the typical species has 00:25:59.060 --> 00:26:01.130 this, where the mean, median, mode are all 00:26:01.130 --> 00:26:02.260 about the same thing. 00:26:02.260 --> 00:26:08.661 So I guess always plot the raw data in an untransformed way. 00:26:08.661 --> 00:26:10.410 There are theoretical reasons why it might 00:26:10.410 --> 00:26:11.618 be nice to plot it like this. 00:26:11.618 --> 00:26:15.120 But be very careful about what you're doing. 00:26:15.120 --> 00:26:17.965 Because then you're left with a mental image of a histogram 00:26:17.965 --> 00:26:19.850 that looks like this. 00:26:19.850 --> 00:26:21.640 And that's very, very dangerous. 00:26:21.640 --> 00:26:22.140 Yeah? 00:26:22.140 --> 00:26:25.570 AUDIENCE: Why does it matter [INAUDIBLE]? 00:26:25.570 --> 00:26:29.000 [INAUDIBLE] the aggregate data in bins like that. 00:26:29.000 --> 00:26:33.150 And I mean, sure, exactly one species is the mode, 00:26:33.150 --> 00:26:35.619 but do you really want the--? 00:26:35.619 --> 00:26:37.410 PROFESSOR: I understand what you're saying. 00:26:42.240 --> 00:26:44.090 It's just that there's a qualitative aspect 00:26:44.090 --> 00:26:49.510 to the data, which is that most species are very rare. 00:26:49.510 --> 00:26:51.630 And this is something that I think is surprising. 00:26:51.630 --> 00:26:53.410 I think it's deep. 00:26:53.410 --> 00:26:56.120 And it's something that you do not get realized. 00:26:56.120 --> 00:27:00.368 AUDIENCE: Most species have more than 16. 00:27:00.368 --> 00:27:03.024 I mean, it depends what you mean by rare. 00:27:03.024 --> 00:27:03.690 PROFESSOR: Yeah. 00:27:03.690 --> 00:27:06.108 AUDIENCE: Look at the way that the distribution is away 00:27:06.108 --> 00:27:07.268 from trend. 00:27:07.268 --> 00:27:08.518 AUDIENCE: That's a good point. 00:27:08.518 --> 00:27:10.687 But the species density is clustered 00:27:10.687 --> 00:27:11.962 around the low numbers. 00:27:11.962 --> 00:27:12.670 PROFESSOR: Right. 00:27:12.670 --> 00:27:14.920 AUDIENCE: But actually most species have more than 30. 00:27:18.679 --> 00:27:20.220 PROFESSOR: Maybe the surprising thing 00:27:20.220 --> 00:27:24.260 is that just if you take-- the mean is 100. 00:27:24.260 --> 00:27:33.440 And so I would've thought that, if you plot number of species 00:27:33.440 --> 00:27:41.650 as a function of the number of individuals, 00:27:41.650 --> 00:27:44.290 given those numbers, I would have guessed, OK, here's 100. 00:27:44.290 --> 00:27:46.480 I would have guessed-- here's 50, so just 00:27:46.480 --> 00:27:49.130 to highlight that this is 150. 00:27:49.130 --> 00:27:53.620 So linear scale, I would have guessed 00:27:53.620 --> 00:27:56.820 it would look something like that, maybe larger than Rudin 00:27:56.820 --> 00:27:57.889 or something. 00:27:57.889 --> 00:28:00.055 AUDIENCE: What would that look like in a log2 scale? 00:28:00.055 --> 00:28:03.951 It would look like It's like the log of [INAUDIBLE]? 00:28:03.951 --> 00:28:06.460 So it goes up really fast and then-- 00:28:06.460 --> 00:28:11.730 PROFESSOR: So this thing would be kind of like shoom. 00:28:11.730 --> 00:28:14.070 I mean all the weight would be in. 00:28:14.070 --> 00:28:16.988 It would be like all here plus a little bit on each of these. 00:28:16.988 --> 00:28:18.884 AUDIENCE: But yeah. 00:28:18.884 --> 00:28:21.730 I don't think it's actually that different. 00:28:21.730 --> 00:28:24.340 The only thing that's different is the tail on the left. 00:28:24.340 --> 00:28:26.092 PROFESSOR: And the tail on the right. 00:28:26.092 --> 00:28:27.800 AUDIENCE: Yeah, it's a little bit longer. 00:28:27.800 --> 00:28:29.383 PROFESSOR: No, it's lot longer, right? 00:28:29.383 --> 00:28:33.530 Because this thing, all of the weight is between 50 and 150, 00:28:33.530 --> 00:28:35.740 which means that all of the counts 00:28:35.740 --> 00:28:41.124 are basically going to be these two, basically. 00:28:41.124 --> 00:28:42.790 Because this thing comes out either way. 00:28:42.790 --> 00:28:46.240 So in this case, if you take that histogram put it 00:28:46.240 --> 00:28:48.560 on this kind of scale, you end up with two bars 00:28:48.560 --> 00:28:50.717 up high, nothing outside. 00:28:50.717 --> 00:28:52.300 So it's a very different distribution. 00:28:52.300 --> 00:28:55.330 And it's not to say that this is a ridiculous thing to do. 00:28:55.330 --> 00:28:57.230 It's just that. 00:28:57.230 --> 00:28:59.610 But the problem is that your mental image 00:28:59.610 --> 00:29:01.930 of what the distribution looks like ends up 00:29:01.930 --> 00:29:03.910 being incorrect, in the sense that you 00:29:03.910 --> 00:29:07.040 have a qualitatively different sense of what's of what's 00:29:07.040 --> 00:29:08.690 going on. 00:29:08.690 --> 00:29:14.260 And if you go up to 10 species, here, and 10 is way down here. 00:29:17.589 --> 00:29:19.130 If this is what it looked like, there 00:29:19.130 --> 00:29:22.970 would be essentially no species with fewer than 10 individuals. 00:29:22.970 --> 00:29:25.780 But if you come over here and you add it up here. 00:29:25.780 --> 00:29:31.740 It's like a mean of 6 times 10 is 60 out of 200. 00:29:31.740 --> 00:29:35.830 A quarter or a third of the species on this plot of land 00:29:35.830 --> 00:29:37.450 have fewer than 10 individuals. 00:29:37.450 --> 00:29:39.515 And 10 is really a very small number. 00:29:43.810 --> 00:29:47.590 Well, rare species are common. 00:29:47.590 --> 00:29:52.170 I think it's a true description of the observed distribution 00:29:52.170 --> 00:29:53.180 here and elsewhere. 00:29:53.180 --> 00:29:56.620 And it's not something that you appreciate 00:29:56.620 --> 00:29:58.911 or realize when you plot it in that way. 00:29:58.911 --> 00:30:01.522 AUDIENCE: But you can get this information from that plot. 00:30:01.522 --> 00:30:02.480 PROFESSOR: No, I agree. 00:30:02.480 --> 00:30:03.440 You can get it. 00:30:03.440 --> 00:30:04.290 You can get it. 00:30:04.290 --> 00:30:07.280 But it was only 10% of the group got it. 00:30:07.280 --> 00:30:10.430 Right, the fact that you can get it-- right, it's possible. 00:30:10.430 --> 00:30:13.370 But you don't get it. 00:30:13.370 --> 00:30:15.680 That is a practical statement. 00:30:15.680 --> 00:30:18.150 Yeah, I'm not dead set against this distribution. 00:30:18.150 --> 00:30:20.737 It's just that it makes everybody think 00:30:20.737 --> 00:30:21.820 something that's not true. 00:30:21.820 --> 00:30:24.430 So if you think that that's OK, then I can't help you. 00:30:27.670 --> 00:30:29.690 It's OK, but it's just you have to be 00:30:29.690 --> 00:30:33.300 careful is my only statement. 00:30:33.300 --> 00:30:36.550 And I very much want you to take away. 00:30:36.550 --> 00:30:38.800 Because I this is an accurate description of the data. 00:30:38.800 --> 00:30:40.270 Rare species are common. 00:30:40.270 --> 00:30:43.360 And one of the readings-- I think it was in this paper, 00:30:43.360 --> 00:30:45.640 maybe it was a different one that I was reading. 00:30:45.640 --> 00:30:49.300 Even Darwin, when talking about this, commented on this fact 00:30:49.300 --> 00:30:54.820 that rarity of species is somehow a typical event. 00:30:54.820 --> 00:30:57.700 AUDIENCE: And common species are rare. 00:30:57.700 --> 00:31:01.160 PROFESSOR: And common species are rare, that's right. 00:31:01.160 --> 00:31:04.720 This distribution is hugely, hugely skewed. 00:31:14.986 --> 00:31:16.110 These are the measurements. 00:31:18.731 --> 00:31:20.730 It's good to look at them in both of these ways. 00:31:20.730 --> 00:31:23.950 Because you can't even plot the data on a linear scale. 00:31:23.950 --> 00:31:26.020 So that's a good reason for doing it. 00:31:26.020 --> 00:31:29.350 But I think it's good to have both of these pictures in mind. 00:31:38.680 --> 00:31:42.960 What we want to do is to talk about two classes of models 00:31:42.960 --> 00:31:45.110 that give something that's essentially 00:31:45.110 --> 00:31:46.620 this log normal distribution. 00:31:46.620 --> 00:31:50.350 So on a log scale it looks normally distributed, 00:31:50.350 --> 00:31:53.130 approximately. 00:31:53.130 --> 00:31:54.640 And those two models are going to be 00:31:54.640 --> 00:31:58.610 kind of a niche-based model and a neutral model. 00:31:58.610 --> 00:32:01.460 Can somebody, in words, explain what they maybe 00:32:01.460 --> 00:32:04.200 see as the difference between this niche 00:32:04.200 --> 00:32:05.800 and a neutral kind of approach? 00:32:21.768 --> 00:32:22.349 Yeah? 00:32:22.349 --> 00:32:23.265 AUDIENCE: [INAUDIBLE]. 00:32:26.255 --> 00:32:27.505 PROFESSOR: Every species is--? 00:32:27.505 --> 00:32:28.430 AUDIENCE: [INAUDIBLE]. 00:32:28.430 --> 00:32:29.846 PROFESSOR: In which one? 00:32:29.846 --> 00:32:31.130 AUDIENCE: In niche. 00:32:31.130 --> 00:32:34.340 PROFESSOR: In the niche theory, the species are different. 00:32:34.340 --> 00:32:39.450 So it seems like a ridiculous statement. 00:32:39.450 --> 00:32:41.380 Do you believe that species are different? 00:32:41.380 --> 00:32:43.200 We can vote, yes or no. 00:32:43.200 --> 00:32:46.150 Ready, three, two, one. 00:32:46.150 --> 00:32:47.800 Yeah. 00:32:47.800 --> 00:32:51.330 Well, somebody's been convinced by the neutral theory. 00:32:51.330 --> 00:32:54.760 It's clear that species are different. 00:32:54.760 --> 00:32:59.390 And the question is which patterns in the data 00:32:59.390 --> 00:33:01.340 do you need to invoke differences 00:33:01.340 --> 00:33:03.890 in order to explain? 00:33:03.890 --> 00:33:06.050 And I think that one, maybe, theme that's 00:33:06.050 --> 00:33:08.640 come out of this relative species abundance literature 00:33:08.640 --> 00:33:11.110 and the debates between the neutral and the niche guys 00:33:11.110 --> 00:33:14.390 is just that this distribution is 00:33:14.390 --> 00:33:18.310 less informative of the micro scale 00:33:18.310 --> 00:33:22.240 or individual kind of interactions 00:33:22.240 --> 00:33:25.640 then you might have thought. 00:33:25.640 --> 00:33:29.290 Because multiple models can adequately 00:33:29.290 --> 00:33:32.050 explain such a pattern. 00:33:32.050 --> 00:33:35.220 In all areas, we have to remember 00:33:35.220 --> 00:33:38.530 that you make an observation, and you write down a model 00:33:38.530 --> 00:33:40.892 that explains that observation. 00:33:40.892 --> 00:33:42.600 So what you do is you write down a model. 00:33:42.600 --> 00:33:44.040 And writing down a model, what that means 00:33:44.040 --> 00:33:46.010 is that you make some set of assumptions. 00:33:46.010 --> 00:33:48.176 And then you look to see what happens in that model. 00:33:48.176 --> 00:33:53.555 And if the model is consistent with the data, that's good. 00:33:53.555 --> 00:33:55.680 But it doesn't prove that the assumptions that went 00:33:55.680 --> 00:33:56.804 into the model are correct. 00:33:56.804 --> 00:33:58.475 And this is a trivial statement. 00:33:58.475 --> 00:33:59.475 And I've said it before. 00:34:01.839 --> 00:34:03.880 You have to tell yourself this or remind yourself 00:34:03.880 --> 00:34:05.580 of this kind of once a month. 00:34:05.580 --> 00:34:07.970 Because it's just such an easy thing to forget about. 00:34:15.090 --> 00:34:18.420 Now, the niche models indeed assume 00:34:18.420 --> 00:34:20.190 that the species are different. 00:34:20.190 --> 00:34:22.840 And that's reasonable. 00:34:22.840 --> 00:34:24.687 Because we think it's true. 00:34:24.687 --> 00:34:26.770 But then, of course, there are many different ways 00:34:26.770 --> 00:34:28.550 of capturing those differences. 00:34:28.550 --> 00:34:32.600 And then you have to decide whether the assumptions there 00:34:32.600 --> 00:34:37.030 are reasonable or whether they're necessary, essential. 00:34:37.030 --> 00:34:39.170 In the context of the niche models, 00:34:39.170 --> 00:34:42.500 we're going to think about the so-called broken stick models. 00:34:52.510 --> 00:34:55.219 So basically, you get log normal distributions 00:34:55.219 --> 00:34:58.320 when there's some sort of multiplicative-type random 00:34:58.320 --> 00:35:01.260 process that's being added together. 00:35:01.260 --> 00:35:02.930 You get normal distributions when 00:35:02.930 --> 00:35:05.794 you have sums of random things going together. 00:35:05.794 --> 00:35:07.210 This is the central limit theorem. 00:35:07.210 --> 00:35:09.560 But when you have multiplicative kind 00:35:09.560 --> 00:35:12.895 of errors or random processes coming together, 00:35:12.895 --> 00:35:14.270 you get log normal distributions. 00:35:14.270 --> 00:35:17.650 And I want to highlight that that does not necessarily 00:35:17.650 --> 00:35:23.720 have to tell you so much about the biology of it. 00:35:23.720 --> 00:35:28.540 Because a classic situation where you get log 00:35:28.540 --> 00:35:35.590 normal distributions is if you take a stone and you crush it. 00:35:41.220 --> 00:35:43.350 You can do this experiment at home. 00:35:43.350 --> 00:35:47.930 And then you measure the mass distribution of the resulting 00:35:47.930 --> 00:35:48.430 fragments. 00:35:55.390 --> 00:36:00.050 And the distribution of mass is log normal. 00:36:05.210 --> 00:36:13.280 Just take a stone, grind it under your boot or hammer it, 00:36:13.280 --> 00:36:15.030 just kind rub it right in. 00:36:15.030 --> 00:36:18.230 You'll get you'll get some distribution of fragments. 00:36:18.230 --> 00:36:20.950 For each of the fragments, measure the mass, and, indeed, 00:36:20.950 --> 00:36:22.930 you end up getting a log normal distribution. 00:36:22.930 --> 00:36:24.930 Because there's some sense that what's happening 00:36:24.930 --> 00:36:28.260 is that you take a larger mass, you break it up randomly, 00:36:28.260 --> 00:36:30.490 and then the resulting fragments, at some rate, 00:36:30.490 --> 00:36:32.270 each of them you break up randomly. 00:36:32.270 --> 00:36:33.770 and the small ones are maybe kind of 00:36:33.770 --> 00:36:35.644 less likely to get broken up as the big ones, 00:36:35.644 --> 00:36:38.232 so then the small ones can still get even smaller. 00:36:38.232 --> 00:36:40.690 But then there's going to be, at some rate, some very large 00:36:40.690 --> 00:36:42.060 ones. 00:36:42.060 --> 00:36:47.880 So such a process ends up-- I mean it's not biology. 00:36:47.880 --> 00:36:51.660 This is just something about the nature of the breaking 00:36:51.660 --> 00:36:53.990 up of this physical object. 00:36:53.990 --> 00:36:56.160 And indeed, the basic idea behind many 00:36:56.160 --> 00:36:58.630 of the niche models that give you a log normal distribution 00:36:58.630 --> 00:37:02.270 is equivalent to crushing a stone 00:37:02.270 --> 00:37:05.910 and measuring the resulting distribution. 00:37:05.910 --> 00:37:07.670 I'll describe what I mean by that. 00:37:07.670 --> 00:37:11.120 Typically, the broken stick models, 00:37:11.120 --> 00:37:16.140 they say there's some resource axis. 00:37:16.140 --> 00:37:20.300 This is a resource axis. 00:37:20.300 --> 00:37:24.430 And this could be, for example, where you're getting food from. 00:37:24.430 --> 00:37:26.570 Now, we're going to have to divide up this resource 00:37:26.570 --> 00:37:28.615 access among some number of different species. 00:37:28.615 --> 00:37:29.990 And what we're going to assume is 00:37:29.990 --> 00:37:32.120 that the number of individuals in the species 00:37:32.120 --> 00:37:35.460 is proportional to the length of the resource axis 00:37:35.460 --> 00:37:36.698 that it's able to capture. 00:37:39.450 --> 00:37:43.150 And I want to make sure I find my notes. 00:37:43.150 --> 00:37:44.275 I want to highlight this. 00:37:44.275 --> 00:37:53.600 This comes from MacArthur in the 1950s. 00:37:53.600 --> 00:37:58.480 MacArthur and it's 1957. 00:37:58.480 --> 00:38:01.780 So we imagine there's this homogeneous resource axis. 00:38:01.780 --> 00:38:03.600 We're going to break it up into N segments. 00:38:03.600 --> 00:38:08.860 And the abundances are proportional to the length. 00:38:08.860 --> 00:38:13.690 And the idea is that, if you just break this up randomly, 00:38:13.690 --> 00:38:17.150 so let's say you just draw N minus 1 lines randomly, 00:38:17.150 --> 00:38:20.290 or N minus 1 points randomly here. 00:38:20.290 --> 00:38:27.775 Now you have N species with N different abundances. 00:38:30.480 --> 00:38:32.360 The question is does that give a log normal? 00:38:35.160 --> 00:38:37.495 We'll say N minus 1 random points. 00:38:41.084 --> 00:38:42.377 Do you understand what I mean. 00:38:42.377 --> 00:38:44.460 You sample uniformly once, sample uniformly twice. 00:38:44.460 --> 00:38:50.220 You do that N minus 1 times, and now you have N and deh deh. 00:38:50.220 --> 00:38:52.130 And then we say, OK, the first species 00:38:52.130 --> 00:38:53.230 has this many individuals. 00:38:53.230 --> 00:38:54.790 The second has this one. 00:38:54.790 --> 00:38:58.230 The third is this one, et cetera. 00:38:58.230 --> 00:39:00.589 The question is does random points, 00:39:00.589 --> 00:39:01.880 does that lead to a log normal? 00:39:07.400 --> 00:39:09.380 Yes and no. 00:39:09.380 --> 00:39:12.250 Let's think about this for 10 seconds. 00:39:12.250 --> 00:39:20.030 N minus 1 random points, log normal distribution, 00:39:20.030 --> 00:39:23.115 ready, three, two, one. 00:39:26.320 --> 00:39:30.200 So I'd say that we have a majority are saying no. 00:39:30.200 --> 00:39:31.610 Can somebody say why that is? 00:39:31.610 --> 00:39:34.430 AUDIENCE: [INAUDIBLE]. 00:39:34.430 --> 00:39:36.440 PROFESSOR: Because it's something else. 00:39:36.440 --> 00:39:39.900 That's fair. 00:39:39.900 --> 00:39:41.865 But can you say qualitatively why it is 00:39:41.865 --> 00:39:43.682 that this is not going to work? 00:39:43.682 --> 00:39:48.632 AUDIENCE: You can't have very long gaps. 00:39:48.632 --> 00:39:49.340 PROFESSOR: Right. 00:39:49.340 --> 00:39:51.256 That is it's going to be very unusual that you 00:39:51.256 --> 00:39:53.310 get a very long gap. 00:39:53.310 --> 00:39:55.510 What about the other end? 00:39:55.510 --> 00:39:58.064 AUDIENCE: Also a very long tail. 00:39:58.064 --> 00:39:59.730 PROFESSOR: Now I'm a little bit worried. 00:40:03.040 --> 00:40:04.460 I think that that's true, right? 00:40:10.640 --> 00:40:12.910 Well, I'm going to say that you're not going 00:40:12.910 --> 00:40:16.140 to get this super long ones. 00:40:16.140 --> 00:40:20.810 I think that the distribution might still 00:40:20.810 --> 00:40:24.500 be peaked at short values. 00:40:24.500 --> 00:40:25.230 No? 00:40:25.230 --> 00:40:26.550 AUDIENCE: No. 00:40:26.550 --> 00:40:28.510 PROFESSOR: Random? 00:40:28.510 --> 00:40:32.236 If we were just traveling along this resource axis, 00:40:32.236 --> 00:40:34.360 at a rate that's kind of exponentially distributed, 00:40:34.360 --> 00:40:36.480 like Poisson rate, we just dropped 00:40:36.480 --> 00:40:40.635 points, that's something very similar to this random-- 00:40:40.635 --> 00:40:43.167 AUDIENCE: It said we're limited in the number-- 00:40:43.167 --> 00:40:43.750 PROFESSOR: No. 00:40:43.750 --> 00:40:45.604 Is that not true? 00:40:45.604 --> 00:40:47.020 AUDIENCE: Your sample [INAUDIBLE]. 00:40:53.030 --> 00:40:57.310 PROFESSOR: I'm a little bit worried that I might be-- now, 00:40:57.310 --> 00:40:58.600 I'm not 100% confident. 00:40:58.600 --> 00:41:01.361 Depending on how I look at this, I get different distributions. 00:41:01.361 --> 00:41:01.860 Yeah? 00:41:01.860 --> 00:41:05.716 AUDIENCE: But I think the first thing that he said, 00:41:05.716 --> 00:41:08.465 where you just say, I'm going to pick N minus 1 points-- 00:41:08.465 --> 00:41:09.090 PROFESSOR: Yes. 00:41:09.090 --> 00:41:11.940 AUDIENCE: --is a different thing than going along the axis 00:41:11.940 --> 00:41:14.320 and exponentially dropping ones along. 00:41:14.320 --> 00:41:17.021 PROFESSOR: I agree it's different. 00:41:17.021 --> 00:41:19.395 AUDIENCE: I don't think that would be the idea simulated, 00:41:19.395 --> 00:41:22.093 because you would be very likely to just get 00:41:22.093 --> 00:41:24.134 this giant thing at the end when you're finished. 00:41:24.134 --> 00:41:26.120 AUDIENCE: What you could do, you could go on 00:41:26.120 --> 00:41:29.830 to draft N plus 2 points. 00:41:29.830 --> 00:41:30.830 PROFESSOR: No, I think-- 00:41:30.830 --> 00:41:33.125 AUDIENCE: These scales that are your two end points 00:41:33.125 --> 00:41:33.980 [? are doubled. ?] 00:41:33.980 --> 00:41:35.938 PROFESSOR: Because I think that the probability 00:41:35.938 --> 00:41:37.310 distribution does grow. 00:41:37.310 --> 00:41:40.370 I think that I'm going to side with you. 00:41:40.370 --> 00:41:42.190 So we've decided that there are not 00:41:42.190 --> 00:41:44.315 going to be as many short sticks, 00:41:44.315 --> 00:41:46.190 and there's not going to be as long sticks as 00:41:46.190 --> 00:41:47.231 compared to a log normal. 00:41:47.231 --> 00:41:48.226 Do we agree with that? 00:41:51.205 --> 00:41:53.580 At least we agree that it's not going to be a log normal. 00:41:53.580 --> 00:41:55.496 So you're not going to get this huge variation 00:41:55.496 --> 00:41:59.230 of some very long sticks and some very short ones. 00:41:59.230 --> 00:42:03.710 Now, the question is how would you change this sort of model 00:42:03.710 --> 00:42:05.581 in order to generate a log normal? 00:42:05.581 --> 00:42:07.330 And the answer is that what you have to do 00:42:07.330 --> 00:42:10.980 is you have to what is called some niche hierarchy or so 00:42:10.980 --> 00:42:12.690 some hierarchical breaking. 00:42:12.690 --> 00:42:15.040 Just like what led to the stone giving you a log normal 00:42:15.040 --> 00:42:17.100 is that you have to have some successive process 00:42:17.100 --> 00:42:18.810 of breaking things. 00:42:18.810 --> 00:42:21.190 So this is what they call some hierarchy model. 00:42:27.225 --> 00:42:29.225 And then they key thing is that it's sequential. 00:42:33.260 --> 00:42:35.060 You have your resource axis. 00:42:35.060 --> 00:42:39.661 First, you have some rule for breaking it up. 00:42:39.661 --> 00:42:41.410 It could be that you just sample uniformly 00:42:41.410 --> 00:42:43.034 or some other probability distribution. 00:42:45.089 --> 00:42:46.880 And the way that you might think about this 00:42:46.880 --> 00:42:57.020 is via-- just everything up on the board 00:42:57.020 --> 00:42:58.270 is so nice and useful. 00:42:58.270 --> 00:43:02.430 I feel bad getting rid of it. 00:43:02.430 --> 00:43:06.170 This thing is not true, so I don't mind erasing it. 00:43:06.170 --> 00:43:10.560 So let's imagine some bird community in the forest. 00:43:13.260 --> 00:43:14.770 And we're going to think about where 00:43:14.770 --> 00:43:18.336 is it that the birds are getting their grub or their food 00:43:18.336 --> 00:43:18.836 to eat. 00:43:21.530 --> 00:43:27.620 First, well, now the axis is somehow vertical. 00:43:27.620 --> 00:43:35.210 You could divide them up into the ground foragers 00:43:35.210 --> 00:43:39.730 as compared to the tree foragers in terms of where 00:43:39.730 --> 00:43:40.971 they're getting their food. 00:43:40.971 --> 00:43:43.470 And you say, oh, well, how much of the food is on each side? 00:43:43.470 --> 00:43:48.850 Oh, well, we'll say 30% is on the ground, 70% is on the tree. 00:43:48.850 --> 00:43:50.889 This is along the stick. 00:43:50.889 --> 00:43:52.430 You cut the stick in some way, or you 00:43:52.430 --> 00:43:54.530 break the stick in some way. 00:43:54.530 --> 00:43:56.060 But then within the tree foragers, 00:43:56.060 --> 00:44:00.110 you'd say, well, the resources might be separated. 00:44:00.110 --> 00:44:01.750 And this is really like speciation, 00:44:01.750 --> 00:44:03.690 a species is in the niche, the species 00:44:03.690 --> 00:44:06.300 are focusing on different niches. 00:44:06.300 --> 00:44:09.030 So you'd say, oh, some are going to focus on the trunk, 00:44:09.030 --> 00:44:12.330 some will focus on branches. 00:44:12.330 --> 00:44:13.950 And again, this part of the stick 00:44:13.950 --> 00:44:17.720 is now broken or divided among different resource locations 00:44:17.720 --> 00:44:18.885 with some amount. 00:44:18.885 --> 00:44:21.855 But then also, you're going to get speciation 00:44:21.855 --> 00:44:23.730 in different directions here, because there's 00:44:23.730 --> 00:44:27.590 both the surface-- I don't know if you guys have ever 00:44:27.590 --> 00:44:30.200 eaten grubs-- but there's the surface grubs, 00:44:30.200 --> 00:44:32.665 and then there's also the sub-bark grubs. 00:44:38.100 --> 00:44:42.230 And so you kind of do this process multiple times, 00:44:42.230 --> 00:44:44.650 where you kind of pick different branches 00:44:44.650 --> 00:44:47.020 and break them to divide up the niche. 00:44:47.020 --> 00:44:49.972 And then you end up with a log normal type distribution. 00:44:49.972 --> 00:44:54.840 And this is a similar process to the crushing of the stone, 00:44:54.840 --> 00:45:01.870 because the idea is that there's sequential breaks of the stone. 00:45:01.870 --> 00:45:04.980 So the stone first breaks into maybe simply two 00:45:04.980 --> 00:45:06.474 or it could be three. 00:45:06.474 --> 00:45:07.640 First, there's one breaking. 00:45:07.640 --> 00:45:09.400 And then one of them is broken more. 00:45:09.400 --> 00:45:11.924 So given this process, you end up 00:45:11.924 --> 00:45:13.340 getting a log normal distribution. 00:45:13.340 --> 00:45:14.334 Yeah. 00:45:14.334 --> 00:45:18.640 AUDIENCE: But you also have a distribution of like how far. 00:45:18.640 --> 00:45:20.340 Because I guess there are two questions. 00:45:20.340 --> 00:45:23.620 Like when you break your stick, you assume, somehow, 00:45:23.620 --> 00:45:25.154 that you uniformly break it. 00:45:25.154 --> 00:45:25.820 PROFESSOR: Yeah. 00:45:29.050 --> 00:45:31.290 A lot of work has gone into the question of how it 00:45:31.290 --> 00:45:33.330 is you should break the stick. 00:45:33.330 --> 00:45:35.390 Given that you have this tree foraging stick. 00:45:39.160 --> 00:45:40.840 On a practical level, what they do is 00:45:40.840 --> 00:45:44.140 they ask, well, what probability distribution gives you the best 00:45:44.140 --> 00:45:45.490 agreement with the data? 00:45:45.490 --> 00:45:46.860 Is it uniform? 00:45:46.860 --> 00:45:50.160 Or is it, oh, it's broken like this? 00:45:50.160 --> 00:45:52.490 And in some cases people say, well, it's 00:45:52.490 --> 00:45:54.055 actually tilted on one side. 00:46:00.900 --> 00:46:02.770 Well, in the context of a succession 00:46:02.770 --> 00:46:04.350 and some other environments, there's 00:46:04.350 --> 00:46:06.660 an idea that, if a species first gets somewhere, 00:46:06.660 --> 00:46:08.790 they can kind of monopolize a larger 00:46:08.790 --> 00:46:11.630 fraction of the resources then if it's divided kind 00:46:11.630 --> 00:46:12.922 of an equally at the beginning. 00:46:12.922 --> 00:46:15.505 And that's going to effect where this probability distribution 00:46:15.505 --> 00:46:17.150 is going to break each one. 00:46:17.150 --> 00:46:23.580 But there's always this question about how constrained 00:46:23.580 --> 00:46:25.080 are the notions and so forth. 00:46:25.080 --> 00:46:27.384 And I'm agnostic on that point. 00:46:27.384 --> 00:46:31.016 AUDIENCE: But you also need distribution for how many times 00:46:31.016 --> 00:46:33.440 it breaks [INAUDIBLE]. 00:46:33.440 --> 00:46:35.390 PROFESSOR: Yes. 00:46:35.390 --> 00:46:38.054 It's just that, if you do this process, 00:46:38.054 --> 00:46:39.970 it's like a central limit theorem type result. 00:46:39.970 --> 00:46:42.010 So you have to do it enough times so that you get 00:46:42.010 --> 00:46:43.260 to some limiting distribution. 00:46:43.260 --> 00:46:45.120 And then you could keep on doing it. 00:46:45.120 --> 00:46:47.120 In the end, we always say that species abundance 00:46:47.120 --> 00:46:49.190 is proportional to the size. 00:46:49.190 --> 00:46:51.520 So we're going to scale, ultimately, 00:46:51.520 --> 00:46:53.360 to get the correct number of individuals. 00:46:53.360 --> 00:46:57.130 It's just that you have to do it some reasonable number of times 00:46:57.130 --> 00:46:58.880 so that the randomness kind of washes out, 00:46:58.880 --> 00:47:00.963 and you end up approaching that limiting behavior. 00:47:04.095 --> 00:47:04.970 Does that make sense? 00:47:09.480 --> 00:47:14.690 And indeed I just want to mention a major result 00:47:14.690 --> 00:47:15.840 in this field. 00:47:18.360 --> 00:47:23.760 These niche type models successfully 00:47:23.760 --> 00:47:27.692 explained or predicted another pattern 00:47:27.692 --> 00:47:30.150 that had been observed, which is the so-called species area 00:47:30.150 --> 00:47:30.733 relationships. 00:47:39.440 --> 00:47:43.570 So this is just saying that, here, we looked at 50 hectares, 00:47:43.570 --> 00:47:45.610 and we asked how many species where there. 00:47:45.610 --> 00:47:47.550 225 species in 50 hectares. 00:47:47.550 --> 00:47:50.550 Now, the question is, if instead of looking at 50 hectares, 00:47:50.550 --> 00:47:54.082 we instead looked at 500, do you think 00:47:54.082 --> 00:47:55.790 of that the number of species we observed 00:47:55.790 --> 00:48:01.538 would have gone up, stayed the same, or gone down? 00:48:01.538 --> 00:48:07.936 Up, same, down, ready, three, two, one. 00:48:07.936 --> 00:48:08.880 Up. 00:48:08.880 --> 00:48:09.830 Up. 00:48:09.830 --> 00:48:13.660 If you look at a larger area, you 00:48:13.660 --> 00:48:17.170 expect to see more species in a larger area. 00:48:17.170 --> 00:48:18.890 And people really do this. 00:48:18.890 --> 00:48:23.070 They look in some area, going from, say, they take a meter, 00:48:23.070 --> 00:48:24.450 and they count all the species. 00:48:24.450 --> 00:48:26.530 And then they go and here is 100 meters, 00:48:26.530 --> 00:48:29.060 and they count all the species. 00:48:29.060 --> 00:48:31.190 And they ask, how many species do you 00:48:31.190 --> 00:48:32.440 see as a function of the area? 00:48:32.440 --> 00:48:34.773 And what people have found is that the number of species 00:48:34.773 --> 00:48:42.580 you observe it is proportional to the area to some power, 00:48:42.580 --> 00:48:44.790 where Z is around a 1/4. 00:48:52.036 --> 00:48:55.584 And of course, the area goes as some r squared. 00:48:55.584 --> 00:48:57.000 If you wanted to, you could say it 00:48:57.000 --> 00:49:01.000 goes as the square root of the radius, whatever. 00:49:01.000 --> 00:49:03.670 But the number species in some area, 00:49:03.670 --> 00:49:06.100 it grows, but it grows in a manner 00:49:06.100 --> 00:49:08.870 that is less than linear. 00:49:08.870 --> 00:49:11.349 Does that make sense? 00:49:11.349 --> 00:49:13.390 It definitely makes sense that's less the linear. 00:49:13.390 --> 00:49:17.302 Because linear would be that you sample a bunch of species here, 00:49:17.302 --> 00:49:19.135 and then you look at another identical plot, 00:49:19.135 --> 00:49:20.125 you get some other species. 00:49:20.125 --> 00:49:22.583 And they were saying that, oh, that you really don't expect 00:49:22.583 --> 00:49:24.400 any of those species overlap. 00:49:24.400 --> 00:49:26.420 That would be a weird world. 00:49:26.420 --> 00:49:31.390 So it very much make sense that this is less than 1. 00:49:31.390 --> 00:49:34.550 Of course, it didn't have to be this power law. 00:49:34.550 --> 00:49:39.010 But one thing that has been discovered, around the world, 00:49:39.010 --> 00:49:42.050 is that power laws are very interesting. 00:49:42.050 --> 00:49:45.810 But once again, many different microscopic processes 00:49:45.810 --> 00:49:47.200 can lead to power laws. 00:49:47.200 --> 00:49:49.270 The niche models have successfully 00:49:49.270 --> 00:49:53.290 predicted or explained why it might have this scaling. 00:49:53.290 --> 00:49:56.400 But it turns out that neutral models can also predict it. 00:49:56.400 --> 00:50:00.490 And may just be that lots of spatially explicit models 00:50:00.490 --> 00:50:02.520 will give you some power law type scaling that 00:50:02.520 --> 00:50:04.589 looks kind of like this. 00:50:04.589 --> 00:50:06.130 So once again, it's a question of how 00:50:06.130 --> 00:50:09.162 convinced you should be about microscopic processes 00:50:09.162 --> 00:50:10.870 based on being able to explain some data. 00:50:10.870 --> 00:50:17.880 And I think the best cure for this danger, 00:50:17.880 --> 00:50:20.480 of assuming that the microscopic assumptions are correct, 00:50:20.480 --> 00:50:22.438 because the model is able to explain something, 00:50:22.438 --> 00:50:25.010 is that, if you find some other very different set 00:50:25.010 --> 00:50:29.629 of microscopic assumptions that also explain the patterns, then 00:50:29.629 --> 00:50:31.670 it becomes clear that you have to take everything 00:50:31.670 --> 00:50:33.496 with a grain of salt. 00:50:33.496 --> 00:50:34.870 And that's I think part of what's 00:50:34.870 --> 00:50:38.270 been very valuable about the neutral theory contribution 00:50:38.270 --> 00:50:39.010 to this field. 00:50:39.010 --> 00:50:41.179 AUDIENCE: Does this just come from-- you 00:50:41.179 --> 00:50:44.312 assume that all the individuals are uniformly distributed 00:50:44.312 --> 00:50:45.758 and then [INAUDIBLE]? 00:50:50.997 --> 00:50:53.080 PROFESSOR: There are multiple derivations of this, 00:50:53.080 --> 00:50:54.371 so it's a little bit confusing. 00:51:03.135 --> 00:51:05.500 The neutral models, that I have seen, 00:51:05.500 --> 00:51:07.780 that lead to these patterns, they basically 00:51:07.780 --> 00:51:11.510 have the individuals randomly, either 00:51:11.510 --> 00:51:14.490 with sex or without sex, kind of diffusing around, 00:51:14.490 --> 00:51:16.640 and then they divide, deh-deh. 00:51:16.640 --> 00:51:20.190 And then you can explicitly just do the different spaces 00:51:20.190 --> 00:51:22.040 and see that you get a scaling. 00:51:22.040 --> 00:51:28.210 It seems to be a surprisingly emergent feature 00:51:28.210 --> 00:51:31.190 of many of these models. 00:51:31.190 --> 00:51:32.740 And once again, it may be something 00:51:32.740 --> 00:51:34.492 that tells us less about biology than it 00:51:34.492 --> 00:51:35.700 does about math or something. 00:51:44.940 --> 00:51:47.890 Any other questions about this, the base notion 00:51:47.890 --> 00:51:49.390 of this niche hierarchy type models? 00:51:54.510 --> 00:51:56.060 So I want to spend some time talking 00:51:56.060 --> 00:51:58.830 about this neutral theory in ecology. 00:51:58.830 --> 00:52:00.550 The math, in particular the derivation 00:52:00.550 --> 00:52:02.216 of this particular closed form solution, 00:52:02.216 --> 00:52:04.775 is not really so interesting or relevant. 00:52:04.775 --> 00:52:06.650 But I think it's very important to understand 00:52:06.650 --> 00:52:09.635 what the assumptions are in the model and maybe also something 00:52:09.635 --> 00:52:12.260 about the circumstances in which we think that it should apply. 00:52:25.530 --> 00:52:29.370 So the basic idea is that we have, what we hope, 00:52:29.370 --> 00:52:36.140 is some metacommunity that is large. 00:52:39.530 --> 00:52:40.900 And then we have an island. 00:52:40.900 --> 00:52:44.670 So this has to do with this theory of island biogeography. 00:52:44.670 --> 00:52:47.185 We have an island over here. 00:52:47.185 --> 00:52:51.540 And in the context of the nomenclature of this paper, 00:52:51.540 --> 00:52:54.249 they are some community size, size j here. 00:52:54.249 --> 00:52:56.165 This tells us about the number of individuals. 00:53:01.360 --> 00:53:04.830 And they're distributed across some number of species. 00:53:04.830 --> 00:53:08.710 Now, the neutral theory, the key thing 00:53:08.710 --> 00:53:12.260 is that we assume that all individuals are identical. 00:53:22.780 --> 00:53:26.490 And once again, it's not that the neutral theorists 00:53:26.490 --> 00:53:28.310 believe that this is true. 00:53:28.310 --> 00:53:31.314 It's that they think that it may be sufficient to explain 00:53:31.314 --> 00:53:32.605 the patterns that are observed. 00:53:40.755 --> 00:53:42.880 And when we say that all individuals are identical, 00:53:42.880 --> 00:53:46.470 what we mean is that the demographic parameters are 00:53:46.470 --> 00:53:49.586 the same, birth, death rates. 00:53:53.990 --> 00:53:56.460 And it's even a stronger assumption, in some ways, 00:53:56.460 --> 00:53:57.010 than that. 00:53:57.010 --> 00:53:59.010 It's assuming that the individuals are the same, 00:53:59.010 --> 00:54:02.360 the species are the same, and that there are no interactions 00:54:02.360 --> 00:54:03.560 within the species as well. 00:54:03.560 --> 00:54:07.880 So there's no Alley effect, or no specific competition. 00:54:07.880 --> 00:54:09.480 So the birth, death rates are going 00:54:09.480 --> 00:54:16.120 to be independent of everything, which is an amazingly 00:54:16.120 --> 00:54:16.917 parsimonious model. 00:54:16.917 --> 00:54:19.250 And it's kind of amazing you can get anything out of it. 00:54:24.040 --> 00:54:28.150 And then we have a migration rate m. 00:54:28.150 --> 00:54:29.740 It's either a rate or a probability, 00:54:29.740 --> 00:54:32.230 depending on how you think about it. 00:54:32.230 --> 00:54:35.490 Rate or probability m. 00:54:35.490 --> 00:54:42.330 And can somebody remind us how we handle that? 00:54:48.713 --> 00:54:51.168 AUDIENCE: Both just in a community? 00:54:51.168 --> 00:54:52.150 PROFESSOR: Yeah. 00:54:52.150 --> 00:54:54.114 AUDIENCE: At some probability that 00:54:54.114 --> 00:54:57.167 is proportional to the distribution of the species 00:54:57.167 --> 00:54:58.042 in the metacommunity? 00:55:00.790 --> 00:55:02.040 PROFESSOR: Yeah, that's right. 00:55:02.040 --> 00:55:03.531 AUDIENCE: --transfer an individual 00:55:03.531 --> 00:55:05.919 from the metacommunity to the island. 00:55:05.919 --> 00:55:06.710 PROFESSOR: Perfect. 00:55:06.710 --> 00:55:08.264 AUDIENCE: We do stick to the island 00:55:08.264 --> 00:55:09.820 to make sure that number of individuals. 00:55:09.820 --> 00:55:10.528 PROFESSOR: Right. 00:55:10.528 --> 00:55:15.150 So what we're going to do is we're basically 00:55:15.150 --> 00:55:19.570 going to pick a random individual, here, each cycle. 00:55:19.570 --> 00:55:21.440 This is kind of like a Moran process. 00:55:21.440 --> 00:55:23.320 We're going to pick an individual here. 00:55:23.320 --> 00:55:25.950 And we're going to kill him. 00:55:25.950 --> 00:55:30.740 And then what we're going to do is, with probability m, 00:55:30.740 --> 00:55:32.830 replace that individual with one member 00:55:32.830 --> 00:55:35.030 of the metacommunity at random. 00:55:35.030 --> 00:55:36.930 So the rate coming from here will 00:55:36.930 --> 00:55:40.380 be proportional to the species abundance in the metacommunity. 00:55:40.380 --> 00:55:43.430 And with a probability of 1 minus m, what we're going to do 00:55:43.430 --> 00:55:46.400 is we're going to replace that individual 00:55:46.400 --> 00:55:51.960 with another individual in the island. 00:55:51.960 --> 00:55:56.510 Now, the math kind of gets hairy and complicated. 00:55:56.510 --> 00:56:01.547 But the basic notion is really quite simple. 00:56:01.547 --> 00:56:03.130 You have a metacommunity distribution, 00:56:03.130 --> 00:56:04.980 which is going to end up being the so-called Fisher log 00:56:04.980 --> 00:56:06.030 series in this model. 00:56:13.780 --> 00:56:17.004 This describes the species abundance on the metacommunity. 00:56:17.004 --> 00:56:18.420 But then on the island, we're just 00:56:18.420 --> 00:56:21.100 going to assume that there's birth, death that occurs over 00:56:21.100 --> 00:56:22.210 here at some rate. 00:56:22.210 --> 00:56:25.205 But we don't even have to hardly think about that. 00:56:25.205 --> 00:56:27.330 From the standpoint of, say, a simulation or model, 00:56:27.330 --> 00:56:29.070 we just run multiple cycles of this, 00:56:29.070 --> 00:56:30.260 where we have j individuals. 00:56:30.260 --> 00:56:31.260 And we always have j individuals, 00:56:31.260 --> 00:56:32.830 because it's like the Moran process. 00:56:32.830 --> 00:56:35.066 At every time point, we kill one individual, 00:56:35.066 --> 00:56:37.690 and we replace it, with somebody either from the same community 00:56:37.690 --> 00:56:40.840 or from the island. 00:56:40.840 --> 00:56:47.539 And you can imagine that in the limit of m going to zero, 00:56:47.539 --> 00:56:49.080 what's going to happen on the island? 00:56:52.240 --> 00:56:54.120 Yeah, so you'll end up just one species, just 00:56:54.120 --> 00:56:56.980 because this is just random, like genetic drift. 00:56:56.980 --> 00:57:00.410 It's ecological drift where one species will take over. 00:57:00.410 --> 00:57:04.320 Whereas if m is large, then somehow it's 00:57:04.320 --> 00:57:07.405 more of a reflection of the metacommunity. 00:57:13.680 --> 00:57:15.830 Are there any questions about what this model 00:57:15.830 --> 00:57:18.098 is looking like for now? 00:57:18.098 --> 00:57:20.264 AUDIENCE: Could we talk about the Fisher log series? 00:57:20.264 --> 00:57:20.732 PROFESSOR: Yeah. 00:57:20.732 --> 00:57:22.606 AUDIENCE: So we would put it on the same axis 00:57:22.606 --> 00:57:23.540 as the [INAUDIBLE]? 00:57:23.540 --> 00:57:25.770 PROFESSOR: Yes, this is a very, very good question. 00:57:25.770 --> 00:57:27.319 So we'll do this in just a moment. 00:57:27.319 --> 00:57:28.610 Because this is very important. 00:57:32.210 --> 00:57:35.950 I want to say just a couple things about this model. 00:57:35.950 --> 00:57:37.750 So when I read this paper, what I imagined 00:57:37.750 --> 00:57:39.208 is that it really looked like this. 00:57:39.208 --> 00:57:44.890 This was Panama, and that, 30 kilometers off the coast, 00:57:44.890 --> 00:57:48.610 there was this island, BCI, Barro Colorado Island. 00:57:48.610 --> 00:57:51.560 But that's not maybe an accurate description 00:57:51.560 --> 00:57:54.660 of what the real system looks like. 00:57:54.660 --> 00:57:56.340 Does anybody know where BCI is? 00:57:56.340 --> 00:57:57.907 AUDIENCE: It's in Panama. 00:57:57.907 --> 00:57:58.490 PROFESSOR: Hm? 00:57:58.490 --> 00:57:59.350 AUDIENCE: Panama. 00:57:59.350 --> 00:58:00.600 PROFESSOR: So it is in Panama. 00:58:00.600 --> 00:58:02.430 But it's not off the coast of Panama. 00:58:02.430 --> 00:58:03.820 I guess that was my original. 00:58:03.820 --> 00:58:05.450 AUDIENCE: It's in the canal. 00:58:05.450 --> 00:58:07.050 PROFESSOR: Yeah, it's in the canal. 00:58:07.050 --> 00:58:09.400 So it's an island that was created when 00:58:09.400 --> 00:58:11.540 they made the Panama Canal. 00:58:11.540 --> 00:58:13.520 So this thing was not always an island. 00:58:13.520 --> 00:58:16.240 It's been an island for 100 years. 00:58:16.240 --> 00:58:17.959 And it's in the middle of a canal. 00:58:17.959 --> 00:58:20.250 And they actually have cougars that swim back and forth 00:58:20.250 --> 00:58:22.980 from the mainland. 00:58:22.980 --> 00:58:25.952 But it does make you wonder whether this 00:58:25.952 --> 00:58:32.390 is-- it's much more strongly coupled to the mainland 00:58:32.390 --> 00:58:36.670 then I imagined when I read this paper at first. 00:58:36.670 --> 00:58:39.390 I don't know what that means for all this. 00:58:39.390 --> 00:58:42.680 But certainly, you expect this to be 00:58:42.680 --> 00:58:44.880 a more or less appropriate model depending on this. 00:58:44.880 --> 00:58:46.810 Because, of course, if you went and you 00:58:46.810 --> 00:58:48.990 sampled 50 hectares here, you wouldn't 00:58:48.990 --> 00:58:51.440 believe that it should have the same distribution. 00:58:51.440 --> 00:58:54.706 You'd believe it should be more like the Fisher log series. 00:58:54.706 --> 00:58:57.080 And there's some evidence that things are tilted in a way 00:58:57.080 --> 00:58:57.920 that you would expect. 00:58:57.920 --> 00:58:59.003 And we'll talk about that. 00:59:01.890 --> 00:59:02.470 It's tricky. 00:59:02.470 --> 00:59:04.990 And of course, you have to decide in all this stuff, oh, 00:59:04.990 --> 00:59:07.480 what do you mean by free parameters? 00:59:07.480 --> 00:59:09.750 And actually, it seems like people can't count. 00:59:09.750 --> 00:59:11.541 And we'll talk about this in a moment, too. 00:59:14.680 --> 00:59:16.830 Because, of course, constructing the model, 00:59:16.830 --> 00:59:19.450 there's some sense of free parameters that you have there. 00:59:19.450 --> 00:59:21.360 Because we could have said, oh, it's 00:59:21.360 --> 00:59:23.140 just going to be the Fisher log series, or we could have said, 00:59:23.140 --> 00:59:23.880 oh, it's going to be island. 00:59:23.880 --> 00:59:25.296 Or we could have said, oh, there's 00:59:25.296 --> 00:59:26.660 another island out here. 00:59:26.660 --> 00:59:28.760 And then that would be another distribution. 00:59:28.760 --> 00:59:31.480 And not all of these things introduce more free parameters, 00:59:31.480 --> 00:59:32.550 necessarily, because you could say, 00:59:32.550 --> 00:59:34.049 oh, this is the same migration rate, 00:59:34.049 --> 00:59:35.240 or you could do something. 00:59:35.240 --> 00:59:37.770 But they are going to lead to different distributions, 00:59:37.770 --> 00:59:38.680 and you have that freedom when you're 00:59:38.680 --> 00:59:39.804 trying to explain the data. 00:59:42.180 --> 00:59:46.156 There are a lot of judgment calls in this business. 00:59:46.156 --> 00:59:47.780 But let's talk about Fisher log series, 00:59:47.780 --> 00:59:48.995 because this is relevant. 00:59:54.550 --> 00:59:58.680 So the model is very similar to what we did for the master 00:59:58.680 --> 01:00:01.290 equation in the context of gene expression 01:00:01.290 --> 01:00:03.930 and the number of mRNA. 01:00:03.930 --> 01:00:06.610 So was the equilibrium or steady state 01:00:06.610 --> 01:00:11.540 distribution of mRNA in a cell, was that a Fisher log series? 01:00:11.540 --> 01:00:14.420 Yes or no, five seconds? 01:00:17.060 --> 01:00:20.970 Was the mRNA steady state probability distribution 01:00:20.970 --> 01:00:21.990 a Fisher log series? 01:00:21.990 --> 01:00:24.550 Ready, three, two, one. 01:00:27.450 --> 01:00:28.030 No. 01:00:28.030 --> 01:00:28.529 No. 01:00:28.529 --> 01:00:29.415 What was it? 01:00:29.415 --> 01:00:30.870 It was a Poisson. 01:00:30.870 --> 01:00:34.420 And you guys should review what all these distributions are, 01:00:34.420 --> 01:00:35.790 when you get them, and so forth. 01:00:35.790 --> 01:00:37.300 So what was the Difference why is it 01:00:37.300 --> 01:00:48.770 that we have some probability, P0, P1, P2? 01:00:48.770 --> 01:00:50.490 This could be mRNA or it could be 01:00:50.490 --> 01:00:53.590 number of individuals in some species with some birth 01:00:53.590 --> 01:00:54.440 and death rates. 01:00:54.440 --> 01:00:58.070 What was the key difference between the mRNA model, which 01:00:58.070 --> 01:01:01.670 led to this distribution becoming Poisson, 01:01:01.670 --> 01:01:03.790 and the model that we just studied here, where 01:01:03.790 --> 01:01:05.080 it became a Fisher log series? 01:01:09.200 --> 01:01:13.020 And I should maybe write down what the Fisher log series is. 01:01:13.020 --> 01:01:17.740 So this is the expected number of species with n individuals 01:01:17.740 --> 01:01:19.069 on the metacommunity. 01:01:19.069 --> 01:01:20.360 Here is the Fisher log species. 01:01:20.360 --> 01:01:23.970 There was some theta X to the n divided by n. 01:01:28.782 --> 01:01:29.990 So what's the key difference? 01:01:29.990 --> 01:01:31.244 Yeah. 01:01:31.244 --> 01:01:36.711 AUDIENCE: I think that the birth and death rates are 01:01:36.711 --> 01:01:38.202 both proportional [INAUDIBLE]. 01:01:41.989 --> 01:01:43.780 PROFESSOR: Right, the birth and death rates 01:01:43.780 --> 01:01:44.855 are both proportional. 01:01:44.855 --> 01:01:46.330 AUDIENCE: In the Fisher log series. 01:01:46.330 --> 01:01:47.829 PROFESSOR: In the Fisher log series. 01:01:47.829 --> 01:01:51.630 So what we have is that b0-- and what should we 01:01:51.630 --> 01:01:54.544 call b0 in this model? 01:01:54.544 --> 01:01:56.055 AUDIENCE: [INAUDIBLE]. 01:01:56.055 --> 01:01:57.430 PROFESSOR: Well, right now, we're 01:01:57.430 --> 01:01:59.026 thinking about the metacommunity. 01:01:59.026 --> 01:01:59.900 AUDIENCE: Speciation. 01:01:59.900 --> 01:02:00.816 PROFESSOR: Speciation. 01:02:00.816 --> 01:02:03.870 b0 is speciation, which we're going to assume 01:02:03.870 --> 01:02:06.670 is going to be constant. 01:02:06.670 --> 01:02:11.070 In this model, do we have speciation on the island? 01:02:11.070 --> 01:02:12.046 No. 01:02:12.046 --> 01:02:13.420 The assumption is that the island 01:02:13.420 --> 01:02:16.280 is small enough that the rate of speciation is just negligible. 01:02:16.280 --> 01:02:20.164 So speciation plays a role in forming 01:02:20.164 --> 01:02:22.080 the metacommunity distribution, but it doesn't 01:02:22.080 --> 01:02:25.210 play a role in the model. 01:02:25.210 --> 01:02:27.280 So this is speciation. 01:02:27.280 --> 01:02:30.610 But then what we assume is that b1, here, 01:02:30.610 --> 01:02:35.180 is equal to some fundamental rate b times n, 01:02:35.180 --> 01:02:36.870 but it's b times, in this case, 1. 01:02:36.870 --> 01:02:43.840 So more broadly, bn is equal to some birth rate times n. 01:02:43.840 --> 01:02:46.290 This is saying that the individuals can give birth 01:02:46.290 --> 01:02:49.260 to other individuals. 01:02:49.260 --> 01:02:51.919 Now, we're not assuming anything about sexual reproduction 01:02:51.919 --> 01:02:52.710 necessarily or not. 01:02:52.710 --> 01:02:55.320 We're just saying that the kind of rates 01:02:55.320 --> 01:02:56.800 are proportional to the numbers. 01:02:56.800 --> 01:02:58.520 So if you have twice as many individuals, 01:02:58.520 --> 01:03:00.209 the birth rate will be twice as large. 01:03:00.209 --> 01:03:01.000 This is reasonable. 01:03:07.980 --> 01:03:12.250 This is Pn and this is Pn plus 1. 01:03:12.250 --> 01:03:17.200 So this is d of n plus 1 is equal to some death 01:03:17.200 --> 01:03:19.330 rate times n plus 1. 01:03:19.330 --> 01:03:21.800 So each individual just has some rate of dying. 01:03:21.800 --> 01:03:23.152 It's exponentially distributed. 01:03:23.152 --> 01:03:24.110 This again makes sense. 01:03:26.750 --> 01:03:28.990 What was the key difference between our mRNA model, 01:03:28.990 --> 01:03:30.406 from before that gave the Poisson, 01:03:30.406 --> 01:03:32.670 and this model that gives the Fisher log series? 01:03:32.670 --> 01:03:36.992 AUDIENCE: So with the mRNA, it's with a standard like a chemical 01:03:36.992 --> 01:03:41.952 equation where there's some fixed external input. 01:03:41.952 --> 01:03:44.680 But then the degradation is according 01:03:44.680 --> 01:03:45.920 to the amount that you have. 01:03:45.920 --> 01:03:48.420 So death is proportionate [INAUDIBLE]. 01:03:48.420 --> 01:03:49.790 PROFESSOR: Perfect. 01:03:49.790 --> 01:03:52.400 In both cases, the death rate is proportional to the number 01:03:52.400 --> 01:03:55.140 of either mRNA or individuals. 01:03:55.140 --> 01:03:57.140 However, in the mRNA model, what we assume 01:03:57.140 --> 01:03:59.840 is there some just constant rate of transcription, 01:03:59.840 --> 01:04:02.466 so a constant rate, per unit time, of making more mRNA. 01:04:02.466 --> 01:04:03.840 So just because there's more mRNA 01:04:03.840 --> 01:04:06.950 doesn't mean that you're going to get more mRNA. 01:04:06.950 --> 01:04:10.080 But here, we assume that the birth rate 01:04:10.080 --> 01:04:12.100 is proportional to the number. 01:04:12.100 --> 01:04:13.982 So that's what leads to the difference. 01:04:13.982 --> 01:04:15.690 And so this is one of the few other cases 01:04:15.690 --> 01:04:18.060 that you can simply solve the master equation 01:04:18.060 --> 01:04:19.692 and get an equilibrium distribution. 01:04:19.692 --> 01:04:21.650 And it's the same thing we do from just always, 01:04:21.650 --> 01:04:26.840 where we say, at steady state, the probability fluxes 01:04:26.840 --> 01:04:28.800 or whatever are equal. 01:04:28.800 --> 01:04:32.510 So you get that P1 should be equal to P0. 01:04:32.510 --> 01:04:35.316 and then we have a b0 divided by d1. 01:04:35.316 --> 01:04:39.130 And more broadly, we just cycle through. 01:04:39.130 --> 01:04:41.400 The probability of being in the nth state, 01:04:41.400 --> 01:04:43.060 it's going to be some P0. 01:04:43.060 --> 01:04:48.380 And then basically, it's going to b0 divided by d1, 01:04:48.380 --> 01:04:58.470 b1 divided by d2, b2, d3, dot, dot, dot, up to bn minus 1 dn. 01:04:58.470 --> 01:05:01.780 And indeed, if we just plug in what these things are equal to, 01:05:01.780 --> 01:05:06.650 we end up getting-- there's P0, the fundamental birth 01:05:06.650 --> 01:05:08.191 over death to the nth power. 01:05:08.191 --> 01:05:09.940 And then we just are left with a 1 over n. 01:05:09.940 --> 01:05:13.170 Because we're going to have a 2 here and a 2 here, and those 01:05:13.170 --> 01:05:13.670 cancel. 01:05:13.670 --> 01:05:15.370 A 2 here and 3 here, and those cancel. 01:05:15.370 --> 01:05:20.130 And we're just left with the n at the end, finally. 01:05:20.130 --> 01:05:24.470 So this x, over there, is then, in this model, 01:05:24.470 --> 01:05:27.920 the ratio of the birth and death rates. 01:05:27.920 --> 01:05:29.890 So which one is larger? 01:05:29.890 --> 01:05:32.330 Is it A slash 1? 01:05:32.330 --> 01:05:35.270 Is it b is greater than d? 01:05:35.270 --> 01:05:39.844 Or is it b slash 2, that b is less than d? 01:05:39.844 --> 01:05:41.260 Think about this for five seconds. 01:05:43.770 --> 01:05:47.002 Do you think that birth rates should 01:05:47.002 --> 01:05:48.710 be larger than death rates or death rates 01:05:48.710 --> 01:05:51.168 should be larger than birth rates or do they have to equal? 01:05:54.990 --> 01:05:57.940 Ready, three, two, one. 01:06:00.460 --> 01:06:07.780 So we got a number of-- it's kind of distributed, 1 and 2's. 01:06:07.780 --> 01:06:10.322 Well, it's maybe not that deep, not deep enough. 01:06:10.322 --> 01:06:11.780 Can somebody say why their neighbor 01:06:11.780 --> 01:06:12.988 thinks it's one or the other? 01:06:15.374 --> 01:06:17.290 People are actually turning to their neighbor. 01:06:20.720 --> 01:06:22.336 A justification for one or the other. 01:06:22.336 --> 01:06:26.144 AUDIENCE: So if this problem where b over d 01:06:26.144 --> 01:06:30.242 is greater than 1, then this distribution is not normalized. 01:06:30.242 --> 01:06:30.950 PROFESSOR: Right. 01:06:30.950 --> 01:06:35.380 So if b over d is greater than 1, so if x is greater than 1, 01:06:35.380 --> 01:06:39.480 then this distribution blows up. 01:06:39.480 --> 01:06:41.230 Then it gets more and more likely to have 01:06:41.230 --> 01:06:44.240 all these larger numbers. 01:06:44.240 --> 01:06:48.510 But then if b is less than d, shouldn't everybody be extinct? 01:06:48.510 --> 01:06:50.000 No. 01:06:50.000 --> 01:06:52.640 Can somebody else say why it is that it's OK 01:06:52.640 --> 01:06:53.640 for b to be less than d? 01:06:53.640 --> 01:06:55.348 If birth rates are less than death rates, 01:06:55.348 --> 01:06:56.681 shouldn't everyone be extinct? 01:06:56.681 --> 01:06:59.140 AUDIENCE: Because there's a rate b0. 01:06:59.140 --> 01:07:01.350 PROFESSOR: Because there's a rate b0, exactly. 01:07:01.350 --> 01:07:03.250 So there's a finite rate of speciation. 01:07:03.250 --> 01:07:05.505 So it's true that every species will go extinct. 01:07:08.070 --> 01:07:12.050 But because we have a constant influx of new species, 01:07:12.050 --> 01:07:16.170 we end up with this distribution that's this Fisher log series. 01:07:16.170 --> 01:07:21.230 Now, if you plot the Fisher log series, 01:07:21.230 --> 01:07:23.690 it looks a bit like this. 01:07:23.690 --> 01:07:25.990 But let's think about it a little bit. 01:07:25.990 --> 01:07:28.540 Does the Fisher log series, does it fall off, 01:07:28.540 --> 01:07:32.200 A, faster or slower than this? 01:07:32.200 --> 01:07:41.580 Fisher falls, A, faster-- this is in this direction-- or, B, 01:07:41.580 --> 01:07:42.080 slower? 01:07:48.519 --> 01:07:50.060 AUDIENCE: Faster or slower than what? 01:07:50.060 --> 01:07:54.300 PROFESSOR: Than the island distribution. 01:07:54.300 --> 01:07:56.640 Because you can see that this falls off pretty rapidly. 01:08:01.180 --> 01:08:04.250 Ready, maybe? 01:08:04.250 --> 01:08:06.575 Three, two, one. 01:08:14.850 --> 01:08:18.452 I saw a fair number of people that 01:08:18.452 --> 01:08:20.380 don't want to make a guess. 01:08:20.380 --> 01:08:21.810 Indeed, it's going to be faster. 01:08:21.810 --> 01:08:22.765 Can somebody say why? 01:08:26.639 --> 01:08:27.139 Yeah. 01:08:27.139 --> 01:08:28.055 AUDIENCE: [INAUDIBLE]. 01:08:30.361 --> 01:08:32.569 PROFESSOR: Is it going to be because of the 1 over n? 01:08:32.569 --> 01:08:34.460 I mean the 1 over n is certainly relevant. 01:08:37.399 --> 01:08:38.899 Without the one over n, then we just 01:08:38.899 --> 01:08:40.930 have sort of a geometric series. 01:08:40.930 --> 01:08:43.550 And the log normal is not just a geometric series either. 01:08:47.382 --> 01:08:53.590 AUDIENCE: [INAUDIBLE] Whereas this has a very long tail. 01:08:53.590 --> 01:08:54.590 PROFESSOR: That's right. 01:08:54.590 --> 01:08:55.649 So this falls off. 01:08:58.189 --> 01:08:59.689 This would be kind of exponentially, 01:08:59.689 --> 01:09:01.272 and this is faster than exponentially. 01:09:03.420 --> 01:09:07.600 And indeed, this make sense based on the model. 01:09:07.600 --> 01:09:10.840 Because this community, the reason 01:09:10.840 --> 01:09:15.920 that it has some very, very abundant species 01:09:15.920 --> 01:09:18.090 is partly because it gets migration 01:09:18.090 --> 01:09:19.479 from the abundant species here. 01:09:22.680 --> 01:09:26.200 This falls off pretty quickly. 01:09:26.200 --> 01:09:30.450 But those frequent species still can play a pretty important 01:09:30.450 --> 01:09:33.859 role in the island community, because the migration rate 01:09:33.859 --> 01:09:38.924 is influenced by large numbers. 01:09:38.924 --> 01:09:40.340 And the other thing is, of course, 01:09:40.340 --> 01:09:43.890 that the rare species are going to often go extinct. 01:09:43.890 --> 01:09:45.880 I mean the distribution on the island 01:09:45.880 --> 01:09:50.330 is some complicated process of the dynamics going here, 01:09:50.330 --> 01:09:51.800 plus sampling from here. 01:09:51.800 --> 01:09:54.480 But there's a sense that it's biased towards-- it's not just 01:09:54.480 --> 01:09:56.690 a reflection of the metacommunity, 01:09:56.690 --> 01:09:58.430 because the migration rate is sampled 01:09:58.430 --> 01:09:59.830 towards the abundant species. 01:09:59.830 --> 01:10:02.620 So the migration of these species 01:10:02.620 --> 01:10:05.980 ends up playing a major role in pushing the distribution 01:10:05.980 --> 01:10:08.390 to the right. 01:10:08.390 --> 01:10:14.500 So you have much more frequent, abundant species on the island 01:10:14.500 --> 01:10:18.205 as compared to the mainland. 01:10:18.205 --> 01:10:19.167 AUDIENCE: [INAUDIBLE]? 01:10:19.167 --> 01:10:19.833 PROFESSOR: Yeah. 01:10:19.833 --> 01:10:23.015 AUDIENCE: [INAUDIBLE] measurement of the distribution 01:10:23.015 --> 01:10:24.940 on the-- 01:10:24.940 --> 01:10:26.500 PROFESSOR: Well, I'm sure they have. 01:10:29.810 --> 01:10:32.450 I think the statement that there's a faster fall 01:10:32.450 --> 01:10:34.120 off on mainlands than on the islands 01:10:34.120 --> 01:10:37.580 I think is borne out by the data. 01:10:37.580 --> 01:10:42.730 But I don't know if trees on the Panama side of the canal 01:10:42.730 --> 01:10:45.050 are actually better described by a Fisher log series 01:10:45.050 --> 01:10:47.215 as compared to this, though. 01:10:47.215 --> 01:10:50.185 AUDIENCE: I guess my question was the abundant species 01:10:50.185 --> 01:10:52.907 that we see on the island, is it just 01:10:52.907 --> 01:10:57.120 the result of diffusive drift? 01:10:57.120 --> 01:11:02.630 PROFESSOR: Well, this also has the diffusive drift. 01:11:08.398 --> 01:11:12.462 AUDIENCE: But in the sense that what really pushes. 01:11:12.462 --> 01:11:13.920 PROFESSOR: Well, I mean I think you 01:11:13.920 --> 01:11:16.705 need both, the diffusive drift and the migration. 01:11:16.705 --> 01:11:19.010 But I think that the fact that the migration is 01:11:19.010 --> 01:11:20.720 from the mainland, and it's biased 01:11:20.720 --> 01:11:24.190 towards those abundant things, I think 01:11:24.190 --> 01:11:26.849 is necessary or important. 01:11:26.849 --> 01:11:28.890 AUDIENCE: I guess just in terms of distinguishing 01:11:28.890 --> 01:11:33.133 between the niche and the neutral models, 01:11:33.133 --> 01:11:39.733 as applied to the mainland, does the niche model predict also 01:11:39.733 --> 01:11:40.274 a log normal? 01:11:40.274 --> 01:11:42.644 Because it seemed like, in the discussion earlier, 01:11:42.644 --> 01:11:47.584 the neutral also predicted log normal [INAUDIBLE]. 01:11:47.584 --> 01:11:49.000 PROFESSOR: That's a good question. 01:11:54.200 --> 01:11:58.560 In this whole area, I mean it's a little bit empirical. 01:11:58.560 --> 01:12:01.490 The fact that the niche model kind of predicts 01:12:01.490 --> 01:12:05.790 this, or this broken stick thing predicts a log normal, 01:12:05.790 --> 01:12:08.540 they didn't say anything about islands there, right? 01:12:08.540 --> 01:12:11.630 I guess even Fisher's original log series, 01:12:11.630 --> 01:12:14.752 he used it to describe-- I think maybe 01:12:14.752 --> 01:12:16.210 that was the beetles on the Thames. 01:12:16.210 --> 01:12:19.790 But his original data set, where the Fisher log series 01:12:19.790 --> 01:12:21.970 was supposed to described it, as it 01:12:21.970 --> 01:12:24.060 was sampled better and better, it eventually 01:12:24.060 --> 01:12:26.393 started looking more and more like a log normal anyways. 01:12:29.316 --> 01:12:31.190 I mean it's easy to see the frequent species, 01:12:31.190 --> 01:12:34.560 because you see them. 01:12:34.560 --> 01:12:36.930 This tail can actually be very hard to see, 01:12:36.930 --> 01:12:40.145 because you have to find the individuals. 01:12:43.770 --> 01:12:46.550 It's a good question of to what degree each of the models 01:12:46.550 --> 01:12:49.580 really predicts one thing on one place and another. 01:12:49.580 --> 01:12:53.450 There's always tweaks of each model that adjust things. 01:12:53.450 --> 01:12:56.010 So I think it's a bit muddy. 01:13:01.070 --> 01:13:03.240 But the one thing that I want to highlight. 01:13:03.240 --> 01:13:05.420 So there's a lot of debates, then, 01:13:05.420 --> 01:13:07.150 between these different models. 01:13:07.150 --> 01:13:09.260 And each of the models have some fit. 01:13:09.260 --> 01:13:12.180 They have red and black. 01:13:12.180 --> 01:13:13.860 There's one that kind of goes like this. 01:13:13.860 --> 01:13:15.820 And another one that kind of goes like that. 01:13:15.820 --> 01:13:20.410 And they're not labeled, because they look the same. 01:13:20.410 --> 01:13:23.990 And you can argue about chi squareds and everything, 01:13:23.990 --> 01:13:26.290 but I think it's irrelevant. 01:13:26.290 --> 01:13:28.490 They both fit the data fine. 01:13:28.490 --> 01:13:31.670 And the other thing, just the sampling of kind 01:13:31.670 --> 01:13:35.710 root n sampling, if you expect to see 10 species, 01:13:35.710 --> 01:13:38.355 then if you go and you actually do sampling, 01:13:38.355 --> 01:13:41.330 you expect to have kind of a root n on each one. 01:13:41.330 --> 01:13:44.110 I mean the error bars, I think, around this 01:13:44.110 --> 01:13:46.540 are consistent with both models. 01:13:46.540 --> 01:13:49.259 So I'd say that the exercise of trying 01:13:49.259 --> 01:13:51.550 to distinguish those models based on fit to such a data 01:13:51.550 --> 01:13:55.760 set I think is hopeless from the beginning. 01:13:55.760 --> 01:13:58.640 And then you can talk about the number of parameters. 01:13:58.640 --> 01:14:00.350 And if you read these two papers, 01:14:00.350 --> 01:14:02.124 they both say that they have fewer number 01:14:02.124 --> 01:14:02.915 of free parameters. 01:14:05.610 --> 01:14:07.862 And it is hard to believe that there could 01:14:07.862 --> 01:14:09.070 be a disagreement about this. 01:14:12.534 --> 01:14:14.200 But then, you know, it's like, oh, well, 01:14:14.200 --> 01:14:16.080 what do you call a free parameter? 01:14:16.080 --> 01:14:19.950 And then what they say, any given RSA data set contains 01:14:19.950 --> 01:14:22.110 information about the local community size j. 01:14:24.790 --> 01:14:27.155 So they say, given that, it's not a free parameter, 01:14:27.155 --> 01:14:28.705 because you put that in. 01:14:28.705 --> 01:14:30.080 That's the number of individuals. 01:14:30.080 --> 01:14:32.093 And then outcome is your distribution, right? 01:14:32.093 --> 01:14:33.800 And you say, OK, well, all right, that's 01:14:33.800 --> 01:14:36.008 fine if you don't want to call that a free parameter. 01:14:36.008 --> 01:14:39.510 But then when you fit the log normal to this distribution, 01:14:39.510 --> 01:14:42.720 the overall amplitude is also to give you 01:14:42.720 --> 01:14:46.280 the number of individuals in the metacommunity. 01:14:46.280 --> 01:14:50.880 So if you don't call j a free parameter in this model, 01:14:50.880 --> 01:14:53.840 then you can't call the amplitude a free parameter when 01:14:53.840 --> 01:14:57.970 you fit the log normal, at least in my opinion. 01:14:57.970 --> 01:15:01.540 I think that they both have three. 01:15:01.540 --> 01:15:04.570 Because if you fit a log normal to this, 01:15:04.570 --> 01:15:05.890 you have the overall amplitude. 01:15:05.890 --> 01:15:07.300 That's the number of individuals. 01:15:07.300 --> 01:15:11.350 And then you have the mean and the standard deviation 01:15:11.350 --> 01:15:14.089 or whatever. 01:15:14.089 --> 01:15:16.047 From that standpoint, I think they're the same. 01:15:16.047 --> 01:15:16.547 Yeah. 01:15:16.547 --> 01:15:18.909 AUDIENCE: But I mean how do you fit the log normal 01:15:18.909 --> 01:15:19.863 when you don't impose? 01:15:19.863 --> 01:15:21.294 Do they impose the amplitude? 01:15:24.530 --> 01:15:26.400 I mean it's still a parameter. 01:15:26.400 --> 01:15:26.890 PROFESSOR: No, that's what I was saying. 01:15:26.890 --> 01:15:27.650 It's a parameter. 01:15:27.650 --> 01:15:30.302 I mean the normalized log normal, you integrate, 01:15:30.302 --> 01:15:31.010 and it goes to 1. 01:15:31.010 --> 01:15:32.600 But then you have some measured number 01:15:32.600 --> 01:15:35.020 of individuals in your sample, and then you 01:15:35.020 --> 01:15:37.650 have to multiply by that to give you. 01:15:37.650 --> 01:15:40.337 AUDIENCE: But is that what they do when they do their fit? 01:15:40.337 --> 01:15:41.003 PROFESSOR: Yeah. 01:15:41.003 --> 01:15:43.544 AUDIENCE: Or do they keep that amplitude as also a parameter? 01:15:46.040 --> 01:15:48.320 PROFESSOR: I think that you can argue whether this 01:15:48.320 --> 01:15:49.540 is a free parameter or not. 01:15:49.540 --> 01:15:51.440 But I think that you can just put it 01:15:51.440 --> 01:15:54.149 as the number of individuals, and it's not 01:15:54.149 --> 01:15:55.190 going to affect anything. 01:15:55.190 --> 01:15:58.330 You could actually have it be a free. 01:15:58.330 --> 01:16:00.790 But this gets into this question about what constitutes 01:16:00.790 --> 01:16:02.230 a free parameter or not. 01:16:02.230 --> 01:16:04.740 And actually, there is some subtlety to this. 01:16:04.740 --> 01:16:09.664 But I think, at the end of the day, 01:16:09.664 --> 01:16:11.580 the log normal is not going to look like this. 01:16:11.580 --> 01:16:12.840 You have to. 01:16:12.840 --> 01:16:14.840 You basically put in the number of individuals 01:16:14.840 --> 01:16:15.880 that you measured. 01:16:15.880 --> 01:16:17.840 AUDIENCE: So when you calculate [INAUDIBLE]? 01:16:23.750 --> 01:16:25.380 PROFESSOR: Huge numbers of pages of 01:16:25.380 --> 01:16:27.602 has been written about comparing these things. 01:16:27.602 --> 01:16:30.060 At some point, it comes down to this philosophical question 01:16:30.060 --> 01:16:32.300 about what you think constitutes a null model. 01:16:32.300 --> 01:16:33.990 And this gets to be much more subtle. 01:16:33.990 --> 01:16:37.200 And I think reasonable people can disagree 01:16:37.200 --> 01:16:39.990 about whether the null model that you need to reject 01:16:39.990 --> 01:16:41.800 should be this neutral model or if it 01:16:41.800 --> 01:16:43.160 should be a niche-based model. 01:16:43.160 --> 01:16:45.997 Or maybe it's just that there's some multiplicative type 01:16:45.997 --> 01:16:48.330 process that's going on and gives you distributions that 01:16:48.330 --> 01:16:50.790 look like this, and you need other kinds of information 01:16:50.790 --> 01:16:52.260 to try to distinguish those things. 01:16:52.260 --> 01:16:54.051 And in particular, I'd say that it's really 01:16:54.051 --> 01:16:56.940 the dynamic information in which these models have strikingly 01:16:56.940 --> 01:16:58.398 different predictions, and then you 01:16:58.398 --> 01:17:00.820 can reject neutral-type models. 01:17:00.820 --> 01:17:03.160 Because that neutral models predict 01:17:03.160 --> 01:17:04.820 that these species that are abundant 01:17:04.820 --> 01:17:07.380 are just transiently abundant, and they should go way. 01:17:07.380 --> 01:17:08.630 Whereas the niche-based models would say, 01:17:08.630 --> 01:17:09.760 oh, they're really fixed. 01:17:09.760 --> 01:17:13.324 And indeed, in many cases, the abundant species kind of 01:17:13.324 --> 01:17:14.740 stick around longer than you would 01:17:14.740 --> 01:17:16.320 expect from a neutral model. 01:17:16.320 --> 01:17:19.692 Of course, the neutral model is not true in the sense 01:17:19.692 --> 01:17:21.400 that different individuals are different. 01:17:21.400 --> 01:17:24.500 But it's important to highlight that even 01:17:24.500 --> 01:17:27.210 such a minimal model can give you 01:17:27.210 --> 01:17:29.220 striking patterns that are similar to what 01:17:29.220 --> 01:17:30.756 you observe in nature. 01:17:30.756 --> 01:17:32.130 And so I think we're out of time. 01:17:32.130 --> 01:17:33.560 So with that, I think we'll quit. 01:17:33.560 --> 01:17:36.250 But it's been a pleasure having you guys for this semester. 01:17:36.250 --> 01:17:38.870 And if you have any questions about any systems 01:17:38.870 --> 01:17:41.460 biology things in the future, please, email me. 01:17:41.460 --> 01:17:42.460 I'm happy to meet up. 01:17:42.460 --> 01:17:44.203 Good luck on the final.