1 00:00:00,080 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high quality, educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:20,900 --> 00:00:22,400 PROFESSOR: Today, what we want to do 9 00:00:22,400 --> 00:00:25,720 is talk about something at a much higher scale than what 10 00:00:25,720 --> 00:00:28,100 we've thought about through most of this semester. 11 00:00:28,100 --> 00:00:30,642 And that's probably by design. 12 00:00:30,642 --> 00:00:32,100 Over the course of the semester, we 13 00:00:32,100 --> 00:00:35,300 started with kind of enzyme kinetics 14 00:00:35,300 --> 00:00:37,220 or molecular binding kind of events, 15 00:00:37,220 --> 00:00:40,680 and we slowly built our way up the larger and larger scales. 16 00:00:40,680 --> 00:00:43,110 Now there's always this question about whether we're 17 00:00:43,110 --> 00:00:46,820 claiming that we really understand how the higher 18 00:00:46,820 --> 00:00:49,974 levels of organization result from the lower level 19 00:00:49,974 --> 00:00:50,515 interactions. 20 00:00:50,515 --> 00:00:54,070 And I'd say, we definitely don't understand all of it. 21 00:00:54,070 --> 00:00:56,990 So you shouldn't come away with that as the notion. 22 00:00:56,990 --> 00:00:59,100 But at least one thing that I think 23 00:00:59,100 --> 00:01:01,690 is fascinating about this area of systems biology 24 00:01:01,690 --> 00:01:05,129 is that much of the framework that we use to understand, 25 00:01:05,129 --> 00:01:08,220 let's say, molecular scale interactions or stochastic gene 26 00:01:08,220 --> 00:01:11,420 expression, so these dynamics at the smaller scale, 27 00:01:11,420 --> 00:01:14,304 much of those ideas and such certainly transport up 28 00:01:14,304 --> 00:01:16,470 to these higher scales or translate up to the higher 29 00:01:16,470 --> 00:01:17,990 scales, where, in this case, we're 30 00:01:17,990 --> 00:01:19,900 using kind of master equation type formulas 31 00:01:19,900 --> 00:01:22,314 to try to understand relative species abundance. 32 00:01:22,314 --> 00:01:23,730 And so I think part of what I like 33 00:01:23,730 --> 00:01:28,420 about this topic of neutral theory versus niche theory 34 00:01:28,420 --> 00:01:30,780 and so forth in ecology is that you can just 35 00:01:30,780 --> 00:01:34,530 see how very, very similar ideas, that we applied 36 00:01:34,530 --> 00:01:36,400 for studying stochastic gene expression, 37 00:01:36,400 --> 00:01:38,358 can also be used to try to understand why it is 38 00:01:38,358 --> 00:01:41,800 that some species are more common than others when you go 39 00:01:41,800 --> 00:01:46,610 and you count them, in this case, on an island in Panama. 40 00:01:46,610 --> 00:01:50,230 Now, the subject is, by its nature, 41 00:01:50,230 --> 00:01:52,840 less experimentally focused than much 42 00:01:52,840 --> 00:01:54,840 of what we've done over the course the semester. 43 00:01:54,840 --> 00:01:56,680 And this is really a topic the tends 44 00:01:56,680 --> 00:02:00,200 to be a combination of mathematical theory 45 00:02:00,200 --> 00:02:04,680 with kind of careful counting of species in some different areas 46 00:02:04,680 --> 00:02:07,199 and trying to understand what that means. 47 00:02:07,199 --> 00:02:08,740 But it's an area that there have been 48 00:02:08,740 --> 00:02:11,880 a number of physicists involved in over the last 10 years. 49 00:02:11,880 --> 00:02:13,690 And I think that it's fascinating, 50 00:02:13,690 --> 00:02:17,010 because it does get to the heart of what we are looking 51 00:02:17,010 --> 00:02:19,530 for from a theory, what kind of evidence 52 00:02:19,530 --> 00:02:23,040 do we use to support a theory or to refute it. 53 00:02:23,040 --> 00:02:27,272 So I think there are a lot of very basic issues about science 54 00:02:27,272 --> 00:02:28,730 that come up when we start thinking 55 00:02:28,730 --> 00:02:31,630 about this question of neutral theory in ecology. 56 00:02:31,630 --> 00:02:34,832 And since it's, for many of us, a totally new area 57 00:02:34,832 --> 00:02:36,290 that we don't know very much about, 58 00:02:36,290 --> 00:02:40,570 you can come to it with maybe fresh eyes. 59 00:02:40,570 --> 00:02:42,320 And you don't have the same preconceptions 60 00:02:42,320 --> 00:02:44,981 that you would have for many other models that you might 61 00:02:44,981 --> 00:02:47,230 be more familiar with in the context of molecular cell 62 00:02:47,230 --> 00:02:47,730 biology. 63 00:02:50,410 --> 00:02:53,640 So the basic question that we're going 64 00:02:53,640 --> 00:02:56,610 to try to talk about today is just 65 00:02:56,610 --> 00:02:59,950 the question of why is it that, when you look out at the world, 66 00:02:59,950 --> 00:03:02,510 you see that there are some species that 67 00:03:02,510 --> 00:03:06,850 seem to be abundant and some that seem to be rare? 68 00:03:06,850 --> 00:03:10,350 Are there other patterns that are somehow universal? 69 00:03:10,350 --> 00:03:14,010 And what kind of sort of lower scale processes 70 00:03:14,010 --> 00:03:17,390 might lead to the patterns that we observe? 71 00:03:17,390 --> 00:03:22,220 And I think that this paper that we read is-- I mean, 72 00:03:22,220 --> 00:03:24,510 it's not that it's. 73 00:03:24,510 --> 00:03:30,930 Well, can somebody say what the actual scientific contribution 74 00:03:30,930 --> 00:03:33,801 of this paper was? 75 00:03:33,801 --> 00:03:34,300 Yes? 76 00:03:34,300 --> 00:03:35,330 AUDIENCE: They did a calculation. 77 00:03:35,330 --> 00:03:36,940 PROFESSOR: They did a calculation. 78 00:03:36,940 --> 00:03:39,350 But it's a little bit more specific than that. 79 00:03:39,350 --> 00:03:39,850 What is it? 80 00:03:39,850 --> 00:03:43,160 AUDIENCE: They came up with the closed form equation? 81 00:03:43,160 --> 00:03:44,160 PROFESSOR: That's right. 82 00:03:44,160 --> 00:03:47,230 Basically, there was a model of this neutral theory in ecology 83 00:03:47,230 --> 00:03:49,920 that we're going to explain or try to understand. 84 00:03:49,920 --> 00:03:51,680 You can simulate the model, but then there 85 00:03:51,680 --> 00:03:54,700 are possible issues associated with convergence or something 86 00:03:54,700 --> 00:03:55,410 of those. 87 00:03:55,410 --> 00:03:56,826 Although it's hard to believe that 88 00:03:56,826 --> 00:03:58,200 that's really such a concern. 89 00:03:58,200 --> 00:03:59,970 But you can simulate that model. 90 00:03:59,970 --> 00:04:01,680 What they did is they just showed 91 00:04:01,680 --> 00:04:05,919 that you could get an analytic-y kind of expression for it. 92 00:04:05,919 --> 00:04:07,460 It's not a super analytic expression, 93 00:04:07,460 --> 00:04:09,920 but, at least, it's not a straight up simulation. 94 00:04:09,920 --> 00:04:11,892 You kind of numerically do something, 95 00:04:11,892 --> 00:04:13,600 integrate something, as compared to doing 96 00:04:13,600 --> 00:04:14,683 the stochastic simulation. 97 00:04:17,844 --> 00:04:19,510 So it's not that that, in and of itself, 98 00:04:19,510 --> 00:04:23,469 is what you feel like-- it's not what we necessarily 99 00:04:23,469 --> 00:04:24,260 care so much about. 100 00:04:24,260 --> 00:04:28,880 But I think that it's still just a nice, short description 101 00:04:28,880 --> 00:04:31,681 of the model and the assumptions that go into it. 102 00:04:31,681 --> 00:04:33,180 And you get a little bit of a window 103 00:04:33,180 --> 00:04:34,990 into the debate that's going on between these two 104 00:04:34,990 --> 00:04:36,910 communities of kind of the neutral theory 105 00:04:36,910 --> 00:04:39,780 guys and the niche theory community. 106 00:04:45,650 --> 00:04:48,080 So there's only one figure in this paper. 107 00:04:48,080 --> 00:04:50,880 And it's an example of the kind of data 108 00:04:50,880 --> 00:04:53,110 that we want to try to understand. 109 00:04:53,110 --> 00:04:55,330 So there's a particular pattern in terms 110 00:04:55,330 --> 00:04:57,400 of the relative species abundance. 111 00:04:57,400 --> 00:05:00,150 And we want to understand what kind of models 112 00:05:00,150 --> 00:05:03,280 might lead to that observed pattern. 113 00:05:03,280 --> 00:05:05,940 But given that there's just one figure in the paper, 114 00:05:05,940 --> 00:05:07,640 we have to make sure that we understand 115 00:05:07,640 --> 00:05:09,560 exactly what is being plotted. 116 00:05:09,560 --> 00:05:12,210 And what I've found from experience-- 117 00:05:12,210 --> 00:05:17,000 and, actually, even the answer to the email question 118 00:05:17,000 --> 00:05:19,514 that was sent out, I think, was incorrect on one 119 00:05:19,514 --> 00:05:20,180 of these things. 120 00:05:20,180 --> 00:05:21,880 So we'll talk about that some more. 121 00:05:21,880 --> 00:05:24,587 So beware. 122 00:05:24,587 --> 00:05:25,420 We'll figure it out. 123 00:05:25,420 --> 00:05:28,040 But I think it's actually surprisingly tricky 124 00:05:28,040 --> 00:05:32,560 to understand what this figure is saying. 125 00:05:32,560 --> 00:05:36,020 But first of all, can somebody describe not 126 00:05:36,020 --> 00:05:38,620 what the figure is saying but just what 127 00:05:38,620 --> 00:05:40,160 the data is supposed to be? 128 00:05:56,550 --> 00:05:59,640 Where do they get the data? 129 00:05:59,640 --> 00:06:02,896 Anything that's useful? 130 00:06:02,896 --> 00:06:05,692 AUDIENCE: They were on an island ecosystem. 131 00:06:05,692 --> 00:06:06,900 PROFESSOR: There's an island. 132 00:06:06,900 --> 00:06:12,752 It's called BCI, Barro Colorado Island. 133 00:06:12,752 --> 00:06:13,668 AUDIENCE: [INAUDIBLE]. 134 00:06:25,080 --> 00:06:27,015 PROFESSOR: So it's a 50 hectare plot. 135 00:06:31,380 --> 00:06:33,504 Does anybody know what a hectare is? 136 00:06:33,504 --> 00:06:35,420 AUDIENCE: It's a lot more than a square meter. 137 00:06:35,420 --> 00:06:39,550 PROFESSOR: It's a lot more than a square meter, yes, indeed. 138 00:06:39,550 --> 00:06:40,200 Yeah. 139 00:06:40,200 --> 00:06:44,921 Is this an English unit of measure? 140 00:06:44,921 --> 00:06:46,920 This is the kind of thing that I have to Google. 141 00:06:46,920 --> 00:06:50,970 But it's one hectare is equal to 10 to the 4 meters squared. 142 00:06:54,032 --> 00:06:55,365 That's a good thing to memorize. 143 00:06:59,041 --> 00:06:59,540 I 144 00:06:59,540 --> 00:07:02,180 AUDIENCE: Exactly or approximate? 145 00:07:02,180 --> 00:07:03,430 PROFESSOR: I think it's exact. 146 00:07:03,430 --> 00:07:04,680 I think I think it's an exact. 147 00:07:04,680 --> 00:07:06,805 AUDIENCE: Then it's a metric unit. 148 00:07:06,805 --> 00:07:08,930 PROFESSOR: Yeah, so apparently it is a metric unit. 149 00:07:08,930 --> 00:07:12,150 So the idea is that if you take a 100 meters by 100 meters, 150 00:07:12,150 --> 00:07:13,070 this is a hectare. 151 00:07:13,070 --> 00:07:15,080 And there's 50 of them. 152 00:07:15,080 --> 00:07:17,435 It's about like a half a square kilometer 153 00:07:17,435 --> 00:07:21,110 to give you a sense of what we're talking about. 154 00:07:21,110 --> 00:07:23,331 And what do they do on this plot? 155 00:07:23,331 --> 00:07:28,930 AUDIENCE: They count a certain number as canopy trees. 156 00:07:28,930 --> 00:07:32,209 So the trees that are, like, really big. 157 00:07:32,209 --> 00:07:34,500 PROFESSOR: And how do they decide which trees to count? 158 00:07:34,500 --> 00:07:36,190 Did they count every tree? 159 00:07:36,190 --> 00:07:40,890 AUDIENCE: No, just the ones that like formed the top layer. 160 00:07:40,890 --> 00:07:44,987 PROFESSOR: I think that the way that they decide-- OK. 161 00:07:44,987 --> 00:07:47,070 Does anybody remember how many trees were counted? 162 00:07:50,283 --> 00:07:51,660 AUDIENCE: [INAUDIBLE]. 163 00:07:51,660 --> 00:08:06,390 PROFESSOR: So there are 21,457 trees in this 50 hectare plot. 164 00:08:06,390 --> 00:08:10,770 They identify the species for each one of these 21,000 trees. 165 00:08:10,770 --> 00:08:11,670 And they assign them. 166 00:08:11,670 --> 00:08:14,610 And they found that there were 225 distinct species. 167 00:08:19,590 --> 00:08:23,552 So this is really quite an amazing data set. 168 00:08:23,552 --> 00:08:27,960 Because I can tell you that I would not be able to do this. 169 00:08:31,080 --> 00:08:35,559 This was highly skilled biologists 170 00:08:35,559 --> 00:08:38,880 that can distinguish 225. 171 00:08:38,880 --> 00:08:40,834 If they can identify these 225, that 172 00:08:40,834 --> 00:08:43,250 means they have to be able to identify other ones as well. 173 00:08:43,250 --> 00:08:45,340 And they did it for 20,000 trees. 174 00:08:45,340 --> 00:08:47,620 And indeed, Barro Colorado Island 175 00:08:47,620 --> 00:08:54,560 is one of the major Smithsonian research institutes, 176 00:08:54,560 --> 00:08:56,032 where they've been tracking. 177 00:08:56,032 --> 00:08:57,740 They do this like every five years or so, 178 00:08:57,740 --> 00:09:00,630 where they do a census, where they count all of the trees. 179 00:09:00,630 --> 00:09:05,929 And they're also tracking many other-- it's not just trees. 180 00:09:05,929 --> 00:09:07,220 They're doing everything there. 181 00:09:07,220 --> 00:09:08,512 AUDIENCE: Is there only plants? 182 00:09:08,512 --> 00:09:09,470 PROFESSOR: What's that? 183 00:09:09,470 --> 00:09:10,890 AUDIENCE: Is it only plants? 184 00:09:10,890 --> 00:09:11,473 PROFESSOR: No. 185 00:09:16,090 --> 00:09:18,285 So actually, I visited BCI, and it 186 00:09:18,285 --> 00:09:20,410 seemed like they were studying all sorts of things. 187 00:09:20,410 --> 00:09:22,760 And there were nice looking birds there. 188 00:09:22,760 --> 00:09:24,589 AUDIENCE: No, I mean in this census. 189 00:09:24,589 --> 00:09:26,380 PROFESSOR: In this census, it's only trees. 190 00:09:26,380 --> 00:09:32,150 And the way that they decide which of the trees to do, 191 00:09:32,150 --> 00:09:36,880 it's the ones that are more than 10 centimeters DBH. 192 00:09:36,880 --> 00:09:38,940 Anybody can guess what DBH might mean? 193 00:09:50,050 --> 00:09:52,240 It's actually diameter at breast height. 194 00:10:00,908 --> 00:10:05,925 So what they do is they walk up to the tree with a ruler, 195 00:10:05,925 --> 00:10:07,800 and then, if it's larger than 10 centimeters, 196 00:10:07,800 --> 00:10:11,020 then they count it. 197 00:10:11,020 --> 00:10:13,200 You need to have some threshold at the lower end, 198 00:10:13,200 --> 00:10:17,140 otherwise you're in trouble, right? 199 00:10:17,140 --> 00:10:20,131 And there were plenty of trees that satisfied this requirement 200 00:10:20,131 --> 00:10:20,630 here. 201 00:10:27,380 --> 00:10:30,420 Then what they do, for all of these trees, 202 00:10:30,420 --> 00:10:34,295 it's assigned to some species. 203 00:10:38,020 --> 00:10:42,350 The basic goal of this branch of biology or ecology 204 00:10:42,350 --> 00:10:46,817 is to try to understand the pattern, 205 00:10:46,817 --> 00:10:48,650 from this sort of data, where it comes from. 206 00:10:48,650 --> 00:10:51,170 Or first describe it, and then once you 207 00:10:51,170 --> 00:10:52,950 have a description of it, then you 208 00:10:52,950 --> 00:10:56,377 can try to understand what microscale processes might 209 00:10:56,377 --> 00:10:57,210 lead to the pattern. 210 00:10:57,210 --> 00:11:00,030 And the pattern is what's plotted in figure 1. 211 00:11:00,030 --> 00:11:01,610 It's the only figure in the paper. 212 00:11:01,610 --> 00:11:05,660 I have reconstructed a rough version of it, 213 00:11:05,660 --> 00:11:07,120 here, for you on the board. 214 00:11:07,120 --> 00:11:08,960 But if you want a more accurate version, 215 00:11:08,960 --> 00:11:12,600 you can look at your paper. 216 00:11:12,600 --> 00:11:15,120 Now, we want to make sure that we understand 217 00:11:15,120 --> 00:11:16,430 what the figure is saying. 218 00:11:16,430 --> 00:11:21,880 So we will ask the following question. 219 00:11:21,880 --> 00:11:23,860 What is the most common number of individuals 220 00:11:23,860 --> 00:11:26,066 for a species in this data set? 221 00:11:30,040 --> 00:11:39,660 The most common/frequent number of individuals for a species 222 00:11:39,660 --> 00:11:49,340 to have in this data set. 223 00:11:49,340 --> 00:11:51,594 Now, it's maybe worth just saying 224 00:11:51,594 --> 00:11:52,760 something a little bit more. 225 00:11:52,760 --> 00:11:54,710 So you notice that they were not trying 226 00:11:54,710 --> 00:11:58,780 to count the total number of species, altogether. 227 00:11:58,780 --> 00:12:01,426 And in general, all of this field of relative species 228 00:12:01,426 --> 00:12:03,467 abundance, to try to understand them, what you do 229 00:12:03,467 --> 00:12:05,792 is typically take one trophic level. 230 00:12:05,792 --> 00:12:07,250 So some of the classic studies were 231 00:12:07,250 --> 00:12:11,000 of beetles in the Thames River. 232 00:12:11,000 --> 00:12:13,050 The idea is that it's some set of species 233 00:12:13,050 --> 00:12:14,850 that you think are going to be interacting, 234 00:12:14,850 --> 00:12:16,350 maybe competing, with each other, 235 00:12:16,350 --> 00:12:17,390 in some way, in the sense that they're 236 00:12:17,390 --> 00:12:20,180 maybe eating related things and being eaten by related things. 237 00:12:20,180 --> 00:12:24,070 And so in this case, these are the trees 238 00:12:24,070 --> 00:12:25,230 in Barro Colorado Island. 239 00:12:25,230 --> 00:12:30,537 And you can imagine that this is useful. 240 00:12:30,537 --> 00:12:32,620 The fact that it's trees instead of something else 241 00:12:32,620 --> 00:12:35,251 means that you can actually track the individuals 242 00:12:35,251 --> 00:12:35,750 over time. 243 00:12:35,750 --> 00:12:37,166 And when you go to the island what 244 00:12:37,166 --> 00:12:40,929 you see is that all the trees, they're wrapped by some tag. 245 00:12:40,929 --> 00:12:42,470 And presumably, they have some system 246 00:12:42,470 --> 00:12:45,590 to tell you which species that is so that they 247 00:12:45,590 --> 00:12:47,880 keep records of everything. 248 00:12:47,880 --> 00:12:53,870 But the question is, what's the most common number 249 00:12:53,870 --> 00:12:55,790 of individuals for species in the data set? 250 00:12:55,790 --> 00:12:57,506 Do you understand what I'm trying to ask? 251 00:13:03,960 --> 00:13:07,120 And we're going do approximate, so we'll say. 252 00:13:20,230 --> 00:13:21,640 Or this, can't determine. 253 00:13:28,530 --> 00:13:32,600 We want to know, what is the mode of this distribution 254 00:13:32,600 --> 00:13:36,360 of the number of individuals for each of these species? 255 00:13:39,140 --> 00:13:41,540 Do you understand the question? 256 00:13:41,540 --> 00:13:45,220 I'm going to give you 20 seconds to look at this. 257 00:13:55,659 --> 00:13:58,110 AUDIENCE: Should we just hold a blank piece of paper? 258 00:13:58,110 --> 00:14:00,516 PROFESSOR: Oh, we don't have our-- ah. 259 00:14:00,516 --> 00:14:02,310 AUDIENCE: [INAUDIBLE]? 260 00:14:02,310 --> 00:14:05,530 PROFESSOR: You know, the TA always lets me down. 261 00:14:05,530 --> 00:14:08,310 All right, yeah. 262 00:14:08,310 --> 00:14:13,634 So you can do A, B, C, D, E. Are we ready? 263 00:14:13,634 --> 00:14:14,550 AUDIENCE: [INAUDIBLE]? 264 00:14:21,570 --> 00:14:23,680 PROFESSOR: You can just do this if you're not. 265 00:14:23,680 --> 00:14:26,100 But given this was the only figure in the paper, 266 00:14:26,100 --> 00:14:28,470 and that this is a basic property of the distribution, 267 00:14:28,470 --> 00:14:31,407 I'm sure that you figured that out last night, anyways, right? 268 00:14:31,407 --> 00:14:33,240 Especially since it was one of the questions 269 00:14:33,240 --> 00:14:34,031 in the [INAUDIBLE]. 270 00:14:34,031 --> 00:14:36,410 So you presumably already thought about this question, 271 00:14:36,410 --> 00:14:36,940 right? 272 00:14:36,940 --> 00:14:38,430 OK. 273 00:14:38,430 --> 00:14:39,253 Yes? 274 00:14:39,253 --> 00:14:40,800 AUDIENCE: Yes. 275 00:14:40,800 --> 00:14:43,725 PROFESSOR: Ready, three, two, one. 276 00:14:47,590 --> 00:14:50,060 I'd say we got a lot of B's. 277 00:14:50,060 --> 00:14:51,935 So it seems like B is the most. 278 00:14:55,284 --> 00:14:56,950 So this, we'll put a question mark here. 279 00:15:00,860 --> 00:15:04,440 Can somebody verbally say why their neighbor 280 00:15:04,440 --> 00:15:08,740 said that the mode of the distribution is around 30? 281 00:15:11,810 --> 00:15:12,427 Yeah? 282 00:15:12,427 --> 00:15:13,510 AUDIENCE: The tallest bar. 283 00:15:13,510 --> 00:15:16,620 PROFESSOR: The tallest bar there is around 30. 284 00:15:16,620 --> 00:15:18,270 That's a very practical definition. 285 00:15:18,270 --> 00:15:21,240 So that's normally what we mean by the mode. 286 00:15:21,240 --> 00:15:23,550 There is a slight problem in all of this, 287 00:15:23,550 --> 00:15:25,520 which is that this thing is plotted 288 00:15:25,520 --> 00:15:28,415 in a very kind of funny way. 289 00:15:28,415 --> 00:15:30,290 So if you look at the figure, what you'll see 290 00:15:30,290 --> 00:15:31,748 is that it's number of individuals. 291 00:15:31,748 --> 00:15:33,375 And down here, it says, log2 scale. 292 00:15:40,610 --> 00:15:44,530 Now, when we say the mode, what we're wondering about 293 00:15:44,530 --> 00:15:50,080 is that, if you just take the most typical kind of species 294 00:15:50,080 --> 00:15:52,080 of tree that's there, how many individuals do 295 00:15:52,080 --> 00:15:53,371 we think there should be there? 296 00:15:55,570 --> 00:15:57,560 Of course, typical is hard to define. 297 00:15:57,560 --> 00:16:00,650 We can talk about mode, median, mean, et cetera. 298 00:16:00,650 --> 00:16:03,210 But the most common number of individuals 299 00:16:03,210 --> 00:16:06,890 for a species of the data set ends up not being 30. 300 00:16:06,890 --> 00:16:08,190 It ends up being 1. 301 00:16:12,220 --> 00:16:16,430 And we will try to reconstruct this right now. 302 00:16:16,430 --> 00:16:18,870 Because you have to do a little bit of digging 303 00:16:18,870 --> 00:16:21,330 to figure out what is being plotted here. 304 00:16:21,330 --> 00:16:23,340 But it's not the raw data. 305 00:16:23,340 --> 00:16:26,360 The problem here is that this is on this log scale, where 306 00:16:26,360 --> 00:16:29,720 the bins here are growing kind of geometrically 307 00:16:29,720 --> 00:16:33,310 or exponentially, whatever, as you move to the right. 308 00:16:33,310 --> 00:16:39,975 So over here, this thing only contains one real bin. 309 00:16:39,975 --> 00:16:41,350 And actually, we're about to find 310 00:16:41,350 --> 00:16:44,040 it's half a bin, which is even weirder. 311 00:16:44,040 --> 00:16:46,330 Whereas out here, this is maybe 30 bins. 312 00:16:46,330 --> 00:16:50,550 So the number of species that we're going to put in this bin 313 00:16:50,550 --> 00:16:56,990 is everything between around 20 something up to 50 or so. 314 00:16:56,990 --> 00:16:59,701 The number of kind of true bins that 315 00:16:59,701 --> 00:17:01,200 end up in each of these plotted bins 316 00:17:01,200 --> 00:17:05,480 is going to grow geometrically as we move to the right. 317 00:17:05,480 --> 00:17:09,730 So this is a very funny transform of the data. 318 00:17:09,730 --> 00:17:14,690 And indeed, I think it's always nice to just, in life, 319 00:17:14,690 --> 00:17:17,890 you always plot the raw data first. 320 00:17:17,890 --> 00:17:20,720 And then what you can do is then you can do funny. 321 00:17:20,720 --> 00:17:23,510 There's a reason to plot it this way. 322 00:17:23,510 --> 00:17:27,770 Because this is where they get this idea that this might 323 00:17:27,770 --> 00:17:30,530 described as described as a log normal. 324 00:17:30,530 --> 00:17:32,550 The idea is, if you take a log of the data, 325 00:17:32,550 --> 00:17:34,550 then you get something that looks like a normal. 326 00:17:34,550 --> 00:17:38,105 But you always plot the raw data first. 327 00:17:38,105 --> 00:17:40,480 So let's try to figure out what the raw data looked like. 328 00:17:45,260 --> 00:17:46,680 And now what we're going to do is 329 00:17:46,680 --> 00:17:49,141 we're going to have real scalings, honest to goodness 330 00:17:49,141 --> 00:17:49,640 numbers. 331 00:17:53,130 --> 00:17:55,170 Now the number of species you get still. 332 00:17:57,800 --> 00:18:00,580 So this is asking, how many different species 333 00:18:00,580 --> 00:18:04,100 do we see with one member or with two members 334 00:18:04,100 --> 00:18:05,980 or with three, four, et cetera? 335 00:18:11,715 --> 00:18:13,340 And I don't know how far we're actually 336 00:18:13,340 --> 00:18:14,860 going to be able to get. 337 00:18:14,860 --> 00:18:21,440 But in this one figure, in our paper, 338 00:18:21,440 --> 00:18:23,640 they tell us what the histogram means. 339 00:18:23,640 --> 00:18:26,600 So the first histogram bar represents what 340 00:18:26,600 --> 00:18:29,680 they call phi 1 divided by 2. 341 00:18:29,680 --> 00:18:36,920 Phi 1 was the number of species observed with one member, which 342 00:18:36,920 --> 00:18:42,980 means that even this first plot bar is not 343 00:18:42,980 --> 00:18:45,940 the number of species observed with a single individual. 344 00:18:45,940 --> 00:18:48,810 It's half of that. 345 00:18:48,810 --> 00:18:50,350 You can argue about the consistency 346 00:18:50,350 --> 00:18:52,230 of how these things should be, but that's 347 00:18:52,230 --> 00:18:54,000 what this thing's plotted. 348 00:18:54,000 --> 00:18:58,000 And it looks like it was nine, here, so this should be 18. 349 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. 350 00:19:05,950 --> 00:19:07,140 Right, so here is an 18. 351 00:19:14,510 --> 00:19:17,670 Now, what do they say? 352 00:19:17,670 --> 00:19:23,270 This bin represents phi 1 divided 353 00:19:23,270 --> 00:19:26,894 by 2 plus phi 2 divided by 2. 354 00:19:26,894 --> 00:19:28,310 So they took the number of species 355 00:19:28,310 --> 00:19:30,940 where they saw just a single individual plus the number 356 00:19:30,940 --> 00:19:32,690 of species where they saw two individuals, 357 00:19:32,690 --> 00:19:35,829 and they added those and they divided by 2. 358 00:19:35,829 --> 00:19:36,620 That's this number. 359 00:19:38,799 --> 00:19:40,840 We're not going to go through this whole process, 360 00:19:40,840 --> 00:19:42,350 because it's a little bit tiresome. 361 00:19:42,350 --> 00:19:46,410 But I've already done it for you. 362 00:19:46,410 --> 00:19:50,670 So I'm going to plot a few of things to get you there. 363 00:19:58,240 --> 00:20:02,490 And so I calculated it was 19, 13, 9, 6. 364 00:20:02,490 --> 00:20:05,696 It becomes ill-determined once you get out here, 365 00:20:05,696 --> 00:20:07,320 in the sense that we don't have enough. 366 00:20:07,320 --> 00:20:10,240 It's not uniquely specified going from that to that 367 00:20:10,240 --> 00:20:12,980 as it has to be. 368 00:20:12,980 --> 00:20:13,970 But I calculated it. 369 00:20:13,970 --> 00:20:17,565 It's around 5, in here, for a few. 370 00:20:17,565 --> 00:20:21,120 And somewhere in here, it's going to go into 4. 371 00:20:21,120 --> 00:20:27,095 And then this might go down to 3, and then deh, deh, deh. 372 00:20:32,550 --> 00:20:42,950 Now, if you look at this and the rapid rapid fall-off, 373 00:20:42,950 --> 00:20:46,665 do you think that you're going to find any species that 374 00:20:46,665 --> 00:20:47,915 have more than 20 individuals? 375 00:20:51,440 --> 00:20:52,340 We're going to vote. 376 00:20:52,340 --> 00:20:53,857 So you see this falling-off? 377 00:20:53,857 --> 00:20:55,690 So let's say that I've just showed you this, 378 00:20:55,690 --> 00:20:58,930 and I haven't yet calculated the rest, 379 00:20:58,930 --> 00:21:01,540 do we think that there's going to be any species with more 380 00:21:01,540 --> 00:21:04,230 than 20 individuals? 381 00:21:04,230 --> 00:21:08,430 Greater than 20 individuals, question mark? 382 00:21:08,430 --> 00:21:10,760 1 is yes. 383 00:21:10,760 --> 00:21:12,245 2 is no. 384 00:21:14,945 --> 00:21:16,230 It's going to be yes, no. 385 00:21:16,230 --> 00:21:18,835 Ready, three, two, one. 386 00:21:21,560 --> 00:21:25,190 So we got some 2s. 387 00:21:25,190 --> 00:21:27,300 So I'd say that most people are saying, no. 388 00:21:27,300 --> 00:21:28,532 Look at this fall-off. 389 00:21:28,532 --> 00:21:29,990 They're not going to be any species 390 00:21:29,990 --> 00:21:33,480 with more than 20 individuals. 391 00:21:33,480 --> 00:21:35,630 Although we already know that there 392 00:21:35,630 --> 00:21:38,480 are many species with more than 20 individuals. 393 00:21:38,480 --> 00:21:41,930 So this plot is useful for something. 394 00:21:41,930 --> 00:21:43,560 You can see that there are. 395 00:21:43,560 --> 00:21:45,450 And we know exactly the number of species 396 00:21:45,450 --> 00:21:47,930 that have more than 20 individuals, roughly. 397 00:21:47,930 --> 00:21:50,760 So those ones are all in these. 398 00:21:50,760 --> 00:21:52,985 So you can see that there are hundreds of species 399 00:21:52,985 --> 00:21:54,235 with more than 20 individuals. 400 00:21:57,630 --> 00:22:00,830 And indeed, it looks like there were two or three species that 401 00:22:00,830 --> 00:22:03,186 had more than 1,000 individuals or 1,500 402 00:22:03,186 --> 00:22:04,560 or whatever the cutoff there was. 403 00:22:07,920 --> 00:22:14,050 So this distribution starts out rather high 404 00:22:14,050 --> 00:22:15,590 but then falls quickly. 405 00:22:15,590 --> 00:22:19,019 And out here, it's going to be very, very sparse. 406 00:22:19,019 --> 00:22:21,060 So there's going to be a bunch of numbers in here 407 00:22:21,060 --> 00:22:24,160 where there's not any species in the histogram. 408 00:22:24,160 --> 00:22:26,340 And then out there, there's going to be one, right? 409 00:22:26,340 --> 00:22:28,680 And indeed, you have to go really far out. 410 00:22:28,680 --> 00:22:30,490 Because there's one species out there 411 00:22:30,490 --> 00:22:33,870 that has a couple thousand. 412 00:22:33,870 --> 00:22:40,660 And indeed, the mean number of individuals per species 413 00:22:40,660 --> 00:22:44,880 has to be around 100. 414 00:22:44,880 --> 00:22:47,160 We know how to calculate a mean. 415 00:22:47,160 --> 00:22:50,810 This divided by this is just short of 100. 416 00:22:50,810 --> 00:22:54,150 So the mean number of individuals in a species 417 00:22:54,150 --> 00:22:56,280 is around 100. 418 00:22:56,280 --> 00:22:59,090 The mode is one. 419 00:23:01,446 --> 00:23:02,070 And the median? 420 00:23:08,630 --> 00:23:11,200 Well, ready? 421 00:23:11,200 --> 00:23:14,387 We decided this was the mode. 422 00:23:14,387 --> 00:23:15,720 Where is the median going to be? 423 00:23:15,720 --> 00:23:18,260 Is it going to be A, B, C, D? 424 00:23:18,260 --> 00:23:20,935 Ready, three, two, one. 425 00:23:23,760 --> 00:23:27,060 Indeed, this tells you pretty clear where the median is. 426 00:23:27,060 --> 00:23:28,854 This thing is indeed around the median. 427 00:23:28,854 --> 00:23:31,020 Because you can say, oh, it's about the same numbers 428 00:23:31,020 --> 00:23:32,240 to either side. 429 00:23:32,240 --> 00:23:33,740 So the median is around here. 430 00:23:36,310 --> 00:23:39,370 And I told you where the mean was, again. 431 00:23:39,370 --> 00:23:40,330 You guys remember? 432 00:23:40,330 --> 00:23:42,710 Ready, three, two, one. 433 00:23:42,710 --> 00:23:43,960 Mean, uno. 434 00:23:43,960 --> 00:23:44,460 Mean. 435 00:23:50,220 --> 00:23:53,070 So this is a very, very funny distribution. 436 00:23:53,070 --> 00:23:54,730 I guess I want to highlight that. 437 00:23:54,730 --> 00:23:57,530 And I think it's not at all what you 438 00:23:57,530 --> 00:24:00,070 would have expected somehow. 439 00:24:00,070 --> 00:24:04,799 At least, if you had described this measurement process to me, 440 00:24:04,799 --> 00:24:06,590 if you told me that you went to this island 441 00:24:06,590 --> 00:24:10,545 and you counted 20,000 trees, I don't know how many species 442 00:24:10,545 --> 00:24:11,420 I would have guessed. 443 00:24:11,420 --> 00:24:16,255 But OK, 220, it's reasonable. 444 00:24:16,255 --> 00:24:18,630 Well, I would have guessed it would have looked something 445 00:24:18,630 --> 00:24:22,380 like this on a linear scale, maybe, right? 446 00:24:22,380 --> 00:24:25,430 You know, that there would be a bunch of them around 50 to 100 447 00:24:25,430 --> 00:24:28,340 and some would go couple hundred, some of them. 448 00:24:28,340 --> 00:24:32,480 So I guess I would have thought that the mean, mode, 449 00:24:32,480 --> 00:24:35,070 median would all be kind of a more similar thing. 450 00:24:35,070 --> 00:24:38,130 But this is just not the way the world is. 451 00:24:38,130 --> 00:24:40,230 It's not just on BCI. 452 00:24:40,230 --> 00:24:44,150 People, for hundreds of years, have been 453 00:24:44,150 --> 00:24:45,520 studying these distributions. 454 00:24:45,520 --> 00:24:51,100 And things that look like this, with extremely long tails, 455 00:24:51,100 --> 00:24:54,810 this is what people see. 456 00:24:54,810 --> 00:24:57,720 Now you can argue about exactly how fast it falls off 457 00:24:57,720 --> 00:25:00,040 and whether it's different on a mainland or an island. 458 00:25:00,040 --> 00:25:05,210 But this basic feature, that rare species are common, 459 00:25:05,210 --> 00:25:09,140 this seems to be just that's what you always see. 460 00:25:09,140 --> 00:25:11,220 This is the thing that you have to remember, 461 00:25:11,220 --> 00:25:12,425 rare species are common. 462 00:25:24,520 --> 00:25:28,680 And I think that this is the basic, surprising thing 463 00:25:28,680 --> 00:25:30,530 in this whole field. 464 00:25:30,530 --> 00:25:32,480 And the ironic thing is that even 465 00:25:32,480 --> 00:25:35,580 after spending all this time reading about theories 466 00:25:35,580 --> 00:25:39,140 to describe these distributions, it's still very possible-- 467 00:25:39,140 --> 00:25:41,780 and I would say, based on the statistics, this year 468 00:25:41,780 --> 00:25:44,380 and past years, it's not just possible, 469 00:25:44,380 --> 00:25:46,060 but it is the standard outcome-- is 470 00:25:46,060 --> 00:25:47,476 that after reading this paper, you 471 00:25:47,476 --> 00:25:51,230 do not realize that the distribution looks like this. 472 00:25:51,230 --> 00:25:54,000 You somehow still think that it looks-- you kind of still 473 00:25:54,000 --> 00:25:59,060 think it's like a linear scale, where the typical species has 474 00:25:59,060 --> 00:26:01,130 this, where the mean, median, mode are all 475 00:26:01,130 --> 00:26:02,260 about the same thing. 476 00:26:02,260 --> 00:26:08,661 So I guess always plot the raw data in an untransformed way. 477 00:26:08,661 --> 00:26:10,410 There are theoretical reasons why it might 478 00:26:10,410 --> 00:26:11,618 be nice to plot it like this. 479 00:26:11,618 --> 00:26:15,120 But be very careful about what you're doing. 480 00:26:15,120 --> 00:26:17,965 Because then you're left with a mental image of a histogram 481 00:26:17,965 --> 00:26:19,850 that looks like this. 482 00:26:19,850 --> 00:26:21,640 And that's very, very dangerous. 483 00:26:21,640 --> 00:26:22,140 Yeah? 484 00:26:22,140 --> 00:26:25,570 AUDIENCE: Why does it matter [INAUDIBLE]? 485 00:26:25,570 --> 00:26:29,000 [INAUDIBLE] the aggregate data in bins like that. 486 00:26:29,000 --> 00:26:33,150 And I mean, sure, exactly one species is the mode, 487 00:26:33,150 --> 00:26:35,619 but do you really want the--? 488 00:26:35,619 --> 00:26:37,410 PROFESSOR: I understand what you're saying. 489 00:26:42,240 --> 00:26:44,090 It's just that there's a qualitative aspect 490 00:26:44,090 --> 00:26:49,510 to the data, which is that most species are very rare. 491 00:26:49,510 --> 00:26:51,630 And this is something that I think is surprising. 492 00:26:51,630 --> 00:26:53,410 I think it's deep. 493 00:26:53,410 --> 00:26:56,120 And it's something that you do not get realized. 494 00:26:56,120 --> 00:27:00,368 AUDIENCE: Most species have more than 16. 495 00:27:00,368 --> 00:27:03,024 I mean, it depends what you mean by rare. 496 00:27:03,024 --> 00:27:03,690 PROFESSOR: Yeah. 497 00:27:03,690 --> 00:27:06,108 AUDIENCE: Look at the way that the distribution is away 498 00:27:06,108 --> 00:27:07,268 from trend. 499 00:27:07,268 --> 00:27:08,518 AUDIENCE: That's a good point. 500 00:27:08,518 --> 00:27:10,687 But the species density is clustered 501 00:27:10,687 --> 00:27:11,962 around the low numbers. 502 00:27:11,962 --> 00:27:12,670 PROFESSOR: Right. 503 00:27:12,670 --> 00:27:14,920 AUDIENCE: But actually most species have more than 30. 504 00:27:18,679 --> 00:27:20,220 PROFESSOR: Maybe the surprising thing 505 00:27:20,220 --> 00:27:24,260 is that just if you take-- the mean is 100. 506 00:27:24,260 --> 00:27:33,440 And so I would've thought that, if you plot number of species 507 00:27:33,440 --> 00:27:41,650 as a function of the number of individuals, 508 00:27:41,650 --> 00:27:44,290 given those numbers, I would have guessed, OK, here's 100. 509 00:27:44,290 --> 00:27:46,480 I would have guessed-- here's 50, so just 510 00:27:46,480 --> 00:27:49,130 to highlight that this is 150. 511 00:27:49,130 --> 00:27:53,620 So linear scale, I would have guessed 512 00:27:53,620 --> 00:27:56,820 it would look something like that, maybe larger than Rudin 513 00:27:56,820 --> 00:27:57,889 or something. 514 00:27:57,889 --> 00:28:00,055 AUDIENCE: What would that look like in a log2 scale? 515 00:28:00,055 --> 00:28:03,951 It would look like It's like the log of [INAUDIBLE]? 516 00:28:03,951 --> 00:28:06,460 So it goes up really fast and then-- 517 00:28:06,460 --> 00:28:11,730 PROFESSOR: So this thing would be kind of like shoom. 518 00:28:11,730 --> 00:28:14,070 I mean all the weight would be in. 519 00:28:14,070 --> 00:28:16,988 It would be like all here plus a little bit on each of these. 520 00:28:16,988 --> 00:28:18,884 AUDIENCE: But yeah. 521 00:28:18,884 --> 00:28:21,730 I don't think it's actually that different. 522 00:28:21,730 --> 00:28:24,340 The only thing that's different is the tail on the left. 523 00:28:24,340 --> 00:28:26,092 PROFESSOR: And the tail on the right. 524 00:28:26,092 --> 00:28:27,800 AUDIENCE: Yeah, it's a little bit longer. 525 00:28:27,800 --> 00:28:29,383 PROFESSOR: No, it's lot longer, right? 526 00:28:29,383 --> 00:28:33,530 Because this thing, all of the weight is between 50 and 150, 527 00:28:33,530 --> 00:28:35,740 which means that all of the counts 528 00:28:35,740 --> 00:28:41,124 are basically going to be these two, basically. 529 00:28:41,124 --> 00:28:42,790 Because this thing comes out either way. 530 00:28:42,790 --> 00:28:46,240 So in this case, if you take that histogram put it 531 00:28:46,240 --> 00:28:48,560 on this kind of scale, you end up with two bars 532 00:28:48,560 --> 00:28:50,717 up high, nothing outside. 533 00:28:50,717 --> 00:28:52,300 So it's a very different distribution. 534 00:28:52,300 --> 00:28:55,330 And it's not to say that this is a ridiculous thing to do. 535 00:28:55,330 --> 00:28:57,230 It's just that. 536 00:28:57,230 --> 00:28:59,610 But the problem is that your mental image 537 00:28:59,610 --> 00:29:01,930 of what the distribution looks like ends up 538 00:29:01,930 --> 00:29:03,910 being incorrect, in the sense that you 539 00:29:03,910 --> 00:29:07,040 have a qualitatively different sense of what's of what's 540 00:29:07,040 --> 00:29:08,690 going on. 541 00:29:08,690 --> 00:29:14,260 And if you go up to 10 species, here, and 10 is way down here. 542 00:29:17,589 --> 00:29:19,130 If this is what it looked like, there 543 00:29:19,130 --> 00:29:22,970 would be essentially no species with fewer than 10 individuals. 544 00:29:22,970 --> 00:29:25,780 But if you come over here and you add it up here. 545 00:29:25,780 --> 00:29:31,740 It's like a mean of 6 times 10 is 60 out of 200. 546 00:29:31,740 --> 00:29:35,830 A quarter or a third of the species on this plot of land 547 00:29:35,830 --> 00:29:37,450 have fewer than 10 individuals. 548 00:29:37,450 --> 00:29:39,515 And 10 is really a very small number. 549 00:29:43,810 --> 00:29:47,590 Well, rare species are common. 550 00:29:47,590 --> 00:29:52,170 I think it's a true description of the observed distribution 551 00:29:52,170 --> 00:29:53,180 here and elsewhere. 552 00:29:53,180 --> 00:29:56,620 And it's not something that you appreciate 553 00:29:56,620 --> 00:29:58,911 or realize when you plot it in that way. 554 00:29:58,911 --> 00:30:01,522 AUDIENCE: But you can get this information from that plot. 555 00:30:01,522 --> 00:30:02,480 PROFESSOR: No, I agree. 556 00:30:02,480 --> 00:30:03,440 You can get it. 557 00:30:03,440 --> 00:30:04,290 You can get it. 558 00:30:04,290 --> 00:30:07,280 But it was only 10% of the group got it. 559 00:30:07,280 --> 00:30:10,430 Right, the fact that you can get it-- right, it's possible. 560 00:30:10,430 --> 00:30:13,370 But you don't get it. 561 00:30:13,370 --> 00:30:15,680 That is a practical statement. 562 00:30:15,680 --> 00:30:18,150 Yeah, I'm not dead set against this distribution. 563 00:30:18,150 --> 00:30:20,737 It's just that it makes everybody think 564 00:30:20,737 --> 00:30:21,820 something that's not true. 565 00:30:21,820 --> 00:30:24,430 So if you think that that's OK, then I can't help you. 566 00:30:27,670 --> 00:30:29,690 It's OK, but it's just you have to be 567 00:30:29,690 --> 00:30:33,300 careful is my only statement. 568 00:30:33,300 --> 00:30:36,550 And I very much want you to take away. 569 00:30:36,550 --> 00:30:38,800 Because I this is an accurate description of the data. 570 00:30:38,800 --> 00:30:40,270 Rare species are common. 571 00:30:40,270 --> 00:30:43,360 And one of the readings-- I think it was in this paper, 572 00:30:43,360 --> 00:30:45,640 maybe it was a different one that I was reading. 573 00:30:45,640 --> 00:30:49,300 Even Darwin, when talking about this, commented on this fact 574 00:30:49,300 --> 00:30:54,820 that rarity of species is somehow a typical event. 575 00:30:54,820 --> 00:30:57,700 AUDIENCE: And common species are rare. 576 00:30:57,700 --> 00:31:01,160 PROFESSOR: And common species are rare, that's right. 577 00:31:01,160 --> 00:31:04,720 This distribution is hugely, hugely skewed. 578 00:31:14,986 --> 00:31:16,110 These are the measurements. 579 00:31:18,731 --> 00:31:20,730 It's good to look at them in both of these ways. 580 00:31:20,730 --> 00:31:23,950 Because you can't even plot the data on a linear scale. 581 00:31:23,950 --> 00:31:26,020 So that's a good reason for doing it. 582 00:31:26,020 --> 00:31:29,350 But I think it's good to have both of these pictures in mind. 583 00:31:38,680 --> 00:31:42,960 What we want to do is to talk about two classes of models 584 00:31:42,960 --> 00:31:45,110 that give something that's essentially 585 00:31:45,110 --> 00:31:46,620 this log normal distribution. 586 00:31:46,620 --> 00:31:50,350 So on a log scale it looks normally distributed, 587 00:31:50,350 --> 00:31:53,130 approximately. 588 00:31:53,130 --> 00:31:54,640 And those two models are going to be 589 00:31:54,640 --> 00:31:58,610 kind of a niche-based model and a neutral model. 590 00:31:58,610 --> 00:32:01,460 Can somebody, in words, explain what they maybe 591 00:32:01,460 --> 00:32:04,200 see as the difference between this niche 592 00:32:04,200 --> 00:32:05,800 and a neutral kind of approach? 593 00:32:21,768 --> 00:32:22,349 Yeah? 594 00:32:22,349 --> 00:32:23,265 AUDIENCE: [INAUDIBLE]. 595 00:32:26,255 --> 00:32:27,505 PROFESSOR: Every species is--? 596 00:32:27,505 --> 00:32:28,430 AUDIENCE: [INAUDIBLE]. 597 00:32:28,430 --> 00:32:29,846 PROFESSOR: In which one? 598 00:32:29,846 --> 00:32:31,130 AUDIENCE: In niche. 599 00:32:31,130 --> 00:32:34,340 PROFESSOR: In the niche theory, the species are different. 600 00:32:34,340 --> 00:32:39,450 So it seems like a ridiculous statement. 601 00:32:39,450 --> 00:32:41,380 Do you believe that species are different? 602 00:32:41,380 --> 00:32:43,200 We can vote, yes or no. 603 00:32:43,200 --> 00:32:46,150 Ready, three, two, one. 604 00:32:46,150 --> 00:32:47,800 Yeah. 605 00:32:47,800 --> 00:32:51,330 Well, somebody's been convinced by the neutral theory. 606 00:32:51,330 --> 00:32:54,760 It's clear that species are different. 607 00:32:54,760 --> 00:32:59,390 And the question is which patterns in the data 608 00:32:59,390 --> 00:33:01,340 do you need to invoke differences 609 00:33:01,340 --> 00:33:03,890 in order to explain? 610 00:33:03,890 --> 00:33:06,050 And I think that one, maybe, theme that's 611 00:33:06,050 --> 00:33:08,640 come out of this relative species abundance literature 612 00:33:08,640 --> 00:33:11,110 and the debates between the neutral and the niche guys 613 00:33:11,110 --> 00:33:14,390 is just that this distribution is 614 00:33:14,390 --> 00:33:18,310 less informative of the micro scale 615 00:33:18,310 --> 00:33:22,240 or individual kind of interactions 616 00:33:22,240 --> 00:33:25,640 then you might have thought. 617 00:33:25,640 --> 00:33:29,290 Because multiple models can adequately 618 00:33:29,290 --> 00:33:32,050 explain such a pattern. 619 00:33:32,050 --> 00:33:35,220 In all areas, we have to remember 620 00:33:35,220 --> 00:33:38,530 that you make an observation, and you write down a model 621 00:33:38,530 --> 00:33:40,892 that explains that observation. 622 00:33:40,892 --> 00:33:42,600 So what you do is you write down a model. 623 00:33:42,600 --> 00:33:44,040 And writing down a model, what that means 624 00:33:44,040 --> 00:33:46,010 is that you make some set of assumptions. 625 00:33:46,010 --> 00:33:48,176 And then you look to see what happens in that model. 626 00:33:48,176 --> 00:33:53,555 And if the model is consistent with the data, that's good. 627 00:33:53,555 --> 00:33:55,680 But it doesn't prove that the assumptions that went 628 00:33:55,680 --> 00:33:56,804 into the model are correct. 629 00:33:56,804 --> 00:33:58,475 And this is a trivial statement. 630 00:33:58,475 --> 00:33:59,475 And I've said it before. 631 00:34:01,839 --> 00:34:03,880 You have to tell yourself this or remind yourself 632 00:34:03,880 --> 00:34:05,580 of this kind of once a month. 633 00:34:05,580 --> 00:34:07,970 Because it's just such an easy thing to forget about. 634 00:34:15,090 --> 00:34:18,420 Now, the niche models indeed assume 635 00:34:18,420 --> 00:34:20,190 that the species are different. 636 00:34:20,190 --> 00:34:22,840 And that's reasonable. 637 00:34:22,840 --> 00:34:24,687 Because we think it's true. 638 00:34:24,687 --> 00:34:26,770 But then, of course, there are many different ways 639 00:34:26,770 --> 00:34:28,550 of capturing those differences. 640 00:34:28,550 --> 00:34:32,600 And then you have to decide whether the assumptions there 641 00:34:32,600 --> 00:34:37,030 are reasonable or whether they're necessary, essential. 642 00:34:37,030 --> 00:34:39,170 In the context of the niche models, 643 00:34:39,170 --> 00:34:42,500 we're going to think about the so-called broken stick models. 644 00:34:52,510 --> 00:34:55,219 So basically, you get log normal distributions 645 00:34:55,219 --> 00:34:58,320 when there's some sort of multiplicative-type random 646 00:34:58,320 --> 00:35:01,260 process that's being added together. 647 00:35:01,260 --> 00:35:02,930 You get normal distributions when 648 00:35:02,930 --> 00:35:05,794 you have sums of random things going together. 649 00:35:05,794 --> 00:35:07,210 This is the central limit theorem. 650 00:35:07,210 --> 00:35:09,560 But when you have multiplicative kind 651 00:35:09,560 --> 00:35:12,895 of errors or random processes coming together, 652 00:35:12,895 --> 00:35:14,270 you get log normal distributions. 653 00:35:14,270 --> 00:35:17,650 And I want to highlight that that does not necessarily 654 00:35:17,650 --> 00:35:23,720 have to tell you so much about the biology of it. 655 00:35:23,720 --> 00:35:28,540 Because a classic situation where you get log 656 00:35:28,540 --> 00:35:35,590 normal distributions is if you take a stone and you crush it. 657 00:35:41,220 --> 00:35:43,350 You can do this experiment at home. 658 00:35:43,350 --> 00:35:47,930 And then you measure the mass distribution of the resulting 659 00:35:47,930 --> 00:35:48,430 fragments. 660 00:35:55,390 --> 00:36:00,050 And the distribution of mass is log normal. 661 00:36:05,210 --> 00:36:13,280 Just take a stone, grind it under your boot or hammer it, 662 00:36:13,280 --> 00:36:15,030 just kind rub it right in. 663 00:36:15,030 --> 00:36:18,230 You'll get you'll get some distribution of fragments. 664 00:36:18,230 --> 00:36:20,950 For each of the fragments, measure the mass, and, indeed, 665 00:36:20,950 --> 00:36:22,930 you end up getting a log normal distribution. 666 00:36:22,930 --> 00:36:24,930 Because there's some sense that what's happening 667 00:36:24,930 --> 00:36:28,260 is that you take a larger mass, you break it up randomly, 668 00:36:28,260 --> 00:36:30,490 and then the resulting fragments, at some rate, 669 00:36:30,490 --> 00:36:32,270 each of them you break up randomly. 670 00:36:32,270 --> 00:36:33,770 and the small ones are maybe kind of 671 00:36:33,770 --> 00:36:35,644 less likely to get broken up as the big ones, 672 00:36:35,644 --> 00:36:38,232 so then the small ones can still get even smaller. 673 00:36:38,232 --> 00:36:40,690 But then there's going to be, at some rate, some very large 674 00:36:40,690 --> 00:36:42,060 ones. 675 00:36:42,060 --> 00:36:47,880 So such a process ends up-- I mean it's not biology. 676 00:36:47,880 --> 00:36:51,660 This is just something about the nature of the breaking 677 00:36:51,660 --> 00:36:53,990 up of this physical object. 678 00:36:53,990 --> 00:36:56,160 And indeed, the basic idea behind many 679 00:36:56,160 --> 00:36:58,630 of the niche models that give you a log normal distribution 680 00:36:58,630 --> 00:37:02,270 is equivalent to crushing a stone 681 00:37:02,270 --> 00:37:05,910 and measuring the resulting distribution. 682 00:37:05,910 --> 00:37:07,670 I'll describe what I mean by that. 683 00:37:07,670 --> 00:37:11,120 Typically, the broken stick models, 684 00:37:11,120 --> 00:37:16,140 they say there's some resource axis. 685 00:37:16,140 --> 00:37:20,300 This is a resource axis. 686 00:37:20,300 --> 00:37:24,430 And this could be, for example, where you're getting food from. 687 00:37:24,430 --> 00:37:26,570 Now, we're going to have to divide up this resource 688 00:37:26,570 --> 00:37:28,615 access among some number of different species. 689 00:37:28,615 --> 00:37:29,990 And what we're going to assume is 690 00:37:29,990 --> 00:37:32,120 that the number of individuals in the species 691 00:37:32,120 --> 00:37:35,460 is proportional to the length of the resource axis 692 00:37:35,460 --> 00:37:36,698 that it's able to capture. 693 00:37:39,450 --> 00:37:43,150 And I want to make sure I find my notes. 694 00:37:43,150 --> 00:37:44,275 I want to highlight this. 695 00:37:44,275 --> 00:37:53,600 This comes from MacArthur in the 1950s. 696 00:37:53,600 --> 00:37:58,480 MacArthur and it's 1957. 697 00:37:58,480 --> 00:38:01,780 So we imagine there's this homogeneous resource axis. 698 00:38:01,780 --> 00:38:03,600 We're going to break it up into N segments. 699 00:38:03,600 --> 00:38:08,860 And the abundances are proportional to the length. 700 00:38:08,860 --> 00:38:13,690 And the idea is that, if you just break this up randomly, 701 00:38:13,690 --> 00:38:17,150 so let's say you just draw N minus 1 lines randomly, 702 00:38:17,150 --> 00:38:20,290 or N minus 1 points randomly here. 703 00:38:20,290 --> 00:38:27,775 Now you have N species with N different abundances. 704 00:38:30,480 --> 00:38:32,360 The question is does that give a log normal? 705 00:38:35,160 --> 00:38:37,495 We'll say N minus 1 random points. 706 00:38:41,084 --> 00:38:42,377 Do you understand what I mean. 707 00:38:42,377 --> 00:38:44,460 You sample uniformly once, sample uniformly twice. 708 00:38:44,460 --> 00:38:50,220 You do that N minus 1 times, and now you have N and deh deh. 709 00:38:50,220 --> 00:38:52,130 And then we say, OK, the first species 710 00:38:52,130 --> 00:38:53,230 has this many individuals. 711 00:38:53,230 --> 00:38:54,790 The second has this one. 712 00:38:54,790 --> 00:38:58,230 The third is this one, et cetera. 713 00:38:58,230 --> 00:39:00,589 The question is does random points, 714 00:39:00,589 --> 00:39:01,880 does that lead to a log normal? 715 00:39:07,400 --> 00:39:09,380 Yes and no. 716 00:39:09,380 --> 00:39:12,250 Let's think about this for 10 seconds. 717 00:39:12,250 --> 00:39:20,030 N minus 1 random points, log normal distribution, 718 00:39:20,030 --> 00:39:23,115 ready, three, two, one. 719 00:39:26,320 --> 00:39:30,200 So I'd say that we have a majority are saying no. 720 00:39:30,200 --> 00:39:31,610 Can somebody say why that is? 721 00:39:31,610 --> 00:39:34,430 AUDIENCE: [INAUDIBLE]. 722 00:39:34,430 --> 00:39:36,440 PROFESSOR: Because it's something else. 723 00:39:36,440 --> 00:39:39,900 That's fair. 724 00:39:39,900 --> 00:39:41,865 But can you say qualitatively why it is 725 00:39:41,865 --> 00:39:43,682 that this is not going to work? 726 00:39:43,682 --> 00:39:48,632 AUDIENCE: You can't have very long gaps. 727 00:39:48,632 --> 00:39:49,340 PROFESSOR: Right. 728 00:39:49,340 --> 00:39:51,256 That is it's going to be very unusual that you 729 00:39:51,256 --> 00:39:53,310 get a very long gap. 730 00:39:53,310 --> 00:39:55,510 What about the other end? 731 00:39:55,510 --> 00:39:58,064 AUDIENCE: Also a very long tail. 732 00:39:58,064 --> 00:39:59,730 PROFESSOR: Now I'm a little bit worried. 733 00:40:03,040 --> 00:40:04,460 I think that that's true, right? 734 00:40:10,640 --> 00:40:12,910 Well, I'm going to say that you're not going 735 00:40:12,910 --> 00:40:16,140 to get this super long ones. 736 00:40:16,140 --> 00:40:20,810 I think that the distribution might still 737 00:40:20,810 --> 00:40:24,500 be peaked at short values. 738 00:40:24,500 --> 00:40:25,230 No? 739 00:40:25,230 --> 00:40:26,550 AUDIENCE: No. 740 00:40:26,550 --> 00:40:28,510 PROFESSOR: Random? 741 00:40:28,510 --> 00:40:32,236 If we were just traveling along this resource axis, 742 00:40:32,236 --> 00:40:34,360 at a rate that's kind of exponentially distributed, 743 00:40:34,360 --> 00:40:36,480 like Poisson rate, we just dropped 744 00:40:36,480 --> 00:40:40,635 points, that's something very similar to this random-- 745 00:40:40,635 --> 00:40:43,167 AUDIENCE: It said we're limited in the number-- 746 00:40:43,167 --> 00:40:43,750 PROFESSOR: No. 747 00:40:43,750 --> 00:40:45,604 Is that not true? 748 00:40:45,604 --> 00:40:47,020 AUDIENCE: Your sample [INAUDIBLE]. 749 00:40:53,030 --> 00:40:57,310 PROFESSOR: I'm a little bit worried that I might be-- now, 750 00:40:57,310 --> 00:40:58,600 I'm not 100% confident. 751 00:40:58,600 --> 00:41:01,361 Depending on how I look at this, I get different distributions. 752 00:41:01,361 --> 00:41:01,860 Yeah? 753 00:41:01,860 --> 00:41:05,716 AUDIENCE: But I think the first thing that he said, 754 00:41:05,716 --> 00:41:08,465 where you just say, I'm going to pick N minus 1 points-- 755 00:41:08,465 --> 00:41:09,090 PROFESSOR: Yes. 756 00:41:09,090 --> 00:41:11,940 AUDIENCE: --is a different thing than going along the axis 757 00:41:11,940 --> 00:41:14,320 and exponentially dropping ones along. 758 00:41:14,320 --> 00:41:17,021 PROFESSOR: I agree it's different. 759 00:41:17,021 --> 00:41:19,395 AUDIENCE: I don't think that would be the idea simulated, 760 00:41:19,395 --> 00:41:22,093 because you would be very likely to just get 761 00:41:22,093 --> 00:41:24,134 this giant thing at the end when you're finished. 762 00:41:24,134 --> 00:41:26,120 AUDIENCE: What you could do, you could go on 763 00:41:26,120 --> 00:41:29,830 to draft N plus 2 points. 764 00:41:29,830 --> 00:41:30,830 PROFESSOR: No, I think-- 765 00:41:30,830 --> 00:41:33,125 AUDIENCE: These scales that are your two end points 766 00:41:33,125 --> 00:41:33,980 [? are doubled. ?] 767 00:41:33,980 --> 00:41:35,938 PROFESSOR: Because I think that the probability 768 00:41:35,938 --> 00:41:37,310 distribution does grow. 769 00:41:37,310 --> 00:41:40,370 I think that I'm going to side with you. 770 00:41:40,370 --> 00:41:42,190 So we've decided that there are not 771 00:41:42,190 --> 00:41:44,315 going to be as many short sticks, 772 00:41:44,315 --> 00:41:46,190 and there's not going to be as long sticks as 773 00:41:46,190 --> 00:41:47,231 compared to a log normal. 774 00:41:47,231 --> 00:41:48,226 Do we agree with that? 775 00:41:51,205 --> 00:41:53,580 At least we agree that it's not going to be a log normal. 776 00:41:53,580 --> 00:41:55,496 So you're not going to get this huge variation 777 00:41:55,496 --> 00:41:59,230 of some very long sticks and some very short ones. 778 00:41:59,230 --> 00:42:03,710 Now, the question is how would you change this sort of model 779 00:42:03,710 --> 00:42:05,581 in order to generate a log normal? 780 00:42:05,581 --> 00:42:07,330 And the answer is that what you have to do 781 00:42:07,330 --> 00:42:10,980 is you have to what is called some niche hierarchy or so 782 00:42:10,980 --> 00:42:12,690 some hierarchical breaking. 783 00:42:12,690 --> 00:42:15,040 Just like what led to the stone giving you a log normal 784 00:42:15,040 --> 00:42:17,100 is that you have to have some successive process 785 00:42:17,100 --> 00:42:18,810 of breaking things. 786 00:42:18,810 --> 00:42:21,190 So this is what they call some hierarchy model. 787 00:42:27,225 --> 00:42:29,225 And then they key thing is that it's sequential. 788 00:42:33,260 --> 00:42:35,060 You have your resource axis. 789 00:42:35,060 --> 00:42:39,661 First, you have some rule for breaking it up. 790 00:42:39,661 --> 00:42:41,410 It could be that you just sample uniformly 791 00:42:41,410 --> 00:42:43,034 or some other probability distribution. 792 00:42:45,089 --> 00:42:46,880 And the way that you might think about this 793 00:42:46,880 --> 00:42:57,020 is via-- just everything up on the board 794 00:42:57,020 --> 00:42:58,270 is so nice and useful. 795 00:42:58,270 --> 00:43:02,430 I feel bad getting rid of it. 796 00:43:02,430 --> 00:43:06,170 This thing is not true, so I don't mind erasing it. 797 00:43:06,170 --> 00:43:10,560 So let's imagine some bird community in the forest. 798 00:43:13,260 --> 00:43:14,770 And we're going to think about where 799 00:43:14,770 --> 00:43:18,336 is it that the birds are getting their grub or their food 800 00:43:18,336 --> 00:43:18,836 to eat. 801 00:43:21,530 --> 00:43:27,620 First, well, now the axis is somehow vertical. 802 00:43:27,620 --> 00:43:35,210 You could divide them up into the ground foragers 803 00:43:35,210 --> 00:43:39,730 as compared to the tree foragers in terms of where 804 00:43:39,730 --> 00:43:40,971 they're getting their food. 805 00:43:40,971 --> 00:43:43,470 And you say, oh, well, how much of the food is on each side? 806 00:43:43,470 --> 00:43:48,850 Oh, well, we'll say 30% is on the ground, 70% is on the tree. 807 00:43:48,850 --> 00:43:50,889 This is along the stick. 808 00:43:50,889 --> 00:43:52,430 You cut the stick in some way, or you 809 00:43:52,430 --> 00:43:54,530 break the stick in some way. 810 00:43:54,530 --> 00:43:56,060 But then within the tree foragers, 811 00:43:56,060 --> 00:44:00,110 you'd say, well, the resources might be separated. 812 00:44:00,110 --> 00:44:01,750 And this is really like speciation, 813 00:44:01,750 --> 00:44:03,690 a species is in the niche, the species 814 00:44:03,690 --> 00:44:06,300 are focusing on different niches. 815 00:44:06,300 --> 00:44:09,030 So you'd say, oh, some are going to focus on the trunk, 816 00:44:09,030 --> 00:44:12,330 some will focus on branches. 817 00:44:12,330 --> 00:44:13,950 And again, this part of the stick 818 00:44:13,950 --> 00:44:17,720 is now broken or divided among different resource locations 819 00:44:17,720 --> 00:44:18,885 with some amount. 820 00:44:18,885 --> 00:44:21,855 But then also, you're going to get speciation 821 00:44:21,855 --> 00:44:23,730 in different directions here, because there's 822 00:44:23,730 --> 00:44:27,590 both the surface-- I don't know if you guys have ever 823 00:44:27,590 --> 00:44:30,200 eaten grubs-- but there's the surface grubs, 824 00:44:30,200 --> 00:44:32,665 and then there's also the sub-bark grubs. 825 00:44:38,100 --> 00:44:42,230 And so you kind of do this process multiple times, 826 00:44:42,230 --> 00:44:44,650 where you kind of pick different branches 827 00:44:44,650 --> 00:44:47,020 and break them to divide up the niche. 828 00:44:47,020 --> 00:44:49,972 And then you end up with a log normal type distribution. 829 00:44:49,972 --> 00:44:54,840 And this is a similar process to the crushing of the stone, 830 00:44:54,840 --> 00:45:01,870 because the idea is that there's sequential breaks of the stone. 831 00:45:01,870 --> 00:45:04,980 So the stone first breaks into maybe simply two 832 00:45:04,980 --> 00:45:06,474 or it could be three. 833 00:45:06,474 --> 00:45:07,640 First, there's one breaking. 834 00:45:07,640 --> 00:45:09,400 And then one of them is broken more. 835 00:45:09,400 --> 00:45:11,924 So given this process, you end up 836 00:45:11,924 --> 00:45:13,340 getting a log normal distribution. 837 00:45:13,340 --> 00:45:14,334 Yeah. 838 00:45:14,334 --> 00:45:18,640 AUDIENCE: But you also have a distribution of like how far. 839 00:45:18,640 --> 00:45:20,340 Because I guess there are two questions. 840 00:45:20,340 --> 00:45:23,620 Like when you break your stick, you assume, somehow, 841 00:45:23,620 --> 00:45:25,154 that you uniformly break it. 842 00:45:25,154 --> 00:45:25,820 PROFESSOR: Yeah. 843 00:45:29,050 --> 00:45:31,290 A lot of work has gone into the question of how it 844 00:45:31,290 --> 00:45:33,330 is you should break the stick. 845 00:45:33,330 --> 00:45:35,390 Given that you have this tree foraging stick. 846 00:45:39,160 --> 00:45:40,840 On a practical level, what they do is 847 00:45:40,840 --> 00:45:44,140 they ask, well, what probability distribution gives you the best 848 00:45:44,140 --> 00:45:45,490 agreement with the data? 849 00:45:45,490 --> 00:45:46,860 Is it uniform? 850 00:45:46,860 --> 00:45:50,160 Or is it, oh, it's broken like this? 851 00:45:50,160 --> 00:45:52,490 And in some cases people say, well, it's 852 00:45:52,490 --> 00:45:54,055 actually tilted on one side. 853 00:46:00,900 --> 00:46:02,770 Well, in the context of a succession 854 00:46:02,770 --> 00:46:04,350 and some other environments, there's 855 00:46:04,350 --> 00:46:06,660 an idea that, if a species first gets somewhere, 856 00:46:06,660 --> 00:46:08,790 they can kind of monopolize a larger 857 00:46:08,790 --> 00:46:11,630 fraction of the resources then if it's divided kind 858 00:46:11,630 --> 00:46:12,922 of an equally at the beginning. 859 00:46:12,922 --> 00:46:15,505 And that's going to effect where this probability distribution 860 00:46:15,505 --> 00:46:17,150 is going to break each one. 861 00:46:17,150 --> 00:46:23,580 But there's always this question about how constrained 862 00:46:23,580 --> 00:46:25,080 are the notions and so forth. 863 00:46:25,080 --> 00:46:27,384 And I'm agnostic on that point. 864 00:46:27,384 --> 00:46:31,016 AUDIENCE: But you also need distribution for how many times 865 00:46:31,016 --> 00:46:33,440 it breaks [INAUDIBLE]. 866 00:46:33,440 --> 00:46:35,390 PROFESSOR: Yes. 867 00:46:35,390 --> 00:46:38,054 It's just that, if you do this process, 868 00:46:38,054 --> 00:46:39,970 it's like a central limit theorem type result. 869 00:46:39,970 --> 00:46:42,010 So you have to do it enough times so that you get 870 00:46:42,010 --> 00:46:43,260 to some limiting distribution. 871 00:46:43,260 --> 00:46:45,120 And then you could keep on doing it. 872 00:46:45,120 --> 00:46:47,120 In the end, we always say that species abundance 873 00:46:47,120 --> 00:46:49,190 is proportional to the size. 874 00:46:49,190 --> 00:46:51,520 So we're going to scale, ultimately, 875 00:46:51,520 --> 00:46:53,360 to get the correct number of individuals. 876 00:46:53,360 --> 00:46:57,130 It's just that you have to do it some reasonable number of times 877 00:46:57,130 --> 00:46:58,880 so that the randomness kind of washes out, 878 00:46:58,880 --> 00:47:00,963 and you end up approaching that limiting behavior. 879 00:47:04,095 --> 00:47:04,970 Does that make sense? 880 00:47:09,480 --> 00:47:14,690 And indeed I just want to mention a major result 881 00:47:14,690 --> 00:47:15,840 in this field. 882 00:47:18,360 --> 00:47:23,760 These niche type models successfully 883 00:47:23,760 --> 00:47:27,692 explained or predicted another pattern 884 00:47:27,692 --> 00:47:30,150 that had been observed, which is the so-called species area 885 00:47:30,150 --> 00:47:30,733 relationships. 886 00:47:39,440 --> 00:47:43,570 So this is just saying that, here, we looked at 50 hectares, 887 00:47:43,570 --> 00:47:45,610 and we asked how many species where there. 888 00:47:45,610 --> 00:47:47,550 225 species in 50 hectares. 889 00:47:47,550 --> 00:47:50,550 Now, the question is, if instead of looking at 50 hectares, 890 00:47:50,550 --> 00:47:54,082 we instead looked at 500, do you think 891 00:47:54,082 --> 00:47:55,790 of that the number of species we observed 892 00:47:55,790 --> 00:48:01,538 would have gone up, stayed the same, or gone down? 893 00:48:01,538 --> 00:48:07,936 Up, same, down, ready, three, two, one. 894 00:48:07,936 --> 00:48:08,880 Up. 895 00:48:08,880 --> 00:48:09,830 Up. 896 00:48:09,830 --> 00:48:13,660 If you look at a larger area, you 897 00:48:13,660 --> 00:48:17,170 expect to see more species in a larger area. 898 00:48:17,170 --> 00:48:18,890 And people really do this. 899 00:48:18,890 --> 00:48:23,070 They look in some area, going from, say, they take a meter, 900 00:48:23,070 --> 00:48:24,450 and they count all the species. 901 00:48:24,450 --> 00:48:26,530 And then they go and here is 100 meters, 902 00:48:26,530 --> 00:48:29,060 and they count all the species. 903 00:48:29,060 --> 00:48:31,190 And they ask, how many species do you 904 00:48:31,190 --> 00:48:32,440 see as a function of the area? 905 00:48:32,440 --> 00:48:34,773 And what people have found is that the number of species 906 00:48:34,773 --> 00:48:42,580 you observe it is proportional to the area to some power, 907 00:48:42,580 --> 00:48:44,790 where Z is around a 1/4. 908 00:48:52,036 --> 00:48:55,584 And of course, the area goes as some r squared. 909 00:48:55,584 --> 00:48:57,000 If you wanted to, you could say it 910 00:48:57,000 --> 00:49:01,000 goes as the square root of the radius, whatever. 911 00:49:01,000 --> 00:49:03,670 But the number species in some area, 912 00:49:03,670 --> 00:49:06,100 it grows, but it grows in a manner 913 00:49:06,100 --> 00:49:08,870 that is less than linear. 914 00:49:08,870 --> 00:49:11,349 Does that make sense? 915 00:49:11,349 --> 00:49:13,390 It definitely makes sense that's less the linear. 916 00:49:13,390 --> 00:49:17,302 Because linear would be that you sample a bunch of species here, 917 00:49:17,302 --> 00:49:19,135 and then you look at another identical plot, 918 00:49:19,135 --> 00:49:20,125 you get some other species. 919 00:49:20,125 --> 00:49:22,583 And they were saying that, oh, that you really don't expect 920 00:49:22,583 --> 00:49:24,400 any of those species overlap. 921 00:49:24,400 --> 00:49:26,420 That would be a weird world. 922 00:49:26,420 --> 00:49:31,390 So it very much make sense that this is less than 1. 923 00:49:31,390 --> 00:49:34,550 Of course, it didn't have to be this power law. 924 00:49:34,550 --> 00:49:39,010 But one thing that has been discovered, around the world, 925 00:49:39,010 --> 00:49:42,050 is that power laws are very interesting. 926 00:49:42,050 --> 00:49:45,810 But once again, many different microscopic processes 927 00:49:45,810 --> 00:49:47,200 can lead to power laws. 928 00:49:47,200 --> 00:49:49,270 The niche models have successfully 929 00:49:49,270 --> 00:49:53,290 predicted or explained why it might have this scaling. 930 00:49:53,290 --> 00:49:56,400 But it turns out that neutral models can also predict it. 931 00:49:56,400 --> 00:50:00,490 And may just be that lots of spatially explicit models 932 00:50:00,490 --> 00:50:02,520 will give you some power law type scaling that 933 00:50:02,520 --> 00:50:04,589 looks kind of like this. 934 00:50:04,589 --> 00:50:06,130 So once again, it's a question of how 935 00:50:06,130 --> 00:50:09,162 convinced you should be about microscopic processes 936 00:50:09,162 --> 00:50:10,870 based on being able to explain some data. 937 00:50:10,870 --> 00:50:17,880 And I think the best cure for this danger, 938 00:50:17,880 --> 00:50:20,480 of assuming that the microscopic assumptions are correct, 939 00:50:20,480 --> 00:50:22,438 because the model is able to explain something, 940 00:50:22,438 --> 00:50:25,010 is that, if you find some other very different set 941 00:50:25,010 --> 00:50:29,629 of microscopic assumptions that also explain the patterns, then 942 00:50:29,629 --> 00:50:31,670 it becomes clear that you have to take everything 943 00:50:31,670 --> 00:50:33,496 with a grain of salt. 944 00:50:33,496 --> 00:50:34,870 And that's I think part of what's 945 00:50:34,870 --> 00:50:38,270 been very valuable about the neutral theory contribution 946 00:50:38,270 --> 00:50:39,010 to this field. 947 00:50:39,010 --> 00:50:41,179 AUDIENCE: Does this just come from-- you 948 00:50:41,179 --> 00:50:44,312 assume that all the individuals are uniformly distributed 949 00:50:44,312 --> 00:50:45,758 and then [INAUDIBLE]? 950 00:50:50,997 --> 00:50:53,080 PROFESSOR: There are multiple derivations of this, 951 00:50:53,080 --> 00:50:54,371 so it's a little bit confusing. 952 00:51:03,135 --> 00:51:05,500 The neutral models, that I have seen, 953 00:51:05,500 --> 00:51:07,780 that lead to these patterns, they basically 954 00:51:07,780 --> 00:51:11,510 have the individuals randomly, either 955 00:51:11,510 --> 00:51:14,490 with sex or without sex, kind of diffusing around, 956 00:51:14,490 --> 00:51:16,640 and then they divide, deh-deh. 957 00:51:16,640 --> 00:51:20,190 And then you can explicitly just do the different spaces 958 00:51:20,190 --> 00:51:22,040 and see that you get a scaling. 959 00:51:22,040 --> 00:51:28,210 It seems to be a surprisingly emergent feature 960 00:51:28,210 --> 00:51:31,190 of many of these models. 961 00:51:31,190 --> 00:51:32,740 And once again, it may be something 962 00:51:32,740 --> 00:51:34,492 that tells us less about biology than it 963 00:51:34,492 --> 00:51:35,700 does about math or something. 964 00:51:44,940 --> 00:51:47,890 Any other questions about this, the base notion 965 00:51:47,890 --> 00:51:49,390 of this niche hierarchy type models? 966 00:51:54,510 --> 00:51:56,060 So I want to spend some time talking 967 00:51:56,060 --> 00:51:58,830 about this neutral theory in ecology. 968 00:51:58,830 --> 00:52:00,550 The math, in particular the derivation 969 00:52:00,550 --> 00:52:02,216 of this particular closed form solution, 970 00:52:02,216 --> 00:52:04,775 is not really so interesting or relevant. 971 00:52:04,775 --> 00:52:06,650 But I think it's very important to understand 972 00:52:06,650 --> 00:52:09,635 what the assumptions are in the model and maybe also something 973 00:52:09,635 --> 00:52:12,260 about the circumstances in which we think that it should apply. 974 00:52:25,530 --> 00:52:29,370 So the basic idea is that we have, what we hope, 975 00:52:29,370 --> 00:52:36,140 is some metacommunity that is large. 976 00:52:39,530 --> 00:52:40,900 And then we have an island. 977 00:52:40,900 --> 00:52:44,670 So this has to do with this theory of island biogeography. 978 00:52:44,670 --> 00:52:47,185 We have an island over here. 979 00:52:47,185 --> 00:52:51,540 And in the context of the nomenclature of this paper, 980 00:52:51,540 --> 00:52:54,249 they are some community size, size j here. 981 00:52:54,249 --> 00:52:56,165 This tells us about the number of individuals. 982 00:53:01,360 --> 00:53:04,830 And they're distributed across some number of species. 983 00:53:04,830 --> 00:53:08,710 Now, the neutral theory, the key thing 984 00:53:08,710 --> 00:53:12,260 is that we assume that all individuals are identical. 985 00:53:22,780 --> 00:53:26,490 And once again, it's not that the neutral theorists 986 00:53:26,490 --> 00:53:28,310 believe that this is true. 987 00:53:28,310 --> 00:53:31,314 It's that they think that it may be sufficient to explain 988 00:53:31,314 --> 00:53:32,605 the patterns that are observed. 989 00:53:40,755 --> 00:53:42,880 And when we say that all individuals are identical, 990 00:53:42,880 --> 00:53:46,470 what we mean is that the demographic parameters are 991 00:53:46,470 --> 00:53:49,586 the same, birth, death rates. 992 00:53:53,990 --> 00:53:56,460 And it's even a stronger assumption, in some ways, 993 00:53:56,460 --> 00:53:57,010 than that. 994 00:53:57,010 --> 00:53:59,010 It's assuming that the individuals are the same, 995 00:53:59,010 --> 00:54:02,360 the species are the same, and that there are no interactions 996 00:54:02,360 --> 00:54:03,560 within the species as well. 997 00:54:03,560 --> 00:54:07,880 So there's no Alley effect, or no specific competition. 998 00:54:07,880 --> 00:54:09,480 So the birth, death rates are going 999 00:54:09,480 --> 00:54:16,120 to be independent of everything, which is an amazingly 1000 00:54:16,120 --> 00:54:16,917 parsimonious model. 1001 00:54:16,917 --> 00:54:19,250 And it's kind of amazing you can get anything out of it. 1002 00:54:24,040 --> 00:54:28,150 And then we have a migration rate m. 1003 00:54:28,150 --> 00:54:29,740 It's either a rate or a probability, 1004 00:54:29,740 --> 00:54:32,230 depending on how you think about it. 1005 00:54:32,230 --> 00:54:35,490 Rate or probability m. 1006 00:54:35,490 --> 00:54:42,330 And can somebody remind us how we handle that? 1007 00:54:48,713 --> 00:54:51,168 AUDIENCE: Both just in a community? 1008 00:54:51,168 --> 00:54:52,150 PROFESSOR: Yeah. 1009 00:54:52,150 --> 00:54:54,114 AUDIENCE: At some probability that 1010 00:54:54,114 --> 00:54:57,167 is proportional to the distribution of the species 1011 00:54:57,167 --> 00:54:58,042 in the metacommunity? 1012 00:55:00,790 --> 00:55:02,040 PROFESSOR: Yeah, that's right. 1013 00:55:02,040 --> 00:55:03,531 AUDIENCE: --transfer an individual 1014 00:55:03,531 --> 00:55:05,919 from the metacommunity to the island. 1015 00:55:05,919 --> 00:55:06,710 PROFESSOR: Perfect. 1016 00:55:06,710 --> 00:55:08,264 AUDIENCE: We do stick to the island 1017 00:55:08,264 --> 00:55:09,820 to make sure that number of individuals. 1018 00:55:09,820 --> 00:55:10,528 PROFESSOR: Right. 1019 00:55:10,528 --> 00:55:15,150 So what we're going to do is we're basically 1020 00:55:15,150 --> 00:55:19,570 going to pick a random individual, here, each cycle. 1021 00:55:19,570 --> 00:55:21,440 This is kind of like a Moran process. 1022 00:55:21,440 --> 00:55:23,320 We're going to pick an individual here. 1023 00:55:23,320 --> 00:55:25,950 And we're going to kill him. 1024 00:55:25,950 --> 00:55:30,740 And then what we're going to do is, with probability m, 1025 00:55:30,740 --> 00:55:32,830 replace that individual with one member 1026 00:55:32,830 --> 00:55:35,030 of the metacommunity at random. 1027 00:55:35,030 --> 00:55:36,930 So the rate coming from here will 1028 00:55:36,930 --> 00:55:40,380 be proportional to the species abundance in the metacommunity. 1029 00:55:40,380 --> 00:55:43,430 And with a probability of 1 minus m, what we're going to do 1030 00:55:43,430 --> 00:55:46,400 is we're going to replace that individual 1031 00:55:46,400 --> 00:55:51,960 with another individual in the island. 1032 00:55:51,960 --> 00:55:56,510 Now, the math kind of gets hairy and complicated. 1033 00:55:56,510 --> 00:56:01,547 But the basic notion is really quite simple. 1034 00:56:01,547 --> 00:56:03,130 You have a metacommunity distribution, 1035 00:56:03,130 --> 00:56:04,980 which is going to end up being the so-called Fisher log 1036 00:56:04,980 --> 00:56:06,030 series in this model. 1037 00:56:13,780 --> 00:56:17,004 This describes the species abundance on the metacommunity. 1038 00:56:17,004 --> 00:56:18,420 But then on the island, we're just 1039 00:56:18,420 --> 00:56:21,100 going to assume that there's birth, death that occurs over 1040 00:56:21,100 --> 00:56:22,210 here at some rate. 1041 00:56:22,210 --> 00:56:25,205 But we don't even have to hardly think about that. 1042 00:56:25,205 --> 00:56:27,330 From the standpoint of, say, a simulation or model, 1043 00:56:27,330 --> 00:56:29,070 we just run multiple cycles of this, 1044 00:56:29,070 --> 00:56:30,260 where we have j individuals. 1045 00:56:30,260 --> 00:56:31,260 And we always have j individuals, 1046 00:56:31,260 --> 00:56:32,830 because it's like the Moran process. 1047 00:56:32,830 --> 00:56:35,066 At every time point, we kill one individual, 1048 00:56:35,066 --> 00:56:37,690 and we replace it, with somebody either from the same community 1049 00:56:37,690 --> 00:56:40,840 or from the island. 1050 00:56:40,840 --> 00:56:47,539 And you can imagine that in the limit of m going to zero, 1051 00:56:47,539 --> 00:56:49,080 what's going to happen on the island? 1052 00:56:52,240 --> 00:56:54,120 Yeah, so you'll end up just one species, just 1053 00:56:54,120 --> 00:56:56,980 because this is just random, like genetic drift. 1054 00:56:56,980 --> 00:57:00,410 It's ecological drift where one species will take over. 1055 00:57:00,410 --> 00:57:04,320 Whereas if m is large, then somehow it's 1056 00:57:04,320 --> 00:57:07,405 more of a reflection of the metacommunity. 1057 00:57:13,680 --> 00:57:15,830 Are there any questions about what this model 1058 00:57:15,830 --> 00:57:18,098 is looking like for now? 1059 00:57:18,098 --> 00:57:20,264 AUDIENCE: Could we talk about the Fisher log series? 1060 00:57:20,264 --> 00:57:20,732 PROFESSOR: Yeah. 1061 00:57:20,732 --> 00:57:22,606 AUDIENCE: So we would put it on the same axis 1062 00:57:22,606 --> 00:57:23,540 as the [INAUDIBLE]? 1063 00:57:23,540 --> 00:57:25,770 PROFESSOR: Yes, this is a very, very good question. 1064 00:57:25,770 --> 00:57:27,319 So we'll do this in just a moment. 1065 00:57:27,319 --> 00:57:28,610 Because this is very important. 1066 00:57:32,210 --> 00:57:35,950 I want to say just a couple things about this model. 1067 00:57:35,950 --> 00:57:37,750 So when I read this paper, what I imagined 1068 00:57:37,750 --> 00:57:39,208 is that it really looked like this. 1069 00:57:39,208 --> 00:57:44,890 This was Panama, and that, 30 kilometers off the coast, 1070 00:57:44,890 --> 00:57:48,610 there was this island, BCI, Barro Colorado Island. 1071 00:57:48,610 --> 00:57:51,560 But that's not maybe an accurate description 1072 00:57:51,560 --> 00:57:54,660 of what the real system looks like. 1073 00:57:54,660 --> 00:57:56,340 Does anybody know where BCI is? 1074 00:57:56,340 --> 00:57:57,907 AUDIENCE: It's in Panama. 1075 00:57:57,907 --> 00:57:58,490 PROFESSOR: Hm? 1076 00:57:58,490 --> 00:57:59,350 AUDIENCE: Panama. 1077 00:57:59,350 --> 00:58:00,600 PROFESSOR: So it is in Panama. 1078 00:58:00,600 --> 00:58:02,430 But it's not off the coast of Panama. 1079 00:58:02,430 --> 00:58:03,820 I guess that was my original. 1080 00:58:03,820 --> 00:58:05,450 AUDIENCE: It's in the canal. 1081 00:58:05,450 --> 00:58:07,050 PROFESSOR: Yeah, it's in the canal. 1082 00:58:07,050 --> 00:58:09,400 So it's an island that was created when 1083 00:58:09,400 --> 00:58:11,540 they made the Panama Canal. 1084 00:58:11,540 --> 00:58:13,520 So this thing was not always an island. 1085 00:58:13,520 --> 00:58:16,240 It's been an island for 100 years. 1086 00:58:16,240 --> 00:58:17,959 And it's in the middle of a canal. 1087 00:58:17,959 --> 00:58:20,250 And they actually have cougars that swim back and forth 1088 00:58:20,250 --> 00:58:22,980 from the mainland. 1089 00:58:22,980 --> 00:58:25,952 But it does make you wonder whether this 1090 00:58:25,952 --> 00:58:32,390 is-- it's much more strongly coupled to the mainland 1091 00:58:32,390 --> 00:58:36,670 then I imagined when I read this paper at first. 1092 00:58:36,670 --> 00:58:39,390 I don't know what that means for all this. 1093 00:58:39,390 --> 00:58:42,680 But certainly, you expect this to be 1094 00:58:42,680 --> 00:58:44,880 a more or less appropriate model depending on this. 1095 00:58:44,880 --> 00:58:46,810 Because, of course, if you went and you 1096 00:58:46,810 --> 00:58:48,990 sampled 50 hectares here, you wouldn't 1097 00:58:48,990 --> 00:58:51,440 believe that it should have the same distribution. 1098 00:58:51,440 --> 00:58:54,706 You'd believe it should be more like the Fisher log series. 1099 00:58:54,706 --> 00:58:57,080 And there's some evidence that things are tilted in a way 1100 00:58:57,080 --> 00:58:57,920 that you would expect. 1101 00:58:57,920 --> 00:58:59,003 And we'll talk about that. 1102 00:59:01,890 --> 00:59:02,470 It's tricky. 1103 00:59:02,470 --> 00:59:04,990 And of course, you have to decide in all this stuff, oh, 1104 00:59:04,990 --> 00:59:07,480 what do you mean by free parameters? 1105 00:59:07,480 --> 00:59:09,750 And actually, it seems like people can't count. 1106 00:59:09,750 --> 00:59:11,541 And we'll talk about this in a moment, too. 1107 00:59:14,680 --> 00:59:16,830 Because, of course, constructing the model, 1108 00:59:16,830 --> 00:59:19,450 there's some sense of free parameters that you have there. 1109 00:59:19,450 --> 00:59:21,360 Because we could have said, oh, it's 1110 00:59:21,360 --> 00:59:23,140 just going to be the Fisher log series, or we could have said, 1111 00:59:23,140 --> 00:59:23,880 oh, it's going to be island. 1112 00:59:23,880 --> 00:59:25,296 Or we could have said, oh, there's 1113 00:59:25,296 --> 00:59:26,660 another island out here. 1114 00:59:26,660 --> 00:59:28,760 And then that would be another distribution. 1115 00:59:28,760 --> 00:59:31,480 And not all of these things introduce more free parameters, 1116 00:59:31,480 --> 00:59:32,550 necessarily, because you could say, 1117 00:59:32,550 --> 00:59:34,049 oh, this is the same migration rate, 1118 00:59:34,049 --> 00:59:35,240 or you could do something. 1119 00:59:35,240 --> 00:59:37,770 But they are going to lead to different distributions, 1120 00:59:37,770 --> 00:59:38,680 and you have that freedom when you're 1121 00:59:38,680 --> 00:59:39,804 trying to explain the data. 1122 00:59:42,180 --> 00:59:46,156 There are a lot of judgment calls in this business. 1123 00:59:46,156 --> 00:59:47,780 But let's talk about Fisher log series, 1124 00:59:47,780 --> 00:59:48,995 because this is relevant. 1125 00:59:54,550 --> 00:59:58,680 So the model is very similar to what we did for the master 1126 00:59:58,680 --> 01:00:01,290 equation in the context of gene expression 1127 01:00:01,290 --> 01:00:03,930 and the number of mRNA. 1128 01:00:03,930 --> 01:00:06,610 So was the equilibrium or steady state 1129 01:00:06,610 --> 01:00:11,540 distribution of mRNA in a cell, was that a Fisher log series? 1130 01:00:11,540 --> 01:00:14,420 Yes or no, five seconds? 1131 01:00:17,060 --> 01:00:20,970 Was the mRNA steady state probability distribution 1132 01:00:20,970 --> 01:00:21,990 a Fisher log series? 1133 01:00:21,990 --> 01:00:24,550 Ready, three, two, one. 1134 01:00:27,450 --> 01:00:28,030 No. 1135 01:00:28,030 --> 01:00:28,529 No. 1136 01:00:28,529 --> 01:00:29,415 What was it? 1137 01:00:29,415 --> 01:00:30,870 It was a Poisson. 1138 01:00:30,870 --> 01:00:34,420 And you guys should review what all these distributions are, 1139 01:00:34,420 --> 01:00:35,790 when you get them, and so forth. 1140 01:00:35,790 --> 01:00:37,300 So what was the Difference why is it 1141 01:00:37,300 --> 01:00:48,770 that we have some probability, P0, P1, P2? 1142 01:00:48,770 --> 01:00:50,490 This could be mRNA or it could be 1143 01:00:50,490 --> 01:00:53,590 number of individuals in some species with some birth 1144 01:00:53,590 --> 01:00:54,440 and death rates. 1145 01:00:54,440 --> 01:00:58,070 What was the key difference between the mRNA model, which 1146 01:00:58,070 --> 01:01:01,670 led to this distribution becoming Poisson, 1147 01:01:01,670 --> 01:01:03,790 and the model that we just studied here, where 1148 01:01:03,790 --> 01:01:05,080 it became a Fisher log series? 1149 01:01:09,200 --> 01:01:13,020 And I should maybe write down what the Fisher log series is. 1150 01:01:13,020 --> 01:01:17,740 So this is the expected number of species with n individuals 1151 01:01:17,740 --> 01:01:19,069 on the metacommunity. 1152 01:01:19,069 --> 01:01:20,360 Here is the Fisher log species. 1153 01:01:20,360 --> 01:01:23,970 There was some theta X to the n divided by n. 1154 01:01:28,782 --> 01:01:29,990 So what's the key difference? 1155 01:01:29,990 --> 01:01:31,244 Yeah. 1156 01:01:31,244 --> 01:01:36,711 AUDIENCE: I think that the birth and death rates are 1157 01:01:36,711 --> 01:01:38,202 both proportional [INAUDIBLE]. 1158 01:01:41,989 --> 01:01:43,780 PROFESSOR: Right, the birth and death rates 1159 01:01:43,780 --> 01:01:44,855 are both proportional. 1160 01:01:44,855 --> 01:01:46,330 AUDIENCE: In the Fisher log series. 1161 01:01:46,330 --> 01:01:47,829 PROFESSOR: In the Fisher log series. 1162 01:01:47,829 --> 01:01:51,630 So what we have is that b0-- and what should we 1163 01:01:51,630 --> 01:01:54,544 call b0 in this model? 1164 01:01:54,544 --> 01:01:56,055 AUDIENCE: [INAUDIBLE]. 1165 01:01:56,055 --> 01:01:57,430 PROFESSOR: Well, right now, we're 1166 01:01:57,430 --> 01:01:59,026 thinking about the metacommunity. 1167 01:01:59,026 --> 01:01:59,900 AUDIENCE: Speciation. 1168 01:01:59,900 --> 01:02:00,816 PROFESSOR: Speciation. 1169 01:02:00,816 --> 01:02:03,870 b0 is speciation, which we're going to assume 1170 01:02:03,870 --> 01:02:06,670 is going to be constant. 1171 01:02:06,670 --> 01:02:11,070 In this model, do we have speciation on the island? 1172 01:02:11,070 --> 01:02:12,046 No. 1173 01:02:12,046 --> 01:02:13,420 The assumption is that the island 1174 01:02:13,420 --> 01:02:16,280 is small enough that the rate of speciation is just negligible. 1175 01:02:16,280 --> 01:02:20,164 So speciation plays a role in forming 1176 01:02:20,164 --> 01:02:22,080 the metacommunity distribution, but it doesn't 1177 01:02:22,080 --> 01:02:25,210 play a role in the model. 1178 01:02:25,210 --> 01:02:27,280 So this is speciation. 1179 01:02:27,280 --> 01:02:30,610 But then what we assume is that b1, here, 1180 01:02:30,610 --> 01:02:35,180 is equal to some fundamental rate b times n, 1181 01:02:35,180 --> 01:02:36,870 but it's b times, in this case, 1. 1182 01:02:36,870 --> 01:02:43,840 So more broadly, bn is equal to some birth rate times n. 1183 01:02:43,840 --> 01:02:46,290 This is saying that the individuals can give birth 1184 01:02:46,290 --> 01:02:49,260 to other individuals. 1185 01:02:49,260 --> 01:02:51,919 Now, we're not assuming anything about sexual reproduction 1186 01:02:51,919 --> 01:02:52,710 necessarily or not. 1187 01:02:52,710 --> 01:02:55,320 We're just saying that the kind of rates 1188 01:02:55,320 --> 01:02:56,800 are proportional to the numbers. 1189 01:02:56,800 --> 01:02:58,520 So if you have twice as many individuals, 1190 01:02:58,520 --> 01:03:00,209 the birth rate will be twice as large. 1191 01:03:00,209 --> 01:03:01,000 This is reasonable. 1192 01:03:07,980 --> 01:03:12,250 This is Pn and this is Pn plus 1. 1193 01:03:12,250 --> 01:03:17,200 So this is d of n plus 1 is equal to some death 1194 01:03:17,200 --> 01:03:19,330 rate times n plus 1. 1195 01:03:19,330 --> 01:03:21,800 So each individual just has some rate of dying. 1196 01:03:21,800 --> 01:03:23,152 It's exponentially distributed. 1197 01:03:23,152 --> 01:03:24,110 This again makes sense. 1198 01:03:26,750 --> 01:03:28,990 What was the key difference between our mRNA model, 1199 01:03:28,990 --> 01:03:30,406 from before that gave the Poisson, 1200 01:03:30,406 --> 01:03:32,670 and this model that gives the Fisher log series? 1201 01:03:32,670 --> 01:03:36,992 AUDIENCE: So with the mRNA, it's with a standard like a chemical 1202 01:03:36,992 --> 01:03:41,952 equation where there's some fixed external input. 1203 01:03:41,952 --> 01:03:44,680 But then the degradation is according 1204 01:03:44,680 --> 01:03:45,920 to the amount that you have. 1205 01:03:45,920 --> 01:03:48,420 So death is proportionate [INAUDIBLE]. 1206 01:03:48,420 --> 01:03:49,790 PROFESSOR: Perfect. 1207 01:03:49,790 --> 01:03:52,400 In both cases, the death rate is proportional to the number 1208 01:03:52,400 --> 01:03:55,140 of either mRNA or individuals. 1209 01:03:55,140 --> 01:03:57,140 However, in the mRNA model, what we assume 1210 01:03:57,140 --> 01:03:59,840 is there some just constant rate of transcription, 1211 01:03:59,840 --> 01:04:02,466 so a constant rate, per unit time, of making more mRNA. 1212 01:04:02,466 --> 01:04:03,840 So just because there's more mRNA 1213 01:04:03,840 --> 01:04:06,950 doesn't mean that you're going to get more mRNA. 1214 01:04:06,950 --> 01:04:10,080 But here, we assume that the birth rate 1215 01:04:10,080 --> 01:04:12,100 is proportional to the number. 1216 01:04:12,100 --> 01:04:13,982 So that's what leads to the difference. 1217 01:04:13,982 --> 01:04:15,690 And so this is one of the few other cases 1218 01:04:15,690 --> 01:04:18,060 that you can simply solve the master equation 1219 01:04:18,060 --> 01:04:19,692 and get an equilibrium distribution. 1220 01:04:19,692 --> 01:04:21,650 And it's the same thing we do from just always, 1221 01:04:21,650 --> 01:04:26,840 where we say, at steady state, the probability fluxes 1222 01:04:26,840 --> 01:04:28,800 or whatever are equal. 1223 01:04:28,800 --> 01:04:32,510 So you get that P1 should be equal to P0. 1224 01:04:32,510 --> 01:04:35,316 and then we have a b0 divided by d1. 1225 01:04:35,316 --> 01:04:39,130 And more broadly, we just cycle through. 1226 01:04:39,130 --> 01:04:41,400 The probability of being in the nth state, 1227 01:04:41,400 --> 01:04:43,060 it's going to be some P0. 1228 01:04:43,060 --> 01:04:48,380 And then basically, it's going to b0 divided by d1, 1229 01:04:48,380 --> 01:04:58,470 b1 divided by d2, b2, d3, dot, dot, dot, up to bn minus 1 dn. 1230 01:04:58,470 --> 01:05:01,780 And indeed, if we just plug in what these things are equal to, 1231 01:05:01,780 --> 01:05:06,650 we end up getting-- there's P0, the fundamental birth 1232 01:05:06,650 --> 01:05:08,191 over death to the nth power. 1233 01:05:08,191 --> 01:05:09,940 And then we just are left with a 1 over n. 1234 01:05:09,940 --> 01:05:13,170 Because we're going to have a 2 here and a 2 here, and those 1235 01:05:13,170 --> 01:05:13,670 cancel. 1236 01:05:13,670 --> 01:05:15,370 A 2 here and 3 here, and those cancel. 1237 01:05:15,370 --> 01:05:20,130 And we're just left with the n at the end, finally. 1238 01:05:20,130 --> 01:05:24,470 So this x, over there, is then, in this model, 1239 01:05:24,470 --> 01:05:27,920 the ratio of the birth and death rates. 1240 01:05:27,920 --> 01:05:29,890 So which one is larger? 1241 01:05:29,890 --> 01:05:32,330 Is it A slash 1? 1242 01:05:32,330 --> 01:05:35,270 Is it b is greater than d? 1243 01:05:35,270 --> 01:05:39,844 Or is it b slash 2, that b is less than d? 1244 01:05:39,844 --> 01:05:41,260 Think about this for five seconds. 1245 01:05:43,770 --> 01:05:47,002 Do you think that birth rates should 1246 01:05:47,002 --> 01:05:48,710 be larger than death rates or death rates 1247 01:05:48,710 --> 01:05:51,168 should be larger than birth rates or do they have to equal? 1248 01:05:54,990 --> 01:05:57,940 Ready, three, two, one. 1249 01:06:00,460 --> 01:06:07,780 So we got a number of-- it's kind of distributed, 1 and 2's. 1250 01:06:07,780 --> 01:06:10,322 Well, it's maybe not that deep, not deep enough. 1251 01:06:10,322 --> 01:06:11,780 Can somebody say why their neighbor 1252 01:06:11,780 --> 01:06:12,988 thinks it's one or the other? 1253 01:06:15,374 --> 01:06:17,290 People are actually turning to their neighbor. 1254 01:06:20,720 --> 01:06:22,336 A justification for one or the other. 1255 01:06:22,336 --> 01:06:26,144 AUDIENCE: So if this problem where b over d 1256 01:06:26,144 --> 01:06:30,242 is greater than 1, then this distribution is not normalized. 1257 01:06:30,242 --> 01:06:30,950 PROFESSOR: Right. 1258 01:06:30,950 --> 01:06:35,380 So if b over d is greater than 1, so if x is greater than 1, 1259 01:06:35,380 --> 01:06:39,480 then this distribution blows up. 1260 01:06:39,480 --> 01:06:41,230 Then it gets more and more likely to have 1261 01:06:41,230 --> 01:06:44,240 all these larger numbers. 1262 01:06:44,240 --> 01:06:48,510 But then if b is less than d, shouldn't everybody be extinct? 1263 01:06:48,510 --> 01:06:50,000 No. 1264 01:06:50,000 --> 01:06:52,640 Can somebody else say why it is that it's OK 1265 01:06:52,640 --> 01:06:53,640 for b to be less than d? 1266 01:06:53,640 --> 01:06:55,348 If birth rates are less than death rates, 1267 01:06:55,348 --> 01:06:56,681 shouldn't everyone be extinct? 1268 01:06:56,681 --> 01:06:59,140 AUDIENCE: Because there's a rate b0. 1269 01:06:59,140 --> 01:07:01,350 PROFESSOR: Because there's a rate b0, exactly. 1270 01:07:01,350 --> 01:07:03,250 So there's a finite rate of speciation. 1271 01:07:03,250 --> 01:07:05,505 So it's true that every species will go extinct. 1272 01:07:08,070 --> 01:07:12,050 But because we have a constant influx of new species, 1273 01:07:12,050 --> 01:07:16,170 we end up with this distribution that's this Fisher log series. 1274 01:07:16,170 --> 01:07:21,230 Now, if you plot the Fisher log series, 1275 01:07:21,230 --> 01:07:23,690 it looks a bit like this. 1276 01:07:23,690 --> 01:07:25,990 But let's think about it a little bit. 1277 01:07:25,990 --> 01:07:28,540 Does the Fisher log series, does it fall off, 1278 01:07:28,540 --> 01:07:32,200 A, faster or slower than this? 1279 01:07:32,200 --> 01:07:41,580 Fisher falls, A, faster-- this is in this direction-- or, B, 1280 01:07:41,580 --> 01:07:42,080 slower? 1281 01:07:48,519 --> 01:07:50,060 AUDIENCE: Faster or slower than what? 1282 01:07:50,060 --> 01:07:54,300 PROFESSOR: Than the island distribution. 1283 01:07:54,300 --> 01:07:56,640 Because you can see that this falls off pretty rapidly. 1284 01:08:01,180 --> 01:08:04,250 Ready, maybe? 1285 01:08:04,250 --> 01:08:06,575 Three, two, one. 1286 01:08:14,850 --> 01:08:18,452 I saw a fair number of people that 1287 01:08:18,452 --> 01:08:20,380 don't want to make a guess. 1288 01:08:20,380 --> 01:08:21,810 Indeed, it's going to be faster. 1289 01:08:21,810 --> 01:08:22,765 Can somebody say why? 1290 01:08:26,639 --> 01:08:27,139 Yeah. 1291 01:08:27,139 --> 01:08:28,055 AUDIENCE: [INAUDIBLE]. 1292 01:08:30,361 --> 01:08:32,569 PROFESSOR: Is it going to be because of the 1 over n? 1293 01:08:32,569 --> 01:08:34,460 I mean the 1 over n is certainly relevant. 1294 01:08:37,399 --> 01:08:38,899 Without the one over n, then we just 1295 01:08:38,899 --> 01:08:40,930 have sort of a geometric series. 1296 01:08:40,930 --> 01:08:43,550 And the log normal is not just a geometric series either. 1297 01:08:47,382 --> 01:08:53,590 AUDIENCE: [INAUDIBLE] Whereas this has a very long tail. 1298 01:08:53,590 --> 01:08:54,590 PROFESSOR: That's right. 1299 01:08:54,590 --> 01:08:55,649 So this falls off. 1300 01:08:58,189 --> 01:08:59,689 This would be kind of exponentially, 1301 01:08:59,689 --> 01:09:01,272 and this is faster than exponentially. 1302 01:09:03,420 --> 01:09:07,600 And indeed, this make sense based on the model. 1303 01:09:07,600 --> 01:09:10,840 Because this community, the reason 1304 01:09:10,840 --> 01:09:15,920 that it has some very, very abundant species 1305 01:09:15,920 --> 01:09:18,090 is partly because it gets migration 1306 01:09:18,090 --> 01:09:19,479 from the abundant species here. 1307 01:09:22,680 --> 01:09:26,200 This falls off pretty quickly. 1308 01:09:26,200 --> 01:09:30,450 But those frequent species still can play a pretty important 1309 01:09:30,450 --> 01:09:33,859 role in the island community, because the migration rate 1310 01:09:33,859 --> 01:09:38,924 is influenced by large numbers. 1311 01:09:38,924 --> 01:09:40,340 And the other thing is, of course, 1312 01:09:40,340 --> 01:09:43,890 that the rare species are going to often go extinct. 1313 01:09:43,890 --> 01:09:45,880 I mean the distribution on the island 1314 01:09:45,880 --> 01:09:50,330 is some complicated process of the dynamics going here, 1315 01:09:50,330 --> 01:09:51,800 plus sampling from here. 1316 01:09:51,800 --> 01:09:54,480 But there's a sense that it's biased towards-- it's not just 1317 01:09:54,480 --> 01:09:56,690 a reflection of the metacommunity, 1318 01:09:56,690 --> 01:09:58,430 because the migration rate is sampled 1319 01:09:58,430 --> 01:09:59,830 towards the abundant species. 1320 01:09:59,830 --> 01:10:02,620 So the migration of these species 1321 01:10:02,620 --> 01:10:05,980 ends up playing a major role in pushing the distribution 1322 01:10:05,980 --> 01:10:08,390 to the right. 1323 01:10:08,390 --> 01:10:14,500 So you have much more frequent, abundant species on the island 1324 01:10:14,500 --> 01:10:18,205 as compared to the mainland. 1325 01:10:18,205 --> 01:10:19,167 AUDIENCE: [INAUDIBLE]? 1326 01:10:19,167 --> 01:10:19,833 PROFESSOR: Yeah. 1327 01:10:19,833 --> 01:10:23,015 AUDIENCE: [INAUDIBLE] measurement of the distribution 1328 01:10:23,015 --> 01:10:24,940 on the-- 1329 01:10:24,940 --> 01:10:26,500 PROFESSOR: Well, I'm sure they have. 1330 01:10:29,810 --> 01:10:32,450 I think the statement that there's a faster fall 1331 01:10:32,450 --> 01:10:34,120 off on mainlands than on the islands 1332 01:10:34,120 --> 01:10:37,580 I think is borne out by the data. 1333 01:10:37,580 --> 01:10:42,730 But I don't know if trees on the Panama side of the canal 1334 01:10:42,730 --> 01:10:45,050 are actually better described by a Fisher log series 1335 01:10:45,050 --> 01:10:47,215 as compared to this, though. 1336 01:10:47,215 --> 01:10:50,185 AUDIENCE: I guess my question was the abundant species 1337 01:10:50,185 --> 01:10:52,907 that we see on the island, is it just 1338 01:10:52,907 --> 01:10:57,120 the result of diffusive drift? 1339 01:10:57,120 --> 01:11:02,630 PROFESSOR: Well, this also has the diffusive drift. 1340 01:11:08,398 --> 01:11:12,462 AUDIENCE: But in the sense that what really pushes. 1341 01:11:12,462 --> 01:11:13,920 PROFESSOR: Well, I mean I think you 1342 01:11:13,920 --> 01:11:16,705 need both, the diffusive drift and the migration. 1343 01:11:16,705 --> 01:11:19,010 But I think that the fact that the migration is 1344 01:11:19,010 --> 01:11:20,720 from the mainland, and it's biased 1345 01:11:20,720 --> 01:11:24,190 towards those abundant things, I think 1346 01:11:24,190 --> 01:11:26,849 is necessary or important. 1347 01:11:26,849 --> 01:11:28,890 AUDIENCE: I guess just in terms of distinguishing 1348 01:11:28,890 --> 01:11:33,133 between the niche and the neutral models, 1349 01:11:33,133 --> 01:11:39,733 as applied to the mainland, does the niche model predict also 1350 01:11:39,733 --> 01:11:40,274 a log normal? 1351 01:11:40,274 --> 01:11:42,644 Because it seemed like, in the discussion earlier, 1352 01:11:42,644 --> 01:11:47,584 the neutral also predicted log normal [INAUDIBLE]. 1353 01:11:47,584 --> 01:11:49,000 PROFESSOR: That's a good question. 1354 01:11:54,200 --> 01:11:58,560 In this whole area, I mean it's a little bit empirical. 1355 01:11:58,560 --> 01:12:01,490 The fact that the niche model kind of predicts 1356 01:12:01,490 --> 01:12:05,790 this, or this broken stick thing predicts a log normal, 1357 01:12:05,790 --> 01:12:08,540 they didn't say anything about islands there, right? 1358 01:12:08,540 --> 01:12:11,630 I guess even Fisher's original log series, 1359 01:12:11,630 --> 01:12:14,752 he used it to describe-- I think maybe 1360 01:12:14,752 --> 01:12:16,210 that was the beetles on the Thames. 1361 01:12:16,210 --> 01:12:19,790 But his original data set, where the Fisher log series 1362 01:12:19,790 --> 01:12:21,970 was supposed to described it, as it 1363 01:12:21,970 --> 01:12:24,060 was sampled better and better, it eventually 1364 01:12:24,060 --> 01:12:26,393 started looking more and more like a log normal anyways. 1365 01:12:29,316 --> 01:12:31,190 I mean it's easy to see the frequent species, 1366 01:12:31,190 --> 01:12:34,560 because you see them. 1367 01:12:34,560 --> 01:12:36,930 This tail can actually be very hard to see, 1368 01:12:36,930 --> 01:12:40,145 because you have to find the individuals. 1369 01:12:43,770 --> 01:12:46,550 It's a good question of to what degree each of the models 1370 01:12:46,550 --> 01:12:49,580 really predicts one thing on one place and another. 1371 01:12:49,580 --> 01:12:53,450 There's always tweaks of each model that adjust things. 1372 01:12:53,450 --> 01:12:56,010 So I think it's a bit muddy. 1373 01:13:01,070 --> 01:13:03,240 But the one thing that I want to highlight. 1374 01:13:03,240 --> 01:13:05,420 So there's a lot of debates, then, 1375 01:13:05,420 --> 01:13:07,150 between these different models. 1376 01:13:07,150 --> 01:13:09,260 And each of the models have some fit. 1377 01:13:09,260 --> 01:13:12,180 They have red and black. 1378 01:13:12,180 --> 01:13:13,860 There's one that kind of goes like this. 1379 01:13:13,860 --> 01:13:15,820 And another one that kind of goes like that. 1380 01:13:15,820 --> 01:13:20,410 And they're not labeled, because they look the same. 1381 01:13:20,410 --> 01:13:23,990 And you can argue about chi squareds and everything, 1382 01:13:23,990 --> 01:13:26,290 but I think it's irrelevant. 1383 01:13:26,290 --> 01:13:28,490 They both fit the data fine. 1384 01:13:28,490 --> 01:13:31,670 And the other thing, just the sampling of kind 1385 01:13:31,670 --> 01:13:35,710 root n sampling, if you expect to see 10 species, 1386 01:13:35,710 --> 01:13:38,355 then if you go and you actually do sampling, 1387 01:13:38,355 --> 01:13:41,330 you expect to have kind of a root n on each one. 1388 01:13:41,330 --> 01:13:44,110 I mean the error bars, I think, around this 1389 01:13:44,110 --> 01:13:46,540 are consistent with both models. 1390 01:13:46,540 --> 01:13:49,259 So I'd say that the exercise of trying 1391 01:13:49,259 --> 01:13:51,550 to distinguish those models based on fit to such a data 1392 01:13:51,550 --> 01:13:55,760 set I think is hopeless from the beginning. 1393 01:13:55,760 --> 01:13:58,640 And then you can talk about the number of parameters. 1394 01:13:58,640 --> 01:14:00,350 And if you read these two papers, 1395 01:14:00,350 --> 01:14:02,124 they both say that they have fewer number 1396 01:14:02,124 --> 01:14:02,915 of free parameters. 1397 01:14:05,610 --> 01:14:07,862 And it is hard to believe that there could 1398 01:14:07,862 --> 01:14:09,070 be a disagreement about this. 1399 01:14:12,534 --> 01:14:14,200 But then, you know, it's like, oh, well, 1400 01:14:14,200 --> 01:14:16,080 what do you call a free parameter? 1401 01:14:16,080 --> 01:14:19,950 And then what they say, any given RSA data set contains 1402 01:14:19,950 --> 01:14:22,110 information about the local community size j. 1403 01:14:24,790 --> 01:14:27,155 So they say, given that, it's not a free parameter, 1404 01:14:27,155 --> 01:14:28,705 because you put that in. 1405 01:14:28,705 --> 01:14:30,080 That's the number of individuals. 1406 01:14:30,080 --> 01:14:32,093 And then outcome is your distribution, right? 1407 01:14:32,093 --> 01:14:33,800 And you say, OK, well, all right, that's 1408 01:14:33,800 --> 01:14:36,008 fine if you don't want to call that a free parameter. 1409 01:14:36,008 --> 01:14:39,510 But then when you fit the log normal to this distribution, 1410 01:14:39,510 --> 01:14:42,720 the overall amplitude is also to give you 1411 01:14:42,720 --> 01:14:46,280 the number of individuals in the metacommunity. 1412 01:14:46,280 --> 01:14:50,880 So if you don't call j a free parameter in this model, 1413 01:14:50,880 --> 01:14:53,840 then you can't call the amplitude a free parameter when 1414 01:14:53,840 --> 01:14:57,970 you fit the log normal, at least in my opinion. 1415 01:14:57,970 --> 01:15:01,540 I think that they both have three. 1416 01:15:01,540 --> 01:15:04,570 Because if you fit a log normal to this, 1417 01:15:04,570 --> 01:15:05,890 you have the overall amplitude. 1418 01:15:05,890 --> 01:15:07,300 That's the number of individuals. 1419 01:15:07,300 --> 01:15:11,350 And then you have the mean and the standard deviation 1420 01:15:11,350 --> 01:15:14,089 or whatever. 1421 01:15:14,089 --> 01:15:16,047 From that standpoint, I think they're the same. 1422 01:15:16,047 --> 01:15:16,547 Yeah. 1423 01:15:16,547 --> 01:15:18,909 AUDIENCE: But I mean how do you fit the log normal 1424 01:15:18,909 --> 01:15:19,863 when you don't impose? 1425 01:15:19,863 --> 01:15:21,294 Do they impose the amplitude? 1426 01:15:24,530 --> 01:15:26,400 I mean it's still a parameter. 1427 01:15:26,400 --> 01:15:26,890 PROFESSOR: No, that's what I was saying. 1428 01:15:26,890 --> 01:15:27,650 It's a parameter. 1429 01:15:27,650 --> 01:15:30,302 I mean the normalized log normal, you integrate, 1430 01:15:30,302 --> 01:15:31,010 and it goes to 1. 1431 01:15:31,010 --> 01:15:32,600 But then you have some measured number 1432 01:15:32,600 --> 01:15:35,020 of individuals in your sample, and then you 1433 01:15:35,020 --> 01:15:37,650 have to multiply by that to give you. 1434 01:15:37,650 --> 01:15:40,337 AUDIENCE: But is that what they do when they do their fit? 1435 01:15:40,337 --> 01:15:41,003 PROFESSOR: Yeah. 1436 01:15:41,003 --> 01:15:43,544 AUDIENCE: Or do they keep that amplitude as also a parameter? 1437 01:15:46,040 --> 01:15:48,320 PROFESSOR: I think that you can argue whether this 1438 01:15:48,320 --> 01:15:49,540 is a free parameter or not. 1439 01:15:49,540 --> 01:15:51,440 But I think that you can just put it 1440 01:15:51,440 --> 01:15:54,149 as the number of individuals, and it's not 1441 01:15:54,149 --> 01:15:55,190 going to affect anything. 1442 01:15:55,190 --> 01:15:58,330 You could actually have it be a free. 1443 01:15:58,330 --> 01:16:00,790 But this gets into this question about what constitutes 1444 01:16:00,790 --> 01:16:02,230 a free parameter or not. 1445 01:16:02,230 --> 01:16:04,740 And actually, there is some subtlety to this. 1446 01:16:04,740 --> 01:16:09,664 But I think, at the end of the day, 1447 01:16:09,664 --> 01:16:11,580 the log normal is not going to look like this. 1448 01:16:11,580 --> 01:16:12,840 You have to. 1449 01:16:12,840 --> 01:16:14,840 You basically put in the number of individuals 1450 01:16:14,840 --> 01:16:15,880 that you measured. 1451 01:16:15,880 --> 01:16:17,840 AUDIENCE: So when you calculate [INAUDIBLE]? 1452 01:16:23,750 --> 01:16:25,380 PROFESSOR: Huge numbers of pages of 1453 01:16:25,380 --> 01:16:27,602 has been written about comparing these things. 1454 01:16:27,602 --> 01:16:30,060 At some point, it comes down to this philosophical question 1455 01:16:30,060 --> 01:16:32,300 about what you think constitutes a null model. 1456 01:16:32,300 --> 01:16:33,990 And this gets to be much more subtle. 1457 01:16:33,990 --> 01:16:37,200 And I think reasonable people can disagree 1458 01:16:37,200 --> 01:16:39,990 about whether the null model that you need to reject 1459 01:16:39,990 --> 01:16:41,800 should be this neutral model or if it 1460 01:16:41,800 --> 01:16:43,160 should be a niche-based model. 1461 01:16:43,160 --> 01:16:45,997 Or maybe it's just that there's some multiplicative type 1462 01:16:45,997 --> 01:16:48,330 process that's going on and gives you distributions that 1463 01:16:48,330 --> 01:16:50,790 look like this, and you need other kinds of information 1464 01:16:50,790 --> 01:16:52,260 to try to distinguish those things. 1465 01:16:52,260 --> 01:16:54,051 And in particular, I'd say that it's really 1466 01:16:54,051 --> 01:16:56,940 the dynamic information in which these models have strikingly 1467 01:16:56,940 --> 01:16:58,398 different predictions, and then you 1468 01:16:58,398 --> 01:17:00,820 can reject neutral-type models. 1469 01:17:00,820 --> 01:17:03,160 Because that neutral models predict 1470 01:17:03,160 --> 01:17:04,820 that these species that are abundant 1471 01:17:04,820 --> 01:17:07,380 are just transiently abundant, and they should go way. 1472 01:17:07,380 --> 01:17:08,630 Whereas the niche-based models would say, 1473 01:17:08,630 --> 01:17:09,760 oh, they're really fixed. 1474 01:17:09,760 --> 01:17:13,324 And indeed, in many cases, the abundant species kind of 1475 01:17:13,324 --> 01:17:14,740 stick around longer than you would 1476 01:17:14,740 --> 01:17:16,320 expect from a neutral model. 1477 01:17:16,320 --> 01:17:19,692 Of course, the neutral model is not true in the sense 1478 01:17:19,692 --> 01:17:21,400 that different individuals are different. 1479 01:17:21,400 --> 01:17:24,500 But it's important to highlight that even 1480 01:17:24,500 --> 01:17:27,210 such a minimal model can give you 1481 01:17:27,210 --> 01:17:29,220 striking patterns that are similar to what 1482 01:17:29,220 --> 01:17:30,756 you observe in nature. 1483 01:17:30,756 --> 01:17:32,130 And so I think we're out of time. 1484 01:17:32,130 --> 01:17:33,560 So with that, I think we'll quit. 1485 01:17:33,560 --> 01:17:36,250 But it's been a pleasure having you guys for this semester. 1486 01:17:36,250 --> 01:17:38,870 And if you have any questions about any systems 1487 01:17:38,870 --> 01:17:41,460 biology things in the future, please, email me. 1488 01:17:41,460 --> 01:17:42,460 I'm happy to meet up. 1489 01:17:42,460 --> 01:17:44,203 Good luck on the final.